def test_isotropic_anisotropy(self):
# radial
r = 2.
R = 1.
anisoClass = MamonLokasAnisotropy(anisotropy_model='isotropic')
kwargs = {}
beta = anisoClass.beta_r(r, kwargs=kwargs)
k = anisoClass.K(r, R, kwargs=kwargs)
print(beta, 'test')
anisoClassMamon = MamonLokasAnisotropy(anisotropy_model='const')
kwargs = {'beta': beta}
k_mamon = anisoClassMamon.K(r, R, kwargs=kwargs)
print(k, k_mamon)
npt.assert_almost_equal(k, k_mamon, decimal=5)
def test_K(self):
r = 2.
R = 1.
anisoClass = MamonLokasAnisotropy(anisotropy_model='const_wrong')
kwargs = {'beta': 1.0}
k = anisoClass.K(r, R, kwargs=kwargs)
npt.assert_almost_equal(k, 0.61418484930437822, decimal=5)
anisoClass = MamonLokasAnisotropy(anisotropy_model='const')
kwargs = {'beta': 1.0}
k = anisoClass.K(r, R, kwargs=kwargs)
npt.assert_almost_equal(k, 0.61418484930437822, decimal=5)
anisoClass = MamonLokasAnisotropy(anisotropy_model='Colin')
kwargs = {'r_ani': 1}
k = anisoClass.K(r, R, kwargs=kwargs)
npt.assert_almost_equal(k, 0.91696135187291117, decimal=5)
anisoClass = MamonLokasAnisotropy(anisotropy_model='radial')
kwargs = {}
k = anisoClass.K(r, R, kwargs=kwargs)
npt.assert_almost_equal(k, 0.61418484930437856, decimal=5)
anisoClass = MamonLokasAnisotropy(anisotropy_model='isotropic')
kwargs = {}
k = anisoClass.K(r, R, kwargs=kwargs)
npt.assert_almost_equal(k, 0.8660254037844386, decimal=5)
anisoClass = MamonLokasAnisotropy(anisotropy_model='OsipkovMerritt')
kwargs = {'r_ani': 1}
k = anisoClass.K(r, R, kwargs=kwargs)
npt.assert_almost_equal(k, 0.95827704196894481, decimal=5)
class Galkin(object):
"""
Major class to compute velocity dispersion measurements given light and mass models
The class supports any mass and light distribution (and superposition thereof) that has a 3d correspondance in their
2d lens model distribution. For models that do not have this correspondance, you may want to apply a
Multi-Gaussian Expansion (MGE) on their models and use the MGE to be de-projected to 3d.
The computation follows Mamon&Lokas 2005 and performs the spectral rendering of the seeing convolved apperture with
the method introduced by Birrer et al. 2016.
The class supports various types of anisotropy models (see Anisotropy class) and aperture types (see Aperture class).
Solving the Jeans Equation requires a numerical integral over the 3d light and mass profile (see Mamon&Lokas 2005).
This class (as well as the dedicated LightModel and MassModel classes) perform those integral numerically with an
interpolated grid.
The seeing convolved integral over the aperture is computed by rendering spectra (light weighted LOS kinematics)
from the light distribution.
The cosmology assumed to compute the physical mass and distances are set via the kwargs_cosmo keyword arguments.
D_d: Angular diameter distance to the deflector (in Mpc)
D_s: Angular diameter distance to the source (in Mpc)
D_ds: Angular diameter distance from the deflector to the source (in Mpc)
The numerical options can be chosen through the kwargs_numerics keywords
sampling_number: number of spectral rendering to compute the light weighted integrated LOS dispersion within
the aperture. This keyword should be chosen high enough to result in converged results within the tolerance.
interpol_grid_num: number of interpolation points in the light and mass profile (radially). This number should
be chosen high enough to accurately describe the true light profile underneath.
log_integration: bool, if True, performs the interpolation and numerical integration in log-scale.
max_integrate: maximum 3d radius to where the numerical integration of the Jeans Equation solver is made.
This value should be large enough to contain most of the light and to lead to a converged result.
min_integrate: minimal integration value. This value should be very close to zero but some mass and light
profiles are diverging and a numerically stabel value should be chosen.
These numerical options should be chosen to allow for a converged result (within your tolerance) but not too
conservative to impact too much the computational cost. Reasonable values might depend on the specific problem.
