ld

Limb darkening functions

The available passband names are:

  • ‘CHEOPS’, ‘MOST’, ‘Kepler’, ‘CoRoT’, ‘Gaia’,
  • ‘U’, ‘B’, ‘V’, ‘R’, ‘I’ (Bessell/Johnson)
  • ‘u_’, ‘g_’, ‘r_’, ‘i_’, ‘z_’ (SDSS)

The power-2 limb-darkening law is described in Maxted (2018) [2].

Examples

>>> from pycheops.ld import *
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> T_eff = 5560
>>> log_g = 4.3
>>> Fe_H = -0.3
>>> passband = 'Kepler'
>>> cK = stagger_claret_interpolator(passband)
>>> ac = cK(T_eff, log_g, Fe_H)
>>> mK = stagger_mugrid_interpolator(passband)
>>> I_mu = mK(T_eff, log_g, Fe_H)
>>> p2K = stagger_power2_interpolator(passband)
>>> c2,a2,h1,h2 = p2K(T_eff, log_g, Fe_H)
>>> plt.plot(mK.mu, I_mu,'bo')
>>> mu = np.linspace(0,1)
>>> plt.plot(mu, ld_claret(mu,ac),label='Claret')
>>> plt.plot(mu, ld_power2(mu,[c2, a2]),label='power-2')
>>> plt.xlim(0,1)
>>> plt.ylim(0,1)
>>> plt.xlabel('$\mu$')
>>> plt.ylabel('$I_{\lambda}(\mu)$')
>>> plt.legend()
>>> plt.show()

References

[2]Maxted, P.F.L., 2018, A&A, submitted
pycheops.ld.ld_power2(mu, a)

Evaluate power-2 limb-darkening law

Parameters:
  • mu – cos of angle between surface normal and line of sight
  • a – array or tuple [c, alpha]
Returns:

1 - c * (1-mu**alpha)
pycheops.ld.ld_claret(mu, a)

Evaluate Claret 4-parameter limb-darkening law

Parameters:
  • mu – cos of angle between surface normal and line of sight
  • a – array or tuple [a_1, a_2, a_3, a_4]
Returns:

1 - Sum(i=1,4) a_i*(1-mu**(i/2))
class pycheops.ld.stagger_power2_interpolator(passband='CHEOPS')

Parameters of a power-2 limb-darkening law interpolated from the Stagger grid.

The power-2 limb darkening law is
I_X(mu) = 1 - c * (1-mu**alpha)

It is often better to use the transformed coefficients

  • h1 = 1 - c*(1-0.5**alpha)

and

  • h2 = c*0.5**alpha

as free parameters in a least-squares fit and/or for applying priors.

Returns NaN if interpolation outside the grid range is attempted

__call__(T_eff, log_g, Fe_H)
Parameters:
  • T_eff – effective temperature in Kelvin
  • log_g – log of the surface gravity in cgs units
  • Fe/H – [Fe/H] in dex
Returns:

c, alpha, h_1, h_2
class pycheops.ld.stagger_mugrid_interpolator(passband='CHEOPS')

Limb-darkening as a function of mu from the Stagger grid.

Returns NaN if interpolation outside the grid range is attempted

Variables:mu – grid of mu values for most recent call
__call__(T_eff, log_g, Fe_H, n_mu=None)
Parameters:
  • T_eff – effective temperature in Kelvin
  • log_g – log of the surface gravity in cgs units
  • Fe/H – [Fe/H] in dex
  • n_mu – No. of points in the output intensity grid
Returns:

I(mu) on a grid of mu values from 0 to 1.
class pycheops.ld.stagger_claret_interpolator(passband='CHEOPS')

Coefficients of a Claret 4-parameter limb-darkening law interpolated from the Stagger grid.

Returns NaN if interpolation outside the grid range is attempted

__call__(T_eff, log_g, Fe_H)
Parameters:
  • T_eff – effective temperature in Kelvin
  • log_g – log of the surface gravity in cgs units
  • Fe/H – [Fe/H] in dex
Returns:

array of coefficients
pycheops.ld.ca_to_h1h2(c, alpha)

Transform for power-2 law coefficients h1 = 1 - c*(1-0.5**alpha) h2 = c*0.5**alpha

Parameters:
  • c – power-2 law coefficient, c
  • alpha – power-2 law exponent, alpha

returns: h1, h2

pycheops.ld.h1h2_to_ca(h1, h2)

Inverse transform for power-2 law coefficients c = 1 - h1 + h2 alpha = log2(c/h2)

Parameters:
  • h1 – 1 - c*(1-0.5**alpha)
  • h2 – c*0.5**alpha

returns: c, alpha