Project Cyclops, A Design... - Department of Earth and Planetary ...

magnitude (low brightness) end, the observed curve falls
off sooner than would be the case if the abscissa were
log-luminosity. This is because, with decreasing surface
temperature, the peak of the black-body radiation curve
shifts below the visible part of the spectrum and
brightness drops more rapidly than luminosity. A given
luminosity range is thus spread over a greater magnitude
range, thereby reducing the number of stars per magnitude.
If the abscissa were bolometric magnitude
(proportional to log-luminosity) the low brightness end
falloff would be extended as indicated by the dashed
curve.
If we wish to deduce the relative frequency with
which stars of different masses are formed, we must also
make a correction of the luminosity function at the high
brightness end. G stars and smaller (magnitude
greater than about 4.5) have lifetimes greater than the
age of the Galaxy. All such stars ever formed are still on
the main sequence. For larger stars, only those born not
longer ago than their lifetimes will be found. Older stars
will have left the main sequence thus depleting the
present number. The correction factor becomes rapidly
larger with increasing mass because of the rapid decrease
in lifetime. When this correction is applied, the high
brightness end is extended as indicated by the dashed
line.
We now see that over a considerable range the
corrected luminosity function can be approximated by a
power law-that is, by a straight line on our logarithmic
plot. Using the mass-luminosity relation (2), the corrected
luminosity function over this range can be expressed
in terms of mass as
I I I lq" I I l I I I
MO
K5
KO
G5
GO
F5
FO
A5
dN
--
dM
= constant X M -7/a (3)
This relation, which was first deduced by Salpeter in the
early 1950s, is a description of how a typical gas cloud
will fragment and condense into stars of various masses.
We see that larger stars are less likely to be born than
smaller stars. For main sequence stars the number
relative to solar G2 stars, as given by equation (3) is
shown in Figure 2-6. Since the smallest mass that can
start and sustain thermonuclear reactions is about
O.05M o it is clear that, if equation (3) holds, the
overwhelming majority of stars born onto the main
sequence will be smaller than the Sun. In fact, the
average stellar mass is about 0.2/I,/0 .
If we assume the frequency vs. mass relation of
equation (3) to hold down to substellar masses
(