Suppose you observe a binary system containing a main-sequence star and a brown dwarf. The orbital period of the system is 1 year, and the average separation of the system is 1 AU. You then measure the Doppler shifts of the spectral lines from the main-sequence star and the brown dwarf, finding that the orbital speed of the brown dwarf in the system is 25 times greater than that of the main-sequence star.

How massive is the brown dwarf?

The brown dwarf is 1/26 solar masses

Explanation:

Step 1:

Find the total mass (mass star A + mass star B) from Kepler's 3rd law:

By using Kepler's third law, which is expressed by the formula:

(M₁ + M₂) = d³ / T²

where

(M₁ + M₂) is the total mass of the binary system d is the distance between the stars T is the orbital period

We get,

M₁ + M₂ = (1 AU) ³ / (1 year) ²

= 1 solar mass

Step 2:

Find the proportion of each star's mass to the total mass from the centre of mass:

Let the brown dwarf be "star 1". Thus,

M₁ / M₂ = v₂ / v₁

= v₂ / (25 v₂)

= 1/25

Step 3:

Setting the mass of star 1 = (mass of star 2) * (the fraction of the previous step) and substituting this for the mass of star 1 in the first step (Kepler's 3rd law step), you will find star 2's mass = the total mass / (1 + the fraction from step 2):

M₂ = (M₁ + M₂) / (1 + M₁ / M₂)

= (1 solar mass) / (1 + 1/25)

= 25/26 solar masses

Therefore, the mass of the main-sequence star is 25/26 solar masses.

Step 4:

M₁ = M₂ * (M₁ / M₂)

M₁ = (25/26) * (1/25)

M₁ = 1/26 solar masses

Therefore, the mass of the brown dwarf is 1/26 solar masses.

To check if this is correct, the sum of the two masses must give you the total mass that was calculated in step 1.

M₁ + M₂ = 1/26 + 25/26 = 1 solar mass