Mass–luminosity relation

In astrophysics, the mass–luminosity relation is an equation giving the relationship between a star's mass and its luminosity, first noted by Jakob Karl Ernst Halm. The relationship is represented by the equation:

{\frac {L}{L_{\odot }}}=\left({\frac {M}{M_{\odot }}}\right)^{a}

where L_⊙ and M_⊙ are the luminosity and mass of the Sun and $1 < a < 6$ . The value a = 3.5 is commonly used for main-sequence stars. This equation and the usual value of a = 3.5 only applies to main-sequence stars with masses $2 M ⊙ < M < 55 M ⊙$ and does not apply to red giants or white dwarfs. As a star approaches the Eddington luminosity then a = 1.

In summary, the relations for stars with different ranges of mass are, to a good approximation, as the following:

{\begin{aligned}{\frac {L}{L_{\odot }}}&\approx 0.23\left({\frac {M}{M_{\odot }}}\right)^{2.3}&(M<0.43M_{\odot })\\{\frac {L}{L_{\odot }}}&=\left({\frac {M}{M_{\odot }}}\right)^{4}&(0.43M_{\odot }<M<2M_{\odot })\\{\frac {L}{L_{\odot }}}&\approx 1.4\left({\frac {M}{M_{\odot }}}\right)^{3.5}&(2M_{\odot }<M<55M_{\odot })\\{\frac {L}{L_{\odot }}}&\approx 32000{\frac {M}{M_{\odot }}}&(M>55M_{\odot })\end{aligned}}

For stars with masses less than 0.43M_⊙, convection is the sole energy transport process, so the relation changes significantly. For stars with masses M > 55M_⊙ the relationship flattens out and becomes L ∝ M but in fact those stars don't last because they are unstable and quickly lose matter by intense solar winds. It can be shown this change is due to an increase in radiation pressure in massive stars. These equations are determined empirically by determining the mass of stars in binary systems to which the distance is known via standard parallax measurements or other techniques. After enough stars are plotted, stars will form a line on a logarithmic plot and slope of the line gives the proper value of a.

Another form, valid for K-type main-sequence stars, that avoids the discontinuity in the exponent has been given by Cuntz & Wang; it reads:

{\frac {L}{L_{\odot }}}\approx \left({\frac {M}{M_{\odot }}}\right)^{a(M)}\qquad \qquad (0.20M_{\odot }<M<0.85M_{\odot })

with

a(M)=-141.7\cdot M^{4}+232.4\cdot M^{3}-129.1\cdot M^{2}+33.29\cdot M+0.215

(M in M_⊙). This relation is based on data by Mann and collaborators, who used moderate-resolution spectra of nearby late-K and M dwarfs with known parallaxes and interferometrically determined radii to refine their effective temperatures and luminosities. Those stars have also been used as a calibration sample for Kepler candidate objects. Besides avoiding the discontinuity in the exponent at M = 0.43M_⊙, the relation also recovers a = 4.0 for M ≃ 0.85M_⊙.

The mass/luminosity relation is important because it can be used to find the distance to binary systems which are too far for normal parallax measurements, using a technique called "dynamical parallax". In this technique, the masses of the two stars in a binary system are estimated, usually in terms of the mass of the Sun. Then, using Kepler's laws of celestial mechanics, the distance between the stars is calculated. Once this distance is found, the distance away can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the apparent magnitudes of both stars, the luminosities can be found, and by using the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies as high as 5% can be achieved. The mass/luminosity relationship can also be used to determine the lifetime of stars by noting that lifetime is approximately proportional to M/L although one finds that more massive stars have shorter lifetimes than that which the M/L relationship predicts. A more sophisticated calculation factors in a star's loss of mass over time.

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