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# 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(layout='tight')
    ax.plot(wavelengths, bb_result, '-')

    ax.set(
        xscale="log",
        xlabel=fr"$\lambda$ [{wavelengths.unit}]",
        ylabel=fr"$F(\lambda)$ [{bb_result.unit}]",
    )

    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(layout="tight")
    ax.plot(wavelengths, mod_result, '-')

    ax.set(xlabel=fr"$\lambda$ [{wavelengths.unit}]", ylabel=r"$D(\lambda)$")

    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)

    # Radial distribution for plotting
    radii = range(1,2001,10) * u.kpc

    # Radial NFW density distribution
    n_result = n(radii)

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

    # Density profile subplot
    axs[0].plot(radii, n_result, '-')
    axs[0].set(
        yscale='log',
        xlabel=fr"$r$ [{radii.unit}]",
        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
    axs[1].plot(radii / n.radius_s, n_result / n.density_s, '-')
    axs[1].set(
        xscale='log',
        yscale='log',
        xlabel=r"$r / r_s$",
        ylabel=r"$\rho / \rho_s$",
    )

    # Display plot
    fig.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)

    # 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()

(`png`, `svg`, `pdf`)

## 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 `FittableModel` instance using the argument `format="astropy.model"`. During the conversion, each |Cosmology| `Parameter` is converted to a `astropy.modeling.Model` `Parameter`, while the redshift-method becomes the model’s `__call__` / `evaluate` method. This means cosmologies can now be fit with data!

    >>> from astropy.cosmology import Planck18
    >>> model = Planck18.to_format(format="astropy.model", method="lookback_time")
    >>> model
    <FlatLambdaCDMCosmologyLookbackTimeModel(H0=67.66 km / (Mpc s), Om0=0.30966,
        Tcmb0=2.7255 K, Neff=3.046, m_nu=[0.  , 0.  , 0.06] eV, Ob0=0.04897,
        name='Planck18')>

When finished, e.g. fitting, a model can be turned back into a |Cosmology| using |Cosmology.from_format|.

    >>> from astropy.cosmology import Cosmology
    >>> cosmo = Cosmology.from_format(model, format="astropy.model")
    >>> cosmo == Planck18
    True