The Modern View of the Cosmos

General Astronomy
	Print version	Observational Astronomy

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The Big Picture

The universe is a big place — too big for us to comprehend. But how big? Astronomers have struggled with this question for millennia, and their view of the known universe has steadily grown to immense and incomprehensible sizes. It’s an important question, and a basic part of our grasp of the universe itself. To study astronomy, it’s essential to understand what’s out there, how everything relates, and where we fit in the universe. The problem is that the size scales, the relative general sizes of classes of objects, are too foreign for things much larger than Earth. In a big universe, this can be a challenge. To tackle the problem, let's try to connect the familiar life-size world around us with the unfamiliar cosmic size scales.

If you're a student, you probably watch your instructor write on the chalkboard almost every day. The chalkboard is something you're much more familiar with than the whole universe because you can see it and touch it. You know the size of the board, the chalk, the markings, the eraser, and so on because they’re right at hand. How much bigger is the board than a dot made on the board with a piece of chalk? It turns out that the answer is about a thousand, for an average sized chalkboard, and a fair sized chalk dot.

Now let's consider something that's a thousand times the size of a blackboard. A blackboard is a few meters across, so we want to think about something a few kilometers across. That's something like the size of a small city. If a city is 1000 times larger than a chalkboard and a chalkboard is 1000 times larger than the mark on a chalkboard, that's a useful connection that helps us think about the size of a city: we can say that the chalkboard in the city is like the mark on the chalkboard.

In this way we will now step out from the city into the larger universe. With each step, we will consider something (very roughly) a thousand times larger than the last step. As we move out, each stop in our journey will be much smaller than the next, like a mark on a chalkboard.

A city is much larger than the blackboard we used as a reference point, but it's still something we are very familiar with. Many people drive across part of their home city and back again every day. It's possible to drive through most small cities in a half hour or so, even with stoplights, and it only takes a few hours to walk from one end to the other. As promised, the next step out will be much larger and farther from our everyday experience. Our next stop will have a size of several thousand kilometers, and that's the size of Earth.

In a car, you can drive across a city in less than an hour, even at a slow speed. If you could drive around the Earth at a speed of 60 miles per hour (100 kilometers per hour), going day and night over land and water, it would take a full 17 days. Remember, driving through a small city at 60 miles per hour would only take a few minutes. Seventeen days is much, much longer than a few minutes. The fastest jets, which have a top speed of about 2,000 miles (3,200 km) per hour, can make the trip around the world much faster. At that speed, you could circle the Earth in 11 hours. Even speeds like that will quickly become inadequate as we continue moving out into the universe.

The size of Earth is typical of the sizes of the rocky planets, the terrestrial planets, but planets made mostly of gas such as Jupiter and Saturn are larger by a factor of several to ten. In general, we can expect the same kinds of things to have similar properties. Given no other information about a planet, we might guess its radius to be the same as Earth's. If we knew that our imaginary planet were a much larger gas giant, we might change our estimate and guess that the planet has the same radius as say, Saturn.

This size scale represents the vast majority of human experience. Only a small number of people have ever been in Earth orbit, and these people remained very close to Earth. Most of the satellites launched remain very close to Earth. The shuttle, for example, orbits at an altitude of only a few hundred kilometers — a few percent of the radius of Earth. Some spacecraft are sent to other planets or to the Moon, but the majority stay at the scale of this step in our journey. Only 24 people — the Apollo astronauts — have ever left Earth's orbit to visit the next stop on our journey.

This illustration shows the Earth and the Moon scaled to their relative sizes and distances. The two are very small and far apart, and the Earth-Moon system is surrounded by a huge expanse of empty space. This view is only a small piece of the fifth step in our journey, and Earth is already becoming very tiny.

As we continue to move out, we reach a size of about 1000 times the distance around the Earth. The distance to the Moon is about 30 times the diameter of Earth, so the Moon is easily within reach in this step, but there's little else in the remaining distance. The nearest planets, Mars and Venus, are out of our reach. Aside from the Earth and Moon, we find the space in the neighborhood of Earth to be almost completely empty, with only the occasional passage of an asteroid or comet.

Although the Moon seems nearby when we consider the huge space surrounding the Earth-Moon system, we should remember that the Earth and Moon are really very far apart. If we could get in a car and drive to the Moon, the trip would take five months of driving nonstop — 24 hours a day and seven days a week. If our jet could go to the Moon, it would take five days to get there. These trips are becoming longer now, but they're still manageable. The trip by foot is much longer, though — a walk to the Moon would take nine years! Light travels faster than anything else in the universe. It has a speed of 300,000 kilometers (186,000 miles) per second. At this speed, light can travel to the Moon in just over one second. This distance, the distance light travels in one second, is called a light-second.

Another step out will extend our reach to most of the planets. We now encompass the bulk of the Solar System, the system comprising the Sun and all of the objects orbiting around it. This size scale is about five billion kilometers across, 30 times the distance between the Earth and the Sun. The distance from the Earth to the Sun is a convenient standard for measurement in the Solar System, so astronomers use the average Earth-Sun distance as a standard unit called the astronomical unit (AU). One astronomical unit is equal to about 93 million miles or 150 million kilometers. We can say that we are now working at a size scale of 30 astronomical units, or 30 AU for short. A box this size centered on the Earth would comfortably fit Saturn's orbit, but Uranus, Neptune and Pluto are still too far away. For much of human history, none of the Solar System objects outside of this box were known to exist.

Remember that our size scale has increased from the last step by a huge factor, a factor of 1000. Using our jet to take a trip from Earth to Saturn would take about fifty years. Light takes about 80 minutes to travel from Saturn to Earth, depending on where Earth and Saturn are in their orbits. Because it takes so long for light to make the trip from Saturn, the light we see from the planet at a given moment actually left 80 minutes ago. This means that we don't see Saturn as it is presently, but as it was 80 minutes ago. This also means that looking out into space is like looking back in time. The farther we look, the older the light. This is not very important for Saturn, but it will become more important as the size scale increases.

As before, we can use the speed of light as a measure of distance. A light-second is the distance light travels in one second. Likewise, a light-minute is the distance light travels in one minute. That means that Saturn is 80 light-minutes away. In the same way, we could write that Saturn is 1.3 light-hours away, and that's about the same as Saturn's distance from the Sun. Saturn's orbit is much bigger than Earth's orbit, which is only 8 light-minutes in radius. This is a striking and important fact about the Solar System — the rocky planets, the terrestrial planets, orbit close to the Sun and close to each other, but the giant gas planets, the Jovian planets, orbit at greater distances and have more widely spaced orbits.

The next step in our journey will encompass a distance of 30,000 AU, or half a light-year. Although this step completely encloses the planets, Solar System objects are found at much greater distances. These objects form the Oort cloud — a vast, sparse region of comets surrounding the Sun. The Oort cloud is almost empty, but it's there just the same. At this step we've only covered a piece of the range of influence of the Sun, and plenty of Oort cloud remains out of our reach. The Oort cloud is thought to extend out from the Sun as much as two light-years.

If we look just beyond this step, we find the nearest star, Proxima Centauri, at about four light-years. It will take Voyager 1 and 2 80,000 years to reach this star. (These ships, launched in 1977, have a speed of 51,500 km/hour.) As other stars enter into the picture, the Sun will no longer be the dominant source of gravity. This means that we can expect the solar system to really end as we begin approaching other stars.

Our next step out places us in a 500 light-year box. This size scale is easily large enough to fit the Sun, Alpha Centauri, and many other stars. In fact, about 250,000 stars are within 500 light-years of Earth. Astronomers refer to this region as the Solar Neighborhood. As we've seen, the stars in the Milky Way are very far apart, with vast stretches of mostly empty space separating them.

Stars in the Solar Neighborhood (and throughout space) are mostly small and faint. If these fainter stars were much farther away, they would be too faint to see from Earth. Much brighter stars are more rare, but they can be seen from much farther away. Because of this, two "kinds" of stars fill the sky as viewed from Earth: stars that are intrinsically faint but nearby, and stars that are bright and more distant.

As we continue outward in our journey, we see the random scattering of stars form into a pattern. Spiral structure emerges, and we see that the Earth, the Solar System, and nearby stars are collected together into an orderly system of stars called a galaxy. Our galaxy is called the Milky Way Galaxy.

Like the Solar System, the Galaxy is shaped like a flat disk, but the Galaxy is much bigger. Our galaxy contains hundreds of billions of stars, and the Solar System is only one member contained inside. The Sun is located in the spiral arms of the galaxy about two thirds of the way from the center, and it orbits around the center of the galaxy with all the other stars. If the Milky Way were fifty miles long, the Solar System would only be a dot the size of the tip of a ball point pen. In actuality, the Galaxy is 100,000 light-years across, but only a few thousand light-years thick.

As we start out towards the next step, we see other galaxies like the Milky Way begin to appear. Compared to stars in the Milky Way, galaxies are packed together much more closely, and collisions between galaxies are much more common. The Andromeda Galaxy, the nearest galaxy to the Milky Way, is 2.5 million light-years away and on a collision course for Earth. Don't panic, though, as the collision won't happen for another 3 billion years.

At a size scale not much larger than our last, we see that the galaxies are clumped together. These galaxy clusters typically have hundreds of galaxies and are millions of light years across. Galaxies orbit around the center of their clusters. The Milky Way is a member of the Virgo Cluster. It orbits near the edge of the Cluster, so we can see much of its center in a small region of the sky in the direction of the constellation Virgo. Presently, the Milky Way is moving away from the center of the Virgo Cluster. In the distant future, however, the Cluster's pull will slow the Milky Way and draw it inward.

Although our reach now includes many more objects, the size scale of galaxy clusters still does not represent an expansion over the last size scale by a factor of 1000. We have not fully reached the next step in our journey until we come to the scale of hundreds of millions of light years. At this scale, even the galaxy clusters form clusters. These groups of galaxy clusters are called superclusters. A supercluster may contain hundreds of thousands of galaxies. The Virgo Cluster is a member of a supercluster called the Virgo Supercluster.

Light from the edge of our 500 million light-year reach has traveled for 500 million years before reaching us. This means that we see the Virgo Supercluster as it was 500 million years ago. Five hundred million years can be a long time, but it isn't really long enough for the universe to have changed significantly. Even though the light is old, the universe at the edge of the Virgo Supercluster still looks much like the universe nearby.

As we continue out from the 500 million light-year size scale, we see older and older parts of the universe. As the distance becomes very large, we begin seeing billions of years into the past, and big changes in the Universe as a whole become important. Going further and further back, we see the formation of the first galaxy clusters, galaxies, and stars. Eventually, we see the universe so young that no stars have formed yet. Before the first stars formed, the Universe was cool and dense enough for the loose, unused gas in space to block visible light. Beyond this, we can't see any further. The contents inside this wall are called the Hubble volume, or the Observable Universe. There's no way to observe objects outside of this volume because the light from these distant objects hasn't reached us yet.

The Universe extends out infinitely, but our view is limited to the Hubble volume. Trying to visit the edge of the Hubble volume is impossible. You'll never reach it because it's only an illusion. If you tried to visit the location where we see the edge of the Hubble volume, you would see the universe around you as it is today, not billions of years ago, and you would see the edge of the Hubble volume all around you, billions of light years away.

Extending out further inevitably delves into the pages of theory and crosses the line dividing what we have seen and what we cannot see. Some have theorised that the universe itself does not comprise all of existence, and that there may be other universes with different physical laws existing together in clusters and groups in a multiverse.

As we go further and further out, many note that there must be an edge, an end, a border of reality. Others have speculated that the universe, or the multiverse it sits in, is simply infinite, and has no such boundaries. However, at this point in time, there can be no absolution, no definite answer to these questions.

Much remains unknown about what the Universe is like, but our picture has evolved dramatically since humanity first began asking questions about the world around it. Armed with curiosity and the tools of science, astronomers have investigated the heavens for centuries, and their work continues today.

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Short History of the Universe

For time immemorial, humans have been intrigued by creation. Where did we, and the universe in which we live, come from? In the Rig Veda, it was proposed that before creation there was "neither existence nor non-existence." The Latin phrase ex nihilo nihil fit ("out of nothing comes nothing") sums up current human beliefs about origins.

The Qur'an contains the following verse regarding the origin of the universe: “Do not the Unbelievers see that the heavens and the earth were joined together (as one unit of Creation), before We clove them asunder?” [Al-Qu’ran 21:30]

Many possibilities have been considered by scientists over the millennia. Did the universe "happen" suddenly?… was it created quickly by God?… has it existed forever?… or is it in a constant state of creation, even now?

Just as we can use size scales to make rough comparisons of the sizes of objects in the universe, we can also use time scales to compare the periods of time over which events occur. For example, Comet Halley takes about 75 years to complete its orbit around the Sun, so we can say that the period of Comet Halley's orbit has about the same time scale as a human lifespan.

The Origin of the Universe according to the Big Bang Theory

The universe is big in space, but it's also big in time. The age of the universe appears to be 13.7 billion years. Like huge space scales, a span of 14 billion years is hard to imagine for a person who will have a much shorter lifespan than that. To better understand very long time scales, we can "compress" long periods of time like the age of the universe into shorter periods such as a human lifetime.

A typical person will live around 80 years. That means that the Earth will go around the Sun eighty times over the course of an average lifetime. The human life time scale and the time scale of human history are far smaller than the time scale of changes in the universe. Astronomers have learned about how the universe as a whole came to be the way it is by studying only an instant of its existence. To get a clearer picture of how the cosmological time scales fit together, let's consider an imaginary astronomer whose life is "stretched out" to fill the entire history of the universe.

If a human lifetime were vastly longer than it is, the orbit of the Earth around the Sun might be too fast to be a useful way to measure age. Instead, it might be more practical measure time from something slower. The Sun takes 230 million years to complete an orbit around the Milky Way Galaxy. That means that a "Milky Way year" will be 230 million times longer than an ordinary year.

Let's suppose our old astronomer has a lifespan of 80 "Milky Way years" instead of 80 normal years. That way, the astronomer will live long enough to see events that happen on cosmological time scales rather than human time scales. The astronomer's life happens in extremely slow motion. His growth and his actions are slowed by a factor of 230 million. At this rate, it takes four normal months for the astronomer just to blink.

Astronomers have theorized that the universe was very small, hot, and dense when it began. Since then, it has expanded and cooled. At first, the universe contained almost entirely hydrogen and helium gas. The universe was very uniform, with no galaxies, stars or planets. We'll place our protagonist's birth at the time of the big bang.

Our astronomer was much too young to remember the early development of the universe after the big bang. By the time the astronomer was only two days old, atoms had formed in the universe and the universe had cooled enough for structure to begin forming in the gas that uniformly filled all of space. Remember, the imaginary long-lived astronomer measures time much more slowly — what the astronomer sees as two days amount to a million years of time on a normal clock. While a million years seems like an extremely long time, that's only a tiny part of the history of the universe.

Our astronomer's earliest memories of childhood will include the first formation of stars and galaxies, which began to occur when the astronomer was around five "Milky Way years" old. Galaxies would continue forming and developing well into the astronomer's teenage years, when the universe was billions of years old. Today, galaxies continue to evolve and change.

There are two schools of thought on how the Universe formed: "top down" and "bottom up". Top down theorists think that large clusters formed after the Big Bang, which later broke down into stars and galaxies. "Bottom up" theorists instead pose the theory that matter was originally dispersed fairly evenly by the Big Bang, later accumulating into stars and galaxies. Recent data from Hubble Deep Field photographs appear to support "bottom up" theories. The photos show young galaxies from up to 11 billion light-years away. These young, small galaxies, from early in the universe's history, support the theory that large structures formed out of smaller ones. The galaxies appear as faint blue blobs with vague spiral structures, 2000 to 3000 light-years across.

An important necessity for the appearance of life is heavy elements. Since only hydrogen and Helium gas were formed during the big bang, everything heavier than that had to be made in stars later. These other elements — all of the elements on the periodic table — were made in stars.

