(tess)=

Fitting TESS data

In [ ]:
import exoplanet

exoplanet.utils.docs_setup()
print(f"exoplanet.__version__ = '{exoplanet.__version__}'")

In this tutorial, we will reproduce the fits to the transiting planet in the Pi Mensae system discovered by Huang et al. (2018). The data processing and model are similar to the {ref}joint case study, but with a few extra bits like aperture selection and de-trending.

To start, we need to download the target pixel file:

In [ ]:
import numpy as np
import lightkurve as lk
import matplotlib.pyplot as plt
from astropy.io import fits

lc_file = lk.search_lightcurve(
    "TIC 261136679", sector=1, author="SPOC"
).download(quality_bitmask="hardest", flux_column="pdcsap_flux")
lc = lc_file.remove_nans().normalize().remove_outliers()
time = lc.time.value
flux = lc.flux

# For the purposes of this example, we'll discard some of the data
m = (lc.quality == 0) & (
    np.random.default_rng(261136679).uniform(size=len(time)) < 0.3
)

with fits.open(lc_file.filename) as hdu:
    hdr = hdu[1].header

texp = hdr["FRAMETIM"] * hdr["NUM_FRM"]
texp /= 60.0 * 60.0 * 24.0

ref_time = 0.5 * (np.min(time) + np.max(time))
x = np.ascontiguousarray(time[m] - ref_time, dtype=np.float64)
y = np.ascontiguousarray(1e3 * (flux[m] - 1.0), dtype=np.float64)

plt.plot(x, y, ".k")
plt.xlabel("time [days]")
plt.ylabel("relative flux [ppt]")
_ = plt.xlim(x.min(), x.max())

Now, let's use the box least squares periodogram from AstroPy (Note: you'll need AstroPy v3.1 or more recent to use this feature) to estimate the period, phase, and depth of the transit.

In [ ]:
from astropy.timeseries import BoxLeastSquares

period_grid = np.exp(np.linspace(np.log(1), np.log(15), 50000))

bls = BoxLeastSquares(x, y)
bls_power = bls.power(period_grid, 0.1, oversample=20)

# Save the highest peak as the planet candidate
index = np.argmax(bls_power.power)
bls_period = bls_power.period[index]
bls_t0 = bls_power.transit_time[index]
bls_depth = bls_power.depth[index]
transit_mask = bls.transit_mask(x, bls_period, 0.2, bls_t0)

fig, axes = plt.subplots(2, 1, figsize=(10, 10))

# Plot the periodogram
ax = axes[0]
ax.axvline(np.log10(bls_period), color="C1", lw=5, alpha=0.8)
ax.plot(np.log10(bls_power.period), bls_power.power, "k")
ax.annotate(
    "period = {0:.4f} d".format(bls_period),
    (0, 1),
    xycoords="axes fraction",
    xytext=(5, -5),
    textcoords="offset points",
    va="top",
    ha="left",
    fontsize=12,
)
ax.set_ylabel("bls power")
ax.set_yticks([])
ax.set_xlim(np.log10(period_grid.min()), np.log10(period_grid.max()))
ax.set_xlabel("log10(period)")

# Plot the folded transit
ax = axes[1]
x_fold = (x - bls_t0 + 0.5 * bls_period) % bls_period - 0.5 * bls_period
m = np.abs(x_fold) < 0.4
ax.plot(x_fold[m], y[m], ".k")

# Overplot the phase binned light curve
bins = np.linspace(-0.41, 0.41, 32)
denom, _ = np.histogram(x_fold, bins)
num, _ = np.histogram(x_fold, bins, weights=y)
denom[num == 0] = 1.0
ax.plot(0.5 * (bins[1:] + bins[:-1]), num / denom, color="C1")

ax.set_xlim(-0.3, 0.3)
ax.set_ylabel("de-trended flux [ppt]")
_ = ax.set_xlabel("time since transit")

The transit model in PyMC3

The transit model, initialization, and sampling are all nearly the same as the one in {ref}joint.

