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{{Short description|Astronomic function}}
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 information on the separation and gravitational force between the two components, and hence on the masses of the components.
== Introduction ==
[[File:orbit2.gif
The binary mass function follows from [[Kepler's third law]] when the radial velocity of one
Kepler's third law describes the motion of two bodies orbiting a common [[center of mass]]. It relates the [[orbital period
Because the orbital period and orbital velocities in the binary system are related to the masses of the binary components, measuring these parameters provides some information about the masses of one or both components.<ref name="podsiadlowski">{{cite web |url=http://www-astro.physics.ox.ac.uk/~podsi/binaries.pdf |title=The Evolution of Binary Systems, in Accretion Processes in Astrophysics |first1=Philipp |last1=Podsiadlowski |publisher=[[Cambridge University Press]] |access-date=April 20, 2016 }}</ref>
Radial velocity is the velocity component of orbital velocity in the line of sight of the observer. Unlike true orbital velocity, radial velocity can be determined from [[Doppler spectroscopy]] of [[spectral line]]s in the light of a star,<ref name="radial">{{cite web |url=http://www.planetary.org/explore/space-topics/exoplanets/radial-velocity.html |title=Radial Velocity – The First Method that Worked |publisher=[[The Planetary Society]] |access-date=April 20, 2016 }}</ref> or from [[pulsar timing|variations in the arrival times]] of pulses from a [[radio pulsar]].<ref name="cornell">{{cite web |url=
The true mass and true orbital velocity cannot be determined from the radial velocity because the [[orbital inclination]] is generally unknown. (The inclination is the orientation of the orbit from the point of view of the observer, and relates true and radial velocity.<ref name="karttunen" />) This causes a degeneracy between mass and inclination.<ref name="brown">{{cite journal|doi=10.1088/0004-637X/805/2/188|title=True Masses of Radial-Velocity Exoplanets|year=2015|last1= Brown| first1=Robert A.| journal=[[The Astrophysical Journal]]|bibcode = 2015ApJ...805..188B|volume=805|issue=2|pages=188|arxiv = 1501.02673|s2cid=119294767}}</ref><ref name="larson">{{cite web |url=http://www.physics.usu.edu/shane/classes/astrophysics/lectures/lec08_binaries.pdf |title=Binary Stars |first1=Shane |last1=Larson |publisher=[[Utah State University]] |access-date=April 26, 2016 |url-status=dead |
== Derivation for a circular orbit ==
[[File:Doppler Shift vs Time.
The peak radial velocity <math>K</math> is the semi-amplitude of the radial velocity curve, as shown in the figure. The orbital period <math>P_\
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>
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]],
<math display="block">GM_\text{tot} = \omega_\text{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 display="block">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>
Inserting this expression for <math>a</math> into Kepler's third law, we find
<math display="block">GM_\mathrm{tot} = \omega_\mathrm{orb}^2 \frac{a_1^3 M_\mathrm{tot}^3}{M_2^3}.</math>
which can be rewritten to
<math display="block">\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 |name-list-style=amp}}</ref>
<math display="block">K = v_1 \sin i = \omega_\text{orb} a_1 \sin i.</math>
After substituting <math>a_1</math> we obtain
<math display="block">\frac{M_2^3}{M_\text{tot}^2} = \frac{K^3}{G \omega_\text{orb} \sin^3 i}.</math>
The binary mass function <math>f</math> (with [[Units of measurement|unit]] of mass) is<ref name="bailes">{{cite journal | doi=10.1126/science.1208890|title=Transformation of a Star into a Planet in a Millisecond Pulsar Binary| year=2011| last1=Bailes|first1=M.|author1-link=Matthew Bailes|last2=Bates|first2=S. D.|last3=Bhalerao|first3=V.|last4=Bhat|first4=N. D. R.| last5=Burgay|first5=M.|last6=Burke-Spolaor|first6=S.| last7=d'Amico|first7=N.| last8=Johnston|first8=S.| last9=Keith|first9=M. J.| last10=Kramer|first10=M.|last11=Kulkarni|first11=S. R.| last12=Levin|first12=L.| last13=Lyne|first13=A. G.| last14=Milia|first14=S.| last15=Possenti|first15=A.| last16=Spitler|first16=L.| last17=Stappers|first17=B.| last18=Van Straten|first18=W.|journal=[[Science (journal)|Science]]|bibcode = 2011Sci...333.1717B | pmid=21868629 | volume=333 | issue=6050 | pages=1717–1720 | arxiv = 1108.5201 | s2cid=206535504 | display-authors=8}}</ref><ref name="tauris" /><ref name="podsiadlowski" /><ref name="kerkwijk">{{cite journal| doi=10.1088/0004-637X/728/2/95|title=Evidence for a Massive Neutron Star from a Radial-velocity Study of the Companion to the Black-widow Pulsar PSR B1957+20|year=2011|last1= van Kerkwijk|first1=M. H.|last2=Breton|first2=R. P.| last3=Kulkarni|first3=S. R.|author3-link=Shrinivas Kulkarni|journal=[[The Astrophysical Journal]]|bibcode = 2011ApJ...728...95V | volume=728|issue=2|pages=95|arxiv = 1009.5427|s2cid=37759376}}</ref><ref name="karttunen" /><ref name="larson" /><ref>{{cite web |url=http://astronomy.swin.edu.au/cms/astro/cosmos/b/Binary+Mass+Function |title=Binary Mass Function |publisher=COSMOS – The SAO Encyclopedia of Astronomy, [[Swinburne University of Technology]] |access-date=April 20, 2016 }}</ref>
