Available model components

A quick list of all available model components can be obtained by calling the zoo() function:

>>> from socca.models import zoo
>>> zoo()

Radial models
=============
Beta
gNFW
Power
TopHat
Sersic
Gaussian
Exponential
PolyExponential
PolyExpoRefact
ModExponential

Filaments
============
SimpleBridge
MesaBridge

Disk-like model
===============
Disk

Other models
============
Point
Background

Radial profiles

A major class of model components implemented in socca are those based on radial profiles. These are two-dimensional surface brightness distributions that are axially symmetric around a centroid position, and whose radial dependence is described by a specific functional form. The radial profiles can be further generalized to account for projected ellipticity, boxiness, and position angle, allowing for a more flexible description of the source morphology.

Each radial profile component inherits from the abstract base class Profile, which defines the common interface and geometric parameters shared by all radial profiles. These include the centroid coordinates (xc, yc), position angle (theta, measured east from north), projected ellipticity (e, i.e. 1 - axis ratio), and boxiness (cbox).

>>> from socca.models import Profile
>>> comp = Profile()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness

These geometric parameters are used internally to build a generalized radial distance grid that accounts for ellipticity, position angle, and boxiness. For each pixel in the image, the distance from the centroid is computed as:

\[ r = \left( |x'|^{2+c} + \left|\frac{y'}{1-e}\right|^{2+c} \right)^{\frac{1}{2+c}}, \]

where \((x', y')\) are the rotated coordinates:

\[\begin{split} \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} -\cos(y_c)\sin(\theta) & -\cos(\theta) \\ \hspace{10pt}\cos(y_c)\cos(\theta) & -\sin(\theta) \end{bmatrix} \begin{bmatrix} x - x_c \\ y - y_c \end{bmatrix} \end{split}\]

This formulation ensures that the rotation is correctly applied in spherical coordinates, accounting for the cosine projection factor at the centroid declination \(y_c\). The boxiness parameter \(c\) allows for deviations from pure elliptical isophotes, with \(c=0\) corresponding to perfect ellipses, while \(c > 0\) produces boxy shapes and \(c < 0\) yields disky isophotes (but \(c > -1\) to ensure physical validity).

Caution

Profile is an abstract base class and does not provide an implementation of any specific radial profile. All the radial profile components, however inherit from this base class and therefore share the same set of geometric parameters listed above.

Available models

Below is a list of all the radial profile components currently implemented in socca.

Beta

The Beta class implements a generalized version of the \(\beta\) model (Cavaliere & Fusco-Femiano 1976) commonly used to describe the projected surface brightness profile of galaxy clusters under the assumption of an isothermal intracluster medium. The functional form of the profile is given by:

\[ I(r) = I_c \left[1 + \left(\frac{r}{r_c}\right)^{\alpha}\right]^{-\beta} \]

The parameter \(\alpha\) controls the radial exponent, with \(\alpha = 2\) corresponding to the standard \(\beta\) model.

>>> from socca.models import Beta
>>> comp = Beta()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
rc    [deg] : None       | Core radius
Ic  [image] : None       | Central surface brightness
alpha    [] : 2.0000E+00 | Radial exponent
beta     [] : 5.5000E-01 | Slope parameter

gNFW

The gNFW class implements the generalized Navarro-Frenk-White (gNFW) profile (Nagai et al. 2007, Mroczkowski et al. 2009) commonly used to describe the pressure profile of the intracluster medium in galaxy clusters. The projected two-dimensional profile is obtained by integrating the three-dimensional gNFW profile along the line of sight. In socca, this is implemented as follows:

\[ I(r) = I_c \times 2\int_{r/r_c}^{+\infty} x^{-\gamma}(1+x^{\alpha})^{\frac{\gamma - \beta}{\alpha}} \tfrac{x}{\sqrt{x^2 - x_r^2}} \mathrm{d}x, \]

where \(x\) is a dimensionless radius in units of the core radius \(r_c\). The parameters \(\alpha\), \(\beta\), and \(\gamma\) control the intermediate, outer, and inner slopes of the profile, respectively. The normalization \(I_c\) has units of surface brightness in units of the input data per unit length.

The integral is computed numerically using Gauss-Kronrod adaptive quadrature, as implemented in quadax. To improve computational efficiency, the profile is evaluated over a logarithmically spaced grid of dimensionless radii and then interpolated onto the two-dimensional coordinate grid on the fly during model evaluation.

