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What Is the Mass Luminosity Relation?

Jerry Morrison
Jerry Morrison

The mass luminosity relation is an astrophysical law relating a star's luminosity, or brightness, to its mass. For main sequence stars, the average relationship is given by L = M3.5, where L is the luminosity in solar luminosity units and M is the star's mass measured in solar masses. Main sequence stars account for about 90% of known stars. A small increase in mass results in a great increase in a star's luminosity.

A Hertzsprung-Russell diagram (HRD) is a graph where the luminosity of a star is plotted relative to its surface temperature. The vast majority of known stars fall into a band ranging from hot stars with high luminosity to cool stars with low luminosity. This band is referred to as the main sequence. Although developed before nuclear fusion was found to be the source of a star's energy, the HRD provided theoretical clues for deriving the thermodynamic properties of a star.

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Scientist with beakers

English astrophysicist Arthur Eddington based his development of the mass luminosity relation on the HRD. His approach considered stars as if they were composed of an ideal gas, a theoretical construct that simplifies calculation. A star was also considered to be a black body, or a perfect emitter of radiation. Using the Stefan-Boltzmann law, the luminosity of a star relative to its surface area and thus its volume can be estimated.

Under hydrostatic equilibrium, compression of a star's gas due to gravity is balanced by the internal pressure of the gas, forming a sphere. For a spherical volume of equal mass objects, such as a star composed of an ideal gas, the virial theorem provides an estimate of the total potential energy of the body. This value can be used to derive the approximate mass of a star and relate this value to its luminosity.

Eddington's theoretical approximation for the mass luminosity relation was verified independently by the measurement of nearby binary stars. The mass of the stars can be determined from an examination of their orbits, and their distance established by Kepler's laws. Once their distance and apparent brightness is known, luminosity can be calculated.

The mass luminosity relation can be used to find distance of binaries that are too far away for optical measurement. An iterative technique is applied where an approximation of mass is used in Kepler's laws to yield a distance between the stars. The arc the bodies subtend in the sky and the approximate distance separating the two yield an initial value for their distance from earth. From this value and their apparent magnitude, their luminosity can be determined and, by means of the mass luminosity relation, their masses. The value for mass is then used to recalculate the distance separating the stars and the process is repeated until the desired accuracy is achieved

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