Basics of Astronomy

This is a brief summary of some basic ideas in astronomy. I've explained terms used to describe the brightness and location of stars, stellar distances, star types based on spectra, and other fundamental ideas in astronomy. If you have any suggestions for things to add to this page, or any corrections or modifications, send me a message.

Contents

• Distances Used in Astronomy
• Movement of the Earth
• The Brightness of Stars
• Stellar Spectra

Stellar Distances

Distances to other planets or to other stars are huge, compared to distances we normally use on Earth. The usual units, such as meters and kilometers are not always the best suited for astronomical distances. Some reasons for choosing other, astronomy-specific units may be:

• To use smaller numbers. For example, distances expressed in meters may contain many more digits compared to the same distance expressed in light years. Smaller numbers are more economical and more easily readable.
• To reveal relationships. For example, expressing distances in units which represent known measurements in astronomy can be useful. If the distance from the Earth to the Sun is defined as 1 unit, and then if we say that the distance from Mars to the Sun is 1.5 such units, the scale of the Solar System becomes more intuitively understandable.
• To reveal physical laws. For example, if we say that the distance to some star is 6 light years, then in addition to being a measure of distance, it also implies that the light reaching us now left the star 6 years ago, and we are seeing the star as it appeared 6 years ago to an observer close to the star.
• For methodological reasons. For example, certain methods may be used to measure stellar distances, such as by parallax measurements. In this case, the measurements are calculated in units specifically derived from the technique used, such as parsecs. These units can then be converted to any other unit, but are often quoted in the original unit because that's where they were first measured, and that's what someone might need to know if he decides to repeat the experiment to confirm the original measurement.

Here are some of the units of length used in astronomy.

Astronomical Unit (AU)

This is the mean distance between the Earth and the Sun, during the course of one full orbit. It is equal to about 149,597,870,700 ±3 meters. Technically, it's defined in a few different ways:

• the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with a angular frequency of 0.01720209895 radians per day
• the length for which the heliocentric gravitational constant is equal to 0.01720209895² AU³/d²

These seem to be very precise ways of describing it, but there are a couple of problems with the Astronomical Unit. The first problem is that the mean distance between the Earth and the Sun isn't constant. The Sun is constantly burning hydrogen into helium and therefore losing mass. It also loses mass by throwing off huge amounts of particles (the solar wind). As the Sun loses mass, the orbits of all the planets expand. So the AU is increasing gradually.

The second problem is that the AU does not take into account relativistic effects. This is a problem with many units, including SI units such as the meter. However, it is generally assumed that relativistic effects over such small distances as a meter are insignificant. But over larger distances such as 1 AU, general relativity needs to be taken into account.

For these reasons, the AU is a unit that is currently under revision. Many proposals have been submitted, including fixing the AU at some predefined number of meters, or using a different formula which takes into account the expansion of Earth's orbit and relativistic effects. These are being considered, but it will take some time.

In day to day use, astronomical units do not require such precision, and therefore the AU continues to serve very well. For example, the distances of the different plants to the Sun are listed in the table below.

From the table, it's much easier to get a sense of the scale of the Solar System. The usual distance of a planet from the Sun is about 1.3 - 1.8 times the distance of the next nearer planet from the Sun. This relationship breaks down between Mars and Jupiter (Jupiter is 3.4 times farther from the Sun than Mars), which would be accounted for if we consider the asteroid belt to represent a missing planet.

Parsec

This unit is derived from the way in which we measure distances to nearer stars. The principle is the same as measuring a distance on Earth. If you want to measure the distance to a cell phone tower, for example, you can take a sighting from a certain position (call it point A), then move some distance away (to point B) and take a different sighting to the tower. Consider the diagram below. The observer is moving along a road in a car, and sees a telephone pole (orange bar). He picks some object in the far background (brown pyramid), and takes a sighting on the telephone pole against the pyramid. Then he drives on until he is some distance (r) away from the point where he took the first sighting. Now he takes another sighting on the telephone pole.

As you can see in the diagram below, the perspective of the telephone pole in relation to the background changes between the two sightings. From this change in perspective, the observer can calculate the angle subtended at the telephone pole (alpha) by the two lines from his initial position and his final position, to the telephone pole.

This method works only if distance between the two objects (the foreground object being measured and the background object far away) is large. This is done by picking some distant object for the background - distant enough that its perspective does not change significantly from point A to point B.

Since he can measure the angle (alpha) between his two sightings (using surveying tools such as a transit or theodolite), and the distance (r) between points A and B using the car's odometer, he can calculate the distance between the telephone pole and the road (d) as follows:

alpha = r / d

The angle must be measured in radians (divide degrees by 57.23 to get radians) to use this formula.

This same principle of parallax based distance measurement can be used on stars as well. In fact, this is the only direct way to measure stellar distances. There are many indirect ways as well, such as by relating the spectrum of the star to its temperature and luminosity, and then using formulae to calculate the distance. Often, these indirect methods are calibrated by comparing their results to those obtained by direct methods, such as parallax measurement.

