PROBLEM SET #6

For the problems below, you need to write programs using C language. Turn in your own source programs written independently (by email) together with reports.

# 1.

It has often been customary to describe the initial mass function (IMF) of stars in a cluster using the Salpeter function of the form:

(1)
\begin{align} N(M) dM \propto M^{- \alpha} dM \quad \mathrm{with} \quad \alpha = 2.35 \end{align}

where $N (M )dM$ denotes the number of stars with mass between $M$ and $M + dM .$ Use $10^6$ random numbers to draw a Monte-Carlo sample from the Salpeter IMF with mass in the range of $1 - 100 M_\odot$. Plot the resulting probability distribution function as a function of $\log M$, and fit it to a power law, and compare the fitted power-index with $\alpha = 2.35.$

# 2.

Purely Scattering Atmosphere: Suppose a photon incident on the bottom of a plane-parallel slab with optical depth $\tau_\mathrm{max}$. We assume that the slab is infinite in the $x$ and $y$ directions: $z$-axis is normal to the slab. The photon can be scattered (with no absorption) at any point within the slab. We begin with $z = 0$ (bottom of the slab) and follow the photon’s trajectories up to $z_\mathrm{max} = 1$ (top of the slab) by taking following steps:

1. We inject a photon from below whose flux is isotropic in any direction. The probability for a certain injection angle at $z = 0$ (with respect to the z-axis), is given by $p(\mu) d \mu = 2 \mu d \mu$ with $0 \le \mu \le 1$, where $\mu \equiv \cos \theta$. Use a uniform deviate $\xi_1$ to sample $\mu$. Use another uniform deviate $\xi_2$ to obtain $\phi = 2 \pi \xi_2$. Calculate the initial direction of the photon $( \sin \theta \cos \phi , \sin \theta \sin \phi, \cos \theta).$ Note that the initial position of the photon is $(x, y, z) = (0, 0, 0).$
2. When the optical depth of a photon is $\tau$, the probability $P$ that a photon interacts with the medium is given by $P = 1 − e^{−τ}.$ Noting that $P$ is random over $[0, 1)$, $\tau$ can be sampled from a uniform deviate $\xi_3$ as $\tau = - \ln (\xi_3)$. Pick up a value for $\tau$.
3. The distance traveled by a photon along a ray is given by $L = \tau z_\mathrm{max} / \tau_\mathrm{max}.$ Update the photon’s new position as $x = x + L \sin \theta \cos \phi, y = y + L \sin \theta \sin \phi,$ and $z = z + L \cos \theta$.
4. If $z < 0$, add one to the number of reflected photons. If $z > z_\mathrm{max}$ , add one to the number of transmitted photons. In either case skip to Step 6 below.
5. Assume that the photons are scattered uniformly into $4 \pi$ steradians. Generate the new direction by sampling uniformly for $\phi$ in the range $0$ to $2 \pi$ and $\mu$ in the range $−1$ to $1$: $\phi = 2 \pi \xi_4$ and $\mu = 2 \xi_5 -1$, where $\xi_4$ and $\xi_5$ are two independent uniform deviates in $[0, 1)$.
6. Repeat Steps 2–5 until the fate of the photon has been determined.
7. Repeat Steps 1–6 with additional incident photons until sufficient data has been obtained.

## (a)

Calculate the transmission and reflection probabilities for $\tau_\mathrm{max} = 0.01, 0.1, 1,$ and $10.$ Begin with $10^3$ incident photons and increase this number until satisfactory statistics are obtained. Give a qualitative explanation of your results.

## (b)

Draw some typical paths of photons in the $x–z$ plane.

# 3.

Consider a 2D Poisson equation in the $x \times y = [−L/2, L/2] \times [−L/2, L/2]$ plane

(2)
\begin{align} \left( {{\partial^2} \over {\partial x^2}} + {{\partial^2} \over {\partial y^2}} \right) u(x, y) = f(x, y), \end{align}

subject to the periodic boundary conditions: $u(x+L/2, y) = u(x−L/2, y)$ and $u(x, y + L/2) = u(x, y − L/2).$ Perhaps the most convenient way to solve this equation is to use Fourier transforms.

## (a)

Use the indices $j$ and $k$ to indicate the grid points in the $x-$ and $y-$directions, respectively. (That is, $x_j = (−1/2 + j/N )L$ and $y_k = (−1/2 + k/N )L$ for $j, k = 0, \cdots , N .$) Write a finite difference representation of equation (2).

## (b)

Define

(3)
\begin{align} u_{j,k} = \sum_{m=0}^{N-1} \sum_{m=0}^{N-1} U_{m,n} e^{- 2 \pi i (m x_j + n y_k ) / L}, \end{align}
(4)
\begin{align} f_{j,k} = \sum_{m=0}^{N-1} \sum_{m=0}^{N-1} F_{m,n} e^{- 2 \pi i (m x_j + n y_k ) / L}, \end{align}

with integers $m$ and $n$. Note that i is the imaginary unit (i.e., $i^2 = −1$). Show that the result of Part (a) is reduced to

(5)
\begin{align} U_{m,n} = F_{m,n} / \lambda_{m, n}, \end{align}

where $\lambda_{m, n}$ is the 2D gravitational kernel. Find an expression for $\lambda_{m, n}$.

## (c)

Calculate $u(x, y)$ for $f = e^{ −(x +y ) }$ on $x \times y = [−1, 1] \times [−1, 1].$ Make a contour plot of $u$ in the $x–y$ plane as well as a cut profile $u(x, 0)$ along the $x$-axis $(y = 0)$.

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