Chromosphere model

A stellar photosphere is limb darkened. Looking at the centre of the disk we see deep, hot layers, while a line of sight near the limb reaches only shallower, cooler ones. An optically thin emitting shell, such as a chromosphere seen in an emission line like H\(\alpha\) or Ca II K, does the opposite: a line of sight near the limb passes through a longer path of emitting gas, so the disk is brightest at its edge. For a thin shell the surface brightness grows as \(1/\mu\), without limit at the limb.

ChromosphereModel computes a transit over such a shell using the analytic solution of Schlawin et al. (ApJL 722, L75, 2010). The profile follows from the geometry of the shell, so the model takes no limb darkening coefficients.

An inverted transit

The same planet transiting a limb-brightened shell, a uniform disk, and a limb-darkened photosphere:

from numpy import pi, linspace
from pytransit import ChromosphereModel, RoadRunnerModel

time = linspace(-0.1, 0.1, 1500)
k, t0, p, a, i = 0.1, 0.0, 4.0, 13.0, 0.5 * pi

tm = ChromosphereModel()
tm.set_data(time)
fluxes = {'shell': tm.evaluate(k, t0, p, a, i)}

for law, ldc in (('uniform', []), ('quadratic', [0.4, 0.25])):
    tm = RoadRunnerModel(law)
    tm.set_data(time)
    fluxes[law] = tm.evaluate(k, ldc, t0, p, a, i)

plot_profiles_and_transits(time, fluxes, [0.4, 0.25])

With the profiles normalised to the same total flux, the shell is half as bright as a uniform disk at its centre and brighter than it beyond about 0.87 stellar radii. The transit follows. At mid-transit the planet covers the faintest part of the shell and blocks half as much light as it would from a uniform disk, 5000 ppm instead of \(k^2 = 10\,000\) ppm. Just inside the limb it covers the brightest part, and the transit reaches its greatest depth, 1.6 times \(k^2\). A limb-darkened transit is deepest in the middle; a transit over a shell is deepest at its edges.

Why a separate model

RoadRunner accepts any radially symmetric profile, so a separate model for one profile needs a reason. The reason is the limb, where the shell’s brightness diverges.

Limb darkening laws cannot follow it. Fitting the shell transit with the standard laws, with the radius ratio, orbit, impact parameter, transit centre, and limb darkening all free, and the coefficients allowed to make the disk brighten towards the limb:

The best fits leave residuals of up to 1260 ppm for the quadratic law, 1860 ppm for the power-2 law, and 530 ppm for the four-coefficient non-linear law, concentrated where the planet crosses the limb. Even RoadRunner given the exact profile and its disk integral does not help: its quadrature assumes a smooth profile, and the singularity at the limb leaves errors of more than 1% even at a high resolution.

The analytic solution has neither problem. It agrees with a direct numerical integration of the shell’s brightness over the planet to within \(10^{-8}\), and it is fast, evaluating a 1500-point light curve in 0.08 ms.

See also

Chromosphere model for the API.