Since astronomical distances are so large, we can't actually use the car driving technique described above. The distance you could travel in a car or airplane is too small, being bounded by the size of the Earth. Distances to even the closest stars are much larger in comparison. If points A and B were 12,800 kilometers apart (the farthest straight line distance you can go, and still remain on Earth), the angle subtended (alpha) at even the nearest star would be too small for accurate measurement. For this reason, we need to use a larger distance between A and B. This is provided by the rotation of the Earth around the Sun. The Earth's orbit is 2 AU in diameter, or about 300 million kilometers. If we take a measurement on a star at some point in time, and then take the second measurement on the same star 6 months later (when the Earth has moved to the opposite point in orbit), our baseline is 300 million kilometers. This large baseline subtends angles which are big enough to measure.

From this methodology, we can define a unit called a parsec. A parsec is the distance at which a star would subtend an angle of 2 arc seconds, if measured from two opposite points in Earth's orbit. This number is chosen because half of the angle is exactly unity (1 arc second) and half of the baseline is exactly 1 AU. So a parsec is the distance at which the angle subtended by two points 1 AU apart is exactly 1 arc second.

An arc second is 1/3600 of a degree. A full circle has 360 degrees. Each degree is divided into 60 minutes of arc, and each minute of arc is further subdivided into 60 seconds of arc. The precision of our measuring instruments is currently in the range of milliarc seconds (1/1000 of an arc second). For example, the Hipparcos Mission launched by the European Space Agency in 1989 cataloged about 120,000 of the brightest stars, with a mean accuracy of about 1 milliarc second. There are better instruments now available for terrestrial use, but the Hipparcos catalog is still among the best and most widely used catalogs in astronomy. There is a plan by the European Space Agency to launch the Gaia Mission in 2012, which will update the Hipparcos catalog. The instruments on Gaia will be capable of measuring down to 20 microarc seconds, for stars with magnitude 15 or brighter. In all, it will catalog stars down to magnitude 20 - about a billion stars.

Because of the limited accuracy of angular measurements, we can only use the parallax method for measuring distances to relatively close stars. Stars which are very distant are measured with indirect methods. As the instruments improve, we should be able to extend the parallax method to more distant stars.

The nearest star, Proxima Centauri, is about 4.243 light years from the Sun, or about 1.3 parsecs. Two other stars in the alpha Centauri group are also within the < 5 light year range. Barnard's Star is at a distance of 1.834 parsecs, Sirius is 2.64 parsecs, and Procyon is 3.496 parsecs from the Sun. The closest star with planets is probably Epsilon Eridani, at a distance of 3.218 parsecs. The Gliese Catalog of Nearby Stars (known as GJ for short, which stands for Gliese Jahreiß) lists the stars within 25 parsecs of Earth.

Light Years

This is the most popular measure for describing distances to stars. A light year is simply the distance traveled by light in vacuum, over the course of 1 year. For the purpose of definition, a year is defined as a Julian year of exactly 365.25 days. So a light year is:

Light Year = (days in a year) * (hours per day) * (seconds per hour) * (speed of light in vacuum in m/s)

Light Year = 365.25 * 24 * 3600 * 299792458 = 9.46073 * 10¹⁵ meters.

This comes to about 9.46 trillion kilometers, or about 0.3 parsecs, or about 63,240 Astronomical Units. Light years are very convenient because the distance in light years automatically tells us how long the star's light took to reach the Earth. When we look at Proxima Centauri, we see it as it appeared slightly over 4 years ago, to someone who might have been observing it from a planet around Proxima Centauri (this is just an example, there are no known planets around Proxima Centauri). On the other hand, the Andromeda Galaxy is about 2.5 million light years away, so the light reaching us from Andromeda now left the galaxy 2.5 million years ago. Our view of Andromeda is therefore as it appeared 2.5 million years ago to an observer close to Andromeda. The events we see happening in stars in Andromeda happened 2.5 million years ago relative to us. With today's powerful telescopes, we can see objects that are billions of light years away from us. The farther the star, the longer the light took to reach us, and therefore the older the view we see today. One of the farthest objects seen from Earth was a gamma ray burst from a star located about 13.1 billion years away from Earth. This means that the the light we see today originated when the universe was only 630 million years old.

The Movement of the Earth

The Earth revolves around the Sun in a slightly elliptical orbit, with the Sun at one of the foci. Check the diagram below to see what it looks like.

Because of the eccentricity of Earth's orbit, the Earth has a point of closest approach to the Sun (perihelion) and a point of farthest distance from the Sun (aphelion), once a year. These are simply the endpoints of the major axis of the ellipse.

The plane of the Earth's orbit is known as the ecliptic plane. This is very important to remember, because a lot of astronomical measurements are made in reference to the ecliptic plane.

