The Buchdahl EOS for the TOV solver. More...

#include <eos_tov.h>

Inheritance diagram for o2scl::eos_tov_buchdahl:

Detailed Description

The Buchdahl EOS is

$\varepsilon = 12 \sqrt{P_{*} P}- 5 P$

which can be inverted to give

$P = - \frac{\varepsilon}{5} + \frac{72 P^{*}}{25} \left[1+\sqrt{1-\frac{5 \varepsilon}{36 P^{*}}} \right]$

Physical solutions are obtained only for $P< 25 P_{*}/144$ (ensuring that the argument to the square root is positive) and $\beta=G M/R<1/6$ (ensuring that the EOS is not acausal).

The baryon chemical potential is

$\mu = \mu_1 \left(\sqrt{P_1}-3\sqrt{P^{*}}\right)^{1/2} \left(\sqrt{P}-3\sqrt{P^{*}}\right)^{-1/2}$

The baryon density is

$n_B = n_{B,1} + 12 \frac{\sqrt{P^{*} P}}{\mu_1} \left(1 - \frac{\sqrt{P}}{3 \sqrt{P^{*}}} \right)^{3/2} \left(1 - \frac{\sqrt{P_1}}{3 \sqrt{P^{*}}}\right)^{-1/2}$

In the case that one assumes $\mu_1 = m_n$ and $ P_1 = 0 $ , the baryon density can be simplified to

$n m_n = 12 \sqrt{P p_{*}} \left( 1-\frac{1}{3} \sqrt{P/p_{*}} \right)^{3/2}$

c.f. Eq. 10 in Lattimer01.

The mass-radius curve is the solution of the equation

$R=\pi (1-\beta)(1-2 \beta)^{-1} A^{-1}$

where $\beta = GM/R$ and $ A $ (which has units of inverse km) is defined by

$A^2 = \frac{288 \pi G P^{*}}{1-2 \beta}$

The central pressure and energy density are

$P_c = 36 p_{*} \beta^2$

${\varepsilon}_c = 72 p_{*} \beta (1-5 \beta/2)$

To obtain energy density and pressure profiles can be obtained by solving

$r=r^{\prime} \left(\frac{1-\beta+u}{1-2 \beta}\right)$

for the new coordinate $r^{\prime}$ where $ u $ is defined by

$u = \beta \frac{\sin(A r^{\prime})}{A r^{\prime}}$

Using these, the profiles are

$P(r) = A^2 (1- 2 \beta) u^2 \left[ 8 \pi \left(1 - \beta+u\right)^2\right]^{-1}$

and

$\varepsilon(r) = 2 A^2 (1- 2 \beta) u \left( 1 - \beta + 3 u/2\right) \left[ 8 \pi \left(1 - \beta+u\right)^2\right]^{-1}$

Based on Lattimer01 .

Definition at line 194 of file eos_tov.h.

Public Member Functions
void	set_baryon_density (double nb, double ed)
	Set the baryon density.

virtual double	ed_from_pr (double pr)
	From the pressure, return the energy density.

virtual double	pr_from_ed (double ed)
	From the energy density, return the pressure.

virtual double	nb_from_ed (double ed)
	From the energy density, return the baryon density.

virtual double	nb_from_pr (double pr)
	From the pressure, return the baryon density.

virtual double	ed_from_nb (double nb)
	From the baryon density, return the energy density.

virtual double	pr_from_nb (double nb)
	From the baryon density, return the pressure.

virtual void	ed_nb_from_pr (double pr, double &ed, double &nb)
	Given the pressure, produce the energy and number densities. More...

virtual double	rad_from_gm (double gm)
	Given the gravitational mass, compute the radius.

virtual double	ed_from_r_gm (double r, double gm)
	Compute the energy density at radius `r` for a fixed gravitational mass. More...

virtual double	pr_from_r_gm (double r, double gm)
	Compute the pressure at radius `r` for a fixed gravitational mass. More...

Public Member Functions inherited from o2scl::eos_tov
bool	has_baryons ()
	Return true if a baryon density is available.

void	check_nb (double &avg_abs_dev, double &max_abs_dev)
	Check that the baryon density is consistent with the $P(\varepsilon)$ .

Public Attributes
double	Pstar
	The parameter with units of pressure in units of solar masses per km cubed (default value $3.2 \times 10^{-5}$ )

Public Attributes inherited from o2scl::eos_tov
int	verbose
	Control for output (default 1)

Protected Member Functions
virtual double	solve_rad (double rad, double gm)
	Solve for the radius at fixed gravitational mass.

virtual double	solve_rp (double rp, double r, double gm, double rad)
	Solve for $r^{\prime}$ as a function of at fixed gravitational mass.

Protected Attributes
double	nb1
	The baryon density at `ed1`.

double	ed1
	The energy density for which the baryon density is known.

double	pr1
	The pressure at `ed1`.

root_brent_gsl	rbg
	Solver.

Protected Attributes inherited from o2scl::eos_tov
bool	baryon_column
	Set to true if the baryon density is provided in the EOS (default false)

Member Function Documentation

◆ ed_from_r_gm()

virtual double o2scl::eos_tov_buchdahl::ed_from_r_gm	(	double	r,
		double	gm
	)

virtual

Idea for Future:: This function is inefficient because it solves for the radius each time. Improve this.

◆ ed_nb_from_pr()

virtual void o2scl::eos_tov_buchdahl::ed_nb_from_pr	(	double	pr,
		double &	ed,
		double &	nb
	)

virtual

If the baryon density is not specified, it should be set to zero or baryon_column should be set to false

Implements o2scl::eos_tov.

◆ pr_from_r_gm()

virtual double o2scl::eos_tov_buchdahl::pr_from_r_gm	(	double	r,
		double	gm
	)

virtual

Idea for Future:: This function is inefficient because it solves for the radius each time. Improve this.

The documentation for this class was generated from the following file:

/home/awsteiner/wcs/o2scl/src/eos/eos_tov.h

Detailed Description

Public Member Functions

Public Attributes

Protected Member Functions

Protected Attributes

Member Function Documentation

◆ ed_from_r_gm()

◆ ed_nb_from_pr()

◆ pr_from_r_gm()