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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.

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.

Parameters:

Name Type Description Default
u array - like

Baseline u coordinates in wavelength units.

 required
v array - like

Baseline v coordinates in wavelength units.

 required
dra float or array - like

Right-ascension offset of the companion in milliarcseconds, positive to the East.

 required
ddec float or array - like

Declination offset of the companion in milliarcseconds, positive to 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 visibility amplitude follows the classic uniform-disk form 2 * J1(x) / x, with x = pi * diam_rad * base_norm the product of the disk diameter (in radians) and the baseline length in wavelength units (base_norm = hypot(u, v), i.e. baseline length divided by wavelength).

Parameters:

Name Type Description Default
u array - like

Baseline u coordinates in wavelength units (cycles / rad).

 required
v array - like

Baseline v coordinates in wavelength units (cycles / rad).

 required
diam float or array - like

Diameter of the uniform disk in milliarcseconds.

 required
dra float or array - like

Right-ascension offset in milliarcseconds.

 0.0
ddec float or array - like

Declination offset in milliarcseconds.

 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)

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

For \(I(\mu) = \sum_\nu a_\nu \mu^\nu\), with \(\mu = \sqrt{1 - (r/R)^2}\), Quirrenbach et al. (1996, A&A 312, 160, eqs. 1-4) show that

\[ V(x) = \frac{1}{C} \sum_\nu a_\nu\, 2^{\nu/2}\, \Gamma\!\left(\frac{\nu}{2} + 1\right) \frac{J_{\nu/2+1}(x)}{x^{\nu/2+1}}, \qquad C = \sum_\nu \frac{a_\nu}{\nu + 2}, \]

with \(x = \pi\,\theta\,|b| / \lambda\) for diameter \(\theta\), normalized to 1 at zero baseline. A uniform disk (\(a_0 = 1\)) gives \(2 J_1(x)/x\). The powers need not be integers: the square-root law has \(\nu = 1/2\), which needs order \(5/4\), from jaxbessel's bessel_jv_over_xv, which takes orders up to 12, so \(-2 < \nu \le 22\) (the lower limit keeps the flux finite). harmonix (Dholakia & Pope 2025) generalizes the result to polynomial limb darkening of spherical-harmonic maps.

Parameters:

Name Type Description Default
u array - like

Baseline u coordinates in wavelength units (cycles / rad).

 required
v array - like

Baseline v coordinates in wavelength units (cycles / rad).

 required
diam float or array - like

Limb-darkened diameter in milliarcseconds.

 required
coeffs array - like

Coefficients \(a_\nu\), one per power (traceable).

 required
powers sequence of float

Powers \(\nu\) of \(\mu\), each with \(-2 < \nu \le 22\) (static).

 required
dra float or array - like

Right-ascension offset in milliarcseconds.

 0.0
ddec float or array - like

Declination offset in milliarcseconds.

 0.0

Returns:

Type Description
array - like

Complex visibility samples.

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

Compute the complex visibility for an azimuthally modulated radial dirac delta ring. The image intensity can be described in polar image coordinates as \(I(r, \theta) \propto \delta(r-r_0) \left( 1 + \sum_{m=1}^{n} A_m \cos{(m(\theta - \phi_m))} \right)\), where \(r_0\) is the ring's radial position, \(A_m\) the amplitude and \(\phi_m\) the position phase angle (defined counter-clockwise , North to East) for the m-th order modulation.

Parameters:

Name Type Description Default
u array - like

Baseline u coordinates in wavelength units (cycles / rad).

 required
v array - like

Baseline v coordinates in wavelength units (cycles / rad).

 required
r0 float or array - like

Scalar with the radial position of the ring in milliarcseconds.

 required
az_amps array - like

1D array containing amplitude coefficients for cosine azimuthal modulations. The first element is seen as the amplitude for the first-order modulation, the second as the amplitude for the second-order modulation, etc.

 required
az_phis array - like

1D array containing offset angles of the cosine azimuthal modulations, relative to the position angle of the rim's projected major axis, in degrees. The first element is seen as the offset for the first-order modulation, the second for the second-order modulation, etc.

 required

Returns:

Type Description
array - like

Complex visibility samples.
Notes

This function does not account for rotation and geometric stretching (e.g. due to inclination). A separate transformation of \(uv\) coordinates should account for this. The phase angles of the cosine modulations are defined relative to the spatial y-axis (North), turning counterclockwise to the x-axis (East). This means that a single 1st order modulation with a phase angle of \(0 \, \mathrm{deg}\) results in a bright peak towards the North, and a faint peak towards the South. A phase angle of \(90 \, \mathrm{deg}\) would result in a bright peak towards the East, and a faint one towards the West.