Now consider that the Earth's axis of rotation is inclined at an angle of about 23.4° to the plane of its orbit. Therefore, we can draw another plane passing through the Earth's equator, which will also be inclined at an angle of 23.4° to the ecliptic plane. This plane passing through the Earth's equator is called the equatorial plane. This is important when we observe stars from the surface of the Earth. Since we are standing on the surface of the Earth, we share the Earth's axial tilt, and our horizon is therefore parallel to the equator. It's the equatorial plane, therefore, that sets a zero point for our observational horizon (at the equator this is literally true, but at latitudes farther north or south, our horizon is not precisely the equatorial plane, but a plane parallel to the equatorial plane and slightly above or below it).

In short, the equatorial plane is our day-to-day reference for the horizon during astronomical observations, since the Earth spins on its axis during the course of a day, and the axis of this spin is exactly perpendicular to the equatorial plane.

On the other hand, the ecliptic plane follows the Earth's orbit around the Sun, and is therefore important when describing the changes in the sky seen over the course of the year. As the year goes past, different constellations appear in the sky. Other constellations change their relative position, rising higher or lower in the sky each night. These yearly changes are produced by our changing perspective as the Earth moves around the Sun with each orbit. To understand these changes, we need to refer to the ecliptic plane.

Since ancient times, people have noted that the constellations change seasonally, and that these observations can be used to produce a calendar. Such calendars were very important, specially for farming communities, since they were guides to the planting season. One way in which people told the seasons was by seeing where in the sky the Sun rose each morning. The Sun always rises in the east for us, but the background of stars against which it rises changes from day to day and month to month. For example, during a certain month, the Sun may appear to rise in the midst of stars which belong to the constellation Scorpio. A month later, as the Earth moves on in its orbit around the Sun, the background of stars against the rising Sun changes, and the Sun may appear to rise in a different constellation. Through observation, ancient people discovered that over the course of a year, the Sun rises in 12 different constellations. These are the constellations of the Zodiac: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Serpentarius, Sagittarius, Capricorn, Aquarius and Pisces.

Constellations are known by different names in different cultures, so the "signs" of the Zodiac are different in, for example, the Hindu, or Chinese, or Maya cultures. However, they all refer to the same groups of stars, no matter how we subdivide them and no matter what we call them. To see how this works out, look at the diagram above. The red arcs represent the constellations of the Zodiac in a top-down view. The line joining the Earth to the Sun, if projected beyond the Sun to the pattern of stars behind the Sun, represents the "sign" of the Zodiac for the current month. This imaginary line lies on the ecliptic plane, the plane of Earth's orbit around the Sun. As the Earth moves over the course of a month, the line now points to a different constellation, which is the "sign" for the next month.

There are a few other factors that determine the stellar background, and therefore our star-based calendars. Some of them include:

Precession of the Earth's Axis

Like a spinning top, the Earth's axis describes a circular arc, once every 26,000 years. From our vantage point on the surface of the Earth, this appears as if the axis moves in a circle against the background of stars in the sky. Since the axis points towards the northern and southern celestial poles, the location of these poles appears to move in circles. This is why the "North Star" and "South Star" change over time.

Currently, Polaris is the North Star, since it's closest to the North Celestial Pole. About 4,000-5,000 years ago, the star Thuban (Alpha Draconis) was the North Star. The Great Pyramid in Egypt, which was built at this time, contains passages that are aligned to this star. About 10,000 years ago, Vega was the North Star, and it will again be the North Star in another 10,000 years or so.

The precession of the Earth's axis is one of the factors responsible for the precession of the equinoxes. This was discovered thousands of years ago in Babylonia, India and Greece. The position of the Sun at vernal equinox appears to move over a very long cycle, and ancient people had some interesting speculations about why this happened.

Axial Tilt

The Earth's axis is currently tilted about 23.4° from the ecliptic plane. The amount of tilt is not constant - it varies between about 22.1° and 24.5° over a 41,000 year cycle. This may seem like a small change, but some scientists believe that this might be the reason for the 41,000 cycle we see in ice ages. There are several "ice age cycles", of which the 41,000 year cycle is one. I wrote a separate section about the ice ages and their cycles here.

Orbital Sweep

Since the Earth's orbit isn't perfectly circular, it has a major and and a minor axis (in an ellipse, the longer diameter is the major axis and the smaller diameter is the minor axis). The major axis sweeps through a full circle once every 112,000 years. This has been shown greatly exaggerated in the "Orbital Sweep" part of the diagram above. One way to think of it is as follows. Imagine the Earth at some specific point in its orbit. What would happen exactly one year (one orbit) later? Would it return to the exact same spot in space? For the moment, ignore the fact that the Sun is circling the Milky Way and carrying the Earth and other planets along with it. The truth is that even if we ignore the motion of the Solar System as a whole, the Earth would not return to the exact same spot after a full orbit, because the inclination of the major axis of Earth's Orbit has changed. The arrow in the diagram shows a change in position of 45°, which is 1/8th of a circle. The position of Earth in those two orbits is aphelion, meaning farthest from the Sun. Since the full sweep takes 112,000 years, 1/8th of a sweep would take 14,000 years. Therefore, after 14,000 years, the Earth at the same position in its orbit (aphelion) would have moved from where it's shown at the beginning of the arrow to where it's shown at the end of the arrow.

