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EXOFAST: A fast exoplanetary fitting suite in IDL

Jason Eastman, B. Scott Gaudi, Eric Agol

arXiv:1206.5798v3astro-ph.IMastro-ph.EP

TL;DR

EXOFAST addresses the need for fast, consistent exoplanet fitting and uncertainty characterization across transit and radial-velocity data. It combines joint modeling with DE-MC, analytical and computational optimizations, and extensible routines, achieving a well-mixed HAT-P-3b fit in under five minutes and a roughly 100-fold transit-model speedup. The paper also identifies parameterization and MCMC implementation choices that can bias eccentricity estimates.

  • Problem

    Exoplanet data require careful, homogeneous extraction of planetary and stellar information from transit and radial-velocity measurements.

  • Method

    EXOFAST jointly or separately fits transit and RV data using DE-MC, realistic priors, orbital parameterization analysis, analytical limb-darkening fits, and optimized transit models.

  • Results

    The HAT-P-3b example was well-mixed in under five minutes, while transit-model execution time improved by a factor of ∼100.

  • Takeaways & Limitations

    EXOFAST provides a modular, extensible framework for self-consistent exoplanet fitting and adaptation to additional effects or other data sets.

  • Takeaways & Limitations

    A Metropolis-Hastings implementation mistake can bias intrinsically circular orbits toward significantly non-zero eccentricities, and detrending can ignore covariance and produce spurious TTV claims.

Abstract

from arXiv · show

We present EXOFAST, a fast, robust suite of routines written in IDL which is designed to fit exoplanetary transits and radial velocity variations simultaneously or separately, and characterize the parameter uncertainties and covariances with a Differential Evolution Markov Chain Monte Carlo method. We describe how our code incorporates both data sets to simultaneously derive stellar parameters along with the transit and RV parameters, resulting in more self-consistent results on an example fit of the discovery data of HAT-P-3b that is well-mixed in under five minutes on a standard desktop computer. We describe in detail how our code works and outline ways in which the code can be extended to include additional effects or generalized for the characterization of other data sets -- including non-planetary data sets. We discuss the pros and cons of several common ways to parameterize eccentricity, highlight a subtle mistake in the implementation of MCMC that could bias the inferred eccentricity of intrinsically circular orbits to significantly non-zero results, discuss a problem with IDL's built-in random number generator in its application to large MCMC fits, and derive a method to analytically fit the linear and quadratic limb darkening coefficients of a planetary transit. Finally, we explain how we achieved improved accuracy and over a factor of 100 improvement in the execution time of the transit model calculation. Our entire source code, along with an easy-to-use online interface for several basic features of our transit and radial velocity fitting, are available online at http://astroutils.astronomy.ohio-state.edu/exofast .

1. INTRODUCTION

EXOFAST is introduced as a modular IDL suite for fitting exoplanet transits and radial velocities, with self-consistent stellar characterization and uncertainty estimation. The paper also describes speed improvements, extensibility, and practical implementation choices.

  • Motivation: Over 500 exoplanets had been confirmed, with four times as many candidates identified, motivating improved and consistent extraction of planetary information.The paper places EXOFAST within the rapid expansion of exoplanet discovery and characterization.
  • Motivation: Transit photometry measures planetary radius and orbital inclination, while combining it with RV minimum mass yields true mass and average density.These measurements constrain planetary structure and illustrate why joint analyses matter.
  • EXOFAST and its goals: EXOFAST simultaneously fits radial velocities, transits, and stellar parameters while providing DE-MC uncertainties and covariances.The suite is designed for separate or combined RV and transit analyses, with realistic priors and pipeline-friendly outputs.
  • EXOFAST and its goals: EXOFAST is designed as a modular framework that can incorporate effects such as TTVs and secondary eclipses or analyze other data sets.The authors cite Supernovae and Cepheids as examples of non-planetary applications.
  • Implementation: IDL was selected for its existing code library and development ease; vectorization helps offset its limitations for model evaluation.The authors note that GDL may support core features with user adjustments, but the code does not work with it out of the box.
  • Paper scope: The paper discusses fitting procedures, eccentricity parameterizations, MCMC pitfalls, limb darkening, transit-model acceleration, and an HAT-P-3b example.The source code and online interfaces are also made available.

2. OVERVIEW OF THE PROBLEM

The paper formulates fitting as likelihood or χ2 optimization followed by posterior uncertainty estimation. It uses numerical strategies for nonlinear models, DE-MC for correlated uncertainties, explicit priors, and caution around hybrid fitting.

