How to Use the Lightkurve Web Interface for Exoplanet Data Analysis

Design matrices encode beliefs about noise behavior through multiple approaches. Polynomial bases use Vandermonde matrices to create columns of monomial terms, capturing low-order trends from illumination or thermal effects. Co-trending Basis Vectors (CBVs) apply Principal Component Analysis to light curves of many stars, extracting common modes representing detector systematics. Pixel Level Decorrelation (PLD) exploits the fact that transits appear uniformly across all pixels while detector motion creates different pixel-specific trends; dividing each pixel by total flux cancels the transit signal, leaving only systematics. Linearized Field Deep Blending extends this to crowded fields by modeling full PSFs of all sources.

Eleanor provides co-trending basis vectors (CBVs) from 2-minute targets, downsampled to 30-minute cadence, for detrending FFI light curves. These CBVs capture long-term trends from TESS and other systematics. The pipeline measures Combined Differential Photometric Precision (CDPP) from each light curve by measuring standard deviation outside transit regions after applying sigma cuts. This metric helps identify light curves with acceptable precision for exoplanet searches. The corrected flux combines co-trending basis vectors with regression against centroid motion, background, and time.

Time series variable detrending is a statistical technique used to remove trends from time series data, making the data stationary by eliminating systematic patterns over time, which is essential for accurate time series analysis and forecasting.

To decorrelate a pair of correlated random variables X and Y: find their covariance matrix, find its eigenvectors and eigenvalues, normalize eigenvectors to unit length, arrange them as columns in matrix A, and use A^T as the linear transformation. The resulting covariance matrix of transformed variables u and v will be diagonal with eigenvalues on the principal axis. If the original transformation was linear, the reverse transformation can recover the original variables. If the original transformation was nonlinear, a linear decorrelation transformation may not recover the original variables but will produce a new pair of uncorrelated variables. In practice, more than two random variables are often measured simultaneously, such as in medical screening (height, weight, cholesterol levels, family history, smoking status) or image processing (extent, boundary length, centroid, projections). These observations are arranged in a vector form, creating a random vector X = [X1, X2, ..., Xn]^T. The joint PDF of a random vector is a function from R^N to R, and marginal densities can be found by integrating with respect to unwanted variables.

This tutorial teaches how to detect exoplanets using NASA's TESS mission data through the transit method, where planets passing in front of their host stars cause temporary dips in stellar brightness. The process involves accessing TESS data products (Full Frame Images, Target Pixel Files, and Light Curve Files) using the Lightkurve Python package, processing light curves by applying corrections like PDCSAP flux to remove instrumental noise, masking bad data, and using techniques such as flattening, binning, and folding around known orbital periods to reveal planetary transits. For cases without pre-processed light curves, custom aperture photometry can be performed on raw Full Frame Images combined with Pixel Level Decorrelation (PLD) to remove systematic trends from spacecraft motion.

From transit observations, astronomers can determine: (1) Planet radius from transit depth (ΔF/F = (R_planet/R_star)²), (2) Orbital period from transit timing, (3) Semi-major axis from Kepler's third law, and (4) Orbital inclination from transit duration and shape.

BATMAN is a Python package that enables rapid computation of exoplanet transit light curves, capable of generating one million model light curves in under 10 minutes for any radially symmetric stellar limb darkening law; the package allows users to define transit parameters including time of inferior conjunction, orbital period, planet radius, semi-major axis, inclination, eccentricity, and limb darkening coefficients, supporting various limb darkening models such as uniform, linear, quadratic, square root, logarithmic, exponential, power2, and non-linear profiles to simulate how different planetary characteristics affect observed light curves.

Transit depth (percentage brightness decrease) reveals planet size through the relationship: transit depth = (planet radius / star radius)². For a 3% depth, the planet is about 17% the star's radius (Jupiter-sized for Sun-like stars). Transit duration, combined with Kepler's laws, reveals orbital period and distance—closer planets have shorter transits. The light curve shape indicates orbital inclination: flat bottoms indicate edge-on orbits, while triangular shapes indicate tilted orbits.

