Binary mass function: Difference between revisions

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Revision as of 19:18, 28 August 2017

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In [[astronomy]], the '''binary mass function''' or simply '''mass function''' is a [[Function (mathematics)|function]] that constrains the [[mass]] of the unseen component (typically a [[star]] or [[exoplanet]]) in a single-lined spectroscopic [[binary star]] or in a [[planetary system]]. It can be calculated from [[Observation|observable]] quantities only, namely the [[orbital period]] of the binary system, and the peak [[radial velocity]] of the observed star. The velocity of one binary component and the orbital period provide (limited) information on the separation and gravitational attractionforce between the two components, and hence on the masses of the components.

== Introduction ==

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The observed object of which the radial velocity can be measured is taken to be object 1 in this article, its unseen companion is object 2.

Let <math>M_{1}</math> and <math>M_{2}</math> be the stellar masses, with <math>M_{1} + M_{2} = M_\mathrm{tot}</math> the total mass of the binary system, <math>v_{1}</math> and <math>v_{2}</math> the orbital velocities, and <math>a_{1}</math> and <math>a_{2}</math> the distances of the objects to the center of mass, such that. <math>a_{1}+a_{2} = a</math> is the [[semi-major axis]] (orbital separation) of the binary system.

We start out with Kepler's third law, with <math>\omega_\mathrm{orb} = 2 \pi/P_\mathrm{orb}</math> the [[Angular frequency|orbital frequency]] and <math>G</math> the [[gravitational constant]],

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<math>GM_\mathrm{tot} = \omega_\mathrm{orb}^{2} a^{3}.</math>

Using the definition of the center of mass ___location, <math>M_{1} a_{1} = M_{2} a_{2}</math>,<ref name="karttunen" /> we can write

<math>a = a_{1} + a_{2} = a_{1} \left(1 + \frac{a_{2}}{a_{1}}\right) = a_{1} \left(1 + \frac{M_{1}}{M_{2}}\right) = \frac{a_{1}}{M_{2}} (M_{1} + M_{2}) = \frac{a_{1} M_\mathrm{tot}}{M_{2}}.</math>

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<math>f \approx M_{2}\ \mathrm{sin}^{3}i,</math>

and hencesince <math>0 \leq \sin(i) \leq 1</math> for <math>0^{\circ} \leq i \leq 90^{\circ}</math>, the mass function gives a lower limit on the mass of the unseen object 2.<ref name="larson" />

In general, for any <math>i</math> or <math>M_{1}</math>,