The Highly Accurate Relation between the Radius and Mass of the White Dwarf Star from Zero to Finite Temperature

In this research, first considering the electron-electron interaction in the high-density Fermi electron gas at T=0 K, this interaction causes the pressure 2/137 time less than the original value. However, the pressure of the Fermi electron gas should have something to do with temperature. Then we estimate the temperature effect using statistical mechanics and find the complicated form of the pressure P depends on temperature at the given particle number N and volume V. According to this, the central density-mass (ρc-M), central density-radius (ρc-R), and mass-radius (M-R) relations of the white dwarf star are obtained by considering the equation of state (EOS). Traditional formula gives the problematic mass-radius relation R∝M^((-1)⁄3) for the low-density white dwarf stars because it leads to R→∞ and P→0 when M→0. We correct this relation and obtain two reasonable relations in the relativistic and nonrelativistic regions. In our EOS calculations, the central density is divided into the high-, middle-, and low-density regions. All three relations are almost unchanged until 10^8 K in the high-density region. The temperature effect mainly affects the middle- and low-density regions and it becomes explicitly above 10^7 K. Our calculations can explain Sloan Digital Sky Survey observations where some white dwarf stars with a radius more than 8x10^3 km have larger mass than the predictions by the relativistic EOS at T=0 K. This result tells us that the temperature effect is important for the low and middle central-density white dwarf star and also useful to estimate the inner temperature of a white dwarf star.