# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Galactic radial source distribution probability density functions."""
import numpy as np
from astropy.modeling import Fittable1DModel, Parameter
from astropy.units import Quantity
from gammapy.utils.coordinates import D_SUN_TO_GALACTIC_CENTER, cartesian, polar
from gammapy.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):
r"""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: https://ui.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().__init__(amplitude=amplitude, r_exp=r_exp, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, r_exp):
"""Evaluate model."""
return amplitude * r_exp ** -2 * np.exp(-r / r_exp)
[docs]class CaseBattacharya1998(Fittable1DModel):
r"""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: https://ui.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=1.0, alpha=2, beta=3.53, **kwargs):
super().__init__(amplitude=amplitude, alpha=alpha, beta=beta, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, alpha, beta):
"""Evaluate model."""
d_sun = D_SUN_TO_GALACTIC_CENTER.value
term1 = (r / d_sun) ** alpha
term2 = np.exp(-beta * (r - d_sun) / d_sun)
return amplitude * term1 * term2
[docs]class YusifovKucuk2004(Fittable1DModel):
r"""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: https://ui.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().__init__(amplitude=amplitude, a=a, b=b, r_1=r_1, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, a, b, r_1):
"""Evaluate model."""
d_sun = D_SUN_TO_GALACTIC_CENTER.value
term1 = ((r + r_1) / (d_sun + r_1)) ** a
term2 = np.exp(-b * (r - d_sun) / (d_sun + r_1))
return amplitude * term1 * term2
[docs]class YusifovKucuk2004B(Fittable1DModel):
r"""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: https://ui.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().__init__(amplitude=amplitude, a=a, b=b, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, a, b):
"""Evaluate model."""
d_sun = D_SUN_TO_GALACTIC_CENTER.value
return amplitude * (r / d_sun) ** a * np.exp(-b * (r / d_sun))
[docs]class FaucherKaspi2006(Fittable1DModel):
r"""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: https://ui.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().__init__(amplitude=amplitude, r_0=r_0, sigma=sigma, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, r_0, sigma):
"""Evaluate model."""
term1 = 1.0 / np.sqrt(2 * np.pi * sigma)
term2 = np.exp(-((r - r_0) ** 2) / (2 * sigma ** 2))
return amplitude * term1 * term2
[docs]class Lorimer2006(Fittable1DModel):
r"""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: https://ui.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().__init__(amplitude=amplitude, B=B, C=C, **kwargs)
[docs] @staticmethod
def evaluate(r, amplitude, B, C):
"""Evaluate model."""
d_sun = D_SUN_TO_GALACTIC_CENTER.value
term1 = (r / d_sun) ** B
term2 = np.exp(-C * (r - d_sun) / d_sun)
return amplitude * term1 * term2
[docs]class Exponential(Fittable1DModel):
r"""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().__init__(amplitude=amplitude, z_0=z_0, **kwargs)
[docs] @staticmethod
def evaluate(z, amplitude, z_0):
"""Evaluate model."""
return amplitude * np.exp(-np.abs(z) / z_0)
[docs]class LogSpiral:
"""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:
raise ValueError("Specify only one of: theta, radius")
theta = np.radians(theta)
x = radius * np.cos(theta)
y = radius * np.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 * np.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 * np.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: https://ui.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"])
@staticmethod
def _blur(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)
@staticmethod
def _gc_correction(
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 * np.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)
# Choose spiral arm
N = random_state.randint(0, 4, radius.size)
theta = self.k[N] * np.log(radius / self.r_0[N]) + self.theta_0[N]
spiralarm = self.spiralarms[N]
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: https://ui.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(1.0 / 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 distribution (dict mapping names to classes)."""
radial_distributions = {
"CB98": CaseBattacharya1998,
"F06": FaucherKaspi2006,
"L06": Lorimer2006,
"P90": Paczynski1990,
"YK04": YusifovKucuk2004,
"YK04B": YusifovKucuk2004B,
}