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 a

Question

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 AUAU . 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 22 times greater than that of the main-sequence star.

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Thu Nguyệt 5 years 2021-07-20T18:36:26+00:00 1 Answers 451 views 1

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  1. bonexptip
    -1
    2021-07-20T18:38:10+00:00
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    Complete Question:

    How massive is the brown dwarf

    Answer:

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

    Explanation:

    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 average separation of the system r₁ = r₂ = 1 AU

    Orbital speed of the brown dwarf is 22 times that of the main – sequence star

    That is, v₂ = 22 v₁

    Since the sun is a main sequence star

    Mass of the main – sequence star = 2 * 10³⁰ kg

    Centripetal force, F = Mv²/r

    M₁v₁²/r₁² = M₂v₂²/r²

    M₁v₁² = M₂v₂²

    M₂/M₁ = v₁²/v₂²

    M₂/M₁ = (1/22)²

    M₂/M₁ = 0.002066

    M₂ = 0.002066M₁

    M₂ = 0.002066 * 2 * 10³⁰

    M₂ = 4.132 * 10²⁷ kg

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