PhD Thesis (PDF) - Department of Astronomy - University of Virginia

Sample is only ∼ 1.01, we do not apply completeness corrections for this fit. We
adopted the background LF from Kim et al. (2004); however, we have assumed that
background sources exhibit the same level of variability as the LMXBs and reduced
the expected background number from 11.7 to 8.2.
We modeled the LMXB populations with a single power law, a cutoff power law,
and a broken power law.
Single :
Cutoff :
Broken :
dN
dL37
dN
dL37
dN
dL37
180
= N0,s L −αs
=
37 ;
⎧
−αc ⎪⎨ LX
Lc
N0,c
⎪⎩ 0
if LX ≤ Lc;
otherwise;
(5.8a)
(5.8b)
=
⎧
−αl ⎪⎨ LX
Lb
N0,b −αh ⎪⎩ LX
if LX ≤ Lb;
otherwise,
(5.8c)
where L37 is the X-ray luminosity in units of 10 37 ergs s −1 . We used the maximum
likelihood method to determine the best fits to the cumulative LF and Monte Carlo
techniques to determine the errors (90% confidence interval). A K-S test against
the cumulative distribution function of our best-fit LF indicated only a 12% chance
that the single power law is a proper fit. Much better fits were achieved for a cutoff
power-law (∆χ 2 = −10.6 for one less dof) with N0,c = (9.5 +22.5
− 5.6)×10 −2 , αc = 1.48 +0.21
−0.27,
and Lc = (6.0 +3.8
−2.7) × 10 38 ergs s −1 and for a broken power-law (∆χ 2 = −14.3 for two
less dof) with N0,b = 2.2 +3.0
−1.2, αl = 1.02 +0.30
−0.55, αh = 2.91 +3.14
−0.59, and Lb = (10.6 +5.8
−4.4) ×
10 37 ergs s −1 . Although the broken power-law is the best fit according to ∆χ 2 , we
note that the one-sided K-S test indicated the cutoff power law model was acceptable
(at the 50% confidence level). All three fits are overlaid in Figure 5.8.
Since each of the five observations is an independent measure of the instantaneous
Lb