.. _predef_physicalmodels: *************** Physical Models *************** These are models that are physical motivated, generally as solutions to physical problems. This is in contrast to those that are mathematically motivated, generally as solutions to mathematical problems. .. _blackbody-planck-law: BlackBody ========= The :class:~astropy.modeling.physical_models.BlackBody model provides a model for using Planck's Law _. The blackbody function is .. math:: B_{\nu}(T) = A \frac{2 h \nu^{3} / c^{2}}{exp(h \nu / k T) - 1} where :math:\nu is the frequency, :math:T is the temperature, :math:A is the scaling factor, :math:h is the Plank constant, :math:c is the speed of light, and :math:k is the Boltzmann constant. The two parameters of the model the scaling factor scale (A) and the absolute temperature temperature (T). If the scale factor does not have units, then the result is in units of spectral radiance, specifically ergs/(cm^2 Hz s sr). If the scale factor is passed with spectral radiance units, then the result is in those units (e.g., ergs/(cm^2 A s sr) or MJy/sr). Setting the scale factor with units of ergs/(cm^2 A s sr) will give the Planck function as :math:B_\lambda. The temperature can be passed as a Quantity with any supported temperature unit. An example plot for a blackbody with a temperature of 10000 K and a scale of 1 is shown below. A scale of 1 shows the Planck function with no scaling in the default units returned by :class:~astropy.modeling.physical_models.BlackBody. .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import BlackBody import astropy.units as u wavelengths = np.logspace(np.log10(1000), np.log10(3e4), num=1000) * u.AA # blackbody parameters temperature = 10000 * u.K # BlackBody provides the results in ergs/(cm^2 Hz s sr) when scale has no units bb = BlackBody(temperature=temperature, scale=10000.0) bb_result = bb(wavelengths) fig, ax = plt.subplots(ncols=1) ax.plot(wavelengths, bb_result, '-') ax.set_xscale('log') ax.set_xlabel(fr"$\lambda$ [{wavelengths.unit}]") ax.set_ylabel(fr"$F(\lambda)$ [{bb_result.unit}]") plt.tight_layout() plt.show() The :meth:~astropy.modeling.physical_models.BlackBody.bolometric_flux member function gives the bolometric flux using :math:\sigma T^4/\pi where :math:\sigma is the Stefan-Boltzmann constant. The :meth:~astropy.modeling.physical_models.BlackBody.lambda_max and :meth:~astropy.modeling.physical_models.BlackBody.nu_max member functions give the wavelength and frequency of the maximum for :math:B_\lambda and :math:B_\nu, respectively, calculated using Wien's Law _. Drude1D ======= The :class:~astropy.modeling.physical_models.Drude1D model provides a model for the behavior of an electron in a material (see Drude Model _). Like the :class:~astropy.modeling.functional_models.Lorentz1D model, the Drude model has broader wings than the :class:~astropy.modeling.functional_models.Gaussian1D model. The Drude profile has been used to model dust features including the 2175 Angstrom extinction feature and the mid-infrared aromatic/PAH features. The Drude function at :math:x is .. math:: D(x) = A \frac{(f/x_0)^2}{((x/x_0 - x_0/x)^2 + (f/x_0)^2} where :math:A is the amplitude, :math:f is the full width at half maximum, and :math:x_0 is the central wavelength. An example of a Drude1D model with :math:x_0 = 2175 Angstrom and :math:f = 400 Angstrom is shown below. .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Drude1D import astropy.units as u wavelengths = np.linspace(1000, 4000, num=1000) * u.AA # Parameters and model mod = Drude1D(amplitude=1.0, x_0=2175. * u.AA, fwhm=400. * u.AA) mod_result = mod(wavelengths) fig, ax = plt.subplots(ncols=1) ax.plot(wavelengths, mod_result, '-') ax.set_xlabel(fr"$\lambda$ [{wavelengths.unit}]") ax.set_ylabel(r"$D(\lambda)$") plt.tight_layout() plt.show() .. _NFW: NFW ========= The :class:~astropy.modeling.physical_models.NFW model computes a 1-dimensional Navarro–Frenk–White profile. The dark matter density in an NFW profile is given by: .. math:: \rho(r)=\frac{\delta_c\rho_{c}}{r/r_s(1+r/r_s)^2} where :math:\rho_{c} is the critical density of the Universe at the redshift of the profile, :math:\delta_c is the over density, and :math:r_s is the scale radius of the profile. This model relies on three parameters: mass : the mass of the profile (in solar masses if no units are provided) concentration : the profile concentration redshift : the redshift of the profile As well as two optional initialization variables: massfactor : tuple or string specifying the overdensity