def deriv(rm_vector, enthalpy, eos):
''' The TOV equations in terms of pseudoenthalpy. Derivatives of vector
(radius, mass) as a function of enthalpy for given eos.'''
(rad, mass) = rm_vector
drdenth = - rad * (rad - 2 * mass) \
/ (mass + 4 * np.pi * rad **3 * eos.pressure(enthalpy)*Gcc)
dmdenth = 4 * np.pi * eos.energy(enthalpy)*Gcc * rad **2 * drdenth
return np.array([drdenth,dmdenth])
def deriv(rm_vector, enthalpy, eos):
''' The TOV equations in terms of pseudoenthalpy. Derivatives of vector
(radius, mass) as a function of enthalpy for given eos.'''
(rad, mass) = rm_vector
drdenth = - rad * (rad - 2 * mass) \
/ (mass + 4 * np.pi * rad **3 * eos.pressure(enthalpy)*Gcc)
dmdenth = 4 * np.pi * eos.energy(enthalpy) * Gcc * rad**2 * drdenth
return np.array([drdenth, dmdenth])
def rotderiv(rm_vector, enthalpy, eos):
''' The TOV equations augmented for a linearly perturbed, slowly
rotating neutron star. Also includes rest mass of star.'''
(rad, mass, restmass, alpha, omega) = rm_vector
pr = eos.pressure(enthalpy) * Gcc
en = eos.energy(enthalpy) * Gcc
rho = eos.density(enthalpy)**Gcc
factor = rad - 2 * mass
coef = 4.0 * np.pi
drdenth = -rad * factor / (mass + coef * rad**3 * pr)
dmdenth = coef * en * rad**2 * drdenth
dmadenth = coef * rho * rad**2 * drdenth / (factor / rad)**0.5
domega = alpha * drdenth
dalpha = drdenth * (- 4.0 * alpha / rad \
+ coef * rad * (pr + en) * (4.0 * omega + rad * alpha ) / factor )
return np.array([drdenth, dmdenth, dmadenth, dalpha, domega])
def rotderiv(rm_vector, enthalpy, eos):
''' The TOV equations augmented for a linearly perturbed, slowly
rotating neutron star. Also includes rest mass of star.'''
(rad, mass, restmass, alpha, omega) = rm_vector
pr = eos.pressure(enthalpy) * Gcc
en = eos.energy(enthalpy) * Gcc
rho = eos.density(enthalpy) ** Gcc
factor = rad - 2 * mass
coef = 4.0 * np.pi
drdenth = - rad * factor / (mass + coef * rad**3 * pr)
dmdenth = coef * en * rad**2 * drdenth
dmadenth = coef * rho * rad**2 * drdenth / ( factor / rad )**0.5
domega = alpha * drdenth
dalpha = drdenth * (- 4.0 * alpha / rad \
+ coef * rad * (pr + en) * (4.0 * omega + rad * alpha ) / factor )
return np.array([drdenth,dmdenth, dmadenth, dalpha, domega])
def energy_of_mass(eos, mass, lowenergy, highenergy=None):
''' for a given eos and a stellar mass in solar masses, calculate the
central energy density in g/cm^3. Include a lower bound on the energy
to avoid convergence problems especially with crusts. About 10**14
g/cm^3 works well for most neutron stars.
Optional params:
highenergy = upper bound for search. If this is not specified, use
the limiting enthalpy range of the EOS.
'''
if highenergy == None:
highenergy = eos.energy(eos.enthrange[1])
def massdiff(x, eos, mass):
return profile(eos, x)[2,-1] - mass*Solarmass_km
try:
return optimize.brenth(massdiff, lowenergy, highenergy, (eos,mass))
except RuntimeError:
warnings.warn("Problem getting convergence in optimze.brenth")
return 0
def energy_maxmass(eos, lowenergy, highenergy=None):
''' Determine properties of the maximum mass star supported by the EOS.
Returns (central energy density in g/cm^3, mass in Solar masses, and
radius in km for the star).
Include a lower bound on the energy to
avoid convergence problems especially with crusts. About 10**14 g/cm^3
works well for most neutron stars.
Optional params:
highenergy = upper bound for search. If this is not specified, use
the limiting enthalpy range of the EOS.
