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\centerline{\medbf Astronomy 124}
\medskip
\centerline{\lilrm Homework \#2, due in class on Feb. 17}
\bigskip

\bigskip
{\bf 1.}~ 
Using figures 7 and 14 in Scalo's (1986) IMF review, estimate the power-law
relations between the stellar mass and luminosity, and mass and main sequence
lifetime; you can ignore the low-mass end, make a good estimate for the upper
main sequence.  

Consider a stellar system producing stars with a constant SFR = $1
~M_\odot$/yr, over a long period of time.  Assume an IMF in the standard
power-law form, $dN/dM \sim M^{-(1+x)}$, between lumiting masses $M_L$ and
$M_H$, and zero outside those limits.  Compute the luminosity and stellar
$(M/L)$ of this system, as a function of the IMF parameters, after 10 Gyr.  You
can ignore the post-main sequence phases.  Assume as fiducial numbers $x =
1.5$, $M_L = 0.1 ~M_\odot$, and $M_H = 100 ~M_\odot$.  What happens as you vary
$x$ from 0.5 to 2.5, $M_L$ from 0.01 to 1 $M_\odot$, or $M_H$ from 1 to 100
$M_\odot$, while keeping the other two parameters fixed? 

\bigskip
{\bf 2.}~ 
Assume that the dark matter associated with galaxies consists of brown dwarfs
with masses $M_{bd} = 0.03 ~M_\odot$, and absolute magnitudes: $M_K = 12$ 
at 2.2 $\mu$m, and $M_{12} = 11$ in the IRAS 12 $\mu$m band (don't confuse the
masses and the absolute mags here!).  Note also that $m_{12} = 0$ corresponds
to a flux density of 28.3 Jy.

(a) Assume that Bahcall is right, and that the disk dark matter is distributed
uniformly locally.  How far should the nearest brown dwarf be?  What is the
chance of detecting a brown dwarf in a random K band exposure, which goes down
to $m_K = 20$ over a 1 arcmin square field (this is the state of the art
today)?  How many brown dwarfs should there be in the IRAS Point Source
Catalog, which reaches the depth of $m_{12} = 4$?  In the IRAS Faint Point
Source Catalog, which will reach the depth of $m_{12} = 6$?  In the forthcoming
2-micron all-sky survey, which will reach the depth of $m_K = 14$?  For each of
these cases, estimate the proper motion of a typical detected brown dwarf (if
any). 

(b) Now consider an edge-on, bulgeless disk galaxy 10 Mpc away.  It has a dark
halo mass $M_{halo} = 10^{12} ~M_\odot$, distributed as $\rho \sim r^{-2}$,
with a cutoff at $r_{max} = 30$ kpc.  What is the run of the projected surface
density with radius?  If the halo consists of the brown dwarfs as above, what 
is the run of surface brightness, both in K mags per square arcsec, and in Jy
per square arcmin at IRAS 12 $\mu$m (ignore the divergence near the center)? 
Plot these surface brightness profiles from 1 to 30 kpc in projection.  Comment 
on the detectability of such a halo, considering that the limiting surface
brightness in state-of-the-art ground-based measurements is about 22 mag per
square arcsec in the K band, and about 1 MJy per steradian in the IRAS 12
$\mu$m band (silly, but alas, standard units). 

\vfill\eject 
\end