Stars use light elements like Hydrogen and Helium as their fuel. Like nuclear bombs, they use the power of the atom for energy. Stars are unlike nuclear bombs, however, in that most nuclear bombs get their energy from heavy atoms like Plutonium and Uranium. In a nuclear bomb, the energy comes from converting heavy elements into lighter elements. This process is called fission. In a star, the energy comes from converting the light elements into heavier elements. This process is called fusion. All of the material in the universe heavier than Hydrogen and Helium were made by fusion in a star.

An ordinary star can make many of the elements, especially the most important ones for life. It can't make all of the elements, though; it can't make anything heavier than Iron. The heavier elements were made in supernovae. A supernova occurs when a massive star reaches the end of its life. Supernovae are extremely bright and extremely hot. This is where the heaviest elements in the universe are made.

All of the material needed to make rocky planets was made in stars, and some of it was made in supernovae. This means that planets couldn't exist until there had been enough time for a star to completely go through its life cycle and become a supernova. The material would then be ejected back into space, and would form a new star, possibly with planets. It also means that everything in the world, including you, came from a star. As the astronomer Carl Sagan said, "We are star stuff."

The Sun and the Earth formed about four and a half billion years ago, when our long-lived astronomer was 37 "Milky Way years" old. A solar system forms relatively quickly, and ours probably only took about 100 million years. The Sun and the planets formed from a cloud of sparse gas in the Milky Way, which condensed down and coalesced. After this happened, the Earth was very hot. It was a completely molten ball of rock, and had melted so thoroughly that its consistency was similar to water. Certainly, no life could have existed on the very young Earth.

Soon, however, the conditions on Earth became suitable for life. For our carbon-based type of life to develop, a planet needs to have

organic matter (the material from which DNA is made),
liquid water,
a source of energy, and
appropriate temperatures.

Earth had all of these fairly early in its history. Scientists aren't sure when life first appeared on Earth, but some evidence dates the earliest life at three and a half billion years ago, when our astronomer was 47 "Milky Way years" old.

Evidence suggests that life appeared almost immediately once the necessary conditions were satisfied on the young Earth, but the first inhabitants of the new planet showed no strong predisposition for evolution to more advanced life forms. More than two billion years passed before primitive, single celled microbial organisms advanced to become multicellular life. Once it did, however, advanced life took rapid hold. This period of rapid development in the complexity of life is called the Cambrian Explosion.

The Cambrian Explosion occurred when our astronomer was 56 "Milky Way years" old. Ten years had passed on our slowed cosmological clock. This period of history was, in some sense, the most profound step in the development of intelligence on Earth.

Since the Cambrian Explosion, things have become very interesting on Earth. The astronomer was barely a year older by the time the primitive multicellular creatures had developed to become the dinosaurs, which were unimaginably more complex and advanced by comparison. We have now come into recent times, as the appearance of the dinosaurs occurred only twelve months before this instant in our astronomer's life.

We can see now that we must occupy a tiny, tiny part of the cosmic timeline. By what standard could the time of the dinosaurs possibly be considered recent? By a cosmological standard. The dinosaurs had a short stay on Earth. They disappeared 65 million years ago, or four months ago by the astronomer's clock. This gave mammals the chance to rise to dominance.

The first humans appeared on Earth only eight hours ago by our astronomer's clock. Civilization accounts for only a small fraction of this time. The first system of writing, and the first evidence of civilization, dates to only 6000 years ago, or fifteen minutes ago by our astronomer's clock. In the lifetime of the universe, all of culture and all of written history could fit into a coffee break!

Periods more similar to the human life span become even smaller in our comparison with the cosmic timescale. It was only very recently that people first endeavored to learn about the world using science in the way we understand it today. These first scientists, who considered observation and experimentation to give the last word on the nature of the universe, lived only about four hundred years ago, or one minute ago on our cosmic clock.

As we come progressively closer to the present, our lives seem progressively more ephemeral and fleeting. How long will our society, our culture, and our ideas endure in the long run? Only time will tell. At the current rate, it does not appear to be long before something must happen to stop the downgrade of society.

We can say little about what might happen on Earth in the next thousand years, let alone the next million years or the next billion years. Meanwhile, the Milky Way will continue on its normal course, a course which happens to be headed straight for a collision with the nearest neighboring galaxy in about 12 "Milky Way years." This is unlikely to be the disaster for Earth it might appear to be — a collision between any stars during a galactic merger is actually extremely improbable. The Solar System will simply take up residence in the new galaxy created in the aftermath of the collision.

Five billion years from now, the Sun will have used up its fuel and will die. At this point, about 20 billion years after the big bang, the long-lived astronomer will be 80 years old. The Earth will no longer be able to support life after the Sun dies, and the Solar System will become rather desolate. The universe as a whole will continue long after the death of both the Sun and our astronomer, but it too will slowly die, eventually becoming cold and empty as it continues to expand.

Looking back over the scales of the universe and seeing humanity's role, unimaginably small in both space and time, one might wonder whether science has painted a bleak, depressing, and hopeless picture for humanity. Some would say so, while others find comfort in the majesty and grandeur of the universe, and the possibilities that the future might bring. For now, the future of humanity in this universe, as chaotic and vast as it is, remains uncertain.

Summary of Events
*230 million years = 1 year*
Event	Real time scale	Compressed time scale'
Formation of structure begins	1 million years after big bang	2 days
Earliest stars and galaxies form	2 billion years after big bang	6 years
Sun and Earth form	9 billion years after big bang	37 years
First evidence of life on Earth	10 billion years after big bang	47 years
Advanced life forms on Earth	500 million years ago	57 years
First dinosaurs	230 million years ago	58 years
Dinosaurs become extinct	65 million years ago	4 months ago
Humans appear	200,000 years ago	8 hours ago
Writing is developed	6000 years ago	15 minutes ago
Modern scientific thought	400 years ago	one minute ago
Present day	13.7 billion years after big bang	59 years
Human lifetime	80 years	ten seconds
Milky Way collides with Andromeda	17 billion years after big bang	71 years
The Sun dies	20 billion years after big bang	80 years

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Scientific Notation

In previous sections, we discussed some numbers that were very large. In astronomy, the appearance of such huge numbers is common. This is one reason astronomers and other scientists use scientific notation when working with very large or very small numbers. Scientific notation is a system for writing and working with numbers that makes it much easier to deal with numbers that are very small or very large.

For example, the Milky Way Galaxy contains roughly three duodecillion tons of material. That is a rather cumbersome number. (Astronomers would never actually write this. Instead, they would say that the Milky Way contains one trillion times the mass of the Sun, which is somewhat easier. We'll use this much larger number for our demonstration.) You could also write this number as

3 000 000 000 000 000 000 000 000 000 000 000 000 000 tons,

but that's even worse. Scientific notation makes the number much more compact and readable:

3 × 10³⁹ tons.

This number is verbally expressed as "three times ten to the 39 power tons." This is numerically equivalent to the first two expressions.

A number written correctly in scientific notation has two parts. The first is a number greater than or equal to 1 and less than 10 (but it can be either positive or negative). This is sometimes called the mantissa. The second part is the number ten raised to a whole number power. The exponent of the second number is called the power. Some examples of numbers written correctly in scientific notation are:

2 × 10¹⁸

-1.4 × 10²

7.656 × 10^-4

2.1 × 10⁰

These, on the other hand, are not valid examples of numbers written in scientific notation:

0.1 × 10⁴ is wrong because the mantissa is less than 1

12 × 10³ is wrong because the mantissa is not less than 10

8.4 × 10^2.2 is wrong because the power is not a whole number

Remember that

10ⁿ = 10 × 10 × 10 × ... for n times,

which means that ten raised to n power is the same as 10 multiplied by itself n times, which is the same as a 1 with n zeros written after it. For example, 10³ is 10 × 10 × 10, or 1000. That means that our earlier number, 3 × 10³⁹ tons, is equivalent to

3 000 000 000 000 000 000 000 000 000 000 000 000 000 tons,

which is a three followed by 39 zeros. A number written in scientific notation with a negative power corresponds to a small number. For example, the number 1 × 10⁻³ is written as 0.001 in conventional notation. In general,

10^-n = 1/10 × 1/10 × 1/10 × ... for n times.

Since scientific notation relies on powers of ten, it's simple to convert a number from scientific notation to standard notation or vice versa. To convert a large number (with a positive power) from scientific notation to standard notation, first identify the decimal point in the mantissa, then shift the decimal to the right by the number indicated by the power. To convert a number from standard notation to scientific notation, just reverse these steps. Find the decimal point in the number, and move it until the number is at least 1 but less than 10. Count the number of places you moved the decimal point and use that number as the power. If you moved the decimal point to the left, make the power positive. If you moved the decimal point to the right, make the power negative.

Scientific notation also makes it simpler to do multiplication and division. To multiply two numbers in scientific notation, multiply the mantissas and add the powers:

(3 × 10⁴) × (4 × 10^-2)

(3 × 4) × 10^{4 - 2}

12 × 10²

1.2 × 10³

In some cases, such as the one shown here, you may need to shift the decimal point again ensure that the number is in correct scientific notation. It should never be necessary to shift the decimal point by more than one digit. When dividing numbers in scientific notation, divide the mantissas and subtract the powers:

{\frac {3\times 10^{4}}{4\times 10^{-2}}}

(3/4)\times 10^{4+2}

0.75 × 10⁶

7.5 × 10⁵

Here also, it may be necessary to shift the decimal point and change the exponent.

Scientific notation makes it easy to compare numbers that have very different values because all the zeroes have been replaced with the much more readable exponent. Numbers with a greater exponent are always bigger than numbers with a lesser exponent.

If one of the exponents is bigger than the other by more than a couple, the difference between the two is clearly very big. Recognizing a huge difference between two numbers can sometimes be a very useful insight, so it often makes sense to take a moment to develop an intuitive feel for a math problem before attacking it. In some cases, it's useful to see roughly by how much one number is larger than another. Scientific notation makes this much simpler. For a rough estimate, you only need to find the difference in the exponents. For example, 10⁷ is greater than 10³, since 7 - 3 = 4.

Some tourists in the Chicago Museum of Natural History are marveling at the dinosaur bones. One of them asks the guard, "Can you tell me how old the dinosaur bones are?"

The guard replies, "They are 73 million, four years, and six months old."

"That's an awfully exact number," says the tourist. "How do you know their age so precisely?"

The guard answers, "Well, the dinosaur bones were seventy three million years old when I started working here, and that was four and a half years ago."

(From the Science Jokes Web page [1])

In science, measurements are never perfect and numbers are never exact. As a result, every measurement we make has some uncertainty associated with it. Scientific notation makes it easy to express how precisely a number is known. Suppose a paleontologist discovers ancient dinosaur bones and finds that they are 73 million years old. Of course, the paleontologist doesn't know exactly how old they are. Maybe they're 73,124,987 years old, but the paleontologist only knows the age within 1 million years, so the age is written as 73,000,000 years, or 7.3 × 10⁷ years. Either of these expressions imply that the bones aren't exactly 73 million years old, but are 73 million years old, give or take a million years.

But what if the paleontologist knows the age within 200,000 years, and is sure that the bones aren't, say, 73.4 million years old? In that case, the standard notation is ambiguous — the number is still written as 73,000,000 years. In scientific notation, we can write the number as 7.30 × 10⁷ years. If we write this, we mean that the third digit is significant. The paleontologist might have calculated that the bones are 72,954,332 years old, but it would be useless to report these numbers, since the error on this measurement was 200,000 years. The extra digits are insignificant. The number of significant figures in a number are a reflection of the precision expressed in the number. In this case, the number of significant figures is three. The first significant figure is 7, the second is 3, and the third is 0.

Scientific notation gives the trailing digits written after the decimal point of a number special meaning — they tell that the number is exactly 7.30 × 10⁷ years. This is unlike the usual use of numbers in mathematics, where trailing zeroes after the decimal point have no special meaning.

In the story about the museum guard, the guard hadn't thought about the precision of the age of the bones. It doesn't make sense to add the four years to 73 million years because the uncertainty in the age quoted to the guard was a lot more than 4 years. When working with numbers that have uncertainties, we must be sure not to express better precision in the results of our arithmetic than we had in the first place.

When doing arithmetic, numbers being added or subtracted are treated differently from numbers being multiplied or divided.

When multiplying or dividing numbers with uncertainties, make sure that the answer has as many significant figures as the least precise of the original numbers.

For example, in (2.3 × 10³) × (1.21 × 10²), the number 2.3 × 10³ has two significant figures and the number 1.21 × 10² has three significant figures. The result should have two significant figures: 2.8 × 10⁵. We assume that there was some uncertainty in the measurement that gave us 2.3 × 10³, and this leads to some uncertainty in the result of the computation.

Addition and subtraction work differently. When adding 23.14 and 2.2, for example, the number 2.2 has uncertainty beginning in the tenths place. This uncertainty makes it pointless to report the hundredths place in the sum. To see this, try adding some uncertainty to 2.2 and see how this affects the sum.

When adding or subtracting numbers with uncertainties, round the result to the last significant place of the original number with the greatest uncertainty.

For example, 2.3 × 10³ + 1.1 × 10² can be written as

	2300
+	110

	2410

But we don't know the real value of the tens place in 2.3 × 10³, so we really only know the answer to the hundreds place. We should write

	2300
+	110

	2400

or 2.3 × 10³ + 1.1 × 10² = 2.4 × 10³. This may seem incorrect, but we're really only rounding off. Since we don't know the result to better than two significant figures, it makes no sense to report the extra digits — that would be like the museum guard who tells visitors that the dinosaur bones are 73 million and four years, six months old.

Almost every number has a unit of measurement attached to it. The number we used as our first example carried units of tons, and we expressed the age of the dinosaur bones in years. The units a number carries are part of the number itself. Units can also be multiplied or divided just like numbers.

As an example, consider a simple equation:

distance = velocity × time.

Suppose you drive in a car with a speed of 100 kilometers per hour (60 miles per hour) and you go in a straight line for one hour. The distance you will travel is

\mathrm {distance} =100{\frac {\mathrm {kilometer} }{\mathrm {hour} \!\!\!\!\!\!\!\!/}}\times 1{\mathrm {hour} \!\!\!\!\!\!\!\!/}

or 60 miles. We have cancelled hours as though it were a number.

This trick is also useful when you need to convert units. If you have a result in one system and you'd like to convert to another, you can set up a ratio, such as 1,000 meters/1 kilometer. Since 1,000 meters is equal to 1 kilometer, the ratio 1,000 meters/1 kilometer equals one. Because of this, multiplying any number by 1,000 meters / 1 kilometer will not change the value of the number. If we want to know what 100 kilometers is in meters, we can write

100\mathrm {\ kilometers} \!\!\!\!\!\!\!\!\!\!\!\!\!\!/\,\,\,\,\,\,\,\,\,\,\,\times {\frac {1000\ \mathrm {meters} }{\mathrm {kilometer} \!\!\!\!\!\!\!\!\!\!\!\!\!\!/}},

or 100,000 meters.

Other units of measurement used in astronomy are kilograms (mass), Newtons (force), and Joules (energy).

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The Scientific Method

The results of scientific thought have played an important role in shaping the world into what it is today. This is true not only because of the influence technology has had in our lives, but also because of the change science has brought to the way we think. Scientific thinking has permeated all aspects of our lives, fundamentally changing the way we see the world. It is science's "way of thinking," the scientific method, that forms its heart. Although "scientific thought" or "science" is sometimes taken to refer to the entire body of scientific theories and knowledge including chemistry, biology, physics, and so on, the term science strictly only refers to the process by which these theories and ideas are examined.

An image portraying some of the fundamental aspects of our world that have been thoroughly analysed by the hand of science.

The scientific method is fundamental to the investigation and acquisition of new knowledge based upon physical evidence. Scientists use observations, hypotheses, and logic to propose explanations for natural phenomena in the form of theories. Predictions from these theories that can be reproducibly tested by experiment are the basis for developing new technology.

Although scientific progress is often described as a linear scheme that allows a single scientist to proceed from a state of relative ignorance to a state of knowledge, the scientific method is really much more complicated than this. The scientific method is not a recipe. It requires intelligence and imagination. Science is not the lifeless execution of step-by-step instructions, but a creative and inspired process. Over the past half-century, philosophers, historians, and sociologists of science have established a more complete model describing the ways science is actually practiced.

