Editing Visual binary (section)

==Distance==
In order to work out the masses of the components of a visual binary system, the distance to the system must first be determined, since from this astronomers can estimate the period of revolution and the separation between the two stars. The trigonometric [[parallax]] provides a direct method of calculating a star's mass. This will not apply to the visual binary systems, but it does form the basis of an indirect method called the dynamical parallax.<ref name=double />

===Trigonometric parallax===
In order to use this method of calculating distance, two measurements are made of a star, one each at opposite sides of the Earth's orbit about the Sun. The star's position relative to the more distant background stars will appear displaced.  The parallax value is considered to be the displacement in each direction from the mean position, equivalent to the angular displacement from observations one [[astronomical unit]] apart. The distance <math>d</math>, in [[parsec]]s is found from the following equation,
:<math> d = \frac{1}{\tan(p)} </math>
Where <math>p</math> is the parallax, measured in units of arc-seconds.<ref>{{cite book
 | author = Martin Harwit
 | title = Astrophysical Concepts 
 | date = 20 April 2000 
 | isbn = 0-387-94943-7
 | publisher = Springer
}}</ref>

===Dynamical parallax===
This method is used solely for binary systems. The mass of the binary system is assumed to be twice that of the Sun. Kepler's Laws are then applied and the separation between the stars is determined. Once this distance is found, the distance away can be found via the arc subtended in the sky, providing a temporary 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 a number of times, with accuracies as high as 5% being achieved. A more sophisticated calculation factors in a star's loss of mass over time.<ref name="double">{{cite book|last=Mullaney|first=James|title=Double and multiple stars and how to observe them|publisher=Springer|date=2005|isbn=1-85233-751-6|url=https://archive.org/details/doublemultiplest0000mull|url-access=registration|page=[https://archive.org/details/doublemultiplest0000mull/page/27 27]|quote=Mass–Luminosity relation distance binary.}}</ref>

===Spectroscopic parallax===
Spectroscopic parallax is another commonly used method for determining the distance to a binary system. No parallax is measured, the word is simply used to place emphasis on the fact that the distance is being estimated. In this method, the luminosity of a star is estimated from its spectrum. It is important to note that the spectra from distant stars of a given type are assumed to be the same as the spectra of nearby stars of the same type. The star is then assigned a position on the Hertzsprung-Russel diagram based on where it is in its life-cycle. The star's luminosity can be estimated by comparison of the spectrum of a nearby star. The distance is then determined via the following inverse square law:

:<math> b = \frac{L}{4\pi d^2} </math>

where <math>b</math> is the apparent brightness and <math>L</math> is the luminosity.

Using the Sun as a reference we can write

:<math> \frac{L}{L_{\odot}} = \bigg(\frac{d^{2}_{\odot}}{b}\bigg)\bigg(\frac{d^{2}}{b_{\odot}}\bigg) </math>

where the subscript <math>\odot</math> represents a parameter associated with the Sun.

Rearranging for <math>d^2</math> gives an estimate for the distance.<ref>[http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=35616&fbodylongid=1667 European Space Agency, ''Stellar distances'']</ref>

:<math> d^2 = \bigg(\frac{L}{L_{\odot}}\bigg)\bigg(\frac{b_{\odot}}{b}\bigg) </math>