Hernquist Potential

The Poisson equation ∇² Φ = 4πGρ has a finite number of solutions with potential-density pairs in which both the gravitational potential Φ and the mass density ρ can be expressed as analytic functions. These provide an important touchstone by which more general numerical codes can be checked (i.e., does your code agree with a known analytic solution?)

Use the Poisson equation^* to derive the density ρ(r) associated with the spherically symmetric Hernquist potential
Φ(r) = −GM/(r+a)
where M is a characteristic mass and a is a length scale.
Try this by hand with pencil & paper before resorting to codes like Mathematica, as over-reliance on such tools often leads to a failure to understand their output.

Show enough steps to follow how you got from Φ to ρ.
What are the asymptotic limits of ρ at small and large radii?
I.e., what happens as r → 0 and r → ∞? Express your answer as a power law: ρ ∼ r^-x in each limit.
Think about what your answers mean physically. How does the density vary with radius? Where does it increase/decrease/reach a maximum?

^* Be sure to use the appropriate coordinate system for the Laplacian operator ∇².

Milky Way Kinematics

• the proper motion of the radio source Sgr A* associated with the supermassive black hole at the center of the Galaxy is Ω_A* = 6.379 milliarcseconds/year (Reid & Brunthaler 2004), and
• our distance from the Galactic Center is R₀ = 8.122 kpc (GRAVITY collaboration 2018),
• the rotation curve of the Galaxy can be roughly approximated as V(R) = 229−1.7(R−R₀) km/s for 5 < R < 25 kpc (Eilers et al. 2019).
  For the curious, a more detailed numerical model is tabulated here.

What are the values of

1. Our speed relative to Sgr A* (in km/s)?
2. The solar motion? (the peculiar speed of the sun relative to the local circular speed, taken here to be 229 km/s)
3. The Oort A & B constants?
4. The orbital frequency and the corresponing period of our orbit around the Milky Way?
5. The epicyclcic frequency and corresponding period?
6. Are stellar orbits closed ellipses? Use the orbital and epicyclic periods to explain why or why not.
7. Use the approximation σ_z² ≈ 2 π G Σ z₀ to estimate the local dynamical surface density Σ for a disk scale height of z₀ = 300 pc and the vertical velocity dispersion σ_z = 21 km/s observed for nearby stars above the plane of the Milky Way. How does this compare with the observed surface density of stars (including stellar remnants), 45 M_☉pc^-2? What dominates the mass budget? What other components matter?

Appropriate Galactic units for frequencies are km/s/kpc and orbital time periods in Myr.
See the the useful numbers page for help with unit conversions.

Local Dark Matter Mass and Density

1. The baryonic mass enclosed by the solar circle (r < R₀ = 8.122 kpc) is about 6 x 10¹⁰ M_☉. If that were all of the mass, what would the circular speed at the solar circle V(R₀) be?
   You may assume a spherical mass distribution, which is not a great approximation (see problem 5).
2. The observed circular speed is V(R₀) = 229 km/s. What mass of dark matter is required to make up the difference from part (i)?
   [Hint: masses add linearly, so velocities add in quadrature.]
3. Is the Milky Way baryon dominated or dark matter dominated within the solar circle?
4. What is the local density of dark matter at the solar circle (in solar mass per cubic parsec)?
   Assume an isothermal dark matter halo with density profile ρ(r) = Ar^-2 where A is a constant to be determined.
5. 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.]

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

NFW halos

The dark matter halos formed in numerical structure formation simulations are often approximated as the "NFW halo" with density profile

ρ(r) = ρ_sr_s³ [r (r+r_s)²]⁻¹

where ρ_s is a characteristic density and r_s is a scale radius. See this helpful write-up.
[Note how this is both similar to and different from the Hernquist profile.]

In the following, you may assume spherical symmetry.

1. Obtain an expression for the mass profile M(r).
2. Find the corresponding rotation curve V(r).
   It is conventional to define a "virial" radius R₂₀₀ inside of which the average density is 200 times the critical density of the universe, ρ_crit = 3H₀²/(8πG). The mass enclosed by this radius, M₂₀₀, is often taken as a working definition of the halo mass. The circular velocity of a test particle at the virial radius is V₂₀₀.
3. Show that V₂₀₀ = R₂₀₀h when V₂₀₀ is in km/s and R₂₀₀ is in kpc and h = H₀/(100 km/s/Mpc).
   
4. The relations above mean that the mass, radius, and circular speed of an NFW halo are interchangeable quantities: if you know one, you know the others. But the over-density of 200 seems arbitrary. Why not simply obtain the mass by integrating the density profile to infinity?

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.

1. Is the total luminosity finite as x ⇒ ∞ infinity? If not, what is it?
2. How many scale lengths contain half the total light? (This is known as the half-light radius, R_e.)
3. Plot the cumulative enclosed luminosity L(< x).

   Disk rotation curve

4. 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.
5. The rotation curve of a thin exponential disk is given by Binney & Tremaine eqn. 2-169 (1st edition) or 2.165 (2nd edition) and can be written
   
   Solve this numerically for a reasonable sampling of radii (or use this tabulation) and plot it on the same diagram as the spherical approximation.
   How do the "spherical disk" and thin disk differ?
6. At what radius (in scale lengths) does the rotation curve of an exponential disk peak? How does the location and amplitude of the peak differ between the spherical and thin disk case?

   ASTR 433 only: Maximum Disk

   The condition of maximum disk is that the disk rotation curve meets but does not exceed the observed rotation so that V_disk,max = V_c. This just says that the contribution of the disk cannot exceed the total mass budget. The radius at which this occurs is that determined above.
   Assuming a thin exponential disk,
7. What is the mass (in solar masses) and central mass surface density (in solar masses per square parsec) of a maximum disk with R_d = 5 kpc and V_c = 200 km/s?
8. What is the central mass surface density for a disk of the same mass, but a larger scale length R_d = 10 kpc?
9. How does the peak velocity change for this more extended disk? How would you expect the total velocity V_c to change?