The modern description of scientific progress gives much more importance to the role of the scientific community. It is impossible for a single person working in isolation to engage in science. This is because of the central role of peer review as a means of correcting error, bias, and self interest. Human nature prevents even a well intentioned scientist from doing anything other than confirming preconceived ideas when working in isolation. Because science is closely tied to a community, progress is usually scattered, proceeding from many stages and different directions at once.

In a new area of scientific exploration, scientific progress is slow at first. Investigators reach out in search of a basic description of how their area of inquiry works. This inquiry is motivated by observations of some interesting phenomena manifested in reality. The observations and experiments undertaken by scientists in a developing field are typically dictated by the scientists' interests and by convenience. The scientists have no heritage of successful experiments and little common ground. Because of this, new pursuits in science lack direction and are wanting of a broad common ground that will unite scientists and prescribe the course of scientific research.

As the new field advances, an underlying set of principles begins to take hold and gain widespread acceptance. These principles become the common ground of scientists in the new field, the basis of a paradigm. With a paradigm in place, researchers no longer need to start from scratch when interpreting the results of new experiments. Instead, they address a specialized audience that works under a common paradigm, and they describe their work in the context of that paradigm. They are no longer unguided in the direction of the work — the paradigm dictates which experiments might be useful in further exploring the questions of their field. An established paradigm is therefore an extremely valuable asset in the pursuit of science.

In the presence of an established paradigm, the normal course of scientific research begins. Scientists seek to clarify the understanding of their area of research by formulating theories grounded in their paradigm and by making relevant observations that test the theories and further clarify the paradigm. The observations give greater insight into the paradigm and test the agreement of theories and paradigms with reality. The theories discern previously unseen details in the paradigm and in nature, and guide the method of new observations.

For the normal course of scientific progress, these methods are extremely efficient. In some cases, however, a scientific discipline might be founded on a paradigm that is fundamentally incorrect. This presents a problem, since the paradigm underlying a discipline is deeply valued by the practitioners in the discipline. Such was the case in astronomy during the scientific revolution of the Renaissance. From the times of Aristotle, it had been thought that the Earth was at the center of the Universe, with everything else moving around it in circles. Eventually, observations of the positions of the planets made Aristotle's model inconsistent with reality. As time passed, it became increasingly difficult to modify Aristotle's paradigm to agree with observations. The prospect of a new and completely different paradigm became ever more attractive, and the new idea of a Sun-centered universe took root.

It is almost always the case that some observations cannot be explained by existing scientific theories, but a gap between knowledge and theory is hardly sufficient to produce a paradigm shift. In The Structure of Scientific Revolutions, Kuhn claims that scientific revolution is only possible when the dominant paradigm of a field is in a state of crisis. Only once it becomes apparent that the ideas tied to the paradigm are irreconcilable with observations does it begin to appear sensible to pursue completely new explanations.

When scientists consider theories, either as candidates for a new paradigm or as elaboration for an existing paradigm, they evaluate theories on the basis of a number of criteria that have proven over the years to be relevant indicators of usefulness. These criteria were described by the philosopher of science Karl Popper in his book, The Logic of Scientific Discovery.

The central goal of science is to produce theories that are consistent with reality. Science approaches this goal by testing a theory for validity, and discarding or modifying the theory if observations can't be shown to agree with it. The use of an approach like this requires that theories be falsifiable. In order for a theory to be scientific, it must be relevant to observable aspects of reality. It must be possible, at least in principle, to make observations that would be inconsistent with the logic of the theory. Said otherwise, the theory must make non-trivial predictions that can be tested by a set of observations. If scientists find that a theory's predictions are incorrect, the theory is falsified. A central property of scientific theories is that they can never be proven, but they can eventually be disproved. This fact makes it essential that a theory make specific, useful predictions before it can be examined scientifically.

One of the greatest triumphs of modern physics was the prediction and subsequent observation of the existence of subatomic particles called top quarks. Einstein's equations predicted that these particles should exist and described some of their properties. One of the goals of the Collision Detector at Fermi National Laboratories in Batavia, Illinois was to test the theory by searching for the top quark. In 1995, scientists at Fermi announced their successful observation of the top quark.

The need for falsifiability implies a need for theories that make clear predictions. Theories that make strong predictions are said to have a high predictive power. A theory with high predictive power makes bold, clear and testable predictions. Theories with high predictive power are more highly valued in science because they are more easily falsified. Scientists consider theories that have not been falsified despite strong predictions to be more reliable, whereas a theory that makes fewer predictions or less bold predictions tends to be received with more skepticism.

The requirement that scientific theories have strong predictive power leads to the surprising notion that an implausible theory should be preferred over a plausible theory. This is because the claims made by the implausible theory are bolder and more subject to being falsified. In order to succeed, a scientific theory must make a prediction that would not be expected otherwise. If a theory makes a seemingly wild claim that turns out to be correct, the theory is very well supported. This mechanism for developing scientific ideas by creating and testing the predictions of a model is the foundation of science.

A final requirement of scientific theories is that they be as simple as possible while still accurately describing nature. This requirement was famously stated by the English logician and Franciscan friar William of Ockham. He stated the requirement, in Latin, as

Numquam ponenda est pluritas sine necessitate,

which translates roughly as

Assertions should not be multiplied unnecessarily.

This maxim is widely known as Ockham's Razor. It is a very useful principle, but also one of the most misused doctrines in science.

Ockham's razor warns us that, when two theories make identical predictions, the theory that relies on fewer assumptions is more likely to be correct. The razor is based on the principle that reality is unlikely to conform to our presuppositions. In the absence of evidence, it's often wiser not to speculate.

When creating theories or distinguishing between competing theories, scientists rely on two distinct types of reasoning: inductive reasoning and deductive reasoning.

Inductive reasoning works from the specific to general. It involves making observations and building generalizations on the basis of observations. For instance, you might observe the sunrise every day for a year, and you notice that this Sun rises more or less in the east every time. You might conclude that the Sun always rises in the east. Inductive reasoning involves drawing conclusions from a limited sample of information. You have no way of knowing that the Sun won't rise in the west tomorrow. Still, a pattern will become apparent as observation continues, and year after year of observation makes it compelling to imagine that the Sun must always rise in the east. If the pattern remains very consistent, it can be considered reliable even if the underlying cause isn't obvious. Should a theory ever be presented that predicts that the Sun will always rise in the east, that theory will be well supported. On the other hand, if a theory predicts that the Sun will sometimes rise in the east and sometimes in the west, that theory will be disfavored by the observations even though it isn't disproved in a strict sense. Notice that it is possible, though unlikely, for inductive reasoning to discredit a correct theory or support an incorrect theory. Although it is a powerful and essential tool in science, inductive reasoning must be treated with skepticism when based on a very limited sample of observations.

Deductive reasoning works from the general to the specific. It is based on logical arguments (syllogisms). An example of deductions invented by the author Lewis Carroll is provided below:

All lions are fierce.

Some lions do not drink coffee.

Therefore, some fierce creatures do not drink coffee.

Unlike inductive reasoning, deductive reasoning is perfectly reliable if you have made correct assumptions and applied correct logic. Because deductive reasoning is absolutely reliable when used with good assumptions and proper method, it is easy to place undue trust in claims made on the basis of deductive reasoning. It is important to remember that these claims also require careful examination to check that the assumptions are good and the reasoning is valid. In reality, deductive reasoning is as much subject to error as inductive reasoning. Only the sources of error differ.

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What People do in Astronomy

Astronomy like all sciences is a social activity in which people are constantly discussing new ideas, interpreting data, and arguing with each other over what observations mean. Astronomers can broadly be divided into two groups. Observational astronomers specialize in building instruments such as telescopes, and spacecraft, and take raw data and process them into meaningful results. Theoretical astronomers often also known as astrophysicists take the results that observational astronomers provide and attempt to create physical models which explain the data that observers see and provide ideas of the directions that observers should go into. Theoretical astronomers increasingly rely on computer models and often are skilled at programming.

Astronomy is rather unique in that a lot of the data is provided by amateur astronomers. The data needed in some fields such as variable star astronomy or comet discovery can be gathered by instruments well within the budget of an interested hobbyist.

Peer review

A central part of the scientific process is peer review which occurs at several stages in the process, and creates what H.H. Bauer calls a knowledge filter. In the peer review process, a proposal or a journal article is given to a group of referees who anonymously submit their comments on the proposal. While the referees will sometimes communicate with each other, they are not intended to reach a consensus on the quality of work. In addition, the referees usually do not have the final authority to decide on the fate of a proposal, but instead given their opinions to an editor or project director who does have the final authority and on occasion overrides the opinions of the referees.

The opinions of the referees are usually made available to the submitter, and often contain suggestions for improvement to the submitter. This is considered crucial for the scientific process as it allows the submitter to receive feedback on his or her proposal and improve it. In some cases, the submitter is encouraged to resubmit their proposal after making changes, and this often develops into an anonymous communication between the referees and the submitter.

Getting what you need

A lot of the work in astronomy involves getting access to resources, which include:

money
for observers, telescope and equipment time
for theorists, computer time

In order to get these resources, astronomers typically write grant proposals which outline the amount of money, telescope, and computer time needed.

Grant proposals typically undergo peer review by the granting agency which includes feedback on how well it fits the priorities of the funding agency, how likely the committees think it is to advance the frontiers of knowledge, and how essential the resource being allocated is to the researcher. Typically, a set portion of telescope or supercomputer time will be made available to the institution that hosts or funds the resource, giving scientists affiliated with that institution priority use over that resource. The remaining time is then made open for research proposals from researchers from other institutions. In the case of ground based telescope time, the most precious and highly sought after time is dark time during which the moon is new and the dimmest objects can be seen.

In some cases, such as building a new telescope, supercomputer center, or funding a new spacecraft, astronomers must lobby funders such as charitable foundations and legislators for money to finance a certain activity. The lobbying for research facilities can be extremely intense as having a facility sited at your institution gives your institution priority access to the facility, as well as prestige, and makes your institution a destination for researchers.

The life of an observational astronomer

Typically, in an observing run, you wake up at about 3 p.m. During the day the technicians will have installed the instruments that you need for a nights observing run. You go to the telescope at 3 p.m., check to see that everything is installed correctly, since you don't want to wake someone up at 3 a.m. if something breaks, you then spend the next two hours before sunset taking some calibration shots.

After the sun starts setting, your first goal is to find the object that you are trying to photograph. You can punch in the coordinates into the computer, but that will only point the telescope in the general area of the sky that you are interested in. The next thing that you have to do is to take out your star atlas, and look for a pattern of stars that is close to the thing that you are looking for. This is a lot like driving in a strange city when you are looking at the monitor and then trying to match the patterns you see with the patterns on the chart.

So you've now found the object you are looking for. In between these measurements, you take some snapshots of a calibration device. If you are looking at spectra, you take picture of a fluorescent lamp that has lines in certain known positions. If you are measuring brightness, then you need to take some pictures of a star whose brightness is known.

So after a night of all of this, you now have some data on hard disk, and you go to sleep. The next few weeks is where the hard part comes in. You see you have a lot of raw data, but it's not very useful to anyone. The problem is that none of the data has been calibrated. So you spend the next few weeks taking the data, subtracting the black levels, correcting the white levels, stretching and shrinking the picture so that you know what the frequencies of your spectra are. You might also be spending your time doing things like trying to correct for the effects of dust in the galaxy. Through it all, you are probably using an astronomy package called IRAF, which like all big software packages has its cute bugs and idiosyncrasies. At the end of all of this you have a paper, and are ready to publish.

Observational astronomers are often at the mercy of things that are outside of their control. Weeks if not months of effort at setting up an observing run can be destroyed if it happens to be cloudy or raining on the night of the run.

The life of a theorist

Unlike observers, theorists are creatures of the day. The typical theorist spends their days reading papers trying to understand how to model a particular type of phenomenon. Once they have a model, the goal then is to try to get testable predictions from that model, and this often means programming a computer to calculate the consequences of that model. There are occasional flashes of inspiration, but most of the time is spent very slowly and methodically trying to understand the consequences of a model, and to slowly and methodically program the model into a computer and systematically remove the bugs from the model.

There are also a lot of social interactions as theorists argue and debate what a particular observation means, and as theorists and observationalists share ideas about the latest data.

Letting people know about your research

There are a number of channels through which scientific results are made known. The primary means that astronomers use to make others aware of current research are through preprints which are papers uploaded through web servers such as the Los Alamos Preprint Server at http://www.arxiv.org/ or through conference proceedings in which scientists announce their results either through lectures or poster papers. Astronomers are also constantly travelling between departments to give talks on their research at seminars, astronomy lunches, and journal clubs.

Although it has been supplemented by preprints, peer reviewed publication in the primary literature is still considered an essential part of publicizing research. This literature consists of articles in journal such as Astrophysical Journal or Astronomy and Astrophysics. Because of the length of time, typically several months, necessary to go through a peer review, research results are typically shared with the community through preprints before peer review is complete. Nevertheless, astronomers still generally submit their papers to peer review even after the results have been released to the research community, because the interaction between anonymous referees and the paper submitter improves the quality of the work and insures the community that the paper does not have any obvious errors.

Once an astronomy paper is available for publication, it can be accessed through the Astrophysics Data System at http://adswww.harvard.edu/ Increasingly, raw data from sky surveys is being made available on the web.

One shortcoming of the primary literature is that it reports on individual research results without providing context. As such it is difficult for someone who is not actively involved in research in a particular area to understand the relevance of the work. To deal with this problem, primary literature is summarized and combined into the secondary literature, where it will be read by a broader audience of scientists. The secondary literature is a synthesis of the results of recent research in a field. The body of secondary literature includes periodic reviews of progress, such as the Annual Review of Astronomy & Astrophysics, professional books, and other research summaries.

How to be an astronomer

Most astronomers major in either physics or astronomy as an undergraduate and then go to graduate school where they work on the Ph.D. under the supervision of a dissertation advisor. The main challenges in becoming an astronomer are to master the language of mathematics and physics, and to gain experience in working through the scientific process.

After graduate school, an astronomer typically works as a post-doctoral fellow before getting a job either as a professor at a university or a researcher at a laboratory. Because of the large number of graduates gaining Ph.D.'s, people with astronomy degrees are increasingly found outside academia. They work in science related fields, as computer programmers for software companies and even on Wall Street.

Astronomy also is open for amateur astronomers. Most of the data for variable stars is accessible via a telescope which is affordable by hobbyists, and amateurs provide important observational data.

Notes

[2] There are some exceptions to the practice of releasing research results before peer review is complete, and this involves a trade-off between speed and completeness.

When a result is believed likely to be controversial (such as the possible discovery of microfossils on Mars), the researchers may choose to keep the finding secret until peer review is complete so that the result will likely withstand challenges after it is released. Another case where results are kept until peer review is complete involves releasing large datasets such as sky surveys. In this case, the delay introduced by peer review is small in comparison to the benefits of having a through review before announcing the results.

Discussion questions

1) Visit either the Los Alamos preprint site or the Astrophysics Data Service and find a paper. How is the paper structured and what concepts in this text book do you find in the papers?

2) Look at the schedule of several university astronomy departments on the web. What is being discussed? How are different astronomy departments different from each other and how are they the same?

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Current Unsolved Mysteries

Dark Matter and Dark Energy

Dark matter is invisible, but has been postulated from its apparent influence on visible matter. It is one explanation for the observed strength of gravity needed to hold galaxies and clusters of galaxies together. Without considerably more mass than can be detected with telescopes, roughly 10 times more, these systems should simply fly apart. The dark matter theory hypothesizes that matter exists that emits little or no radiation and therefore is not observable with telescopes. Dark matter might also be needed to explain the cosmic microwave background (CMB) power spectrum. Some proposals for explaining dark matter are, for example, particles like weakly interacting massive particles (hypothetical WIMPs) or neutrinos, or massive compact halo objects (MACHOS), or ordinary matter that is hidden somehow, or modified gravity (MOND, MOG, f(R)), or some combination of these things.

