Radiation Transfer, Stellar Atmospheres, Hydrostatic Equilibrium l. points) Phot
ID: 3161256 • Letter: R
Question
Radiation Transfer, Stellar Atmospheres, Hydrostatic Equilibrium l. points) Photons scatter off free electrons by a process of Thomson scattering. The cross section for Thomson scattering has a wavelength-independent cross section 6.65 x 10 (Note: photons can scatter off protons as well, but the cross section is much smaller than that of electrons since mp me). In the core of the Sun, essentially all of the hydrogen gas is ionized, and the mass density is roughly 1.5 x 105 kg m (a) Assuming the Sun were made out of purely hydrogen, estimate the number density of electrons, ne, in the core of the Sun. (b) Estimate the mean-free path of a photon, l, in the core of the Sun due to Thomson scattering. (c) The Sun isn't pure hydrogen, but contains helium and other -metals." It turns out the opacity in the core is dominated by these metals, and the overall opacity is roughly 4 times higher than that due to Thomson scattering. Estimate the mean-free path of a photon due to all forms of opacity. (d) The number of random-walk steps a photon would take on average to diffuse out of the core is given by N (Rore/)2, where the radius of the Sun's core is Roore 0.2R Estimate how long it takes for a photon to diffuse from the center of the core to the core's boundary. 2. (5 points) In class we derived the radiative transfer equation with respect to optical depth and found dla IA Sa where Sa Nx/ix is the source function. Consider a beam of light of specific intensity IA(o) that is incident on a uniform cloud having a constant value of SA (with respect to optical depth) (a) Solve the radiative transfer equation to determine (TA). (b Suppose A(0/SA 2.0. Plot IA(TA)/IA (0) vs. TA for the range 0 s TA S 3 (7 points) Consider a stellar atmosphere comprised of electrically neutral hydrogen gas. In this simple situation, the number density of free electrons must equal the number density of H II ions: ne nm. Also, the total number density of hydrogen atoms (both neutral and ionized), nr, is related to the density of the gas by nr p/(mp me) s p/mp, where mp is the mass of the proton. Let the density of the gas be p 10-6 kg m 3, typical of the photosphere of an A star.Explanation / Answer
i) the electron density= density/mass of photon =(1.5*105 /1.67*10-27 )=0.898*1032 m-3
ii) mean free path of photon =1/(electron density*thomson scattering)= 1/0.898*1032 *6.65*10-29 =0.167*103 m
iii) the mean free path is calculate in terms of opacity and density with the followinf formula
1/(opacity* density)
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