Fitting TESS data

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 Putting it all together case study, but with a few extra bits like aperture selection and de-trending.

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

[3]:

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
m = lc.quality == 0
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())
../../_images/tutorials_tess_4_0.png

The transit model in PyMC3

The transit model, initialization, and sampling are all nearly the same as the one in Putting it all together.

[5]:

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


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")

        # 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)
        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]),
        )
        period = pm.Deterministic("period", tt.exp(log_period))
        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 = pm.Deterministic(
            "light_curves",
            xo.LimbDarkLightCurve(u_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))

        # 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_r_pl])
        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_r_pl])
        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)

    return model, map_soln


model0, map_soln0 = build_model()

optimizing logp for variables: [log_rho_gp, log_sigma_gp, log_sigma_lc]
100.00% [18/18 00:00<00:00 logp = 1.202e+04] 


message: Optimization terminated successfully.
logp: 11803.534996510281 -> 12022.319792604483
optimizing logp for variables: [log_r_pl]
100.00% [10/10 00:00<00:00 logp = 1.203e+04] 


message: Optimization terminated successfully.
logp: 12022.319792604483 -> 12031.806646036945
optimizing logp for variables: [b, log_r_pl, r_star]
100.00% [21/21 00:00<00:00 logp = 1.231e+04] 


message: Optimization terminated successfully.
logp: 12031.806646036945 -> 12312.641607547512
optimizing logp for variables: [t0, log_period]
100.00% [20/20 00:00<00:00 logp = 1.232e+04] 


message: Optimization terminated successfully.
logp: 12312.641607547515 -> 12320.062773383888
optimizing logp for variables: [u_star]
100.00% [11/11 00:00<00:00 logp = 1.232e+04] 


message: Optimization terminated successfully.
logp: 12320.06277338388 -> 12323.976548867786
optimizing logp for variables: [log_r_pl]
100.00% [8/8 00:00<00:00 logp = 1.232e+04] 


message: Optimization terminated successfully.
logp: 12323.97654886779 -> 12324.153307400622
optimizing logp for variables: [b, log_r_pl, r_star]
100.00% [16/16 00:00<00:00 logp = 1.232e+04] 


message: Optimization terminated successfully.
logp: 12324.153307400622 -> 12324.272909174417
optimizing logp for variables: [ecs]
100.00% [14/14 00:00<00:00 logp = 1.232e+04] 


message: Optimization terminated successfully.
logp: 12324.272909174417 -> 12324.828200366446
optimizing logp for variables: [mean]
100.00% [4/4 00:00<00:00 logp = 1.233e+04] 


message: Optimization terminated successfully.
logp: 12324.828200366446 -> 12325.183875986868
optimizing logp for variables: [log_rho_gp, log_sigma_gp, log_sigma_lc]
100.00% [14/14 00:00<00:00 logp = 1.235e+04] 


message: Optimization terminated successfully.
logp: 12325.183875986868 -> 12349.80589909514
optimizing logp for variables: [log_sigma_gp, log_rho_gp, log_sigma_lc, ecs, b, log_r_pl, log_period, t0, r_star, m_star, u_star, mean]
100.00% [113/113 00:00<00:00 logp = 1.241e+04] 


message: Desired error not necessarily achieved due to precision loss.
logp: 12349.805899095156 -> 12410.253939054232

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

[6]:

def plot_light_curve(soln, 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 = soln["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 = soln["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(soln["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)
../../_images/tutorials_tess_10_0.png

As in Putting it all together, we can do some sigma clipping to remove significant outliers.

[7]:

mod = (
    map_soln0["gp_pred"]
    + map_soln0["mean"]
    + np.sum(map_soln0["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())
../../_images/tutorials_tess_12_0.png

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

[8]:

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

optimizing logp for variables: [log_rho_gp, log_sigma_gp, log_sigma_lc]
100.00% [16/16 00:00<00:00 logp = 1.293e+04] 


message: Optimization terminated successfully.
logp: 12909.297590015649 -> 12927.002817354894
optimizing logp for variables: [log_r_pl]
100.00% [6/6 00:00<00:00 logp = 1.293e+04] 


message: Optimization terminated successfully.
logp: 12927.002817354894 -> 12927.026010601196
optimizing logp for variables: [b, log_r_pl, r_star]
100.00% [13/13 00:00<00:00 logp = 1.293e+04] 


message: Optimization terminated successfully.
logp: 12927.026010601196 -> 12927.027380355064
optimizing logp for variables: [t0, log_period]
100.00% [45/45 00:00<00:00 logp = 1.293e+04] 


message: Desired error not necessarily achieved due to precision loss.
logp: 12927.027380355064 -> 12927.02938955412
optimizing logp for variables: [u_star]
100.00% [8/8 00:00<00:00 logp = 1.293e+04] 


message: Optimization terminated successfully.
logp: 12927.029389554118 -> 12927.02995706821
optimizing logp for variables: [log_r_pl]
100.00% [5/5 00:00<00:00 logp = 1.293e+04] 