Orbital Eccentricity

The eccentricity (the degree to which the Earth's orbit resembles an ellipse rather than a perfect circle) also changes over time. As you can see in the diagram, two things happen. First, the Earth's orbit slightly flattens out (the minor axis grows a bit, so the orbit is more circular). Second, as a result of this, the location of the two foci within the ellipse change. Since the Sun must be located at one of the foci (and obviously the Sun doesn't move, it's too heavy - the Earth is what does the moving), the major axis of the ellipse moves in one direction (shown moving upwards in the diagram). Note the the magnitude of the major axis doesn't change - which means that the period of the Earth's orbit doesn't change - only the location of the major axis changes, as a result of the widening of the minor axis with the corresponding change in the location of the foci. The net result of this is that aphelion moves farther away from the Sun and perihelion moves closer to the Sun. This exaggerates the effect of the seasons - summers are hotter and winters colder than they would otherwise be. Which hemisphere these effects predominate in depends upon the axial tilt at the time. The full eccentricity cycle takes about 100,000 years, while the precession cycle takes only 26,000 years. Therefore, during one cycle of change in orbital eccentricity, the Earth's axis precesses 4 times, making summers warmer (and winters colder) in the northern and southern hemispheres 4 times each. At least, this is the simple theoretical view.

In practice, these cycles interfere with each other, and much like electromagnetic waves, periodically reinforce or attenuate each other. This produces stronger cycles of different periodicity, which are produced by some combination of these cycles described above.

Orbital Plane

Finally, the plane of the Earth's orbit moves up and down, as shown in the last part (bottom right) of the figure. Let's understand more clearly what this means. As mentioned above, the plane of Earth's orbit is known as the ecliptic, or ecliptic plane. This is what's moving. What is it moving in relation to? The reference plane in this case is something called the "Invariable Plane", which is basically the plane of the entire Solar System. This is calculated by calculating the average angular momentum of the Solar System as a whole. It's a mathematical construct, and does not correspond to the actual orbit of any particular planet. However, since Jupiter has the bulk of the mass in the Solar System, the "Invariable Plane" is very close (though not identical) to the ecliptic plane of Jupiter.

The ecliptic plane of Earth moves up and down in relation to the Invariable Plane with a cycle of about 70,000 years. The reason for this movement is the gravitational attraction of other planets on the Earth. When planets line up on the same side of the Sun, the attraction between them tends to bring them closer to the Invariable Plane. This produces a cycle of up and down movement of each planet, mainly in accordance with its distance from Jupiter. This movement is very small, and has been greatly exaggerated in the diagram.

Summary

As we can see, the Earth's motion is fairly complex, not the simple clock-like motion we learned about in elementary school books. This is one reason why it took so long to figure out the Solar System, and why weird theories of geocentricity and the like weren't thoroughly rejected even earlier. It may seem like some of these complex motions are very subtle, and occur over such long periods that humans would never perceive them in their short lifetimes. However, this is not true. The magnitudes of the effects of many of these motions were perceivable and known to ancient astronomers throughout the world, in Babylonian, Egyptian, Indian, Greek and Chinese civilizations, as well as by the Mayans in later periods. The precession of the equinoxes was written about by Hipparcus in the 2nd century BC, and calculated to be about 36,000 years. The Indians were even more accurate, calculating the period to be 21,636 years, which is astonishingly close to the modern value if we take anomalistic precession into account (anomalistic periods are created by the combination of two or more proper motions as described above).

Since the motions are large enough for humans to observe even over a single lifetime (with just their naked eyes and no telescopes), obviously they are large enough to be of interest when we consider astronomy through the ages. Paleoastronomers (people who try to figure out what ancient astronomers saw, and use astronomical events described in history to calculate dates) are very much interested in what the sky looked like back in some period of history. Archeological investigations of monuments such as Stonehenge or the Pyramids, which appear to be astronomically aligned, very much depend upon understanding these complex motions of the Earth, because only when we can account for them can we begin to calculate what the skies must have looked like when the Great Pyramid was being built, the timing of equinoxes and solstices in the past, the dates and times of lunar and solar eclipses, etc. These are the kinds of events that are recorded in history and archeology, and there's a lot of information that can be gained from these records if we can recreate the sky at some given date.

Today, this is done by computers. There are many free and commercial planetarium programs that allow the user to recreate the sky at a given date. However, their accuracy is not always the best, specially at dates very far removed from the present. In general, the accuracy is very good for periods of plus or minus 1,000 years from the present. Over longer periods, errors tend to accumulate and accuracy falls. Nevertheless, we know enough about these motions to calculate such things with great precision, regardless of whether a specific program is able to do so. In short, I'm saying - yes, there are dozens of programs these days which will recreate the sky in 2,500 BC and show you what the Pharaoh's saw, and for the most part they are pretty good. However, if you are doing scholarly research and need high accuracy, you need to look into the specific program you are using and ensure that it really offers the accuracy you need for the dates you are interested in.