In [ ]:
import exoplanet as xo
import pymc3 as pm
import aesara_theano_fallback.tensor as tt

import pymc3_ext as pmx
from celerite2.theano import terms, GaussianProcess

phase_lc = np.linspace(-0.3, 0.3, 100)


def build_model(mask=None, start=None):
    if mask is None:
        mask = np.ones(len(x), dtype=bool)
    with pm.Model() as model:

        # Parameters for the stellar properties
        mean = pm.Normal("mean", mu=0.0, sd=10.0)
        u_star = xo.QuadLimbDark("u_star")
        star = xo.LimbDarkLightCurve(u_star)

        # Stellar parameters from Huang et al (2018)
        M_star_huang = 1.094, 0.039
        R_star_huang = 1.10, 0.023
        BoundedNormal = pm.Bound(pm.Normal, lower=0, upper=3)
        m_star = BoundedNormal(
            "m_star", mu=M_star_huang[0], sd=M_star_huang[1]
        )
        r_star = BoundedNormal(
            "r_star", mu=R_star_huang[0], sd=R_star_huang[1]
        )

        # Orbital parameters for the planets
        t0 = pm.Normal("t0", mu=bls_t0, sd=1)
        log_period = pm.Normal("log_period", mu=np.log(bls_period), sd=1)
        period = pm.Deterministic("period", tt.exp(log_period))

        # Fit in terms of transit depth (assuming b<1)
        b = pm.Uniform("b", lower=0, upper=1)
        log_depth = pm.Normal("log_depth", mu=np.log(bls_depth), sigma=2.0)
        ror = pm.Deterministic(
            "ror",
            star.get_ror_from_approx_transit_depth(
                1e-3 * tt.exp(log_depth), b
            ),
        )
        r_pl = pm.Deterministic("r_pl", ror * r_star)

        #         log_r_pl = pm.Normal(
        #             "log_r_pl",
        #             sd=1.0,
        #             mu=0.5 * np.log(1e-3 * np.array(bls_depth))
        #             + np.log(R_star_huang[0]),
        #         )
        #         r_pl = pm.Deterministic("r_pl", tt.exp(log_r_pl))
        #         ror = pm.Deterministic("ror", r_pl / r_star)
        #         b = xo.distributions.ImpactParameter("b", ror=ror)

        ecs = pmx.UnitDisk("ecs", testval=np.array([0.01, 0.0]))
        ecc = pm.Deterministic("ecc", tt.sum(ecs ** 2))
        omega = pm.Deterministic("omega", tt.arctan2(ecs[1], ecs[0]))
        xo.eccentricity.kipping13("ecc_prior", fixed=True, observed=ecc)

        # Transit jitter & GP parameters
        log_sigma_lc = pm.Normal(
            "log_sigma_lc", mu=np.log(np.std(y[mask])), sd=10
        )
        log_rho_gp = pm.Normal("log_rho_gp", mu=0, sd=10)
        log_sigma_gp = pm.Normal(
            "log_sigma_gp", mu=np.log(np.std(y[mask])), sd=10
        )

        # Orbit model
        orbit = xo.orbits.KeplerianOrbit(
            r_star=r_star,
            m_star=m_star,
            period=period,
            t0=t0,
            b=b,
            ecc=ecc,
            omega=omega,
        )

        # Compute the model light curve
        light_curves = (
            star.get_light_curve(orbit=orbit, r=r_pl, t=x[mask], texp=texp)
            * 1e3
        )
        light_curve = tt.sum(light_curves, axis=-1) + mean
        resid = y[mask] - light_curve

        # GP model for the light curve
        kernel = terms.SHOTerm(
            sigma=tt.exp(log_sigma_gp),
            rho=tt.exp(log_rho_gp),
            Q=1 / np.sqrt(2),
        )
        gp = GaussianProcess(kernel, t=x[mask], yerr=tt.exp(log_sigma_lc))
        gp.marginal("gp", observed=resid)
        #         pm.Deterministic("gp_pred", gp.predict(resid))

