Deriving the Radial Velocity Equation | Exoplanet Detection Series

Radial velocity analysis for exoplanet detection requires understanding probabilistic frameworks including likelihood, prior, and posterior distributions. Noise modeling is critical, with traditional stationary assumptions creating computationally expensive O(n³) covariance matrices. The Celerity model offers faster linear scaling through modified covariance decay. Leaf models exploit block correlation structures for efficient calibration noise modeling. Multi-component noise includes nominal bars, jitter, calibration noise, exponential decay, and quasi-periodic terms. Linear base models combine Keplerian orbits with polynomial trends for stellar activity and polynomial terms for long-term variations. Activity indicators require smoothing before serving as linear predictors. These foundational techniques enable robust exoplanet detection from radial velocity data.

Astronomers analyze radial velocity data by first removing all known stellar activity signals using their activity model. After subtracting the stellar noise, any remaining periodic signals are examined for potential planetary candidates. Each candidate is then modeled using Keplerian orbital parameters, and the model is subtracted from the data to reveal additional signals. This iterative process allows astronomers to identify multiple planets in a system.

Stellar jitter introduces noise that masks the tiny radial velocity signals from low-mass planets. Key sources include: (1) P-mode oscillations from pressure-driven waves that cause the Sun to pulse like a gong over ~5-minute periods; (2) Granulation from convective cells where hot gas rises and cool gas falls, creating bright granules and dark lanes that produce convective blue shift; (3) Supergranulation operating on larger spatial scales; (4) Magnetic activity including sunspots (which block light asymmetrically) and faculae (bright regions that affect convection patterns). These effects create wiggles in the data much larger than the planet signals we seek to detect.

EPRV data processing involves extracting ~117 spectral orders from complex 2D focal plane images containing science fibers, laser frequency comb calibrators, and sky fibers. Processing includes image corrections, spectral extraction, wavelength calibration, and treatment of sky contamination. Radial velocity is calculated using cross-correlation with stellar masks—selecting stable spectral lines to build average line profiles. True planetary motion causes all lines to shift identically without changing shape. As instruments improved, instrumental noise decreased, revealing stellar activity as the dominant noise source. Key stellar noise sources occur at different timescales: p-modes (~minutes), granulation (~hours-days), spots/faculae (~months), and magnetic cycles (~years-decades). Traditional approaches include targeting quiet stars, averaging over p-mode oscillations, and avoiding activity-contaminated periods.

Stellar activity (spots, flares, convection cells) creates background noise that can mask or mimic exoplanet signals. The radial velocity method detects exoplanets by measuring how a planet's gravity causes its star to wobble, creating Doppler shifts in spectral lines. Stars moving toward us show blue shifts, while those moving away show red shifts. This method works from any viewing angle, unlike transit observations. Jupiter causes the Sun to wobble at about ±12.5 m/s, Saturn at ±2.8 m/s, and Earth at only ±0.10 m/s.

Two primary methods now detect exoplanets. The radial velocity method measures stellar wobble caused by planetary gravity using the Doppler effect—detecting velocity changes as small as 1 m/s. The transit method observes periodic brightness dips when planets pass in front of stars. Each method has strengths and limitations: radial velocity reveals mass, transit reveals size, and combining both determines planetary density and composition. The Kepler Space Telescope revolutionized transit detection, finding hundreds of planets and revealing extraordinary system diversity.

The HD 14082b discovery combined radial velocity and transit methods for confirmation. Astronomers collected over 3,000 spectra from 111 young stars using the MPG/ESO 2.2-meter telescope at La Silla Observatory. For HD 14082b specifically, approximately 60 spectra were obtained. The radial velocity method detected a signal, while TESS space telescope data revealed a transit signal. When both methods showed consistent results, the planetary nature was confirmed. This combination provides both mass (from radial velocity) and radius (from transit), enabling calculation of density and internal structure.

The radial velocity method detects exoplanets by measuring the star's wobble caused by orbiting planets through spectral line shifts, while the transit method detects planets by observing periodic dimming of starlight as planets pass in front of their host stars; the transit method has proven most successful, discovering the majority of the 6,316 confirmed exoplanets as of July 2026, and can determine planet radius, density, and atmospheric composition when combined with radial velocity data, enabling the search for biomarkers in exoplanet atmospheres.

