# 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¶

The BlackBody model provides a model for using Planck’s Law. The blackbody function is

$B_{\nu}(T) = A \frac{2 h \nu^{3} / c^{2}}{exp(h \nu / k T) - 1}$

where $$\nu$$ is the frequency, $$T$$ is the temperature, $$A$$ is the scaling factor, $$h$$ is the Plank constant, $$c$$ is the speed of light, and $$k$$ 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 $$B_\lambda$$. 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 BlackBody.

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()


(png, svg, pdf)

The bolometric_flux() member function gives the bolometric flux using $$\sigma T^4/\pi$$ where $$\sigma$$ is the Stefan-Boltzmann constant.

The lambda_max() and nu_max() member functions give the wavelength and frequency of the maximum for $$B_\lambda$$ and $$B_\nu$$, respectively, calculated using Wien’s Law.

## Drude1D¶

The Drude1D model provides a model for the behavior of an electron in a material (see Drude Model). Like the Lorentz1D model, the Drude model has broader wings than the 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 $$x$$ is

$D(x) = A \frac{(f/x_0)^2}{((x/x_0 - x_0/x)^2 + (f/x_0)^2}$

where $$A$$ is the amplitude, $$f$$ is the full width at half maximum, and $$x_0$$ is the central wavelength. An example of a Drude1D model with $$x_0 = 2175$$ Angstrom and $$f = 400$$ Angstrom is shown below.

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()


(png, svg, pdf)

## NFW¶

The NFW model computes a 1-dimensional Navarro–Frenk–White profile. The dark matter density in an NFW profile is given by:

$\rho(r)=\frac{\delta_c\rho_{c}}{r/r_s(1+r/r_s)^2}$

where $$\rho_{c}$$ is the critical density of the Universe at the redshift of the profile, $$\delta_c$$ is the over density, and $$r_s$$ 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 = $$2.0 x 10^{15} M_{sun}$$

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.

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)

# Plot creation
fig, ax = plt.subplots(2)
fig.suptitle('1 Dimensional NFW Profile')

# Density profile subplot
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)

radii = np.logspace(np.log10(1e-5), np.log10(2), num=1000) * u.Mpc

# Scaled density / scaled radius subplot
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()


(png, svg, pdf)

The circular_velocity() member provides the circular velocity at each position r via the equation:

$v_{circ}(r)^2=\frac{1}{x}\frac{\ln(1+cx)-(cx)/(1+cx)}{\ln(1+c)-c/(1+c)}$

where x is the ratio r$$/r_{vir}$$. 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 = $$2.0 x 10^{15} M_{sun}$$

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.

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)

# NFW circular velocity distribution

# Plot creation
fig,ax = plt.subplots()
ax.set_title('NFW Profile Circular Velocity')
ax.set_xlabel(fr"$r$ [{radii.unit}]")
ax.set_ylabel(r"$v_{circ}$" + f" [{n_result.unit}]")