Another hypothesis, dark energy, has been proposed to explain the surprising observation that the expansion of the universe is getting faster. This acceleration of the expansion was discovered by measuring a certain type of supernova, called type Ia, in surveys of galaxies. Type Ia supernova are used since they all have the same absolute brightness. (We will discuss why they have the same absolute brightness, and how they are used to measure distances, in a later chapter, General Astronomy/The Death of High Mass Stars). This makes them very useful for measuring distances. It had been anticipated, before the measurements were made, that the expansion of the universe would be decelerating, due to the gravity of all the matter. Dark energy is postulated to explain the apparent repulsive force pushing things apart. Ongoing research efforts are to measure the expansion rate more accurately, and discover the nature of dark energy.

Dark matter is generally accepted to exist, though questions remain. It is probably not regular matter made of protons and neutrons (baryonic matter). That is because a higher density of ordinary matter during the nuclear reactions occurring in the first few minutes of the universe (known as big bang nucleosynthesis), would be expected to produce different abundances of light elements and their isotopes, like deuterium, than are actually observed. Another problem for ordinary matter is that the "clumpiness" of the observed galaxies and the cosmic background radiation is not what would be expected from predictions. Other considerations tend to disfavor neutrinos as dark matter as well.

Alternatives to the dark matter hypothesis are new forces or modified gravity theories. There is speculation that there is another large-scale force that is keeping our universe together. ^{[citation needed]} Another possible explanation is to think of space as a gas-and-space solid. If you place two objects apart from each other then pressurize the area, the two objects will be forced towards each other. This reverses our current ideas of gravity from an object having a pull on other object, to an object being pushed from all directions. (An object alone has no movement, but two objects create an uneven pressure pushing the objects together.)

An estimated 23% of the matter in the universe is dark matter. Ordinary matter only makes up 4% of the universe. The remaining 73% is an even more mysterious, repulsive "dark vacuum energy".

The most popular theory right now is that the repulsive force is actually a property of space itself: it is caused by waves of energy, created by particles and anti-particles popping into existence and then annihilating each other with no net effect. Early in the universe, when there was not much space, the effect was small compared to gravity. But as the galaxies moved apart, the effect became greater. [2]

History and Ideas of Composition

Dark matter was first proposed in 1933 by Swiss astrophysicist Fritz Zwicky to explain the orbital motions of galaxies in clusters. He observed that there was apparently much more mass in a cluster of galaxies than just the visible objects, like stars, gas, and dust. So there was something unseen adding to the mass of the cluster. Later, when X-ray telescopes became operational, they revealed a cloud of hot hydrogen gas between the galaxies, accounting for part of the missing mass. Beginning in the 1960's, Vera Rubin discovered that contrary to Kepler's law that objects orbiting around a central body move slower the farther away they are, that in fact the orbital speed of stars in galaxies remains roughly the same beyond a certain distance from the galaxy core. So there has to be some extra matter, either in the flat disk of galaxies or in a spherical halo around the galactic core. Building on Zwicky’s work she concluded that this extra mass was dark matter. The term dark matter refers to matter that is perceived to be present because of its effect on the objects around it. While the composition of dark matter is still unknown, scientist have proposed some possible candidates that dark matter could be. They are:

Ionized gas — Emits thermal free radiation which cannot be observed.
Dust — Emits radiation and is made up of elements heavier that helium.
Main Sequence Stars — Could be an ingredient, but could not be sole component of dark matter because a great portion of them would be visible.
Black Holes — Are highly unlikely because they would disrupt the binary separations of dark matter. However, not much is known about the explosions that produce black holes, so it is still an option.
White Dwarfs — When forming, white dwarfs produce many intermediate-mass elements (He, N, Ne, C, O) or halo gas, which is not visible.
Neutrinos — Unlikely, but they do have enough mass to be a candidate.
WIMPS or Cold Dark Matter — Weak interacting particles, though they do move at nonrelativistic speeds.

A more in-depth flow chart depicting how the above suggestions are connected is picture below taken from Modern Cosmological Observations And Problems (140). http://ned.ipac.caltech.edu/level5/Bothun2/Figures/dm1.gif

Reionization

The cosmic background radiation was formed when protons and electrons combined to form atoms. The trouble is that we know that the matter between galaxies today is ionized (i.e. it's separate protons and electrons) with clumps of hydrogen atoms. We know this because when we look at all but the most distant galaxies, we don't see the spectra lines of hydrogen. So at some point the hydrogen in the universe reionized. The notion was that starlight caused the hydrogen in the universe to reionize, but the latest observations seem to indicate that this reionization occurred before the first stars were there.

Galaxy Formation

The idea is that galaxies started from tiny fluctuations in density that formed after the big bang. By assuming that the universe consists mainly of cold dark matter, you can almost get the clumpiness that you see with the current galaxies. But there are still puzzles. There is an annoying lack of tiny galaxies, and the rotation curve that cold dark matter predicts, isn't quite the one that we see.

Before the Big Bang

Now to get really speculative, there have been some papers written recently that try to figure out what happened before the Big Bang. One of the strange ideas is that the universe is merely one plane in a multidimensional space, and that what happened was that two membranes in a multidimensional space collided causing a massive expansion in three of the dimensions. This is all really speculative, but the weird thing is that it isn't totally disconnected from observation. The idea is that you can use this model to predict the initial expansion of the universe, and this might have some effects on the ripples that you see in the cosmic microwave background. The big problem is that the matter that began expanding had to have always existed, yet, because of the predictable nature of the elements, it had to have had a definite, external force to set it in motion that could decide when to start the "chain reaction". Something cannot just be in a stable form, or even an unstable form, forever and finally explode, it has to go in a cycle. In other words, consider the following. Out of nothing, a theretofore nonexistent dense mass spontaneously emerged, which erupted in an enormously powerful fireball by its own theretofore nonexistent energy to spontaneously and immediately create from this chaos the defined fundamental forces of physics and the subatomic fundamental particles, which eventually organized themselves into a variety of atomic species, then into molecules, and then into a diverse assortment of inorganic matter that gravitationally assembled itself into this highly structured and precisely ordered universe. We all know that this is ridiculous, but it is equally ridiculous to say "a theretofore stable mass spontaneously became unstable".

With all of these puzzles, its not clear what is going to happen next. There is a lot of data coming in, and it may be that with new data, it will be possible to make our models of the universe work with minor tweaks here and there, and we can go on in the mode of what Kuhn calls "normal science." It's also possible that one day there will be some observation which is like Galileo seeing the phases of Venus — some observation that makes absolutely no sense in the current paradigm of things, and this will force people to fundamentally change how they view the universe.

Discussion Questions

1) Find an old astronomy textbook, and compare it with a very recent one. What mysteries in the old astronomy textbook are now believed to be resolved, and what facts and statements in the old astronomy textbook are now believed to be incorrect?

General Astronomy
Table of Contents	The Modern View of the Cosmos	Observational Astronomy

Observational Astronomy

General Astronomy
The Modern View of the Cosmos	Print version	Motion and Gravity

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The Celestial Sphere

If you look out from an empty field into a dark sky, you will get the impression that you are standing on a flat plate, enclosed by a giant dome. Depth perception fails us for the distant objects we see in the sky. This creates the appearance that all of the stars have the same distance.

The stars appear to move together across the sky during the night, rising in the east and setting in the west, as if they are affixed to the inside of a dome. Because of this, many ancient civilizations believed that a dome really did enclose the Earth. Only a few centuries ago astronomers came to realize that the stars are actually very far away, scattered throughout the Milky Way Galaxy, rather than attached to the inside of a vast sphere.

The old idea remains useful, however. The concept of the celestial sphere provides a simple way of thinking about the appearance of the stars from Earth without the complication of a more realistic model of the universe. Working with the celestial sphere offers a convenient way of describing what we see from Earth. When we refer to the celestial sphere, we are imagining that everything we see on the sky is set on the inside of a huge spherical shell that surrounds the Earth. We will use the reference points of the celestial sphere as the basis for several coordinate systems used to place celestial locations with respect to one another and to us.

The celestial sphere is an imaginary hollow globe that encloses the Earth. The sphere has no defined size. It can be taken to be infinite (or at least really big), with an infinitesimal Earth at the center. The observer is always taken to be at the center of the celestial sphere, even though the observer isn't at the center of the Earth. Our particular position among the stars gives us a particular view. Brighter stars appear closer; stars in nearly the same direction appear nearby each other, even if they are separated by great distances. Our first and most basic look out into the universe is completely stripped of any depth perception.

The celestial sphere can be seen from either of two perspectives. In one perspective, the celestial sphere itself remains still while the Earth turns inside it. In the other perspective, the Earth stands still and the celestial sphere rotates once per day. To an observer on Earth, these two perspectives appear the same. As we think about how we would expect to perceive the rotation of the Earth, we can use this second perspective to guide us.

Everything we see in the sky, we see as though projected onto the celestial sphere. The stars in the constellation Orion, for example, are at a variety of distances, but the differences are imperceptible to us on Earth. Orion's pattern would disappear if we could view it from any other angle or if we could perceive the depth, because the stars would project differently.

Because depth perception is lost, measurements of size are much more difficult. The Sun and the Moon look about the same size in the sky, even though the Sun is really much larger. The Sun appears to be the same size as the moon because the Sun is much farther away simply because the Sun is both 400 times larger in diameter and 400 times farther away than the Moon.

Although we can't easily measure the physical sizes of celestial objects, we can measure their apparent sizes. We do this by measuring the angle an object subtends in the sky. The Sun and the Moon, for example, subtend an angular diameter of half a degree. Most objects in the sky are smaller than this, so it is often convenient to use a smaller measure of angle. For this purpose, astronomers use arc minutes and arc seconds. There are sixty arc minutes in a degree, and sixty arc seconds in an arc minute. Angles this small are near or beyond the limits of ordinary human vision, but they become useful when using a telescope to make observations.

For casual stargazing, observers think about much larger angles. You can easily measure these angles when stargazing by using your hand as your ruler. From arm's length, your index finger has a width of about one degree, your palm measures about ten degrees across, and your full finger-span, including your thumb, is about 25°. This can be useful for estimating the position of a star in the sky, or for gauging the angular separation of two stars.

While the apparent movement of a star across the sky each night, with the celestial sphere, is great, the measurement of an object's movement across the Celestial Sphere as the object drifts through space, is called proper motion, and is measured in arc seconds per year.

To begin thinking about the view of the sky from Earth, we will identify a few points of reference that are fixed to the ground and of importance to astronomers. Some of these are widely known from common experience.

A great circle is a circle drawn on the celestial sphere (or any sphere) which has the center of the Earth as its center. On the Earth, the equator is an example of a great circle. Other lines of latitude are not great circles because their center is not at the center of Earth. Lines of longitude are all great circles because they are always centered on Earth's center. A great circle is the largest possible circle that can be drawn on a sphere.
The horizon is where earth and sky meet. It is the boundary between the portion of the sky that is blocked by the Earth and the portion that is visible. There is a distinction between the local horizon, which is defined by real objects specific to the observer's location such as trees or buildings, and the idealized horizon, which is what the local horizon would be if the ground were completely flat and there were no obstructions. For hypothetical or idealized cases, astronomers use the idealized horizon.
A star's altitude is the angle between it and the horizon.
The cardinal points are points on the celestial sphere that are on the horizon and due north, south, east and west. The North point, for example, is the point due north on the horizon.
The zenith is the point in the sky directly overhead. It is necessarily true that any point on the horizon is 90° from the zenith.
The meridian is the great circle that passes through the North point, the South point, and the zenith and lies on the celestial sphere.

To any observer, regardless of location, these markers stay in the same positions relative to the observer. The zenith is always directly overhead, the horizon is always level, and so on. Observers standing at different places on Earth will have a different view of the sky. An observer in Singapore might see the Sun at the zenith while another observer in New York would not see the Sun at all. These reference points change with the location of the observer.

There are also reference points that are fixed in the sky. These fixed reference points don't move with respect to the stars, but different observers see them in different positions. They are the basis for the fixed coordinate systems that we discuss later. For now, we will identify only the two most useful of these — the celestial poles and the celestial equator.

The celestial equator is an extension of the Earth's equator onto the celestial sphere. If you stand on Earth's equator, the celestial equator will always be directly overhead and pass through the zenith. It will run from the East point up to the zenith and down again to the West point. Anywhere you stand on Earth, the celestial equator will intersect the East and West points on the horizon. The nearer you are to the equator, the nearer the celestial equator come to the zenith. At the North Pole or the South Pole, the celestial equator lines up with the horizon.

Like the celestial equator, the celestial poles are an extension of the Earth's pole onto the celestial sphere. The North Pole extends out into space to create the North Celestial Pole. Likewise the South Pole creates the South Celestial Pole. In the Northern Hemisphere, only the North Celestial Pole is visible because the South Celestial Pole is below the horizon. In the Southern Hemisphere, only the South Celestial Pole is visible. At the equator, the North Celestial and South Celestial Poles would lie on the horizon where the meridian intersects the horizon.

Polaris is called the "North Star." It can be found at the last star on the "handle" of the Small Dipper. The last two stars of the "cup" of the Big Dipper are called the Guardians (or "Pointers"), and point to Polaris in the sky. Polaris is special because the Earth's North Pole points almost exactly towards it. This means that Polaris will always appear to be due north to any observer, and it will always stay in the same position on the sky.

Often, beginning stargazers assume that Polaris must be a very bright or prominent star. This is not really the case. Polaris is only remarkable because it is almost exactly in line with Earth's axis of rotation. Because of this, Polaris always remains at nearly the same place in the sky. For example, Shakespeare made reference to Polaris in the play Julius Caesar:

I am constant as the northern star,

Of whose true fixed and resting quality

There is no fellow in the firmament.

Julius Caesar, William Shakespeare's Julius Caesar, III.1.65-68

Though it must be pointed out that Shakespeare actually got it wrong. At the time Shakespeare wrote Julius Caesar Polaris was indeed the pole star but in Julius Ceasar's time Polaris was not the pole star.

The fact that Polaris always stays in the same position due north has given it much fame. It also makes Polaris a useful reference point for navigation — Using geometry, it is easy to show that the angle Polaris or the celestial pole makes with the horizon is equal to the observer's latitude. In the diagram, the angle $\angle d$ is the observer's latitude. The pole and the equator are at right angles, so

\angle d+\angle a=90^{\circ }.

or $\angle a=90^{\circ }-\angle d.$ Since the angles in a triangle add to 180°, we know that

\angle a+\angle b+90^{\circ }=180^{\circ }.

When we combine these two equations, we have $\angle b=\angle d$ . The angles $\angle b$ and $\angle c$ are alternate interior angles, so

\angle b=\angle c

and

\angle d=\angle c,

which means that the angle between the pole and the horizon is the same as the observer's latitude. This fact was once used by navigators at sea, who could easily find their latitude by measuring the position of Polaris.

Like many things in astronomy, the celestial sphere can be very difficult to visualize because of its three dimensional geometry. A visit to a planetarium or a session under the night sky can be very helpful to you in developing a conceptual understanding of the celestial sphere. In the absence of the opportunity for these, it can be helpful to try to draw diagrams such as the one at the beginning of this section for yourself.

To begin drawing a celestial sphere such as the one above, you only need to know the latitude of the observer. Then imagine that the spot where the observer is standing is the "top of the world"; draw circle for the earth, and draw an observer standing at the top. Now draw a much larger circle around that; this represents the celestial sphere.

Since our observer is always on top of the Earth, the features on the celestial sphere that are defined relative to the ground will always be in the same position on the sphere. The zenith is the point directly above the observer's head, at the top of the celestial sphere.

The next important reference is the horizon. The horizon will be horizontal on the diagram. Remember that the celestial sphere has no specific size relative to the Earth, regardless of how you've drawn it. Draw the horizon across the middle of the celestial sphere, so that it's center is the same as the center of Earth. Markers such as the horizon are always idealized, so it doesn't matter whether your observer's view of the sky is actually cut off at the position marked by the horizon.

