숙제 03

1.

The radiation flux at the surface of a blackbody can be obtained by integrating the Planck function over wavelength

(1)
\begin{align} F = \int_0^\infty B_\lambda d \lambda = {{2(kT)^4} \over {c^2 h^3}} I, \end{align}

where

(2)
\begin{align} I = \int_0^\infty {{x^3 dx} \over {\exp[x] -1}} = {{\pi^4} \over {15}}. \end{align}

Evaluate $I$ numerically using the Trapezoidal rule. Make sure that the relative error is less than 10 −6 .

2.

The exact limb darkening formula normalized at $\mu = 1$ is given by

(3)
\begin{align} H(\mu) = {{1} \over {1 + \mu}} \exp \left[ {{1} \over {\pi}} \int_0^{\pi \over 2} {{\phi \arctan ( \mu \tan \phi ) } \over {1 - \phi \cot \phi}} d \phi \right]. \end{align}

Use Simpson’s rule to integrate the above equation and plot $H(\mu)$ as a function of $\mu$ for $0 \le \mu \le 1$. (Hint: use atan() function for the arctangent function and 1/tan() for the cotangent function. What is the value of $\phi \cot \phi$ at $\phi = 0$?)

3.

Consider a 6-level model of the N+ (which produces the [N II] lines). The statistical equilibrium between transitions into and out of any level $i$ requires

(4)
\begin{align} n_i \left[ n_e \sum_{k \ne i } q_{ik} + \sum_{k < i} A_{ik} \right] = n_e \sum_{k \ne i} n_k q_{ki} + \sum_{k > i} n_k A_{ki}. \end{align}

This leads to six equations in the six unknowns $n_1, \cdots, n_6$, but it can be shown that this system is degenerate, i.e., any one of the equations can be constructed as a linear combination of the other five. We thus introduce another equation, which is the equation of normalization: $\sum_{i=1}^6 n_i = 1$. This plus any five of the others may then be
solved for the relative populations $n_i$.

In this problem, you are asked to develop a computer routine to solve for $n_i$ of the N+ ions. The required values of the atomic parameters are statistical weights $\omega_i$ , differences between the energy levels $E_{ji}$, Einstein A-values $A_{ji}$ , and collisional parameters $\Omega_{ji}$, which are given in Figure 1 and Table 1. The temperature dependence of the collision parameter $\Omega_{ji}$ can be fit to a power law: $\Omega_{ji} (t)= t^{\beta_{ji}} \Omega_{ji}$ (1.0), where the temperature
is $t \equiv T / (10^4\ \mathrm{T})$. The rate of downward collisions $j \longrightarrow i$ is given by

(5)
\begin{align} q_{ji} = {{8.629 \times 10^{-8}} \over t^{1/2}} {{\Omega_{ji}(t)} \over {\omega_j}} \mathrm{cm^3 s^{-1}}, \qquad ( j > i ), \end{align}

while the rate of upward collisions can be found from

(6)
\begin{align} q_{ji} = {{\omega_j} \over {\omega_i}} q_{ji} \exp \left[ {{−1.1605 E_{ij}} \over {t}} \right] \end{align}

where the energy separation $E_{ij}$ is in $\mathrm{eV}$.

  • Write a computer program to calculate the populations for any given $n_e , T$ pair. You may want to make use of the subroutines gaussj or ludcmp contained in “Numerical Recipes”.
  • Find the populations for the following values of $n_e$ and $T : T = 10^4\ \mathrm{K}$ and $n_e = 10, 10^2 , 10^3 , 10^4 , 10^5\ \mathrm{cm^{−3}}$; also $n_e = 10^4\ \mathrm{cm^{−3}}$ and $T = 5,000,\ 7,000,\ 12,000,\ 15,000,\ 20,000\ \mathrm{K}$. Make sure that all $n_i$ ’s should be positive.
  • The ratio of the $\lambda 6584$ line to the $\lambda 5755$ line of [N II] can be an important indicator of the electron temperature in ionized nebulae. Plot this ratio over the temperature range $5,000 \mathrm{K} < T < 20,000 \mathrm{K}$ for densities $n_e = 10, 10^2 , 10^3 , 10^4 ,$ and $10^5\ \mathrm{cm^{−3}}$.
comastro-hw03-figure1.png

Figure 1: The first six energy levels and statistical weights of N II

$i$ $j$ $\lambda_{ij}$ $E_{ij}\ \mathrm{(eV)}$ $A_{ji}$ $\Omega_{ji}\ (1.0)$ $\beta_{ji}$
1 2 203.53 ㎛ 0.00609 2.08×10−6 0.4080 0.125
1 3 76.136 ㎛ 0.01628 1.16×10−12 0.2720 0.210
1 4 6527.8 Å 1.89879 5.35×10−7 0.2934 0.048
1 5 ―― 4.05244 0.0 0.0326 0.050
1 6 ―― 5.80061 0.0 0.1323 0.025
2 3 121.64 ㎛ 0.01019 7.46×10−6 1.1200 0.170
2 4 6548.8 Å 1.89270 1.01×10−3 0.8803 0.048
2 5 3063.2 Å 4.04635 3.38×10−2 0.0977 0.050
2 6 2139.0 Å 5.79452 4.80×10+1 0.3968 0.025
3 4 6584.3 Å 1.88251 2.99×10−3 1.4672 0.048
3 5 3070.9 Å 4.03616 1.51×10−4 0.1628 0.050
3 6 2142.8 Å 5.78433 1.07×10+2 0.6613 0.025
4 5 5755.3 Å 2.15365 1.12 0.8338 −0.18
4 6 ―― 3.90182 0.0 0.0 0.0
5 6 ―― 1.74817 0.0 0.0 0.0

Table 1: Data for all possible transitions between the levels