By Carroll B.W., Ostlie D.A.

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6) implies that Betelgeuse would be detectable from a distance of d D 1 Mpc. 16 Given the number of telescopes the student is asked to research, it may be useful to have the class split into teams and have each team subdivide the research. 1. (b) See Fig. 2. (c) See Fig. 3. 1: The wavelength ranges and energies of some ground-based and space-based telescopes. , 3(-7) D 3 10 7 . 2: The photon wavelengths covered by some ground-based and space-based telescopes. 17 (a) See Fig. 4. (b) See Fig. 4. 3: The photon energies covered by some ground-based and space-based telescopes.

Near the edge of the shell, where your line of sight passes through more gas, the shell appears brighter and you see a ring. 14 From Eq. 34), 1 dI DI Ä ds S ; 3 sr 1 : Solutions for An Introduction to Modern Astrophysics 51 with Ä , , and S independent of position, can be integrated as Z I where I ;0 I ;0 Z dI 0 D 0 I S Ä s 0 ds 0 ; is the value of the specific intensity at s D 0. This results in Â ln S I I ;0 S Ã D Ä s; which simplifies to Eq. 35). 16 (a) Begin with the transfer equation, Eq. 34), 1 dI DI Ä ds S : Here, s is a distance measured along the light ray’s direction of travel.

For these two points, log10 Ä 2 log10 Ä 1 D 3:43 ' 3:5; log10 T2 log10 T1 as expected for a Kramers opacity law. 1:5 105 kg m 3 / D3 10 5 m: (b) From Eq. 29), the number of random-walk steps of this size from the center to the surface of the Sun is Â Ã2 Â Ã d Rˇ 2 N D D D 5 1026: ` ` The time for a photon to cover this many steps of size ` is tD N` D5 c 1013 s; almost two million years! 12 Because you always look back to an optical depth of about D 2=3, Eq. 17) implies that you see down to a depth s into the star given by Z s 2 Ä ds D : 3 0 At wavelengths where the opacity is greatest, the value of s is smallest.