10.6: Problems

1. Find the general expression for \[\int_0^{\infty} \frac{(t-\tau)^n E_1|t-\tau|}{n!} d t\nonumber\]
2. Find the eigenvalues \(k_\alpha\) and \(\mathrm{L_{+\alpha}}\) for the discrete ordinate solution to the semi-infinite plane-parallel gray atmosphere for n = 8.
3. Repeat Problem 2 for the double-gauss quadrature scheme for n = 8.
4. If there is an arbitrary iterative function Φ(x) such that \[x_{k+1}=\Phi\left(x_k\right)\nonumber\] then an iterative sequence defined by Φ(x_k) will converge to a fixed point x₀ if and only if \[\left|\frac{\partial \Phi(x)}{\partial x}\right|<1 \quad\left|x_k\right| \leq|x| \leq\left|x_0\right|\nonumber\] Use this theorem to prove that any fixed-point iteration scheme will provide a solution for \[B(\tau)=\frac{1}{2} \int_0^{\infty} B(t) E_1|t-\tau| d t\nonumber\]
5. Find a general interpolative scheme for \(\mathrm{I}(\tau,\mu)\) when \(\mu<0\) for the discrete ordinate approximation. The interpolative formula should have the same degree of precision as the quadrature scheme used in the discrete ordinate solution.
6. Consider a pure scattering plane-parallel gray atmosphere of optical depth \(\mathrm{t_0}\), illuminated from below by \(\mathrm{I}(\tau_0,+\mu)=\mathrm{I}_0\). Further assume that the surface is not illuminated [that is, \(\mathrm{I}(0,-\mu)=0\). Use the Eddington approximation to find \(\mathrm{F}(\tau)\), \(\mathrm{J}(\tau)\), and \(\mathrm{I}(0,+\mu)\) in terms of \(\mathrm{I}_0\) and \(\tau_0\).
7. Show that in a gray atmosphere \[\frac{d P_r}{d T}=\frac{16 \sigma}{3 c} \frac{T^3}{1+d q(\tau) / d \tau}\nonumber\]
8. Use the first of the Schwarzschild-Milne integral equations for the source function in a gray atmosphere [equation (10.2.6)] to derive an integral equation for the Hopf function \(\mathrm{q}(\tau)\).
9. Show that no self-consistent solution to the equation of radiative transfer exists for a pure absorbing plane-parallel gray atmosphere in radiative equilibrium where the source function has the form \[S(\tau)=a+b \tau\nonumber\]
10. Show that the equation of transfer in spherical coordinates \[\cos \theta \frac{\partial I_\nu(r, \theta)}{\partial r}-\frac{\sin \theta}{r} \frac{\partial I_\nu(r, \theta)}{\partial \theta}=\rho\left(\kappa_\nu+\sigma_\nu\right)\left[S_\nu(r)-I_\nu(r, \theta)\right]\nonumber\] transforms to \[\pm \frac{\partial I_{ \pm}(p, z)}{\partial z}=\rho\left(\kappa_\nu+\sigma_\nu\right)\left[S_\nu(p, z)-I_{ \pm}(p, z)\right]\nonumber\] in impact space where \(\mathrm{r^2}=(p^2+\mathrm{z^2})\), and \(|\mu|=\mathrm{z/r}\).
11. Derive an integral equation for the mean intensity \(\mathrm{J}_\nu(\tau_\nu)\) when the source function is given by \[S_\nu\left(\tau_\nu\right)=\kappa_\nu B_\nu\left(\tau_\nu\right)+\frac{\sigma_\nu}{2} \int_{-1}^{+1}\left[1-\left(\mu^{\prime}\right)^2\right] I_\nu\left(\mu^{\prime}, \tau_\nu\right) d \mu^{\prime}\nonumber\]
12. Numerically obtain a solution for the Schwarzschild-Milne integral equation for the source function in a gray atmosphere by solving equation (10.2.11) for the ratio of the source function at eight points in the atmosphere to its value at one point. Describe why you picked the points as you did, and compare your result with that obtained from the Eddington approximation.
13. Using equation (10.2.21), show that equations (10.2.22) and (10.2.23) follow from the discrete ordinate equation of transfer [equation (10.2.20)].
14. Show that equations (10.2.29) and (10.2.30) follow from the substitution of the solution for the discrete ordinate method [equation (10.2.26)] into the definition for the moments of the radiation field, \(\mathrm{J}(\tau)\), and \(\mathrm{F}\).
15. Show that equation (10.2.36) is indeed the eigen-equation for the nonconservative gray atmosphere.
16. Use the Feautrier method to solve the problem of radiative transfer in a gray atmosphere.