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.
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