Binary mass function: Difference between revisions

Revision as of 05:12, 24 August 2020

Revision as of 14:35, 13 October 2020

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== Introduction ==

[[File:orbit2.gif|160px|frame|Two bodies orbiting a common center of mass, indicated by the red plus. The larger body has a higher mass, and therefore a smaller orbit and a lower orbital velocity than its lower-mass companion.]]

The binary mass function follows from [[Kepler's third law]] when the radial velocity of one (observed) binary component is introduced.<ref name="karttunen">{{cite book |editor1-last=Karttunen |editor1-first=Hannu |editor2-last=Kröger |editor2-first=Pekka |editor3-last=Oja |editor3-first=Heikki |editor4-last=Poutanen |editor4-first=Markku |editor5-last=Donner |editor5-first=Karl J. |title=Fundamental Astronomy |publisher=[[Springer Verlag]] |date=2007 |orig-year=1st pub. 1987 |pages=221–227 |chapter=Chapter 9: Binary Stars and Stellar Masses |chapterurl=https://books.google.com/books?id=DjeVdb0sLEAC&pg=PA221|isbn=978-3-540-34143-7 |lastauthorampname-list-style=yamp}}</ref>

Kepler's third law describes the motion of two bodies orbiting a common [[center of mass]]. It relates the orbital period (the time it takes to complete one full orbit) with the distance between the two bodies (the orbital separation), and the sum of their masses. For a given orbital separation, a higher total system mass implies higher [[Orbital speed|orbital velocities]]. On the other hand, for a given system mass, a longer orbital period implies a larger separation and lower orbital velocities.

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<math>\frac{M_{2}^{3}}{M_\mathrm{tot}^{2}} = \frac{\omega_\mathrm{orb}^{2} a_{1}^{3}}{G}.</math>

The peak radial velocity of object 1, <math>K</math>, depends on the orbital inclination <math>i</math> (an inclination of 0° corresponds to an orbit seen face-on, an inclination of 90° corresponds to an orbit seen edge-on). For a circular orbit ([[orbital eccentricity]] = 0) it is given by<ref name="tauris">{{cite book |last1=Tauris |first1=T.M. |last2=van den Heuvel |first2=E.P.J. |author2-link=Ed van den Heuvel |editor1-last=Lewin |editor1-first=Walter |editor1-link=Walter Lewin |editor2-last=van der Klis |editor2-first=Michiel |editor2-link=Michiel van der Klis |title=Compact stellar X-ray sources |url=https://archive.org/details/compactstellarxr00whgl |url-access=limited |publisher=Cambridge, UK: [[Cambridge University Press]] |date=2006 |pages=[https://archive.org/details/compactstellarxr00whgl/page/n640 623]–665 |chapter=Chapter 16: Formation and evolution of compact stellar X-ray sources |arxiv=astro-ph/0303456 |isbn=978-0-521-82659-4 |doi=10.2277/0521826594 |lastauthorampname-list-style=yamp|doi-broken-date=2020-08-24 }}</ref>

<math>K = v_{1} \mathrm{sin} i = \omega_\mathrm{orb} a_{1} \mathrm{sin} i.</math>

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== Eccentric orbit ==

In an orbit with eccentricity <math>e</math>, the mass function is given by<ref name="tauris" /><ref name="boffin">{{cite book |last1=Boffin |first1=H. M. J. |editor1-last=Arenou |editor1-first=F. |editor2-last=Hestroffer |editor2-first=D. |title=Proceedings of the workshop "Orbital Couples: Pas de Deux in the Solar System and the Milky Way" |journal=Orbital Couples: Pas de Deux in the Solar System and the Milky Way |date=2012 |pages=41–44 |chapter=The mass-ratio distribution of spectroscopic binaries |isbn=978-2-910015-64-0 |lastauthorampname-list-style=yamp|bibcode=2012ocpd.conf...41B }}</ref>

<math>f = \frac{M_{2}^{3}\ \mathrm{sin}^{3}i }{(M_{1} + M_{2})^{2}} = \frac{P_\mathrm{orb}\ K^{3}}{2 \pi G} (1 - e^{2})^{3/2}.</math>