1
Ay124: Spring 2005
Homework #1 Solutions
David R. Law (drlaw(AT)astro.caltech.edu)
Problem #1:
Referring to Figure 1, basic trigonometry tells us that
tan(radians/year)t(years) =
VT (pc/year)t(years)
d(pc)
or in the small angle approximation, and cancel

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Ay124: Spring 2005
Homework #8 Solutions
David R. Law, E. S. Phinney
Problem #1:
(a) Assuming that deviations from a circular disk are caused only by projection eects, then
b
by inspection a = cos(i) = 0.5, and so the disk inclination i 60 .
(b) Fitting

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Ay124: Spring 2005
Homework #7 Solutions
David R. Law, E. S. Phinney
Problem #1:
Since the central mass is changing slowly (adiabatically), the radial action Jr =
is an invariant. To work out this integral, we substitute E =
1
Jr =
rmax
2E +
rmin
GM
=

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Ay124: Spring 2005
Homework #6 Solutions
David R. Law, E. S. Phinney
Problem #1:
The tensor virial theorem was proven by Binney & Tremaine (BT 8-10) to be
1 d2 Ijk
= 2Kjk + Wjk
2 dt2
We want to nd the form of the scalar virial theorem for the case where

1
Ay124: Spring 2005
Homework #5 Solutions
David R. Law, E. S. Phinney
Problem #1:
(a) The zero-velocity curve is the curve bounding the surface of section, in this particularly
simple case the star never moves out of the z = 0 meridional plane, so using

1
Ay124: Spring 2005
Homework #4 Solutions
David R. Law (drlaw(AT)astro.caltech.edu)
Problem #1:
1 2 |r v |2
Q(r , v ) = (|r |) + |v | +
2
2
2 ra
At a given radius r , |r | = constant, so = constant.
Dene a spherical coordinate system where = 0 along the

1
Ay124: Spring 2005
Homework #3 Solutions
David R. Law (drlaw(AT)astro.caltech.edu)
Problem #1:
This is an exercise in the moving-cluster method of distance determination, in which the
basic assumption is that the radius of the cluster remains constant (

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Ay124: Spring 2005
Homework #2 Solutions
David R. Law (drlaw(AT)astro.caltech.edu)
Problem #1:
Partial solution courtesy Sean Moran (TA 04)
The total proper motion is just the quadrature combination of the individual components, although we must be care