Luminosity and Mass Profiles

Abstract type for luminosity profiles for extended sources.

Any concrete subtype f of this abstract type must implement a method f(ϱ²) to compute the surface brightness of the profile. The argument ϱ² is the square elliptical radius of the source.

GaussianProfile()

A bivariate Gaussian profile.

It is associated to a volume density profile of the form

\[ \rho(r) = \mathrm{e}^{-r^2 / 2} \; .\]

That corresponds to a projected density profile of the form

\[ \Sigma(r) = \mathrm{e}^{-r^2 / 2} \; ,\]

and to a squared velocity profile of the form

\[ v^2(r) = 3 \left(\sqrt{\frac{\pi}{2}} rac{\mathrm{erf}(r / \sqrt{2})}{r} - \mathrm{e}^{-r^2 / 2}\right) \; .\]

ExponentialProfile()

An flat exponential profile.

The surface brightness (or surface density) is of the form

\[ \Sigma(r) = \mathrm{e}^{-r} \; .\]

The associated velocity profile is

\[ v^2(r) = \frac{r}{2} \left(\mathrm{I}_0(r / 2) \mathrm{K}_0(r / 2) - \mathrm{I}_1(r / 2) \mathrm{K}_1(r / 2)\right) \; ,\]

where $I_n$ and $K_n$ are the modified Bessel functions of the first and second kind.

IsothermalProfile()

A singular isothermal profile.

The volume density is of the form

\[ \rho(r) = \frac{1}{r^2} \; ,\]

which correspond to a projected density of the form

\[ \Sigma(r) = \frac{1}{\sqrt{r^2}} \; .\]

Note that this distribution is unnormalized (and unnormalizable).

The associated velocity profile is just

\[ v^2(r) = 1 \; .\]

See also: [SoftIsothermalProfile], [JaffeProfile]

SoftIsothermalProfile()

A singular isothermal profile.

The volume density is of the form

\[ \rho(r) = \frac{1}{r^2 + 1} \; ,\]

which correspond to a projected density of the form

\[ \Sigma(r) = \frac{1}{\sqrt{r^2 + 1}} \; .\]

Note that this distribution is unnormalized (and unnormalizable).

The associated velocity profile is

\[ v^2(r) = 1 - \frac{\arctan(r)}{r} \; .\]

See also: [SoftIsothermalProfile], [JaffeProfile]

PlummerProfile()

A projected Plummer profile.

This profile is often used to describe the surface brightness of globular clusters.

The volume density is of the form

\[ \rho(r) = \frac{1}{(1 + r^2)^{5/2}} \; ,\]

which correspond to a projected density of the form

\[ \Sigma(r^2) = \left(\frac{1}{1 + r^2}\right)^2 \; .\]

The associated velocity profile is

\[ v^2(r) = \frac{r^2}{(1 + r^2)^{3/2}} \; .\]

JaffeProfile()

A pseudo-Jaffe profile.

This is the difference of a singular and a non-singular isothermal profiles.

The volume density is of the form

\[ \rho(r) = \frac{1}{r^2} - \frac{1}{r^2 + 1} = \frac{1}{r^2 (r^2 + 1)} \; ,\]

which correspond to a projected density of the form

\[ \Sigma(r) = \frac{1}{r} - \frac{1}{\sqrt{1 + r^2}} \; .\]

The associated velocity profile is

\[ v^2(r) = \frac{\arctan(r)}{r} \; .\]

HernquistProfile()

An Hernquist profile.

The volume density is of the form

\[ \rho(r) = \frac{1}{r (1 + r)^3} \; .\]

The surface density has a closed analytical form, but it is somewhat complicated.

The associated velocity profile is

\[ v^2(r) = \frac{r}{(1 + r)^2} \; .\]

NFWProfile()

A Navarro-Frenk-White (NFW) profile.

The volume density is of the form

\[ \rho(r) = \frac{1}{r (1 + r)^2} \; .\]

The surface density has a closed analytical form, but it is somewhat complicated.

The associated velocity profile is

\[ v^2(r) = \frac{\log(1 + r)}{r} - \frac{1}{1 + r} \; .\]

PowerLawProfile(γ)

A power-law profile.

The volume density is of the form

\[ \rho(r) = r^{-\gamma} \; ,\]

which correspond to a surface brightness of the form

\[ \Sigma(r) = r^{1 - \gamma} \; .\]

The associated velocity profile is

\[ v^2(r) = r^{2 - \gamma} \; .\]

SoftPowerLawProfile(γ)

A softened power-law profile.

The volume density is of the form

\[ \rho(r^2) = (r^2 + 1)^{-\gamma/2} \; ,\]

which correspond to a surface brightness of the form

\[ \Sigma(r^2) = (r^2 + 1)^{(1 - \gamma)/2} \; .\]

The associated velocity profile is

\[ v^2(r) = r^2 {}_2 \mathrm{F}_1(3/2, \gamma/2, 5/2, -r^2) \; ,\]

where ${}_2 \mathrm{F}_1$ is the hypergeometric function.

SersicProfile([n=4])

A Sersic profile, a surface brightness profile only.

The surface brightness is defined as

\[ \Sigma(r) = \mathrm{e}^{- b_n (r^{1/n} - 1)} \; .\]

The parameter $b_n$ is computed from the Sersic index $n$ using the approximation given by Ciotti & Bertin (1999). The default value for $n$ is 4, corresponding to a de-Vaucouleurs profile.

There is no velocity associated to this profile.

Profile(f)

A generic profile.

This profile is defined by a function f(r²) that computes both the projected mass distribution and the squared velocity as a function of the squared (possibly elliptical) radius r².

The function f must be a callable object that accepts a single argument of returns a single value of the same type. The function must be defined for non-negative values of r².

MagnifiedProfile(profile, m)

A magnified profile.

This profile is the same as the input profile, but magnified in projected density by a factor m:

\[ \Sigma(r) = m \times \Sigma_0(r) \; .\]

where $\Sigma_0$ is the projected density of the input profile.

The same scaling factor applies to the squared velocity:

\[ v^2(r) = m \times v_0^2(r) \; .\]

Therefore, the velocity is rescaled by a factor $\sqrt{m}$.

The operator * can be used to create a magnified profile.

ScaledProfile(profile, ξ)

A scaled profile.

This profile is the same as the input profile, but scaled in size by a factor $r_0$. This is useful to model sources with different scale radii.

The projected density is defined as

\[ \Sigma(r) = \Sigma_0(r / r_0) \; .\]

where $\Sigma_0$ is the projected density of the input profile.

The scaling factor $r_0$ applies to he squared velocity through the relation

\[ v^2(r) = v_0^2(r / r_0) r_0^2 \; .\]

These equations ensure that the size of isophotes of the profile are modified by a factor $r_0$, and also that the size of the full object increases correspondingly: as a result the mass increases as $r_0^3$ and the squared velocity as $r_0^2$.

Note that, for exponential profiles, the scaling rule is

\[ v^2(r) = v_0^2(r / r_0) r_0 \; ,\]

since it represents a flat object (so that $M \propto r_0^2$).

The operator % can be used to create a scaled profile.

CompositeProfile(profiles...)

A composite profile.

This profile is the sum of set of profiles: both the projected density and the squared velocity are the sum of the corresponding quantities of the input profiles.

The operator + can be used to create a composite profile.