`virgil.models` Functions

Analytic complex visibilities, in spatial-frequency units (baselines divided by wavelength). Likelihoods are in virgil.likelihood.

Source models: sky-brightness distributions and their visibilities.

Components (PointSource, GaussianDisk, UniformDisk, the limb-darkened disks such as QuadraticLimbDarkenedDisk, ModulatedGaussianRim, MultipoleDisk (a radial profile, with its profiles such as GaussianRing), and the flared scattered-light disks such as FlaredDiskPowerLaw) are single shapes, combined with flux weights in a System.
BinaryModelCartesian and BinaryModelAngular are fast forms of a primary plus a point-source companion.
The cvis_* functions are the analytic visibilities behind them.

Every flux is relative: for a companion it is the companion/primary flux ratio. Likelihoods of these models are in virgil.likelihood.

`cvis_binary(u, v, dra, ddec, flux)`

Compute complex visibilities for a Cartesian-parameterized binary model.

Source: Martinache (2010), xara (formerly pysco), references; closure-phase contrast limits: Ireland (2013), references.

Parameters:

Name	Type	Description	Default
`u`	`array - like`	Baseline coordinates in wavelength units (cycles / rad).	required
`v`	`array - like`	Baseline coordinates in wavelength units (cycles / rad).	required
`dra`	`float or array - like`	Offset of the companion in mas, positive to the East and the North.	required
`ddec`	`float or array - like`	Offset of the companion in mas, positive to the East and the North.	required
`flux`	`float or array - like`	Companion/primary flux ratio.	required

Returns:

Type	Description
`array - like`	Complex visibility samples, normalized to 1 at zero baseline.

`cvis_uniform_disk(u, v, diam, dra=0.0, ddec=0.0)`

Compute complex visibilities for a uniform (tophat) disk.

The amplitude is 2 * J1(x) / x with x = pi * diam_rad * base_norm, the disk diameter in radians times hypot(u, v), the baseline length in wavelength units.

Parameters:

Name	Type	Description	Default
`u`	`array - like`	Baseline coordinates in wavelength units (cycles / rad).	required
`v`	`array - like`	Baseline coordinates in wavelength units (cycles / rad).	required
`diam`	`float or array - like`	Diameter of the uniform disk in mas.	required
`dra`	`float or array - like`	Right-ascension and declination offsets in mas.	`0.0`
`ddec`	`float or array - like`	Right-ascension and declination offsets in mas.	`0.0`

Returns:

Type	Description
`array - like`	Complex visibility samples.

`cvis_limb_darkened_disk(u, v, diam, coeffs, powers, dra=0.0, ddec=0.0)`

Visibilities of a disk whose brightness is a sum of powers of \(\mu\).

For \(I(\mu) = \sum_\nu a_\nu \mu^\nu\), \(\mu = \sqrt{1 - (r/R)^2}\), the visibility is a sum of \(J_{\nu/2+1}(x)/x^{\nu/2+1}\) terms with \(x = \pi\theta|b|/\lambda\), normalized to 1 at zero baseline. Powers need not be integers, with \(-2 < \nu \le 22\) (the lower limit keeps the flux finite). Formula: Limb-darkened stars.

Source: Quirrenbach et al. (1996), eqs. 1-4, references; generalized by harmonix, references.

Parameters:

Name	Type	Description	Default
`u`	`array - like`	Baseline coordinates in wavelength units (cycles / rad).	required
`v`	`array - like`	Baseline coordinates in wavelength units (cycles / rad).	required
`diam`	`float or array - like`	Limb-darkened diameter in mas.	required
`coeffs`	`array-like, sequence of float`	Coefficients \(a_\nu\) (traceable) and the powers \(\nu\) of \(\mu\) (static), one each.	required
`powers`	`array-like, sequence of float`	Coefficients \(a_\nu\) (traceable) and the powers \(\nu\) of \(\mu\) (static), one each.	required
`dra`	`float or array - like`	Right-ascension and declination offsets in mas.	`0.0`
`ddec`	`float or array - like`	Right-ascension and declination offsets in mas.	`0.0`

Returns:

Type	Description
`array - like`	Complex visibility samples.

`cvis_radial_dirac_delta_modulated(u, v, r0, az_amps, az_phis)`

Complex visibility of an azimuthally modulated radial Dirac-delta ring.

The intensity is \(I(r, \theta) \propto \delta(r-r_0)(1 + \sum_m A_m \cos[m(\theta - \phi_m)])\) with \(\theta\) counter-clockwise from North to East. Rotation and inclination are not applied: transform the \(uv\) coordinates separately. One first-order modulation with phase 0 is bright to the North and faint to the South; with phase 90, bright to the East.

