# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Dark matter profiles."""
import abc
import html
import numpy as np
import astropy.units as u
from gammapy.modeling import Parameter, Parameters
from gammapy.utils.integrate import trapz_loglog
__all__ = [
"BurkertProfile",
"DMProfile",
"EinastoProfile",
"IsothermalProfile",
"MooreProfile",
"NFWProfile",
"ZhaoProfile",
]
[docs]
class DMProfile(abc.ABC):
"""DMProfile model base class."""
LOCAL_DENSITY = 0.3 * u.GeV / (u.cm**3)
"""Local dark matter density as given in reference 2"""
DISTANCE_GC = 8.33 * u.kpc
"""Distance to the Galactic Center as given in reference 2"""
[docs]
def __call__(self, radius):
"""Call evaluate method of derived classes."""
kwargs = {par.name: par.quantity for par in self.parameters}
return self.evaluate(radius, **kwargs)
def _repr_html_(self):
try:
return self.to_html()
except AttributeError:
return f"<pre>{html.escape(str(self))}</pre>"
[docs]
def scale_to_local_density(self):
"""Scale to local density."""
scale = (self.LOCAL_DENSITY / self(self.DISTANCE_GC)).to_value("")
self.parameters["rho_s"].value *= scale
def _eval_substitution(self, radius, separation, squared):
"""Density at given radius together with the substitution part."""
exponent = 2 if squared else 1
return (
self(radius) ** exponent
* radius
/ np.sqrt(radius**2 - (self.DISTANCE_GC * np.sin(separation)) ** 2)
)
[docs]
def integral(self, rmin, rmax, separation, ndecade, squared=True):
r"""Integrate dark matter profile numerically.
.. math::
F(r_{min}, r_{max}) = \int_{r_{min}}^{r_{max}}\rho(r)^\gamma dr \\
\gamma = 2 \text{for annihilation} \\
\gamma = 1 \text{for decay}
Parameters
----------
rmin, rmax : `~astropy.units.Quantity`
Lower and upper bound of integration range.
separation : `~numpy.ndarray`
Separation angle in radians.
ndecade : int, optional
Number of grid points per decade used for the integration.
Default is 10000.
squared : bool, optional
Square the profile before integration.
Default is True.
"""
integral = self.integrate_spectrum_separation(
self._eval_substitution, rmin, rmax, separation, ndecade, squared
)
inegral_unit = u.Unit("GeV2 cm-5") if squared else u.Unit("GeV cm-2")
return integral.to(inegral_unit)
[docs]
def integrate_spectrum_separation(
self, func, xmin, xmax, separation, ndecade, squared=True
):
"""Squared dark matter profile integral.
Parameters
----------
xmin, xmax : `~astropy.units.Quantity`
Lower and upper bound of integration range.
separation : `~numpy.ndarray`
Separation angle in radians.
ndecade : int
Number of grid points per decade used for the integration.
squared : bool
Square the profile before integration.
Default is True.
"""
unit = xmin.unit
xmin = xmin.value
xmax = xmax.to_value(unit)
logmin = np.log10(xmin)
logmax = np.log10(xmax)
n = np.int32((logmax - logmin) * ndecade)
x = np.logspace(logmin, logmax, n) * unit
y = func(x, separation, squared)
val = trapz_loglog(y, x)
return val.sum()
[docs]
class ZhaoProfile(DMProfile):
r"""Zhao Profile.
It is a generalization of the NFW profile. The volume density
is parametrized with a double power-law. Scale radii smaller than the scale radius are described with a slope of
:math:`-\gamma` and scale radii larger than the scale radius are described with a slope of :math:`-\beta`.
:math:`\alpha` is a measure for the width of the transition region.
.. math::
\rho(r) = \rho_s \left(\frac{r_s}{r}\right)^\gamma \left(1 + \left(\frac{r}{r_s}\right)^\frac{1}{\alpha} \right)^{(\gamma - \beta) \alpha}
Equation (1) from [1]_.
Parameters
----------
r_s : `~astropy.units.Quantity`
Scale radius, :math:`r_s`.
alpha : `~astropy.units.Quantity`
:math:`\alpha`.
beta: `~astropy.units.Quantity`
:math:`\beta`.
gamma : `~astropy.units.Quantity`
:math:`\gamma`.
rho_s : `~astropy.units.Quantity`
Characteristic density, :math:`\rho_s`.
