1. The Power-Law Tail in the Initial Mass Function (IMF) of Stars
Basu & Jones, 2004, MNRAS Letters, in press
Fundamental questions:
Can the IMF be described by a lognormal distribution? Or, is there a power-law tail?
What is the generation mechanism of the IMF distribution? If multiplicative factors determine stellar mass, central limit theorem => lognormal distribution
probability distribution function

2. The Orion Nebula
A great laboratory for IMF studies
nearby, away from Galactic plane
lots of young stellar objects, low mass stars detectable
wide range of stellar density
Trapezium stars near center => symptomatic of more general phenomenon of mass segregation?

3. IMF of Orion Nebula Cluster (ONC)
Squares = histogram of masses of 696 stars (Hillenbrand 1997) binned in increments D log m = 0.2.
Dash-dotted line : best fit lognormal to complete histogram – c2 minimized but implied probability = 4.75 x 10-12!
Dotted line: best fit lognormal for M < MO – c2 implies probability
Solid line: best fit straight line for M > MO – slope =

4. IMF of Orion Nebula Cluster (ONC)
Look for region dependent variations – mass segregation?
Lognormal good fit for M < Msun. Power-law present for M > Msun in all regions. Most massive stars make slope shallower in some regions.
Bottom line: only significant evidence for mass segregation comes from the five stars near the center (Trapezium stars plus one other).

5. A Minimum-Hypothesis Model
Reed (2002,2003) – similar model for distribution of incomes, city sizes.
Protostellar condensation masses initially drawn from a lognormal distribution (circles)
Accretion growth dm(t)/dt = gm. Fixed time of growth => continued lognormal distribution (diamonds)
Above accretion law but an exponential distribution of accretion times t, i.e.,

6. A Minimum-Hypothesis Model
Solid line is analytically derivable:
Expect
since both rates controlled by external medium.

7. Another Growth Law
Dashed line: sequence of bounded isothermal equilibria (Bonnor-Ebert) for external pressure Pext. Dash-dot line: same but for higher external pressure.
Dashed line well described by
Geometric accretion =>

8. Alternate Growth Law
Squares => a power-law tail with varying index; average value –1.6.
Protostellar condensation masses initially drawn from a lognormal distribution (circles)
Accretion growth dm(t)/dt =g1m2/3. Fixed time of growth (diamonds).
Above accretion law but an exponential distribution of accretion times t, i.e.,

9. Conclusion
Even if the central limit theorem ensures a lognormal distribution of protostellar condensation masses at birth – subsequent accretion growth (proportional to some power of mass) coupled with a distribution of accretion times, will skew the distribution toward having a power-law tail.