# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Galactic radial source distribution probability density functions."""
from __future__ import absolute_import, division, print_function, unicode_literals
from collections import OrderedDict
import numpy as np
from numpy import exp, pi, log, abs, cos, sin
from astropy.units import Quantity
from astropy.modeling import Fittable1DModel, Parameter
from ...utils.coordinates import cartesian, polar, D_SUN_TO_GALACTIC_CENTER
from ...utils.random import get_random_state
__all__ = [
'CaseBattacharya1998',
'FaucherKaspi2006',
'Lorimer2006',
'Paczynski1990',
'YusifovKucuk2004',
'YusifovKucuk2004B',
'Exponential',
'LogSpiral',
'FaucherSpiral',
'ValleeSpiral',
'radial_distributions',
]
# Simulation range used for random number drawing
RMIN, RMAX = Quantity([0, 20], 'kpc')
ZMIN, ZMAX = Quantity([-0.5, 0.5], 'kpc')
[docs]class Paczynski1990(Fittable1DModel):
"""Radial distribution of the birth surface density of neutron stars - Paczynski 1990.
.. math ::
f(r) = A r_{exp}^{-2} \\exp \\left(-\\frac{r}{r_{exp}} \\right)
Reference: http://adsabs.harvard.edu/abs/1990ApJ...348..485P (Formula (2))
Parameters
----------
amplitude : float
See formula
r_exp : float
See formula
See Also
--------
CaseBattacharya1998, YusifovKucuk2004, Lorimer2006, YusifovKucuk2004B,
FaucherKaspi2006, Exponential
"""
amplitude = Parameter()
r_exp = Parameter()
evolved = False
def __init__(self, amplitude=1, r_exp=4.5, **kwargs):
super(Paczynski1990, self).__init__(amplitude=amplitude,
r_exp=r_exp, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, r_exp):
"""One dimensional Paczynski 1990 model function"""
return amplitude * r_exp ** -2 * np.exp(-r / r_exp)
[docs]class CaseBattacharya1998(Fittable1DModel):
"""Radial distribution of the surface density of supernova remnants in the galaxy - Case & Battacharya 1998.
.. math ::
f(r) = A \\left( \\frac{r}{r_{\\odot}} \\right) ^ \\alpha \\exp
\\left[ -\\beta \\left( \\frac{ r - r_{\\odot}}{r_{\\odot}} \\right) \\right]
Reference: http://adsabs.harvard.edu//abs/1998ApJ...504..761C (Formula (14))
Parameters
----------
amplitude : float
See model formula
alpha : float
See model formula
beta : float
See model formula
See Also
--------
Paczynski1990, YusifovKucuk2004, Lorimer2006, YusifovKucuk2004B,
FaucherKaspi2006, Exponential
"""
amplitude = Parameter()
alpha = Parameter()
beta = Parameter()
evolved = True
def __init__(self, amplitude=1., alpha=2, beta=3.53, **kwargs):
super(CaseBattacharya1998, self).__init__(amplitude=amplitude,
alpha=alpha, beta=beta, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, alpha, beta):
"""One dimensional Case & Battacharya 2006 model function"""
term1 = (r / D_SUN_TO_GALACTIC_CENTER.value) ** alpha
term2 = np.exp(-beta * (r - D_SUN_TO_GALACTIC_CENTER.value) / D_SUN_TO_GALACTIC_CENTER.value)
return amplitude * term1 * term2
[docs]class YusifovKucuk2004(Fittable1DModel):
"""Radial distribution of the surface density of pulsars in the galaxy - Yusifov & Kucuk 2004.
.. math ::
f(r) = A \\left ( \\frac{r + r_1}{r_{\\odot} + r_1} \\right )^a \\exp
\\left [-b \\left( \\frac{r - r_{\\odot}}{r_{\\odot} + r_1} \\right ) \\right ]
Used by Faucher-Guigere and Kaspi. Density at ``r = 0`` is nonzero.