References
----------
.. [1] `Zhao (1996), "Analytical models for galactic nuclei"
<https://ui.adsabs.harvard.edu/abs/1996MNRAS.278..488Z>`_
* `Marco et al. (2011), "PPPC 4 DM ID: a poor particle physicist cookbook for dark matter indirect detection"
<https://ui.adsabs.harvard.edu/abs/2011JCAP...03..051C>`_
"""
DEFAULT_SCALE_RADIUS = 24.42 * u.kpc
DEFAULT_ALPHA = 1
DEFAULT_BETA = 3
DEFAULT_GAMMA = 1
"""
(alpha, beta, gamma) = (1,3,1) is NFW profile.
Default scale radius as given in reference 2 (same as for NFW profile)
"""
def __init__(
self, r_s=None, alpha=None, beta=None, gamma=None, rho_s=1 * u.Unit("GeV / cm3")
):
r_s = self.DEFAULT_SCALE_RADIUS if r_s is None else r_s
alpha = self.DEFAULT_ALPHA if alpha is None else alpha
beta = self.DEFAULT_BETA if beta is None else beta
gamma = self.DEFAULT_GAMMA if gamma is None else gamma
self.parameters = Parameters(
[
Parameter("r_s", u.Quantity(r_s)),
Parameter("rho_s", u.Quantity(rho_s)),
Parameter("alpha", alpha),
Parameter("beta", beta),
Parameter("gamma", gamma),
]
)
[docs]
@staticmethod
def evaluate(radius, r_s, alpha, beta, gamma, rho_s):
"""Evaluate the profile."""
rr = radius / r_s
return rho_s / (rr**gamma * (1 + rr ** (1 / alpha)) ** ((beta - gamma) * alpha))
[docs]
class NFWProfile(DMProfile):
r"""NFW Profile.
.. math::
\rho(r) = \rho_s \frac{r_s}{r}\left(1 + \frac{r}{r_s}\right)^{-2}
Parameters
----------
r_s : `~astropy.units.Quantity`
Scale radius, :math:`r_s`.
rho_s : `~astropy.units.Quantity`
Characteristic density, :math:`\rho_s`.
References
----------
* `Navarro et al. (1997), "A Universal Density Profile from Hierarchical Clustering"
<https://ui.adsabs.harvard.edu/abs/1997ApJ...490..493N>`_
* `Marco et al. (2011), "PPPC 4 DM ID: a poor particle physicist cookbook for dark matter indirect detection"
<https://ui.adsabs.harvard.edu/abs/2011JCAP...03..051C>`_
"""
DEFAULT_SCALE_RADIUS = 24.42 * u.kpc
"""Default scale radius as given in reference 2"""
def __init__(self, r_s=None, rho_s=1 * u.Unit("GeV / cm3")):
r_s = self.DEFAULT_SCALE_RADIUS if r_s is None else r_s
self.parameters = Parameters(
[Parameter("r_s", u.Quantity(r_s)), Parameter("rho_s", u.Quantity(rho_s))]
)
[docs]
@staticmethod
def evaluate(radius, r_s, rho_s):
"""Evaluate the profile."""
rr = radius / r_s
return rho_s / (rr * (1 + rr) ** 2)
[docs]
class EinastoProfile(DMProfile):
r"""Einasto Profile.
.. math::
\rho(r) = \rho_s \exp{
\left(-\frac{2}{\alpha}\left[
\left(\frac{r}{r_s}\right)^{\alpha} - 1\right] \right)}
Parameters
----------
r_s : `~astropy.units.Quantity`
Scale radius, :math:`r_s`.
alpha : `~astropy.units.Quantity`
:math:`\alpha`.
rho_s : `~astropy.units.Quantity`
Characteristic density, :math:`\rho_s`.
References
----------
* `Einasto (1965), "On the Construction of a Composite Model for the Galaxy and on the Determination of the System
of Galactic Parameters" <https://ui.adsabs.harvard.edu/abs/1965TrAlm...5...87E>`_
* `Marco et al. (2011), "PPPC 4 DM ID: a poor particle physicist cookbook for dark matter indirect detection"
<https://ui.adsabs.harvard.edu/abs/2011JCAP...03..051C>`_
"""
DEFAULT_SCALE_RADIUS = 28.44 * u.kpc
"""Default scale radius as given in reference 2"""
DEFAULT_ALPHA = 0.17
"""Default scale radius as given in reference 2"""
def __init__(self, r_s=None, alpha=None, rho_s=1 * u.Unit("GeV / cm3")):
alpha = self.DEFAULT_ALPHA if alpha is None else alpha
r_s = self.DEFAULT_SCALE_RADIUS if r_s is None else r_s
self.parameters = Parameters(
[
Parameter("r_s", u.Quantity(r_s)),
Parameter("alpha", u.Quantity(alpha)),
Parameter("rho_s", u.Quantity(rho_s)),
]
)
[docs]
@staticmethod
def evaluate(radius, r_s, alpha, rho_s):
"""Evaluate the profile."""
rr = radius / r_s
exponent = (2 / alpha) * (rr**alpha - 1)
return rho_s * np.exp(-1 * exponent)
[docs]
class IsothermalProfile(DMProfile):
r"""Isothermal Profile.
