The mass luminosity relationship relates a star’s luminosity to its mass, with main sequence stars having a mean relationship of L = M3.5. The Hertzsprung-Russell diagram plots a star’s brightness against its surface temperature, with most stars falling into the main sequence. Arthur Eddington used the HRD to develop the mass luminosity relationship, which was later verified by measuring nearby binary stars. The relationship can also be used to find the distance between stars using an iterative technique.
The mass luminosity relationship is an astrophysical law that relates the luminosity, or luminosity, of a star to its mass. For main sequence stars, the mean relationship is given by L = M3.5, where L is the luminosity in units of solar luminosity and M is the mass of the star measured in solar masses. Main sequence stars account for about 90% of known stars. A small increase in mass results in a large increase in a star’s luminosity.
A Hertzsprung-Russell diagram (HRD) is a graph in which the brightness of a star is plotted against its surface temperature. The vast majority of known stars fall into a range 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, HRD has provided theoretical clues for deriving a star’s thermodynamic properties.
The British astrophysicist Arthur Eddington based his development of the mass luminosity relationship on the HRD. His approach considered stars as if they were composed of an ideal gas, a theoretical construct that simplifies the calculation. A star was also considered a black body, or a perfect emitter of radiation. Using the Stefan-Boltzmann law, one can estimate the luminosity of a star with respect to its surface area and thus its volume.
In hydrostatic equilibrium, the 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 objects of equal mass, such as a star composed of an ideal gas, the virial theorem gives 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 relationship was independently verified by the measurement of nearby binary stars. The mass of stars can be determined by examining their orbits and their distance established by Kepler’s laws. Once their distance and apparent brightness are known, the brightness can be calculated.
The mass luminosity relationship can be used to find the distance of tracks that are too far away for optical measurement. An iterative technique is applied in which a mass approximation in Kepler’s laws is used to obtain a distance between stars. The arc that the bodies subtend in the sky and the approximate distance separating the two give an initial value for their distance from the earth. From this value and from their apparent magnitude, their luminosity can be determined and, by means of the mass-luminosity ratio, their masses. The mass value is then used to recalculate the distance between the stars and the process is repeated until the desired accuracy is achieved
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