Binary mass function: Difference between revisions

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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 force between the two components, and hence on the masses of the components.

== Introduction ==

[[File:orbit2.gif|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 |chapter-url=https://books.google.com/books?id=DjeVdb0sLEAC&pg=PA221|isbn=978-3-540-34143-7 |name-list-style=amp}}</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=http://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 |url-status=dead |archive-url=https://web.archive.org/web/20150412200552/http://www.physics.usu.edu/shane/classes/astrophysics/lectures/lec08_binaries.pdf |archive-date=April 12, 2015 }}</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).