  • Best-fit estimation: For Gaussian uncertainties, maximum likelihood is equivalent to minimizing χ2 when uncertainties are fixed.Linear models can be minimized analytically, whereas nonlinear transit and RV models require numerical optimization.
  • Best-fit estimation: Nonlinear fitting requires numerical searches, with EXOFAST using problem-specific parameter-space restrictions before robust local minimization.AMOEBA and Levenberg-Marquardt are discussed as local optimization methods with different robustness and evaluation-cost trade-offs.
  • Uncertainty estimation: MCMC step densities are proportional to posterior probability, supporting median values and 68% confidence intervals once priors are specified.The absolute evidence normalization is not evaluated for comparing alternative parameterizations.
  • Uncertainty estimation: DE-MC runs parallel chains whose pairwise differences estimate step scales, directions, and parameter covariances, reducing links needed for mixing.A small uniform deviate helps the chains reach the full parameter space rather than becoming cyclic.
  • Priors: Uniform priors apply to stepped parameters, so parameterization must reflect theoretical expectations or be corrected using Jacobians or importance sampling.A parameterization that excludes viable regions cannot be repaired by post-processing.
  • Hybrid fitting: Hybrid fitting can underestimate uncertainties when analytically fitted parameters covary with parameters of physical interest.The authors therefore fit all parameters nonlinearly, leaving analytical compensation for future work.
  • Hybrid fitting: Pre-fitting detrending can ignore uncertainty and covariance in trend parameters, potentially producing spurious transit-timing-variation claims.The paper identifies covariance between linear time or airmass trends and transit time as a specific risk.

3. RADIAL VELOCITY

The radial-velocity framework models orbital motion through Keplerian quantities and uses parameterizations intended to improve convergence and constrain transit-relevant timing. The preferred choices also expose prior and boundary considerations.

  • Radial-velocity model: The RV model uses orbital quantities including semi-amplitude K, systemic velocity γ, acceleration γdot, eccentricity e, and stellar argument of periastron ω*.The true anomaly varies with time, while γdot can represent a long-period companion or data systematics.
  • Radial-velocity model: Kepler’s equation must be solved numerically for eccentric anomaly E(t), using an improved routine with Newton-Raphson refinement at high eccentricity.The implementation also handles inputs near angular differences of 2π that can prevent convergence.
  • Radial-velocity model: EXOFAST reverses orbital phase calculations to obtain times of periastron, transit, eclipse, ascending node, descending node, and other specified true anomalies.The EXOFAST_GETPHASE routine supports both named orbital phases and arbitrary true anomalies.
  • RV parameterization: Parameterization matters because uniform priors are imposed on stepped variables, and physically unmotivated choices can bias inferred parameters.Highly covariant parameter sets are also inefficient for MCMC exploration.
  • RV parameterization: The favored RV parameterization uses logP, logK, √e cosω*, √e sinω*, TC, γ, and γdot, while avoiding overly restrictive bounds.The authors state that unbounded posteriors would appear through non-convergent chains when data provide no constraint.
  • RV parameterization: TC is preferred because it is generally better constrained than TP, avoids system-specific tuning, and is directly relevant to transit searches.For circular orbits, the convention ω*=π/2 makes predicted conjunction coincide with periastron passage.

3.3. Radial Velocity Best Fit

EXOFAST accelerates radial-velocity fitting by reducing circular-orbit searches to a one-dimensional period scan with analytic solutions for the remaining linear parameters. It also evaluates eccentricity parameterizations and convergence behavior to select a practical, prior-consistent approach.

  • Radial-velocity best fit: For circular orbits, EXOFAST reduces radial-velocity fitting to a single nonlinear parameter, P, while solving the other parameters analytically at each step.The circular-orbit simplification removes the ecosω* term and permits a rapid period scan.
  • Radial-velocity best fit: The period spacing criterion is dP ≤ P^2/(4πT), where T is the observation duration, to limit phase changes between successive trial periods.Scanning typically reveals likelihood peaks from the planetary period and aliases, which seed more rigorous nonlinear minimization.
  • Radial-velocity best fit: Alias-induced multiple periods require separate investigation because AMOEBA cannot find minima widely separated from the initial values.Adequate phase coverage over one or more complete orbits supports robust fitting, while highly eccentric cases may require searching many periodogram peaks.
  • Eccentricity prior: The ecosω* and esinω* parameterization produces a linear prior in e unless Jacobian weighting is applied, with slight overcorrection at e = 0 after correction.The Jacobian correction preferentially rejects steps toward higher eccentricity and nearly restores a uniform prior.
  • Eccentricity prior: The √e cosω* and √e sinω* parameterization naturally recovers a uniform prior in e and ω*, while direct stepping in e and ω* also yields a uniform prior.The square-root parameterization takes smaller eccentricity steps near zero, compensating for the smaller coordinate-space area there.

