Caption: A logarithmic plot of the mass-luminosity relation for main-sequence stars. The line in the plot is NOT quantitatively accurate for any stellar mass range, but can be considered as a fiducial single-power mass-luminosity relation: the coefficient C is 1 and the power is 3.5 (see below for the meanings of coefficients and powers).

Features:

1. The mass-luminosity relation for a stellar mass range is approximately a power-law relation:

     L/L_☉ ≅ C*(M/M_☉)p  ,  

   where the Sun symbol ☉ indicates solar values, L is luminosity, M is stellar mass, C is a coefficient, and p is a power.

   Note, on a logarithmic plot a power-law relation becomes straight line with slope equal to the power.

   The coefficients C and powers p for the stellar mass ranges are given in below Table: Stellar Mass-Luminosity Relation Parameters.

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   Table:  Stellar Mass-Luminosity Relation
   Parameters
   
   coefficient C  power p  Mass Range (M_☉)    Comment
   
       0.23         2.3       0.08--0.43  
       1            4         0.43--2           If M_2 = 2M_1, then L_2 = 16L_1.
       1.4          3.5       2--55
    3200            1         55--∞
       1            3.5       Fiducial and plot 
   

   Reference: Wikipedia: Mass-luminosity relation.

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2. An important consequence of the mass-luminosity relation is that main-sequence stellar lifetime decreases rapidly with stellar mass even though the amount of hydrogen fuel increases with stellar mass. The more massive the star, the more efficient its hydrogen burning and this more than negates having more hydrogen fuel to use.

3. To be quantitative, we can assume that the amount of hydrogen fuel is proportional to stellar mass M. In which case:

          t_lifetime ∝∼  M/L ∝∼ M**(-2.5) = 1/M**2.5 

   assuming the fiducial mass-luminosity relation L/L_☉ ≅ (M/M_☉)**3.5.

   From this formula, if stellar mass increases by 10, then main-sequence stellar lifetime decreases by 10**2.5 ≅ 300. By this result, a star of 10 M_☉ would have a main-sequence stellar lifetime of 1/300 that of the Sun. Thus, its main-sequence stellar lifetime would be 10 Gyr/300 = 10**4 Myr/300 = 30 Myr which is roughly correct (see Star file: star_lifetimes.html).