<math display="block">f = \frac{M_2^3 \sin^3 i }{(M_1 + M_2)^2} = \frac{P_\mathrm{orb}\ K^3}{2 \pi G}.</math>
For an estimated or assumed mass <math>M_1</math> of the observed object 1, a [[minimum mass]] <math>M_\mathrm{2, min}</math> can be determined for the unseen object 2 by assuming <math>i = 90^{\circ}</math>. The true mass <math>M_2</math> depends on the orbital inclination. The inclination is typically not known, but to some extent it can be determined from observed [[Transit (astronomy)|eclipses]],<ref name="podsiadlowski" /> be constrained from the non-observation of eclipses,<ref name="bailes" /><ref name="kerkwijk" /> or be modelled using ellipsoidal variations (the non-spherical shape of a star in binary system leads to variations in brightness over the course of an orbit that depend on the system's inclination).<ref>{{cite web |url=http://cmi2.yale.edu/bh/week4/pages/page5.html |title=The Orbital Inclination |publisher=[[Yale University]] |access-date=February 17, 2017 |archive-date=May 14, 2020 |archive-url=https://web.archive.org/web/20200514093553/http://cmi2.yale.edu/bh/week4/pages/page5.html |url-status=dead }}</ref>
===Limits===
In the case of <math>
<math display="block">f \approx \frac{M_2^3\ \sin^3 i }{M_1^2}.</math>
In the other extreme, when <math>M_1 \ll M_2</math> (for example, when the unseen object is a high-mass [[black hole]]), the mass function becomes<ref name="podsiadlowski" />
<math display="block">f \approx M_2 \sin^3 i,</math>
and since <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>
<math display="block">M_2 > \max\left(f, f^{1/3} M_1^{2/3}\right).</math>
== 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"
<math display="block">f = \frac{M_2^3 \sin^3 i }{(M_1 + M_2)^2} = \frac{P_\mathrm{orb}\ K^3}{2 \pi G} \left(1 - e^2\right)^{3/2}.</math>
== Applications ==
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=== Exoplanets ===
An [[exoplanet]] causes its host star to move in a small orbit around the center of mass of the star-planet system. This 'wobble' can be observed if the radial velocity of the star is sufficiently high. This is the [[Methods of detecting exoplanets|radial velocity method]] of detecting exoplanets.<ref name="brown" /><ref name="radial" /> Using the mass function and the radial velocity of the host star, the minimum mass of an exoplanet can be determined.<ref>{{cite web |url=http://exoplanets.org/methodology.html |title=Documentation and Methodology |publisher=[[Exoplanet Data Explorer]] |access-date=April 25, 2016 }}</ref><ref>{{cite journal|doi=10.1086/504701|title=Catalog of Nearby Exoplanets | year=2006 | last1=Butler|first1=R.P.|author1-link=R. Paul Butler|last2=Wright|first2=J. T.|last3=Marcy|first3=G. W.|author3-link=Geoffrey Marcy|last4=Fischer|first4=D. A.|author4-link=Debra Fischer|last5=Vogt|first5=S. S.|author5-link=Steven S. Vogt|last6=Tinney|first6=C. G.|last7= Jones|first7=H. R. A.|last8= Carter|first8=B. D.|last9=Johnson|first9=J. A.| last10=McCarthy|first10=C.|last11=Penny|first11=A. J.|journal=[[The Astrophysical Journal]]|bibcode = 2006ApJ...646..505B | volume=646|issue=1|pages=505–522|arxiv = astro-ph/0607493 |s2cid=119067572|display-authors=8}}</ref>{{rp|p=9}}<ref name="boffin" /><ref>{{cite web |url=http://www.phy.duke.edu/~kolena/invisible.html |title=Detecting Invisible Objects: a guide to the discovery of Extrasolar Planets and Black Holes |first1=John |last1=Kolena |publisher=[[Duke University]] |access-date=April 25, 2016 }}</ref> Applying this method on [[Proxima Centauri]], the closest star to the
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=== Pulsar planets ===
[[Pulsar planet]]s are planets orbiting [[pulsar]]s, and [[List of exoplanets detected by timing|several]] have been discovered using [[pulsar timing]]. The radial velocity variations of the pulsar follow from the varying intervals between the arrival times of the pulses.<ref name="cornell" /> The first exoplanets were discovered this way in 1992 around the [[millisecond pulsar]] [[PSR 1257+12]].<ref>{{cite journal |last1=Wolszczan|first1=D. A.|author1-link=Aleksander Wolszczan|last2=Frail|first2=D.|author2-link=Dale Frail| title = A planetary system around the millisecond pulsar PSR1257+12 |journal = [[Nature (journal)|Nature]]|volume= 355 |issue = 6356 |pages= 145–147 |date = 9 January 1992 |url = http://www.nature.com/physics/looking-back/wolszczan/index.html |bibcode = 1992Natur.355..145W |doi = 10.1038/355145a0 |s2cid=4260368|url-access= subscription}}</ref> Another example is [[PSR J1719-1438]], a millisecond pulsar whose companion, [[PSR J1719-1438 b]], has a minimum mass approximate equal to the mass of [[Jupiter]], according to the mass function.<ref name="bailes" />
== References ==
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