# In a binary star system, a small white dwarf orbits a companion with a period of 52 years at a distance of 20 A.U. What is the mass of the white dwarf assuming the companion star has mass of 1.5 solar masses? Many thanks if anyone can help!?

Jul 30, 2015

Using the third Kepler law (simplified for this particular case), which establishes a relation between distance between stars and their orbital period, we shall determine the answer.

#### Explanation:

Third Kepler law establishes that:

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

where $T$ represents orbital period and $a$ represents the semi-major axis of star orbit.
Assuming that stars are orbiting on the same plane (i.e., the inclination of the axis of rotation relative to the orbital plane is 90º), we can affirm that proportionality factor between ${T}^{2}$ and ${a}^{3}$ is given by:

$\frac{G \left({M}_{1} + {M}_{2}\right)}{4 {\pi}^{2}} = \frac{{a}^{3}}{{T}^{2}}$

or, giving ${M}_{1}$ and ${M}_{2}$ on solar masses, $a$ on A.U. and $T$ on years:

${M}_{1} + {M}_{2} = \frac{{a}^{3}}{{T}^{2}}$

Introducing our data:

${M}_{2} = \frac{{a}^{3}}{{T}^{2}} - {M}_{1} = \frac{{20}^{3}}{{52}^{2}} - 1.5 = 1.46 {M}_{\odot}$