Homework # 10 Solutions
1. Compute the deflection angle of a star whose light just grazes the limb of the
Sun. Also compute the deflection angle of a star whose light just grazes the
limb of a 1.4M neutron star, if the neutron star was at the same distance
from the Earth as the Sun. State assumptions.
Assume that dLens << dS , then α = 4GM/(bc2 ). b will be equal to the
radius of the lens. For the Sun, we have
α=
4GM
= 8.6 × 10−6 = 1.76 arcsec.
R c2
For a neutron star,
4GM
= 0.98 = 56.4◦ .
2
R c
α=
However, this assumes that the deflection angle is small, which is invalid for
this case. A full calculation in GR yields a deflection about 21% larger!
2. Use the Plummer potential model, with a value of a = 100 kpc, together with
the Viral theorem to infer a value for galaxy masses in Fig. 7.3 assuming that
all the galaxies have the same mass. Do not include Stephan’s Quintet in this
estimation.
The Virial theorem says
1
3π GM2
3
Mσr2 = KE = − PE =
2
2
64 a
for the Plummer potential. Recall that
Z
1
GM
PE =
ρ (r) Φ (r) d3 r, Φ (r) = − √
,
2
r2 + a2
ρ (r) =
3a2
M
.
4π (r2 + a2 )5/2
Therefore, the masses of individual galaxies should be
Mg =
32aσr2
M
=
N
πGN
√
where N is the number of galaxies in a group. Fitting the relation σr = K N
implied by this equation to Fig. 7.3, one finds K ∼ 60 km/s and
Mg '
32aK 2
= 8.5 × 1011 M .
πG
3. From dynamical friction, determine that the LMC will sink into the Milky
Way’s center in about 3 Gyr. About how long will it take a typical Galactic
globular cluster to sink into the Milky Way’s center? State your assumptions.
The sinking time is
tsink =
r2 V
2GFM ln Λ
where F ∼ 0.4 reflects a reduction factor due to the motion of dark halo masses.
For the LMC, we have M ' 2 × 1010 M , V ' 200 km/s, r ∼ 50 kpc, and
strong encounter radius
rs =
2GM
' 4300 pc,
V2
so Λ ∼ ln(r/rs ) ∼ 2.5. (Note the text is wrong is suggesting Λ ∼ 20.) Then
tsink ≈ 3 Gyr.
For globular clusters, with masses 104 times smaller and orbital radii 10
times smaller, the sinking time is 100 times greater or 300 Gyr.
4. There is a figure in the text showing that the luminosity in X-rays LX increases as the cube of the X-ray temperature TX . Can you provide an analytic
argument showing why this is?
For a virialized gas, the mass is proportional to rσ 2 ∝ rTX where r is
the cluster size. Under the assumption that the average density of clusters or
3/2
groups is roughly the same, then r ∝ (M/n)1/3 ; hence M ∝ TX /n1/2 . The
1/2
cooling formula says the cooling rate of a hot gas is proportional to n2 TX , so
the total luminosity of X-rays must be proportional to n3/2 TX2 . Perhaps the
density in galaxy groups and clusters increases slowly with mass, in which case
the power of 3 can be realized.