        # Compute and save the phased light curve models
        pm.Deterministic(
            "lc_pred",
            1e3
            * star.get_light_curve(
                orbit=orbit, r=r_pl, t=t0 + phase_lc, texp=texp
            )[..., 0],
        )

        # Fit for the maximum a posteriori parameters, I've found that I can get
        # a better solution by trying different combinations of parameters in turn
        if start is None:
            start = model.test_point
        map_soln = pmx.optimize(
            start=start, vars=[log_sigma_lc, log_sigma_gp, log_rho_gp]
        )
        map_soln = pmx.optimize(start=map_soln, vars=[log_depth])
        map_soln = pmx.optimize(start=map_soln, vars=[b])
        map_soln = pmx.optimize(start=map_soln, vars=[log_period, t0])
        map_soln = pmx.optimize(start=map_soln, vars=[u_star])
        map_soln = pmx.optimize(start=map_soln, vars=[log_depth])
        map_soln = pmx.optimize(start=map_soln, vars=[b])
        map_soln = pmx.optimize(start=map_soln, vars=[ecs])
        map_soln = pmx.optimize(start=map_soln, vars=[mean])
        map_soln = pmx.optimize(
            start=map_soln, vars=[log_sigma_lc, log_sigma_gp, log_rho_gp]
        )
        map_soln = pmx.optimize(start=map_soln)

        extras = dict(
            zip(
                ["light_curves", "gp_pred"],
                pmx.eval_in_model([light_curves, gp.predict(resid)], map_soln),
            )
        )

    return model, map_soln, extras


model0, map_soln0, extras0 = build_model()

Here's how we plot the initial light curve model:

In [ ]:
def plot_light_curve(soln, extras, mask=None):
    if mask is None:
        mask = np.ones(len(x), dtype=bool)

    fig, axes = plt.subplots(3, 1, figsize=(10, 7), sharex=True)

    ax = axes[0]
    ax.plot(x[mask], y[mask], "k", label="data")
    gp_mod = extras["gp_pred"] + soln["mean"]
    ax.plot(x[mask], gp_mod, color="C2", label="gp model")
    ax.legend(fontsize=10)
    ax.set_ylabel("relative flux [ppt]")

    ax = axes[1]
    ax.plot(x[mask], y[mask] - gp_mod, "k", label="de-trended data")
    for i, l in enumerate("b"):
        mod = extras["light_curves"][:, i]
        ax.plot(x[mask], mod, label="planet {0}".format(l))
    ax.legend(fontsize=10, loc=3)
    ax.set_ylabel("de-trended flux [ppt]")

    ax = axes[2]
    mod = gp_mod + np.sum(extras["light_curves"], axis=-1)
    ax.plot(x[mask], y[mask] - mod, "k")
    ax.axhline(0, color="#aaaaaa", lw=1)
    ax.set_ylabel("residuals [ppt]")
    ax.set_xlim(x[mask].min(), x[mask].max())
    ax.set_xlabel("time [days]")

    return fig


_ = plot_light_curve(map_soln0, extras0)

As in {ref}joint, we can do some sigma clipping to remove significant outliers.

In [ ]:
mod = (
    extras0["gp_pred"]
    + map_soln0["mean"]
    + np.sum(extras0["light_curves"], axis=-1)
)
resid = y - mod
rms = np.sqrt(np.median(resid ** 2))
mask = np.abs(resid) < 5 * rms

plt.figure(figsize=(10, 5))
plt.plot(x, resid, "k", label="data")
plt.plot(x[~mask], resid[~mask], "xr", label="outliers")
plt.axhline(0, color="#aaaaaa", lw=1)
plt.ylabel("residuals [ppt]")
plt.xlabel("time [days]")
plt.legend(fontsize=12, loc=3)
_ = plt.xlim(x.min(), x.max())

And then we re-build the model using the data without outliers.

