Public Types | Public Member Functions | Public Attributes | Protected Member Functions | Protected Attributes | List of all members
o2scl::tov_love Class Reference

Determination of the neutron star Love number. More...

#include <tov_love.h>

Public Types

typedef boost::numeric::ublas::vector< double > ubvector
 

Public Member Functions

void set_ODE (o2scl::ode_iv_solve<> &ois_new)
 Set ODE integrator.
 
void calc_y (double &yR, double &beta, double &k2, double &lambda_km5, double &lambda_cgs, bool tabulate=false)
 Compute the love number using y.
 
void calc_H (double &yR, double &beta, double &k2, double &lambda_km5, double &lambda_cgs)
 Compute the love number using H.
 

Public Attributes

o2scl::table_units results
 A table containing the solution to the differential equation(s)
 
double eps
 The first radial point in $ \mathrm{km} $ (default 0.02)
 
std::shared_ptr< o2scl::table_units<> > tab
 Pointer to the input profile.
 

Protected Member Functions

int y_derivs (double r, size_t nv, const ubvector &vals, ubvector &ders)
 The derivative $ y^{\prime}(r) $.
 
int H_derivs (double r, size_t nv, const ubvector &vals, ubvector &ders)
 The derivatives $ H^{\prime \prime}(r) $ and $ H^{\prime}(r) $.
 
double eval_k2 (double beta, double yR)
 Compute $ k_2(\beta,y_R) $ using the analytic expression. More...
 

Protected Attributes

o2scl::ode_iv_solve def_ois
 The default ODE integrator.
 
o2scl::ode_iv_solveoisp
 The ODE integrator.
 
double schwarz_km
 Schwarzchild radius in km (set in constructor)
 

Detailed Description

We use $ c=1 $ but keep factors of $ G $, which has units $ \mathrm{km}/\mathrm{M_{\odot}} $.

Following the notation in Postnikov10, define the function $ H(r) $, which is the solution of

\[ H^{\prime \prime} (r) + H^{\prime}(r) \left\{ \frac{2}{r} + e^{\lambda(r)} \left[ \frac{2 G m(r)}{r^2} + 4 \pi G r P(r) - 4 \pi G r \varepsilon(r) \right] \right\} + H(r) Q(r) = 0 \]

where (now surpressing the dependence on $ r $),

\[ \nu^{\prime} \equiv 2 G e^{\lambda} \left(\frac{m+4 \pi P r^3}{r^2}\right) \, \]

which has units of $ 1/\mathrm{km} $ ,

\[ e^{\lambda} \equiv \left(1-\frac{2 G m}{r}\right)^{-1} \, , \]

and

\[ Q \equiv 4 \pi G e^{\lambda} \left( 5 \varepsilon + 9 P + \frac{\varepsilon+P}{c_s^2}\right) - 6 \frac{e^{\lambda}}{r^2} - \nu^{\prime 2} \]

which has units of $ 1/\mathrm{km}^2 $ . The boundary conditions on $ H(r) $ are that $ H(r) = a_0 r^2 $ and $ H^{\prime} = 2 a_0 r $ for an arbitrary constant $ a_0 $ ( $ a_0 $ is chosen to be equal to 1). Internally, $ P $ and $ \varepsilon $ are stored in units of $ \mathrm{M}_{\odot}/\mathrm{km}^3 $ .

From this we can define another (unitless) function $ y(r) \equiv r H^{\prime}(r)/H(r) $, which obeys

\[ r y^{\prime} + y^2 + y e^{\lambda} \left[ 1+4 \pi G r^2 \left( P-\varepsilon \right) \right] + r^2 Q = 0 \]

with boundary condition is $ y(0) = 2 $ . Solving for $ y^{\prime} $,

\[ y^{\prime} = \frac{1}{r} \left\{-r^2 Q - y e^{\lambda} \left[ 1+ 4 \pi G r^2 \left( P - \varepsilon \right) \right] -y^2 \right\} \]

Define $ y_R = y(r=R) $. This form for $ y^{\prime}(r) $ is specified in y_derivs() .

The unitless quantity $ k_2[\beta,y_R] $ (the Love number) is defined by (this is the expression from Postnikov10 )

\begin{eqnarray*} k_2[\beta,y(r=R)] &\equiv& \frac{8}{5} \beta^5 \left(1-2 \beta\right)^2 \left[ 2 - y_R + 2 \beta \left( y_R - 1 \right) \right] \nonumber \\ && \times \left\{ 2 \beta \left( 6 - 3 y_R + 3 \beta ( 5 y_R - 8) + 2 \beta^2 \left[ 13 - 11 y_R + \beta (3 y_R-2) \right.\right.\right. \nonumber \\ && + \left.\left.\left. 2 \beta^2 (1+y_R) \right] \right) + 3 (1-2 \beta)^2 \left[ 2 - y_R + 2 \beta (y_R - 1) \right] \log (1-2 \beta) \right\}^{-1} \end{eqnarray*}

Hinderer10 writes the differential equation for $ H(r) $ in a slightly different (but equivalent) form,

\[ H^{\prime \prime}(r) = 2 \left( 1 - \frac{2 G m}{r}\right)^{-1} H(r) \left\{ - 2 \pi G \left[ 5 \varepsilon + 9 P + \frac{\left( \varepsilon+P\right)}{c_s^2} \right] + \frac{3}{r^2} + 2 \left( 1 - \frac{2 G m}{r}\right)^{-1} \left(\frac{G m}{r^2}+4 \pi G r P\right)^2 \right\} +\frac{2 H^{\prime}(r)}{r} \left( 1 - \frac{2 G m}{r}\right)^{-1} \left[ -1+\frac{G m}{r} + 2 \pi G r^2 \left(\varepsilon-P\right) \right] \, . \]

This is the form given in H_derivs() .

The tidal deformability is then

\[ \lambda \equiv \frac{2}{3} k_2 R^5 \]

and has units of $ \mathrm{km}^5 $ or can be converted to $ \mathrm{g}~\mathrm{cm}^2~\mathrm{s}^2 $ .

It is assumed that tab has been specified before-hand and has (at least) the following columns

Idea for Future:
Use o2scl::ode_iv_solve instead of several steps of type o2scl::astep_gsl .

Definition at line 140 of file tov_love.h.

Member Function Documentation

◆ eval_k2()

double o2scl::tov_love::eval_k2 ( double  beta,
double  yR 
)
protected

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

Documentation generated with Doxygen. Provided under the GNU Free Documentation License (see License Information).