'''
if highenergy == None:
highenergy = eos.energy(eos.enthrange[1])
# Calculate the maximum mass energy density
massen = lambda x: 10.0 - profile(eos, x)[2, -1]
maxen = optimize.brent(massen, brack=(lowenergy, highenergy))
# Calculate the maximum mass model
maxprofile = profile(eos, maxen)[:, -1]
return maxen, maxprofile[2] / Solarmass_km, maxprofile[1]
def energy_of_mass(eos, mass, lowenergy, highenergy=None):
''' for a given eos and a stellar mass in solar masses, calculate the
central energy density in g/cm^3. Include a lower bound on the energy
to avoid convergence problems especially with crusts. About 10**14
g/cm^3 works well for most neutron stars.
Optional params:
highenergy = upper bound for search. If this is not specified, use
the limiting enthalpy range of the EOS.
'''
if highenergy == None:
highenergy = eos.energy(eos.enthrange[1])
def massdiff(x, eos, mass):
return profile(eos, x)[2, -1] - mass * Solarmass_km
try:
return optimize.brenth(massdiff, lowenergy, highenergy, (eos, mass))
except RuntimeError:
warnings.warn("Problem getting convergence in optimze.brenth")
return 0
def energy_maxmass(eos, lowenergy, highenergy=None):
''' Determine properties of the maximum mass star supported by the EOS.
Returns (central energy density in g/cm^3, mass in Solar masses, and
radius in km for the star).
Include a lower bound on the energy to
avoid convergence problems especially with crusts. About 10**14 g/cm^3
works well for most neutron stars.
Optional params:
highenergy = upper bound for search. If this is not specified, use
the limiting enthalpy range of the EOS.
'''
if highenergy == None:
highenergy = eos.energy(eos.enthrange[1])
# Calculate the maximum mass energy density
massen = lambda x: 10.0 - profile(eos, x)[2,-1]
maxen = optimize.brent(massen, brack=(lowenergy,highenergy))
# Calculate the maximum mass model
maxprofile = profile(eos,maxen)[:,-1]
return maxen, maxprofile[2]/Solarmass_km, maxprofile[1]
def tidederiv(rm_vector, enthalpy, eos):
(rad, mass, restmass, beta, h) = rm_vector
pr = eos.pressure(enthalpy) * Gcc
en = eos.energy(enthalpy) * Gcc
func = 1.0 / eos.dprden(enthalpy)
factor = rad - 2 * mass
coef = 4.0 * np.pi
drdenth = -rad * factor / (mass + coef * rad**3 * pr)
dmdenth = coef * en * rad**2 * drdenth
dbeta = 2 * drdenth *\
( h * ( \
(- 2. * np.pi * func * rad * (pr + en))/ factor + \
(2. * mass**2 + \
rad**2 * (3.0 + 2.0 * np.pi * rad**2 * \
( pr * (16.0 * np.pi * pr * rad**2 - 9.0 ) - 5. * en ))\
+ mass * ( - 6 * rad + 4 * np.pi * rad**3 * (13 *pr + 5*en) ) )\
/ rad**2 / factor**2 )
+ beta / rad / factor * ( mass - rad + 2 * np.pi * rad**3 * (en -
pr)))
dh = beta * drdenth
return np.array([drdenth, dmdenth, dbeta, dh])
def tidederiv(rm_vector, enthalpy, eos):
(rad, mass, restmass, beta, h) = rm_vector
pr = eos.pressure(enthalpy) * Gcc
en = eos.energy(enthalpy) * Gcc
func = 1.0/eos.dprden(enthalpy)
factor = rad - 2 * mass
coef = 4.0*np.pi
drdenth = - rad * factor / (mass + coef * rad**3 * pr)
dmdenth = coef * en * rad**2 * drdenth
dbeta = 2 * drdenth *\
( h * ( \
(- 2. * np.pi * func * rad * (pr + en))/ factor + \
(2. * mass**2 + \
rad**2 * (3.0 + 2.0 * np.pi * rad**2 * \
( pr * (16.0 * np.pi * pr * rad**2 - 9.0 ) - 5. * en ))\
+ mass * ( - 6 * rad + 4 * np.pi * rad**3 * (13 *pr + 5*en) ) )\
/ rad**2 / factor**2 )
+ beta / rad / factor * ( mass - rad + 2 * np.pi * rad**3 * (en -
pr)))
dh = beta * drdenth
return np.array([drdenth,dmdenth, dbeta, dh])