jesterTOV.tov module

jesterTOV.tov module#

The TOV (Tolman-Oppenheimer-Volkoff) module solves the stellar structure equations for neutron stars.

Tolman-Oppenheimer-Volkoff (TOV) equation solver for neutron star structure.

This module provides functions to solve the TOV equations, which describe the hydrostatic equilibrium of a spherically symmetric, static star in General Relativity. The implementation follows the approach used in the NMMA code.

Units: All calculations are performed in geometric units where \(G = c = 1\).

Reference: NMMA code nuclear-multimessenger-astronomy/nmma

calc_k2(R, M, H, b)[source]#

Calculate the second Love number k₂ for tidal deformability.

The Love number k₂ relates the tidal deformability to the neutron star’s mass and radius. It is computed from the auxiliary variables H and β obtained from the TOV integration.

The tidal deformability is given by:

\[\Lambda = \frac{2}{3} k_2 C^{-5}\]

where \(C = M/R\) is the compactness.

Parameters:

  • R (float) – Neutron star radius [geometric units].

  • M (float) – Neutron star mass [geometric units].

  • H (float) – Auxiliary tidal variable at surface.

  • b (float) – Auxiliary tidal variable β at surface.

Returns:

float – Second Love number k₂.

tov_ode(h, y, eos)[source]#

TOV ordinary differential equation system.

This function defines the coupled ODE system for the TOV equations plus tidal deformability. The system is solved in terms of the enthalpy h as the independent variable (decreasing from center to surface).

The TOV equations are:

\[\begin{split}\frac{dr}{dh} &= -\frac{r(r-2m)}{m + 4\pi r^3 p} \\ \frac{dm}{dh} &= 4\pi r^2 \varepsilon \frac{dr}{dh} \\ \frac{dH}{dh} &= \beta \frac{dr}{dh} \\ \frac{d\beta}{dh} &= -(C_0 H + C_1 \beta) \frac{dr}{dh}\end{split}\]

where H and \(\beta\) are auxiliary variables for tidal deformability.

Parameters:

  • h (float) – Enthalpy (independent variable).

  • y (tuple) – State vector (r, m, H, β).

  • eos (dict) – EOS interpolation data.

Returns:

tuple – Derivatives (dr/dh, dm/dh, dH/dh, dβ/dh).

tov_solver(eos, pc)[source]#

Solve the TOV equations for a given central pressure.

This function integrates the TOV equations from the center of the star (where r=0, m=0) outward to the surface (where p=0), using the enthalpy as the integration variable. The integration starts slightly off-center to avoid singularities.

The solver uses the Dormand-Prince 5th order adaptive method (Dopri5) with proper error control for numerical stability.

Parameters:

  • eos (dict) –

    EOS interpolation data containing:

    • p: Pressure array [geometric units]

    • h: Enthalpy array [geometric units]

    • e: Energy density array [geometric units]

    • dloge_dlogp: Logarithmic derivative array

  • pc (float) – Central pressure [geometric units].

Returns:

tuple –

A tuple containing:

  • M: Gravitational mass [geometric units]

  • R: Circumferential radius [geometric units]

  • k2: Second Love number for tidal deformability

Note

The integration is performed from center to surface, with the enthalpy decreasing from h_center to 0. Initial conditions are set using series expansions valid near the center.

Mathematical Background#

The TOV equations describe the structure of a spherically symmetric, static star in general relativity:

\[\frac{dP}{dr} = -\frac{G(\varepsilon + P)(M + 4\pi r^3 P)}{r(r - 2GM)}\]
\[\frac{dM}{dr} = 4\pi r^2 \varepsilon\]

where: - \(P(r)\) is the pressure at radius \(r\) - \(\varepsilon(r)\) is the energy density at radius \(r\) - \(M(r)\) is the mass enclosed within radius \(r\) - \(G\) is the gravitational constant (set to 1 in geometric units)

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