3.5. Eccentricity Bias

EXOFAST examines biases that can make intrinsically circular orbits appear eccentric, including the Lucy–Sweeney effect, parameterization priors, and incorrect Metropolis–Hastings boundary handling. Simulations show that linear priors and implementation errors can substantially bias inferred eccentricities, while MCMC reduces bootstrap-related bias.

  • Lucy-Sweeney bias: Low-eccentricity measurements are biased upward because e = 0 has zero phase space, so noise favors positive best-fit eccentricities.Lucy and Sweeney recommend e > 2.45σ_e for a 95% significant non-zero eccentricity.
  • Lucy-Sweeney bias: 100,000 AMOEBA fits of intrinsically circular orbits reproduce a deficit at e = 0, demonstrating bootstrap bias for bounded parameters.The resulting best-fit eccentricity histogram is compared with combined MCMC posterior PDFs from 100 simulated trials.
  • Parameterization effects: MCMC PDFs suffer far less bootstrap bias, but stepping in e cosω* and e sinω* introduces a linear prior that biases eccentricity high.The corrected parameterization has only a slight residual bias, negligible relative to the Gaussian width.
  • Parameterization effects: The linear-prior bias can reach ∼5σ, whereas differences among parameterizations are otherwise negligible relative to the posterior width.The authors therefore step in √e cosω* and √e sinω* rather than the uncorrected e-based variables.
  • Metropolis-Hastings algorithm: Incorrect Metropolis–Hastings rejection handling biases fits away from hard boundaries such as e = 0 by failing to copy the previous step.The obsolete IDL Astronomy Library implementation made this mistake, and the bias depends on parameterization and step size.
  • Metropolis-Hastings algorithm: Rejected boundary-crossing proposals must be retained as repeated copies of the previous step; otherwise prior probability is attenuated near boundaries.The affected region scales with step size and remains significant within roughly three step sizes of a boundary.

4. TRANSIT FITTING

EXOFAST models transit light curves from planetary geometry while accounting for stellar limb darkening and parameterization biases. It also supports simultaneous stellar, transit, and radial-velocity constraints, with analytic and computational refinements that improve fitting efficiency.

  • 4.1. Transit Model: Transit modeling computes relative stellar flux from the planet-to-star radius ratio and projected separation, with both quantities varying through the orbit.The projected separation is calculated over time from the planetary path and supplied to the Mandel & Agol transit model.
  • 4.2. Limb Darkening: Limb darkening makes stellar brightness decrease toward the limb, so realistic transit models require more than a uniformly bright stellar disk.The linear law can be adequate for some ground-based data but is insufficient for high-precision light curves, motivating quadratic modeling.
  • 4.2. Limb Darkening: Quadratic and linear limb-darkened transit fluxes can be expressed as linear combinations of analytic functions, allowing limb-darkening coefficients to be solved analytically for fixed geometry.This reduces the dimensionality of the nonlinear solver and can substantially accelerate fitting.
  • 4.3. Planetary Path: The planetary path is converted into projected separation z in stellar-radius units, while the sign of line-of-sight coordinate Z distinguishes primary transits from secondary eclipses.Primary transit occurs when Z > 0; secondary eclipse occurs when Z < 0.
  • 4.3. Planetary Path: EXOFAST does not fit secondary eclipses, although their model flux follows the transit calculation with substitutions p → 1/p and z → z/p and without planetary limb darkening.The eclipse depth is represented by one additional normalization parameter per observed bandpass.
  • 4.5. Other biases: Allowing negative planetary radii removes the bounded-parameter bias in p and helps distinguish absent transits, strong systematics, and statistically significant shallow transits.A small negative tail alongside a significant result is treated as evidence that the nonzero transit measurement is less likely to arise from fitting bias.

5. RADIAL VELOCITY AND TRANSIT

Simultaneous radial-velocity and transit fitting links shared parameters and stellar constraints to produce a more self-consistent system model. EXOFAST addresses the resulting stellar-parameter degeneracy with empirical relations and priors, while acknowledging assumptions that can affect uncertainties.

