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. 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, together with the mass-luminosity relation, the distance to the binary star can then be determined.

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. 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.

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.

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.

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

$F=mv^{2}/r$

Then using the fact that the orbital velocity is given as

$v=2\pi r/T$

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"

$F_{12}=-F_{21}$

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 equation 4 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, $F=Gm_{1}m_{2}/a^{2}$), and solving for the period squared yields the required result.

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

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.

See also

• Photometric parallax method
• Spectroscopic parallax