Limb darkening functions

Limb darkening functions#

pytransit.models.limb_darkening holds the Numba-compiled limb darkening profiles that RoadRunnerModel uses. See Limb darkening for the narrative description and for how to supply your own.

Each law is defined by up to three functions:

ld_<model>(mu, pv)

The intensity profile, evaluated at the cosine of the viewing angle mu for the coefficients pv.

ldi_<model>(pv)

The disk-integrated stellar intensity \(2\pi \int_0^1 I(\mu)\, z\, dz\), where an analytic solution exists. RoadRunner uses it to normalise the model; without one, it integrates the profile numerically on every evaluation.

ldd_<model>(mu, pv)

The derivatives of the profile, where implemented.

A model is passed to RoadRunner either by name or as the (ld_<model>, ldi_<model>) pair.

from pytransit import RoadRunnerModel
from pytransit.models.limb_darkening import ld_power_2, ldi_power_2

tm = RoadRunnerModel('power-2')                     # by name
tm = RoadRunnerModel((ld_power_2, ldi_power_2))     # equivalently, by function pair

Note

These are Numba CPUDispatcher objects rather than plain Python functions. They can be called from Python and from other Numba-compiled functions alike, and a profile you write yourself must be @njit-decorated to be usable by the model.

Uniform#

pytransit.models.limb_darkening.ld_uniform(mu, pv)[source]#

Uniform (no limb darkening) intensity profile.

\[I(\mu) = 1\]
Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight.

  • pv (ndarray) – Unused, present for interface compatibility.

Returns:

ndarray – Stellar intensity, unity everywhere.

pytransit.models.limb_darkening.ldi_uniform(pv)[source]#

Disk-integrated intensity of the uniform profile, \(\pi\).

Linear#

pytransit.models.limb_darkening.ld_linear(mu, pv)[source]#

Linear limb darkening profile.

\[I(\mu) = 1 - u(1 - \mu)\]
Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight.

  • pv (ndarray) – Limb darkening coefficients [u].

Returns:

ndarray – Stellar intensity.

pytransit.models.limb_darkening.ldi_linear(pv)[source]#

Disk-integrated intensity of the linear profile.

\[2\pi \int_0^1 I(\mu)\, z\, dz = \frac{\pi}{3}(3 - u)\]
pytransit.models.limb_darkening.ldd_linear(mu, pv)[source]#

Quadratic#

pytransit.models.limb_darkening.ld_quadratic(mu, pv)[source]#

Quadratic limb darkening profile (Kopal 1950).

\[I(\mu) = 1 - u(1 - \mu) - v(1 - \mu)^2\]

The most widely used limb darkening law in transit modelling. It is flexible enough for most optical photometry but systematically misrepresents the profile of cool stars near the limb; see ld_power_2() for an alternative with the same number of coefficients.

Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight.

  • pv (ndarray) – Limb darkening coefficients [u, v].

Returns:

ndarray – Stellar intensity.

pytransit.models.limb_darkening.ldi_quadratic(pv)[source]#

Disk-integrated intensity of the quadratic profile.

\[\frac{\pi}{6}(6 - 2u - v)\]
pytransit.models.limb_darkening.ldd_quadratic(mu, pv)[source]#

Quadratic limb darkening model derivatives.

Quadratic limb darkening model derivatives as an array [di/dmu, di/da, di/db].

Multiply di/dmu by -z/sqrt(1-z**2) to get di/dz

Quadratic, triangular parametrisation#

pytransit.models.limb_darkening.ld_quadratic_tri(mu, pv)[source]#

Quadratic limb darkening profile with the triangular sampling parametrisation.

The quadratic law reparametrised following Kipping (MNRAS 435, 2152, 2013) as

\[u = 2\sqrt{q_1} q_2, \qquad v = \sqrt{q_1}(1 - 2 q_2)\]

A uniform prior on \((q_1, q_2) \in [0,1]^2\) maps to a uniform prior over the whole region of \((u, v)\) space that gives a physically valid (everywhere positive, monotonically decreasing) intensity profile. This makes it the right parametrization for sampling limb darkening coefficients with uninformative priors.

Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight.

  • pv (ndarray) – Triangular sampling coefficients [q1, q2], both in [0, 1].

Returns:

ndarray – Stellar intensity.

pytransit.models.limb_darkening.ldi_quadratic_tri(pv)[source]#

Disk-integrated intensity of the triangularly parametrised quadratic profile.

Power-2#

pytransit.models.limb_darkening.ld_power_2(mu, pv)[source]#

Power-2 limb darkening profile (Hestroffer, A&A 327, 199, 1997).

\[I(\mu) = 1 - c(1 - \mu^{\alpha})\]

A two-coefficient law that reproduces the limb darkening of cool stars considerably better than the quadratic law, especially in the near-infrared, and is the recommended two-parameter law for M dwarfs.

Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight.

  • pv (ndarray) – Limb darkening coefficients [c, alpha].

Returns:

ndarray – Stellar intensity.