  • 5. RADIAL VELOCITY AND TRANSIT: Simultaneous RV and transit fitting improves both data-set fits by exploiting shared parameter constraints and enables additional effects such as stellar-reflex light-travel time.The combined approach also exposes covariances between parameters that separate fits do not model jointly.
  • 5.1. Combined Parameterization: The combined parameterization forms a one-parameter family of solutions, so an estimate of stellar mass, radius, or surface gravity can determine the full physical system.Stellar surface gravity provides a useful mass-radius combination for breaking the degeneracy.
  • 5.1. Combined Parameterization: Transit and RV data alone do not constrain stellar logg, so adding it without external information causes inefficient exploration of a degeneracy.Iterating between light-curve, RV, and isochrone fitting does not properly account for covariances between stellar parameters.
  • 5.2. Stellar Parameters: EXOFAST uses Torres mass-radius relations and spectroscopic priors to derive self-consistent stellar mass and radius at each Markov-chain step.The model applies prior penalties using scatter values σ_logM∗ = 0.027 and σ_logR∗ = 0.014.
  • 5.2. Stellar Parameters: Stellar constraints feed back into planetary mass and radius, while transit density and spectroscopy can sometimes constrain logg more precisely than the alternative source.The stronger constraint propagates through the linked stellar and transit parameters.
  • 5.2. Stellar Parameters: The method assumes Torres mass and radius relations are independent; if they are correlated, their prior contribution can be double-counted and uncertainties artificially reduced.Discarding one relation instead assumes perfect correlation and loses information when that assumption is false.
  • 5.2. Limb Darkening: Nonlinear limb-darkening models are currently impractically slow for large data sets, and accurate coefficient uncertainties and 3D-based tables are unavailable.These constraints limit the precision and scalability of the limb-darkening treatment.

6. AN EXAMPLE

The HAT-P-3b example demonstrates EXOFAST’s end-to-end workflow for preparing, fitting, and validating combined transit and radial-velocity data. Its results broadly agree with prior work while enforcing self-consistent stellar parameters and providing diagnostics for uncertainties and covariances.

  • 6. AN EXAMPLE: The example combines HAT-P-3b transit and radial-velocity data from Torres et al. (2007) to explain EXOFAST’s workflow and outputs.The system was selected because sufficient public data were available from a single RV source.
  • 6. AN EXAMPLE: The fitting procedure converts input times and fluxes to BJDTDB and normalized flux, then sequentially fits RV, transit, and combined data.EXOFAST scales uncertainties for each data set before beginning the combined Markov-chain fit.
  • 6. AN EXAMPLE: The example fixes eccentricity to zero and omits an RV slope because the available RV phase coverage cannot constrain them.The photometric period is used as a prior because the RV time span constrains it less strongly.
  • 6. AN EXAMPLE: The output includes publication-ready best-fit plots, residuals, parameter distributions, derived quantities, uncertainties, and covariance diagnostics.PDFs and covariance plots are intended primarily to diagnose problematic distributions and parameter relationships.
  • 6.2. Validation: All quoted values agree with Torres et al. (2007) within 1σ, with most agreeing to better than 0.25σ.The main methodological difference is EXOFAST’s simultaneous enforcement of consistency between stellar, transit, and RV parameters.
  • 6.2. Validation: 2.78 g cm−3 is the EXOFAST stellar density from the two parameter sets, compared with 2.36 and 2.67 g cm−3 from the two methods used by Torres et al. (2007).The paper also reports that the prior transit-time uncertainty is almost an order of magnitude larger than EXOFAST’s estimate.

7. ONLINE TOOLS

EXOFAST provides online tools for fitting, scheduling observations, retrieving exoplanet information, and estimating limb-darkening coefficients. These interfaces expose useful basic functionality but have explicit precision and research-use limitations.

  • 7. ONLINE TOOLS: The authors caution that poorly understood selection effects in the exoplanet sample limit interpretation of results derived from the database.The warning applies when using these tools to make population-level inferences.
  • 7.2. Ephemerides: The ephemeris calculator identifies planets transiting or eclipsing from a specified observatory and date using a database synchronized daily.Predicted times can be substantially less precise than fits to original data because published parameter covariances are omitted.
  • 7. ONLINE TOOLS: The online interface fits transit data, radial velocities, or both, includes arbitrary systematics, and reports fitted and derived parameters.Stellar-temperature and surface-gravity priors can be pulled from the daily-updated exoplanets.org database.
  • 7. ONLINE TOOLS: The online fitter reports only best-fit values because server-side execution cannot support the full DE-MC code, so it is not intended for research-quality fits.This is an explicit scope boundary of the online service rather than the full EXOFAST code.
  • 7. ONLINE TOOLS: A separate online tool interpolates quadratic limb-darkening tables across optical, infrared, space-based, and other listed photometric bands.The tool can also retrieve stellar parameters from exoplanets.org when available.

