Question

A binary star system with an invisible star?

When a star is in motion relative to Earth, such as when it is part of a two-star system, we detect a
shift in its light (due to the Doppler Effect) as it periodically moves toward and away from us.
Roughly half of all stars are part of binary systems. When one of such star orbits another, but the
other is invisible (emits no radiation), the visible star could be orbiting what used to be a star and
has now collapsed into a black hole.
As in the figure, suppose a visible star has a measured orbital speed of 340 km/s, an orbital period
of 1.9 days and an approximate mass of 6 M, (where Ms is the mass of the sun, which is 1.99 x
1030
kg). Assume both the visible star and the invisible companion object have circular orbits
around their common center of mass (labeled O). Calculate the approximate mass of the dark star
(in multiples of Ms, so that an answer of 1.0 would correspond to a star with the same mass as the
sun).
Hints:
Distances r₁ and r2 are related by the star masses (by center of mass calculation). Gravitational
force provides the centripetal force for circular orbit (but the distances are not the same). Gravity
depends on the separation of the stars whereas the radius of the orbit is used for the centripetal
force magnitude. You can substitute 6 for m₁/M, and use "x" for m2/Ms. You should end up with a
third-order polynomial (a cubic equation). Look up the cubic formula or use an online calculator to
solve for "x". A guide to solution of the problem is provided.
Since your answer is a ratio, it will not have units:
m2
M₂
-
Binary_Star_Problem_Notes.pdf [+]
m₁
r₁
r2
m.

Answer 1

To calculate the mass of the invisible star in the binary system, we can use Kepler's third law and Newton's reformulation of Kepler's law. By setting up an equation and solving for the unknown mass, we can find the approximate mass of the dark star.

In order to calculate the mass of the invisible star in the binary system, we can use Kepler's third law and Newton's reformulation of Kepler's law. Kepler's third law states that the cube of the semimajor axis of the orbit is equal to the sum of the masses of the stars multiplied by the square of the orbital period. We can set up an equation using this law and the given values for the visible star's orbital speed, period, and mass. By solving this equation, we can find the approximate mass of the invisible star.

Let's denote the mass of the invisible star as 'm2' and the mass of the visible star as 'm1'. We can substitute the mass of the visible star (6M, where M is the mass of the sun) and use 'x' for m2/Ms (where Ms is the mass of the sun) to set up the equation.

D³ = (m1 + m2)P²

Plug in the values of the visible star's orbital speed (340 km/s) and period (1.9 days) and solve for x using an online cubic equation solver or the cubic formula. The resulting value of x will give us the approximate mass of the dark star in multiples of the mass of the sun.