The next reference points we'd like to place are the North Celestial Pole and the South Celestial Pole. Think about what the orientation of the pole should be given the observer's latitude. If the observer is at the equator, the pole should go horizontally through the Earth. If the observer is at one of the poles, the pole should go through the Earth vertically. Extend the Earth's poles out to the celestial sphere and mark the intersections as the North Celestial Pole and the South Celestial Pole.

If we're in the northern hemisphere, the North Celestial Pole will be above the northernmost point on the horizon, and the South Celestial Pole will be on the opposite side of the celestial sphere, below the horizon. If we're in the southern hemisphere, the situation is reversed. Remember to check that the angle the horizon makes with the pole is about the same as the observer's latitude.

For any given latitude, one can build an appropriate celestial sphere. First, consider the sky in relation to the earth. Take the north and south poles and extend them into the sky; these become the north and south celestial poles. The Earth's equator can be projected outward to form the celestial equator. We'll get something that looks like the picture above.

When you're done, you should have a celestial sphere very like the one at the top of this section.

A celestial sphere forms the basis for the application of many coordinate systems. For example, the horizon and the celestial meridian together form the reference circles for giving the position of stars in terms of altitude and azimuth, making it easier for one to find them on the night sky. The celestial sphere is also a natural system for describing the motion of the sun. In order to explore these concepts, however, it is necessary to understand just how the celestial sphere changes for an observer at a given latitude. As we consider the daily rotation of the Earth, we'll see that your perception of the daily motion depends very much on your latitude.

As you look at the sky, your mind will naturally identify obvious patterns. The Big Dipper and Orion are two very prominent groupings of stars, and others stand out all over the celestial sphere. These asterisms are guideposts to the night sky. You can use them to keep your bearings when you look at the sky. The appearance of the night sky has remained much the same for millennia. Many of the ancient civilizations across the globe invented stories about the sky.

Often, the groups of stars are called constellations. Constellations have a very long history in astronomy, dating back thousands of years. Early in the twentieth century, a list of constellations was formally established by the International Astronomical Union, a widely recognized body of astronomers. The IAU identified constellations that would be used in astronomy and defined specific boundaries to unambiguously establish which constellations each star belonged to. It's easy to learn a few of the most prominent constellations so that you can find your way around the night sky. Beginning with a few easy-to-find landmarks you can find the rest by using familiar stars as guideposts.

Another useful guide in the sky is the ecliptic. The ecliptic is an imaginary line in the sky that the sun draws. The ecliptic is even with the plane of the Earth's orbit around the sun; thus, all of the main planets and the moon should be found relatively close or on the ecliptic, because the solar system is mostly flat. Also, along the ecliptic are the 12 constellations of the zodiac. Thus, by finding some of the main zodiac constellations in the night sky, one can determine if certain objects they see may or may not be planets by whether or not they lie on the ecliptic.

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Phases of the Moon

Like the stars and planets, the Moon doesn't stay fixed in the sky but slowly moves as the Earth rotates and as the Moon moves through its orbit about the Earth. To someone taking a casual glance at the Moon, it seems as fixed as the stars. But observation of either the Moon or the stars over a period of several hours will reveal their diurnal (daily) motion across the sky. The Moon rises and sets each day. An observer who watches the Moon over the course of many days will notice the Moon moving not only with the stars, but among them. Every month, the Moon completes one fewer pass across the sky than the stars have completed. We see this because of the Moon's orbit about the Earth. As the Moon progresses through its orbit, its rising and setting times change. Each day, the Moon rises and sets fifty minutes later than the day before.

The moon usually takes 27 days to rotate once on its axis. So any place on the surface of the moon experiences about 13 days of sunlight, followed by 13 days of darkness. Temperatures on the Moon range from -153 C at night to 253 C during the day. For example if you were standing on the surface of the moon during sunlight hours it would be blazing hot. When the sun goes down, the temperature automatically drop 250 degrees in just a matter of moments. Furthermore, there are craters around the North and South poles of the moon which never seen the sunlight. These dark places would always be as cool as -153 C. However, there are nearby mountain peaks that are covered in continuous sunlight, and would always be hot.

The "dividing line" between the light and dark halves of the globe is called the terminator, as it terminates the area of darkness (and also that of daylight).

Typically, one-half of the Moon will be lit up by the Sun, while the half facing away from the Sun remains dark. (The only exception occurs during a lunar eclipse, when the Earth blocks the light falling on the lit side of the Moon.) The part illuminated by the Sun is not, it should be emphasized, always the same portion of the Moon's surface! Like the Earth, the Moon turns on its axis, exposing different areas at different times. In combination with the orbital revolution of the Moon around the Earth, this phenomenon creates the phases of the Moon as seen from Earth. The phrase "Dark side of the Moon" arose before the age of artificial satellites and the back side of the Moon could not be observed. Hence, the that one side was unknown or "dark".

The tilt of the moon’s spin axis is only 1.54 degrees and as a result, lunar seasons are barely noticeable in most locations on the Moon. However, at the North and South poles, the height of the sun above the horizon varies by more than 3 degrees over the course of the year. In other words, it affects the percentage of sunlit regions and surface temperatures at the poles. Furthermore, the coldest areas are located in doubly shadowed regions inside small craters, in which they are located within the permanently shadowed regions of larger craters. Temperatures are as low as 35K (-238 C or -397 F) in these areas, even at noon on the warmest day of the year.

Some half of the Moon is always illuminated but the fraction of the illuminated part or the Moon's phases directly depend on the relative positions of the Earth, Moon, and Sun. Simply put, it's a matter of how much of the daylight side of the Moon we can see from our current viewing angle. The phase will depend on how much of the side facing toward us is illuminated at any given time. The sketch below illustrates the phases of the moon for various Earth-Moon-Sun positions (the Sun is presumed to be off of the diagram to the right):

Next to each "Moon" is a black-and-white sketch of the phase as it would be seen from Earth when the Moon is in that position. When the Moon is between the Earth and the Sun, the sunlit side of the Moon is facing completely away from us, and therefore we have the dark "New Moon". When the Moon reaches the other side of the Earth, the sunlit side will be fully toward us, and we have the "Full Moon". As the Moon moves from New to Full and the sunlit side grows increasingly large, we say the Moon is waxing; as we see less, in the decline from Full to New Moon, we say it is waning.

Midway between the Full Moon and New Moon, half of the sunlit side of the Moon is visible from the Earth. Because a half of the half illuminated Moon can be seen, this is referred to as a "quarter Moon". When the Moon is waxing and reaches this position, it's called the "first quarter Moon"; when waning, the "third quarter Moon." When less than a quarter-moon is visible, it's referred to as a "crescent Moon" - waxing crescent or waning crescent, as appropriate. When more than a quarter-moon is visible, it's referred to as a "gibbous moon", again, waxing or waning.

The Moon's orbit and rotation speed is just such that the Moon always shows the same side to Earth, aside from only a slight "wobble." The pattern of markings on the side facing Earth is very familiar in history and culture. Western society has long imagined a face in the markings — the "Man in the Moon." Other cultures have seen a woman, a rabbit, a frog or other creatures. The Moon always shows this same face to Earth because its rotation is "locked" with its orbit, for reasons we will see later, when we discuss gravity. More precisely, the time it takes for the Moon to complete a trip in its orbit is the same as the time it takes for the Moon to rotate once around its axis. Because we see the Moon moving around us, it appears as though the Moon isn't turning at all.

If you stood on the Moon and looked up at Earth in the sky, you would see that it never rises, never sets, and never moves in the sky at all. Imagine, for example, standing at the middle of the face of the Moon that we see. From there, the Earth would always remain straight overhead. If you stood at the edge of the face we see from Earth — the "limb" — you would always see the Earth on your horizon.

Up to now, we have considered the time for the Moon to complete one orbit around the Earth to be the same as the time for it to pass once through its series of phases, but this is not quite right. The Moon's phase at a particular point in its orbit changes as the Earth goes around the Sun. Once the Earth has gone halfway around the Sun, the position of the Moon for a given phase has also moved halfway around the orbit, since the Sun is on the opposite side. Thus, it takes a little longer for the Moon to go through its phases than it does for it to go through its orbit about the Earth.

Suppose a full moon marks the beginning of both the period of the orbit of the Moon about the Earth and the period of the orbit of the Earth about the Sun. At the time of full Moon, the Sun, Earth, and Moon are aligned. Once the Moon returns to that position in its orbit, the Earth has moved a little around the Sun. Now, the Moon is not aligned with the Earth and the Sun. It takes about two days before the Moon has moved back into alignment with the Earth and Sun line (synodic month). The time for the Moon to complete an orbit, called a sidereal month, is about 27 days and 8 hours. The time to move through its phases, a synodic month, is about 29 days and 12 hours.

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Eclipses

An eclipse occurs when one casts a shadow on another object. A lunar eclipse happens when the shadow of the Earth falls on the Moon, because the Earth is in a position where it blocks the light from the Sun. Likewise, when the Moon interposes itself between the Earth and Sun, so that the Moon's shadow falls on the Earth (and the Sun is temporarily blocked from view), a solar eclipse is the result.

Eclipses occur when the Sun, Moon, and Earth align. An eclipse blocks sunlight headed toward either the Moon or the Earth, depending on whether the eclipse is a solar eclipse or a lunar eclipse. Such an alignment can only happen when the Moon is in new phase or full phase. If the Moon is new, then it's possible for a solar eclipse to occur. During a full moon, there might be a lunar eclipse. During a solar eclipse, the Moon's shadow falls on the Earth, and the Moon blocks the Earth's view of the Sun. During a lunar eclipse, the Earth's shadow falls on the Moon, and the Moon appears to be darkened as we see it from Earth.

As one of the rare and very spectacular celestial events, a solar eclipse is well worth seeing if you should ever have the opportunity. When viewing a solar eclipse, however, it is important that you avoid damage to your vision by taking steps to view the eclipse safely. (There is no danger in viewing a lunar eclipse, during which the observer never looks at the Sun.) The simplest way to view a solar eclipse is to project the sunlight through a pinhole onto a card or a sheet of paper. The sunlight on the paper will form an image of the Sun. As the eclipse progresses, you will see a round "bite" that the Moon's shadow has taken out of the Sun. If possible, you might also find a special filter, which you can look through directly to watch the eclipse. Solar filters make it possible to look directly at the Sun without harming your vision. Be sure to use a filter specifically designed for looking at the Sun — sunglasses, welder's glass, and other filters are generally not sufficient. Don't use a telescope or binoculars unless the filter was designed for that purpose. Even near totality, when the Sun is almost completely covered, you risk damage to your eyes by staring into the Sun. Once totality arrives, however, it is safe to look directly. It is also safe (though not advisable) to look for a brief moment at the uneclipsed Sun. When viewing an eclipse, there's no need to be paranoid about the danger of accidentally glimpsing the Sun's surface for a split second. If your eyes catch the bright surface of the Sun, simply turn your head away.

The shadow of the Earth on the Moon (or any shadow cast by the Sun) has two parts. The inner part of the shadow, called the umbra, is the darkest part. If you were to stand on the Moon under the umbra, your view of the Sun would be completely blocked. Under the outer part, the penumbra, the Sun is only partially blocked. This is why the Moon is more darkened under the umbra than under the penumbra. While the location of the boundary of the umbra and penumbra are precisely defined geometrically, there is no sharp shadow line between the two when looking at an eclipsed Moon. At the innermost part of the penumbra, only a tiny fraction of the sunlight reaches the Moon, and this part is almost as dark as the totally eclipsed umbra. During a solar eclipse, when the Moon's shadow falls on the Earth, we stand in the shadow rather than looking at it. Observers standing in the umbra see a total eclipse of the Sun, while observers under the penumbra see a partial eclipse.

The present average distance of the Moon from the Earth happens to be just right for the Sun and the Moon to have almost the same apparent size as viewed from Earth. The Sun is 400 times larger than the Moon, but it's 400 times farther from Earth. Because of this, the Moon can obscure the bright surface of the Sun while leaving the fainter outer part, called the corona, exposed. The result is dramatic. During a total solar eclipse, the seething outer layers of the Sun are visible around the Moon. An annular eclipse is possible when the Moon is near the most distant part of its orbit, a point called the apogee. In an annular eclipse, the Moon passes directly in front of the Sun but fails to block it completely, creating the appearance of a bright ring, or an annulus.

Since it's possible for an eclipse to occur twice a month, once of the Sun and once of the Moon, at each new and full moon, students often wonder why eclipses don't occur twice a month, each time the Moon passes through its new and full phase. Months often pass with neither a solar or lunar eclipse. Eclipses don't occur in these months because the Moon passes just north or south of the Earth's shadow during the full moon. Likewise, the Moon's shadow misses the Earth as it passes through new moon. This happens because the orbit of the Moon is not perfectly aligned with the Earth's orbit. The Moon's path is tilted 5° to the ecliptic. It only passes through the ecliptic at two points in its orbit. These points are called the nodes.

Whenever the Moon isn't located at a node, it is either north or south of the Earth-Sun line. Eclipses can only occur when the Moon is at one of the nodes and at new or full phase — when the Moon's orbit is positioned so that the nodes are aligned with the Earth-Sun line. This happens about twice a year, at intervals called eclipse seasons. The time of eclipse season changes over the course of many years, but consecutive eclipse seasons are about six months apart. In any year there are at least two eclipses, and there may be as many as seven, all occurring during the eclipse seasons. There are typically around four eclipses in a year.

Eclipses can only happen when the moon is at a node in its orbit - a point where the Moon's orbit around the Earth intersects with the Earth's orbit around the Sun. If the Moon crosses one of these nodes while it's in the full or new moon position, a lunar or solar eclipse (respectively) can occur.

In a lunar eclipse, the observer watches the Earth's shadow fall on the Moon. Observers don't need to be in any special place to see the eclipse, so long as the Moon is visible. Consequently, an observer on the side of Earth facing the Moon will be able to see the eclipse. Fully half of the Earth will have a view. In a solar eclipse, the observer stands in the shadow of the Moon. The umbra of this shadow is small, so only a small portion of the Earth will see a total eclipse. A much larger portion will see the Moon only partially blocked, and will have a partial eclipse.

The Moon will often become reddish or orange in color during a lunar eclipse. The color of eclipses varies, with the Moon sometimes becoming very red and at others only darkening with no change in color at all. The Moon reddens during the eclipse because sunlight filters through the Earth's atmosphere and is scattered into the Earth's shadow. Sunlight reddens as it is scattered by dust and smoke in the Earth's atmosphere; this is why sunsets are red. The reddened sunlight illuminates the eclipsed Moon, making the Moon brighter and redder. The color and darkness of an eclipse depends on the geometry of the eclipse and the weather on Earth during the eclipse.

Two phenomena caused by eclipses are the Diamond Ring and Baily's Beads.

Throughout history, astronomers have relied on cycles in their observations. The year, for example, is the cycle of the Earth's orbit around the Sun, and the seasons follow with this cycle, as do the motions of the stars. Since ancient times, astronomers have known about an eclipse cycle, called the Saros cycle. they can be used to predict in advance when an eclipse will occur and what the eclipse will be like.

The Saros cycle depends on three other cycles of the Moon: the time from one full moon to the next — the synodic month; the time it takes for the Moon to pass through a node and come all the way around — the draconic month; and the time it takes for the Moon to come close to the Earth in its orbit, move away, and come close again — the anomalistic month. In one Saros cycle, there are almost exactly 242 draconic months, almost exactly 223 synodic months, and almost exactly 239 anomalistic months. The Saros cycle is the smallest length of time that evenly contains all of these cycles. This is why eclipses repeat within the Saros cycle.

Stonehenge is an ancient structure in Britain. It is made up of 30 monoliths in the Sarsen Circle. There are 56 Aubrey Holes outside of Stonehenge. It is believed that Stonehenge was used for predicting eclipses.

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Daily Motions

Every day, the Earth turns once on its axis. At the Earth's equator, we move around the center of the Earth with speeds near a thousand kilometers per hour because of this rotation. We feel nothing, but we can see the effect of Earth's rotation. An observer on Earth sees everything in the sky appear to rotate around Earth at the rate of once per day. This motion is not apparent to an observer who steps outside for a moment, since it is so slow. For the motion to become apparent, an observer must watch the sky for a period of hours. During the day, the motion manifests itself in the movement of the Sun, which proceeds from the East in the morning towards the west in the evening. At night, the constellations move, seeming to circle about the pole.

