Note

This tutorial was generated from an IPython notebook that can be downloaded here.

# Quick fits for TESS light curves¶

Note

You will need exoplanet version 0.2.6 or later to run this tutorial.

In this tutorial, we will fit the TESS light curve for a known transiting planet. While the Fitting TESS data tutorial goes through the full details of an end-to-end fit, this tutorial is significantly faster to run and it can give pretty excellent results depending on your goals. Some of the main differences are:

1. We start from the light curve rather than doing the photometry ourselves. This should pretty much always be fine unless you have a very bright, faint, or crowded target.
2. We assume a circluar orbit, but as you’ll see later, we can approximately relax this assumption later.
3. We only fit the data near transit. In many cases this will be just fine, but if you have predictable stellar variability (like coherent rotation) then you might do better fitting more data.

We’ll fit the planet in the HD 118203 (TIC 286923464) system that was found to transit by Pepper et al. (2019) because it is on an eccentric orbit so assumption #2 above is not valid.

import numpy as np
import lightkurve as lk
import matplotlib.pyplot as plt

lc = lcfs.PDCSAP_FLUX.stitch()
lc = lc.remove_nans().remove_outliers()

x = np.ascontiguousarray(lc.time, dtype=np.float64)
y = np.ascontiguousarray(1e3 * (lc.flux - 1), dtype=np.float64)
yerr = np.ascontiguousarray(1e3 * lc.flux_err, dtype=np.float64)

texp = np.min(np.diff(x))

plt.plot(x, y, "k", linewidth=0.5)
plt.xlabel("time [days]")
plt.ylabel("relative flux [ppt]"); Then, find the period, phase and depth of the transit using box least squares:

import exoplanet as xo

pg = xo.estimators.bls_estimator(x, y, yerr, min_period=2, max_period=20)

peak = pg["peak_info"]
period_guess = peak["period"]
t0_guess = peak["transit_time"]
depth_guess = peak["depth"]

plt.plot(pg["bls"].period, pg["bls"].power, "k", linewidth=0.5)
plt.axvline(period_guess, alpha=0.3, linewidth=5)
plt.xlabel("period [days]")
plt.ylabel("bls power")
plt.yticks([])
plt.xlim(pg["bls"].period.min(), pg["bls"].period.max()); Then, for efficiency purposes, let’s extract just the data within 0.25 days of the transits:

transit_mask = (
np.abs((x - t0_guess + 0.5 * period_guess) % period_guess - 0.5 * period_guess)
< 0.25
)

plt.figure(figsize=(8, 4))
x_fold = (x - t0_guess + 0.5 * period_guess) % period_guess - 0.5 * period_guess
plt.scatter(x_fold, y, c=x, s=3)
plt.xlabel("time since transit [days]")
plt.ylabel("relative flux [ppt]")
plt.colorbar(label="time [days]")
plt.xlim(-0.25, 0.25); That looks a little janky, but it’s good enough for now.

## The probabilistic model¶

Here’s how we set up the PyMC3 model in this case:

import pymc3 as pm
import theano.tensor as tt

with pm.Model() as model:

# Stellar parameters
mean = pm.Normal("mean", mu=0.0, sigma=10.0)
star_params = [mean, u]

# Gaussian process noise model
sigma = pm.InverseGamma("sigma", alpha=3.0, beta=2 * np.median(yerr))
log_Sw4 = pm.Normal("log_Sw4", mu=0.0, sigma=10.0)
log_w0 = pm.Normal("log_w0", mu=np.log(2 * np.pi / 10.0), sigma=10.0)
kernel = xo.gp.terms.SHOTerm(log_Sw4=log_Sw4, log_w0=log_w0, Q=1.0 / 3)
noise_params = [sigma, log_Sw4, log_w0]

# Planet parameters
log_ror = pm.Normal("log_ror", mu=0.5 * np.log(depth_guess * 1e-3), sigma=10.0)
ror = pm.Deterministic("ror", tt.exp(log_ror))

# Orbital parameters
log_period = pm.Normal("log_period", mu=np.log(period_guess), sigma=1.0)
t0 = pm.Normal("t0", mu=t0_guess, sigma=1.0)
log_dur = pm.Normal("log_dur", mu=np.log(0.1), sigma=10.0)
b = xo.distributions.ImpactParameter("b", ror=ror)

period = pm.Deterministic("period", tt.exp(log_period))
dur = pm.Deterministic("dur", tt.exp(log_dur))

# Set up the orbit
orbit = xo.orbits.KeplerianOrbit(period=period, duration=dur, t0=t0, b=b)

# We're going to track the implied density for reasons that will become clear later
pm.Deterministic("rho_circ", orbit.rho_star)

