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I am trying to model a fully relativistic white dwarf using different central densities. Using the below equations: $\frac{dP}{dm} = -\frac{Gm}{4\pi r^4}$ $\frac{dr}{dm} = -\frac{1}{4\pi\rho r^2}$ and equation of state for fully relativistic case: $P_e = (\frac{3}{\pi})^\frac{1}{3} \frac{hc}{8m_u^\frac{4}{3}} (\frac{\rho}{\mu})^\frac{4}{3}$

Using a similar approach by @Kyle on: TOV equation and the equation of state, solving numerically, and rk4 to calculate mass, pressure and radius when pressure reaches 0 value.Unfortunately pressure suddenly becomes negative instead of reaching 0 value and iteration ends abruptly. Hence I get incorrect final mass and radius. Is this due to numerical instability? If yes, how can I fix it?

ProfRob
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A.M
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1 Answers1

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The pressure becomes negative because $dP/dm$ in your last step is negative. That would seem to be inevitable!

All you can do to minimise the effect is make your step sizes smaller and smaller or at least small enough, that an error in mass smaller than $\Delta m$ does not matter.

In practice, your model is physically inaccurate in any case, so I would not worry too much about it. The outer $\sim 1$% by mass of your white dwarf will not have an equation of state that follows ideal degeneracy pressure. That is because white dwarfs are roughly isothermal in their interiors, so as you approach the surface, the ratio of the density-dependent Fermi energy to $k_BT$ ceases to be large and the gas becomes partially degenerate. This feature will not greatly affect the mass-radius relation because it only affects a very thin outer skin of a white dwarf (as long as it isn't too hot).

A more serious issue is that your model is also totally unrealistic because you have assumed the entire star is governed by relativistic degeneracy pressure. There is no equilibrium solution for hydrostatic equilibrium for a white dwarf governed by that equation of state - it is what defines the Chandrasekhar mass. The equation of state actually varies with density such that it only approaches the equation you have written at very high density. At lower densities and towards the outside of the star, there must be a lower density region where $P \propto \rho^{5/3}$ and then a partially degenerate outer layer as I explained above.

ProfRob
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