Note

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

Fitting TESS data

theano version: 1.0.4
pymc3 version: 3.7
exoplanet version: 0.2.1.dev0

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 Case study: K2-24, putting it all together tutorial, 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
from astropy.io import fits
import matplotlib.pyplot as plt

tpf_url = "https://archive.stsci.edu/missions/tess/tid/s0001/0000/0002/6113/6679/tess2018206045859-s0001-0000000261136679-0120-s_tp.fits"
with fits.open(tpf_url) as hdus:
    tpf = hdus[1].data
    tpf_hdr = hdus[1].header

texp = tpf_hdr["FRAMETIM"] * tpf_hdr["NUM_FRM"]
texp /= 60.0 * 60.0 * 24.0
time = tpf["TIME"]
flux = tpf["FLUX"]
m = np.any(np.isfinite(flux), axis=(1, 2)) & (tpf["QUALITY"] == 0)
ref_time = 0.5 * (np.min(time[m])+np.max(time[m]))
time = np.ascontiguousarray(time[m] - ref_time, dtype=np.float64)
flux = np.ascontiguousarray(flux[m], dtype=np.float64)

mean_img = np.median(flux, axis=0)
plt.imshow(mean_img.T, cmap="gray_r")
plt.title("TESS image of Pi Men")
plt.xticks([])
plt.yticks([]);
../../_images/tess_4_0.png

Aperture selection

Next, we’ll select an aperture using a hacky method that tries to minimizes the windowed scatter in the lightcurve (something like the CDPP).

from scipy.signal import savgol_filter

# Sort the pixels by median brightness
order = np.argsort(mean_img.flatten())[::-1]

# A function to estimate the windowed scatter in a lightcurve
def estimate_scatter_with_mask(mask):
    f = np.sum(flux[:, mask], axis=-1)
    smooth = savgol_filter(f, 1001, polyorder=5)
    return 1e6 * np.sqrt(np.median((f / smooth - 1)**2))

# Loop over pixels ordered by brightness and add them one-by-one
# to the aperture
masks, scatters = [], []
for i in range(10, 100):
    msk = np.zeros_like(mean_img, dtype=bool)
    msk[np.unravel_index(order[:i], mean_img.shape)] = True
    scatter = estimate_scatter_with_mask(msk)
    masks.append(msk)
    scatters.append(scatter)

# Choose the aperture that minimizes the scatter
pix_mask = masks[np.argmin(scatters)]

# Plot the selected aperture
plt.imshow(mean_img.T, cmap="gray_r")
plt.imshow(pix_mask.T, cmap="Reds", alpha=0.3)
plt.title("selected aperture")
plt.xticks([])
plt.yticks([]);
../../_images/tess_6_0.png

This aperture produces the following light curve:

plt.figure(figsize=(10, 5))
sap_flux = np.sum(flux[:, pix_mask], axis=-1)
sap_flux = (sap_flux / np.median(sap_flux) - 1) * 1e3
plt.plot(time, sap_flux, "k")
plt.xlabel("time [days]")
plt.ylabel("relative flux [ppt]")
plt.title("raw light curve")
plt.xlim(time.min(), time.max());
../../_images/tess_8_0.png

The transit model in PyMC3

The transit model, initialization, and sampling are all nearly the same as the one in Case study: K2-24, putting it all together, but we’ll use a more informative prior on eccentricity.

import exoplanet as xo
import pymc3 as pm
import theano.tensor as tt

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.distributions.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
        logP = pm.Normal("logP", mu=np.log(bls_period), sd=1)
        t0 = pm.Normal("t0", mu=bls_t0, sd=1)
        b = xo.distributions.UnitUniform("b")
        logr = pm.Normal("logr", 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(logr))
        ror = pm.Deterministic("ror", r_pl / r_star)

        # This is the eccentricity prior from Kipping (2013):
        # https://arxiv.org/abs/1306.4982
        BoundedBeta = pm.Bound(pm.Beta, lower=0, upper=1-1e-5)
        ecc = BoundedBeta("ecc", alpha=0.867, beta=3.03, testval=0.1)
        omega = xo.distributions.Angle("omega")

        # Transit jitter & GP parameters
        logs2 = pm.Normal("logs2", mu=np.log(np.var(y[mask])), sd=10)
        logw0_guess = np.log(2*np.pi/10)
        logw0 = pm.Normal("logw0", mu=logw0_guess, sd=10)

        # We'll parameterize using the maximum power (S_0 * w_0^4) instead of
        # S_0 directly because this removes some of the degeneracies between
        # S_0 and omega_0
        logpower = pm.Normal("logpower",
                             mu=np.log(np.var(y[mask]))+4*logw0_guess,
                             sd=10)
        logS0 = pm.Deterministic("logS0", logpower - 4 * logw0)

        # Tracking planet parameters
        period = pm.Deterministic("period", tt.exp(logP))

        # 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 using starry
        light_curves = xo.LimbDarkLightCurve(u_star).get_light_curve(
            orbit=orbit, r=r_pl, t=x[mask], texp=texp)*1e3
        light_curve = pm.math.sum(light_curves, axis=-1) + mean
        pm.Deterministic("light_curves", light_curves)