In [ ]:
model, map_soln, extras = build_model(mask, map_soln0)
_ = plot_light_curve(map_soln, extras, mask)

Now that we have the model, we can sample:

In [ ]:
import platform

with model:
    trace = pm.sample(
        tune=1500,
        draws=1000,
        start=map_soln,
        # Parallel sampling runs poorly or crashes on macos
        cores=1 if platform.system() == "Darwin" else 2,
        chains=2,
        target_accept=0.95,
        return_inferencedata=True,
        random_seed=[261136679, 261136680],
        init="adapt_full",
    )
In [ ]:
import arviz as az

az.summary(
    trace,
    var_names=[
        "omega",
        "ecc",
        "r_pl",
        "b",
        "t0",
        "period",
        "r_star",
        "m_star",
        "u_star",
        "mean",
    ],
)

Results

After sampling, we can make the usual plots. First, let's look at the folded light curve plot:

In [ ]:
flat_samps = trace.posterior.stack(sample=("chain", "draw"))

# Compute the GP prediction
gp_mod = extras["gp_pred"] + map_soln["mean"]  # np.median(
#     flat_samps["gp_pred"].values + flat_samps["mean"].values[None, :], axis=-1
# )

# Get the posterior median orbital parameters
p = np.median(flat_samps["period"])
t0 = np.median(flat_samps["t0"])

# Plot the folded data
x_fold = (x[mask] - t0 + 0.5 * p) % p - 0.5 * p
plt.plot(x_fold, y[mask] - gp_mod, ".k", label="data", zorder=-1000)

# Overplot the phase binned light curve
bins = np.linspace(-0.41, 0.41, 50)
denom, _ = np.histogram(x_fold, bins)
num, _ = np.histogram(x_fold, bins, weights=y[mask])
denom[num == 0] = 1.0
plt.plot(
    0.5 * (bins[1:] + bins[:-1]), num / denom, "o", color="C2", label="binned"
)

# Plot the folded model
pred = np.percentile(flat_samps["lc_pred"], [16, 50, 84], axis=-1)
plt.plot(phase_lc, pred[1], color="C1", label="model")
art = plt.fill_between(
    phase_lc, pred[0], pred[2], color="C1", alpha=0.5, zorder=1000
)
art.set_edgecolor("none")

# Annotate the plot with the planet's period
txt = "period = {0:.5f} +/- {1:.5f} d".format(
    np.mean(flat_samps["period"].values), np.std(flat_samps["period"].values)
)
plt.annotate(
    txt,
    (0, 0),
    xycoords="axes fraction",
    xytext=(5, 5),
    textcoords="offset points",
    ha="left",
    va="bottom",
    fontsize=12,
)

plt.legend(fontsize=10, loc=4)
plt.xlim(-0.5 * p, 0.5 * p)
plt.xlabel("time since transit [days]")
plt.ylabel("de-trended flux")
_ = plt.xlim(-0.15, 0.15)

And a corner plot of some of the key parameters:

In [ ]:
import corner
import astropy.units as u

trace.posterior["r_earth"] = (
    trace.posterior["r_pl"].coords,
    (trace.posterior["r_pl"].values * u.R_sun).to(u.R_earth).value,
)

_ = corner.corner(
    trace,
    var_names=["period", "r_earth", "b", "ecc"],
    labels=[
        "period [days]",
        "radius [Earth radii]",
        "impact param",
        "eccentricity",
    ],
)

These all seem consistent with the previously published values.

Citations

As described in the citation tutorial, we can use citations.get_citations_for_model to construct an acknowledgement and BibTeX listing that includes the relevant citations for this model.

In [ ]:
with model:
    txt, bib = xo.citations.get_citations_for_model()
print(txt)
In [ ]:
print(bib.split("\n\n")[0] + "\n\n...")
In [ ]: