Selection of Homework Questions
Topic 6: Theory I : Disks
(1) Epicyclic Motion : Theory
- 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
(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 = 4 G .
- Use solar neighborhood values for
, , and
, to estimate the local density in the MW disk.
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 Vr = -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
Vz = 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:
Consider a three armed spiral with pattern angular velocity p = - 1/3 . How does the star's epicyclic motion interact with the pattern?
- - ½
- - 1/3
- + ½
- - 0.49
- A galaxy has the following rotation curve:
Vc = 200 sin(/2 × Rkpc/2) km/s, 0 < R < 2 kpc
Vc = 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; 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 × Rkpc/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?