Chromosphere model

Chromosphere model#

ChromosphereModel implements the optically thin shell model of Schlawin et al. (ApJL 722, L75, 2010), for a transit observed in a chromospheric emission line.

Limb brightening#

A stellar photosphere is limb darkened: the disk is brightest at the centre. An optically thin, spherically symmetric emitting shell – a chromosphere seen in an emission line such as Hα or Ca II K – behaves in the opposite way. A line of sight near the limb passes through a longer path of emitting material than one through the disk centre, so the projected disk is brightest at its edge.

This inverts the transit shape. A model built for a limb-darkened photosphere cannot reproduce it, which is why this model exists as a separate implementation rather than as a limb darkening law.

The effect is large, not subtle. For a planet with \(k = 0.1\) on a central transit, the shell model blocks half as much light at mid-transit as a uniform disk would, and 1.6 times as much just inside the limb: the transit is shallower in the middle and deeper at the edges than the naive \(k^2\) expectation.

The model assumes an infinitesimally thin shell, whose surface brightness grows as \(1/\mu\) towards the limb, and evaluates the transit analytically. The profile follows from the geometry of the shell, so the model takes no limb darkening coefficients. Chromosphere model shows why limb darkening laws, and RoadRunner’s numerical integration, cannot reproduce it.

Usage#

from numpy import pi, linspace
from pytransit import ChromosphereModel

time = linspace(-0.1, 0.1, 1000)

tm = ChromosphereModel()
tm.set_data(time)

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

API#

class pytransit.models.ma_chromosphere.ChromosphereModel[source]#

Bases: TransitModel

Transit over an optically thin shell (Schlawin et al., ApJL 722, L75, 2010).

A transit model for an optically thin, spherically symmetric emitting shell such as a stellar chromosphere observed in a chromospheric emission line. The surface brightness of such a shell is limb brightened rather than limb darkened: the line of sight near the limb passes through a longer path of emitting material, so the projected disk is brightest at its edge and faintest at its centre. This inverts the usual transit shape, and a model built for a limb-darkened photosphere cannot reproduce it.

The profile is fixed by the geometry of the shell, so the model takes no limb darkening coefficients.

Examples

from pytransit import ChromosphereModel

tm = ChromosphereModel()
tm.set_data(time)
flux = tm.evaluate(k=0.1, t0=0.0, p=1.0, a=3.0, i=0.5*pi)
evaluate(k, t0, p, a, i, e=None, w=None, copy=True)[source]#

Evaluates the uniform 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.

  • 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 – Orbital eccentricity as a float or a 1D vector.

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

  • copy

Notes

The model can be evaluated either for one set of parameters or for many sets of parameters simultaneously. The orbital parameters can be given either as a float or a 1D array-like (preferably ndarray for optimal speed.)

Returns:

Transit model