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R (for red), and I (for infrared) have been added. (There are actually a couple of filters in different parts of the infrared.)

For example, the B - V color is defined by

B - V = 2.5 log10 [I(AV) / I(AB)] + constant where I(AV) and I(AB) are the intensities averaged over the filter ranges. (The constant is adjusted so that B — V is zero for a particular temperature star, designated A0. These designations will be discussed in the next chapter.) As the temperature of an object increases, the ratio of blue to visible increases. This means that the B — V color decreases (again because the magnitude scale runs backwards.)

2.6 I Stellar distances

So far we have discussed how bright stars appear as seen from Earth. However, the apparent brightness depends on two quantities: the intrinsic luminosity of the star and its distance from us. (As we will see in Chapter 14, starlight is also dimmed when it passes through clouds of interstellar dust.) Two identical stars at different distances will have different apparent brightnesses. If we want to understand how stars work, we must know their total luminosities. This requires correcting the apparent brightness for the distance to the star.

If we have a star of luminosity L, we can calculate the observed energy flux at a distance d. If no radiation is absorbed along the way, all the energy per second leaving the surface of the star will cross a sphere at a distance d in the same time. It will just be spread over a larger area. Therefore, the energy per second reaching d is still L, but it is spread over an area of 4-^d2 so the energy flux, f, is f = L/4wd2 (2.13)

The received flux falls off inversely as the square of the distance.

Unfortunately, distances to astronomical objects are generally hard to determine. There is a direct method for determining distances to nearby stars. It is called trigonometric parallax, and amounts to triangulation from two different observing points. You can demonstrate parallax for yourself by holding out a finger at arm's length and viewing it against a distant background. Look at the finger alternately using your left and right eye. The finger appears to shift against the distant background. Bring the finger closer and repeat the experiment. The shift now appears larger. If you could move your eyes farther apart, the effect would be even greater.

Even the closest stars are too far away to demonstrate parallax when we just use our eyes. However, we can take advantage of the fact that the Earth orbits the Sun at a distance defined to be one astronomical unit (AU). Therefore, if we observe a star and then observe it again six months later, we have viewing points separated by 2 AU. The situation is illustrated in Fig. 2.6. We note the position of the star against the background of distant stars, and then six months later we note the angle by which the position has shifted. If we take half of the value of this angle, we have the parallax angle, p.

Once we know the value of p, we can construct a right triangle with a base of 1 AU and the other leg being the length, d, the unknown distance to the star. From the right triangle, we can see that tan p = 1 AU/d (2.14)

Since p is small, tan(p) = p (rad), which is the value of p, measured in radians. Equation (2.14) then gives us p(rad) = 1 AU/d (2.15)

It is not very convenient measuring such small angles in radians, so we convert to arc seconds (see Box 2.1):

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