Parameters:

Name	Type	Description	Default
`u`	`array - like`	Baseline coordinates in wavelength units (cycles / rad).	required
`v`	`array - like`	Baseline coordinates in wavelength units (cycles / rad).	required
`r0`	`float or array - like`	Radial position of the ring in mas.	required
`az_amps`	`array - like`	1D amplitudes \(A_m\) of the cosine modulations, and their offset angles \(\phi_m\) in degrees relative to the position angle of the projected major axis, first order first.	required
`az_phis`	`array - like`	1D amplitudes \(A_m\) of the cosine modulations, and their offset angles \(\phi_m\) in degrees relative to the position angle of the projected major axis, first order first.	required

Returns:

Type	Description
`array - like`	Complex visibility samples.

`multipole_disk_cvis(u, v, r, w, inc=0.0, pa=0.0, blur_fwhm=None, c=None, s=None, height=None, stretch=None)`

Unit-flux visibility of a disk from radial quadrature.

Provisional

May change in any 0.4.x release without a deprecation cycle.

A sum of thin rings at nodes r with weights w, with inc and pa undone, normalized so \(V(0) = 1\). The Bessel series, the height phase and the envelope are in Model notes.

Source: Jennings et al. (2020), references, and Kennedy (2025), references; ideas only.

Parameters:

Name	Type	Description	Default
`u`	`array - like`	Baseline coordinates in wavelength units (cycles/rad); any shape.	required
`v`	`array - like`	Baseline coordinates in wavelength units (cycles/rad); any shape.	required
`r`	`array - like`	Quadrature radii in mas and weights with \(I_0(r_k)\,r_k\,dr\) folded in, shape `(n_r,)` (`w` may be `u.shape + (n_r,)` for a chromatic profile); each `w` is normalized to unit sum.	required
`w`	`array - like`	Quadrature radii in mas and weights with \(I_0(r_k)\,r_k\,dr\) folded in, shape `(n_r,)` (`w` may be `u.shape + (n_r,)` for a chromatic profile); each `w` is normalized to unit sum.	required
`inc`	`float or array - like`	Inclination and position angle in degrees; FWHM in mas, in the disk plane, of a smoothing Gaussian.	`0.0`
`pa`	`float or array - like`	Inclination and position angle in degrees; FWHM in mas, in the disk plane, of a smoothing Gaussian.	`0.0`
`blur_fwhm`	`float or array - like`	Inclination and position angle in degrees; FWHM in mas, in the disk plane, of a smoothing Gaussian.	`0.0`
`c`	`array - like or None`	Linear harmonic coefficients at the nodes, shape `(M, n_r)` (row `m - 1` is order `m`; keeping the brightness non-negative is the caller's task); ring heights above the disk plane in mas, shape `(n_r,)` (`None` is flat); minor-axis foreshortening on the sky (default `cos(inc)`; negative mirrors the azimuth).	`None`
`s`	`array - like or None`	Linear harmonic coefficients at the nodes, shape `(M, n_r)` (row `m - 1` is order `m`; keeping the brightness non-negative is the caller's task); ring heights above the disk plane in mas, shape `(n_r,)` (`None` is flat); minor-axis foreshortening on the sky (default `cos(inc)`; negative mirrors the azimuth).	`None`
`height`	`array - like or None`	Linear harmonic coefficients at the nodes, shape `(M, n_r)` (row `m - 1` is order `m`; keeping the brightness non-negative is the caller's task); ring heights above the disk plane in mas, shape `(n_r,)` (`None` is flat); minor-axis foreshortening on the sky (default `cos(inc)`; negative mirrors the azimuth).	`None`
`stretch`	`array - like or None`	Linear harmonic coefficients at the nodes, shape `(M, n_r)` (row `m - 1` is order `m`; keeping the brightness non-negative is the caller's task); ring heights above the disk plane in mas, shape `(n_r,)` (`None` is flat); minor-axis foreshortening on the sky (default `cos(inc)`; negative mirrors the azimuth).	`None`

Returns:

Type	Description
`array - like`	Complex visibilities with the shape of `u`.

virgil.models Functions

cvis_binary(u, v, dra, ddec, flux)

cvis_uniform_disk(u, v, diam, dra=0.0, ddec=0.0)

cvis_limb_darkened_disk(u, v, diam, coeffs, powers, dra=0.0, ddec=0.0)

cvis_radial_dirac_delta_modulated(u, v, r0, az_amps, az_phis)

multipole_disk_cvis(u, v, r, w, inc=0.0, pa=0.0, blur_fwhm=None, c=None, s=None, height=None, stretch=None)

`virgil.models` Functions

`cvis_binary(u, v, dra, ddec, flux)`

`cvis_uniform_disk(u, v, diam, dra=0.0, ddec=0.0)`

`cvis_limb_darkened_disk(u, v, diam, coeffs, powers, dra=0.0, ddec=0.0)`

`cvis_radial_dirac_delta_modulated(u, v, r0, az_amps, az_phis)`

`multipole_disk_cvis(u, v, r, w, inc=0.0, pa=0.0, blur_fwhm=None, c=None, s=None, height=None, stretch=None)`