Answer:
The brown dwarf is <em>1/26 solar masses</em>
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 <em>mass of the brown dwarf is 1/26 solar masses</em>.
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