See also

ld_power_2_pm

the same law in the better-behaved (h1, h2) parametrisation.

pytransit.models.limb_darkening.ldi_power_2(pv)[source]#

Disk-integrated intensity of the power-2 profile.

pytransit.models.limb_darkening.ldd_power_2(mu, pv)[source]#

Power-2, (h1, h2) parametrisation#

pytransit.models.limb_darkening.ld_power_2_pm(mu, pv)[source]#

Power-2 limb darkening profile in the (h1, h2) parametrisation.

The power-2 law reparametrised following Maxted (A&A 622, A33, 2018) using the intensity at two fixed points on the disk,

\[h_1 = I(\mu = 1/2), \qquad h_2 = h_1 - I(\mu = 1/\sqrt{2}),\]

which are far less correlated than \((c, \alpha)\) and therefore much better behaved as free parameters in a fit.

Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight.

  • pv (ndarray) – Limb darkening coefficients [h1, h2].

Returns:

ndarray – Stellar intensity.

pytransit.models.limb_darkening.ldi_power_2_pm(pv)[source]#

Disk-integrated intensity of the (h1, h2)-parametrised power-2 profile.

Square root#

pytransit.models.limb_darkening.ld_square_root(mu, pv)[source]#

Square-root limb darkening profile (Diaz-Cordoves & Gimenez, A&A 259, 227, 1992).

\[I(\mu) = 1 - u(1 - \mu) - v(1 - \sqrt{\mu})\]
Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight.

  • pv (ndarray) – Limb darkening coefficients [u, v].

Returns:

ndarray – Stellar intensity.

pytransit.models.limb_darkening.ldi_square_root(pv)[source]#

Disk-integrated intensity of the square-root profile.

Logarithmic#

pytransit.models.limb_darkening.ld_logarithmic(mu, pv)[source]#

Logarithmic limb darkening profile (Klinglesmith & Sobieski, AJ 75, 175, 1970).

\[I(\mu) = 1 - u(1 - \mu) - v\,\mu \ln \mu\]
Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight. The profile diverges as \(\mu \to 0\), so the exact stellar limb needs care.

  • pv (ndarray) – Limb darkening coefficients [u, v].

Returns:

ndarray – Stellar intensity.

pytransit.models.limb_darkening.ldi_logarithmic(pv)[source]#

Disk-integrated intensity of the logarithmic profile.

Exponential#

pytransit.models.limb_darkening.ld_exponential(mu, pv)[source]#

Exponential limb darkening profile (Claret & Hauschildt, A&A 412, 241, 2003).

\[I(\mu) = 1 - u(1 - \mu) - \frac{v}{1 - e^{\mu}}\]
Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight.

  • pv (ndarray) – Limb darkening coefficients [u, v].

Returns:

ndarray – Stellar intensity.

pytransit.models.limb_darkening.ldi_exponential(pv)[source]#

Non-linear#

pytransit.models.limb_darkening.ld_nonlinear(mu, pv)[source]#

Four-coefficient non-linear limb darkening profile (Claret, A&A 363, 1081, 2000).

\[I(\mu) = 1 - c_1(1 - \mu^{1/2}) - c_2(1 - \mu) - c_3(1 - \mu^{3/2}) - c_4(1 - \mu^2)\]

The standard law for tabulated theoretical limb darkening coefficients. Its four coefficients are strongly correlated, so it is usually a poor choice for fitting but a good one for fixing limb darkening to model predictions.

Parameters:

  • mu (ndarray) – Cosine of the angle between the surface normal and the line of sight.

  • pv (ndarray) – Limb darkening coefficients [c1, c2, c3, c4].

Returns:

ndarray – Stellar intensity.

pytransit.models.limb_darkening.ldi_nonlinear(pv)[source]#

Disk-integrated intensity of the four-coefficient non-linear profile.

General#

pytransit.models.limb_darkening.ld_general(mu, pv)[source]#

Giménez (2006) general limb darkening model: I(mu) = 1 - sum c_i (1 - mu^(i+1)).

pytransit.models.limb_darkening.ldi_general(pv)[source]#

Evaluation helpers#

These evaluate a limb darkening model for 1D, 2D, or 3D coefficient arrays, normalising the result to the (npv, npb, nmu) layout the models expect.

pytransit.models.limb_darkening.evaluate_ld(ldm, mu, pvo)[source]#

Evaluate a limb darkening model for 1D, 2D, or 3D parameter arrays.

The parameters can be given as a 1D array with a shape (nldc), a 2D array with a shape (npb, nldc), or a 3D array with a shape (npv, npb, nldc), where nldc is the number of limb darkening coefficients, npb is the number of passbands, and npv is the number of parameter vectors.

Returns the limb darkening profiles as a 3D array with a shape (npv, npb, nmu).

pytransit.models.limb_darkening.evaluate_ldi(ldi, pvo)[source]#

Evaluate a disk-integrated stellar intensity for 1D, 2D, or 3D parameter arrays.

The parameters follow the same conventions as in evaluate_ld.

Returns the disk-integrated stellar intensities as a 2D array with a shape (npv, npb).