def dimensionless_velocity_dispersion(self, x, conc):
"""
Parameters
-----------
x : array_like
Halo-centric distance scaled by the halo boundary, so that
:math:`0 <= x <= 1`. Can be a scalar or numpy array
conc : float
Concentration of the halo.
Returns
-------
result : array_like
Radial velocity dispersion profile scaled by the virial velocity.
The returned result has the same dimension as the input ``x``.
"""
x = convert_to_ndarray(x)
x = x.astype(float)
result = np.zeros_like(x)
prefactor = conc*(conc*x)*(1. + conc*x)**2/self.g(conc)
lower_limit = conc*x
upper_limit = float("inf")
for i in range(len(x)):
term1, _ = quad_integration(self._jeans_integrand_term1,
lower_limit[i], upper_limit, epsrel=1e-5)
term2, _ = quad_integration(self._jeans_integrand_term2,
lower_limit[i], upper_limit, epsrel=1e-5)
result[i] = term1 - term2
return np.sqrt(result*prefactor)
def dimensionless_radial_velocity_dispersion(scaled_radius,
halo_conc,
gal_conc,
profile_integration_tol=1e-4):
r"""
Analytical solution to the isotropic jeans equation for an NFW potential,
rendered dimensionless via scaling by the virial velocity.
:math:`\tilde{\sigma}^{2}_{r}(\tilde{r})\equiv\sigma^{2}_{r}(\tilde{r})/V_{\rm vir}^{2} = \frac{c^{2}\tilde{r}(1 + c\tilde{r})^{2}}{g(c)}\int_{c\tilde{r}}^{\infty}{\rm d}y\frac{g(y)}{y^{3}(1 + y)^{2}}`
See :ref:`nfw_jeans_velocity_profile_derivations` for derivations and implementation details.
Parameters
-----------
scaled_radius : array_like
Length-Ngals numpy array storing the halo-centric distance
*r* scaled by the halo boundary :math:`R_{\Delta}`, so that
:math:`0 <= \tilde{r} \equiv r/R_{\Delta} <= 1`.
halo_conc : float
Concentration of the halo.
Returns
-------
result : array_like
Radial velocity dispersion profile scaled by the virial velocity.
The returned result has the same dimension as the input ``scaled_radius``.
"""
x = np.atleast_1d(scaled_radius).astype(np.float64)
result = np.zeros_like(x)
prefactor = gal_conc * gal_conc * x * (
1. + gal_conc * x)**2 / _g_integral(halo_conc)
extra_args = halo_conc / np.atleast_1d(gal_conc).astype('f4')
lower_limit = gal_conc * x
upper_limit = float("inf")
for i in range(len(x)):
term1, _ = quad_integration(_jeans_integrand_term1,
lower_limit[i],
upper_limit,
epsrel=profile_integration_tol,
args=extra_args)
term2, _ = quad_integration(_jeans_integrand_term2,
lower_limit[i],
upper_limit,
epsrel=profile_integration_tol,
args=extra_args)
result[i] = term1 - term2
return np.sqrt(result * prefactor)
def cumulative_mass_PDF(self, scaled_radius, *prof_params):
r"""
The fraction of the total mass enclosed within dimensionless radius,
:math:`P_{\rm prof}(<\tilde{r}) \equiv M_{\Delta}(<\tilde{r}) / M_{\Delta},`
where :math:`\tilde{r} \equiv r / R_{\Delta}`.
Parameters
-----------
scaled_radius : array_like
Halo-centric distance *r* scaled by the halo boundary :math:`R_{\Delta}`, so that
:math:`0 <= \tilde{r} \equiv r/R_{\Delta} <= 1`. Can be a scalar or numpy array.
*prof_params : array_like, optional
Any additional array(s) necessary to specify the shape of the radial profile,
e.g., halo concentration.
Returns
-------------
p: array_like
The fraction of the total mass enclosed
within radius x, in :math:`M_{\odot}/h`;
has the same dimensions as the input ``x``.
Notes
------
See :ref:`halo_profile_definitions` for derivations and implementation details.
"""
x = np.atleast_1d(scaled_radius).astype(np.float64)
enclosed_mass = np.zeros_like(x)
for i in range(len(x)):
enclosed_mass[i], _ = quad_integration(
self._enclosed_dimensionless_mass_integrand,
0.0,
x[i],
epsrel=1e-5,
args=prof_params,
)
total, _ = quad_integration(
self._enclosed_dimensionless_mass_integrand,
0.0,
1.0,
epsrel=1e-5,
args=prof_params,
)
return enclosed_mass / total
def dimensionless_radial_velocity_dispersion(scaled_radius, halo_conc, gal_conc,
profile_integration_tol=1e-4):
r"""
Analytical solution to the isotropic jeans equation for an NFW potential,
rendered dimensionless via scaling by the virial velocity.
