Accurate and efficient photo-eccentric transit modeling

A planet's orbital eccentricity is fundamental to understanding the present dynamical state of a system and is a relic of its formation history. There is high scientific value in measuring eccentricities of Kepler and TESS planets given the sheer size of these samples and the diversity of their planetary systems. However, Kepler and TESS lightcurves typically only permit robust determinations of planet-to-star radius ratio $r$, orbital period $P$, and transit mid-point $t_0$. Three other orbital properties, including impact parameter $b$, eccentricity $e$, and argument of periastron $\omega$, are more challenging to measure because they are all encoded in the lightcurve through subtle effects on a single observable -- the transit duration $T_{14}$. In Gilbert, MacDougall,&Petigura (2022), we showed that a five-parameter transit description $\{P, t_0, r, b, T_{14}\}$ naturally yields unbiased measurements of $r$ and $b$. Here, we build upon our previous work and introduce an accurate and efficient prescription to measure $e$ and $\omega$. We validate this approach through a suite of injection-and-recovery experiments. Our method agrees with previous approaches that use a seven-parameter transit description $\{P, t_0, r, b, \rho_\star, e, \omega\}$ which explicitly fits the eccentricity vector and mean stellar density. The five-parameter method is simpler than the seven-parameter method and is"future-proof"in that posterior samples can be quickly reweighted (via importance sampling) to accommodate updated priors and updated stellar properties. This method thus circumvents the need for an expensive reanalysis of the raw photometry, offering a streamlined path toward large-scale population analyses of eccentricity from transit surveys.


INTRODUCTION
Out of more than 5,300 confirmed planets to date, ∼75% were discovered via the transit method. These discoveries have paved the way for keystone scientific advancements in our understanding of planet formation, evolution, and demographics. To ensure the reliability of inferences based on the transiting planet population, we must also ensure that characterizations of individual transiting planets are consistently and accurately derived. Previously, uncertainties on stellar parameters significantly limited the achievable precision of planet properties (e.g. σ(R ⋆ ) ∼ 27% and σ(ρ ⋆ ) ∼ 51%; Thompson et al. 2018). Now, in the era of Gaia (Gaia Collaboration et al. 2018) and high-precision stellar characterizations (e.g. σ(R ⋆ ) ≲ 2% and σ(ρ ⋆ ) ≲ 10%), the determination of key planet properties is limited by light curve modeling (see, e.g. Petigura 2020).
A variety of methods exist for modeling transit signals, including various parameterizations (e.g. Sea-ger & Mallén-Ornelas 2003;Carter et al. 2008;Dawson & Johnson 2012;Eastman et al. 2013; Thompson et al. 2018;Gilbert et al. 2022) and sampling techniques (e.g. Feroz & Hobson 2008;Foreman-Mackey et al. 2013;Foreman-Mackey et al. 2021;Speagle 2020;Gilbert 2022). Differences in posterior inference which arise from adopting a particular model parameterization and sampling method are often assumed to be insignificant relative to other sources of uncertainty. However, if one wishes to achieve percent-level precision on all quantities, one must also carefully consider the strengths and weaknesses of competing model/sampler implementations (see, e.g., Gilbert 2022. Although substantial effort has been put into vetting methods for transit signal detection (see, e.g., Christiansen et al. 2015), far less effort has been devoted to validating subsequent methods for transit signal modeling. A key aim of this work -which builds directly upon our previous work in Gilbert et al. 2022, hereafter G22 -is to place the transit modeling problem on the same secure foundation as the transit detection problem. Our primary focus here is on the effects of model parameterization, with a secondary focus on the role of the sampler.
A popular and straightforward method for transit model parameterization is to use a seven-parameter basis which includes orbital period P , transit epoch t 0 , planet-to-star radius ratio r, impact parameter b, eccentricity e, argument of periastron ω, and either stellar density ρ ⋆ or scaled orbital separation a/R ⋆ , these latter two parameters being related via Kepler's third law 1 (see, e.g. Eastman et al. 2013). This eccentricityexplicit basis {P, t 0 , r, b, e, ω, ρ ⋆ }, e-ω-ρ hereafter, benefits from being fully characterized by properties of the star, planet, and planetary orbit. However, realworld photometric transit lightcurves typically only include enough information to constrain four or five out of the seven parameters. More precisely, in most realworld cases the signal-to-noise S/N of observations is low enough that one cannot precisely measure the duration and curvature of ingress/egress nor can one detect any transit asymmetry (see Barnes 2007). Without resolved ingress/egress or transit asymmetry, the problem remains unconstrained, with b, e, ω, and ρ ⋆ each imprinting themselves on the lightcurve indirectly via the transit duration T 14 (r imprints itself via the transit depth; P and t 0 via the ephemeris). More explicitly, for a given ρ ⋆ , b influences the transit chord length, e and ω influence the speed of the planet during transit, and the ratio of transit chord length to orbital speed produces T 14 .
