gefera 0.1

Light curves for mutual transits of limb-darkened stars

gefera

Installation

Basic Usage

The Likelihood

Installation

To install gefera begin by cloning the github repository.

git clone https://www.github.com/tagordon/gefera

Once the repository has downloaded cd into the main directory and install with pip.

cd gefera; pip install .

Basic Usage

Gefera has two components: the dynamical model which is used to compute the positions of two bodies at a given time or array of times, and the photometric model which computes the observed flux given the positions of the bodies. Gefera implements two dynamical systems. The first is a HierarchicalSystem, which describes an exomoon/exoplanet pair. To instantiate this system let's start by defining the dynamical parameters:

For the planet:

ap = 215.0			# semimajor axis in stellar radii
tp = -91.25			# starting epoch in days
ep = 0.0			# eccentricity 
pp = 365			# period in days
wp = 0.1 * np.pi / 180		# longitude of periastron in degrees
ip = 89.8 * np.pi / 180		# inclination in degrees

and for the moon:

am = 2			# semimajor axis of the moon's orbit around the planet in stellar radii
tm = -4.2			# starting epoch in days
em = 0.0			# eccentricity
pm = 8				# period in days
om = 90 * np.pi / 180		# longitude of the ascending node in degrees
wm = -90 * np.pi / 180		# longitude of periastron in degrees
im = 90.0 * np.pi / 180		# inclination in degrees
mm = 0.01			# mass of the moon in units of the mass of the planet

We can then instantiate the system as follows:

import gefera as gf

po = gf.orbits.PrimaryOrbit(ap, tp, ep, pp, wp, ip)
mo = gf.orbits.SatelliteOrbit(am, tm, em, pm, om, wm, im, mm)
sys = gf.systems.HierarchicalSystem(po, mo)

To compute the light curve we also need the radii of both bodies and the quadratic limb darkening parameters (right now gefera only supports quadratic limb darkening)

rp = 0.1
rm = 0.05
u1 = 0.5
u2 = 0.3

We can then compute the light curve:

t = np.linspace(-0.6, 0.3, 10000)
# The out-of-transit flux is set to zero by default, so we add 1 to get the normalized flux.
flux = sys.lightcurve(t, u1, u2, rp, rm) + 1

When plotted, the lightcurve for this system looks like this:

If the gradient is required we use

flux, grad = sys.lightcurve(t, u1, u2, rp, rm, grad=True)

This will return the gradient of the light curve with respect to each of the input parameters as a dictionary keyed by the name of the parameter. The keys are a1, t1, e1... a2, t2, e2... where the 1 parameters are for the larger body and 2 refers to the smaller body.

The Likelihood

In order to conduct inference using the gefera model we need to compute the likelihood with respect to some observations. If t is an array containing the times of the observations and y is an array containing flux measurements (normalized to an out-of-transit flux of 1) then we can compute the likelihood for the model as follows:

ll = sys.loglike(y - 1, t, u1, u2, r1, r2, sigma)

where sigma is the uncertainty, assumed to be the same for each measurement. If we want to make use of the gradient in our inference then we can call:

ll, dll = sys.loglike(y - 1, t, u1, u2, r1, r2, sigma, grad=True)

The gradient is returned as an array of arrays containing the gradients with respect to each parameter in the order a1, t1, e1, p1, w1, i1, a2, t2, e2, p2, om2, w2, i2, mm2, r1, r2, u1, u2. If func is a callable function that returns -ll, -dll and init is an array of initial parameter guesses in the same order as the gradients are returned then we can use scipy.optimize.minimize to optimize the model as follows:

res = minimize(func, init, jac=True, method='TNC')