message: Optimization terminated successfully.
logp: 12927.02995706821 -> 12927.02995961484
optimizing logp for variables: [b, log_r_pl, r_star]
100.00% [13/13 00:00<00:00 logp = 1.293e+04] 


message: Optimization terminated successfully.
logp: 12927.02995961484 -> 12927.030127938131
optimizing logp for variables: [ecs]
100.00% [7/7 00:00<00:00 logp = 1.293e+04] 


message: Optimization terminated successfully.
logp: 12927.030127938135 -> 12927.03012832856
optimizing logp for variables: [mean]
100.00% [4/4 00:00<00:00 logp = 1.293e+04] 


message: Optimization terminated successfully.
logp: 12927.030128328564 -> 12927.032487170087
optimizing logp for variables: [log_rho_gp, log_sigma_gp, log_sigma_lc]
100.00% [9/9 00:00<00:00 logp = 1.293e+04] 


message: Optimization terminated successfully.
logp: 12927.032487170087 -> 12927.032654515917
optimizing logp for variables: [log_sigma_gp, log_rho_gp, log_sigma_lc, ecs, b, log_r_pl, log_period, t0, r_star, m_star, u_star, mean]
100.00% [55/55 00:00<00:00 logp = 1.293e+04] 


message: Desired error not necessarily achieved due to precision loss.
logp: 12927.032654515917 -> 12927.032851659491
../../_images/tutorials_tess_14_34.png

Now that we have the model, we can sample:

[9]:

np.random.seed(261136679)
with model:
    trace = pmx.sample(
        tune=2500,
        draws=2000,
        start=map_soln,
        cores=2,
        chains=2,
        initial_accept=0.8,
        target_accept=0.95,
        return_inferencedata=True,
    )

Multiprocess sampling (2 chains in 2 jobs)
NUTS: [log_sigma_gp, log_rho_gp, log_sigma_lc, ecs, b, log_r_pl, log_period, t0, r_star, m_star, u_star, mean]
100.00% [9000/9000 57:04<00:00 Sampling 2 chains, 0 divergences] 
Sampling 2 chains for 2_500 tune and 2_000 draw iterations (5_000 + 4_000 draws total) took 3427 seconds.
The number of effective samples is smaller than 25% for some parameters.

[10]:

import arviz as az

[11]:

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

[11]:
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
omega 0.712 1.692 -2.668 3.135 0.052 0.037 1246.0 2703.0 1.0
ecc 0.224 0.145 0.001 0.499 0.004 0.004 1479.0 1037.0 1.0
r_pl 0.018 0.001 0.017 0.019 0.000 0.000 1202.0 727.0 1.0
b 0.387 0.220 0.006 0.731 0.008 0.007 652.0 496.0 1.0
t0 -13.733 0.001 -13.735 -13.731 0.000 0.000 1771.0 1151.0 1.0
period 6.268 0.000 6.267 6.268 0.000 0.000 2898.0 2704.0 1.0
r_star 1.098 0.023 1.053 1.139 0.000 0.000 3850.0 2539.0 1.0
m_star 1.096 0.038 1.028 1.167 0.001 0.000 4708.0 2999.0 1.0
u_star[0] 0.160 0.147 0.000 0.435 0.003 0.003 2253.0 1959.0 1.0
u_star[1] 0.430 0.257 -0.056 0.901 0.007 0.005 1478.0 1813.0 1.0
mean 0.003 0.006 -0.007 0.014 0.000 0.000 3863.0 2420.0 1.0

Results

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

[12]:

flat_samps = trace.posterior.stack(sample=("chain", "draw"))

# Compute the GP prediction
gp_mod = 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
inds = np.argsort(x_fold)
inds = inds[np.abs(x_fold)[inds] < 0.3]
pred = np.percentile(
    flat_samps["light_curves"][inds, 0], [16, 50, 84], axis=-1
)
plt.plot(x_fold[inds], pred[1], color="C1", label="model")
art = plt.fill_between(
    x_fold[inds], 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)
../../_images/tutorials_tess_20_0.png

And a corner plot of some of the key parameters:

[13]:

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",
    ],
)
../../_images/tutorials_tess_22_0.png

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.

[14]:

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

This research made use of \textsf{exoplanet} \citep{exoplanet} 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}.

[15]:

print("\n".join(bib.splitlines()[:10]) + "\n...")


@misc{exoplanet,
  author = {Daniel Foreman-Mackey and Arjun Savel and Rodrigo Luger  and
            Eric Agol and Ian Czekala and Adrian Price-Whelan and
            Christina Hedges and Emily Gilbert and Luke Bouma and Tom Barclay
            and Timothy D. Brandt},
   title = {exoplanet-dev/exoplanet v0.5.0},
   month = may,
    year = 2021,
     doi = {10.5281/zenodo.1998447},
...
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