This quantity is sometimes labelled . The following adiabatic equation therefore holds true for the entire interior: where is the adiabatic gamma, which is 5/3 for an ideal monatomic gas. The ideal gas law says: where is the molecular weight per particle and H is (to a very good approximation) the mass of a hydrogen atom. This equation represents a polytrope of index 1.5, since a polytrope is defined by , where n=1.5 is the polytropic index. Applying the equation to the center of the star gives: . We can solve for C: But for any polytrope, and . are all constants independent of pressure and density, and the average density is defined as . Plugging this 2 equations into the equation for C, we have: where all multiplicative constants have been ignored. Recall that our original definition of C was: We therefore have, for any star of mass M and radius R: We need another relationship between P, T, M, and R, in order to eliminate P. This relationship will come from the atmosphere model. The atmosphere is assumed to be thin, with average opacity k. Opacity is defined to be optical depth divided by density. Thus, by definition, the optical depth of the stellar surface, also called the photosphere, is: where R is the stellar radius, also known as the position of the photosphere. The pressure at the surface is: The optical depth at the photosphere turns out to be . By definition, the temperature of the photosphere is where effective temperature is given by . Therefore, the pressure is: We can approximate the opacity to be: where a=1, b=3. Plugging this into the pressure equation, we get: Finally, we need to eliminate R and introduce L, the luminosity. This can be done with the equation: Equation and can now be combined by setting and in Equation 1, then eliminating . R can be eliminated using Equation . After some algebra, and after setting , we get: where In cool stellar atmospheres (T < 5000 K) like those of newborn stars, the dominant source of opacity is the H- ion, for which and , we get and . Since A is much smaller than 1, the Hayashi track is extremely steep: if the luminosity changes by a factor of 2, the temperature only changes by 4 percent. The fact that B is positive indicates that the Hayashi track