>>> from socca.models import gNFW
>>> comp = gNFW()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
rc    [deg] : None       | Scale radius
Ic  [image] : None       | Characteristic surface brightness
alpha    [] : 1.0510E+00 | Intermediate slope
beta     [] : 5.4905E+00 | Outer slope
gamma    [] : 3.0810E-01 | Inner slope

Power

The Power class implements a simple power-law surface brightness profile. The functional form of the profile is given by:

\[ I(r) = I_c \left(\frac{r}{r_c}\right)^{-\alpha} \]

where \(I_c\) is the characteristic surface brightness at the scale radius \(r_c\), and \(\alpha\) is the power-law slope. For positive \(\alpha\) values, the profile decreases with radius.

>>> from socca.models import Power
>>> comp = Power()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
rc    [deg] : None       | Scale radius
Ic  [image] : None       | Characteristic surface brightness
alpha    [] : 2.0000E+00 | Power law slope

TopHat

The TopHat class implements a uniform (boxcar) surface brightness profile within a cutoff radius. The functional form of the profile is given by:

\[\begin{split} I(r) = \begin{cases} 1 & \text{if } |r| < r_c \\ 0 & \text{otherwise} \end{cases} \end{split}\]

This profile is useful for modeling flat-topped emission regions or as a simple masking function. Note that unlike other radial profiles, the TopHat does not have an amplitude parameter — it simply returns unity within the cutoff radius.

>>> from socca.models import TopHat
>>> comp = TopHat()
>>> comp.parameters()

Model parameters
================
rc    [deg] : None       | Cutoff distance

Note

The TopHat profile does not inherit the standard geometric parameters (xc, yc, theta, e, cbox) from the Profile base class. It only defines the cutoff radius rc.

Sersic

The Sersic class implements the Sérsic profile (Sérsic 1963), a widely used empirical model for describing the surface brightness distribution of galaxies and other extended sources. The functional form of the profile is given by:

\[ I(r) = I_e \exp\left\{-b_n \left[ \left( \frac{r}{r_e} \right)^{1/n} - 1 \right]\right\}, \]

where \(r_e\) is the effective radius enclosing half of the total flux, \(I_e\) is the surface brightness at \(r_e\), and \(n\) is the Sérsic index controlling the concentration of the profile. The constant \(b_n\) is defined such that \(\Gamma(2n) = 2\gamma(2n, b_n)\), where \(\Gamma\) and \(\gamma\) are the complete and incomplete Gamma functions, respectively. For computational efficiency, the value of \(b_n\) is interpolated at each step of the inference from a precomputed lookup table covering the range \(n \in [0.25, 10]\).

The Sérsic profile encompasses several commonly used models as special cases, including the Gaussian (\(n=0.5\)), the exponential (\(n = 1\)), and the de Vaucouleurs (\(n = 4\)) profiles.

>>> from socca.models import Sersic
>>> comp = Sersic()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
re    [deg] : None       | Effective radius
Ie  [image] : None       | Surface brightness at re
ns       [] : 5.0000E-01 | Sersic index

Gaussian

The Gaussian class implements a Gaussian surface brightness profile. The functional form of the profile is given by:

\[ I(r) = I_s \exp\left\{-\frac{1}{2}\left(\frac{r}{r_s}\right)^2\right\}, \]

where \(I_s\) is the central surface brightness and \(r_s\) is the scale radius (standard deviation of the Gaussian). The Gaussian profile is equivalent to a Sérsic profile with Sérsic index \(n = 0.5\). The half-width at half-maximum (HWHM) is approximately \(1.177 \, r_s\). Still, it is provided as a separate component for computational convenience and efficiency.

>>> from socca.models import Gaussian
>>> comp = Gaussian()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
rs    [deg] : None       | Scale radius
Is  [image] : None       | Central surface brightness

Exponential

The Exponential class implements a simple exponential surface brightness profile, commonly used to describe galactic disks and other extended systems with smoothly declining radial profiles:

\[ I(r) = I_s \exp\left(-\frac{r}{r_s}\right), \]

where \(I_s\) is the central surface brightness and \(r_s\) is the exponential scale radius. The exponential profile is equivalent to a Sérsic profile with Sérsic index \(n = 1\), \(I_e=I_s \exp\left({-b_{1}}\right)\), and \(r_e = b_{1} r_s\), where \(b_{1} \approx 1.67835\). As for Gaussian, this is implemented as a separate component for convenience.