The Brightness of Stars

A person with good eyesight viewing the night sky on a moonless night, and far from any source of light pollution, can theoretically see up to about 6000 stars. In practice, this number is far lower, because there is almost always some source of light pollution. In a brightly lit city, you might only see half a dozen stars. The reason for this small number is that stars are really not very bright, and even minor sources of light nearby can be enough to overwhelm the light from the star and obscure it.

The brightness of stars depends upon a number of factors, including:

• how far the star is from us - nearer stars appear much brighter than stars which are more distant, following the inverse square law
• how much energy the star produces - large stars which produce a lot of energy are brighter than smaller stars which produce less energy
• how much of the star's energy is in the visible spectrum - some stars might produce a lot of energy, but if most of that energy is in the form of x-rays, for example, the star may appear dim to us because our eyes don't detect x-rays.

In astronomy, the brightness of a star is referred to as its magnitude, and a number is assigned to it for measurement purposes. This system of assigning magnitudes is very old, and was originated by Hip parch us in ancient Greece. It was formalized by Ptolemy, who categorized all visible stars into 6 magnitudes, with magnitude 1 being the brightest and magnitude 6 being the faintest. Each magnitude was supposed to be twice as bright as the previous, so for example, magnitude 3 was twice as bright as magnitude 4.

Since the ancient Greeks had no instruments to measure the brightness, the scale relied on the human eye to decide which stars were brighter than others. This is not easy, since you can't bring two arbitrary stars together to compare them directly. This system was modified in 1856 by an English astronomer named Norman Pogson, who redefined the magnitude scale so that a magnitude 1 star was 100 times brighter than a magnitude 6 star.

On this new scale, the old Greek system of each successive magnitude being twice as bright as the previous, was no longer true. If you define magnitude 1 as being 100 times as bright as magnitude 6, then you have 5 steps going from 1 to 6. If you want each of these steps to be the same size (an even scale), then each step must be the 5th root of 100 larger than the previous. This number, the 5th root of 100 comes to 2.51189, and is termed Pogson's Ratio.

Magnitude is often represented in the superscript "m" notation, thus a magnitude of 4 is represented as 4^m or a magnitude of 6 represented as 6^m, for brevity.

Now if you consider the brightness of the faintest star you can see (which would be 6^m) to be unity, a 5^m star will be 2.512 times as bright. The next magnitude (4^m) will be 2.512 times brighter than 5^m, or 2.512² times brighter than 6^m. In the same way, a 3^m star will be 2.512³ times as bright as a 6^m star; a 2^m star will be 2.512⁴ times as bright as a 6^m star, and a 1^m star will be 2.512⁵ times as bright as a 6^m star, which comes to exactly 100 times brighter.

This is the same system we use currently, except that we are no longer limited to 6^m being the faintest stars. On a dark night, a good pair of binoculars can easily allow a person to see stars as faint as 9^m. A basic 6 inch reflector telescope, under good observing conditions, will allow the observer to see down to about 13^m, and the best amateur telescopes can take that down to about 15^m or 16^m. Professional telescopes are even more sensitive. The best Earth-based telescopes can see objects down to about 24^m, while the Hubble Space Telescope can observe objects to about 30^m. The magnitude of the faintest object visible through a given telescope is called that telescope's limiting magnitude. If you have a telescope and would life to calculate its limiting magnitude, here is a handy calculator. The general rule is that the limiting magnitude depends upon the diameter of the objective lens in a refractor, or the diameter of the mirror in a reflector telescope. Each time you double the diameter, you increase the limiting magnitude by 1.5^m. Of course this is only a generic rule, a specific telescope may vary depending upon the quality of the optics. Also, limiting the field of view increases the contrast, and so that may increase the limiting magnitude beyond what you might expect simply by considering the objective or mirror size.

The other modern change in the magnitude scale is that we no longer consider 1m to be the brightest objects. Pogson initially set his baseline with the star Polaris, the north star, by assigning it magnitude 2. Other stars were therefore calibrated to Polaris. Later, it was discovered that Polaris is a variable star with significant fluctuations in its magnitude, so for a time Vega was selected instead. Today, astronomers no longer use any fixed star to define magnitude, instead it's a calibrated scale based on hundreds of stars.

There are in fact many stars brighter than the 1m stars defined by earlier astronomers. We have better instruments today to measure these differences. Therefore, the magnitude scale has been extended into the negative number range, with increasingly negative numbers representing increasingly brighter objects. We can even assign magnitudes to very bright objects such as the Sun and the Moon. The table above shows some of them.

