The stars are EXTREMELY far away. If you created a scale model, in which the Sun (which is about a million miles across) is shrunk down to the size of a pin-top (says, 2 millimeters), then the earth would be a dust speck 20 centimeters away, Jupiter would be a pinhead about a meter away, Pluto, the most distant planet in our Solar System would be a bit over 8 meters away (at the back of the classroom), and the nearest star would be 44 kilometers away (in Lewistown). That's a long ways -- if you want to write it in kilometers, its about 3 followed by 13 zeros! No wonder the parallax of stars wasn't found until 1848!

Fortunately, parallax was found, for without parallax, we wouldn't know which stars are nearby and which are far away. With parallax, we can measure the distances to (at least the nearest) stars. The motion of the earth around the Sun defines a baseline; the displacement of a star (with respect to background stars) gives an angle. For the nearest stars, this angle is about one arcsecond, that is, 1/60 or 1/60 or one degree. For comparison the angular size of the Moon is 1/2 of a degree. Thus, if you divide the Moon into 1800 pieces, you'll get 1 arcsecond. One arcsecond is also close to the amount that the atmosphere blurs out the (point-like) images of stars.

Through the trigonometry of the extremely skinny triangle, we can then measure a distance. Fortunately, because the triangle is so skinny, the formula for this distance is easy D = r(earth) / p where D is the distance to the star, r(earth) is the size of the earth's orbit (which is 1 A.U.), and p is the parallax angle.

Astronomers always try to make things as easy as possible for themselves. To make things easy, they define a new unit of distance, called the parsec. One parsec is the distance a star would have if it had a parallax angle of 1 arcsecond. (For those who insist on numbers, is about 206265 Astronomical Units, or 3 x 10^{13} kilometers. It's also the distance light travels in a bit over 3.25 years.) Note that by defining a parcsec in this way, the relationship between the measured parallax angle and the implied distance is trivial. If a star has a parallax of 1 arcsec, its distance is 1 parcsec. If it has a parallax of 1/2 arcsec, its distance is 2 parcsec. If it has a parallax of 1/10 arcsec, its distance is 10 parcsecs. And so on.

If we measure the distance to a star, and we measure how bright a star appears in the sky, we can then figure out the star's intrinsic brightness, i.e., how bright the star really and truly is. (Is the star a matchstick in front of our eyes, or a searchlight far away.) This is done using the inverse square law of light, which relates the intrinsic luminosity, L to its apparent luminosity l by l = L / r^2 where r is the distance.

Astronomers don't measure brightness in terms of watts (or even gigawatts). Sometimes they use solar luminosities, which is how bright a star is compared to the Sun. (For reference, the Sun is equivalent to a 4 x 10^{26} watt light bulb!) More often (unfortunately), they use absolute magnitude.

You have already encountered apparent magnitude which describes how bright stars appear in the sky. Vega has an apparent magnitude of zero. Deneb's apparent magnitude is close to 1; Polaris is close to 2. The faintest star you can generally see from State College has an apparent magnitude of around 3 or 4. The faintest star you can see out in the desert far from the lights of a city has an apparent magnitude of 6. Note that the magnitude scale goes backward --- big numbers represent faint stars. Also note that magnitude scale works the way the human eye does, which is logarithmically. This means that each magnitude is 2.5 times fainter than the previous magnitude. A star with apparent magnitude m = 1 is 2.5 times fainter than a star with m = 0. A star with m = 2 is 2.5 x 2.5 = 6.25 times fainter than one with m = 0. And so on. A difference of 5 magnitudes is equivalent to a factor of 100 in brightness.

The apparent magnitude of a star describes how bright the star appears. The absolute magnitude of a star describes the star's intrinsic brightness. Absolute magnitude is defined in this way. A star's absolute magnitude is the apparent magnitude the star WOULD have IF it were at a distance of 10 parcsec. The Sun has an apparent magnitude of -26 . However, if the Sun were 10 parsecs away from the earth, it would be much fainter -- its apparent magnitude would be about 5. (You wouldn't even be able to see it from State College.) So the Sun's absolute magnitude is 5. (There is a formula which relates apparent magnitude, absolute magnitude, and distance, just likes there's a formula relating apparent luminosity, absolute luminosity and distance (see above). However, the magnitude scale involves logarithms, so the formula is complicated. We won't use it.)

If we can measure a star's parallax, then we know its distance. If we know a star's distance (and we measure how bright the star appears), then we can derive its absolute magnitude (or, equivalently, its absolute luminosity). So we know one property of the star.

Absolute luminosity is not the only stellar property that we can measure. For example, stellar surface temperatures are relatively easy to estimate. We already know one way to do this --- estimate the star's color. The redder the star, the cooler it is; the bluer the star, the hotter it is. But there's another way to measure a star's temperature, which works even when (for one reason or another), the color method doesn't work well. The method involves looking at the star's spectrum.

Stars are made mostly of hydrogen and helium. Nine out of ten atoms in the universe are hydrogen atoms. Nine out of ten of what's left is helium. So, when we observe the spectrum of a star, we should see mostly hydrogen and helium. We don't, and the reason comes from the atomic physics of the individual elements.

