# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Pulsar source models."""
import numpy as np
from astropy.units import Quantity
__all__ = ["Pulsar", "SimplePulsar"]
DEFAULT_I = Quantity(1e45, "g cm2")
"""Pulsar default moment of inertia"""
DEFAULT_R = Quantity(1e6, "cm")
"""Pulsar default radius of the neutron star"""
B_CONST = Quantity(3.2e19, "gauss s^(-1/2)")
"""Pulsar default magnetic field constant"""
[docs]class SimplePulsar:
"""Magnetic dipole spin-down model for a pulsar.
Reference: http://www.cv.nrao.edu/course/astr534/Pulsars.html
Parameters
----------
P : `~astropy.units.Quantity`
Rotation period (sec)
P_dot : `~astropy.units.Quantity`
Rotation period derivative (sec sec^-1)
I : `~astropy.units.Quantity`
Moment of inertia (g cm^2)
R : `~astropy.units.Quantity`
Radius of the pulsar (cm)
"""
def __init__(self, P, P_dot, I=DEFAULT_I, R=DEFAULT_R):
self.P = Quantity(P, "s")
self.P_dot = P_dot
self.I = I
self.R = R
@property
def luminosity_spindown(self):
r"""Spin-down luminosity (`~astropy.units.Quantity`).
.. math:: \dot{L} = 4\pi^2 I \frac{\dot{P}}{P^{3}}
"""
return 4 * np.pi ** 2 * self.I * self.P_dot / self.P ** 3
@property
def tau(self):
r"""Characteristic age (`~astropy.units.Quantity`).
.. math:: \tau = \frac{P}{2\dot{P}}
"""
return (self.P / (2 * self.P_dot)).to("yr")
@property
def magnetic_field(self):
r"""Magnetic field strength at the polar cap (`~astropy.units.Quantity`).
.. math:: B = 3.2 \cdot 10^{19} (P\dot{P})^{1/2} \text{ Gauss}
"""
return B_CONST * np.sqrt(self.P * self.P_dot)
[docs]class Pulsar(SimplePulsar):
"""Magnetic dipole spin-down pulsar model.
Reference: http://www.cv.nrao.edu/course/astr534/Pulsars.html
Parameters
----------
P_0 : float
Period at birth
B : `~astropy.units.Quantity`
Magnetic field strength at the poles (Gauss)
n : float
Spin-down braking index
I : float
Moment of inertia
R : float
Radius
"""
def __init__(
self, P_0="0.1 s", B="1e10 G", n=3, I=DEFAULT_I, R=DEFAULT_R, age=None, L_0=None
):
P_0 = Quantity(P_0, "s")
B = Quantity(B, "G")
self.I = I
self.R = R
self.P_0 = P_0
self.B = B
self.P_dot_0 = (B / B_CONST) ** 2 / P_0
self.tau_0 = P_0 / (2 * self.P_dot_0)
self.n = float(n)
self.beta = -(n + 1.0) / (n - 1.0)
if age is not None:
self.age = Quantity(age, "yr")
if L_0 is None:
self.L_0 = 4 * np.pi ** 2 * self.I * self.P_dot_0 / self.P_0 ** 3
[docs] def luminosity_spindown(self, t):
r"""Spin down luminosity.
.. math::
\dot{L}(t) = \dot{L}_0 \left(1 + \frac{t}{\tau_0}\right)^{-\frac{n + 1}{n - 1}}
Parameters
----------
t : `~astropy.units.Quantity`
Time after birth of the pulsar
"""
t = Quantity(t, "yr")
return self.L_0 * (1 + (t / self.tau_0)) ** self.beta
[docs] def energy_integrated(self, t):
r"""Total energy released by a given time.
Time-integrated spin-down luminosity since birth.
.. math:: E(t) = \dot{L}_0 \tau_0 \frac{t}{t + \tau_0}
Parameters
----------
t : `~astropy.units.Quantity`
Time after birth of the pulsar.
"""
t = Quantity(t, "yr")
return self.L_0 * self.tau_0 * (t / (t + self.tau_0))
[docs] def period(self, t):
r"""Rotation period.
.. math::
P(t) = P_0 \left(1 + \frac{t}{\tau_0}\right)^{\frac{1}{n - 1}}
Parameters
----------
t : `~astropy.units.Quantity`
Time after birth of the pulsar
"""
t = Quantity(t, "yr")
return self.P_0 * (1 + (t / self.tau_0)) ** (1.0 / (self.n - 1))
[docs] def period_dot(self, t):
r"""Period derivative at age t.
P_dot for a given period and magnetic field B, assuming a dipole
spin-down.
.. math:: \dot{P}(t) = \frac{B^2}{3.2 \cdot 10^{19} P(t)}
Parameters
----------
t : `~astropy.units.Quantity`
Time after birth of the pulsar.
"""
t = Quantity(t, "yr")
return self.B ** 2 / (self.period(t) * B_CONST ** 2)
[docs] def tau(self, t):
r"""Characteristic age at real age t.
.. math:: \tau = \frac{P}{2\dot{P}}
Parameters
----------
t : `~astropy.units.Quantity`
Time after birth of the pulsar.
"""
t = Quantity(t, "yr")
return self.period(t) / 2 * self.period_dot(t)
[docs] def magnetic_field(self, t):
r"""Magnetic field at polar cap (assumed constant).
.. math::
B = 3.2 \cdot 10^{19} (P\dot{P})^{1/2} \text{ Gauss}
Parameters
----------
t : `~astropy.units.Quantity`
Time after birth of the pulsar.
"""
t = Quantity(t, "yr")
return B_CONST * np.sqrt(self.period(t) * self.period_dot(t))