Because the daily motion of the stars is driven by the same mechanism that drives the motion of the Sun, the stars move in almost exactly the same way that the Sun moves. Indeed, everything in the sky moves in almost exactly the same way over the course of a day. The motion of the Sun and the planets only differ at all because the planets have their own motion around the Sun along their orbits, and is noticeable from Earth.

As the Earth spins, it appears to us that the celestial sphere is spinning. It spins about the Earth's pole, so that the celestial poles appear stationary and the stars and planets seem to move in circles around the poles. Stars move at 15° an hour.

The schematic illustrates the daily motion of the celestial sphere resulting from the spin of the Earth. The Sun, Moon, planets and stars make circles around the Earth. Since the observer is tilted with respect to Earth's rotation axis, the daily motion also appears tilted. As a result, celestial objects appear to rise and set at an angle.

Looking at the diagram, we see that stars should rise roughly in the East and come up at an angle. In the Northern Hemisphere, the stars move to the south as they rise, while stars move northward in the Southern Hemisphere as they rise. To an observer standing at the equator, the axis of rotation is horizontal, and stars rise in the East perpendicular to the horizon.

Since the stars seem to rotate around the celestial pole, some of the stars very near the pole never rise or set. Certainly, Polaris never sets — it remains fixed. Only an observer very close to the equator will see any of the stars in the Little Dipper rise or set. The stars close enough to the pole never seen to rise or set are called circumpolar stars. These stars always remain above an observer's horizon. Different stars are circumpolar to different observers.

Imagine standing at the North Pole. You would see Polaris overhead with all the other stars spinning around it. In reality, you are the one who is spinning. From the North Pole, all of the motion of the stars is horizontal. Stars at the horizon skim along the horizon, never rising or setting. The stars higher in the sky also move horizontally, never moving up or down. From the poles, all visible stars are circumpolar.

At the Earth's equator, the situation is different. The celestial poles appear at the horizon on the north and south points. As stars move around the poles, they all rise and set, no matter how close to the pole they are. From here, there are no circumpolar stars. As you move from the equator to the pole, you will see gradually more and more stars become circumpolar, until on finally reaching the pole you find that all stars are circumpolar.

For centuries, the day has been the most fundamental unit of time in the calendar. Measuring the passage of days is as easy as counting sunrises or sunsets. The earliest clocks, the predecessors to the sundial, worked by tracking the daily motion of the Sun across the sky.

A sundial uses the position of the Sun to give the time. (The gnomon of a sundial can be used to find north: the shortest shadow cast by the sun (at noon) points north.) Of course, not all clocks work by measuring the Sun, although all work by measuring some reliably periodic and regular process. A wristwatch, for example, measures the oscillations of a quartz crystal. Atomic clocks use the natural period of oscillation of cesium atoms to measure time. Other systems of time are tied to the motion of the Earth, but there are a variety of ways to measure time in this way. The time measured by any particular method is not guaranteed to agree with the time measured by another, so it is sometimes necessary to convert between different "kinds" of time.

The time measured by sundials is called local solar time. The local solar time progresses according to the Sun's motion around the Earth. Since the Earth's speed in its orbit around the Sun changes slightly over the course of a year, this motion is not completely uniform. Sometimes, when the Earth is moving faster in its orbit, the solar day is shorter. Early timekeepers never noticed this slight difference, but the appearance of accurate mechanical clocks made it possible to measure the small changes in the length of the day.

To deal with this problem, astronomers invented mean solar time. The mean solar time averaged the length of the day so that each day was of the same length. This standard is up to fifteen minutes off from the true solar time, but is much more convenient for clocks that are accurate enough to see the difference. Over the course of a year, the local solar time drifts, but the two always agree after a full year has passed.

For much of history, every town in the world kept a slightly different "correct time" from every other town. Under local solar time, noon occurs at the moment the Sun passes through the observer's meridian. Two observers in different locations will observe local noon to occur at different times. This is why local solar time is called "local." This difference wasn't a problem when transportation and communication were slow, but the advance of trains and telegraphs made even a small difference between nearby towns important. To deal with this, the time zones were developed. The standard time was defined to be the time at a nearby line of longitude. For most locations, the time zone was offset by an round number of hours from the time measured at Earth's prime meridian.

For historical reasons, the prime meridian is based on the line of longitude through the observatory in Greenwich, England. That time zone is known as Greenwich Mean Time, and is often used as a standard for when one wishes to compare times without taking local time zones into account. For this reason (since it is used universally) it is also called Universal Time. This is the most commonly used form of solar time, and the most commonly used measure of time in general.

It's very natural to use the Sun as a standard of time, since the cycles of day and night are so important to life on Earth. For this reason, most people think of a day as the time it takes the Sun to move through the sky once. The apparent motion of the Sun in the sky is similar to the motion of everything else, but not exactly the same. The daily motion is driven almost entirely by the rotation of the Earth, but the Sun's motion differs from the stars' motion. This happens because the Earth is moving around the Sun. The Earth moves a little in its orbit while a it rotates and day passes. Because the Earth's position has changed, a full rotation doesn't quite bring the Sun all the way around the sky. This means that the Earth makes a little more than a full rotation over the course of a solar day.

A sidereal day is the amount of time it takes for the stars to go once around the sky, equal to 23 hours and 56 minutes. The word sidereal means "relating to the stars." The difference in length between the solar and sidereal day causes the rising and setting times of stars to change throughout the year. If the star Rigel, for example, rises at noon today, it will rise at 11:56 tomorrow. In six months, it will rise at midnight. Because the difference between solar days is tied to the orbit of the Earth, there is exactly one more sidereal day in a year than there are solar days.

A sidereal month is the period of the moon in relation to the stars; approximately 27 (1/3) days (13 degrees a day). Ancient peoples used this period to track time, as evidenced by the Big Horn Medicine Wheel in Sheridan, Wyoming.

Sidereal time is widely used in astronomy because it can be used to tell which stars will be up. At a given sidereal time, the stars in the sky will always be the same. Accurately finding the position of stars is easier than finding the position of the Sun, so measurements of the solar time are usually based indirectly on the sidereal time. Modern techniques measure time from atomic clocks, which are then tied to the motion of the Earth by measurement of the sidereal time.

A synodic month is one new moon to the next; approximately 29.5 days. This is about two days longer than the sidereal month. The reason for this is that while the Moon is orbiting the Earth, the Earth is orbiting the Sun. Therefore, the Moon has to travel more than 360 degrees to return to its original position above the same meridian on the Earth.

For anyone living on the coast, tides play an important role in everyday life. Despite this, the cause of the tides was a mystery for centuries, before gravity was well understood. People have always suspected that tides are related to the moon, since high tides always occur when the Moon is highest in the sky and again when the Moon is lowest below the horizon. This turns out to be correct.

Tides occur because the gravitational pull from the Moon is greater on the side of the Earth facing the Moon than on the side facing away. As a result, the gravitational pull "stretches" the Earth. The tidal pull from the Moon gives the Earth an oblong shape. The pull affects the ocean more strongly than the ground, because the ocean is liquid and less resistant to movement.

The stretching effect from the tides creates two places on Earth where the tides are high, with one high tide on the side of the Earth facing the Moon and the other on the side opposite the Moon. As the Earth rotates, the locations under the positions of high tides also move. If the moon didn't orbit around the Earth, high tide would occur exactly twice a day, every 12 hours. The orbit of the moon changes the position of the high tides over the course of the month, which lengthens the time between high tides to about 12½ hours.

The Sun also has a tidal influence on the Earth, although this is a much smaller effect than the Moon's tidal influence. Because of the difference in the gravitational pull from the Sun on opposite sides of the Earth is much smaller, the Sun's tendency to make the Earth bulge is much less. Still, the contribution to tides from the Sun is noticeable. When the Sun, Moon and Earth are aligned, the Sun adds to the Moon's tidal pull, making the tides greater. This is called the spring tide. (The spring tide has no connection with the season of spring.) When the Moon is at a right angle with the Sun, the Sun's tidal pull interferes with the Moon's, making the tides weaker. This configuration is called the neap tide.

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Yearly Motions

Why do we have seasons? A little thought will suggest that it can't have much to do with the Earth's distance from the sun, as that would affect the Southern and Northern Hemispheres at the same time. (In fact, the Earth is slightly nearer to the sun around December than at other times of the year.) Why, then are there seasons?

Every year the Earth completes one orbit of the Sun. We see this observe the effect as the change of the seasons and the movement of the constellations. Over the course of a year, the Sun moves through a great circle on the celestial sphere, tracing out the same path year after year. This path is called the ecliptic. The ecliptic is not only the path of the Sun in the sky, it also marks the plane of the Earth's orbit of the Sun. The planets orbit the Sun in different planes but near to the ecliptic.

The Earth is tilted with respect to the ecliptic plane by 23.5°

The axis of the Earth's rotation is tilted by 23½° with respect to the plane of the ecliptic. Globes are typically built with an inclined rotation axis. The 23½° North latitude is marked as the Tropic of Cancer and the 23½° South latitude is marked as the Tropic of Capricorn. In the Northern Hemisphere, the Sun will pass directly over head only between June 20 and 22th along the Tropic of Cancer. That day is called the day of the summer maximum or Solstice in the Northern Hemisphere. In the Southern Hemisphere, the Sun will pass directly over head only once between December 20 and 23rd along the Tropic of Capricorn. Anywhere between the Tropic of Cancer and the Tropic of Capricorn, the Sun will pass directly overhead at least twice during the year, but the Sun will never pass overhead for people living outside the tropics. Within the Tropics, over the course of a year, the Sun's position in the sky changes, beginning in the southern sky about December 21, moving to the northern sky in mid-year, and ending the year back in the southern sky.

The tilt of Earth's rotation axis causes a 23½° tilt of the great circle of the ecliptic with respect to the Earth's equator. The ecliptic and the equator intersect at two points, but are otherwise separated by up to 23½°. When the sun lies at one of the intersections, it is directly overhead somewhere on the equator. This occurs at the equinox, and the points on the sky where the equinox intersects the equator are also called equinoxes. Once every year, the Sun passes through the equator going north. This happens in late March — the "vernal" or "spring" equinox. The "autumnal" equinox occurs when the Sun passes through the equator in late September.

On the equinox days, the day and night are equally long. This is the origin of the name equinox, which is from Latin for "equal night." On the day of the equinox, the Sun rises due east and sets due west. It doesn't rise to directly overhead, though, except for observers on the Equator. The equinoxes are the only days of the year that have twelve hours of daylight and twelve hours of dark. After the vernal equinox, moving into Northern summertime, the Sun begins rising in the northeast and sets in the northwest. Days in the Northern Hemisphere become longer, while days in the Southern Hemisphere become shorter.

The points at which the Sun is at its greatest distance from the equator are called the solstices. The solstices mark the longest and shortest day of the year. The longest day of the year is the summer solstice and the shortest day is the winter solstice. In the Northern Hemisphere, the summer solstice occurs when the Sun is farthest north, while the winter solstice occurs at the Sun's southernmost point. In the Southern Hemisphere, the solstices are reversed.

Viewed from space, we see that the Earth's tilt changes the exposure of different parts of the Earth to the Sun. Observers in the Northern Hemisphere will see the Sun at its lowest position in the southern sky, about December 21. They see it this way because the Southern Hemisphere is tilted towards the Sun and the Northern Hemisphere is tilted away. About June 22, the situation is reversed, with the Northern Hemisphere pointed toward the Sun, and the Sun will be in its extreme high point in the sky at solar noon. For an observer in the Southern Hemisphere, the Sun will appear at its lowest point in the sky in the north, about June 22, while the Sun will appear at its high point in the sky about December 21. One effect of this phenomenon is that during the months of Northern Hemisphere summer, the North Pole will be able to receive sunlight twenty-four hours a day. The Sun will remain visible through much of the autumn, passing below the horizon at the autumnal equinox. As winter sets in at the North Pole, the Sun will not be seen for six months, while that portion of the Earth is tilted away from the Sun.

As one moves toward the Earth's equator from either pole, this effect becomes less severe. The nearer one is to the equator, the less difference there will be between the number of hours of illumination and night hours. At the equator, there's practically no difference between the length of day all through the year.

Clearly, the annual motion of the Earth around the Sun is the cause of Earth's seasons. What effect gives rise to this seasonal change is less obvious. At first glance, one might think that winter occurs when the Earth is farther from the Sun. If we realize the Northern and Southern Hemispheres have winter at different times of year, we see that this can't be right. Also, the Earth's orbit is very nearly circular. The change in the Earth's orbital distance is much too small to have a noticeable effect on Earth's climate.

Certainly the length of time each day during which sunlight falls on a particular location has a great deal to do with the seasonal changes in temperature. However, another effect less obvious, but more influential is the angle at which the sunlight hits a region. At the equator, there is little difference throughout the year as the Sun varies by 23.5 degrees on either side of the vertical. The length of the ray's path on Dec 21 at solar noon is increased by a factor of only 1.1 from a direct vertical path and the reduction of the sunlight is small. The more direct radiation gives the maximum amount of heat and energy to the earth where it falls, and therefore these areas will receive the most warmth. Away from the equator, however, the Earth's tilt means that sunlight is not received so directly and a greater amount of the Sun's energy is blocked by the longer path it takes through the atmosphere. At 50 degrees north latitude, the path the Sun's rays travel through the atmosphere on Dec 21, at solar noon, will be increased by a factor of 3.5 from a direct vertical path. In general, the rays will come at an angle that depends on the time of day, the latitude of the region from the equator, and the position of Earth in its orbit.

The constellations in the ecliptic, the zodiac, have a long history in the tradition of astrology. In most newspapers, you can read a (completely unscientific) prediction of your future or some personal advice, specific to your birthday. Each entry is associated with one of the constellations of the zodiac and a range of birth dates. In the tradition of astrology, the constellation the Sun occupied on your birthday, your "sign," reveals information about your personality and your future.

Interestingly, the dates given for each constellation in the newspaper don't match the Sun's position in the sky for those dates. There is a mismatch between the date in the newspaper and the real position of the Sun of a little more than a month. The mismatch appears because the dates corresponding to each sign were set thousands of years ago. Over the course of thousands of years, the Earth "wobbles" on its axis, causing the calendar and the positions of the stars in the sky to shift. This wobble is caused by the pull on the equator by the sun and moon, and is called precession. It affects the positions of all the constellations with respect to the equinoxes and the pole.

The precession of the Earth is like the movement of a top. If you spin a top with the axis tilted, the axis will slowly rotate as the top spins. Likewise, the Earth's axis remains tilted at 23½°, but the orientation of this tilt changes over the course of thousands of years.

Since precession changes the direction in which Earth's pole points, it also changes which star is the North Star, if any. Earlier, we quoted Shakespeare, who referenced Polaris in Julius Caesar, describing it as the northern star. Strictly, this would be incorrect. Polaris was not "fixed" in the sky in Julius Caesar's time because Earth's axis was pointed differently, toward the Big Dipper.

Precession is a slow drift, and a difficult motion to detect. The motion of the stars from precession only becomes noticeable to the unaided eye after many, many years of careful observation, although it becomes very quickly noticeable through a telescope. The Greek astronomer Hipparchus was the first to measure the precession by comparing his own observations to observations collected a century and a half before.

Precession changes the position of the Earth in its orbit for the solstices and the equinoxes. As the Earth's axis turns over, the moment when it points most closely towards the Sun changes, and so the seasons change. If a calendar didn't account for this, the seasons would drift as the axis precessed. Eventually, the Northern Hemisphere would be cold in July and warm in January, and the Southern Hemisphere would have warm July weather and cold January weather. The calendar takes the extra motion of precession into account by using the tropical year as its basis. The year as we usually define it is a sidereal year, the time it takes for the Earth to make one orbit around the Sun. In one year, as we usually measure it, the Earth really completes a little more than a full orbit around the Sun. During a sidereal year, the Sun moves fully around the sky and back into the same position with respect to the stars. In a tropical year, the Sun goes from the Vernal Equinox, around the sky, and back to the Vernal Equinox again. During this time, the equinox has shifted slightly in its position, so that a tropical year is a bit shorter than a sidereal year.

