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    If the exponent, α, in the mass-luminosity relationship is 3.0, estimate the lifetime mof the above star. (You can assume that the Sun’s lifetime is 1 x 10^10 years).


    Relevant equations

    L*/Lsun = (M*/Msun)^α
    lifetime ∝ M/L ∝ M^-2


    The solution is given as:

    For a 20Msun star

    Lifetime = 20^-2 x 1 x 10^10 yrs
    = 25 x 10^6 yrs

    Could anyone explain to me how this solution has been arrived at?

    Thank you.
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    I should add:

    L stands for luminosity

    L* the luminosityof the star in question. Lsun that of the sun.

    'M' I believe stands for the mass.
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    Since lifetime is proportional to M/L and the lifetime of the sun is 10^10 years we can write:

     10^{10} = A\frac{M_{sun}}{L_{sun}}

    Use this to get an expression for the constant of proportionality A. Now we know the lifetime of the star is given by an expression of the same form, with the same constant A.

    Using M* = 20Msun, you should get:


    lifetime* = 10^{10}\times20\frac{L_{sun}}{L*  }


    Which gives you the desired solution if you use:

    L*/Lsun = (M*/Msun)^3 to make a substitution and plug numbers in.
 
 
 
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