:math:`\tilde{\sigma}^{2}_{r}(\tilde{r})\equiv\sigma^{2}_{r}(\tilde{r})/V_{\rm vir}^{2} = \frac{c^{2}\tilde{r}(1 + c\tilde{r})^{2}}{g(c)}\int_{c\tilde{r}}^{\infty}{\rm d}y\frac{g(y)}{y^{3}(1 + y)^{2}}`
See :ref:`nfw_jeans_velocity_profile_derivations` for derivations and implementation details.
Parameters
-----------
scaled_radius : array_like
Length-Ngals numpy array storing the halo-centric distance
*r* scaled by the halo boundary :math:`R_{\Delta}`, so that
:math:`0 <= \tilde{r} \equiv r/R_{\Delta} <= 1`.
halo_conc : float
Concentration of the halo.
Returns
-------
result : array_like
Radial velocity dispersion profile scaled by the virial velocity.
The returned result has the same dimension as the input ``scaled_radius``.
"""
x = np.atleast_1d(scaled_radius).astype(np.float64)
result = np.zeros_like(x)
prefactor = gal_conc*gal_conc*x*(1. + gal_conc*x)**2/_g_integral(halo_conc)
extra_args = halo_conc/gal_conc
lower_limit = gal_conc*x
upper_limit = float("inf")
for i in range(len(x)):
term1, _ = quad_integration(_jeans_integrand_term1,
lower_limit[i], upper_limit, epsrel=profile_integration_tol,
args=extra_args)
term2, _ = quad_integration(_jeans_integrand_term2,
lower_limit[i], upper_limit, epsrel=profile_integration_tol,
args=extra_args)
result[i] = term1 - term2
return np.sqrt(result*prefactor)
def dimensionless_radial_velocity_dispersion(self, scaled_radius, *conc):
"""
Analytical solution to the isotropic jeans equation for an NFW potential,
rendered dimensionless via scaling by the virial velocity.
:math:`\\tilde{\\sigma}^{2}_{r}(\\tilde{r})\\equiv\\sigma^{2}_{r}(\\tilde{r})/V_{\\rm vir}^{2} = \\frac{c^{2}\\tilde{r}(1 + c\\tilde{r})^{2}}{g(c)}\int_{c\\tilde{r}}^{\infty}{\\rm d}y\\frac{g(y)}{y^{3}(1 + y)^{2}}`
See :ref:`nfw_jeans_velocity_profile_derivations` for derivations and implementation details.
Parameters
-----------
scaled_radius : array_like
Length-Ngals numpy array storing the halo-centric distance
*r* scaled by the halo boundary :math:`R_{\\Delta}`, so that
:math:`0 <= \\tilde{r} \\equiv r/R_{\\Delta} <= 1`.
total_mass: array_like
Length-Ngals numpy array storing the halo mass in :math:`M_{\odot}/h`.
conc : float
Concentration of the halo.
Returns
-------
result : array_like
Radial velocity dispersion profile scaled by the virial velocity.
The returned result has the same dimension as the input ``scaled_radius``.
"""
x = np.atleast_1d(scaled_radius).astype(np.float64)
result = np.zeros_like(x)
prefactor = conc*(conc*x)*(1. + conc*x)**2/self.g(conc)
lower_limit = conc*x
upper_limit = float("inf")
for i in range(len(x)):
term1, _ = quad_integration(self._jeans_integrand_term1,
lower_limit[i], upper_limit, epsrel=1e-5)
term2, _ = quad_integration(self._jeans_integrand_term2,
lower_limit[i], upper_limit, epsrel=1e-5)
result[i] = term1 - term2
return np.sqrt(result*prefactor)
def dimensionless_radial_velocity_dispersion(self, scaled_radius, *conc):
"""
Analytical solution to the isotropic jeans equation for an NFW potential,
rendered dimensionless via scaling by the virial velocity.
:math:`\\tilde{\\sigma}^{2}_{r}(\\tilde{r})\\equiv\\sigma^{2}_{r}(\\tilde{r})/V_{\\rm vir}^{2} = \\frac{c^{2}\\tilde{r}(1 + c\\tilde{r})^{2}}{g(c)}\int_{c\\tilde{r}}^{\infty}{\\rm d}y\\frac{g(y)}{y^{3}(1 + y)^{2}}`
See :ref:`nfw_jeans_velocity_profile_derivations` for derivations and implementation details.
