Download PHYS-4240 — General Relativity ASTR-4240 — Gravitation & Cosmology Spring, 2016 Class 1

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PHYS-4240 — General Relativity
ASTR-4240 — Gravitation & Cosmology
Spring, 2016
Class 1
The Field Equation for Newtonian Gravity
Exercise (30 pts)
The Sun is one of ∼ 1011 stars which populate the Milky Way galaxy. The stars in the
Milky Way (aka “the Galaxy”) reside in several different structural components. The Sun
lies in a thin disk of stars and gas which orbit the center of the Galaxy and extend to several
tens of kiloparsecs1 from the Galactic center. In this exercise we will model the thin disk
as an infinite sheet of thickness H and uniform mass density ρ.
1. (10 pts) — Let z be vertical distance measured from z = 0 at the midplane of the
disk. Find the gravitational field g (magnitude and direction) as a function of z. Hint: Use
Gauss’s Law.
2. (10 pts) — A star is initially at rest at some position z0 . Find its trajectory z(t) in
terms of the quantities given.
3. (10 pts) — What is the period of the motion in years if the disk contains 50 M⊙ in a
1 pc2 column? Assume that H = 50 pc.
1
1 parsec (pc) = 3.086 × 1018 cm which is about 3.3 light years.
Solution
1. By symmetry, we must have g = −g(z) ẑ, where g denotes the magnitude of g (a positive
number). Now apply Gauss’s Law to a cylindrical surface with endcaps (area A) at ±z:
Z
I
ρ dV.
(1)
g·n̂ dA = −4πG
V
S
The sides of the cylinder give no contribution and each endcap gives −g(z)A. Thus
−2g(z)A = −4πG × 2zρA
(2)
g(z) = −4πGρz ẑ.
(3)
and
2. If
then the equation of motion for z(t) is
p
4πGρ,
(4)
d2 z
= −ω 2 z.
dt2
(5)
ω≡
The solution subject to the boundary conditions z(0) = z0 and ż(0) = 0 is
z(t) = z0 cos ωt.
(6)
The data imply a mass density of ρ = 6.8 × 10−23 g cm−3 .
3. The period is
T =
2π
= 8.3 × 1014 s = 26 Myr.
ω
2
(7)