An alternative approach that improves upon these limitations of the seven parameter method is to model the lightcurve assuming a circular orbit, e = 0, regardless of what the true underlying eccentricity might be (see, e.g., Seager & Mallén-Ornelas 2003;Dawson & Johnson 2012). This shortcut reduces the total number of model parameters by two with a trade-off that the transit is now explicitly assumed to by symmetric. Fortunately, for virtually all Kepler and TESS class photometry, this assumption does not introduce measurable biases into the analysis. In G22, we explored the effectivity of two different five-parameter bases: {P, t 0 , r, b, T 14 } vs {P, t 0 , r, b,ρ}, whereρ is the stellar pseudo-density, i.e. the stellar density inferred from the transit photometry under the (probably false) assumption of a circular orbit. We found that the two bases are equivalent when an appropriate Jacobian transformation is properly applied, but that the latter basis introduces complex, nonintuitive covariances between b andρ. These covariances 1 In practice, other parameters related to the stellar limb darkening (i.e. quadratic limb darkening coefficients u 1 , u 2 ) and to the properties of the photometry (i.e. flux zero-point F 0 and photometric noise σ F ) are usually also needed, but these complicating details are not the focus of this paper.
artificially disfavor b ≳ 0.7, which propagates through to other parameters, shifting e toward higher values and r toward lower ones. Historically, the use of parameter bases which includeρ has resulted in biased inference, and we consequently recommend avoiding the use ofρ altogether. For the remainder of this work, we therefore do not consider any parameterizations which includeρ. Our preferred model parameterization {P, t 0 , r, b, T 14 }, hereafter the T 14 basis, benefits from being intuitive and close to quantities which can be directly measured from the transit photometry, which minimizes the risk of introducing unintended bias. In G22, we demonstrated that this parameterization yields unbiased posteriors on both b and T 14 . In this work, we build on G22 to develop a post-hoc importance sampling routine that enables indirect recovery of e and ω from direct measurements of T 14 and an independent external constraint on ρ ⋆ (e.g. from asteroseismology or spectroscopy). To validate our methods, perform injection-and-recovery tests using simulated transit photometry over a grid of transit parameters and compare the performance of our proposed T 14 + importance sampling approach to the performance of the standard e-ω-ρ modeling basis. We find that the two methods yield equivalent posterior inferences on b, e, and ω, with significant improvements to speed and efficiency when using our new approach. Another major advantage of our proposed technique is that it is "future-proof" in that it allows us to update estimates of e and ω as stellar characterization is inevitably updated in the future (e.g. from new Gaia data releases) without requiring a computationally expensive re-run of the transit fits. In comparison, the usual sevenparameter e-ω-ρ basis "bakes in" a particular value of ρ ⋆ at the time of transit modeling.
We lay out our methodology for lightcurve synthesis and transit injection-and-recovery in §2. We then highlight the procedural differences between the e-ω-ρ method ( §3) and our T 14 method ( §4). In §5, we analyze the results of our injection-and-recovery tests and compare the performances of the two parameterizations. We provide a summary of our conclusions in §6.

SYNTHETIC LIGHTCURVE CONSTRUCTION
Our objective is to compare the performance of the physical e-ω-ρ parameter basis to the simpler T 14 basis. We aim to demonstrate whether or not these methods return equivalent and accurate posterior results and determine their relative efficiencies. To achieve these objectives, we perform a suite of injection-and-recovery tests over a grid of parameters which spans a wide range of values of eccentricity e, argument of periastron ω, inclination (parameterized as impact parameter b), and signal-to-noise (see Figure 1). Injection-and-recovery is a standard tool used to evaluate transit signal detection methods (see, e.g., Christiansen et al. 2015), but it has not been applied to transit model validation on nearly the same scale. Here, we construct a set of synthetic lightcurves, then we proceed to use two distinct transit modeling methods to recover the injected transit properties and compare the relative model performances.
