stellar density as a function of magnitude

[Stupendous Man <richmond@a188-l009.rit.edu>]

  We've been discussing the effects of crowding.  Tom Droege and I
have considered it from the following point of view:

        Choose a magnitude limit M such that any isolated star
        brighter than M will be clearly and accurately measured.

        Question: what fraction of the stars brighter than M
        are blended with other stars brighter than M?

  Andrew Bennett looks it from a slightly different perspective:

        Question: what fraction of the stars brighter than M
        are blended with other stars of ANY magnitude, including
        those too faint to be detected reliably in isolation?

  For some purposes, blends with faint stars _can_ be important.
Astronomers who study microlensing, for instance, are very 
interested in the perturbations to magnitude and position measurements
due to faint (but close) companions.

  Suppose we restrict the use of TASS images to making a catalog
of stars bright enough to appear reliably in all (or almost all)
images taken under good conditions.  Suppose further we are 
interested only in making photometric measurements.  Then, as I 
see it, close companions present two problems:

         1. "Confusion": two stars of nearly comparable brightness,
            separated by the typical FWHM of the camera,
            may be detected as one star in some images,
            as two in another.  

            This causes problems in assigning detected objects
            to entries in a catalog of stars.  

         2. "Contamination": two stars which may be of 
            different brightness, down to some limit (see below),
            and which are closer than the FWHM, will appear
            as a single star.

            This causes the measured magnitude (which is really
            a combination of the light of two stars) to be 
            assigned to the brighter star.

  I believe that one can place some constraints on the brightness
of the companion for both cases.  In case 1, the companion must
be bright enough to be detected well above the noise in any
image -- otherwise, even if the conditions permit it to be separated
from the primary star, it probably won't be detected, and presents
no problem.  This means the companion must be brighter than "M".

  On the other hand, even a star too faint to be detected on its
own can contribute some fraction of light to a nearby, brighter
star.  Thus, in case 2, the companion can be fainter than "M".
Suppose we are interested in 1 percent photometry.  Then, in the
worst case, a primary star just on the edge of the reliable
plate limit (magnitude M) can be perturbed significantly by
a star 100 times fainter (magnitude M+5).

  Okay, let's look at some numbers.  For the Mark IV, at a suburban
site, let's say that M = 15.  How many stars do we expect
to find inside a field of 4.2 by 4.2 degrees = 17 square degrees,
with magnitudes around 15?  Data from Allen.  The "magnitude"
is some photographic passband, much closer to V than I.

                      number of stars in field of Mark IV
 magnitude         galactic equator            galactic poles
    m             mag = m    mag <= m         mag = m    mag <= m
--------------------------------------------------------------------
   13             3,800       6,700            330          830

   14             9,800      18,000            630        1,700

   15            24,000      44,700          1,100        3,200

   16            49,000     100,000          2,000        5,900

   17           120,000     230,000          3,200       10,000

   18           270,000     540,000          5,400       18,000


  Note the different distributions: at the equator, roughly half of
all stars visible are in the faintest magnitude bin; whereas at
the poles, only about one-third of all stars visible are in the 
faintest magnitude bin.

  I can see why Arne described the FASTT fields on the galactic 
equator as "white".

  Actual Mark IV data frames are the best source of information
on crowding, but perhaps these numbers might help us plan 
observing programs.  It's pretty clear that no matter where
we look, we cannot avoid blending (at the 1% level) most 
measured stars with faint, but significant companions.  Note
that this is NOT necessarily true of other surveys (such as 
USNO A2 and POSS), since they have much higher resolution than
the Mark IV.

  On the other hand, it appears from the table above that fields
far from the galactic equator will not have a significant fraction
of bright stars blended with other bright stars.  For example,
near the poles, we may expect 3,200 stars in the field down to
M = 15.  Tech Note 66 shows that about 8 percent of those stars
would have companions within 5 pixels, and about 4 percent
would have companions within 3 pixels.  Hmmmm.  Actually, even 
8 percent seems like it could be annoying ...

                                               Michael Richmond