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\centerline{\medbf Astronomy 124}
\medskip
\centerline{\lilrm Basic stuff, sophomoric, self-help, optional homework}
\bigskip

\bigskip
\pp {\bf 1.}~ What are the apparent visual magnitudes of: (a) a Sun-like
star 50 pc away? (b) a globular cluster containing $\sim 10^5$ stars of an
average luminosity $\sim L_\odot$ 10 kpc away?  (c) a galaxy containing $\sim 3
\times 10^{10}$ stars of an average luminosity $\sim 0.5 L_\odot$ 20 Mpc away? 
(Assume that all these objects have spectra identical to that of the Sun.)

\bigskip
\pp {\bf 2.}~ Consider a globular cluster 10 kpc away.  The cluster moves with
a tangential velocity of 123 km/s.  The tangential velocity dispersion (i.e.,
in the plane of the sky) of stars in the cluster is 10 km/s.  What is the
proper motion of the cluster (in arcsec/yr)? Of its typical stars, relative to
each other?  Estimate how large time baselines (in yrs) would be needed to
detect these motions from the ground. (Assume the limiting accuracy of the
proper motions measurement similar to that of the parallaxes.) 

\bigskip
\pp {\bf 3.}~
Compute the total luminosity and mass of the stellar disk of our Galaxy,
assuming that both mass and light are distributed in a biexponential disk with
a scale height $h = 300$ pc, and a scale length $a = 4$ kpc.  Assume that at
the $R_\odot = 8$ kpc, the mean number density in the Galactic plane is 0.1
stars/pc$^3$; an average star mass of $0.3 M_\odot$, and an average luminosity
of $0.1 L_\odot$.  Assuming that the bulge and the stellar halo have the same
mass and luminosity, compute the total absolute magnitude in the $V$ band of
our Galaxy.  Given a flat rotation curve with $V_{circ} = 220$ km/s and a sharp
cutoff at a radius of 30 kpc, compute the total dynamical mass; compare the
luminous mass (i.e., in stars) with the total dynamical mass. 

\bigskip
\pp {\bf 4.}~
Assuming the disk mass computed in problem (3), compute the kinetic energy of
the Galactic disk, assuming $V_c = 220$ km/s, and the velocity dispersion of 30
km/s in $each$ coordinate.  What is the ratio of the kinetic energy in the
ordered (rotational) and random components? Assume that the bulge has the same
mass as the disk, no rotation, and a velocity dispersion of 120 km/s in each
coordinate.  What is its total kinetic energy?  Assume that the dark halo has
the same velocity dispersion as the bulge stars, and the dark halo mass as
evaluated in problem (3) inside 30 kpc.  Compute its total kinetic energy, and
compare with those of the luminous matter. 

\bigskip
\pp {\bf 5.}~ Globular cluster M15 has typical observed horizontal branch
star magnitudes $\langle V_{HB} \rangle = 15.86$, observed main sequence
turnoff color $(B-V)_{TO} = 0.5$, and the foreground extinction estimated from
spectroscopy $A_V = 3.2 E_{B-V}= 0.32$ magnitudes.  Assuming the true absolute
magnitudes for the HB stars of $M_V = +0.7$, derive the distance to M15. 
Another cluster, NGC 6553, has the observed HB star magnitudes $\langle V_{HB}
\rangle = 16.95$, and the observed main sequence turnoff color $(B-V)_{TO} =
1.2$.  Derive the distance and the foreground extinction to NGC 6553. 

\vfill\eject 
\end