"""
def __init__(self,
mass_profile_list,
light_profile_list,
aperture_type='slit',
anisotropy_model='isotropic',
fwhm=0.7,
kwargs_cosmo={
'D_d': 1000,
'D_s': 2000,
'D_ds': 500
},
sampling_number=1000,
interpol_grid_num=500,
log_integration=False,
max_integrate=10,
min_integrate=0.001):
"""
:param mass_profile_list: list of lens (mass) model profiles
:param light_profile_list: list of light model profiles of the lensing galaxy
:param aperture_type: type of slit/shell aperture where the light is coming from. See details in Aperture() class.
:param anisotropy_model: type of stellar anisotropy model. See details in MamonLokasAnisotropy() class.
:param fwhm: full width at half maximum seeing condition
:param kwargs_numerics: keyword arguments that control the numerical computation
:param kwargs_cosmo: keyword arguments that define the cosmology in terms of the angular diameter distances involved
"""
self.massProfile = MassProfile(mass_profile_list,
kwargs_cosmo,
interpol_grid_num=interpol_grid_num,
max_interpolate=max_integrate,
min_interpolate=min_integrate)
self.lightProfile = LightProfile(light_profile_list,
interpol_grid_num=interpol_grid_num,
max_interpolate=max_integrate,
min_interpolate=min_integrate)
self.aperture = Aperture(aperture_type)
self.anisotropy = MamonLokasAnisotropy(anisotropy_model)
self._fwhm = fwhm
self.cosmo = Cosmo(**kwargs_cosmo)
self._num_sampling = sampling_number
self._interp_grid_num = interpol_grid_num
self._log_int = log_integration
self._max_integrate = max_integrate # maximal integration (and interpolation) in units of arcsecs
self._min_integrate = min_integrate # min integration (and interpolation) in units of arcsecs
def vel_disp(self, kwargs_mass, kwargs_light, kwargs_anisotropy,
kwargs_apertur):
"""
computes the averaged LOS velocity dispersion in the slit (convolved)
:param kwargs_mass: mass model parameters (following lenstronomy lens model conventions)
:param kwargs_light: deflector light parameters (following lenstronomy light model conventions)
:param kwargs_anisotropy: anisotropy parameters, may vary according to anisotropy type chosen.
We refer to the Anisotropy() class for details on the parameters.
:param kwargs_apertur: Aperture parameters, may vary depending on aperture type chosen.
We refer to the Aperture() class for details on the parameters.
:return: integrated LOS velocity dispersion in units [km/s]
"""
sigma2_R_sum = 0
for i in range(0, self._num_sampling):
sigma2_R = self.draw_one_sigma2(kwargs_mass, kwargs_light,
kwargs_anisotropy, kwargs_apertur)
sigma2_R_sum += sigma2_R
sigma_s2_average = sigma2_R_sum / self._num_sampling
# apply unit conversion from arc seconds and deflections to physical velocity disperison in (km/s)
sigma_s2_average *= 2 * const.G # correcting for integral prefactor
return np.sqrt(sigma_s2_average /
(const.arcsec**2 * self.cosmo.D_d**2 *
const.Mpc)) / 1000. # in units of km/s
def draw_one_sigma2(self, kwargs_mass, kwargs_light, kwargs_anisotropy,
kwargs_aperture):
"""
:param kwargs_mass: mass model parameters (following lenstronomy lens model conventions)
:param kwargs_light: deflector light parameters (following lenstronomy light model conventions)
:param kwargs_anisotropy: anisotropy parameters, may vary according to anisotropy type chosen.
We refer to the Anisotropy() class for details on the parameters.
:param kwargs_apertur: Aperture parameters, may vary depending on aperture type chosen.
We refer to the Aperture() class for details on the parameters.
:return: integrated LOS velocity dispersion in angular units for a single draw of the light distribution that
falls in the aperture after displacing with the seeing
"""
while True:
R = self.lightProfile.draw_light_2d(kwargs_light) # draw r
x, y = util.draw_xy(R) # draw projected R
x_, y_ = util.displace_PSF(x, y, self._fwhm) # displace via PSF
bool = self.aperture.aperture_select(x_, y_, kwargs_aperture)
if bool is True:
break
sigma2_R = self.sigma2_R(R, kwargs_mass, kwargs_light,
kwargs_anisotropy)
return sigma2_R
def sigma2_R(self, R, kwargs_mass, kwargs_light, kwargs_anisotropy):
"""
returns unweighted los velocity dispersion for a specified projected radius
:param R: 2d projected radius (in angular units)
:param kwargs_mass: mass model parameters (following lenstronomy lens model conventions)
:param kwargs_light: deflector light parameters (following lenstronomy light model conventions)
:param kwargs_anisotropy: anisotropy parameters, may vary according to anisotropy type chosen.