type and factor (default ("critical", 200)) cosmo : the cosmology for density calculation (default default_cosmology) .. note:: Initialization of NFW profile object required before evaluation (in order to set mass overdensity and cosmology). Sample plots of an NFW profile with the following parameters are displayed below: mass = :math:2.0 x 10^{15} M_{sun} concentration = 8.5 redshift = 0.63 The first plot is of the NFW profile density as a function of radius. The second plot displays the profile density and radius normalized by the NFW scale density and scale radius, respectively. The scale density and scale radius are available as attributes rho_s and r_s, and the overdensity radius can be accessed via r_virial. .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import NFW import astropy.units as u from astropy import cosmology # NFW Parameters mass = u.Quantity(2.0E15, u.M_sun) concentration = 8.5 redshift = 0.63 cosmo = cosmology.Planck15 massfactor = ("critical", 200) # Create NFW Object n = NFW(mass=mass, concentration=concentration, redshift=redshift, cosmo=cosmo, massfactor=massfactor) # Radial distribution for plotting radii = range(1,2001,10) * u.kpc # Radial NFW density distribution n_result = n(radii) # Plot creation fig, ax = plt.subplots(2) fig.suptitle('1 Dimensional NFW Profile') # Density profile subplot ax[0].plot(radii, n_result, '-') ax[0].set_yscale('log') ax[0].set_xlabel(fr"$r$ [{radii.unit}]") ax[0].set_ylabel(fr"$\rho$ [{n_result.unit}]") # Create scaled density / scaled radius subplot # NFW Object n = NFW(mass=mass, concentration=concentration, redshift=redshift, cosmo=cosmo, massfactor=massfactor) # Radial distribution for plotting radii = np.logspace(np.log10(1e-5), np.log10(2), num=1000) * u.Mpc n_result = n(radii) # Scaled density / scaled radius subplot ax[1].plot(radii / n.radius_s, n_result / n.density_s, '-') ax[1].set_xscale('log') ax[1].set_yscale('log') ax[1].set_xlabel(r"$r / r_s$") ax[1].set_ylabel(r"$\rho / \rho_s$") # Display plot plt.tight_layout(rect=[0, 0.03, 1, 0.95]) plt.show() The :meth:~astropy.modeling.physical_models.NFW.circular_velocity member provides the circular velocity at each position r via the equation: .. math:: v_{circ}(r)^2=\frac{1}{x}\frac{\ln(1+cx)-(cx)/(1+cx)}{\ln(1+c)-c/(1+c)} where x is the ratio r:math:/r_{vir}. Circular velocities are provided in km/s. A sample plot of circular velocities of an NFW profile with the following parameters is displayed below: mass = :math:2.0 x 10^{15} M_{sun} concentration = 8.5 redshift = 0.63 The maximum circular velocity and radius of maximum circular velocity are available as attributes v_max and r_max. .. plot:: :include-source: import matplotlib.pyplot as plt from astropy.modeling.models import NFW import astropy.units as u from astropy import cosmology # NFW Parameters mass = u.Quantity(2.0E15, u.M_sun) concentration = 8.5 redshift = 0.63 cosmo = cosmology.Planck15 massfactor = ("critical", 200) # Create NFW Object n = NFW(mass=mass, concentration=concentration, redshift=redshift, cosmo=cosmo, massfactor=massfactor) # Radial distribution for plotting radii = range(1,200001,10) * u.kpc # NFW circular velocity distribution n_result = n.circular_velocity(radii) # Plot creation fig,ax = plt.subplots() ax.set_title('NFW Profile Circular Velocity') ax.plot(radii, n_result, '-') ax.set_xscale('log') ax.set_xlabel(fr"$r$ [{radii.unit}]") ax.set_ylabel(r"$v_{circ}$" + f" [{n_result.unit}]") # Display plot plt.tight_layout(rect=[0, 0.03, 1, 0.95]) plt.show() .. _Cosmologies: Cosmologies =========== The instances of the |Cosmology| class (and subclasses) include |Cosmology.to_format|, a method to convert a Cosmology to another python object. Specifically, any redshift method can be converted to a :class:~astropy.modeling.FittableModel instance using the argument format="astropy.model". During the conversion, each |Cosmology| :class:~astropy.cosmology.Parameter is converted to a :class:astropy.modeling.Model :class:~astropy.modeling.Parameter, while the redshift-method becomes the model's __call__ / evaluate method. This means cosmologies can now be fit with data! .. code-block:: >>> from astropy.cosmology import Planck18 >>> model = Planck18.to_format(format="astropy.model", method="lookback_time") >>> model When finished, e.g. fitting, a model can be turned back into a |Cosmology| using |Cosmology.from_format|. .. code-block:: >>> from astropy.cosmology import Cosmology >>> cosmo = Cosmology.from_format(model, format="astropy.model") >>> cosmo == Planck18 True