Plummer Potential
Derive the density ρ(r) associated with the
spherically symmetric Plummer potential
Φ(r) = -GM/(r2+b2)1/2.
Show enough steps to follow how you got from Φ to ρ.
What are the limiting behaviors of ρ at small and large radii?
I.e., what happens as r → 0?
Does the density fall off exponentially as r → ∞, or what?
Local Dark Matter Density
Assume that most of the mass of the Milky Way interrior to the
solar radius Ro is in a spherical dark matter halo.
- (a) If the rotation curve is flat with V = 235 km/s and Ro = 8 kpc,
what is the local density of dark matter?
[Hint: what must the density profile be to obtain a flat rotation curve?]
- (b) Use your result to estimate the mass of the dark matter within the solar
system (out to Neptune's orbit of 30 AU).
Do you expect the dark matter have a noticeable impact on planetary dynamics?
[It may be helpful to express the dark mass in terms of planetary objects.]
- (c) The baryonic mass of the Milky Way is about 6 x 1010 M☉. Assuming this is interior to the solar circle, what would the rotation
velocity at Ro = 8 kpc be in the absence of dark matter?
- (d) Repeat the estimate of the local dark matter density from (a), but now
accounting for the mass in baryons from (c).
[Hint: velocities add and subtract in quadrature.]
[See the the useful
numbers page for Newton's constant in convenient units.]
Oort limit
This dataset
contains radial velocities, and apparent and absolute magnitudes
for a sample of K giant stars at high galactic latitude (looking
"down" out of the plane of the Milky Way's disk). We are going
to use this dataset to measure the mass density of the Milky Way's disk
in the solar neighborhood using the Oort limit. To do this,
we need the velocity dispersion and scale height of the stars.
Scale Height
- Calculate the distance to each star from its apparent and absolute magnitude.
[If it isn't immediately obvious to you how to do this, see
"distance modulus" on
the useful
numbers page.]
Because these stars are at high galactic latitude, their Z coordinate is
essentially the same as their distance, once we correct for the Z distance of
the Sun from the disk plane (Z☉ = +30pc).
Then find the number of stars (N) as a function of Z by counting
the number of stars in bins of Z.
[Hint: 400 pc is a good bin size.]
Make a plot of ln(N) (y axis) vs Z (x axis)
and find the scale height of the stars.
Velocity Dispersion
- Just as the distance is roughly the
Z coordinate (once we corrected for the Sun's Z coordinate),
the radial velocity is roughly the Z velocity once we correct for the
Sun's Z velocity.
From the distribution of velocities, calculate
the Sun's Z velocity. Then subtract that from all the velocities
to get the Z velocity of each star. Then calculate the velocity dispersion.
- ASTR 433 only:
Compute the velocity dispersion in each bin for the same bins you used
to measure the scale height. How does the velocity dispersion vary with height
above the plane? Does this make sense? [i.e., what is σ(Z) responding
to?]
Surface Density
- Given the scale height and velocity dispersion, estimate
the mass density of the disk, Σ.
You may assume σ2 = 2πGΣZ0.
Discuss sources of error and how they might affect your result.
Exponential Disks
Spiral and irregular galaxies have azimuthally averaged radial light
profiles that are tolerably well approximated as "exponential disks":
Σ(r) = Σ0 e-r/Rd,
where Σ0 is the central surface brightness (in solar
luminosities per square parsec) and Rd
is the scale length (in kpc) of a galaxy fit by this description.
Integrate Σ(r) from 0 to 2π and r = 0 to
x scale lengths (r = xRd) to obtain an expression for the
luminosity enclosed by x scale lengths.
- Is the total luminosity finite as x ⇒ ∞
infinity? If not, what is it?
- How many scale lengths contain half the total light? (This is known
as the half-light radius, Re.)
- Plot the cumulative enclosed luminosity L(< x).
Disk rotation curve
- Assuming M/L is constant with radius and making the fudge of a
"spherical" disk, use the expression for L(x) to compute and
plot Vd(r) for an exponential disk.
- ASTR 433 only:
The actual rotation curve of a thin exponential disk is given by
Binney & Tremaine eqn. 2-169 (1st edition) or 2.165 (2nd edition).
Solve this numerically for a reasonable sampling or radii and
plot it on the same diagram as the spherical approximation.
Why are the two different? Which rotates faster?
- At what radius (in scale lengths) does the rotation curve of an
exponential disk peak?
- ASTR 433 only: do this for the thin disk as well
as the spherical approximation.
Maximum Disk
Consider a galaxy with a flat rotation curve V(r) = Vc.
The condition of maximum disk is Vd,max = Vc
(the contribution of the disk cannot exceed the total mass budget).
- What is the central mass density (in solar masses per square parsec)
of a maximum disk with Rd = 5 kpc and Vc = 200 km/s?
- ASTR 433 only: do this for the thin disk as well
as the spherical approximation.
Representations of Sky Surveys
Use the Digital Sky Survey
to examine the galaxy UGC 303. Under "Retrieve from" select POSS1 (red or blue)
Find the same galaxy on the print of the Palomar Sky Survey (POSS1) in the cabinet
to the left of the printers in the library.
What do you see on the print that you did not see on the digitized version
of the same image?
[Hint: it is really obvious, but not about UGC 303.]