>>> from socca.models import Exponential
>>> comp = Exponential()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
rs    [deg] : None       | Scale radius
Is  [image] : None       | Central surface brightness

PolyExponential

The PolyExponential class implements a generalized exponential surface brightness profile in which the radial decay is modulated by a polynomial function of radius. This profile was introduced by Mancera Piña et al. (2024). The functional form of this model is given by:

\[ I(r) = I_s \left[ 1 + \sum_{k=1}^{4} c_k \left(\frac{r}{r_c}\right)^k \right] \exp\!\left(-\frac{r}{r_s}\right). \]

where \(I_s\) is the central surface brightness, \(r_s\) is a characteristic scale radius, and the coefficients \(\{c_k\}\) define the polynomial modulation of the exponential decay. This parameterization allows for flexible deviations from a pure exponential profile while preserving a smooth radial behavior.

>>> from socca.models import PolyExponential
>>> comp = PolyExponential()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
rs    [deg] : None       | Scale radius
Is  [image] : None       | Central surface brightness
c1       [] : 0.0000E+00 | Polynomial coefficient 1
c2       [] : 0.0000E+00 | Polynomial coefficient 2
c3       [] : 0.0000E+00 | Polynomial coefficient 3
c4       [] : 0.0000E+00 | Polynomial coefficient 4
rc    [deg] : 2.7778E-04 | Reference radius for polynomial terms

PolyExpoRefact

The PolyExpoRefact class provides a refactored polynomial–exponential profile that is mathematically equivalent to PolyExponential but expressed in a form that improves parameter interpretability and reduces the degeneracies among the different polynomial terms:

\[ I(r) = \left[I_s+\sum_{k=1}^{4} I_k \left(\frac{r}{r_s}\right)^k\right] \exp\!\left(-\frac{r}{r_s}\right), \]

>>> from socca.models import PolyExpoRefact
>>> comp = PolyExpoRefact()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
rs    [deg] : None       | Scale radius
Is  [image] : None       | Central surface brightness
I1  [image] : 0.0000E+00 | Polynomial intensity coefficient 1
I2  [image] : 0.0000E+00 | Polynomial intensity coefficient 2
I3  [image] : 0.0000E+00 | Polynomial intensity coefficient 3
I4  [image] : 0.0000E+00 | Polynomial intensity coefficient 4
rc    [deg] : 2.7778E-04 | Reference radius for polynomial terms

ModExponential

The ModExponential class implements a modified exponential surface brightness profile in which the exponential term is altered by a power-law modification at large radii. The functional form of this model is given by:

\[ I(r) = I_s \exp\!\left(-\frac{r}{r_s}\right) \left(1 + \frac{r}{r_m}\right)^{\alpha}. \]

>>> from socca.models import ModExponential
>>> comp = ModExponential()
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
rs    [deg] : None       | Scale radius
Is  [image] : None       | Central surface brightness
rm    [deg] : None       | Modification radius
alpha    [] : None       | Modification exponent

Truncation

Any radial profile can be multiplied by a smooth truncation function that suppresses emission beyond a characteristic radius \(r_t\). This is controlled by attaching a Truncation object to a profile via its truncation attribute. When set, two additional parameters are automatically registered on the profile: the truncation radius rt and the truncation width wt.

socca provides two built-in truncation functions, both housed in socca.models.radial.truncation:

Exponential truncation

The Exponential model applies a one-sided exponential roll-off:

\[ T(r) = \frac{1 - e^{(r-r_t)/w_t}}{1 - e^{-r_t/w_t}}, \]

clipped to zero for \(r > r_t\). The denominator ensures \(T(0) = 1\).

Hyperbolic-tangent truncation

HyperTangent uses a \(\tanh\) roll-off:

\[ T(r) = \frac{1 - \tanh\!\left((r - r_t)/w_t\right)}{1 - \tanh\!\left(-r_t/w_t\right)}. \]

Unlike the exponential variant the weight never reaches exactly zero, but decays rapidly for \(r \gg r_t\).

In both cases the final surface brightness is \(I(r) \times T(r)\). The parameter \(w_t\) controls the sharpness of the transition, such that smaller values yield a steeper cut-off.

