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photosphere. As the line of sight passes farther from the center of the Sun, the opacity decreases because (1) the path length through the Sun is less, and (2) it passes through less dense regions. Since the amount of light getting through is proportional to e~T, the effect of t changing from line of sight to line of sight is enhanced by the exponential behavior. Therefore the transition from the Sun being mostly opaque to being mostly transparent takes place over a region that is small compared with our resolution, and the edge looks sharp.

6.3.2 Temperature distribution

Another interesting phenomenon near the solar limb can be seen in the photograph in Fig. 6.1. The Sun does not appear as bright near the limb as near the center. This limb darkening is also an optical depth effect, as shown in Fig. 6.10. We compare two lines of sight: (1) toward the center of the Sun from the observer, and (2) offset from

H Limb darkening. Line of sight 1 is directed from the observer toward the center of the Sun, so it takes the shortest path through the atmosphere.This line allows us to see the deepest into the photosphere, and the base of the photosphere is defined to be where the optical depth t along this line reaches unity. Line of sight 2 is closer to the edge, so it doesn't allow us to see as deep into the photosphere. If the temperature decreases with increasing height in the photosphere, then line of sight 1 allows us to see hotter material than does line of sight 2, and the edge of the Sun appears darker than the center.

H Limb darkening. Line of sight 1 is directed from the observer toward the center of the Sun, so it takes the shortest path through the atmosphere.This line allows us to see the deepest into the photosphere, and the base of the photosphere is defined to be where the optical depth t along this line reaches unity. Line of sight 2 is closer to the edge, so it doesn't allow us to see as deep into the photosphere. If the temperature decreases with increasing height in the photosphere, then line of sight 1 allows us to see hotter material than does line of sight 2, and the edge of the Sun appears darker than the center.

the center of the Sun. In each line of sight, we can only see down to t = 1. Line of sight 1 is looking straight down into the atmosphere, so it gets closer to the center of the Sun before an optical depth of unity is reached than does line of sight 2, which has a longer path through any layer. We see deeper into the Sun on line of sight 1 than we do on line of sight 2. If the temperature decreases with increasing height in the photosphere, we are seeing hotter material on line of sight 1 than on 2, so line of sight 1 appears brighter than line of sight 2. Since line of sight 1 takes us the deepest into the Sun, we define the point at which t reaches unity on this line as being the base of the photosphere. When we talk about the temperature of the Sun, we are talking about the temperature at the base of the photosphere.

When we look at the Sun, it appears brighter along line of sight 1 than it does along line of sight 2. This means that the Sun is hotter at the end of 1 than at the end of 2. From this, we conclude that the photosphere cools as one moves farther from the center of the Sun. If the photosphere became hotter as one moves farther from the center, we would see limb brightening.

We obtain more useful information about the photosphere by studying its spectral lines (Fig. 6.11). The spectrum shows a few strong absorption lines and a myriad of weaker ones. The stronger lines were labeled A through K by Fraunhofer in 1814. These lines have since been identified. For example, the C line is the first Balmer line (Ha); the D line is a pair of lines belonging to neutral sodium (NaI); and the H and K lines belong to singly ionized calcium (Call). Sodium and calcium are much

The solar spectrum. [NOAO/AURA/NSF]

less abundant than hydrogen but their absorption lines are as strong as Ha. We have already seen in Chapter 3 that this can result from the combined effects of excitation and ionization.

6.3.3 Doppler broadening of spectral lines

When we study the lines with good spectral resolution, we can look at the details of the line profile. The lines are broadened by Doppler shifts due to the random motions of the atoms and ions in the gas (Figs. 6.12 and 6.13). If all the atoms were at rest, all the photons from a given transition would emerge with a very small spread in wavelength. However, the atoms are moving with random speeds in random directions. We therefore see a spread in the Doppler shifts, and the line is broadened. This process is called Doppler broadening. If the gas is hotter, the spread in speeds is greater, and thus the Doppler broadening is also greater. If in addition to these random motions all the objects

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