Sky Surveys - what you see and what you get

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.]
Why is the on-line image so different from the original from which it was scanned?

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?

Milky Way Kinematics

• our distance from the Galactic Center is R₀ = 8.122 kpc,
• the proper motion of Sgr A* is Ω = 6.38 milliarcseconds/year
• the solar motion is 12 km/s ahead of the Local Standard of Rest, and
• the measured radial and tangential velocity dispersions of local stars are related by σ_X² = 2.2 σ_Y²

What are the values of

• Our reflex speed relative to Sgr A*?
• The orbital velocity of the Local Standard of Rest around the center of the Milky Way?
• The Oort A & B constants? Is the rotation curve rising or falling?
• The period of our orbit around the Milky Way?
• The period of one solar epicycle?
• Are stellar orbits closed ellipses? Use the orbital and epicyclic periods to explain why or why not.

Local Dark Matter Density

• If the rotation curve is flat at the LSR orbital speed found in the preceeding problem, what is the local density of dark matter?
  [Hint: what must the density profile ρ(r) be to obtain a flat rotation curve?
  Note that ρ(r) = R_o is not the same as the average density within R_o.]

• Use your result to estimate the mass of the dark matter within the solar system (a sphere 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.]

• The baryonic mass of the Milky Way is about 7 x 10¹⁰ M_☉. Assuming this is interior to the solar circle, what would the rotation velocity at R_o = 8.122 kpc be in the absence of dark matter? How does this compare with the observed value?

• ASTR 433 only: Repeat the estimate of the local dark matter density, but now accounting for the mass in baryons.
  [Hint: velocities add and subtract in quadrature.]

[See the the useful numbers page for Newton's constant in convenient units.]

Exponential Disks

Integrate Σ(r) from 0 to 2π and r = 0 to x scale lengths (r = xR_d) 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, R_e.)
• 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 V_d(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

• What is the central mass density (in solar masses per square parsec) of a maximum disk with R_d = 5 kpc and V_c = 200 km/s?
  
• ASTR 433 only: do this for the thin disk as well as the spherical approximation.