Truncation can be attached at construction time or set afterwards:

>>> from socca.models import Beta
>>> from socca.models.radial.truncation import Exponential, HyperTangent
>>>
>>> # Attach at construction, using the default HyperTangent
>>> comp = Beta(truncation=True)
>>>
>>> # Or pass an explicit truncation object
>>> comp = Beta(truncation=Exponential(rt=1.0e-2, wt=5.0e-4))
>>>
>>> # Or assign after construction
>>> comp = Beta()
>>> comp.truncation = HyperTangent()
>>> comp.rt = 1.0e-2
>>> comp.wt = 5.0e-4
>>>
>>> comp.parameters()

Model parameters
================
xc    [deg] : None       | Right ascension of centroid
yc    [deg] : None       | Declination of centroid
theta [rad] : 0.0000E+00 | Position angle (east from north)
e        [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
cbox     [] : 0.0000E+00 | Projected boxiness
rc    [deg] : None       | Core radius
Ic  [image] : None       | Central surface brightness
alpha    [] : 2.0000E+00 | Radial exponent
beta     [] : 5.5000E-01 | Slope parameter
rt    [deg] : 1.0000E-02 | Truncation radius
wt    [deg] : 5.0000E-04 | Truncation width

Passing truncation=True is shorthand for using HyperTangent with its default parameter values. To remove a truncation that was previously set, assign truncation=None.

Note

Truncation parameters rt and wt are treated as ordinary profile parameters: they can be fixed to a numerical value, assigned a prior distribution for fitting, or tied to another parameter via a lambda function.

Custom radial profiles

In case none of the built-in Profile subclasses adequately describe the desired surface-brightness distribution, socca provides a simple interface for defining custom radial profiles through the CustomProfile class. In such a case, the user must provide two key ingredients:

• A list of parameter specifications (parameters), each in the form of a dictionary comprising the name, unit, and description fields for the corresponding variable. Each parameter specified in this way is automatically registered as an attribute of the model and integrated into the standard socca parameter handling.

• A callable profile function (profile). The signature of the callable profile function must include the radial coordinate r as the first argument, followed by all the model parameters with name matching the argument names matching those specified in the parameter list. The function must return the surface brightness at radius r given the input parameter values. For consistency with the modelling framework, the profile function must be implemented using JAX operations to ensure compatibility with JAX’s just-in-time compilation and automatic differentiation capabilities.

>>> import jax.numpy as jp
>>> from socca.models import CustomProfile
>>>
>>> def custom_profile(r,scale):
>>>     return scale * jp.exp(-r)
>>>
>>> parameters = [{'name':'scale',
>>>                'unit':'image',
>>>                'description':'Scaling factor'}]
>>>
>>> comp = CustomProfile(parameters=parameters, profile=custom_profile)
>>> comp.parameters()

Model parameters
================
xc      [deg] : None       | Right ascension of centroid
yc      [deg] : None       | Declination of centroid
theta   [rad] : 0.0000E+00 | Position angle (east from north)
e          [] : 0.0000E+00 | Projected axis ratio
cbox       [] : 0.0000E+00 | Projected boxiness
scale [image] : None       | Scaling factor

Disk with finite thickness

The Disk class implements a three-dimensional model for describing the emission from disk galaxies, including their finite thickness and inclination effects, particularly relevant when modeling edge-on systems. This is achieved by modeling the disk component by combining a radial surface brightness profile with a vertical structure model. The radial profile \(s(r)\) describes the distribution of light in the disk plane, while a vertical profile \(h(r,z)\) accounts for the thickness and inclination of the disk along the line of sight. The overall surface brightness distribution is obtained by integrating the three-dimensional model along the line of sight:

\[ I(r) = \int_{-l_{\mathrm{max}}}^{+l_{\mathrm{max}}} s(R) \, h(R,z) \, \mathrm{d}l, \]

where \(R\) and \(z\) are the cylindrical coordinates in the disk frame, while \(r\) and \(l\) denote the projected radius and line-of-sight coordinate in the observer frame, respectively. The transformation between the two coordinate systems depends on the inclination angle of the disk with respect to the line-of-sight direction. A detailed discussion of the coordinate transformations involved in this calculation can be found in van Asselt et al. (2026).