Apparent Magnitude and Absolute Magnitude

So far we've talked only about how bright a star appears to us, that is, the Apparent Magnitude. However, this greatly depends upon how close the star is to us. If you look at the list above of the brightest objects visible from the Earth, you can see stars like Alpha Centauri A on the list, which are really quite dim, but only appear so bright because they are so close. Our Sun is also a somewhat unremarkable medium-sized star, but is very bright because it's close. There are many stars that are millions of times brighter than the Sun (in that they produce millions of times more energy per second than the Sun does), but appear dim because they are very far.

The Absolute Magnitude is a measure of how much energy the star produces in the visible spectrum, independent of its distance from us. Absolute magnitude is defined as how bright a star would appear to us if it were located exactly 10 parsecs away from us. Obviously, we can't just move a star to a fixed distance of 10 parsecs, so absolute magnitude is a calculated number. In order to determine a star's absolute magnitude, we need to know its distance. This is not easy to determine. We covered some of this problem in the earlier section on distance to stars, where we talked about parallax based measurements. The parallax method works fine for nearby stars, but for very distant stars (and for objects outside our galaxy), other methods are used.

The table on the right shows the absolute magnitudes of some of the brightest objects that have been observed so far. These are some incredibly bright objects. Consider that a parsec is about 200,000 astronomical units. One astronomical unit (AU) is the distance from the Earth to the Sun. Therefore, when you consider a star with an absolute magnitude of -12.5 (as you can see for the brightest star on the list), it means that if this star was over 2 million times farther from us than the Sun is (standard 10 parsec distance), it would still appear to us to be as bright as the full moon.

We can calculate the Sun's absolute magnitude in the same way, by moving the Sun from its current position to the standard 10 parsecs distance used for calculating absolute magnitudes. The Sun's current distance from Earth is 1 astronomical unit (AU), and 10 parsecs would therefore be 2,062,648.06 AUs. From the inverse square law, we know that doubling the distance decreases the apparent brightness of a light source by 4 times; therefore the apparent brightness of the Sun would be 2,062,648.06² times less, or about 4.25 trillion times less.

The apparent magnitude is:

log_2.512(F)

where 2.512 is Pogson's Ratio mentioned earlier in this section, and F is the flux of sunlight. Since we already know the apparent magnitude of the Sun at its current position (-26.74 according to the table above), we can express the flux as a ratio to that (4.25 trillion times less) and get a result which can be added to the known apparent magnitude at its current position.

log_2.512(4.25452*10¹²) = 31.5721

So the absolute magnitude of the Sun would be:

-26.74 + 31.5721 = 4.83

At 4.83^m, the Sun would be among the faintest stars you could see from Earth. In fact, without good viewing conditions away from light pollution, you would not be able to see it with your naked eyes. Compared to that, the brightest stars on the list would be as bright as the full moon as seen from Earth.

Dependence on Wavelength

As mentioned earlier, we talk of "brightness" in terms of light visible to the human eye. Many stars output more of their energy in some part of the spectrum that is not visible to the human eye; therefore Absolute and Apparent Magnitudes are defined in terms of the visible spectrum only.

However, even for light that is visible to the human eye, there is a dependence upon the wavelength. Human eyes are most sensitive to green, probably due to some evolutionary reason to select for the easy detection of chlorophyll (green plants for food). This is known as the scotopic function of the eye, and is different from the photopic function, which is the actual brightness of different wavelengths in the visible spectrum. Green stuff looks brighter to us than objects with other colors, even though photometrically the same amount of light energy from the visible spectrum may be emitted from both.

For this reason, measures of Absolute or Apparent Magnitudes must be specified by including the wavelength at which the measurement was made. A further complication is the widespread use of cameras in astronomical observations. Photographic film is more sensitive to blue light than to yellow-green light, compared to human eyes. Many cameras no longer use film, but instead use electronic sensors such as CCDs or CMOS devices. All of these have different sensitivities to different wavelengths, so the raw results must be calibrated to some standard in order to calculate the magnitude of stars being observed.

Typically, the standard used is the normal (not color blind) human eye, and this is what we can assume when we see numbers listed for stellar magnitudes, unless specified otherwise. It's important to keep this in mind when switching between different modes of observation (watching the sky through binoculars or a telescope, versus photographing the sky through a telescope). The source of the variability, of course, is the spectra of stars - the fact that different stars emit their light in different bands of the spectrum. That is the next topic.

The Spectra of Stars

Stars produce electromagnetic energy through nuclear fusion. This energy is produced across the entire spectrum, from radio waves through infrared and the visible spectrum, to ultraviolet, x-rays, and beyond. The highest (and most energetic frequencies) depend upon the size and power of the star. Larger stars are hotter and produce more energy in the higher frequencies. Extremely energetic events, such as novas, can produce gamma ray bursts which have the highest frequencies.

Although stars produce energy through a wide range of the electromagnetic spectrum, typically when we refer to the "spectra of stars" we only refer to the visible part of the spectrum. There is a historical reason for this. At the time the word "spectrum" was introduced into optics in the 1600's, astronomers were not aware of the non-visible part of the electromagnetic spectrum. Therefore, "spectrum" referred to the range of visible frequencies and their distribution.

