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.

Parallax

There is a simple formula relating distance to parallax. Firstly, two measurements are made at both sides of the Earth's orbit. For a nearby star, it will appear displaced relative to background stars. The distance to this nearby star is given by

d=1AU/tan(p)

Where p is the parallax, defined as half the apparent angular displacement, and AU refers to the astronomical unit. (1AU = 1.49*10^8km)^[5]

Visual Binary

A visual binary is a gravitationally bound system that can be resolved into two stars with the aid of a telescope. These stars are estimated, via Kepler's 3rd law, to have periods ranging from a number of years to thousands of years. A visual binary consists of two stars, usually of a different brightness. Because of this, the brightest star is called the primary and the fainter one is called the companion. If the primary is too bright, relative to the companion, this can cause a glare making it difficult to resolve the two components. ^[6] However, it is possible to resolve the system if observations of the brighter star show it to wobble about a centre of mass. ^[7] In general, a visual binary is resolved into two stars with a telescope if their centres are separated by a value greater than or equal to one arcsecond.

For a visual binary system, measurements taken need to specify, in arc-seconds, the apparent angular separation on the sky and the position angle- which is the angle measured eastward from North in degrees- of the companion star relative to the primary star. Taken over a period of time, the apparent relative orbit of the visual binary system will appear on the celestial sphere.^[8]

Kepler's Laws

The two stars orbiting each other, as well as there centre of mass, must obey Kepler's laws. This means that the orbit is an ellipse with the centre of mass at one of the two focii (Kepler's 1st law) and the orbital motion satisfies the fact that a line joining the star to the centre of mass sweeps out equal areas over equal time intervals (Kepler's 2nd law). The orbital motion must also satisfy Kepler's 3rd law.^[9]

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.^[9]

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.^[10]

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.^[11]

Applying the definition of centripetal acceleration to Newton's second law gives a force of

F=mv^{2}/r

^[12]

Then using the fact that the orbital velocity is given as

v=2\pi r/T

^[12]

we can state the force on each star as

F_{1}=4\pi ^{2}m_{1}r_{1}/T^{2}

and

F_{2}=4\pi ^{2}m_{2}r_{2}/T^{2}

If we apply Newton's 3rd law- "For every action there is an equal and opposite reaction"

F_{12}=-F_{21}

^[11]

We can set the force on each star equal to each other.

4\pi ^{2}m_{1}r_{1}/T^{2}=4\pi ^{2}m_{2}r_{2}/T^{2}

This reduces to

r_{1}m_{1}=r_{2}m_{2}

If we assume that the masses are not equal, then this equation tells us that the smaller mass remains farther from the centre of mass than does the larger mass.

The separation $a$ of the two objects is

a=r_{1}+r_{2}

Since $r_{1}$ and $r_{2}$ would form a line starting from opposite directions and joining at the centre of mass.

Now we can substitute this expression into one of the equations describing the force on the stars and rearrange for $r_{1}$ to find an expression relating the position of one star to the masses of both and the separation between them. Equally, this could have been solved for $r_{2}$ . We find that

r_{1}=m_{2}a/(m_{1}+m_{2})

Substituting this equation into the equation for the force on one of the stars, setting it equal to Newton's Universal Law of Gravitation (namely, $F=Gm_{1}m_{2}/a^{2}$ )^[11], and solving for the period squared yields the required result.

T^{2}=4\pi ^{2}a^{3}/G(m_{1}+m_{2})

^[11]

This is Newton's version of Kepler's 3rd Law. It requires that the mass is measured in solar masses, the orbital period is measured in years, and the orbital semi-major axis is measured in astronomical units.

Determining Stellar Masses of a Binary System

The Mass-Luminosity Relationship

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.
^ Martin Harwit. Astrophysical concepts. springer. ISBN 0-387-94943-7.
^ The Binary Stars, Robert Grant Aitken, New York: Dover, 1964, p. 41.
^ ,"Binary Systems and Stellar Parameters" (PDF). Retrieved 2013-11-02.
^ Michael Zeilik, Stephan A. Gregory, and Elske V. P. Smith. Introductory Astronomy and Astrophysics. Brooks/Cole. ISBN 978-0030062285.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ ^a ^b Leanard Susskind and Hrabovsky (2013). The Theoretical Minimum: What You Need To Know To Start Doing Physics. the Penguin Group. ISBN 978-1846147982.
^ "The Physics of Binary Stars". Retrieved 15/10/2013. {{cite web}}: Check date values in: |accessdate= (help)
^ ^a ^b ^c ^d Bradley W. Carroll and Dale A. Ostlie (2013). An Introduction to Modern Astrophysics. Pearson. ISBN 978-1292022932.
^ ^a ^b Hugh D. Young (2010). University Physics. Bertrams. ISBN 0321501306.