Power-2 model

Power-2 model#

QPower2Model implements the transit model of Maxted & Gill (A&A 622, A33, 2019) for a star with a power-2 limb darkening profile,

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

Why the power-2 law#

The quadratic law is a poor description of the limb darkening of cool stars. It has the wrong shape near the limb, and the mismatch grows towards the infrared. The power-2 law uses the same number of coefficients but follows the profiles predicted by model atmospheres for M dwarfs and other late-type stars much more closely.

The model is also fast: the qpower2 approximation is an analytic expression that avoids the elliptic integrals of the Mandel & Agol solution.

Since the approximation is built into the model, it is at its best for small planets. Consult Maxted & Gill (2019) for the accuracy as a function of radius ratio before using it for a large planet.

Usage#

from numpy import pi, array, linspace
from pytransit import QPower2Model

time = linspace(-0.1, 0.1, 1000)

tm = QPower2Model()
tm.set_data(time)

flux = tm.evaluate(k=0.1, ldc=array([0.6, 0.5]), t0=0.0, p=1.0, a=3.0, i=0.5*pi)

Note

This is the one model that needs ldc as an ndarray rather than a list. evaluate passes it straight to a Numba kernel, which cannot handle a Python list, so ldc=[0.6, 0.5] raises a TypingError.

The coefficients are \((c, \alpha)\). When fitting them, consider RoadRunner with the 'power-2-pm' law instead: it implements the same profile in the \((h_1, h_2)\) parametrisation of Maxted (2018), whose parameters are far less correlated and therefore much easier to sample. See Limb darkening.

API#

class pytransit.models.qpower2.QPower2Model[source]#

Bases: TransitModel

QPower2 transit model by Maxted & Gill (A&A, 622, A33, 2019).

evaluate(k, ldc, t0, p, a, i, e=None, w=None, copy=True)[source]#

Evaluate the transit model for a set of scalar or vector parameters.

Parameters:

  • k – Radius ratio(s) either as a single float, 1D vector, or 2D array.

  • ldc – Limb darkening coefficients as a 1D or 2D array.

  • t0 – Transit center(s) as a float or a 1D vector.

  • p – Orbital period(s) as a float or a 1D vector.

  • a – Orbital semi-major axis (axes) divided by the stellar radius as a float or a 1D vector.

  • i – Orbital inclination(s) as a float or a 1D vector.

  • e (optional) – Orbital eccentricity as a float or a 1D vector.

  • w (optional) – Argument of periastron as a float or a 1D vector.

Notes

The model can be evaluated either for one set of parameters or for many sets of parameters simultaneously. In the first case, the orbital parameters should all be given as floats. In the second case, the orbital parameters should be given as a 1D array-like.

Returns:

ndarray – Modelled flux either as a 1D or 2D ndarray.