APPENDIX

The appendix derives an analytic treatment of linear and quadratic limb darkening that reduces nonlinear fitting dimensionality. This can accelerate fitting, but weak data may produce non-physical coefficients requiring alternative constraints.

  • APPENDIX: Transit flux is reformulated using geometry-dependent quantities x0, x1, and x2 that can be calculated independently of limb darkening.These quantities are optional outputs of EXOFAST_OCCULTQUAD.
  • APPENDIX: The limb-darkening coefficients c0, c1, and c2 are solved analytically through standard linear χ2 minimization, after which u1 and u2 are obtained from them.The same relations simplify for linear limb darkening by setting u2 = 0.
  • APPENDIX: The analytic fit can greatly increase speed by reducing the nonlinear solver’s dimensionality, especially for multi-band data.The appendix also discusses fitting both linear and quadratic laws to assess whether the additional parameter improves χ2.
  • APPENDIX: In low-signal-to-noise data, the analytic fit can yield non-physical limb-darkening values and should be replaced or constrained using alternative approaches.Suggested alternatives include fixing tabulated values, fitting with priors, or using a linear law.
  • APPENDIX: Hybrid analytic/nonlinear fitting must be used with care, as discussed in the paper’s treatment of the fitting procedure.The appendix explicitly flags this as a limitation of the analytic approach.

EXOFAST_OCCULTQUAD

EXOFAST_OCCULTQUAD accelerates analytical calculation of the flux decrement during planetary transit by exploiting IDL’s vector-language capabilities and correcting bugs.

  • EXOFAST_OCCULTQUAD: A factor of 100-500 speedup over the prior IDL implementation was achieved for analytical transit flux-decrement calculations.The routine also fixed many bugs in the earlier implementation.

Speed Enhancements

EXOFAST’s speed enhancements restructure transit calculations, replace elliptic-integral routines, and provide IDL and Fortran implementations with substantially faster execution. Further gains are constrained by the remaining elliptic-integral bottleneck and portability issues.

  • Speed Enhancements: Vector operations and optimal case ordering supplied the majority of the speed enhancement.
  • Speed Enhancements: The elliptic integral of the third kind remained the primary computational bottleneck after replacing the numerical recipes routines.The replacement used an IDL translation of an ALGOL program by Bulirsch.
  • Speed Enhancements: Combining polynomial calculations for the first- and second-kind elliptic integrals avoided repeating the expensive computation of log(1 −k).
  • Speed Enhancements: ∼100 times faster than the previous IDL version, the revised transit routine substantially reduced the runtime of the overall program.The Fortran rewrite was twice as fast as its predecessor and 2-5x faster than the current IDL version, depending on the compiler.
  • Speed Enhancements: 200-500 times faster than the original IDL version, the CALL_EXTERNAL Fortran wrapper matched native Fortran speed after first use.It was also 2-5x faster than the current IDL code, depending on the Fortran compiler.
  • Speed Enhancements: The Fortran wrapper was tested only on a 32 bit Linux machine, and some compilers refuse to compile its technically non-legal Fortran syntax.
  • Speed Enhancements: The Python version was 5 times slower than the IDL version despite mirroring its vector structure.The authors note that this may reflect either Python generally or their relative programming expertise.

Bug fixes and conceptual enhancements

EXOFAST addresses numerical accuracy and modeling flexibility through corrected special-case behavior, broader parameter handling, and analytic limb-darkening support.

  • Bug fixes and conceptual enhancements: Near transit-contact special cases, the previous elliptic-integral routines could differ from the true value by as much as 10%.The Bulirsch algorithm reduced these differences to less than 10^-5.
  • Bug fixes and conceptual enhancements: Typos affecting p > 1, including secondary-eclipse cases, were fixed along with other precision and Fortran-function bugs.The p > 1 errors would have caused catastrophic failures but had sufficiently non-physical behavior to be readily detected.
  • Bug fixes and conceptual enhancements: EXOFAST now permits negative p values and returns coefficients needed to fit quadratic limb-darkening parameters analytically.
  • Bug fixes and conceptual enhancements: Testing across special cases and hundreds of p values found no other errors.The tests included values within ±10−13 of every special case and 1,000 uniformly spaced z values.