Reference: http://adsabs.harvard.edu/abs/2004A%26A...422..545Y (Formula (15))
Parameters
----------
amplitude : float
See model formula
a : float
See model formula
b : float
See model formula
r_1 : float
See model formula
See Also
--------
CaseBattacharya1998, Paczynski1990, Lorimer2006, YusifovKucuk2004B,
FaucherKaspi2006, Exponential
"""
amplitude = Parameter()
a = Parameter()
b = Parameter()
r_1 = Parameter()
evolved = True
def __init__(self, amplitude=1, a=1.64, b=4.01, r_1=0.55, **kwargs):
super(YusifovKucuk2004, self).__init__(amplitude=amplitude,
a=a, b=b, r_1=r_1, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, a, b, r_1):
"""One dimensional Yusifov & Kucuk 2004 model function"""
term1 = ((r + r_1) / (D_SUN_TO_GALACTIC_CENTER.value + r_1)) ** a
term2 = np.exp(-b * (r - D_SUN_TO_GALACTIC_CENTER.value) / (D_SUN_TO_GALACTIC_CENTER.value + r_1))
return amplitude * term1 * term2
[docs]class YusifovKucuk2004B(Fittable1DModel):
"""Radial distribution of the surface density of OB stars in the galaxy - Yusifov & Kucuk 2004.
.. math ::
f(r) = A \\left( \\frac{r}{r_{\\odot}} \\right) ^ a
\\exp \\left[ -b \\left( \\frac{r}{r_{\\odot}} \\right) \\right]
Derived empirically from OB-stars distribution.
Reference: http://adsabs.harvard.edu/abs/2004A%26A...422..545Y (Formula (17))
Parameters
----------
amplitude : float
See model formula
a : float
See model formula
b : float
See model formula
See Also
--------
CaseBattacharya1998, Paczynski1990, YusifovKucuk2004, Lorimer2006,
FaucherKaspi2006, Exponential
"""
amplitude = Parameter()
a = Parameter()
b = Parameter()
evolved = False
def __init__(self, amplitude=1, a=4, b=6.8, **kwargs):
super(YusifovKucuk2004B, self).__init__(amplitude=amplitude,
a=a, b=b, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, a, b):
"""One dimensional Yusifov & Kucuk 2004 model function"""
return amplitude * (r / D_SUN_TO_GALACTIC_CENTER.value) ** a * np.exp(-b * (r / D_SUN_TO_GALACTIC_CENTER.value))
[docs]class FaucherKaspi2006(Fittable1DModel):
"""Radial distribution of the birth surface density of pulsars in the galaxy - Faucher-Giguere & Kaspi 2006.
.. math ::
f(r) = A \\frac{1}{\\sqrt{2 \pi} \sigma} \\exp
\\left(- \\frac{(r - r_0)^2}{2 \sigma ^ 2}\\right)
Reference: http://adsabs.harvard.edu/abs/2006ApJ...643..332F (Appendix B)
Parameters
----------
amplitude : float
See model formula
r_0 : float
See model formula
sigma : float
See model formula
See Also
--------
CaseBattacharya1998, Paczynski1990, YusifovKucuk2004, Lorimer2006,
YusifovKucuk2004B, Exponential
"""
amplitude = Parameter()
r_0 = Parameter()
sigma = Parameter()
evolved = False
def __init__(self, amplitude=1, r_0=7.04, sigma=1.83, **kwargs):
super(FaucherKaspi2006, self).__init__(amplitude=amplitude,
r_0=r_0, sigma=sigma, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, r_0, sigma):
"""One dimensional Faucher-Giguere & Kaspi 2006 model function"""
term1 = 1. / np.sqrt(2 * pi * sigma)
term2 = np.exp(-(r - r_0) ** 2 / (2 * sigma ** 2))
return amplitude * term1 * term2
[docs]class Lorimer2006(Fittable1DModel):
"""Radial distribution of the surface density of pulsars in the galaxy - Lorimer 2006.
.. math ::
f(r) = A \\left( \\frac{r}{r_{\\odot}} \\right) ^ B \\exp
\\left[ -C \\left( \\frac{r - r_{\\odot}}{r_{\\odot}} \\right) \\right]
Reference: http://adsabs.harvard.edu/abs/2006MNRAS.372..777L (Formula (10))
Parameters
----------
amplitude : float
See model formula
B : float
See model formula
C : float
See model formula
See Also
--------
CaseBattacharya1998, Paczynski1990, YusifovKucuk2004, Lorimer2006,
YusifovKucuk2004B, FaucherKaspi2006
"""
amplitude = Parameter()
B = Parameter()
C = Parameter()
evolved = True
def __init__(self, amplitude=1, B=1.9, C=5.0, **kwargs):
super(Lorimer2006, self).__init__(amplitude=amplitude, B=B, C=C, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, B, C):
"""One dimensional Lorimer 2006 model function"""
term1 = (r / D_SUN_TO_GALACTIC_CENTER.value) ** B
term2 = np.exp(-C * (r - D_SUN_TO_GALACTIC_CENTER.value) / D_SUN_TO_GALACTIC_CENTER.value)
return amplitude * term1 * term2
[docs]class Exponential(Fittable1DModel):
"""Exponential distribution.
