숙제 04

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.

Consider the data in Table 1.

  1. Interpolating linearly the data, calculate the expected value of the function at $x = 3.2, 0.4, −0.128,$ and $−2.0.$
  2. Interpolate the above data using 10-th order polynomials to calculated the expected values at the $x$ positions given in (1).
  3. Now use cubic splines to interpolate the data and plot the results together with the data points. What are the expected values at the $x$ positions given in (1)?
x y
−2.1 0.012155
−1.45 0.122151
−1.3 0.184520
−0.2 0.960789
0.1 0.990050
0.15 0.977751
0.8 0.527292
1.1 0.298197
1.5 0.105399
2.8 3.936690 × 10−4
3.8 5.355348 × 10−7

Table 1: Data points for Problem 1

2.

It has been well established that all galaxies contain supermassive black holes at their centers, and that the black hole mass $M_{BH}$ is related to the stellar velocity dispersion $\sigma_e$ in the bulge of its host galaxy such that $\log (M_{BH} / 1 M_\odot ) = a + b\ \log ( \sigma_e /1\ \mathrm{km\ s^{-1}} )$, with $a$ and $b$ being constants. The BlackHall.txt file in the class web page contains data for 67 galaxies: the first and second columns give $M_{BH}$ and its measurement error $\Delta M_{BH}$ in units of $M_\odot$, respectively; the third and fourth columns give $\sigma_e$ and its error $\Delta \sigma_e$ in units of km s−1, respectively.

  • First, ignore $\Delta(\sigma_e)$ and fit the data using fit.c in the Numerical Recipes, and plot the data together with your fit. What are the fitting coefficients $a$ and $b$? What is the value of $\chi^2 / \nu$?
  • Now, allowing for $\Delta(\sigma_e)$, fit the data using fitexy.c in the Numerical Recipes, and plot the data together with your fit. What are the fitting coefficients? What is the value of $\chi^2 / \nu$?

3.

The Bondi or Parker Wind Problem: The Bondi (stellar wind) problem involves the spherical accretion (outflow) of gas by (from) a gravitating point mass $M$. In spherical symmetry, the steady flow of an isothermal gas with density $\rho$ and velocity $v$ obeys the following equations

(1)
\begin{align} \left( u - {1 \over u} \right) {{du} \over {dx}} = {2 \over x} - {1 \over x^2}, \end{align}

where $u$ and $x$ are the dimensionless velocity and distance from the point mass.

  • Solve Equation (1) starting inward from $x = 5$ to $x = 0.1$ for three values of $u = 3, 0.1$, and $0.01$ at $x = 5$, and plot the results on the $u–x$ plane. You will see that the case with $u(x = 5) = 0.1$ does not give the solution you want. Why does this happen? How can you circumvent this problem?
  • Now you want to have the solutions with a sonic transition (that is, solutions with $u = 1$ at some point). In view of Equation (1), u = 1 should occur at $x = 1/2$ for regular solutions. Show that the transonic solutions satisfy
(2)
\begin{align} \Delta u = \pm 2 \Delta x, \end{align}

where $\Delta u$ and $\Delta x$ denote the small changes in $u$ and $x$ around 1 and 1/2, respectively. (That is, $x = 0.5 + \Delta x$ and $u = 1 + \Delta u$ for $\left\vert \Delta x \right\vert , \left\vert \Delta u \right\vert \ll 1.$) The minus sign in Equation (2) corresponds to the Bondi accretion, while the plus sign is for the isothermal Parker winds.

  • Draw the transonic solutions for winds and accretion on the $u–x$ plane, with $x$ in the range between 0.1 and 5.