Deriving the Core Mass Function

1
Deriving the Core Mass Function
Di Li Jet Propulsion Laboratory Xin Guan Yu
Dai Yuefang Wu Beijing University
2
Initial Mass Function (IMF)

• Salpeter (1955)
• dN / dM M-2.35
• (0.4 10 M? )
• Scalo (1986)
• dN / dM M-2.7
• (1.2 20 M? )
• Massey (2001)
• dN / dM M-2.4 (extends high mass limit to 120
  M? )
• Scalo (2004)
• divergence in the high mass slope
• And between field stars and clusters

(Krouper 2002)
3
High Mass Core Sample
4
Dynamical State

• MBE-CR critical mass of a Bonnor Ebert Sphere
• Teq ??mH?V2/(8ln2kB )
• M? c? ?Br2/G0.5 mass supported by static
  magnetic field
• Li et al. 2007, ApJ 655 351

5
Core Mass Function Current View

• A break point in CMF
• Higher mass part mimics Salpeter IMF
• (Johnstone Bally 2006
• Reid Wilson 2005
• Mookerjea et al. 2004
• Testi Sargent 1998,
• Motte et al.1998)

Ward-Thompson et al. 2006, Review article for PPV
6
Orion CMF 2 vs 1 !
7
Two Power Laws -- Really?
8
Least Square Fit of Discrete Cumulative Function

• Data sample S
• S(mi) , i 1,2, , n
• Kernel Function F(x)
• generate a sample according to an assumed
  distribution, e.g., a power law with index ?
• calculate the discrete CF
• Interpolate CF to calculate F(x)
• Minimize ?2 to derive ?
• ?2 ???i (F(mi) S(mi)), i 1, 2, n

9
Bootstrap

• Traditional Inference
• Normally distributed population
• ( mean M SD ?)
• Draw random samples of size n ?normally
  distributed sample mean
• (mean M SD ?/n0.5 )
• Bootstrap Inference
• One sample with unknown distribution
• draw resamples ? the distribution of a certain
  sample statistic usually approximates that of the
  population

10
Forward Fitting
Orion B
r Oph
11
Conclusions

• The core mass function (CMF) in Orion is
  significantly flatter than the IMF, suggesting
  environmental factors shaping the IMF.
• Direct multiple power law fit to accumulative
  function does not work.
• Forward fitting of past works (Motte et al. 1998
  Jonhstone et al. 2000 Kerton et al. 2001
  Johnstone et al. 2001 Tothill et al. 2002
  Beuther et al. 2001 Mookerjea et al. 2004 Reid
  Wilson 2005 Johnstone Bally 2006) indicates
  that the majority of existing core samples is
  consistent with a single power law CMF with
  indices between 0.8 and 1.6.