Gravity-darkened star model

Every other model in PyTransit assumes that the star is a circular disk whose brightness depends only on the distance from its centre. A rapidly rotating star is neither. Rotation flattens it, and the flattening lowers the effective surface gravity at the equator. By von Zeipel’s theorem the local effective temperature follows the gravity,

\[T_\mathrm{eff} \propto g^{\beta},\]

so the equator is cooler and dimmer than the poles. This is gravity darkening.

GravityDarkenedModel implements the model of Barnes (ApJ 705, 683, 2009) for a transit across such a star. Because the star’s brightness depends on where its spin axis points, so does the transit: a planet whose orbit is misaligned with the stellar equator crosses regions of different temperature, and the light curve becomes asymmetric. The spin-orbit angle is then measurable from photometry alone, without the spectroscopy a Rossiter-McLaughlin measurement needs.

The model integrates numerically over the stellar surface, and converts the local temperature into flux in the observed passband using either blackbody spectra or model atmospheres (see Stellar spectra and Contamination).

What gravity darkening does to a transit

An A-type star of 1.8 solar masses and radii, with a polar temperature of 8500 K and a rotation period of 11 hours, is flattened by 9% and rotates at 194 km/s at its equator. A planet with a radius ratio of 0.1 transits it at an impact parameter of 0.3, observed in a blue and a red passband, for three projected spin-orbit angles \(\lambda\):

from numpy import pi, linspace, arccos
from pytransit import GravityDarkenedModel
from pytransit.contamination import BoxcarFilter

time = linspace(-0.13, 0.13, 800)
rho, rperiod, tpole, phi, beta = 0.48, 0.47, 8500.0, 0.5, 0.22
ldc, t0, p, a, i = [0.3, 0.2], 0.0, 3.0, 6.0, arccos(0.3 / 6.0)

models = {}
for band, wl in (('blue', (400, 500)), ('red', (800, 900))):
    models[band] = GravityDarkenedModel(filters=BoxcarFilter(band, *wl), rstar=1.8,
                                        model='blackbody', tmin=5000, tmax=9500,
                                        sres=160, pres=40, tres=240)
    models[band].set_data(time)

fluxes = {(band, l): tm.evaluate(0.1, rho, rperiod, tpole, phi, beta, ldc, t0, p, a, i, l)
          for band, tm in models.items() for l in (0.0, pi / 4, pi / 2)}

plot_gravity_darkening_signal(time, fluxes, models, rho, rperiod, tpole, phi, beta,
                              ldc, t0, p, a, i)

The upper row shows the star as the model sees it in the blue passband. Its spin axis is tilted by \(\phi = 0.5\) towards the observer, so the hot pole at the top is visible, and the planet moves along the dashed line. The lower row shows what is left after fitting the light curve with a standard transit model: a spherical, limb-darkened star, with the radius ratio, orbit, impact parameter, transit centre, and limb darkening all free.

When the orbit is aligned with the stellar equator, the standard model absorbs the gravity darkening almost entirely: less than 10 ppm remains. A misaligned orbit is different. The planet first crosses the cooler, dimmer part of the disk and then passes near the hot pole, so the transit is shallower than average early and deeper late. No spherical star produces that, and the residual reaches about 580 ppm for \(\lambda = 45°\) and 1500 ppm for \(\lambda = 90°\) in the blue passband.

The residual is about half as large in the red passband, because a given temperature difference changes the surface brightness more at short wavelengths than at long ones. Observations in several passbands therefore help to separate gravity darkening from other effects, and the model evaluates each passband with its own temperature-to-flux conversion.

Accuracy and resolution

The model’s accuracy is set by the resolution of the stellar surface, the planet, and the orbit. With the default resolution, the model error is 30 to 50 ppm for a radius ratio of 0.1, which is small compared with the signals above. The figure uses sres=160, pres=40, tres=240, which brings the error to 10 to 25 ppm, so that the small residuals of the aligned orbit show the physics rather than the model:

sres, pres, tres	Maximum error	Time per 1000 points
160, 20, 60 (default)	30 to 50 ppm	3 ms
160, 40, 240	10 to 25 ppm	24 ms

evaluate_brute computes the same transit by rendering the stellar disk as an image and summing the pixels behind the planet. It is much slower, but is a useful check when the model is used in a new region of parameter space.

See also

Gravity-darkened model for the parameters, the accuracy settings, and the API; Stellar spectra and Contamination for the stellar spectra and passbands.