A visual binary star system has been observed for fifty years. In that time the

Question

A visual binary star system has been observed for fifty years. In that time the two stars have completed ! of their orbit around “each other” (actually around their mutual center of mass). They are just far apart enough that our observations indicate that they are in circular orbits - they have stayed a constant 2.0 arcseconds apart since they were first observed. The fainter of the two stars is a G2V star, identical to our Sun, and therefore has a mass of 1 Solar Mass. From the apparent magnitude observed for this star we can calculate the distance to the binary system: these stars are 30 parsecs from us. what is the mass of its companion star? (in units of Solar Mass)

Explanation / Answer

Now we have to find out the semi-major axis of their orbit:
AU = D x a
AU is semi-major axis of the systems orbit in AU
D is the distance to the binary pair in parsecs, which we now know is 30 parsecs
a is the angular separation in arc seconds, which is 2 arc seconds

AU = 30 x 2
semi-major axis or orbit = 60 AU

Now that we know the semi-major axis of the orbit, we can use Newtons version of Keplers third law to find the mass of the second star.

Newtons Version of Keplers Third Law:
(M1 + M2) = A^3/P^2
M1 is the mass of one of the stars, which is 1 solar mass
M2 is the mass of the second star, ?
A is the semi-major axis of their orbit, 60 AU
P is the orbital period, 50 years

M1+ M2 = 60^3/50^2
M1 + M2 = 86.4 solar masses

since M1 is 1 solar masses, M2 must be 85.4 solar masses.

M2 = 85.4 solar masses

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