Dynamical parallax

In astronomy, the distance to a visual binary star may be estimated from the masses of its two components, the size of their orbit, and the period of their revolution around one another.^[1] A dynamical parallax is an (annual) 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 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]

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 magnitudes 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.^[4] 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.

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

T^{2}\propto a^{3},

where $T$ is the orbital period of the planet and $a$ is the semi-major axis of the orbit.^[5]

Newton's Generalisation

Consider a binary star system. This consists of two objects, of mass $m_{1}$ and $m_{2}$ , orbiting around their centre of mass. $m_{1}$ has position vector $r_{1}$ and orbital velocity $v_{1}$ , and $m_{2}$ has position vector $r_{2}$ and orbital velocity $v_{2}$ relative to the centre of mass. The separation between the two stars is denoted $a$ , 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.^[6]

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."

F_{net}=\sum \,F_{i}=ma,

where $F_{net}$ is the net force acting on the object of mass $m$ , and $a$ is the acceleration of the object.^[7]

References

^ Patrick Moore (2002). Philip's Astronomy Encyclopedia (revised and expanded ed. ed.). London: Philip's. p. 120. ISBN 0-540-07863-8. {{cite book}}: |edition= has extra text (help)
^ "dynamical parallax". Astronomy Knowledge Base. 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. ISBN 1-85233-751-6.
^ Leanard Susskind and Hrabovsky (2013). The Theoretical Minimum: What You Need To Know To Start Doing Physics. the Penguin Group. p. 256. ISBN 978-1846147982.
^ "The Physics of Binary Stars". Retrieved 15/10/2013. {{cite web}}: Check date values in: |accessdate= (help)
^ Bradley W. Carroll and Dale A. Ostlie (2013). An Introduction to Modern Astrophysics. Pearson. p. 1478. ISBN 978-1292022932.

Kepler's 3rd Law

Newton's Generalisation

References

See also