To understand this, let's consider the hydrogen atom as an example. Like all atoms, hydrogen has multiple levels for its (one, lone) electron. It turns out that for hydrogen, the distance between the first level and the second level is huge -- it's equivalent to a far ultraviolet photon. Optical absorptions for hydrogen only occur when an electron in the second level grabs a photon and goes up to a higher level. This atomic structure has an interesting consequence.

Consider hydrogen in the atmophere of a cool, red star. Virtually all the hydrogen will have its electrons in the ground (lowest) state. In order for one of these electrons to be in the second level, where it can grab an optical photon, it either has to a) absorb an ultraviolet photon of the proper energy, or b) be hit by something that has enough energy to push it up. But, in the case of a cool star, there are hardly any ultraviolet photons to absorb, and the atoms are moving so slowly that none of the collisions are hard enough to move an electron to a higher energy level. As a result, in cool stars, there are no hydrogen atoms that have their electrons in the second level, and therefore is no optical absorption lines from hydrogen.

Now, consider a very hot star. This hot star emits many high energy photons, and many of these are energetic enough kick a hydrogen electron completely out of the atom (i.e., to ionize the atom.) If all the hydrogen atoms have lost their electrons, then there won't be any hydrogen electrons in the second energy level, and again, there will be no absorption from hydrogen. So hot stars, like cool stars, will show no hydrogen absorption lines in the optical.

In fact, hydrogen absorption occurs only over a range of temperatures. If the star is too cool, hydrogen doesn't absorb. If the star is too hot, hydrogen doesn't absorb. Thus, if one sees strong hydrogen absorption, the star must be of intermediate temperature. (It is strongest at about 10,000 degrees.)

Similar arguments apply to other elements as well. For example, helium absorption only occurs in the very hottest stars. Calcium and other metal lines are strong at cooler temperatures. Absorption by molecules only happens in the very coolest stars. By examining the spectrum of a star and noticing which absorptions are present, it is possible to estimate the star's surface temperature extremely accurately.

When astronomers began studying stellar spectra about 100 years ago, atomic physics hadn't been invented yet. So they just classified the stars without regard to the physics. Stars with the strongest hydrogen lines were A stars. Stars with slightly weaker hydrogen lines were B stars. And the sequence continued through K, M, and O. Nowadays, we know that stars with weak hydrogen lines can be either hot or cold. From hot to cold, the temperature sequence of stellar spectra is O   B   A   F   G   K   M; the traditional (but politically incorrect) mnemonic is "Oh Be A Fine Girl Kiss Me." For reference, the Sun is a G-star; Vega is an A-star; Arcturus is a K-star.

[Note: in the last few years, solid-state physicists have made great strides developing efficient instruments for detecting and measuring infrared light. Recently, astronomers have begun using these using these next-generation detectors on telescopes to explore how stars look at these wavelengths. One result has been the discovery of a brand-new class of stars, L stars. L-stars are not easily visible in the optical because are they so very, very cool, but thanks to infrared detectors, we now know they exist. As a result, the cool-end of the spectral classification now goes K-M-L-T. (And, if you're looking for a mnemonic, recall the name of a recent White House intern.]

So far, we have discussed how to derive the intrinsic luminosity of a star and the star's temperature. When astronomers find out two properties of an object (such as a star), the first thing they usually do is plot one versus the other. This is the Hertzprung-Russell diagram, otherwise known as the HR diagram.

The x-axis of the HR diagram gives the stars' temperatures (or, equivalently their color, or spectral type). Because astronomers like to do things backwards, hot blue, O-stars are plotted on the left, and cool, red, M-stars are plotted on the right. The y-axis of the HR diagram gives the stars intrinsic luminosity (or absolute magnitude). Bright stars are at the top of the diagram, cool stars are near the bottom.

An overwhelming majority of stars (more than 90%) fall in a band across the diagram, going from cool and faint to hot and bright. This band is called the main sequence and it makes sense. Recall that from the blackbody law, hot objects radiate alot more energy than cool objects (by temperature to the fourth power). So, it is reasonable that cools stars are faint and hot stars are bright. However, there are some peculiar stars that do not fall on the main sequence. In particular, some stars are both very red and very bright, while others are very blue and very faint. Let's consider the red stars first. Each inch of these stars must be relatively faint, since cool objects do not radiate much light. The only way that these stars can be bright is to be enormous (i.e., they must have alot of square inches). We will call these red giant stars. Conversely, the only way very hot stars can be faint is to be exceedingly small. These are white dwarf stars (though a better name would be blue dwarf stars).

Note that in reality, the brightness of a star depends on two factors: its temperature and how much surface area the star has. In other words, L = b R^2 T^4 where L is the absolute luminosity of the star, R is the star's radius, T is the star's surface temperature, and b is a number to make the units come out all right. Note also that stellar size (i.e., radius) is plotted implicitly in the HR diagram --- the lower left (blue) of the diagram is small, while the upper right (red) end of the diagram is large.