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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 23 times greater than that of the main-sequence star.1. How massive is the brown dwarf?
Mdwarf = ? KG



Answer :

Mass of the brown. dwarf., M₂ = 4.132 * 10²⁷ kg

Let M₁ = Mass of the main-sequence star

M₂ = Mass of the brown dwarf

v₁ = speed of the main-sequence star

v₂ = speed of the brown dwarf

The system's average distance apart r₁ = r₂ = 1 AU

The brown dwarf's orbital speed is 22 times that of a star in the main sequence.

Consequently, v2 = 22 v1.

Considering that the Sun is a main-sequence star

2 * 1030 kg is the main-sequence star's mass.

Centripetal force: F = Mv2/r

M1v12 = M2v22 M2/M1 = v12/v22 M2/M1 = (1/22) M1v12 = M2v22/r2 M2v22/r2

2 M2/M1 = 0.002066 M2 = 0.002066M1 M2 = 0.002066*2*1030 M2 = 4.132*1027 kg

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