Later, astronomers realized that they could use the spectrum to classify stars. Different stars produce light of different colors. Some are easily distinguishable just with the naked eye - some stars look redder than others, some look more yellow, some look more blue-white. As telescopes became more powerful and opticians were able to create good prisms to split the spectrum into its constituent parts, astronomers were able to observe the spectra of stars in greater detail, and built various classification systems to categorize stars based on their light.

Generally, the spectrum of a star depends upon the star's mass. The bigger the star, the greater the temperature at its core, and therefore the hotter it is. Hot stars fall towards the blue-white end of the spectrum, while cooler stars fall towards the red end. However, there are a couple important caveats to remember:

First, the temperature at the core of the star (where the fusion occurs) is very different from the temperature at the surface of the star. The temperature at the core can be millions or even billions of degrees Kelvin. The temperature at the surface is measured in a few thousands of degrees Kelvin. We do not see the light from the core, only from the surface. In fact, a photon can take a long time to travel from the core to the surface, and during this time, it loses energy due to collisions and its frequency decreases. In our own Sun, a photon can take a million years to travel from the core to the surface, and it can take even longer in larger stars. This is not because the photon is slow (it travels at the speed of light after all), but because it does not follow a straight path, and goes through a lot of collisions with other particles before it reaches the surface. When we refer to the color of a star, we are only talking about the surface of the star.

Second, these colors and categories only represent stars in the main sequence. The main sequence is an important concept to understand in the life cycle of stars. There's a more detailed description of what it is here. There is also a briefer description below. But for now, remember that stars outside the main sequence, such as red giants or white dwarfs do not fall into these categories. A red giant may be huge, much larger in terms of radius than the biggest stars in the main sequence. But it will be much cooler than large stars in the main sequence, and therefore its color will not correlate with the color of large stars in the main sequence. Similarly, white dwarfs are tiny, but their color is not what you would expect for a star that small. Again, it's because they are outside the main sequence.

The table above shows the modern spectral classification of stars into 7 categories: O, B, A, F, G, K and M. They are in order of decreasing size and temperature. The units of mass, radius and luminosity are in Solar units, where one unit represents the mass of the Sun, the radius of the Sun, and the luminosity of the Sun, respectively.

As you can see, the largest stars (Class O stars) are huge - 16 solar masses and higher. The cores of these stars are very hot, and they may be fusing elements other than hydrogen. There is a more detailed description of the fusion reactions that occur in stars of different sizes here. Generally, the hotter the core of the star, the hotter its surface will be. Since larger stars burn hotter, the larger the star, the higher the temperature of its surface. Again, remember, this only refers to main sequence stars. There can be stars outside the main sequence, such as red giants, which are very large but their surfaces are quite cool in comparison.

There are two columns for color in the table - Conventional and Apparent. The reason for this is historical. Conventionally, Class A stars were considered to be white, and all other classes were adjusted accordingly. The Apparent color is what an observer on Earth would actually see, if observing on a very dark night through a good pair of binoculars. The colors used in the table are exaggerated - they represent what the color would look like if it were greatly magnified and then observed in daylight. In reality, most stars tend to look white when observed with the naked eye. This is because star light is very faint, and incapable of exciting the color receptors in our eyes (the cones), and we tend to observe it with the rod cells of the retina which are only sensitive to light intensity and not to color.

In addition to the modern O, B, A, F, G, K, M classification (which is sometimes also called the Harvard Spectral Classification), there are other systems of classification as well. An older system, devised by Angelo Secchi in the 1860's, classified stars into 5 categories based on spectra. This is no longer used. Another system was devised in the 1940's at Yerkes Observatory, and is known as the Yerkes System. This system categorizes stars based on a two dimensional scale of temperature and luminosity, and divides stars into 7 categories. However, the Harvard Spectral Classification is the most commonly used.

Some extensions to the Harvard classification have been made recently. Most notably, a new class "W" was added, which represents Wolf-Rayet stars. These are superluminous old and dying stars which have had the hydrogen blown away, and therefore have predominantly helium in their atmospheres. Class W would be among the most massive stars, and therefore would be placed before class O in the Harvard system. There are also a number of categories intermediate between W and O, which are sometimes referred to as OC and ON, depending upon the prevalence of carbon or nitrogen lines in the spectra of O type stars. Some Class B stars can also show these lines, hence there are also classes BC and BN. At the other end of the spectrum, there is a new class L, which is cooler than class M. There are also classes T (methane stars), Y (very small brown dwarfs), class C (red giants near the end of their lives, with an abundance of carbon), S (with lots of zirconium), etc. Additionally, there are many subcategories of each, and many individual peculiarities (usually referred to by adding a lowercase letter to the classification) which can represent various peculiarities of a given star.

The Main Sequence

In 1910, Ejnar Hertzsprung and Henry Russell discovered that if they plotted the spectral classes (colors) of stars against their absolute luminosity or absolute magnitude on a 2D scatterplot, the vast majority of stars clustered along a somewhat wavy curved line. Such scatterplots are called Hertzsprung-Russell diagrams. Stars that cluster along the curved line are stars that are in the main sequence. The physical significance of the main sequence is that such stars are fusing hydrogen at their cores.

