Source code for gammapy.astro.population.spatial

# 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
import numpy as np
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): """Evaluate model.""" 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): """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): """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): """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): """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): """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): """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): """Evaluate model.""" term1 = 1. / np.sqrt(2 * np.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): """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): """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): """Evaluate model.""" 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: 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: 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 * 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: 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 distribution (dict mapping names to classes).""" radial_distributions = { "CB98": CaseBattacharya1998, "F06": FaucherKaspi2006, "L06": Lorimer2006, "P90": Paczynski1990, "YK04": YusifovKucuk2004, "YK04B": YusifovKucuk2004B, }