We refer to the Anisotropy() class for details on the parameters.
:return:
"""
I_R_sigma2 = self.I_R_simga2(R, kwargs_mass, kwargs_light,
kwargs_anisotropy)
I_R = self.lightProfile.light_2d(R, kwargs_light)
return I_R_sigma2 / I_R
def I_R_simga2(self, R, kwargs_mass, kwargs_light, kwargs_anisotropy):
"""
equation A15 in Mamon&Lokas 2005 as a logarithmic numerical integral (if option is chosen)
modulo pre-factor 2*G
:param R: 2d projected radius (in angular units)
:param kwargs_mass: mass model parameters (following lenstronomy lens model conventions)
:param kwargs_light: deflector light parameters (following lenstronomy light model conventions)
:param kwargs_anisotropy: anisotropy parameters, may vary according to anisotropy type chosen.
We refer to the Anisotropy() class for details on the parameters.
:return: integral of A15 in Mamon&Lokas 2005
"""
R = max(R, self._min_integrate)
if self._log_int is True:
min_log = np.log10(R + 0.001)
max_log = np.log10(self._max_integrate)
r_array = np.logspace(min_log, max_log, self._interp_grid_num)
dlog_r = (np.log10(r_array[2]) - np.log10(r_array[1])) * np.log(10)
IR_sigma2_dr = self._integrand_A15(
r_array, R, kwargs_mass, kwargs_light,
kwargs_anisotropy) * dlog_r * r_array
else:
r_array = np.linspace(R + 0.001, self._max_integrate,
self._interp_grid_num)
dr = r_array[2] - r_array[1]
IR_sigma2_dr = self._integrand_A15(
r_array, R, kwargs_mass, kwargs_light, kwargs_anisotropy) * dr
IR_sigma2 = np.sum(
IR_sigma2_dr
) * const.arcsec * self.cosmo.D_d # integral from angle to physical scales
return IR_sigma2
def _integrand_A15(self, r, R, kwargs_mass, kwargs_light,
kwargs_anisotropy):
"""
integrand of A15 (in log space) in Mamon&Lokas 2005
:param r: 3d radius
:param R: 2d projected radius
:param kwargs_mass: mass model parameters (following lenstronomy lens model conventions)
:param kwargs_light: deflector light parameters (following lenstronomy light model conventions)
:param kwargs_anisotropy: anisotropy parameters, may vary according to anisotropy type chosen.
We refer to the Anisotropy() class for details on the parameters.
:return:
"""
k_r = self.anisotropy.K(r, R, kwargs_anisotropy)
l_r = self.lightProfile.light_3d_interp(r, kwargs_light)
m_r = self.massProfile.mass_3d_interp(r, kwargs_mass)
out = k_r * l_r * m_r / r
return out
class Galkin(object):
"""
major class to compute velocity dispersion measurements given light and mass models
"""
def __init__(self,
mass_profile_list,
light_profile_list,
aperture_type='slit',
anisotropy_model='isotropic',
fwhm=0.7,
kwargs_numerics={},
kwargs_cosmo={
'D_d': 1000,
'D_s': 2000,
'D_ds': 500
}):
self.massProfile = MassProfile(mass_profile_list,
kwargs_cosmo,
kwargs_numerics=kwargs_numerics)
self.lightProfile = LightProfile(light_profile_list,
kwargs_numerics=kwargs_numerics)
self.aperture = Aperture(aperture_type)
self.anisotropy = MamonLokasAnisotropy(anisotropy_model)
self.FWHM = fwhm
self.cosmo = Cosmo(kwargs_cosmo)
#kwargs_numerics = {'sampling_number': 10000, 'interpol_grid_num': 5000, 'log_integration': False,
# 'max_integrate': 500}
self._num_sampling = kwargs_numerics.get('sampling_number', 1000)
self._interp_grid_num = kwargs_numerics.get('interpol_grid_num', 500)
self._log_int = kwargs_numerics.get('log_integration', False)
