Fit a frozen TESS transit without losing its assumptions

Begin with the actual product and columns

The input is the SPOC Sector 1 light-curve product for TIC 25155310, WASP-126: tess2018206045859-s0001-0000000025155310-0120-s_lc.fits. Its data release is 42 and processing version spoc-5.0.10-20200904. The original file has 20076 rows. Its exact source checksum, original header blocks and deterministic extraction recipe accompany the small observed input.

Five windows are centered on BTJD 1327.52 plus successive periods of 3.2888 days, each extending 0.28 days on either side. These windows were specified in time; their selection does not clip low or high flux. The working period agrees with the discovery paper’s Table 6 initialization. The actual fit will refine it.

The CSV retains 2017 original rows, their source row and window, cadence number, quality, SAP and PDCSAP flux/errors, removed background/errors and centroid/position information. Blank nonfinite values remain blank. The accepted set has 2007 rows after requiring zero quality, finite time and PDC flux/error, and positive error. All ten rejected rows remain in the original CSV.

Reading: TESS Science Data Products Description, Revision F · Maxted et al. 2016, Five WASP-South Planets, arXiv v1

Check time, background and the correction already applied

The TIME column is in days, Barycentric Dynamical Time, with reference Julian date 2457000 and a midpoint time stamp. The listed TIMECORR has already been applied to TIME. Adding it again would shift the event instead of improving the timestamp.

Flux is in electrons per second. SAP_BKG records the background that the pipeline removed; it is not another sky subtraction. PDCSAP also corrects captured aperture fraction and known-source crowding. The original CROWDSAP is 0.97252083 and FLFRCSAP 0.88525611. Preserve them as provenance; do not repeat those corrections on PDC flux.

The cadence span is 120 seconds. Single-frame accumulation, readout and onboard rejection make the live time different; the original header records these separately. The fit approximates the measurement with a uniform 120-second window. Frame-level rejection details remain unresolved within that approximation.

Reading: TESS Science Data Products Description, Revision F · MAST TESS Data Product Overview

Run the package in an ordinary folder

Save the linked scientific files together. The observed input and frozen outputs are ordinary public downloads. An optional original-FITS refresh additionally needs Astropy and checks the original file hash before extracting the same windows. The ordinary fit uses the retained CSV, not Astropy or a network service.

The first command fits observations. The second independently generates analytic synthetic references and dense exposure averages. Those are distinct calculations with different labels. Detached runs on the recorded host reproduce the saved JSON. Across numerical libraries, compare the values with stated tolerances and inspect any changes in the optimizer path.

                  python3 -m venv .venv
source .venv/bin/activate
python3 -m pip install -r requirements.txt
python3 -W error replay-transits.py --data . --out transit-fit-rerun.json
python3 -W error reference-transits.py --input synthetic-input.json --out analytic-reference-rerun.json

Reading: batman author tutorial and pinned package 2.5.3 · TESS Science Data Products Description, Revision F

Fit the transit and a baseline for each window

Normalize accepted PDC values and their errors by one saved median flux. Fit radius ratio, impact parameter, scaled orbital axis, epoch and period. The default brightness coefficients are fixed to 0.45 and 0.20 as an explicit teaching assumption. Additional contaminant light is set to zero after the pipeline correction.

At each trial, compute the circular transit, average it over each two-minute window, and solve a separate baseline intercept and linear trend for every observed window. These nuisance coefficients are solved by weighted linear least squares within the nonlinear fit. They help describe local baseline drift; they do not guarantee removal of stellar or instrumental structure.

The objective uses the supplied pointwise errors. Four starting paths check for alternative minima. The resulting conditional best fit is approximately $p=0.07673$ , $b=0$ , $a/R_\star=7.858$ , with $P=3.288756\,\mathrm d$ and $T_{14}=3.454\,\mathrm h$ . Its circularly inferred density is about $848\,\mathrm{kg\,m^{-3}}$ . The fixed stellar-brightness, circular-orbit and dilution assumptions accompany these results.

Reading: Mandel & Agol 2002, Analytic Lightcurves, arXiv v1 · TESS Science Data Products Description, Revision F

Read the path profile before quoting an error

The saved profile fixes 25 impact parameters from zero to 1.2, refitting the other parameters each time. The best point reaches the nonnegative boundary. At fixed impacts 0.20, 0.35 and 0.50, the objective rises by about 0.86, 3.40 and 10.46. The curve in the figure is a conditional objective profile, not a posterior probability density.

A naive high-impact start can produce no overlap, leaving the optimizer on a flat light curve with no useful gradient. The script retains that failed-start objective and also starts from a physically overlapping radius. This matters in the high-impact tail: an optimization success flag by itself does not establish that the desired transit minimum was explored.

The residual RMS is about 0.00131 in normalized flux. Adjacent residual correlation is near minus 0.0024; 124 complete fifteen-cadence blocks have scatter about 1.04 times the white-noise scaling. Those diagnostics describe this sample; independent residuals and calibrated parameter confidence levels require further testing.

