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>

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 supressing 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 stores a stellar profile (such as one computed with tov_solve::fixed(), tov_solve::fixed_pr(), or tov_solve::max() ) has been specified before-hand and contains (at least) the following columns

This class handles the inner boundary by starting from the small non-zero radius stored in eps instead of at $ r=0 $. The value of eps defaults to 0.2 km.

If there is a discontinuity in the EOS (i.e. a jump in the energy density at some radius $ r_d $), then the function $ y(r) $ must satisfy (see Damour09 and Postnikov10)

\[ y(r_d+\delta) - y(r_d-\delta) = \frac{ \varepsilon(r_d+\delta)-\varepsilon(r_d-\delta)}{m(r_d)/(4 \pi r_d^3) + p} \]

Note
The function calc_H() cannot yet handle discontinuities (if there are any then the error handler is called).
This class does not yet handle strange quark stars which have an energy discontinuity at the surface (this currently causes the error handler to be called).

Definition at line 166 of file tov_love.h.

Public Types

typedef std::function< int(double, size_t, const std::vector< double > &, std::vector< double > &)> ode_funct2
 

Public Member Functions

void set_ODE (o2scl::ode_iv_solve< ode_funct2, std::vector< double > > &ois_new)
 Set ODE integrator.
 
int calc_y (double &yR, double &beta, double &k2, double &lambda_km5, double &lambda_cgs, bool tabulate=false)
 Compute the love number using y.
 
void add_disc (double rd)
 Add a discontinuity at radius rd (in km)
 
void clear_discs ()
 Remove all discontinuities.
 
int calc_H (double &yR, double &beta, double &k2, double &lambda_km5, double &lambda_cgs)
 Compute the love number using H.
 

Public Attributes

int show_ode
 If greater than zero, show the ODE output (default 0)
 
bool addl_testing
 Additional testing if the ODE solver fails.
 
bool err_nonconv
 If true, call the error handler if the solution does not converge (default true)
 
o2scl::table_units results
 A table containing the solution to the differential equation(s)
 
double delta
 The radial step for resolving discontinuities in km (default $ 10^{-4} $)
 
double eps
 The first radial point in $ \mathrm{km} $ (default 0.02)
 
o2scl::ode_iv_solve< ode_funct2, std::vector< double > > def_ois
 The default ODE integrator.
 
std::shared_ptr< o2scl::table_units<> > tab
 Pointer to the input profile.
 

Protected Member Functions

int y_derivs (double r, size_t nv, const std::vector< double > &vals, std::vector< double > &ders)
 The derivative $ y^{\prime}(r) $.
 
int H_derivs (double r, size_t nv, const std::vector< double > &vals, std::vector< double > &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< ode_funct2, std::vector< double > > * oisp
 The ODE integrator.
 
double schwarz_km
 Schwarzchild radius in km (set in constructor)
 
std::vector< double > disc
 List of discontinuities.
 

Member Function Documentation

◆ eval_k2()

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

Used in both tov_love::calc_y() and tov_love::calc_H().

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).