The radial velocity method detects exoplanets by measuring Doppler shifts in stellar light as stars move toward and away from us due to planetary gravitational tugs. Jupiter-like planets produce signals of 10-13 m/s while Earth-like planets produce only about 9 cm/s. Transit photometry measures starlight dimming during planetary passages, revealing planetary sizes. The Kepler mission used this method to discover thousands of planets, with detection probability decreasing significantly for planets at larger orbital distances.

The transit method reveals a planet's size but not its mass, while the radial velocity method detects gravitational effects that indicate mass. By combining data from both methods, astronomers can calculate a planet's density, determining whether it is gaseous or rocky. This multi-method approach provides more complete information about exoplanets, including their composition and internal structure, making it essential for comprehensive planetary characterization.

EXPRESS at Lowell Discovery Telescope targets 20 cm/s precision through key innovations: laser frequency comb calibration with 10^-12 precision, spectral resolution of 140,000, deployable tertiary mirror for rapid instrument switching, and fiber agitator for scrambling. Initial observations revealed beat frequencies (solved by reprogramming SLM to 55 Hz) and modal noise (addressed with low-torque motors). EXCALIBUR software uses PCA to model slow instrumental drifts over nights. This demonstrates that systematic error identification and calibration are critical for extreme precision, with post-redesign achieving 5 cm/s RMS over one hour.

Laser frequency combs are precision optical rulers that generate hundreds of thousands of equally spaced frequency lines, enabling absolute wavelength calibration of astronomical spectrographs with unprecedented accuracy (10^-11 precision). These combs, developed using Ti:sapphire lasers and optical parametric oscillators, provide dense, stable calibration references across the visible to near-infrared spectrum (500-2200 nm), essential for detecting exoplanets and measuring cosmic phenomena like variations in fundamental constants. The technology was demonstrated on the Southern African Large Telescope (SALT), achieving two-fold improvement in wavelength solution precision compared to traditional thorium-argon lamps.

Frequency combs provide traceable, ultra-precise wavelength calibration for astronomical spectrographs, enabling detection of exoplanets and tests of fundamental physics by achieving radial velocity precision down to centimeters per second; the technology involves generating broadband combs from Ti:Sapphire lasers expanded via supercontinuum generation and mode-filtered with Fabry-Pérot etalons, with recent advances including hybrid systems combining visible and infrared combs for coverage from 550-2200 nm and novel locking techniques that reduce noise by factors of 5-8 compared to conventional methods.

Laser frequency combs revolutionized precision spectroscopy by generating trains of ultrashort femtosecond pulses at nanosecond intervals. These coherent pulses create interference patterns producing thousands of equally spaced spectral lines ('comb teeth') across the visible spectrum. When deployed at telescope sites, these combs serve as ultra-stable calibration references, enabling correction of spectrograph drifts. This advancement achieved unprecedented sensitivity—detecting Doppler shifts corresponding to Earth-mass planets at one astronomical unit from Sun-like stars. Combined with transit photometry (which measures planetary diameter via light dips during transits), this dual approach enables determination of planetary density and composition.

Modern high-resolution spectroscopy pursues unprecedented precision to address fundamental questions in physics and cosmology. Two primary science drivers demand one-meter-per-second accuracy and one-centimeter-per-second stability: measuring fine structure constant variation across cosmic history using quasar absorption lines, and observing real-time universe expansion through Lyman-alpha forest redshift measurements. These ambitious goals require revolutionary advances in wavelength calibration technology. Fabry-Perot etalons exhibit chromatic drift—wavelength-dependent systematic errors requiring sophisticated data reduction corrections. Meanwhile, laser frequency combs offer SI-second-locked super-narrow lines but face practical implementation challenges, particularly in the blue spectral region. Success requires understanding both calibration sources and instrumental responses, including proper line spread function modeling that accounts for asymmetric profiles and spatial variations across detectors. The BLUEVIS project exemplifies ongoing efforts to develop blue laser frequency comb technology through interdisciplinary collaboration, addressing the technological barriers that stand between current capabilities and the extreme precision needed for fundamental physics discovery.