Fitting TESS data

import exoplanet

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

exoplanet.__version__ = '0.5.4.dev14+gc7cfa9a'

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 Joint RV & transit fits case study, but with a few extra bits like aperture selection and de-trending.

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

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())

Transit search

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.

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 Joint RV & transit fits.

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()

optimizing logp for variables: [log_rho_gp, log_sigma_gp, log_sigma_lc]

100.00% [44/44 00:00<00:00 logp = 3.460e+03]

message: Optimization terminated successfully.
logp: 3338.181795537269 -> 3460.39834113472

optimizing logp for variables: [log_depth]

100.00% [8/8 00:00<00:00 logp = 3.466e+03]

message: Optimization terminated successfully.
logp: 3460.39834113472 -> 3465.9617441726405

optimizing logp for variables: [b]

100.00% [11/11 00:00<00:00 logp = 3.501e+03]

message: Optimization terminated successfully.
logp: 3465.9617441726405 -> 3501.4417887424706

optimizing logp for variables: [t0, log_period]

100.00% [60/60 00:00<00:00 logp = 3.507e+03]

message: Desired error not necessarily achieved due to precision loss.
logp: 3501.4417887424725 -> 3507.325904702138

optimizing logp for variables: [u_star]

100.00% [10/10 00:00<00:00 logp = 3.511e+03]

message: Optimization terminated successfully.
logp: 3507.325904702136 -> 3510.723293096469

optimizing logp for variables: [log_depth]

100.00% [8/8 00:00<00:00 logp = 3.514e+03]

message: Optimization terminated successfully.
logp: 3510.723293096469 -> 3514.131734483311

optimizing logp for variables: [b]

100.00% [10/10 00:00<00:00 logp = 3.515e+03]

message: Optimization terminated successfully.
logp: 3514.131734483311 -> 3515.137731047199

optimizing logp for variables: [ecs]

100.00% [17/17 00:00<00:00 logp = 3.516e+03]

message: Optimization terminated successfully.
logp: 3515.137731047199 -> 3515.6567965737236

optimizing logp for variables: [mean]

100.00% [4/4 00:00<00:00 logp = 3.516e+03]

message: Optimization terminated successfully.
logp: 3515.6567965737236 -> 3515.6766245047024

optimizing logp for variables: [log_rho_gp, log_sigma_gp, log_sigma_lc]

100.00% [12/12 00:00<00:00 logp = 3.516e+03]

message: Optimization terminated successfully.
logp: 3515.6766245047024 -> 3516.274095529281

optimizing logp for variables: [log_sigma_gp, log_rho_gp, log_sigma_lc, ecs, log_depth, b, log_period, t0, r_star, m_star, u_star, mean]

100.00% [117/117 00:00<00:00 logp = 3.722e+03]

message: Desired error not necessarily achieved due to precision loss.
logp: 3516.274095529281 -> 3722.200730721436

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

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 Joint RV & transit fits, we can do some sigma clipping to remove significant outliers.

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.

model, map_soln, extras = build_model(mask, map_soln0)
_ = plot_light_curve(map_soln, extras, mask)

optimizing logp for variables: [log_rho_gp, log_sigma_gp, log_sigma_lc]

100.00% [15/15 00:00<00:00 logp = 3.886e+03]

message: Optimization terminated successfully.
logp: 3879.897528212497 -> 3886.071717475003

optimizing logp for variables: [log_depth]

100.00% [6/6 00:00<00:00 logp = 3.886e+03]

message: Optimization terminated successfully.
logp: 3886.071717475003 -> 3886.073633480184

optimizing logp for variables: [b]

100.00% [34/34 00:00<00:00 logp = 3.886e+03]

message: Desired error not necessarily achieved due to precision loss.
logp: 3886.073633480184 -> 3886.0741001509227

optimizing logp for variables: [t0, log_period]

100.00% [13/13 00:00<00:00 logp = 3.886e+03]

message: Optimization terminated successfully.
logp: 3886.0741001509205 -> 3886.0741690080004

optimizing logp for variables: [u_star]

100.00% [6/6 00:00<00:00 logp = 3.886e+03]

message: Optimization terminated successfully.
logp: 3886.0741690080026 -> 3886.0743208152003

optimizing logp for variables: [log_depth]

100.00% [5/5 00:00<00:00 logp = 3.886e+03]

message: Optimization terminated successfully.
logp: 3886.0743208152003 -> 3886.0743230249277

optimizing logp for variables: [b]