From transit light curves, astronomers can derive several planetary parameters: (1) Planet radius from the depth of the brightness drop (proportional to the square of the planet-to-star radius ratio), (2) Orbital period from the time between transits, (3) Semi-major axis (orbital distance) using Kepler's third law, (4) Orbital inclination from the transit duration and geometry. The transit method alone cannot determine the planet's mass, which requires the velocity radial method. Combining both methods allows calculation of planetary density, revealing whether the planet is rocky or gaseous.

Transit light curve analysis allows astronomers to determine multiple physical properties of exoplanets. By fitting mathematical models to the observed brightness data, astronomers can calculate the planet's orbital distance (in units of stellar diameters), planet size (in units of Jupiter radii), and orbital inclination (the angle at which we view the planet's orbit relative to the star). For example, a planet with an orbital distance of 6.4 stellar diameters and a size of 1.4 Jupiter radii can be precisely characterized from transit data alone. These measurements provide fundamental information about the planet's physical characteristics without requiring direct imaging.

The radial velocity method detects exoplanets by measuring stellar wobble through Doppler shifts in spectral lines. The first exoplanet around a Sun-like star (51 Pegasi b) was discovered this way in 1995. Combining transit data (size) with radial velocity data (mass) reveals exoplanet density and composition—some are rocky, some are water worlds, some are gas giants. This diversity was not predicted by our solar system's structure, showing planetary formation is more varied than previously thought.

Before Kepler, the most effective exoplanet detection method was radial velocity or Doppler spectroscopy. This method is based on the fact that a star is not completely stationary when it has a planet orbiting it. Although planets are much smaller than stars, they exert a tiny gravitational pull on their parent star. When a planet is behind the star from our perspective, it pulls the star away from us; when in front, it pulls the star toward us, causing the star to oscillate. Astronomers use powerful spectrographs to examine starlight, decomposing it into its component colors. As starlight passes through a star's atmosphere, some wavelengths are absorbed, creating dark lines in the spectrum. When the star moves toward us, these lines shift toward the blue end; when moving away, they shift toward the red end. Astronomers detect orbiting planets by observing the movement of spectral lines caused by the star's wobble. The transit method, used by Kepler, is based on observing small variations in star brightness when a planet's orbit passes in front of its star. The amount of light lost depends on the relative sizes of the star and planet, and the duration depends on the distance between the planet and star. Combining transit photometry with radial velocity measurements reveals a planet's radius, mass, and density.

Transit photometry detects exoplanets by measuring brightness decreases when planets pass in front of their stars. Transit depth reveals planet-to-star radius ratios, enabling size measurements. Combined with radial velocity data, this allows density calculations revealing composition. Transit spectroscopy analyzes stellar light passing through planetary atmospheres, detecting molecules like water, methane, and sodium. Secondary eclipse observations reveal planetary thermal emission and albedo.

Astronomers determine exoplanet compositions using multiple techniques: (1) Transmission spectroscopy—measures how planetary atmospheres absorb starlight during transits, revealing atmospheric composition; (2) Direct imaging—separates planet light from host star light for massive planets far from their stars; (3) Bulk density analysis—combines transit radius measurements with radial velocity mass measurements to infer average density, indicating whether planets are rocky, gaseous, or icy. Surveys reveal that super-Earths and mini-Neptunes are the most common planet types, with compositions unlike anything in our solar system.

Transit light curves show periodic brightness dips corresponding to planet-to-star area ratios, enabling radius calculation when stellar size is known. Radial velocity creates Doppler shifts in stellar spectral lines as planets induce stellar wobble, with shift amplitude indicating planet mass. Combined transit and radial velocity detections provide both radius and mass, enabling density calculation and composition inference. These methods reveal different planetary populations due to inherent observational biases in detection capabilities.