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

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

found in the nucleus

has mass of one amu

the minimum amperage that the current in this wire would have to have so that its weight is completely supported by the magnetic force due to earth's field, assuming that no other forces except gravity act on it