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The Stats of Couples
From Pairing Algorithms to Binary Mass-Ratio Distributions
Yutong Shan


Binary Formation: Theory vs. Observation
How do we think most Binaries form? Fragmentation? What can we obser ve to give us clues?


Binary Fraction Orbital Element Distribution Mass-Ratio (q) Distribution



Multi-Body Dynamical Evolution?





Capture?

(Goodwin et al. 2007)


Theory Ю Pairing Algorithms Observation Ю Mass-Ratio Stats
What are Popular Pairing Algorithms? Fragmentation?
Primary-Constrained Pairing (PCP) Split-Core Pairing (SCP)


What can we obser ve for constraint?


Binary Fraction Orbital Element Distribution Mass-Ratio (q) Distribution

Multi-Body Dynamical Evolution? Capture?
Random Pairing (RP) Primary-Constrained Random Pairing (PCRP)





(Kouwenhoven et al. 2009)


Mass-Ratio Distribution: жpik's Law
Observed binary mass-ratio distribution is often `fitted' with a power law: жpik's Law: f(q) ~ q
q = Msecondary/Mpri
mary



жpik's Exponent,

Equal-mass favoured

Unequal-mass favoured Primary Mass (Mprim)

Flat

Often stated as a function of Primary Mass
q = Msec / Mprim (Raghavan et al. 2010) (DuchЙne & Kraus 2013)


Simulating a Binary Population
1.

Choose a generating mass function (e.g. IMF) Ю Chabrier 2003 Choose a minimal stellar mass, Mmin ! Mmin = 0.08 (substellar limit) Choose a pairing algorithm
i. ii. iii. iv. Random Pairing (RP) Primary-Constrained RP (PCRP) Primary-Constrained Pairing (PCP) Split-Core Pairing (SCP)

Mmin
IMF

2.

f(q) ~ q

3.

Primary

4.

Choose a generating f(q) ! жpik's Law, f(q) ~ q

gen

Secondar y


Maximum-Likelihood & Kolmogorov-Smirnov
Which pairing algorithm and generating жpik-produces simulated mass-ratio statistics consistent with observation?
Maximum Likelihood Fit: What mass-ratio power law best describes the simulated population? Is it consistent with observed?
N

Kolmogorov-Smirnov Test: Are the two populations (simulated & observed) drawn from the same distribution?

L ( ) =


i

C qi

D = sup CDF (q ) - CDF obs (q )


For F3 Ю K6 Field Primaries (Raghavan+10), жpik- is observed to be ~0:
Pairing Mechanism RP PCRP PCP-III Generating n/a n/a -0.5 0. +0.5 SCP-III -0.5 0. +0.5 Max-Likelihood (=0.05) -0.026 [-0.285, -0.167] -0.384 [-0.441, -0.327] -0.312 [-0.37, -0.253] 0.113 [0.047, 0.18] 0.515 [0.438, 0.593] -0.354 [-0.408, -0.3] 0.082 [0.02, 0.144] 0.494 [0.423, 0.567] K-S Test ( variable) Rejected @ 0.001 Rejected @ 0.001 Rejected @ 0.001 Consistent w/ 0.1 Rejected @ 0.001 Rejected @ 0.001 Consistent w/ 0.1 Rejected @ 0.001

N.B. Universal input can result in new and mass-dependent !


Which Mechanism is Ruled Out?


Random Pairing and Primar y-Constrained Random Pairing cannot reproduce observed mass-ratio statistics for solar-type stars Capture is probably not the main channel of binary formation! Should not use RP to simulate realistic binary pop! PCP and SCP can both reproduce Raghavan+10's observation, but require (gen) ~ (obs) Future Work:
Compare simulation w/ observation in more mass ranges! Vary the IMF (and I-Core-F) and minimal stellar mass accepted to include BDs Consider observational biases Investigate more, physically-motivated binary pairing mechanisms! Incorporate generating binary fraction into simulation and compare with binary rate statistics






References


Chabrier, 2003, PASP, 115, 763 Clauset, Shalizi, & Newman, 2007, arXiv: 0706.1062v2 DuchЙne & Kraus, 2013, ARAA, 51, 269 Goodwin, Kroupa, Goodman, & Burkert, 2007 Goodwin, 2013, MNRAS, 430, L6-L9 Kouwenhoven, Brown, Goodwin, et al., 2009, A&A, 493, 979 Raghavan, McAlister, Henry, Latham, et al., 2010, ApJS, 190, 1 Reggiani & Meyer, 2011, ApJ, 738, 60