For all injection-recovery tests in this work, we inject the transit signal of a sub-Neptune-size planet orbiting a Sun-like star with an orbital period close to the average among Kepler planets. We synthesize 10 transits per lightcurve with a photometric zero-point flux of µ flux = 0 and a fixed photometric noise σ flux , consistent with raw photometry that has been accurately prewhitened. We calculate the duration of each injected transit signal T 14 according to the following equation from Winn (2010): We construct our synthetic lightcurves at three different signal-to-noise ratio (SNR) levels: SNR ∼ [20,40,80]. We show example lightcurves for each SNR level in Figure 1. At an SNR of 20, the injected signal has a slightly lower significance compared to the median Kepler planet signal. At the higher SNR levels of 40 and 80, we seek to identify any differences that emerge between our two models as the transit ingress and egress become more distinct from the photometric noise, making b measurements more precise. From the selected SNR and other injected lightcurve properties, we generate Gaussian white noise per lightcurve centered on σ flux , which we calculate according to: where N transits is the number of injected transits and t exp is the simulated exposure time of our synthetic lightcurve. The random seed used to generate the synthetic white noise is unique to each injection-recovery test.
We also assign a unique set of transit parameter values {b, e, ω} for each injection-recovery test, where each of these inputs is drawn from a grid of discrete parameter values (see Figure 1). We specifically choose a parameter grid that emphasizes the region of parameter space where the e−ω−b degeneracy is strongest (see, e.g., Van Eylen & Albrecht 2015) since this is where the two parameterizations are more likely to yield differing results. As a result, our injected planet signals do not exactly mirror the distribution of Kepler planets, but they do include a broad range of realistic planet characteristics. Since the transit shape is more sensitive to small changes in b at high values, we select injected values of b with tighter spacing towards higher values, spanning the non-grazing parameter space. We construct an array of b values that are evenly spaced on a reversed log scale: b ∼ [0.1, 0.48, 0.7, 0.83, 0.9]. We also prefer to use e values that span the range of eccentricities with tighter spacing towards low-to-moderate values, since these are more common. We select an array of possible e values which are evenly spaced on a log scale: e ∼ [0.05, 0.1, 0.2, 0.4, 0.8]. Additionally, the ω values that we draw upon for our grid of injected parameters are intentionally selected to include the inflection points of periastron (π/2 or 90 • ) and apastron (3π/2 or 270 • ) along with three roughly evenly spaced values in be- We construct a set of 375 unique transit lightcurves from all combinations of {b, e, ω, SNR} using the batman transit modeling package (Kreidberg 2015). We synthesize these injected lightcurve models with an oversampling rate of 11 and t exp = 30 minutes, similar to real Kepler photometry. These lightcurves serve as inputs to the two modeling methods that we are comparing, described below, in order to demonstrate similarities and differences in model performance across a range of potential transit signals (see Figure 2 for an overview).

METHOD #1: DIRECT SAMPLING IN e-ω-ρ
We first model our synthetic transit lightcurves using the e-ω-ρ model, which serves as our baseline model and standard reference when evaluating the per-formance of our proposed T 14 + umb model. This physically-motivated transit model is parameterized by {P, t 0 , r, b, ρ ⋆ , e, ω}, along with quadratic limb darkening parameters {u 1 , u 2 }. Since we simulate lightcurves with white noise, we fix µ flux and σ flux which would otherwise be directly sampled parameters when modeling real transit photometry.
We construct the e-ω-ρ model using uninformative priors that are of standard use in transit fitting literature or drawn directly from G22, summarized in Table 1. We apply a normal prior on ρ ⋆ which assumes that the stellar density is known with 10% uncertainty through independent measurements. To mitigate boundary issues that can occur when sampling e and ω directly, we use a common redefinition of these parameters { √ e sin ω, √ e cos ω} (see, e.g. Eastman et al. 2013), with implicit uniform priors on both e and ω. These priors do not account for transit probability or other astrophysically motivated considerations (see Barnes 2007). We implement this model using exoplanet (Foreman-Mackey et al. 2021), with sampling performed by the NUTS algorithm via PyMC3 (Salvatier et al. 2016). We use 3,000 tuning steps with an additional 4,000 sampler draws to ensure that the sampler converges with an effective sample size N eff ≈ 10 3 . We also set a high target acceptance fraction of 0.99 to encourage the sampler to adequately explore complex topologies in the posterior parameter space, such as the b − r and e − ω degeneracies. We follow the standard practice of oversampling the light curve model in order to mitigate binning artifacts (see, e.g., Kipping 2010), using an oversampling factor of 11. We fit our transit models via two sampler chains across two CPU cores per injection-recovery test.