It's easy to identify the progression of the calendar if you take careful notice of the sky. Next time you see sunrise or sunset, take notice of whether the Sun is setting due west or just north or south of west. Many ancient cultures watched the motion of the Sun carefully and over long periods of time. Using simple techniques and tools, they were able to measure periods like the length of a year very accurately. Ancient people who took notice of celestial motion would have found that the summer solstice occurred every 365 days. They would also notice that the solstice was delayed an extra day every four years. This is the reason for the leap year in the modern calendar. The delay occurs because the length of the year is a little more than 365 days — closer to 365¼ days long. By taking some simple observations over a period of a few years, it is possible to measure the length of a year to surprising accuracy using this technique.

Solar calendars have been used throughout history. The ancient Babylonians thought the year had only 360 days, and made their calendar accordingly. The Islamic calendar is lunar, and is 11 (1/2) days different than the solar calendar. The Hebrew calendar is lunisolar.

Our modern calendar is handed down to us from the Ancient Roman civilization. The calendar took its first mature form as the Julian calendar, almost exactly the same as the one used today. It had 365 days in a year, with a 366-day leap year every four years. In the Julian calendar, years divisible by 4 — such as 1992, 1996, and 2008 — are leap years. This gave the Julian calendar an average of 365¼ days per year, which is very close to the true 365.2422 day length for a tropical year.

Although the drift of the Julian calendar is slow, the error in the calendar had accumulated enough by the sixteenth century that the Catholic Church became concerned about the drift's effect on the date of the celebration of Easter. The Italian chronologer Aloisius Lilius invented modifications to the Julian calendar to correct the difference. Pope Gregory XI instituted the new calendar, now named the Gregorian calendar, in the year 1582.

The Gregorian calendar was identical to the Julian calendar except that the leap year was skipped on years not divisible by 400. In the years 1600, 2000, and 2400, there would be a leap year in the Gregorian calendar, but not 1800, 1900, or 2100. This produced a year of average length 365.2425 days, much closer to the correct value than the Julian calendar. The Gregorian calendar accumulates only 3 days of error over 10,000 years.

Discussion questions

1) On the date of the summer solstice, the sun is overhead on the Tropic of Cancer, and on the date of the winter solstice the sun is overhead on the Tropic of Capricorn. Draw a quick sketch that shows the relative positions of the sun and the earth on those dates.

2) On the date when the sun is overhead on the Tropic of Capricorn, the sun is actually located in the constellation of Sagittarius. So why did the Greeks name Tropic of Capricorn the Tropic of Capricorn instead of the Tropic of Sagittarius?

Suppose you are an astronomer in America. You observe an exciting event (say, a supernova) in the sky and would like to tell your colleagues in Europe about it. Suppose the supernova appeared at your zenith. You can't tell astronomers in Europe to look at their zenith because their zenith points in a different direction. You might tell them which constellation to look in. This might not work, though, because it might be too hard to find the supernova by searching an entire constellation. The best solution would be to give them an exact position by using a coordinate system.

On Earth, you can specify a location using latitude and longitude. This system works by measuring the angles separating the location from two great circles on Earth (namely, the equator and the prime meridian). Coordinate systems in the sky work in the same way.

The equatorial coordinate system is the most commonly used. The equatorial system defines two coordinates: right ascension and declination, based on the axis of the Earth's rotation. The declination is the angle of an object north or south of the celestial equator. Declination on the celestial sphere corresponds to latitude on the Earth. The right ascension of an object is defined by the position of a point on the celestial sphere called the vernal equinox. The further an object is east of the vernal equinox, the greater its right ascension.

A coordinate system is a system designed to establish positions with respect to given reference points. The coordinate system consists of one or more reference points, the styles of measurement (linear measurement or angular measurement) from those reference points, and the directions (or axes) in which those measurements will be taken. In astronomy, various coordinate systems are used to precisely define the locations of astronomical objects.

Latitude and longitude are used to locate a certain position on the Earth's surface. The lines of latitude (horizontal) and the lines of longitude (vertical) make up an invisible grid over the earth. Lines of latitude are called parallels. Lines of longitude aren't completely straight (they run from the exact point of the north pole to the exact point of the south pole) so they are called meridians. 0 degrees latitude is the Earth's middle, called the equator. 0 degrees longitude was tricky because there really is no middle of the earth vertically. It was finally agreed that the observatory in Greenwich, U.K. would be 0 degrees longitude due to its significant role in scientific discoveries and creating latitude and longitude. 0 degrees longitude is called the prime meridian.

Latitude and longitude are measured in degrees. One degree is about 69 miles. There are 60 minutes (') in a degree and 60 seconds (") in a minute. These tiny units make GPS's (Global Positioning Systems) much more exact.

There are a few main lines of latitude:the Arctic Circle, the Antarctic Circle, the Tropic of Cancer, and the Tropic of Capricorn. The Antarctic Circle is 66.5 degrees south of the equator and it marks the temperate zone from the Antarctic zone. The Arctic Circle is an exact mirror in the north. The Tropic of Cancer separates the tropics from the temperate zone. It is 23.5 degrees north of the equator. It is mirrored in the south by the Tropic of Capricorn.

Horizontal coordinate system

One of the simplest ways of placing a star on the night sky is the coordinate system based on altitude and azimuth, thus called the Alt-Az or horizontal coordinate system. The reference circles for this system are the horizon and the celestial meridian, both of which may be most easily graphed for a given location using the celestial sphere.

In simplest terms, the altitude is the angle made from the position of the celestial object (e.g. star) to the point nearest it on the horizon. The azimuth is the angle from the northernmost point of the horizon (which is also its intersection with the celestial meridian) to the point on the horizon nearest the celestial object. Usually azimuth is measured eastwards from due north. So east has az=90°, south has az=180°, west has az=270° and north has az=360° (or 0°). An object's altitude and azimuth change as the earth rotates.

Equatorial coordinate system

The equatorial coordinate system is another system that uses two angles to place an object on the sky: right ascension and declination.

Ecliptic coordinate system

The ecliptic coordinate system is based on the ecliptic plane, i.e., the plane which contains our Sun and Earth's average orbit around it, which is tilted at 23°26' from the plane of Earth's equator. The great circle at which this plane intersects the celestial sphere is the ecliptic, and one of the coordinates used in the ecliptic coordinate system, the ecliptic latitude, describes how far an object is to ecliptic north or to ecliptic south of this circle. On this circle lies the point of the vernal equinox (also called the first point of Aries); ecliptic longitude is measured as the angle of an object relative to this point to ecliptic east. Ecliptic latitude is generally indicated by $\phi$ , whereas ecliptic longitude is usually indicated by $\lambda$ .

Galactic coordinate system

As a member of the Milky Way Galaxy, we have a clear view of the Milky Way from Earth. Since we are inside the Milky Way, we don't see the galaxy's spiral arms, central bulge and so forth directly as we do for other galaxies. Instead, the Milky Way completely encircles us. We see the Milky Way as a band of faint starlight forming a ring around us on the celestial sphere. The disk of the galaxy forms this ring, and the bulge forms a bright patch in the ring. You can easily see the Milky Way's faint band from a dark, rural location.

Our galaxy defines another useful coordinate system — the galactic coordinate system. This system works just like the others we've discussed. It also uses two coordinates to specify the position of an object on the celestial sphere. The galactic coordinate system first defines a galactic latitude, the angle an object makes with the galactic equator. The galactic equator has been selected to run through the center of the Milky Way's band. The second coordinate is galactic longitude, which is the angular separation of the object from the galaxy's "prime meridian," the great circle that passes through the Galactic center and the galactic poles. The galactic coordinate system is useful for describing an object's position with respect to the galaxy's center. For example, if an object has high galactic latitude, you might expect it to be less obstructed by interstellar dust.

Transformations between coordinate systems

A spherical triangle solved by the law of cosines.

One can use the principles of spherical trigonometry as applied to triangles on the celestial sphere to derive formulas for transforming coordinates in one system to those in another. These formulas generally rely on the spherical law of cosines, known also as the cosine rule for sides. By substituting various angles on the celestial sphere for the angles in the law of cosines and by thereafter applying basic trigonometric identities, most of the formulas necessary for coordinate transformations can be found. The law of cosines is stated thus:

\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C)

To transform from horizontal to equatorial coordinates, the relevant formulas are as follows:

Dec=\arcsin {\Big (}\sin(Alt)\sin(Lat)+\cos(Alt)\cos(Lat)\cos(Az){\Big )}

RA=LST-\arccos \left({\frac {\sin(Alt)-\sin(Dec)\sin(Lat)}{\cos(Dec)\cos(Lat)}}\right)

where $RA$ is the right ascension, $Dec$ is the declination, $LST$ is the local sidereal time, $Alt$ is the altitude, $Az$ is the azimuth, and $Lat$ is the observer's latitude. Using the same symbols and formulas, one can also derive formulas to transform from equatorial to horizontal coordinates:

Alt=\arcsin {\Big (}\sin(Dec)\sin(Lat)+\cos(Dec)\cos(Lat)\cos(LST-RA){\Big )}

Az=\arccos \left({\frac {\sin(Dec)-\sin(Alt)\sin(Lat)}{\cos(Alt)\cos(Lat)}}\right)

Transformation from equatorial to ecliptic coordinate systems can similarly be accomplished using the following formulae:

\phi =\arcsin {\Big (}\sin(Dec)\cos(\epsilon )-\cos(Dec)\sin(\epsilon )\sin(RA){\Big )}

\lambda =\arcsin \left({\frac {\sin(Dec)-\sin(\phi )\cos(\epsilon )}{\cos(\phi )\sin(\epsilon )}}\right)=\arctan \left({\frac {\sin(RA)\cos(\epsilon )+\tan(Dec)\sin(\epsilon )}{\cos(RA)}}\right)

where $RA$ is the right ascension, $Dec$ is the declination, $\phi$ is the ecliptic latitude, $\lambda$ is the ecliptic longitude, and $\epsilon$ is the tilt of Earth's axis relative to the ecliptic plane. Again, using the same formulas and symbols, new formulas for transforming ecliptic to equatorial coordinate systems can be found:

Dec=\arcsin {\Big (}\sin(\phi )\cos(\epsilon )+\cos(\phi )\sin(\epsilon )\sin(\lambda ){\Big )}

RA=\arcsin \left({\frac {\sin(Dec)\cos(\epsilon )-\sin(\phi )}{\cos(Dec)\sin(\epsilon )}}\right)

General Astronomy
The Modern View of the Cosmos	Observational Astronomy	Motion and Gravity

Motion and Gravity

General Astronomy
Observational Astronomy	Print version	Principles of Light

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The Early Origins of Astronomy

Thousands of years ago, the night sky was much more a part of everyday life than it is today. Without light pollution from streetlights, anyone could see thousands of stars or the path of the Milky Way through the sky. The motion of the stars provided an essential calendar for predicting the weather, so ancient people were accordingly very attentive and observant of the heavens and the rest of their surroundings. These ancient observations were the foundation for what would become modern astronomy, and were the first manifestations of scientific thought in society.

A central principle in scientific reasoning is that the world is inherently orderly and capable of being understood, and that observation is the means by which we can learn about how the universe works. There is, however, nothing that requires the universe to make sense or be accessible to human understanding. Humanity has placed trust in the ideas behind the scientific method because science has been so successful, and so much of the world really does obey a set of rules.

The ingredients that make up a successful philosophy of science were not conceived all at once, but rather in a gradual process of development over the entire course of history. The ancients could never have known in advance which principles of scientific method would eventually prove to be reliable. In its adolescence, society stumbled through the dark to learn about the world, to discover how their experiences could teach them what the universe was like, and to identify the boundaries between reason and superstition. The story of the development of science in cultures throughout the world is revealing not only of the nature of the world and of truth, but also of human nature more generally.

Ancient artifacts going as far back as 20 thousand years, to the paleolithic age, may arguably be related to astronomy, although the interpretation of such artifacts is a subject of controversy. The best-known examples are found in the caves of Lascaux in France, where a variety of paleolithic artwork are found. Some specimens found in the caves might possibly depict the Pleiades star cluster or the zodiac. Animal bones nearby have markings such that they could have been used as a lunar calendar. Although artifacts from the paleolithic era are scarce and ambiguous, this scarcity does not necessarily imply that the peoples of the period took no interest in the heavens. Modern aboriginal cultures, for example, pass down ceremonies with strong astronomical symbolism that would not reflect themselves in archaeological artifacts.

The character of artifacts of the neolithic period is markedly different, since understanding of the skies and the calendar clearly bear deep meaning for agricultural cultures in even the earliest times. Observers gained the ability to manage the planning of significant annual events. The motions of the heavens develop a profound influence on the affairs of humanity, and this influence comes to manifest itself in religious interpretations of heavenly phenomena and worship of the planets, practices that form the origins of both astrology and astronomy.

Ancient monuments and markers demonstrate societies' early interest in the heavens to archaeologists and historians. Many graves of this period are aligned with the cardinal directions. Ancient monuments and altars of sacrifice face east, south, or west to within a few degrees — a clear indication that neolithic peoples had begun very early to identify the most fundamental concepts of astronomy. The most famous of the ancient monuments is Stonehenge. The stones of the monument mark the positions of sunrise on the summer solstice, among other things.

Several of the planets have been known since prehistoric times. These bodies are very bright, and they wander among the fixed stars, through the constellations. These were known to the Greeks as planētēs, "wanderers". The ancients knew of 7 "planets": the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn. The names for the seven days of the week come from the Norse gods: Sunday (the Sun), Monday (the Moon), Tuesday (Mars/Tiw), Wednesday (Mercury/Wodan), Thursday (Jupiter/Thor), Friday (Venus/Frigg), and Saturday (Saturn/Soeternes).

Highly systematic and carefully recorded observation of the heavens appears as early as the third millennium BC, in the ancient Mesopotamian civilization of Babylon. The Babylonians also developed the earliest timekeeping instruments. Of perhaps even greater significance, however, was the birth of astrology in Mesopotamia. The astrologers of Babylon were the first to suppose a cause and effect relationship between heavenly and earthly events, and the first to imagine an underlying order in nature — though no real connection between human and celestial events existed, the prophecies of astrology were the ancestors of modern scientific prediction.

The early astronomers understood a great deal about the world around them. By Aristotle's time, many people believed that the world was round. They knew this from several sources of evidence:

When ships sailed beyond the horizon, those watching from land would see the hull disappear first, and then would watch the sail disappear. Sailors see the land sink below the horizon bottom-first. From this, they gathered that the surface of the world was curved, and that the ship was moving around the curvature of the Earth.
Ancient people supposed eclipses of the Moon to be the shadow of the Earth. Shadows seen during lunar eclipses were always round. The only shape that always casts a round shadow is a sphere. This suggested that the Earth was spherical.
Travelers noticed that new stars became visible as they moved in their journeys. When travelers move north, the North Star and the northern constellations gain altitude in the sky. The ancients recognized that this implied that the Earth is curved.

Aristotle mentions in his writings that some thinkers of his time hold that the Earth is flat, while others believe it to be spherical. He himself argues firmly that the Earth is a sphere. After Aristotle's time, nearly all Western writers claim that the Earth is a sphere.

Eratosthenes knew that obelisks in the city of Syene cast no shadow at midday on the summer solstice. In Alexandria, however, obelisks cast a shadow. This reveals that the Earth is not flat and makes it possible to measure its circumference.

In the third century BC, the Hellenistic Egyptian scholar Eratosthenes served as head of the library of Alexandria, where he studied and wrote on a variety of subjects. During his studies, Eratosthenes learned that the Sun cast no shadow on midday of the summer solstice from the city of Syene. From Alexandria, Eratosthenes observed that the Sun did cast a shadow, and he measured that the angle between the Sun and the zenith was about seven degrees when the Sun was highest in the sky.

Eratosthenes knew he could measure the circumference of the Earth by measuring the distance between Alexandria and Syene, assuming that the Sun was at the highest point in both cities at the same time. Since 7 degrees is 7/360 of a circle, the distance between Syene and Alexandria must be 7/360 (or, equivalently, about 1/50) of the circumference of Earth.