        # GP model for the light curve
        kernel = xo.gp.terms.SHOTerm(log_S0=logS0, log_w0=logw0, Q=1/np.sqrt(2))
        gp = xo.gp.GP(kernel, x[mask], tt.exp(logs2) + tt.zeros(mask.sum()), J=2)
        pm.Potential("transit_obs", gp.log_likelihood(y[mask] - light_curve))
        pm.Deterministic("gp_pred", gp.predict())

        # 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 = xo.optimize(start=start, vars=[logs2, logpower, logw0])
        map_soln = xo.optimize(start=map_soln, vars=[logr])
        map_soln = xo.optimize(start=map_soln, vars=[b])
        map_soln = xo.optimize(start=map_soln, vars=[logP, t0])
        map_soln = xo.optimize(start=map_soln, vars=[u_star])
        map_soln = xo.optimize(start=map_soln, vars=[logr])
        map_soln = xo.optimize(start=map_soln, vars=[b])
        map_soln = xo.optimize(start=map_soln, vars=[ecc, omega])
        map_soln = xo.optimize(start=map_soln, vars=[mean])
        map_soln = xo.optimize(start=map_soln, vars=[logs2, logpower, logw0])
        map_soln = xo.optimize(start=map_soln)

    return model, map_soln

model0, map_soln0 = build_model()
optimizing logp for variables: ['logw0', 'logpower', 'logs2']
message: Optimization terminated successfully.
logp: 12376.808570986894 -> 12639.99697023041
optimizing logp for variables: ['logr']
message: Optimization terminated successfully.
logp: 12639.99697023041 -> 12678.437541150912
optimizing logp for variables: ['b_logodds__']
message: Optimization terminated successfully.
logp: 12678.437541150908 -> 12766.212035563012
optimizing logp for variables: ['t0', 'logP']
message: Desired error not necessarily achieved due to precision loss.
logp: 12766.21203556302 -> 12775.280265954581
optimizing logp for variables: ['u_star_quadlimbdark__']
message: Optimization terminated successfully.
logp: 12775.28026595457 -> 12785.94497650719
optimizing logp for variables: ['logr']
message: Optimization terminated successfully.
logp: 12785.94497650719 -> 12808.53381146776
optimizing logp for variables: ['b_logodds__']
message: Optimization terminated successfully.
logp: 12808.533811467745 -> 12810.245024497122
optimizing logp for variables: ['omega_angle__', 'ecc_interval__']
message: Optimization terminated successfully.
logp: 12810.245024497122 -> 12830.745429464801
optimizing logp for variables: ['mean']
message: Optimization terminated successfully.
logp: 12830.745429464805 -> 12830.776889876774
optimizing logp for variables: ['logw0', 'logpower', 'logs2']
message: Desired error not necessarily achieved due to precision loss.
logp: 12830.776889876774 -> 12841.612045222775
optimizing logp for variables: ['logpower', 'logw0', 'logs2', 'omega_angle__', 'ecc_interval__', 'logr', 'b_logodds__', 't0', 'logP', 'r_star_interval__', 'm_star_interval__', 'u_star_quadlimbdark__', 'mean']
message: Desired error not necessarily achieved due to precision loss.
logp: 12841.612045222775 -> 13051.734914861225

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

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/tess_20_0.png

As in the Case study: K2-24, putting it all together tutorial, we can do some sigma clipping to remove significant outliers.

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/tess_22_0.png

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

model, map_soln = build_model(mask, map_soln0)
plot_light_curve(map_soln, mask);
optimizing logp for variables: ['logw0', 'logpower', 'logs2']
message: Optimization terminated successfully.
logp: 13706.240110867482 -> 13737.199585810293
optimizing logp for variables: ['logr']
message: Optimization terminated successfully.
logp: 13737.199585810293 -> 13737.219737712667
optimizing logp for variables: ['b_logodds__']
message: Optimization terminated successfully.
logp: 13737.219737712663 -> 13737.22061065265
optimizing logp for variables: ['t0', 'logP']
message: Desired error not necessarily achieved due to precision loss.
logp: 13737.220610652654 -> 13737.229447374608
optimizing logp for variables: ['u_star_quadlimbdark__']
message: Optimization terminated successfully.
logp: 13737.229447374597 -> 13737.253295836774
optimizing logp for variables: ['logr']
message: Optimization terminated successfully.
logp: 13737.253295836774 -> 13737.256455957235
optimizing logp for variables: ['b_logodds__']
message: Optimization terminated successfully.
logp: 13737.256455957246 -> 13737.265292970169
optimizing logp for variables: ['omega_angle__', 'ecc_interval__']
message: Optimization terminated successfully.
logp: 13737.265292970169 -> 13737.265332559527
optimizing logp for variables: ['mean']
message: Optimization terminated successfully.
logp: 13737.265332559527 -> 13737.268483697037
optimizing logp for variables: ['logw0', 'logpower', 'logs2']
message: Optimization terminated successfully.
logp: 13737.268483697037 -> 13737.26848655426
optimizing logp for variables: ['logpower', 'logw0', 'logs2', 'omega_angle__', 'ecc_interval__', 'logr', 'b_logodds__', 't0', 'logP', 'r_star_interval__', 'm_star_interval__', 'u_star_quadlimbdark__', 'mean']
message: Desired error not necessarily achieved due to precision loss.
logp: 13737.26848655426 -> 13737.280285382401
../../_images/tess_24_1.png