>>> from socca.models import Disk
>>> comp = Disk()
>>> comp.parameters()

Model parameters
================
radial.xc         [deg] : None       | Right ascension of centroid
radial.yc         [deg] : None       | Declination of centroid
radial.theta      [rad] : 0.0000E+00 | Position angle (east from north)
radial.e             [] : 0.0000E+00 | Projected ellipticity (1 - axis ratio)
radial.cbox          [] : 0.0000E+00 | Projected boxiness
radial.re         [deg] : None       | Effective radius
radial.Ie       [image] : None       | Surface brightness at re
radial.ns            [] : 5.0000E-01 | Sersic index
vertical.inc      [rad] : 0.0000E+00 | Inclination angle (0=face-on)
vertical.zs       [deg] : None       | Scale height

Hyperparameters
===============
vertical.losdepth [deg] : 2.7778E-03 | Half line-of-sigt extent for integration
vertical.losbins     [] : 2.0000E+02 | Number of points for line-of-sight integration

If compared to any of the radial profile components described in the previous section, it is possible to observe many differences in the model and parameter interface. First of all, the parameters are split between radial and vertical subgroups. The former contains all the parameters defining the radial profile on the disk plane. These are inherited from the reference radial profile class specified when initializing the Disk component. As it will be shown later on in more details, any of the radial profiles described in the previous section can be used. The vertical subset instead includes the parameters related to the vertical structure (including the inclination of the disk plane with respect to the line-of-sight direction, inc). More details on the specific meaning of each parameter will be provided below. As for the radial component, the vertical structure can be customized by the user.

Second, the Disk class implements a set of hyperparameters that do not enter the inference process, but tune key aspects of the model generation. In this specific case, they control the numerical accuracy of the line-of-sight integration. To ensure computational efficiency, the integration is currently computed via a trapezoid method (as implemented in jax.numpy.trapezoid) over a arbitrary finite extent \([-l_{\mathrm{max}}, +l_{\mathrm{max}}]\). The Height hyperparameters include the half line-of-sight extent losdepth (i.e., the \(l_{\mathrm{max}}\) parameter in the equation above) and the number of integration points losbins. Increasing these values improves the accuracy of the model, at the cost of increased computational time.

Important

When accessing a specific variable, it is important to remember that both the radial and vertical parameters are stored as sub-attributes of the radial and vertical attributes of the Disk component, respectively. For instance, to access the effective radius parameter of the radial profile, one must use comp.radial.re, while the scale height of the vertical structure is stored in comp.vertical.zs.

Vertical profiles

The hyperparameters introduced above as well as the inclincation angle inc are inherited by Disk from the model for the vertical structures. These derived from a dedicated abstract base class, Height, which defines the common interface for all vertical profiles. It is stored in the socca.models.disk.vertical submodule:

>>> from socca.models.disk.vertical import Height
>>> comp = Height()
>>> comp.parameters()

Model parameters
================
inc      [rad] : 0.0000E+00 | Inclination angle (0=face-on)

Hyperparameters
===============
losdepth [deg] : 2.7778E-03 | Half line-of-sigt extent for integration
losbins     [] : 2.0000E+02 | Number of points for line-of-sight integration

Currently, socca implements two vertical profiles: the power of a hyperbolic secant profile (HyperSecantHeight) and the exponential profile (ExponentialHeight).

Caution

Please note that the Height subclasses can not be used as standalone model components, as they do not provide a complete two-dimensional surface brightness distribution. They can only be used in combination with a radial profile within the Disk component.

HyperSecantHeight

The HyperSecantHeight class implements a hyperbolic secant vertical profile raised to a power \(\alpha\):

\[ h(z) = \mathrm{sech}^{\alpha}\left(|z|/z_s\right). \]

Here, \(z_s\) is the scale height of the disk, and \(\alpha\) is an exponent that controls the sharpness of the vertical profile. By default, \(\alpha = 2\), corresponding to the squared hyperbolic secant profile commonly used to describe the vertical structure of disk galaxies (van der Kruit & Searle 1981).

