Fig. 5. (A) (Upper panel) The alpha plot compiles measurements of the
power-law index, , as a function of the logarithmic stellar mass and so
measures the shape of a MF. [Notation: lm  log10(m/M), l  log10(/year),
lL log10(L/L).] The shape of the MF is mapped in the upper panel by
plotting measurements of at < lm > = (lm2  lm1)/2 obtained by fitting
power-laws, (m)  m, to logarithmic mass ranges lm1 to lm2 (not indicated
here for clarity). Many of the green circles and blue triangles are
pre-1998 data compiled by Scalo (58, 117) for MW (green filled circles)
and Large Magellanic Cloud clusters and OB associations (blue solid
triangles). Newer data are also plotted using the same symbols, but some
are emphasized using different symbols and colors such as by yellow
triangles for globular cluster MFs. [Web table 3 (18)]. Unresolved
multiple systems are not corrected for in all these data including the
MW-bulge data. The average solar-neighborhood IMFs (Eq. 5 in Web table
1) are the red thick short-dashed lines together with the associated
uncertainty ranges. Other binary-star-corrected solar-neighborhood-IMF
measurements are indicated as magenta dotted error-bars (Web table 3). The
quasi-diagonal black lines are analytical forms summarized in Web table
1. The vertical dotted lines delineate the four mass ranges (Eq. 5 in
Web table 1), and the shaded areas highlight those stellar mass regions
where the derivation of the IMF is additionally complicated especially
for Galactic field stars: for 0.08 < m/M < 0.15 long pre-main sequence
contraction times (42) make the conversion from an empirical LF to an
IMF (Eq. 1) dependent on the precise knowledge of stellar ages and the
SFH. For 0.8 < m/M < 2.5 uncertain main-sequence evolution, Galactic-disk
age and the SFH of the MW disk do not allow accurate IMF determinations
(31). (Lower panel) The bolometric MLR, lL(lm), and stellar main-sequence
lifetime, l, are plotted as a function of logarithmic stellar mass. The
uncertainty in the age of the Milky-Way disk is shown as the shaded
region. Stellar spectral types are written between the panels. (B) The
histogram of MF power-law indices () for massive stars (lm > 0.40). If
the  measurements are not distributed like a Gaussian function then this
may imply that some of the data are different from the mean because of
true IMF variations. The green histogram shows the observational data
from (A). The blue shaded histogram shows theoretical values from an
ensemble of 12 star clusters containing initially 800 to 104 stars that
are snapshots at 3 and 70 Ma (58). Stellar companions in binaries are
merged to give the system MFs, which are used to measure . The assumed
IMF is Eq. 5 in Web table 1. The dotted curves are Gaussians with mean
and standard deviation, , obtained from the histograms. The theoretical
data give    = 2.20  = 0.63 (magenta dotted curve), and thus arrive at
the input Salpeter value. The empirical data from (A) give    = 2.36,
= 0.36 which is the Salpeter value. Fixing f =    and using only     2
for the observational data gives the narrow thin red dotted Gaussian
distribution which describes the Salpeter peak (f = 2.36, ,f = 0.08)