Now that we have the model, we can sample it using a exoplanet.PyMC3Sampler:

np.random.seed(42)
with model:
    trace = pm.sample(tune=5000, draws=3000, start=map_soln, chains=4,
                      step=xo.get_dense_nuts_step(target_accept=0.9))
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [logpower, logw0, logs2, omega, ecc, logr, b, t0, logP, r_star, m_star, u_star, mean]
Sampling 4 chains: 100%|██████████| 32000/32000 [2:24:43<00:00,  1.01draws/s]
The number of effective samples is smaller than 25% for some parameters.
pm.summary(trace, varnames=["logw0", "logpower", "logs2", "omega", "ecc", "r_pl", "b", "t0", "logP", "r_star", "m_star", "u_star", "mean"])
mean sd mc_error hpd_2.5 hpd_97.5 n_eff Rhat
logw0 1.177015 0.132107 1.567361e-03 0.912019 1.429859 8882.308481 0.999924
logpower -2.101866 0.310381 3.262105e-03 -2.714826 -1.500485 10641.049654 0.999853
logs2 -4.382845 0.010665 1.014718e-04 -4.404241 -4.362731 11414.227651 0.999905
omega 0.704976 1.686367 3.259006e-02 -2.795366 3.139100 2878.999094 1.001957
ecc 0.190323 0.137737 2.286425e-03 0.000042 0.464601 3190.391506 1.000581
r_pl 0.017238 0.000637 1.191213e-05 0.016043 0.018488 2980.279422 1.001719
b 0.406784 0.210295 4.474942e-03 0.000098 0.698977 2138.193986 1.001838
t0 -1.197369 0.000631 9.770836e-06 -1.198579 -1.196129 3664.920285 0.999977
logP 1.835411 0.000070 8.153155e-07 1.835271 1.835540 7610.644475 0.999975
r_star 1.098267 0.022690 1.949378e-04 1.054767 1.143082 10711.660137 1.000014
m_star 1.095327 0.038089 3.457118e-04 1.022662 1.172617 10865.550306 1.000206
u_star__0 0.200215 0.167950 1.821083e-03 0.000053 0.536966 6145.241801 1.000589
u_star__1 0.456185 0.271050 3.168628e-03 -0.063919 0.952691 5521.056955 1.000634
mean -0.001211 0.008997 8.911611e-05 -0.018408 0.017063 9442.547002 0.999941

Results

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

# Compute the GP prediction
gp_mod = np.median(trace["gp_pred"] + trace["mean"][:, None], axis=0)

# Get the posterior median orbital parameters
p = np.median(trace["period"])
t0 = np.median(trace["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 = trace["light_curves"][:, inds, 0]
pred = np.percentile(pred, [16, 50, 84], axis=0)
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(trace["period"]), np.std(trace["period"]))
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/tess_29_0.png

And a corner plot of some of the key parameters:

import corner
import astropy.units as u
varnames = ["period", "b", "ecc", "r_pl"]
samples = pm.trace_to_dataframe(trace, varnames=varnames)

# Convert the radius to Earth radii
samples["r_pl"] = (np.array(samples["r_pl"]) * u.R_sun).to(u.R_earth).value

corner.corner(
    samples[["period", "r_pl", "b", "ecc"]],
    labels=["period [days]", "radius [Earth radii]", "impact param", "eccentricity"]);
../../_images/tess_31_0.png

These all seem consistent with the previously published values and an earlier inconsistency between this radius measurement and the literature has been resolved by fixing a bug in exoplanet.

Citations

As described in the Citing exoplanet & its dependencies tutorial, we can use exoplanet.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} and its
dependencies citep{exoplanet:astropy13, exoplanet:astropy18,
exoplanet:exoplanet, exoplanet:foremanmackey17, exoplanet:foremanmackey18,
exoplanet:kipping13, exoplanet:luger18, exoplanet:pymc3, exoplanet:theano}.
print("\n".join(bib.splitlines()[:10]) + "\n...")
@misc{exoplanet:exoplanet,
  author = {Daniel Foreman-Mackey and Ian Czekala and Eric Agol and
            Rodrigo Luger and Geert Barentsen and Tom Barclay},
   title = {dfm/exoplanet: exoplanet v0.2.0},
   month = aug,
    year = 2019,
     doi = {10.5281/zenodo.3359880},
     url = {https://doi.org/10.5281/zenodo.3359880}
}
...