>>> from socca.models.disk.vertical import HyperSecantHeight
>>> comp = HyperSecantHeight()
>>> comp.parameters()

Model parameters
================
inc      [rad] : 0.0000E+00 | Inclination angle (0=face-on)
zs       [deg] : None       | Scale height
alpha       [] : 2.0000E+00 | Exponent to the hyperbolic secant

Hyperparameters
===============
losdepth [deg] : 2.7778E-03 | Half line-of-sigt extent for integration
losbins     [] : 2.0000E+02 | Number of points for line-of-sight integration

ExponentialHeight

The ExponentialHeight class implements an exponential vertical profile. The functional form of this profile is given simply by:

\[ h(z) = \exp\left(-|z|/z_s\right). \]

As in the previous case, \(z_s\) is the scale height of the disk.

>>> from socca.models.disk.vertical import ExponentialHeight
>>> comp = ExponentialHeight()

Model parameters
================
inc      [rad] : 0.0000E+00 | Inclination angle (0=face-on)
zs       [deg] : None       | Scale height

Hyperparameters
===============
losdepth [deg] : 2.7778E-03 | Half line-of-sigt extent for integration
losbins     [] : 2.0000E+02 | Number of points for line-of-sight integration

Customizing the radial and vertical components

By default, the radial profile is modeled using a Sérsic profile, while the vertical structure is described by a squared hyperbolic secant function (HyperSecantHeight with \(\alpha=2\)). As mentioned above, both the radial and vertical profiles can however be customized by providing alternative profile classes when instantiating the Disk component.

>>> from socca.models import Disk, Exponential
>>> from socca.models.disk.vertical import ExponentialHeight
>>> comp = Disk(radial=Exponential(), vertical=ExponentialHeight())

Bridge/Filament models

The bridge models in socca are designed for modeling elongated emission structures such as intracluster bridges and filaments connecting galaxy clusters. These models describe the surface brightness distribution by combining a radial profile (perpendicular to the bridge axis) with a parallel profile (along the bridge axis), allowing for flexible modeling of asymmetric, elongated emission.

All bridge models share a common set of geometric parameters inherited from the Bridge base class:

• xc, yc: Centroid coordinates of the bridge in degrees

• rs: Scale radius shared by both radial and parallel profiles

• Is: Scale intensity of the bridge

• theta: Position angle of the bridge major axis (measured east from north)

• e: Projected ellipticity (1 - axis ratio) controlling the relative extent of radial vs. parallel profiles

The scale radius rs is applied directly to the parallel profile, while the radial profile uses rs * (1 - e) to create an elongated structure.

SimpleBridge

The SimpleBridge class combines radial and parallel profiles multiplicatively, producing a surface brightness distribution of the form:

\[ I(r, z) = I_s ~ f_{\mathrm{radial}}(r) ~ f_{\mathrm{parallel}}(z) \]

This creates structures where the emission is the product of the two independent profile functions. By default, the radial profile is a Beta model and the parallel profile is a TopHat, producing a uniform distribution along the bridge length with a smooth transverse profile.

>>> from socca.models import SimpleBridge
>>> comp = SimpleBridge()
>>> comp.parameters()

Model parameters
================
xc              [deg] : None       | Right ascension of the bridge centroid
yc              [deg] : None       | Declination of the bridge centroid
rs              [deg] : None       | Scale radius of the bridge profiles
Is            [image] : None       | Scale intensity of the bridge profiles
theta           [rad] : 0.0000E+00 | Position angle of the bridge major axis
e                  [] : 5.0000E-01 | Projected ellipticity of the bridge profiles
radial.alpha       [] : 2.0000E+00 | Radial exponent
radial.beta        [] : 5.5000E-01 | Slope parameter

MesaBridge

The MesaBridge class combines radial and parallel profiles using a harmonic mean, producing a mesa-like (flat-topped) surface brightness distribution:

\[ I(r, z) = \frac{I_s}{\frac{1}{f_{\mathrm{radial}}(r)} + \frac{1}{f_{\mathrm{parallel}}(z)}} \]

This creates smooth transitions between a flat central region and declining edges, resembling a mesa or table-top shape. The default configuration uses a Beta profile for the radial direction and a Power law profile for the parallel direction, with steep slopes chosen to produce characteristic mesa-like edges.

