# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Pulsar velocity distribution models."""
import numpy as np
from astropy.modeling import Fittable1DModel, Parameter
from astropy.units import Quantity
__all__ = [
"FaucherKaspi2006VelocityBimodal",
"FaucherKaspi2006VelocityMaxwellian",
"Paczynski1990Velocity",
"velocity_distributions",
]
# Simulation range used for random number drawing
VMIN, VMAX = Quantity([0, 4000], "km/s")
[docs]class FaucherKaspi2006VelocityMaxwellian(Fittable1DModel):
r"""Maxwellian pulsar velocity distribution.
.. math::
f(v) = A \sqrt{ \frac{2}{\pi}} \frac{v ^ 2}{\sigma ^ 3 }
\exp \left(-\frac{v ^ 2}{2 \sigma ^ 2} \right)
Reference: https://ui.adsabs.harvard.edu/abs/2006ApJ...643..332F
Parameters
----------
amplitude : float
Value of the integral.
sigma : float
Velocity parameter (km s^-1).
"""
amplitude = Parameter()
sigma = Parameter()
def __init__(self, amplitude=1, sigma=265, **kwargs):
super().__init__(amplitude=amplitude, sigma=sigma, **kwargs)
[docs] @staticmethod
def evaluate(v, amplitude, sigma):
"""One dimensional velocity model function."""
term1 = np.sqrt(2 / np.pi) * v**2 / sigma**3
term2 = np.exp(-(v**2) / (2 * sigma**2))
return term1 * term2
[docs]class FaucherKaspi2006VelocityBimodal(Fittable1DModel):
r"""Bimodal pulsar velocity distribution. - Faucher & Kaspi (2006).
.. math::
f(v) = A\sqrt{\frac{2}{\pi}} v^2 \left[\frac{w}{\sigma_1^3}
\exp \left(-\frac{v^2}{2\sigma_1^2} \right) + \frac{1-w}{\sigma_2^3}
\exp \left(-\frac{v^2}{2\sigma_2^2} \right) \right]
Formula (7) [FaucherKaspi2006]_.
Parameters
----------
amplitude : float
Value of the integral.
sigma1 : float
See model formula.
sigma2 : float
See model formula.
w : float
See model formula.
References
----------
.. [FaucherKaspi2006] https://ui.adsabs.harvard.edu/abs/2006ApJ...643..332F
"""
amplitude = Parameter()
sigma_1 = Parameter()
sigma_2 = Parameter()
w = Parameter()
def __init__(self, amplitude=1, sigma_1=160, sigma_2=780, w=0.9, **kwargs):
super().__init__(
amplitude=amplitude, sigma_1=sigma_1, sigma_2=sigma_2, w=w, **kwargs
)
[docs] @staticmethod
def evaluate(v, amplitude, sigma_1, sigma_2, w):
"""One dimensional Faucher-Guigere & Kaspi 2006 velocity model function."""
A = amplitude * np.sqrt(2 / np.pi) * v**2
term1 = (w / sigma_1**3) * np.exp(-(v**2) / (2 * sigma_1**2))
term2 = (1 - w) / sigma_2**3 * np.exp(-(v**2) / (2 * sigma_2**2))
return A * (term1 + term2)
[docs]class Paczynski1990Velocity(Fittable1DModel):
r"""Distribution by Lyne 1982 and adopted by Paczynski and Faucher.
.. math::
f(v) = A\frac{4}{\pi} \frac{1}{v_0 \left[1 + (v / v_0) ^ 2 \right] ^ 2}
Formula (3) [Paczynski1990]_.
Parameters
----------
amplitude : float
Value of the integral.
v_0 : float
Velocity parameter (km s^-1).
References
----------
.. [Paczynski1990] https://ui.adsabs.harvard.edu/abs/1990ApJ...348..485P
"""
amplitude = Parameter()
v_0 = Parameter()
def __init__(self, amplitude=1, v_0=560, **kwargs):
super().__init__(amplitude=amplitude, v_0=v_0, **kwargs)
[docs] @staticmethod
def evaluate(v, amplitude, v_0):
"""One dimensional Paczynski 1990 velocity model function."""
return amplitude * 4.0 / (np.pi * v_0 * (1 + (v / v_0) ** 2) ** 2)
"""Velocity distributions (dict mapping names to classes)."""
velocity_distributions = {
"H05": FaucherKaspi2006VelocityMaxwellian,
"F06B": FaucherKaspi2006VelocityBimodal,
"F06P": Paczynski1990Velocity,
}