By B. Carroll, D. Ostlie
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Extra resources for An Introduction to Modern Astrophysics 2nd ed - SOLUTIONS MANUAL
A C b v / sec Â e D a sec Â Z 1 e v sec Â v sec Â d subscripts for convenience) v Z d 0 v C b sec Â 1 0 v e v sec Â d v: The first integral on the right-hand side is 1= sec Â, while the second can be integrated by parts, resulting in ˇ1 ! 24 Assuming that there is zero flux received from the center of the line (as in the Schuster–Schwarzschild model), the equivalent width is equal to the area of the half-ellipse divided by the semimajor axis. From Eq. 25 Heisenberg’s uncertainty principle, Eq. 20), relates E, the uncertainty in the energy of an atomic orbital, to t, the time an electron occupies the orbital before making a downward transition: E t D h 2 t : When an electron makes a downward transition from an initial to a final orbital, the energy of the emitted photon is (Eq.
2/2 D 8, E1 D 13:6 eV, and E2 D 3:40 eV. Then, with N2 =N1 D 0:01, T D 1:97 104 K. If the number of atoms in the first excited state is only 10% of the number of atoms in the ground state, then T D 3:21 104 K. 6 (a) We solve the Boltzmann equation, Eq. 3/3 D 18 and E3 D N3 =N1 D 1, then T D 6:38 104 K. (b) From the Boltzmann equation using T D 85,400 K, N3 =N1 D N3 =N D 1:74. 42 1:51 eV, we find that if Solutions for An Introduction to Modern Astrophysics (c) 43 As T ! 1, the exponential in the Boltzmann equation goes to unity and Nb =Na !
4 The most probable speed, vmp , occurs at the peak of the Maxwell–Boltzmann distribution, Eq. 1). Setting d nv =dv D 0, we find Á d nv m Á3=2 d 2 e mv =2kT v 2 D 0: D4 n dv 2 kT dv This leads to Â Ã mv 3 2 C 2v e mv =2kT D 0; kT p so that vmp D 2kT =m, which is Eq. 2). 5 Solving the Boltzmann equation, Eq. 2/2 D 8, E1 D 13:6 eV, and E2 D 3:40 eV. Then, with N2 =N1 D 0:01, T D 1:97 104 K. If the number of atoms in the first excited state is only 10% of the number of atoms in the ground state, then T D 3:21 104 K.
An Introduction to Modern Astrophysics 2nd ed - SOLUTIONS MANUAL by B. Carroll, D. Ostlie