From initial experimentation, we found that sampler limitations exist which restrict the valid parameter space of eccentricity modeling when applying the e-ω-ρ parameterization via NUTS sampling with exoplanet. When sampling e ≳ 0.92, this implementation of the eω-ρ model can have convergence issues due to the high curvature of the posterior parameter space being traversed. This also roughly corresponds with the upper eccentricity limit where we expect transit duration approximations to begin breaking down (see, e.g. Kipping 2014). Given that only 5 known planets have e > 0.9 and only one of these was discovered via transit modeling, we choose to restrict our eccentricity sampling to e < 0.92 for all modeling approaches considered in this work. By doing so, we avoid conflating our primary interest -differences in modeling methods -with rare edge cases that are beyond the scope of this work. Our alternative transit modeling approach, the T 14 + umb model, has a parameter basis that includes the observable transit duration T 14 as an explicit parameter. This parameterization avoids explicitly sampling the complex degeneracies introduced by e and ω, allowing us to instead measure these parameters post-hoc via importance sampling (see 4.2). We couple this duration-based parameterization with umbrella sampling (see Gilbert 2022) to ensure that our model accurately samples the complicated topology of the high-b "grazing" parameter space. Based on the arguments made in both G22 and Gilbert (2022), we expect that our T 14 + umb approach should achieve results that are consistent with those from the e-ω-ρ model with a potential boost in efficiency.

Transit fitting
Similar to our implementation of the baseline e-ωρ model, we also construct our T 14 + umb model via exoplanet with NUTS sampling and use it to model our synthetic transit signals. This parameterization is motivated by observable transit properties and characterized by the basis {P, t 0 , r, b, T 14 }. Like the e-ω-ρ model, the T 14 + umb model also includes quadratic limb darkening parameters {u 1 , u 2 } as well as fixed values of µ flux and σ flux . The priors used here are identical to those used in our e-ω-ρ model, summarized in Table 1. Neither e nor ω is explicitly constrained during the sampling process here, and their values are instead estimated from post-model importance sampling. This parameterization is thus agnostic to orbital eccentricity, except for the implicit assumption of a symmetric transit. This is a reasonable approximation since the acceleration of an eccentric planet during its transit is unlikely to introduce detectable asymmetry given modern photometry (Barnes 2007).
To improve both the sampling convergence and the exploration of complex posterior topologies, we follow Gilbert (2022) to implement umbrella sampling. We separate our NUTS sampler into three windows (i.e. "umbrellas") defined within the joint {r, b} parameter space, which allows us to sample the full posterior parameter space in smaller pieces that are easier to explore. The resulting posteriors can later be stitched together by applying the appropriate umbrella weights. The three umbrella windows that we use correspond to non-grazing and grazing orbits separated by a region that we refer to as the transition umbrella, which partially overlaps with the other two (see Gilbert 2022 for full description). In our implementation, we apply the three umbrella models in series but emphasize that this task can easily be parallelized to reduce the apparent wall-clock run-time. In the Appendix, we also discuss a potential alternative to umbrella sampling, known as dynamic nested sampling (see, e.g., Skilling 2004;Skilling 2006), which achieves roughly comparable results.

Importance sampling
To recover {e, ω} samples from the T 14 + umb modeling approach, we apply post-hoc importance sampling to the combined umbrella model posterior distributions. Importance sampling (see, e.g., Oh & Berger 1993;Gilks et al. 1995;Madras & Piccioni 1999) allows one to measure the properties of a given parameter's probability distribution based on samples generated from a different (typically easier to sample) parameter's distribution. This method was first incorporated into exoplanet characterization models by Ford (2005) and Ford (2006), used in combination with MCMC sampling to improve radial velocity model efficiency. Such methods can be useful to correct for observational biases post-hoc or derive the distributions of more complicated distributions outside of the MCMC sampling routine. Importance sampling is closely related to umbrella sampling, and the former can be thought of as a single-window special case of the latter. In our implementation, importance sampling only marginally increase the total run-time of the T 14 + umb approach by a few seconds.
We first compute the relative weights of the three umbrella models following Gilbert (2022) and combine our posterior chains into a single set of weighted posterior distributions. Since the umbrella weights effectively reduce the total number of samples, we up-sample the merged posterior distributions via random resampling to generate a total of 10 5 samples per parameter for convenience. We then perform importance sampling to weigh how well the measured values of {P, r, b, T 14 } at each sampler step can be described by an independently measured density of the host star. We will refer to this independent stellar density as ρ ⋆,true , with some uncertainty σ ρ⋆,true . To determine the appropriate importance weights, we first calculate the sampler-derived stellar density, ρ ⋆,samp , at each point in the umbrella-weighted posterior. This calculation directly follows from the transit duration equation described by Winn (2010): We note that Equation 3 explicitly includes e and ω, for which we do not yet have any information. We substitute these parameters with random draws of {e, ω} from uniform priors e ∼ U (0, 0.92) and ω ∼ U (− π 2 , 3π 2 ) -recall that the upper limit e = 0.92 was chosen to circumvent sampling issues at high e in the e-ω-ρ basis. By deriving ρ ⋆,samp from measured values of {P, r, b, T 14 } and random uniform values of {e, ω}, we ensure that ρ ⋆,samp reflects a true stellar density as opposed to the pseudo-stellar density parameterization which assumes e = 0 and was deemed unreliable by G22.