100.00% [6/6 00:00<00:00 logp = 3.886e+03]

message: Optimization terminated successfully.
logp: 3886.0743230249277 -> 3886.0743421943985

optimizing logp for variables: [ecs]

100.00% [80/80 00:00<00:00 logp = 3.886e+03]

message: Optimization terminated successfully.
logp: 3886.074342194397 -> 3886.0743426810814

optimizing logp for variables: [mean]

100.00% [4/4 00:00<00:00 logp = 3.886e+03]

message: Optimization terminated successfully.
logp: 3886.0743426810814 -> 3886.0765813141725

optimizing logp for variables: [log_rho_gp, log_sigma_gp, log_sigma_lc]

100.00% [9/9 00:00<00:00 logp = 3.886e+03]

message: Optimization terminated successfully.
logp: 3886.0765813141725 -> 3886.076590737357

optimizing logp for variables: [log_sigma_gp, log_rho_gp, log_sigma_lc, ecs, log_depth, b, log_period, t0, r_star, m_star, u_star, mean]

100.00% [131/131 00:00<00:00 logp = 3.886e+03]

message: Desired error not necessarily achieved due to precision loss.
logp: 3886.076590737356 -> 3886.0768930260997

Now that we have the model, we can sample:

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",
    )

Auto-assigning NUTS sampler...

Initializing NUTS using adapt_full...

Multiprocess sampling (2 chains in 2 jobs)

NUTS: [log_sigma_gp, log_rho_gp, log_sigma_lc, ecs, log_depth, b, log_period, t0, r_star, m_star, u_star, mean]

100.00% [5000/5000 10:52<00:00 Sampling 2 chains, 0 divergences]

Sampling 2 chains for 1_500 tune and 1_000 draw iterations (3_000 + 2_000 draws total) took 655 seconds.

The number of effective samples is smaller than 25% for some parameters.

import arviz as az

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

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
omega	0.437	1.790	-2.715	3.138	0.071	0.050	794.0	1440.0	1.00
ecc	0.213	0.144	0.000	0.450	0.006	0.005	641.0	388.0	1.00
r_pl	0.018	0.001	0.017	0.020	0.000	0.000	702.0	860.0	1.01
b	0.480	0.220	0.032	0.794	0.012	0.008	356.0	529.0	1.00
t0	-13.735	0.002	-13.739	-13.731	0.000	0.000	1078.0	861.0	1.00
period	6.269	0.001	6.267	6.270	0.000	0.000	1531.0	1943.0	1.00
r_star	1.100	0.023	1.053	1.140	0.001	0.000	1674.0	1158.0	1.00
m_star	1.095	0.040	1.021	1.169	0.001	0.001	1523.0	1320.0	1.00
u_star[0]	0.311	0.235	0.000	0.733	0.006	0.004	1434.0	1280.0	1.00
u_star[1]	0.205	0.320	-0.344	0.799	0.009	0.006	1263.0	1243.0	1.00
mean	0.002	0.007	-0.012	0.015	0.000	0.000	1361.0	1069.0	1.00

Results

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

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:

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.

with model:
    txt, bib = xo.citations.get_citations_for_model()
print(txt)

This research made use of \textsf{exoplanet} \citep{exoplanet:joss,
exoplanet:zenodo} and its dependencies \citep{celerite2:foremanmackey17,
celerite2:foremanmackey18, exoplanet:agol20, exoplanet:arviz,
exoplanet:astropy13, exoplanet:astropy18, exoplanet:kipping13,
exoplanet:kipping13b, exoplanet:luger18, exoplanet:pymc3, exoplanet:theano}.

print(bib.split("\n\n")[0] + "\n\n...")

@article{exoplanet:joss,
       author = {{Foreman-Mackey}, Daniel and {Luger}, Rodrigo and {Agol}, Eric
                and {Barclay}, Thomas and {Bouma}, Luke G. and {Brandt},
                Timothy D. and {Czekala}, Ian and {David}, Trevor J. and
                {Dong}, Jiayin and {Gilbert}, Emily A. and {Gordon}, Tyler A.
                and {Hedges}, Christina and {Hey}, Daniel R. and {Morris},
                Brett M. and {Price-Whelan}, Adrian M. and {Savel}, Arjun B.},
        title = "{exoplanet: Gradient-based probabilistic inference for
                  exoplanet data \& other astronomical time series}",
      journal = {arXiv e-prints},
         year = 2021,
        month = may,
          eid = {arXiv:2105.01994},
        pages = {arXiv:2105.01994},
archivePrefix = {arXiv},
       eprint = {2105.01994},
 primaryClass = {astro-ph.IM},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2021arXiv210501994F},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

...