As long as we are only concerned with the vertical structure on scales z which are much smaller than the distance from the center R, we can ignore radial gradients and pretend that our disk has infinite extent. In this approximation, the vertical structure of a stellar system becomes a 1-D problem.
The integrals in the following problems are fairly straight-forward. You may use Mathematica or a similar system to evaluate them, but I recommend that you explicitly check each indefinite integral by differentiating to recover the integrand.
Each of these problems is worth 10 points.
Due Date: 3/21/95.
16. Using Gauss's theorem, calculate the gravitational force produced by a sheet of matter with infinite extent and zero thickness, assuming the surface density Sigma is the same everywhere.
17. Now consider a sheet with infinite extent and finite thickness. Let the mass density be
2 z rho(z) = rho_0 sech (-----) , 2 z_0where z is the direction perpendicular to the sheet, rho_0 is the density at the midplane, and z_0 is the vertical scale-height. Calculate the vertical force a(z) and potential Phi(z), adopting the boundary condition Phi(0)=0.
18. Suppose the distribution function has the form
-1/2 f(E_z) = rho_0 (2 pi sigma^2) exp(-E_z / sigma^2) ,where sigma is the vertical velocity dispersion and
1 2 E_z = - v_z + Phi(z) 2is the energy in vertical motion. Calculate the mass density rho(Phi). Note that the density is expressed as a function of Phi.
19. Now compare the expression for rho(Phi) from problem 18 with the implicit relation between rho and Phi you got from problem 17. These relations should have the same functional form. Set them equal to derive a value for z_0 in terms of rho_0, sigma, and the gravitational constant G. Congratulations! You have now solved the CBE for a thin disk with a Gaussian velocity distribution.
Hint: See BT87, Problem 4-25 for another way to approach this solution.
Last modified: March 7, 1995