ASTR221 HW#4  Due Oct 21 2005
1. Saturn's Rings
We will make a rough estimate of the mass
contained in Saturn's rings. Assume that the rings have a constant mass
density and that the rings are 30 meters thick with an inner radius of
1.5 R_{Saturn} and an outer radius of 3 R_{Saturn}.
Assume also that the ring particles are waterice spheres of radius 1
cm and that the optical depth of the rings is unity (i.e., tau=1, so
you can just barely see through them).
 What is the number density of particles in
the
rings (how many particles per cubic meter)? So how many particles
total
are there?
 If waterice particles have a density of 1000
kg/m^{3}, what is the mass of each particle? So how much total
mass is in the rings?
 If you were to take these ring particles and
merge them together to make a spherical moon, how big would that moon
be? How does
this compare to the size of Mimas, one of Saturn's inner moons?
2. Planetary Accretion
When the solar system first formed, it was
characterized by a disk of dust and gas (called the "solar nebula")
with some larger planetesimals embedded within it. As these
planetesimals orbited in the solar nebula, they continually accreted
material from the nebula, slowly growing in size. In this problem, we
will estimate how long it took these planetesimals to grow into
reasonable sized objects (ie a thousand km or so in size).
 If a planetesimal has a crosssectional area of
piR^{2} (where R is the planetesimal's radius) and is sweeping
through a cloud of smaller particles of fixed size with a velocity V,
show that the number of collisions per second will be
where rho_{N}=the space density
(kg/m^{3}) of particles in the cloud and m=the mass of each
particle.
 Show that if each collision results in the
target particles sticking to the planetesimal, the planetesimal will
gain mass at a rate of
where M=the planetesimal mass.
 Show that the time to grow to radius R is
where rho_{p}=the density of the planetesimal
itself. (Assume that rho_{N} and V stay constant as particles
are
swept up.) Assuming that a reasonable value
for the density of accretable material in inner part of the early solar
nebula is rho_{N} = 10^{7} kg/m^{3}, estimate
the
time to accrete a body of 1,000 km radius. Assume a reasonable rho_{p}.
Is the timescale resonable given the formation time of the planets (50
million
years or so)? How does the timescale change if the accretion process is
inefficient,
so that not every particle sticks to the planetesimal?
3. Magnitudes and Distances
 Fill in the blanks for the stars on this table:
App
Mag

Distance
(pc)

Parallax
(")

Abs
Mag

5.0

100



10.0

1



6.5

250



3.0


0.2


1.0

500



6.5


0.004


 Let's consider a (rather strange) cluster of 10,000 stars.
100 of the stars have an absolute magnitude of M=0 and the rest have an
absolute magnitude of M=5. What is the total (also
known as the "integrated")
absolute magnitude of the cluster? If the
cluster is 3 kiloparsecs away, what is its apparent integrated
magnitude?
4. Textbook Problems
 Problem 2.5
 A variation on Problem 2.30: If we have an uncertainty (call
it "dm") in our measurement of the magnitude of a star, derive an
analytic expression for the resulting relative
uncertainty in the distance we derive to the star (assuming we know its
absolute magnitude exactly). Note that this expression only holds if
the uncertainty is relatively small.
(Be careful, the textbook has
Questions, Problems, and Computer Problems at the end of each chapter.
I am asking for the "Problems", not the "Questions" or "Computer
Problems")