- Derive the radial oscillation frequency, ,
for a star perturbed from a circular orbit in an arbitrary axisymmetric
potential (R). Express your result first
in terms of the angular velocity, (R), and then
in terms of the rotation curve, V(R).
- Show that a disk in which the angular momentum (per unit mass)
decreases outwards cannot support stable circular rotation. [Hint: find the condition that perturbations
to circular motion
**cannot**yield small epicyclic oscillations.] (B&T-2 Q 3.8) - Starting with Poisson's equation in cylindrical
coordinates:
^{2}= 4 G (see B&T-2 Eq B.52 p 777), show that an axisymmetric galaxy has epicycle, vertical and orbital frequencies which obey:^{2}+^{2}- 2^{2}= 4 G . - Use solar neighborhood values for , , and , to estimate the local density in the MW disk. (Adapted from B&T-2 Q 3.15).

**(2) Solar Epicyclic Motion : **

For the sun, assume a current galactocentric
distance R_{} = 8.5 kpc; Oort's constants
A = 15 km/s/kpc and B = -12 km/s/kpc; and a current solar motion relative
to the local circular velocity of V_{r} = -10 km/s (ie towards the
galactic center) and V_{} = +5.2 km/s
(ie faster than circular).

- Using the epicycle approximation, what are the Sun's minimum and maximum distances from the Galactic center?
- Assuming the Sun currently resides in the plane and has
V
_{z}= 7 km/s, what is the maximum excursion above and below the plane (assume a local mass density of 0.2 M_{}pc^{-3}, which extends well above the excursion height).

**(3) Disk Resonances : **

- Use psm units (Topic 1.3e) to quickly show that a velocity gradient of km/s/kpc has associated angular velocity radians/Gyr, frequency
/2 Gyr
^{-1}, and period P = 2 / Gyr. - Consider circular orbital motion of angular velocity viewed in a frame rotating with angular velocity F (same, CCW, direction). What is the
**apparent**angular velocity and period of the star? Now add retrograde epicyclic motion of angular velocity . For what values of F does the new orbit appear closed after one revolution? Sketch (or write a program to plot) the shape of the orbit and the guiding circle as seen from the rotating frame when F is:- -
- - ½
- -
^{1}/_{3} - + ½
- - 0.49

_{p}= -^{1}/_{3}. How does the star's epicyclic motion interact with the pattern? - A galaxy has the following rotation curve:

V_{c}= 200 sin(/2 × R_{kpc}/2) km/s, 0 < R < 2 kpc

V_{c}= 200 km/s, R > 2 kpc.

The galaxy has a bar and spiral pattern which have constant slow angular velocity of 20 km/s/kpc.On a single plot, show and label clearly the following functions of R: ; - ½; + ½;

_{p}. On the same plot with the same x-axis (but with different y-axis), show the rotation curve, V(R). [Hint: it is easiest to evaluate (R) numerically rather than algebraically]. - Identify, if present, the locations of the ILR, CR and OLR resonances.

**(4) Estimating Pattern Speeds : **
Express all frequencies in km/s/kpc, and in Myr^{-1}

- For a galaxy with a flat rotation curve at 250 km/s, what's the epicyclic frequency at R = 7 kpc?
- If corotation is at R = 6 kpc, what's this galaxy's pattern speed ?
- For a two-armed spiral, is R = 7 kpc a resonance radius ?
- Assume the outer Lindblad
resonance is at R = 20 kpc. What's the galaxy's pattern speed now (assume
the pattern has m = 2) ?

**(5) Disk Stability : **

- Derive an approximate expression for local disk instability to gravitational clumping, the so-called Toomre Q parameter (for stars).
- A galaxy has rotation curve V = 200 × sin(/2 × R
_{kpc}/3) out to 3 kpc, and is flat (V = 200 km/s) beyond. The disk itself has an exponential scale length of 3 kpc, and surface mass density of 100 M_{}pc^{-2}at 6 kpc. Assume the disk has uniform velocity dispersion = 20 km/s and uniform M/L ratio (i.e. the surface density is also exponential).Plot a graph of Q vs R to find which parts of the disk are locally unstable (it is probably easiest to evaluate Q numerically).

- If the disk is "heated" by the passage of orbiting satellites, what is the minimum value of that will supress local instabilities (and associated star formation) throughout the disk?