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ASTR415/688C Spring 2005 Problem Set #3

Due March 18, 2005 (5 pm)

This problem set will give you some experience with root finding and/or least-squares fitting. You may do either question 1 or question 2 (see over; if you do both, you will get a maximum of 2 points of extra credit).

  1. Consider two bodies of mass $m_1$ and $m_2$ separated by a distance $d$ travelling in circular orbits around their mutual center of mass. Take the angular momentum vector of the system to be pointing in the $+z$ direction. In a frame that corotates with the orbital motion there are five locations (called the Lagrange points) where the effective acceleration felt by a test particle vanishes. The acceleration arises from the gravity due to $m_1$ and $m_2$ plus the rotation of the system as a whole. If the masses are at points $x_1 < x_2$ on the $x$ axis (so $d = x_2 - x_1$), then by convention $L_3 < x_1 < L_1 < x_2 < L_2$. The $L_4$ and $L_5$ points lie off the $x$ axis and form equilateral triangles with the points $x_1$ and $x_2$. Conventionally, $L_4$ is taken to be in the $+y$ direction, $L_5$ in the $-y$ direction.

    1. Write a program to explore this system by computing the effective gravity (a vector) and potential (a scalar) at specific grid points in the $xy$-plane. Use units such that the gravitational constant $G \equiv 1$. The program should take as input the mass of both bodies and their separation. You may ``hardwire'' the grid dimensions into your code if you wish. The output should be the potential and $x$ and $y$ components of the acceleration at each grid point.

    2. Use your favorite plotting program to plot vectors (for the effective acceleration) and contours (for the effective potential) for the cases where $m_1 = 3$, $m_2 = 1$, $d = 1$ and $m_1 = 100$, $m_2 = 1$, $d = 1$.

    3. Use a simple root solver to determine the locations of the Lagrange points for both these cases, using your a priori knowledge of their approximate locations (i.e., first do a search along the $x$ axis for the $L_1$, $L_2$, and $L_3$ points, then along the $x = (x_1 + x_2)/2$ axis for the $L_4$ and $L_5$ points. Watch out for singularities!...).

  2. Write a program to fit first a Lorentzian
\phi(\nu) = \frac{1}{\pi} \frac{\alpha_L}{(\nu - \nu_0)^2 + \alpha_L^2}
\end{displaymath} 1
    and then a Gaussian
\phi(\nu) = \frac{1}{\alpha_D} \sqrt{\frac{\ln 2}{\pi}} e^{-(\ln
2)(\nu - \nu_0)^2/\alpha_D^2}
\end{displaymath} 2
    to the data in
    which is in the format $\nu$ $\phi$ $e$, where $\nu$ is the frequency, $\phi$ is the line strength, and $e$ is the estimated error in each $\phi$.

    1. What values of $\alpha_L$ and $\nu_0$ do you get for the Lorentzian, and what values of $\alpha_D$ and $\nu_0$ do you get for the Gaussian? Also report the error estimates for the fit parameters. Which model is a better fit to the data?

    2. Plot the data and the fits. Be sure to include the errorbars.

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Derek C. Richardson 2005-03-03