>>> from socca.models import MesaBridge
>>> comp = MesaBridge()
>>> comp.parameters()

Model parameters
================
xc              [deg] : None       | Right ascension of the bridge centroid
yc              [deg] : None       | Declination of the bridge centroid
rs              [deg] : None       | Scale radius of the bridge profiles
Is            [image] : None       | Scale intensity of the bridge profiles
theta           [rad] : 0.0000E+00 | Position angle of the bridge major axis
e                  [] : 5.0000E-01 | Projected ellipticity of the bridge profiles
radial.alpha       [] : 8.0000E+00 | Radial exponent
radial.beta        [] : 1.0000E+00 | Slope parameter
parallel.alpha     [] : 8.0000E+00 | Power law slope

This model is an adaptation of the intracluster bridge parameterization introduced by Hincks et al. (2022).

Customizing bridge profiles

Both SimpleBridge and MesaBridge allow customization of the radial and parallel profiles by providing alternative profile classes at instantiation:

>>> from socca.models import SimpleBridge, Gaussian, Exponential
>>> comp = SimpleBridge(radial=Gaussian(), parallel=Exponential())

Note

The PolyExpoRefact, Background, and Point profiles are not supported as bridge components due to their special parameter handling requirements.

Other components

socca also includes a few additional model components that do not fall into the radial profile category or its finite-thickness disk generalization.

Point source

The Point class implements a point-like source, representing an unresolved object whose intrinsic spatial extent is negligible compared to the instrumental resolution. The surface brightness distribution is modeled as a Dirac delta function centered at the source position,

\[ I(x_c,y_c) = I_c\,\delta(x-x_c)\delta(y-y_c), \]

where \(I_{c}\) is the peak surface brightness (or total flux) and \((x_c,y_c)\) denote the source coordinates.

>>> from socca.models import Point
>>> comp = Point()
>>> comp.parameters()

Model parameters
================
xc   [deg] : None       | Right ascension
yc   [deg] : None       | Declination
Ic [image] : None       | Peak surface brightness

Caution

To allow for sub-pixel positioning of the point source, the Point component is computed directly in Fourier space as a constant function across all spatial frequencies of amplitude \(I_c\) multipltied by a complex phase factor encoding the source position. The resulting model is then multiplied by the Fourier transform of the instrumental PSF, and transformed back to image space for comparison with the data. For this reason, fitting a point source without providing a PSF model might lead to artifacts in the resulting image.

Background

The Background class implements a two-dimensional polynomial background model defined on the image plane. This is intended to model large-scale background variations due to instrumental effects, sky emission, or residual systematics not captured by other model components. The surface brightness is modeled as a Cartesian polynomial in the sky coordinates \((x, y)\), expressed in units of a characteristic scale \(r_s\):

\[\begin{split} \begin{aligned} I(x,y) =\;& a_0 + a_{1x}\,\frac{x}{r_s} + a_{1y}\,\frac{y}{r_s} \\ &+ a_{2xx}\,\left(\frac{x}{r_s}\right)^2 + a_{2xy}\,\frac{x y}{r_s^2} + a_{2yy}\,\left(\frac{y}{r_s}\right)^2 \\ &+ a_{3xxx}\,\left(\frac{x}{r_s}\right)^3 + a_{3xxy}\,\frac{x^2 y}{r_s^3} + a_{3xyy}\,\frac{x y^2}{r_s^3} + a_{3yyy}\,\left(\frac{y}{r_s}\right)^3. \end{aligned} \end{split}\]

>>> from socca.models import Background
>>> comp = Background()
>>> comp.parameters()

Model parameters
================
rs [deg] : 2.7778E-04 | Reference radius for polynomial terms
a0    [] : None       | Polynomial coefficient 0
a1x   [] : 0.0000E+00 | Polynomial coefficient 1 in x
a1y   [] : 0.0000E+00 | Polynomial coefficient 1 in y
a2xx  [] : 0.0000E+00 | Polynomial coefficient 2 in x*x
a2xy  [] : 0.0000E+00 | Polynomial coefficient 2 in x*y
a2yy  [] : 0.0000E+00 | Polynomial coefficient 2 in y*y
a3xxx [] : 0.0000E+00 | Polynomial coefficient 3 in x*x*x
a3xxy [] : 0.0000E+00 | Polynomial coefficient 3 in x*x*y
a3xyy [] : 0.0000E+00 | Polynomial coefficient 3 in x*y*y
a3yyy [] : 0.0000E+00 | Polynomial coefficient 3 in y*y*y

Warning

By construction, the Background model is not convolved with the instrumental PSF, as it is assumed to vary smoothly on scales much larger than the PSF size.