.. math ::
f(z) = A \\exp \\left(- \\frac{|z|}{z_0} \\right)
Usually used for height distribution above the Galactic plane,
with 0.05 kpc as a commonly used birth height distribution.
Parameters
----------
amplitude : float
See model formula
z_0 : float
Scale height of the distribution
See Also
--------
CaseBattacharya1998, Paczynski1990, YusifovKucuk2004, Lorimer2006,
YusifovKucuk2004B, FaucherKaspi2006, Exponential
"""
amplitude = Parameter()
z_0 = Parameter()
evolved = False
def __init__(self, amplitude=1, z_0=0.05, **kwargs):
super(Exponential, self).__init__(amplitude=amplitude, z_0=z_0, **kwargs)
[docs] @staticmethod
def evaluate(z, amplitude, z_0):
"""One dimensional exponential model function"""
return amplitude * np.exp(-np.abs(z) / z_0)
[docs]class LogSpiral(object):
"""Logarithmic spiral.
Reference: http://en.wikipedia.org/wiki/Logarithmic_spiral
"""
[docs] def xy_position(self, theta=None, radius=None, spiralarm_index=0):
"""Compute (x, y) position for a given angle or radius.
Parameters
----------
theta : array_like
Angle (deg)
radius : array_like
Radius (kpc)
spiralarm_index : int
Spiral arm index
Returns
-------
x, y : array_like
Position (x, y)
"""
if (theta is None) and not (radius is None):
theta = self.theta(radius, spiralarm_index=spiralarm_index)
elif (radius is None) and not (theta is None):
radius = self.radius(theta, spiralarm_index=spiralarm_index)
else:
ValueError('Specify only one of: theta, radius')
theta = np.radians(theta)
x = radius * cos(theta)
y = radius * sin(theta)
return x, y
[docs] def radius(self, theta, spiralarm_index):
"""Radius for a given angle.
Parameters
----------
theta : array_like
Angle (deg)
spiralarm_index : int
Spiral arm index
Returns
-------
radius : array_like
Radius (kpc)
"""
k = self.k[spiralarm_index]
r_0 = self.r_0[spiralarm_index]
theta_0 = self.theta_0[spiralarm_index]
d_theta = np.radians(theta - theta_0)
radius = r_0 * exp(d_theta / k)
return radius
[docs] def theta(self, radius, spiralarm_index):
"""Angle for a given radius.
Parameters
----------
radius : array_like
Radius (kpc)
spiralarm_index : int
Spiral arm index
Returns
-------
theta : array_like
Angle (deg)
"""
k = self.k[spiralarm_index]
r_0 = self.r_0[spiralarm_index]
theta_0 = self.theta_0[spiralarm_index]
theta_0 = np.radians(theta_0)
theta = k * log(radius / r_0) + theta_0
return np.degrees(theta)
[docs]class FaucherSpiral(LogSpiral):
"""Milky way spiral arm used in Faucher et al (2006).
Reference: http://adsabs.harvard.edu/abs/2006ApJ...643..332F
"""
# Parameters
k = Quantity([4.25, 4.25, 4.89, 4.89], 'rad')
r_0 = Quantity([3.48, 3.48, 4.9, 4.9], 'kpc')
theta_0 = Quantity([1.57, 4.71, 4.09, 0.95], 'rad')
spiralarms = np.array(['Norma', 'Carina Sagittarius', 'Perseus', 'Crux Scutum'])
def _blur(self, radius, theta, amount=0.07, random_state='random-seed'):
"""Blur the positions around the centroid of the spiralarm.
The given positions are blurred by drawing a displacement in radius from
a normal distribution, with sigma = amount * radius. And a direction
theta from a uniform distribution in the interval [0, 2 * pi].
Parameters
----------
radius : `~astropy.units.Quantity`
Radius coordinate
theta : `~astropy.units.Quantity`
Angle coordinate
amount: float, optional
Amount of blurring of the position, given as a fraction of `radius`.
random_state : {int, 'random-seed', 'global-rng', `~numpy.random.RandomState`}
Defines random number generator initialisation.