There is some context needed to understand this. For most of their life cycles, stars fuse hydrogen at their cores. In time, however, the concentration of hydrogen at their cores falls to a level that hydrogen fusion cannot be sustained at their cores, and the hydrogen fusion moves out into a region that forms a ring around the core. Since fusion is now occurring in a more superficial layer and not at the core, the star expands in volume and becomes larger. Fusion is still occurring, and hydrogen is still the fuel, but the region in which fusion is occurring is no longer the core. Such a star has moved out of the main sequence. These stars will typically grow to very large volumes and become red giants (or even red supergiants if they had a large mass to begin with). Eventually, hydrogen in the ring around the core is also exhausted, and fusion stops. At this point, there is no longer energy being generated to force expansion, so the star suffers gravitational collapse.

Gravitational collapse produces a lot of heat (since large amounts of material falls down a gravitational potential gradient, and gravitational potential energy is converted to heat energy). The heat may raise the temperature of the core far beyond what it was even when the star was fusing hydrogen at the core. To give some numbers to this, typical temperatures at the core of a medium sized star that is fusing hydrogen at the core are around 20-50 million °K. Under gravitational collapse, they can rise to billions of °K. This immense heat initiates the fusion of helium, and then for a while the star continues to produce energy, now fusing helium at the core. Under the influence of this energy, the star expands again, and grows in volume.

Eventually, helium in the core may also be exhausted, and the star may move to fusing helium in a ring around the core, just as it had previously moved from fusing hydrogen in the core to fusing hydrogen in a ring around the core. Again, the star will expand to a red giant, because fusion has moved closer to the surface. Such oscillations between its regular size and an expanded red giant state may occur many times, with intervening collapses, as the star moves up the fusion chain, from hydrogen to helium to carbon, and other elements. The exact fusion reactions depend upon many factors, such as the initial mass of the star (smaller stars will never reach the helium fusion stage, since they don't have enough mass to raise temperatures sufficiently high at their cores), the constitution of the star (older stars that were formed early after the big bang only have hydrogen and helium, because those were the major elements produced by the big bang; younger stars contain many more elements because they were formed from the detritus of earlier generations of stars), etc.

These fusion pathways are described in much more detail in another article I wrote, which can be found here. For now, the thing to remember is that stars can fuse different elements in different regions of the star, but for a large part of their life cycle, they fuse hydrogen at their cores. Stars that are fusing hydrogen at their cores, no matter what the star's size or color, are in the main sequence. When they are not fusing hydrogen at their cores, they have moved out of the main sequence.

Hertzsprung-Russell Diagrams

There are various types of H-R diagrams, the most popular being a plot of absolute luminosity versus spectral class. Since "spectral class" is actually a list of categories (O, B, A F, G, K and M, as described above) rather than a continuous variable, the star's color is used as a substitute for the spectral class when plotting the diagram.

The figure on the right shows a typical H-R diagram. The 2D plot is between temperature on the X axis and Luminosity on the Y axis.

Since temperature is difficult to measure directly, more commonly the color is used as an indication of temperature. The color has been plotted as well on the X scale, at the bottom of the plot. Luminosity corresponds to Absolute Magnitude, since a more luminous star would be brighter when placed at some standard distance from the Earth. The Absolute Magnitude is therefore also shown on the Y axis, on the right hand side.

The long, wavy curve going from upper left to bottom right is the main sequence. Each dot on the diagram is a star. These are the stars from the Hipparcos Catalog.

As you can see, stars on the main sequence are naturally sorted by color and size. At the top left are the very large and very hot stars, which are predominantly blue and blue white. At the bottom right are the smaller and cooler stars, which are mostly red. In the middle regions of the curve are the yellow and orange stars, which are intermediate in size and temperature.

All the stars in the main sequence are distinguished by burning hydrogen at their cores. Since an H-R diagram is basically a snapshot of the sky at some given time, it shows different stars in various stages of their life cycles. The main sequence stage, during which stars fuse hydrogen at their cores, forms the much larger part of the life cycle of a star - about 90% of its life, during which it is emitting substantial amounts of energy. Since this stage of the life cycle is so long, it is to be expected that the majority of stars will be seen in this stage in a snapshot such as an H-R diagram.

The stars which don't fall in the main sequence are shown clustered in other parts of the plot. These are mostly either red giants or white dwarfs. The red giant cluster is fairly large, because many stars are old enough to have exhausted the hydrogen in their cores, so fusion is occurring in more superficial layers of the star. White dwarfs are an end-stage in the life cycle of stars, and they would be even more numerous, but their faintness makes detection difficult.

Spectral classes are listed at the top, but remember, these classes only apply to stars along the main sequence curve. The Roman numerals refer to the Yerkes Classification mentioned earlier. In this classification, Class V stars are main sequence stars.

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