# Set up the mean transit model
star = xo.LimbDarkLightCurve(u)

def lc_model(t):
return mean + 1e3 * tt.sum(
star.get_light_curve(orbit=orbit, r=ror, t=t), axis=-1
)

# Finally the GP observation model
gp = xo.gp.GP(kernel, x, yerr ** 2 + sigma ** 2, mean=lc_model)
gp.marginal("obs", observed=y)

# Double check that everything looks good - we shouldn't see any NaNs!
print(model.check_test_point())

# Optimize the model
map_soln = model.test_point
map_soln = xo.optimize(map_soln, [sigma])
map_soln = xo.optimize(map_soln, [log_ror, b, log_dur])
map_soln = xo.optimize(map_soln, noise_params)
map_soln = xo.optimize(map_soln, star_params)
map_soln = xo.optimize(map_soln)

mean                   -3.22
sigma_log__            -0.53
log_Sw4                -3.22
log_w0                 -3.22
log_ror                -3.22
log_period             -0.92
t0                     -0.92
log_dur                -3.22
b_impact__             -1.39
obs                -24037.81
Name: Log-probability of test_point, dtype: float64

optimizing logp for variables: [sigma]
16it [00:01,  9.51it/s, logp=-6.491897e+03]
message: Optimization terminated successfully.
logp: -24060.450789871506 -> -6491.896832790021
optimizing logp for variables: [log_dur, b, log_ror]
21it [00:00, 105.31it/s, logp=-4.926412e+03]
message: Optimization terminated successfully.
logp: -6491.896832790021 -> -4926.412276779751
optimizing logp for variables: [log_w0, log_Sw4, sigma]
79it [00:00, 145.63it/s, logp=-1.872912e+03]
message: Optimization terminated successfully.
logp: -4926.412276779751 -> -1872.9123033179976
optimizing logp for variables: [u, mean]
13it [00:00, 76.15it/s, logp=-1.867587e+03]
message: Optimization terminated successfully.
logp: -1872.9123033179976 -> -1867.58739329712
optimizing logp for variables: [b, log_dur, t0, log_period, log_ror, log_w0, log_Sw4, sigma, u, mean]
134it [00:00, 236.16it/s, logp=-1.381469e+03]
message: Desired error not necessarily achieved due to precision loss.
logp: -1867.58739329712 -> -1381.4688211234925


Now we can plot our initial model:

with model:
gp_pred, lc_pred = xo.eval_in_model([gp.predict(), lc_model(x)], map_soln)

plt.figure(figsize=(8, 4))
x_fold = (x - map_soln["t0"] + 0.5 * map_soln["period"]) % map_soln[
"period"
] - 0.5 * map_soln["period"]
inds = np.argsort(x_fold)
plt.scatter(x_fold, y - gp_pred - map_soln["mean"], c=x, s=3)
plt.plot(x_fold[inds], lc_pred[inds] - map_soln["mean"], "k")
plt.xlabel("time since transit [days]")
plt.ylabel("relative flux [ppt]")
plt.colorbar(label="time [days]")
plt.xlim(-0.25, 0.25); That looks better!

Now on to sampling:

np.random.seed(286923464)
with model:
trace = xo.sample(tune=2000, draws=2000, start=map_soln, chains=4)

Multiprocess sampling (4 chains in 4 jobs)
NUTS: [b, log_dur, t0, log_period, log_ror, log_w0, log_Sw4, sigma, u, mean]
Sampling 4 chains, 0 divergences: 100%|██████████| 16000/16000 [03:07<00:00, 85.25draws/s]


Then we can take a look at the summary statistics:

pm.summary(trace)