Passed to `~gammapy.utils.random.get_random_state`.
"""
random_state = get_random_state(random_state)
dr = Quantity(abs(random_state.normal(0, amount * radius, radius.size)), 'kpc')
dtheta = Quantity(random_state.uniform(0, 2 * np.pi, radius.size), 'rad')
x, y = cartesian(radius, theta)
dx, dy = cartesian(dr, dtheta)
return polar(x + dx, y + dy)
def _gc_correction(self, radius, theta, r_corr=Quantity(2.857, 'kpc'),
random_state='random-seed'):
"""Correction of source distribution towards the galactic center.
To avoid spiralarm features near the Galactic Center, the position angle theta
is blurred by a certain amount towards the GC.
Parameters
----------
radius : `~astropy.units.Quantity`
Radius coordinate
theta : `~astropy.units.Quantity`
Angle coordinate
r_corr : `~astropy.units.Quantity`, optional
Scale of the correction towards the GC
random_state : {int, 'random-seed', 'global-rng', `~numpy.random.RandomState`}
Defines random number generator initialisation.
Passed to `~gammapy.utils.random.get_random_state`.
"""
random_state = get_random_state(random_state)
theta_corr = Quantity(random_state.uniform(0, 2 * pi, radius.size), 'rad')
return radius, theta + theta_corr * np.exp(-radius / r_corr)
[docs] def __call__(self, radius, blur=True, random_state='random-seed'):
"""Draw random position from spiral arm distribution.
Returns the corresponding angle theta[rad] to a given radius[kpc] and number of spiralarm.
Possible numbers are:
* Norma = 0,
* Carina Sagittarius = 1,
* Perseus = 2
* Crux Scutum = 3.
Parameters
----------
random_state : {int, 'random-seed', 'global-rng', `~numpy.random.RandomState`}
Defines random number generator initialisation.
Passed to `~gammapy.utils.random.get_random_state`.
Returns
-------
Returns dx and dy, if blurring= true.
"""
random_state = get_random_state(random_state)
N = random_state.randint(0, 4, radius.size) # Choose Spiralarm
theta = self.k[N] * log(radius / self.r_0[N]) + self.theta_0[N] # Compute angle
spiralarm = self.spiralarms[N] # List that contains in wich spiralarm a postion lies
if blur: # Apply blurring model according to Faucher
radius, theta = self._blur(radius, theta, random_state=random_state)
radius, theta = self._gc_correction(radius, theta, random_state=random_state)
return radius, theta, spiralarm
[docs]class ValleeSpiral(LogSpiral):
"""Milky way spiral arm model from Vallee (2008).
Reference: http://adsabs.harvard.edu/abs/2008AJ....135.1301V
"""
# Model parameters
p = Quantity(12.8, 'deg') # pitch angle in deg
m = 4 # number of spiral arms
r_sun = Quantity(7.6, 'kpc') # distance sun to Galactic center in kpc
r_0 = Quantity(2.1, 'kpc') # spiral inner radius in kpc
theta_0 = Quantity(-20, 'deg') # Norma spiral arm start angle
bar_radius = Quantity(3.0, 'kpc') # Radius of the galactic bar (not equal r_0!)
spiralarms = np.array(['Norma', 'Perseus', 'Carina Sagittarius', 'Crux Scutum'])
def __init__(self):
self.r_0 = self.r_0 * np.ones(4)
self.theta_0 = self.theta_0 + Quantity([0, 90, 180, 270], 'deg')
self.k = Quantity(1. / np.tan(np.radians(self.p.value)) * np.ones(4), 'rad')
# Compute start and end point of the bar
x_0, y_0 = self.xy_position(radius=self.bar_radius, spiralarm_index=0)
x_1, y_1 = self.xy_position(radius=self.bar_radius, spiralarm_index=2)
self.bar = dict(x=Quantity([x_0, x_1]), y=Quantity([y_0, y_1]))
radial_distributions = OrderedDict()
radial_distributions.__doc__ = """Radial distribution (dict mapping names to classes)."""
radial_distributions['CB98'] = CaseBattacharya1998
radial_distributions['F06'] = FaucherKaspi2006
radial_distributions['L06'] = Lorimer2006
radial_distributions['P90'] = Paczynski1990
radial_distributions['YK04'] = YusifovKucuk2004
radial_distributions['YK04B'] = YusifovKucuk2004B