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In astronomy, the distance to a visual binary star may be estimated from the masses of its two components, the angular size of their orbit, and the period of their orbit about one another.^[1] A dynamical parallax is an (annual)^{[clarification needed]} parallax which is computed from such an estimated distance.

To calculate a dynamical parallax, the angular semi-major axis of the orbit of the stars is observed, as is their apparent brightness. By using Newton's generalisation of Kepler's Third Law, which states that the total mass of a binary system multiplied by the square of its orbital period is proportional to the cube of its semi-major axis,^[2] together with the mass–luminosity relation, the distance to the binary star can then be determined.^[3]

With this technique, the masses of the two stars in a binary system are estimated, usually as the mass of the Sun. Then, using Kepler's laws of celestial mechanics, the distance between the stars is calculated. Once this distance is found, their distance from the observer can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the apparent magnitudes of both stars, the luminosities can be found, and from the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies within 5% can be achieved.^[4]

References

^ Patrick Moore (2002). Philip's Astronomy Encyclopedia (revised and expanded ed.). London: Philip's. p. 120. ISBN 0-540-07863-8.
^ "dynamical parallax". Astronomy Knowledge Base. Archived from the original on 2006-07-05. Retrieved 2006-07-18.
^ The Mass–Luminosity Relation, University of Tennessee, Astronomy 162: Stars, Galaxies, and Cosmology, lecture notes. Accessed July 18, 2006.
^ Mullaney, James (2005). Double and multiple stars and how to observe them. Springer. p. 27. ISBN 1-85233-751-6.

[1] Patrick Moore (2002). Philip's Astronomy Encyclopedia (revised and expanded ed.). London: Philip's. p. 120. ISBN 0-540-07863-8.

[2] "dynamical parallax". Astronomy Knowledge Base. Archived from the original on 2006-07-05. Retrieved 2006-07-18.

[3] The Mass–Luminosity Relation, University of Tennessee, Astronomy 162: Stars, Galaxies, and Cosmology, lecture notes. Accessed July 18, 2006.

[double-4] Mullaney, James (2005). Double and multiple stars and how to observe them. Springer. p. 27. ISBN 1-85233-751-6.