.. math:: \rho(r) = \frac{\rho_s}{1 + (r/r_s)^2}
Parameters
----------
r_s : `~astropy.units.Quantity`
Scale radius, :math:`r_s`.
References
----------
* `Begeman et al. (1991), "Extended rotation curves of spiral galaxies : dark haloes and modified dynamics"
<https://ui.adsabs.harvard.edu/abs/1965TrAlm...5...87E>`_
* `Marco et al. (2011), "PPPC 4 DM ID: a poor particle physicist cookbook for dark matter indirect detection"
<https://ui.adsabs.harvard.edu/abs/2011JCAP...03..051C>`_
"""
DEFAULT_SCALE_RADIUS = 4.38 * u.kpc
"""Default scale radius as given in reference 2"""
def __init__(self, r_s=None, rho_s=1 * u.Unit("GeV / cm3")):
r_s = self.DEFAULT_SCALE_RADIUS if r_s is None else r_s
self.parameters = Parameters(
[Parameter("r_s", u.Quantity(r_s)), Parameter("rho_s", u.Quantity(rho_s))]
)
[docs]
@staticmethod
def evaluate(radius, r_s, rho_s):
"""Evaluate the profile."""
rr = radius / r_s
return rho_s / (1 + rr**2)
[docs]
class BurkertProfile(DMProfile):
r"""Burkert Profile.
.. math:: \rho(r) = \frac{\rho_s}{(1 + r/r_s)(1 + (r/r_s)^2)}
Parameters
----------
r_s : `~astropy.units.Quantity`
Scale radius, :math:`r_s`.
References
----------
* `Burkert (1995), "The Structure of Dark Matter Halos in Dwarf Galaxies"
<https://ui.adsabs.harvard.edu/abs/1965TrAlm...5...87E>`_
* `Marco et al. (2011), "PPPC 4 DM ID: a poor particle physicist cookbook for dark matter indirect detection"
<https://ui.adsabs.harvard.edu/abs/2011JCAP...03..051C>`_
"""
DEFAULT_SCALE_RADIUS = 12.67 * u.kpc
"""Default scale radius as given in reference 2"""
def __init__(self, r_s=None, rho_s=1 * u.Unit("GeV / cm3")):
r_s = self.DEFAULT_SCALE_RADIUS if r_s is None else r_s
self.parameters = Parameters(
[Parameter("r_s", u.Quantity(r_s)), Parameter("rho_s", u.Quantity(rho_s))]
)
[docs]
@staticmethod
def evaluate(radius, r_s, rho_s):
"""Evaluate the profile."""
rr = radius / r_s
return rho_s / ((1 + rr) * (1 + rr**2))
[docs]
class MooreProfile(DMProfile):
r"""Moore Profile.
.. math::
\rho(r) = \rho_s \left(\frac{r_s}{r}\right)^{1.16}
\left(1 + \frac{r}{r_s} \right)^{-1.84}
Parameters
----------
r_s : `~astropy.units.Quantity`
Scale radius, :math:`r_s`.
References
----------
* `Diemand et al. (2004), "Convergence and scatter of cluster density profiles"
<https://ui.adsabs.harvard.edu/abs/1965TrAlm...5...87E>`_
* `Marco et al. (2011), "PPPC 4 DM ID: a poor particle physicist cookbook for dark matter indirect detection"
<https://ui.adsabs.harvard.edu/abs/2011JCAP...03..051C>`_
"""
DEFAULT_SCALE_RADIUS = 30.28 * u.kpc
"""Default scale radius as given in reference 2"""
def __init__(self, r_s=None, rho_s=1 * u.Unit("GeV / cm3")):
r_s = self.DEFAULT_SCALE_RADIUS if r_s is None else r_s
self.parameters = Parameters(
[Parameter("r_s", u.Quantity(r_s)), Parameter("rho_s", u.Quantity(rho_s))]
)
[docs]
@staticmethod
def evaluate(radius, r_s, rho_s):
"""Evaluate the profile."""
rr = radius / r_s
rr_ = r_s / radius
return rho_s * rr_**1.16 * (1 + rr) ** (-1.84)