Parameters
-----------
scaled_radius : array_like
Length-Ngals numpy array storing the halo-centric distance
*r* scaled by the halo boundary :math:`R_{\\Delta}`, so that
:math:`0 <= \\tilde{r} \\equiv r/R_{\\Delta} <= 1`.
total_mass: array_like
Length-Ngals numpy array storing the halo mass in :math:`M_{\odot}/h`.
conc : float
Concentration of the halo.
Returns
-------
result : array_like
Radial velocity dispersion profile scaled by the virial velocity.
The returned result has the same dimension as the input ``scaled_radius``.
"""
x = convert_to_ndarray(scaled_radius, dt = np.float64)
result = np.zeros_like(x)
prefactor = conc*(conc*x)*(1. + conc*x)**2/self.g(conc)
lower_limit = conc*x
upper_limit = float("inf")
for i in range(len(x)):
term1, _ = quad_integration(self._jeans_integrand_term1,
lower_limit[i], upper_limit, epsrel=1e-5)
term2, _ = quad_integration(self._jeans_integrand_term2,
lower_limit[i], upper_limit, epsrel=1e-5)
result[i] = term1 - term2
return np.sqrt(result*prefactor)
def cumulative_mass_PDF(self, scaled_radius, *prof_params):
"""
The fraction of the total mass enclosed within dimensionless radius,
:math:`P_{\\rm prof}(<\\tilde{r}) \equiv M_{\\Delta}(<\\tilde{r}) / M_{\\Delta},`
where :math:`\\tilde{r} \\equiv r / R_{\\Delta}`.
Parameters
-----------
scaled_radius : array_like
Halo-centric distance *r* scaled by the halo boundary :math:`R_{\\Delta}`, so that
:math:`0 <= \\tilde{r} \\equiv r/R_{\\Delta} <= 1`. Can be a scalar or numpy array.
*prof_params : array_like, optional
Any additional array(s) necessary to specify the shape of the radial profile,
e.g., halo concentration.
Returns
-------------
p: array_like
The fraction of the total mass enclosed
within radius x, in :math:`M_{\odot}/h`;
has the same dimensions as the input ``x``.
Notes
------
See :ref:`halo_profile_definitions` for derivations and implementation details.
"""
x = convert_to_ndarray(scaled_radius, dt=np.float64)
enclosed_mass = np.zeros_like(x)
for i in range(len(x)):
enclosed_mass[i], _ = quad_integration(
self._enclosed_dimensionless_mass_integrand, 0., x[i], epsrel=1e-5,
args=prof_params)
total, _ = quad_integration(
self._enclosed_dimensionless_mass_integrand, 0., 1.0, epsrel=1e-5,
args=prof_params)
return enclosed_mass / total
def _nfw_velocity_dispersion_table(scaled_radius_table, conc, tol=1e-5):
"""
"""
x = np.atleast_1d(scaled_radius_table).astype(np.float64)
result = np.zeros_like(x)
prefactor = conc * (conc * x) * (1. + conc * x)**2 / _g_integral(conc)
lower_limit = conc * x
upper_limit = float("inf")
for i in range(len(x)):
term1, __ = quad_integration(_jeans_integrand_term1,
lower_limit[i],
upper_limit,
epsrel=tol)
term2, __ = quad_integration(_jeans_integrand_term2,
lower_limit[i],
upper_limit,
epsrel=tol)
result[i] = term1 - term2
dimless_velocity_table = np.sqrt(result * prefactor)
return dimless_velocity_table
def cumulative_mass_PDF(self, x, *args):
"""
The fraction of the total mass enclosed within
dimensionless radius :math:`x = r / R_{\\rm halo}`.
Parameters
-----------
x : array_like
Halo-centric distance scaled by the halo boundary, so that
:math:`0 <= x <= 1`. Can be a scalar or numpy array
args : array_like, optional
Any additional array(s) necessary to specify the shape of the radial profile,
e.g., halo concentration.
Returns
-------------
p: array_like
The fraction of the total mass enclosed
within radius x, in :math:`M_{\odot}/h`;
has the same dimensions as the input ``x``.
"""
x = convert_to_ndarray(x)
x = x.astype(np.float64)
enclosed_mass = np.zeros_like(x)
for i in range(len(x)):
enclosed_mass[i], _ = quad_integration(
self._enclosed_dimensionless_mass_integrand, 0., x[i], epsrel = 1e-5,
args = args)
total, _ = quad_integration(
self._enclosed_dimensionless_mass_integrand, 0., 1.0, epsrel = 1e-5,
args = args)
return enclosed_mass / total