Eratosthenes estimated the distance between Syene and Alexandria to be about 5,000 stadia, and concluded that the circumference of the Earth was about 50 times that, or 250,000 stadia. The stadium was a common unit of distance in the ancient world, comparable to kilometers or miles in modern usage; the exact length meant by a stadium varied over history and from region to region, so it is difficult to compare Eratosthenes's estimate with modern values. Most values of the stadium, however, translate to a circumference in the area of 40,000 kilometers, which is remarkably close to the accepted circumference of 40,070 kilometers.

As science was beginning to grow in the cultures of Western civilization, it was likewise taking root elsewhere in the world. Ancient and medieval China, India, and America in particular have been centers of the scientific world.

The history of science in India is one of the longest and best documented in the world. Largely isolated from the rest of the world through much of its earliest history, India was free to develop its unique flavor of astronomical practice at its own pace and without external influence. Ancient India had no system of astrology, but celestial timekeeping became popular very early. The first astronomers monitored the progression of the year and tracked the movement of the Sun, Moon, and planets through the nakshatra, the Indian zodiac.

The quality of these observations would make it possible for later Indian astronomers to detect the effects of precession, just as Hipparcos had done for Greek astronomy. Indian astronomers of the Middle Ages estimated the speed of precession to be 54 seconds per year. Although this is much more accurate than Hipparcos's measurement, we must note that the number 54 was chosen for mystical rather than scientific reasons. Ancient people throughout the world mixed mysticism with arithmetic in this way, but we see this much more consistently in the Indian culture than in any other.

The Christian era brought great change to Indian astronomy. By about the year AD 519, Babylonian conquest had spread into the Indus basin, and many Greek scholars had already visited or settled. Naturally, cultural exchange took root.

The ancient Chinese valued astronomy very highly. Systematic observation of the heavens may have begun in China as early as the third millennium BC. Science was viewed as a powerful tool and therefore guarded very closely by the state. Under dynastic rule, the perceived purpose of astrology and astronomy was to remind the people of the connection between the emperor and celestial events. Accordingly, astrology was charged specifically to an Astronomical Bureau appointed by the emperor and carried out according to strict rules.

The Chinese regarded celestial events as important omens, particularly in the case of "guest stars," stars that temporarily appeared in the sky and quickly faded away. Today we know that these are stellar explosions called supernovae. In the year AD 1054, a new supernova became bright enough to be visible even during the day. Chinese and Arab astronomers recorded the event, and it appears to be depicted in an American cave painting. Because of the importance the Chinese assigned to supernovae, their observations of ancient supernovae are by far the most complete in the world, and remain useful as a source of scientific data even today.

Meanwhile, astronomy was thriving in pre-Columbian America, where interest in the heavens was widespread. Astronomy held a high position in the Mayan civilization, which dominated Mesoamerica for two thousand years prior to the arrival of Europeans. The Maya people believed that patterns in nature were very important, and that it should be possible to use astrology to predict the future based on past experience. This made the observation of celestial cycles very important, since they would give advance knowledge of future events, and Mayans dutifully kept track of motions of the Sun and Moon. With the accumulation of centuries of observation, Mayan astronomers were able to predict eclipses and measure the lengths of heavenly cycles very well.

Similar ideas were sprouting throughout the Americas. Throughout the Americas, ancient people built monuments, buildings, mounds, and temples aligned with rising and setting points of important stars, or with the northernmost and southernmost positions of the Moon. Calendars and monuments like Stonehenge are found here. Clearly, interest in the heavens was common throughout the ancient world, as were the rudimentary beginnings of science.

Scientific thought has appeared throughout history and all over the world, but it's clear that the scientific enterprise enjoyed more success in some areas and time periods than in others. The reasons for this are entirely unclear. What factors of culture, economics, and philosophy affect the rise of scientific thought? The Babylonian culture, for example, did not permit deviation of any kind from tradition or conventional practice. Such a culture might be expected to hinder the progress of science, yet science and mathematics flourished there. In ancient China, only nobles were permitted to practice science. Some historians have suggested that this culture prevented a renaissance that might otherwise have occurred in China long before the European Renaissance. This question of the interplay between science and sociology has been the subject of great interest in the study of the history of science.

While magic and superstition were a significant part of early thought and difficult to isolate from the emerging pursuit of science, it would be unfair to say that spiritualism completely permeated the first ideas about the nature of the universe. The scientific spirit, the principle of learning about the universe by observing it, was clearly present even from the earliest times, and scientific thought has been remarkably widespread. Anaximander (611-547 BC) postulated that there were five elements: earth, air, fire, water...and quintessence, which heavenly bodies were made of.

Astrology is based on Ptolemy's Tetrabiblos, written in 140 AD. The horoscope is divided up into 12 "houses", 12 constellations along the ecliptic. Ptolemy believed that an "8th Sphere" held up the stars in the sky.

Kinetic energy is the energy of motion. The formula for this is ${\frac {1}{2}}MV^{2}$ ( $M$ is mass, and $V$ is velocity). Before the common acceptance of the heliocentric model in the mid 1500s, the Earth was believed to be the center of the known universe. Many inherent problems in the geocentric model of the universe led Nicolaus Copernicus to formulate an alternate means by which planets and stars moved. Two such problems are the apparent retrograde motion of various planets, specifically Mars, and the ensuing epicycle theory made to compensate for this issue, as well as the phases of Venus that occur.

Planetary Motion and Epicycles

Viewing Earth as the center of our universe, many of the planets in our solar system take on a peculiar motion across the night sky. For instance, Mars appears to move in a retrograde fashion from time to time. Since the geocentric model holds that all the planets in our solar system orbit the Earth, this would mean that from time to time Mars reverses its orbital direction only to resume it later. The Greek astronomer, Ptolemy, sought to explain this phenomenon with the use of epicycles - that is, small circular motions that planets orbiting the Earth followed on top of their circular orbit around the Earth. Astronomers from Greek times to the Enlightenment compensated for flaws in the geocentric model by adding extra layers of epicycles, until the system was so complex it became virtually unusable. Copernicus' heliocentric model, placing the Sun at the center with planets going around it rather than around the Earth, rid the universe of many of the epicycles that plagued astronomy at that time.

Phases of Venus

Another problem with the geocentric model occurred when observing Venus and its relative phases: should Venus and the Sun orbit the Earth, then it would be impossible for Venus to be in the "full" phase, that is complete illumination (similar to a full moon). Thus, the Ptolemaic model predicted only crescent and new phases of Venus. Yet, as Galileo observed, this was clearly not the case; Venus exhibited all phases of the moon, a phenomenon predicted and accounted for by Copernicus' model. Discovered by Galileo, this phenomenon was one of the key factors that would end up supporting the Heliocentric view of the universe.

Note that this is only one of a number of observations made by Galileo that contradicted the ancient beliefs, but this one most directly called into question the idea that planets orbit the Earth.

References

Shea, William R. Galileo in Rome: the Rise and Fall of a Troublesome Genius. New York: Oxford UP, USA, 2004.

The modern heliocentric model of the universe was proposed by Nicolaus Copernicus in his book De revolutionibus orbium coelestium which was published just before his death to avoid prosecution for heresy. It addressed many of the problems associated with the Ptolemaic model which required a multitude of epicycles to describe planetary motion. Heliocentrism places the Sun at the center of the solar system. While some of the ancient civilizations in India and the Middle East had proposed the heliocentric model before, Copernicus' discovery set the groundwork for the modern view of the solar system.

Copernicus' Model

In Copernicus' heliocentric model, there are seven theories he presents:

There exists more than one center of the universe.
There is a center of the universe near the sun.
The planet Earth is not the center of our universe.
The scale of distances between the Earth and the Sun is minuscule compared with the astronomical distances between Earth and the stars in the sky.
The Earth revolves around the sun, thus causing the annual cycles of the seasons that we witness.
The Earth's own rotation causes the stars to appear to shift every night, rather than any motion of the stars themselves.
Any retrograde motion, previously explained by epicycles, is in fact caused by the differing orbital velocities of planets around the sun.

Copernicus harbored his belief of heliocentrism for many years, and did not publish his work until 1542, a year before he died. Some speculate that Copernicus held onto his idea for a considerable time simply because he was fearful of the reception it would have among his peers and more importantly, the Church. Many consider the publication of De revolutionibus orbium coelestium (On the Revolution of the Celestial Sphere) to be the start of the Scientific Revolution.

In the years that followed, Copernicus' theory was refined and independently verified by notable scientists, most prominently Galileo and Kepler. Specifically, Galileo noted that all phases of Venus occurred (similar to our moon), which cannot be adequately explained in a geocentric model. This contradicted the solely new and crescent phases predicted by the geocentric model.

Acceptance by the Church

The principle behind rockets was discovered in 100 BC. An early rocket was the aeolipile, a steam-powered device. Johannes Kepler was a mathematician who attempted to derive a set of fundamental principles which would explain the motions of the planets. He believed in the heliocentric view of the solar system proposed by Copernicus, and he also possessed a rich set of observations of the planets made by Tycho Brahe.

After twenty years of painstaking attempts and various discarded ideas based on geometry, he finally arrived at a mathematical model of orbital motions based on the ellipse. Kepler summarized his findings in the form of three laws of planetary motion, frequently referred to as Kepler's First, Second and Third Laws, respectively:

Kepler's First Law, also known as The Law of Ellipses — The orbits of the planets are ellipses, with the sun at one focus.
Kepler's Second Law, or The Law of Equal Areas in Equal Time — The line between a planet and the sun sweeps out equal areas in the plane of the planet's orbit over equal times.
Kepler's Third Law, or The Law of Harmony — The time required for a planet to orbit the sun, called its period, is proportional to half the long axis of the ellipse raised to the 3/2 power. The constant of proportionality is the same for all the planets. It is often called the Law of Harmony because it shows a harmonic relation between distances and periods.

At that time he developed these laws, there was not yet a developed theory of gravity capable of explaining why the planets moved as they were observed to. Later, Isaac Newton, using his universal inverse-square law theory of gravity, was able to show how Kepler's Laws fit into a scientific theory of celestial mechanics.

Elliptical orbits

An ellipse is a shape formed by taking a diagonal slice through a cone. It is essentially the shape of a circle viewed at an angle.

An ellipse can be drawn by taking a piece of paper, two push-pins, a loop of string, and a pencil. The two pins are pushed through the paper into a suitable surface, providing the two foci for the ellipse. They should be closer together than the loop is long. The loop of string is placed around the base of these pins, leaving some slack. The pencil is now placed so that the pins and the loop form a triangle with a slight tension on the string.

Now try to draw a shape by moving the pencil about the pins while keeping the string taunt. The result should be an ellipse. The shape of the ellipse can be varied either by moving the pins closer together or further apart. This shape, according to Kepler, defines the path that a planet takes when it orbits the Sun.

Kepler's First Law - A planet orbits the Sun on an ellipse with the Sun at one focus.

Kepler's Second Law says in brief that an object speeds up as it gets closer to the Sun and slows down as it moves further away. Where the distance from the Sun to the orbital path is longer, only a smaller arc needs to be traversed to sweep out an area that requires a wider arc near the Sun.

As the planet moves closer to the Sun along its orbit the gravitational force works to increase the velocity. In contrast, as the planet is moving further away, the gravity of the Sun gradually decelerates the body and it is slowed down.

Kepler's Second Law - A planet in orbit about the Sun sweeps out equal areas

A

in the same time interval

t

.

A line that divides an ellipse in half and passes through the widest part of the ellipse is called the major axis. A line perpendicular to this axis and dividing the ellipse in half is called the minor axis. Half the length of the major axis is called the semi-major axis, and is represented by $a$ . The period required for a planet to complete one full orbit is represented by $P$ . The relationship between the period P and the length of the semimajor axis $a$ is known as Kepler's Third Law, and can be represented as follows:

P^{2}\propto a^{3}

where the symbol ∝ means "proportional to", and implies that there is a direct mathematical relationship between the period squared and the length of the semi-major axis cubed.

The Second and Third Laws provide a basis for calculating the period of any planet orbiting the Sun, as well as determining where the planet will be located along the orbital path.

Eccentricity and Orbital Paths

The ratio of the distance of a focus from the center of an ellipse to the semi-major axis is called the eccentricity of the orbit. When the two foci of the ellipse are on top of each other, the eccentricity is exactly 0.0 and the shape is a circle. As the eccentricity increases, the orbiting planet moves much further away than at the closest approach. The orbital eccentricities for planets in our Solar system vary from as much as 0.21 for Mercury down to 0.0068 for Venus.

The scientific name for the point of closest approach is the periapsis, while the most distance separation is the apoapsis. In the case of planets orbiting the Sun, these are called the perihelion and aphelion, respectively. (The -helion suffix comes from the Greek name for the Sun deity, Helios. This word is also the source of the name for the element Helium.)

Two elliptical orbits with the same major axis

a

but different eccentricity.

Perhaps the most counter-intuitive aspect of the Third Law is that for any two identical bodies orbiting the Sun with the same semi-major axis, the orbital period is the same. This is true even if one is orbiting in a perfect circle and the other has an orbit that is highly elliptical (has a relatively high eccentricity). The elliptical shape will fit entirely within the circle except at two points (the ends of the major axis, at which the two curves will be tangent), so it is actually a shorter orbital path. However the aphelion of the ellipse will be located further from the Sun, so the planet will spend more time traversing the distant section of the orbit. The shorter orbit and the slower traverse of the aphelion compensate for each other, resulting in an identical period with the circular orbit.

Some Examples Using Kepler's Third Law

Originally, Kepler's Third Law was used to describe the motions of the planets around the Sun. As it turns out, it also works very well with regards to other two-body orbital systems like the orbits of moons around Jupiter, or the orbits of binary stars about the center of mass of their system. In all of these cases the period of the orbit squared is proportional to the length of the semi-major axis cubed, with the differences between the orbital systems reflected in the constant of proportionality.

In this section we will consider the special case of the planets going around the Sun. If we choose to measure the length of the semi-major axis of an orbit in astronomical units (abbreviated AU, where 1 AU is the distance from the Earth to the Sun) and we measure the orbital period in years (abbreviated as $y$ ), then we can express Kepler's Third Law as

P^{2}=\left({\frac {y^{2}}{AU^{3}}}\right)a^{3}

where $P$ is measured in years and $a$ is measured in astronomical units. What follows are a few example of how to use this equation.

Mars' Orbital Period

Repeated measurements of Mars' orbit have determined that the semi-major axis of its orbit has a length of 1.52 AU. How long does it take for Mars to orbit the Sun once?

Solution: In this question, we are being asked for the orbital period $P$ of Mars. We know from reading the question that the length of the semi-major axis $a$ is 1.52 AU. Solving Kepler's Third Law for the period gives us

P=\left({\frac {y}{\sqrt {AU^{3}}}}\right){\sqrt {a^{3}}}=\left({\frac {y}{\sqrt {AU^{3}}}}\right){\sqrt {(1.52AU)^{3}}}=1.87y

which tells us that it takes 1.87 years for Mars to go around the Sun once.

An Unknown Asteroid's Orbital Semi-Major Axis

An amateur astronomer spends several months tracking an asteroid and is able to determine that it takes approximately 3/4 of a year for it to orbit once around the Sun. What is the semi-major axis of this asteroid's orbit?

Solution: Reading the question tells us that the orbital period of the asteroid is 3/4 of a year, or 0.75 $y$ . We are expected to find the length of the semi-major axis $a$ , so we need to solve Kepler's Third Law for that. Doing so gives us

a=\left({\frac {AU}{\sqrt[{3}]{y^{2}}}}\right){\sqrt[{3}]{P^{2}}}=\left({\frac {AU}{\sqrt[{3}]{y^{2}}}}\right){\sqrt[{3}]{(0.75y)^{2}}}=0.83AU

which means that is has a semi-major axis somewhere between that of Venus' orbit and Earth's orbit.

General Astronomy
Observational Astronomy	Motion and Gravity	Principles of Light

The Early Origins of Astronomy

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