3 [4p.] Vertical motion with respect to the galactic disk Assume that the disk o

Question

3 [4p.] Vertical motion with respect to the galactic disk Assume that the disk of a spiral galaxy like our own has 75 solar masses per square parsec surface density (= 75M/pc2), and is a uniform-density layer of a small total thickness. You are going to consider the vertical motion of a star outside that layer, approximating the galaxy as practically two-dimensional, infinite, uniform- sheet of material. Show that there is only a vertical force, but no sideway forces in this approximation. Using Gauss law calculate the vertical force F) (specific force, or acceleration that is the same on all small test particles- stars) at altitude above the sheet representing the galactic disks. Hint: choose the Gauss surface such that the force is either perpendicular or parallel to parts of the surface, this will make the calculation easy. Gauss law was stated in the lecture notes. You may also be guided by Gauss law as used in electrostatics. Show that the period of vertical motion does depend on the vertical amplitude of motion and explain how. Derive and calculate that period in units of millions of years, if the star travels vertically between -H and +H, from the mid-plane of galaxy. Evaluate the vertical motion period for H-1 kpc.

Explanation / Answer

given density of galaxy = 75M/pc^2
where M is amss of sun = 1.99*10^30 kg

now, consider a gaussean cylinderical surface with axis of cylinder perpendicular to the plane of the galaxy and the circular faces of the cylinder parallel to the central galactic plane

now, for the points on the circular faces of cylinder
ther eis no sideways force as from symmetry, for a star at the axis of this planar approximation of galaxy, the sideways forces are calcelled by symmetry and the only forces that remain are towards the gslactic center
hence
from gauss law

d(g)/dz = -4*pi*G*rho
dg = -4*pi*G*rho*dz
integrating
g = -4*pi*G*rho*z

now if vertical amplitude is z = H
then
g = -4*pi*G*rho*H

so in this case the time period of osscialtoin is given by
T = 2*pI*sqroot(m/4*pi*G*rho)
where m is mass of the star
for H = 1 kPc
w = sqroot(4*pi*G*rho/m)
and
equation of motion of the star is given by
z = Hsin(wt)
H = 1 kpc
z = sin(wt)

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