All 2007 accepted observed cadences, after dividing by each fitted local baseline and folding on the fitted period; every fifteenth sample shows its supplied pointwise error. The solid curve is the same saved conditional fit. No flux clipping or invented observation was added.

All 25 profile points; vertical coordinate is objective increase with an asinh stretch. Each point refits the other parameters. This is a conditional fit profile under supplied Gaussian pointwise errors, not a posterior or a symmetric error bar. The original high-impact initialization failures remain in the downloadable output.

Reading: Dawson & Johnson 2012, Photoeccentric Effect I, v3 · TESS Science Data Products Description, Revision F

Change the assumptions and refine the exposure

A uniform stellar disk gives approximately $p=0.08198$ and $b=0.680$ for these same observations. Adding an assumed extra companion-to-host light ratio of 0.3 after PDC gives $p=0.08750$ . These are sensitivity fits with changed source assumptions. They are not evidence that either assumption is true.

The two-minute window is well resolved here: at fixed best parameters, nine-node and sixty-five-node averages differ by at most about seven hundredths of a millionth in flux. Refitting with seventeen nodes changes the reported radius ratio very little. The instantaneous fit changes the duration only slightly in this two-minute example; a much longer exposure can have a larger effect, as the synthetic long-exposure preset shows.

Keep the full samples, residuals, impact profile, initialization outcomes and sensitivity fits when modifying the experiment. A useful next step would constrain the star’s band-dependent limb brightness or residual contaminating light independently, then revisit the conditional parameter range.

Reading: Kipping 2010, Finite Integration Time, arXiv v1 · MAST TESS Data Product Overview

Exercises and explanations

Why does a best impact parameter at zero not imply a tightly measured perfectly central orbit?
Read the explanation
Zero is the parameter boundary. Nearby paths can produce similar shapes after other parameters change. Inspect the one-sided profile and its assumptions; a symmetric local covariance can hide the shape.
You have only PDCSAP flux and its CROWDSAP keyword. Should you divide the transit depth by that crowding fraction again?
Read the explanation
The selected pipeline product already includes a known-source crowding correction. Repeating it changes the data twice. Additional unknown light is a separate physical assumption to test explicitly.
A high-impact fit reports success but predicts no transit. How does the retained replay diagnose it?
Read the explanation
It keeps the original objective and repeats the fit with an overlapping radius. Comparing the objectives exposes a flat, poorly initialized solution even when an optimizer stopped normally.

Try the prediction

Connect the observed fit to the synthetic calculation

Use the brightness model to predict the direction of the radius change when extra companion light is assumed.
Select the grazing preset and explain why the observed fit’s high-impact starts need an overlapping radius.
Compare the exposure and instantaneous curves without calling either synthetic curve an observation of WASP-126.

Save the scientific files in one folder. Ordinary replay uses the pinned NumPy, SciPy and batman dependencies; optional original-FITS refresh uses a separate Astropy environment.

Sources and calculation scope

Mandel & Agol 2002, Analytic Lightcurves, arXiv v1
3 October 2002, version 1; ApJ 580,L171–L176. Sections 2–4, uniform overlap, quadratic intensity normalization and Table 1 special cases; Section 5 states the small-planet approximation. The current archive PDF has a later typesetting header; its submission version is pinned here.
Kipping 2010, Finite Integration Time, arXiv v1
21 April 2010, version 1. Section 2 follows contact shifts and biased parameter retrieval; Section 3.3 treats numerical integration errors around curved segments and contact discontinuities. A model window is kept distinct from an exact instrument kernel.
TESS Science Data Products Description, Revision F
11 September 2020. Sections 2.4, 2.7, 5.2.2 and quality Table 32: time standards, frame rejection, light-curve columns and flags. The actual DR42/SPOC source header and values are frozen with the observed input; generic manual examples are not substituted for those values.
MAST TESS Data Product Overview
Official archive overview: SAP versus PDC processing, aperture fraction and known-source crowding already corrected in PDCSAP. This is why the tutorial does not apply those corrections twice.
batman author tutorial and pinned package 2.5.3
Author documentation for analytic quadratic limb darkening and orbit conventions. Independent reference uses the separately installed batman-package 2.5.3 distribution, verified on PyPI. Raw boundary failures are retained; published special-case elliptic and independent Cartesian integrals resolve the checked stellar-center crossing. No GPL kernel is copied into the interactive Engine.
Maxted et al. 2016, Five WASP-South Planets, arXiv v1
4 February 2016, version 1; A&A 591,A55. Sections 2–4 and Table 6: WASP-126 discovery photometry, radial velocities, adopted orbital assumptions and initialization period 3.28880 days. The table was also checked in the original arXiv TeX source. This tutorial fits a separate later TESS product, not the discovery data.
Dawson & Johnson 2012, Photoeccentric Effect I, v3
2 November 2012, version 3. Section 2 equations3–7 and Sections 3.1–3.2 with Figures 2–3: short-transit speed factor, conditional density, curved eccentricity/orientation posteriors, stellar uncertainty and transit selection. Posterior illustrations and observed applications are labeled separately.