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+{{Short description|Iterative technique to find the properties of the stars in a binary pair}}
-{{Underconstruction}}
-In [[astronomy]], the [[distance]] to a [[visual binary|visual binary star]] may be estimated from the [[mass]]es of its two components, the size of their [[orbit]], and the period of their revolution around one another.<ref>{{cite book
+In [[astronomy]], the [[distance]] to a [[visual binary|visual binary star]] may be estimated from the [[mass]]es of its two components, the [[angular size]] of their [[orbit]], and the period of their orbit about one another.<ref>{{cite book
  | author = Patrick Moore
  | title = Philip's Astronomy Encyclopedia
+ | url = https://archive.org/details/philipsastronomy00ehrl_838
- | edition = revised and expanded ed.
+ | url-access = limited
+ | edition = revised and expanded
  | year = 2002
  | publisher = Philip's
  | location = London
  | isbn = 0-540-07863-8
+ | page = [https://archive.org/details/philipsastronomy00ehrl_838/page/n128 120]
- | page = 120
-}}</ref>  A '''dynamical parallax''' is an (annual) [[parallax]] which is computed from such an estimated distance.
+}}</ref>  A '''dynamical parallax''' is an (annual){{clarify|reason=How is this annual?|date=November 2024}} [[parallax]] which is computed from such an estimated distance.
-To calculate a dynamical parallax, the angular [[semi-major axis]] of the orbit of the stars around each other is observed, together with their [[apparent brightness]].  By using [[Isaac Newton|Newton's]] generalisation of [[Johannes Kepler|Kepler's]] [[Kepler's third law|Third Law]], which states that the total mass of a binary system multiplied by the [[square (algebra)|square]] of its [[orbital period]] is proportional to the [[cube (arithmetic)|cube]] of its [[semi-major axis]],<ref>{{cite web
+To calculate a dynamical parallax, the angular [[semi-major axis]] of the orbit of the stars is observed, as is their [[apparent brightness]].  By using [[Isaac Newton|Newton's]] generalisation of [[Johannes Kepler|Kepler's]] [[Kepler's third law|Third Law]], which states that the total mass of a binary system multiplied by the [[square (algebra)|square]] of its [[orbital period]] is proportional to the [[cube (arithmetic)|cube]] of its [[semi-major axis]],<ref>{{cite web
-| url = http://www.site.uottawa.ca:4321/astronomy/index.html#dynamicalparallax
+ |url         = http://www.site.uottawa.ca:4321/astronomy/index.html#dynamicalparallax
-| title = dynamical parallax
+ |title       = dynamical parallax
-| accessdate = 2006-07-18
+ |accessdate  = 2006-07-18
-| work = Astronomy Knowledge Base
+ |work        = Astronomy Knowledge Base
+ |url-status     = dead
-}}</ref> together with the [[mass-luminosity relation]], the distance to the binary star can then be determined.<ref>[http://csep10.phys.utk.edu/astr162/lect/binaries/masslum.html The Mass-Luminosity Relation], University of Tennessee, Astronomy 162: Stars, Galaxies, and Cosmology, lecture notes.  Accessed July 18, 2006.</ref>
+ |archiveurl  = https://web.archive.org/web/20060705214728/http://www.site.uottawa.ca:4321/astronomy/index.html#dynamicalparallax
+ |archivedate = 2006-07-05
+}}</ref> together with the [[mass–luminosity relation]], the distance to the binary star can then be determined.<ref>[http://csep10.phys.utk.edu/astr162/lect/binaries/masslum.html The Mass–Luminosity Relation], University of Tennessee, Astronomy 162: Stars, Galaxies, and Cosmology, lecture notes.  Accessed July 18, 2006.</ref>
-In this technique, the masses of the two stars in a binary system are estimated, usually as being the mass of the sun. Then, using [[Kepler's laws]] of [[celestial mechanics]], the distance between the stars is calculated. Once this distance is found, the distance away can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the [[apparent magnitude]]s of both stars, the luminosities can be found, and by using the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies as high as 5% can be achieved.<ref name="double">{{cite book|last=Mullaney|first=James|title=Double and multiple stars and how to observe them|publisher=Springer|year=2005|isbn=1-85233-751-6|url=http://books.google.com/books?id=OM3hozlKt9AC&lpg=PA27&ots=da06NVpIY_&dq=Mass-Luminosity%20relation%20distance%20binary&pg=PA27#v=onepage&q=&f=false}}</ref> The mass–luminosity relationship can also be used to determine the lifetime of stars by noting that lifetime is approximately proportional to M/L. One finds that more massive stars live shorter. A more sophisticated calculation factors in a star's loss of mass over time.
+With this technique, the masses of the two stars in a binary system are estimated, usually as the mass of the Sun. Then, using [[Kepler's laws]] of [[celestial mechanics]], the distance between the stars is calculated. Once this distance is found, their distance from the observer can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the [[apparent magnitude]]s of both stars, the luminosities can be found, and from the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies within 5% can be achieved.<ref name="double">{{cite book|last=Mullaney|first=James|title=Double and multiple stars and how to observe them|publisher=Springer|year=2005|isbn=1-85233-751-6|url=https://archive.org/details/doublemultiplest0000mull|url-access=registration|page=[https://archive.org/details/doublemultiplest0000mull/page/27 27]}}</ref>
-==Kepler's 3rd Law==
-Keplar's 3rd Law can be stated as follows: "The square of the orbital period of a planet is directly proportional to the cube of its semi-major axis." Mathmatically, this translates as
-:<math> T^2 \propto a^3 </math>
-where <math>T</math> is the orbital period of the planet and <math>a</math> is the semi-major axis of the orbit.<ref>{{cite book
- | author = Leanard Susskind and Hrabovsky
- | title = The Theoretical Minimum: What You Need To Know To Start Doing Physics
- | year = 2013
- | isbn = 978-1846147982
- | publisher = the Penguin Group
- | page = 256
-}}</ref>
-===Newton's Generalisation===
-Consider a binary star system. This consists of two objects, of mass <math>m_1</math> and <math>m_2</math>, orbiting around their centre of mass. <math>m_1</math> has position vector <math>r_1</math> and orbital velocity <math>v_1</math>, and <math>m_2</math> has position vector <math>r_2</math> and orbital velocity <math>v_2</math> relative to the centre of mass.  The separation between the two stars is denoted <math>a</math>, and is assumed to be constant. Since the gravitational force acts along a line joining the centers of both stars, we can assume the stars have an equivalent time period around their center of mass, and therefore a constant separation between each other.<ref>{{cite web
- | url = http://www.egglescliffe.org.uk/physics/gravitation/binary/binary.html
- | title = The Physics of Binary Stars
- | accessdate = 15/10/2013
-}}</ref>
-To arrive at Newton's version of Kepler's 3rd law we can start by considering Newton's 2nd law which states: "The net force acting on an object is proportional to the objects mass and resultant acceleration."
-:<math> F_{net} = \sum \, F_{i} = ma </math>
-where <math>F_{net}</math> is the net force acting on the object of mass <math>m</math>, and <math>a</math> is the acceleration of the object.<ref>{{cite book
- | author = Bradley W. Carroll and Dale A. Ostlie
- | title = An Introduction to Modern Astrophysics
- | year = 2013
- | isbn = 978-1292022932
- | publisher = Pearson
- | page = 1478
-}}</ref>
-Applying the definition of centripetal acceleration to Newton's second law gives a force of
-:<math> F = mv^2/r </math>
-Then using the fact that the orbital velocity is given as
-:<math> v = 2\pi r/T </math>
-we can state the force on each star as
-:<math> F_{1} = 4\pi^2 m_{1}r_{1}/T^2 </math> and <math> F_{2} = 4\pi^2 m_{2}r_{2}/T^2 </math>
+==See also==
+* [[Parallax in astronomy]]
+* [[Photometric parallax method]]
+* [[Spectroscopic parallax]]
 ==References==
 {{Reflist}}
-==See also==
-*[[Photometric parallax method]]
-*[[Spectroscopic parallax]]
 {{DEFAULTSORT:Dynamical Parallax}}
 [[Category:Binary stars|-]]
 [[Category:Astrometry]]
-{{Astronomy-stub}}