mean sd hpd_3% hpd_97% mcse_mean mcse_sd ess_mean ess_sd ess_bulk ess_tail r_hat
mean 0.217 0.103 0.015 0.401 0.001 0.001 6905.0 6435.0 7005.0 5079.0 1.0
log_Sw4 6.131 0.339 5.503 6.776 0.004 0.003 8900.0 8807.0 8908.0 5655.0 1.0
log_w0 2.187 0.172 1.861 2.498 0.002 0.001 8009.0 8009.0 8041.0 5592.0 1.0
log_ror -2.913 0.007 -2.926 -2.898 0.000 0.000 6984.0 6984.0 7017.0 5499.0 1.0
log_period 1.814 0.000 1.814 1.814 0.000 0.000 9271.0 9271.0 9276.0 6378.0 1.0
t0 1712.662 0.000 1712.662 1712.662 0.000 0.000 9552.0 9552.0 9546.0 6070.0 1.0
log_dur -1.504 0.002 -1.507 -1.500 0.000 0.000 8222.0 8221.0 8224.0 5774.0 1.0
u 0.173 0.076 0.029 0.309 0.001 0.001 5429.0 5429.0 4913.0 2354.0 1.0
u 0.252 0.104 0.063 0.447 0.001 0.001 5715.0 4661.0 5521.0 3815.0 1.0
sigma 0.159 0.008 0.145 0.173 0.000 0.000 8942.0 8942.0 8949.0 5887.0 1.0
ror 0.054 0.000 0.054 0.055 0.000 0.000 6979.0 6975.0 7017.0 5499.0 1.0
b 0.214 0.094 0.020 0.357 0.002 0.001 2096.0 2096.0 2356.0 1938.0 1.0
period 6.135 0.000 6.135 6.135 0.000 0.000 9271.0 9271.0 9276.0 6378.0 1.0
dur 0.222 0.000 0.221 0.223 0.000 0.000 8224.0 8224.0 8224.0 5774.0 1.0
rho_circ 0.316 0.018 0.285 0.345 0.000 0.000 3136.0 2964.0 2539.0 2750.0 1.0

And plot the posterior covariances compared to the values from Pepper et al. (2019):

import corner
import astropy.units as u

samples = pm.trace_to_dataframe(trace, varnames=["period", "ror", "b"])
corner.corner(samples, truths=[6.134980, 0.05538, 0.125]); ## Bonus: eccentricity¶

As discussed above, we fit this model assuming a circular orbit which speeds things up for a few reasons. First, setting eccentricity to zero means that the orbital dynamics are much simpler and more computationally efficient, since we don’t need to solve Kepler’s equation numerically. But this isn’t actually the main effect! Instead the bigger issues come from the fact that the degeneracies between eccentricity, arrgument of periasteron, impact parameter, and planet radius are hard for the sampler to handle, causing the sampler’s performance to plummet. In this case, by fitting with a circular orbit where duration is one of the parameters, everything is well behaved and the sampler runs faster.

But, in this case, the planet is actually on an eccentric orbit, so that assumption isn’t justified. It has been recognized by various researchers over the years (I first learned about this from Bekki Dawson) that, to first order, the eccentricity mainly just changes the transit duration. The key realization is that this can be thought of as a change in the impled density of the star. Therefore, if you fit the transit using stellar density (or duration, in this case) as one of the parameters (note: you must have a different stellar density parameter for each planet if there are more than one), you can use an independent measurement of the stellar density to infer the eccentricity of the orbit after the fact. All the details are described in Dawson & Johnson (2012), but here’s how you can do this here using the stellar density listed in the TESS input catalog:

from astroquery.mast import Catalogs

star = Catalogs.query_object("TIC 286923464", catalog="TIC", radius=0.001)
tic_rho_star = float(star["rho"]), float(star["e_rho"])
print("rho_star = {0} ± {1}".format(*tic_rho_star))

# Extract the implied density from the fit
rho_circ = np.repeat(trace["rho_circ"], 100)

# Sample eccentricity and omega from their priors (the math might
# be a little more subtle for more informative priors, but I leave
# that as an exercise for the reader...)
ecc = np.random.uniform(0, 1, len(rho_circ))
omega = np.random.uniform(-np.pi, np.pi, len(rho_circ))

# Compute the "g" parameter from Dawson & Johnson and what true
# density that implies
g = (1 + ecc * np.sin(omega)) / np.sqrt(1 - ecc ** 2)
rho = rho_circ / g ** 3

# Re-weight these samples to get weighted posterior samples
log_weights = -0.5 * ((rho - tic_rho_star) / tic_rho_star) ** 2
weights = np.exp(log_weights - np.max(log_weights))

# Estimate the expected posterior quantiles
q = corner.quantile(ecc, [0.16, 0.5, 0.84], weights=weights)
print("eccentricity = {0:.2f} +{1:.2f} -{1:.2f}".format(q, np.diff(q)))

corner.corner(
np.vstack((ecc, omega)).T,
weights=weights,
truths=[0.316, None],
plot_datapoints=False,
labels=["eccentricity", "omega"],
);

rho_star = 0.121689 ± 0.0281776
eccentricity = 0.45 +0.25 -0.14 As you can see, this eccentricity estimate is consistent (albeit with large uncertainties) with the value that Pepper et al. (2019) measure using radial velocities and it is definitely clear that this planet is not on a circular orbit.