self._max_integrate = kwargs_numerics.get(
'max_integrate',
10) # maximal integration (and interpolation) in units of arcsecs
self._min_integrate = kwargs_numerics.get(
'min_integrate',
0.001) # min integration (and interpolation) in units of arcsecs
def vel_disp(self,
kwargs_mass,
kwargs_light,
kwargs_anisotropy,
kwargs_apertur,
r_eff=1.):
"""
computes the averaged LOS velocity dispersion in the slit (convolved)
:param gamma:
:param phi_E:
:param r_eff:
:param r_ani:
:param R_slit:
:param FWHM:
:return:
"""
sigma2_R_sum = 0
for i in range(0, self._num_sampling):
sigma2_R = self.draw_one_sigma2(kwargs_mass,
kwargs_light,
kwargs_anisotropy,
kwargs_apertur,
r_eff=r_eff)
sigma2_R_sum += sigma2_R
sigma_s2_average = sigma2_R_sum / self._num_sampling
# apply unit conversion from arc seconds and deflections to physical velocity disperison in (km/s)
sigma_s2_average *= 2 * const.G # correcting for integral prefactor
return np.sqrt(sigma_s2_average /
(const.arcsec**2 * self.cosmo.D_d**2 *
const.Mpc)) / 1000. # in units of km/s
def draw_one_sigma2(self,
kwargs_mass,
kwargs_light,
kwargs_anisotropy,
kwargs_aperture,
r_eff=1.):
"""
:param kwargs_mass:
:param kwargs_light:
:param kwargs_anisotropy:
:param kwargs_aperture:
:return:
"""
while True:
R = self.lightProfile.draw_light_2d(kwargs_light,
r_eff=r_eff) # draw r
x, y = util.draw_xy(R) # draw projected R
x_, y_ = util.displace_PSF(x, y, self.FWHM) # displace via PSF
bool = self.aperture.aperture_select(x_, y_, kwargs_aperture)
if bool is True:
break
sigma2_R = self.sigma2_R(R, kwargs_mass, kwargs_light,
kwargs_anisotropy)
return sigma2_R
def sigma2_R(self, R, kwargs_mass, kwargs_light, kwargs_anisotropy):
"""
returns unweighted los velocity dispersion
:param R:
:param kwargs_mass:
:param kwargs_light:
:param kwargs_anisotropy:
:return:
"""
I_R_sigma2 = self.I_R_simga2(R, kwargs_mass, kwargs_light,
kwargs_anisotropy)
I_R = self.lightProfile.light_2d(R, kwargs_light)
#I_R = self.lightProfile._integrand_light(R, kwargs_light)
return I_R_sigma2 / I_R
def I_R_simga2(self, R, kwargs_mass, kwargs_light, kwargs_anisotropy):
"""
equation A15 in Mamon&Lokas 2005 as a logarithmic numerical integral
modulo pre-factor 2*G
:param R:
:param kwargs_mass:
:param kwargs_light:
:param kwargs_anisotropy:
:return:
"""
R = max(R, self._min_integrate)
if self._log_int is True:
min_log = np.log10(R + 0.001)
max_log = np.log10(self._max_integrate)
r_array = np.logspace(min_log, max_log, self._interp_grid_num)
dlog_r = (np.log10(r_array[2]) - np.log10(r_array[1])) * np.log(10)
IR_sigma2_dr = self._integrand_A15(
r_array, R, kwargs_mass, kwargs_light,
kwargs_anisotropy) * dlog_r * r_array
else:
r_array = np.linspace(R + 0.001, self._max_integrate,
self._interp_grid_num)
dr = r_array[2] - r_array[1]
IR_sigma2_dr = self._integrand_A15(
r_array, R, kwargs_mass, kwargs_light, kwargs_anisotropy) * dr
IR_sigma2 = np.sum(IR_sigma2_dr)
return IR_sigma2
def _integrand_A15(self, r, R, kwargs_mass, kwargs_light,
kwargs_anisotropy):
"""
integrand of A15 (in log space)
:param r:
:param kwargs_mass:
:param kwargs_light:
:param kwargs_anisotropy:
:return:
"""
k_r = self.anisotropy.K(r, R, kwargs_anisotropy)
l_r = self.lightProfile.light_3d_interp(r, kwargs_light)
m_r = self.massProfile.mass_3d_interp(r, kwargs_mass)
out = k_r * l_r * m_r / r
return out