Binary mass function: Difference between revisions

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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 (limited) information on the separation and gravitational force between the two components, and hence on the masses of the components.

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|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 introducedknown.<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 |chapterurlchapter-url=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 distanceorbital separation 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.

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> But becauseHowever, the true orbital velocity cannotis beoften determined generallyunknown, thisbecause informationvelocities isin the plane of the sky are much more difficult to determine than velocities along the line of limitedsight.<ref name="karttunen" />

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=httphttps://www.astro.cornell.edu/academics/courses/astro201/psr1913.htm |title=The Binary Pulsar PSR 1913+16 |publisher=[[Cornell University]] |access-date=April 26, 2016 }}</ref> A binary system is called a single-lined spectroscopic binary if the radial motion of only one of the two binary components can be measured. In this case, a lower limit on the mass of the ''other'', (unseen) component can be determined.<ref name="karttunen" />

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 |deadurlurl-status=yesdead |archiveurlarchive-url=https://web.archive.org/web/20150412200552/http://www.physics.usu.edu/shane/classes/astrophysics/lectures/lec08_binaries.pdf |archivedatearchive-date=April 12, 2015 |df= }}</ref> For example, if the measured radial velocity is low, this can mean that the true orbital velocity is low (implying low mass objects) and the inclination high (the orbit is seen edge-on), or that the true velocity is high (implying high mass objects) but the inclination low (the orbit is seen face-on).

== Derivation for a circular orbit ==

[[File:Doppler Shift vs Time.jpgsvg|framethumb|500px|right|Radial velocity curve with peak radial velocity ''K''=1 m/s and orbital period 2 years.]]

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_\mathrmtext{orb}</math> is found from the periodicity in the radial velocity curve. These are the two observable quantities needed to calculate the binary mass function.<ref name="podsiadlowski" />

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}M_1</math> and <math>M_{2}M_2</math> be the stellar masses, with <math>M_{1}M_1 + M_{2}M_2 = M_\mathrm{tot}</math> the total mass of the binary system, <math>v_{1}v_1</math> and <math>v_{2}v_2</math> the orbital velocities, and <math>a_{1}a_1</math> and <math>a_{2}a_2</math> the distances of the objects to the center of mass. <math>a_{1}a_1 +a_{2} 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]],

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

<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>

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>

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>

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

<math display="block">K = v_1 \sin i = \omega_\text{orb} a_1 \sin i.</math>

After substituting <math>a_1</math> we obtain

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 |publisher=Cambridge, UK: [[Cambridge University Press]] |date=2006 |pages=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 |lastauthoramp=y|doi-broken-date=2019-08-20 }}</ref>

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

<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>

After substituting <math>a_{1}</math> we obtain

<math>\frac{M_{2}^{3}}{M_\mathrm{tot}^{2}} = \frac{K^{3}}{G \omega_\mathrm{orb} \mathrm{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 |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=M. 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|title-link=Black-widow Pulsar}}</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>f = \frac{M_{2}^{3}\ \mathrm{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 }}</ref>

===Limits===

In the case of <math>M_{1}M_1 \gg M_{2}M_2</math> (for example, when the unseen object is an exoplanet<ref name="bailes" />), the mass function simplifies to

<math display="block">f \approx \frac{M_2^3\ \sin^3 i }{M_1^2}.</math>

<math>f \approx \frac{M_{2}^{3}\ \mathrm{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>f \approx M_{2}\ \mathrm{sin}^{3}i,</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>M_{1}M_1</math>,

<math display="block">M_2 > \max\left(f, f^{1/3} M_1^{2/3}\right).</math>

<math>M_{2} > \mathrm{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" |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 |publisher=Observatoire de Paris |isbn=978-2-910015-64-0 |lastauthorampname-list-style=yamp|bibcode=2012ocpd.conf...41B }}</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} \left(1 - e^2\right)^{3/2}.</math>

<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>

== Applications ==

=== X-ray binaries ===

If the accretor in an [[X-ray binary]] has a minimum mass that significantly exceeds the [[Tolman–Oppenheimer–Volkoff limit]] (the maximum possible mass for a [[neutron star]]), it is expected to be a black hole. This is the case in [[Cygnus X-1]], for example, where the radial velocity of the companion star has been measured.<ref>{{citation | last=Mauder | first=H. | date=1973 | title=On the Mass Limit of the X-ray Source in Cygnus X-1 | bibcode=1973A&A....28..473M | journal=[[Astronomy and Astrophysics]] | volume=28 | pages=473–475}}</ref><ref>{{cite web |url=http://eagle.phys.utk.edu/guidry/astro421/lectures/lecture490_ch15.pdf |title=Observational Evidence for Black Holes |publisher=[[University of Tennessee]] |access-date=November 3, 2016 |archive-url=https://web.archive.org/web/20171010230722/http://eagle.phys.utk.edu/guidry/astro421/lectures/lecture490_ch15.pdf |archive-date=October 10, 2017 |url-status=dead }}</ref>

=== 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 solar[[Solar systemSystem]], led to the discovery of [[Proxima Centauri b]], a [[terrestrial planet]] with a minimum mass of {{Earth mass|sym=y|1.27}}.<ref name="proxima b discovery paper">{{cite journal
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| first21 = José L.

| last22 = Ofir

| first22 = Aviv

| last23 = Paardekooper

| first23 = Sijme-Jan

| last24 = Reiners

| first24 = Ansgar

| last25 = Rodríguez

| first25 = Eloy

| last26 = Rodríguez-López

| first26 = Cristina

| last27 = Sarmiento

| first27 = Luis F.

| last28 = Strachan

| first28 = John P.

| last29 = Tsapras

| first29 = Yiannis

| last30 = Tuomi

| first30 = Mikko

| first31=Mathias |last31=Zechmeister

| display-authors = 3

| year = 2016

| arxiv = 1609.03449

| doi = 10.1038/nature19106

| pmid = 27558064

| s2cid = 4451513

| url=https://www.nature.com/articles/nature19106

}}</ref>

=== 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 ==