RANDOM NUMBER GENERATOR

EXOFAST addresses periodicity concerns in IDL’s random-number generator for large MCMC fits by providing a slower replacement and testing its practical effect with DE-MC.

  • RANDOM NUMBER GENERATOR: IDL’s RANDOMU generator has periodicity of ∼10^9 and is recommended for sequences of roughly 50 million random numbers or fewer.
  • RANDOM NUMBER GENERATOR: The HAT-P-3b fit could generate up to 41.6 million random numbers across 26 chains and 100,000 steps per chain.
  • RANDOM NUMBER GENERATOR: A shortened-periodicity test generator was constructed to make effects from RANDOMU’s cycling easier to detect.
  • RANDOM NUMBER GENERATOR: EXOFAST_RANDOM wraps PG_RAN, whose periodicity is ∼10^57, and returns 8-dimensional uniform and Gaussian random-number arrays as a RANDOMU replacement.
  • RANDOM NUMBER GENERATOR: The replacement generator is ∼120 times slower than RANDOMU but contributes only about 3% of total program runtime.
  • RANDOM NUMBER GENERATOR: No statistical difference was found in the repeated HAT-P-3b fit, while DE-MC made periodicity effects much less significant but still noticeable.
  • RANDOM NUMBER GENERATOR: EXOFAST defaults to its slower generator for generality, while allowing users to select RANDOMU or another compatible generator with a flag.

NEGATIVE ECCENTRICITY

EXOFAST explores allowing negative eccentricity as an alternative parameterization, motivated by orbital-label symmetry, but excludes it from final distributions because it creates a discrete degeneracy.

  • NEGATIVE ECCENTRICITY: Allowing eccentricity from -1 to 1 reflects the possibility that apoapsis and periapsis were incorrectly labeled.The paper motivates negative eccentricity through e = (ra − rp)/(ra + rp).
  • NEGATIVE ECCENTRICITY: Negative eccentricities must be excluded from final distributions because they create a discrete degeneracy between e, ω∗ and −e, ω∗ + π.The parameter transformation also requires recalculating the time of periastron after changing ω∗.
  • NEGATIVE ECCENTRICITY: The resulting prior and posterior distributions were indistinguishable from those obtained with the √esinω∗ parameterization.
  • NEGATIVE ECCENTRICITY: With standard Markov Chain implementation, this parameterization was about 20% faster than the alternatives discussed.With DE-MC, it was about 20% faster than stepping directly in e, but nearly half as fast as parameterizing with ecosω∗ and esinω∗.
  • NEGATIVE ECCENTRICITY: The authors generally recommend DE-MC stepping in √esinω∗ while retaining this alternative parameterization for potential use by others.
  • NEGATIVE ECCENTRICITY: The same sign-and-angle transformation could be applied to projected rotation, allowing negative vsini while shifting λ by π.

PERFORMANCE

EXOFAST achieves rapid fitting through optimized transit calculations, while profiling identifies serial orbital calculations and diagnostics as major performance costs and bounds further speedups.

  • PERFORMANCE: A simulated eccentric transit-and-RV fit was well-mixed in about 5 minutes.The measured runtime depends on the number of data points and model details.
  • PERFORMANCE: The solution to Kepler’s equation consumed 28% of total computation time and was the largest remaining bottleneck.Because the iterative solution is serial, rewriting it in Fortran or C could improve runtime; the calculation is trivial for circular orbits.
  • PERFORMANCE: Covariance and parameter plotting contributed 20% of runtime because EXOFAST generated over 1000 diagnostic plots.Each plot bins chain points into a two-dimensional grid and computes probability contours.
  • PERFORMANCE: Reference-frame conversion contributed 12%, while its total contribution including the called Kepler-equation routine was 32%.The conversion is iterative and runs twice per MCMC step, once for RV and once for photometry.
  • PERFORMANCE: Quadratic limb-darkening interpolation contributed 6.3% because large tables take substantial time to load into memory.
  • PERFORMANCE: Transit-light-curve calculation contributed 12% overall after optimization, compared with an hour by itself before optimization.Non-linear limb darkening was not optimized and would increase total fit time by more than an order of magnitude.
  • PERFORMANCE: An all-Fortran EXOFAST was estimated to be around 5 times faster, but an improvement greater than 50 times was considered unlikely.
  • PERFORMANCE: GPU acceleration could exploit parallel model calculations and chains, but other bottlenecks currently make the overall boost small.GPUs were described as promising for large data sets after further bottleneck reduction.
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