We compare the samples of ρ ⋆,samp against the independently measured ρ ⋆,true by computing the loglikelihood of each i th sample, assuming a Gaussian likelihood function. We then weight each sample from our umbrella-weighted posterior distributions by to produce the final, importance-weighted posterior distributions for each parameter. We apply these same weights to the random uniform {e, ω} samples to derive the final posterior distributions of these two parameters. All analysis in this work regarding the T 14 + umb model is based on these posterior distributions that have been umbrella-weighted, up-sampled, and importanceweighted. The final posterior distribution of e that we measure using our T 14 + umb modeling approach can thus be directly compared to the e posterior from the e-ω-ρ model. We therefore can use the T 14 basis {P, t 0 , b, r, T 14 } along with an independently constrained ρ true to derive posterior distributions for all parameters represented by the e-ω-ρ basis {P, t 0 , b, r, e, ω, ρ ⋆ }. With the T 14 basis, we have the advantage of avoiding introducing significant stellar constraints (i.e. ρ ⋆ ) until after the transit has already been fully modeled. Thus, our T 14 + umb model only needs to be run once while the e-ω-ρ model would have to be re-run for each updated measurement of stellar density. The post-hoc importance sampling step can easily be re-run for an updated ρ ⋆,true value (or different priors on e or ω) within only a few seconds, making our T 14 + umb modeling approach essentially future-proof. In the era of Gaia and high-precision stellar characterization, such future-proofing will become increasingly valuable.

Both methods return equivalent eccentricity constraints
We fit 375 injected transit signals from our grid of injection-recovery tests using both the e-ω-ρ baseline model and our T 14 + umb modeling approach. We measure all transit parameters using both modeling approaches, including e and ω. The posterior distributions of e, ω, and b serve as our primary points of comparison between the baseline model and our alternative modeling approach. Here, we specifically focus our analysis on e, since b (and its relationship with r) was already covered in G22 and ω is often a nuisance parameter in photometric modeling. We use posterior comparisons of ω and b for secondary analysis when necessary.
We perform a quantile-quantile comparison of the posterior values e k at the k = 15 th , 50 th , and 85 th percentiles of the e eωρ and e T14+umb eccentricity distributions. In Figure 3, we present a comparison of e k from both modeling methods at each of the key percentiles for all injection-recovery tests. We see that all tests at each percentile are close to the 1-to-1 line (black), demonstrating that the two modeling methods produce nearly equivalent posterior results for e.
We compute the difference ∆e k (e.g. ∆e 50 = e 50,eωρ − e 50,umb ) and use this as a measure of similarity between the two model results. To estimate the significance of ∆e k for each posterior comparison, we assume a standard eccentricity uncertainty of σ e = 0.05, informed by the typical uncertainty on e measured among all known planets (σ median (e) ≈ 0.05; NASA Exoplanet Science Institute 2020 2 ). For injection-recovery tests where |∆e k | ≲ 0.05 at the 15 th , 50 th , and 85 th percentiles of eccentricity, we assert that the e-ω-ρ and T 14 + umb methods produce equivalent results. Among multiple iterations of our suite of injection-recovery tests, we did not identify any tests which consistently produced posterior measurements for e that differed by |∆e k | ≲ 0.05 (see Figure 3). This suggests that our approach is an excellent alternative to the e-ω-ρ method, since the two methods should converge on identical results (as opposed to ∼68% identical).
We also consider how ∆e k differs as a function of both the lightcurve SNR and the injected transit duration T 14 . Specifically, we consider the ratio between T 14 and the expected duration of the same planet on a circular, centrally transiting orbit (the reference duration, This duration ratio is a more concise metric to interpret the effects of e, ω, and b on the duration of a transit. While we observe no trend in ∆e k with respect to SNR, we do note a marginal trend in ∆e k as a function of T 14 /T 14,ref across our sample. We find that the T 14 + umb model estimates slightly higher e values than the e-ω-ρ model at short transit durations and vice-versa at long transit durations, but the deviations that contribute to this trend are sub-significant. We ultimately conclude that the two modeling methods produce equivalent eccentricity measurements (within a Figure 3. Comparison of e values measured from the T14 + umb and e-ω-ρ modeling methods at the 15 th , 50 th , and 85 th percentiles of their distributions, along with the residuals ∆e k for each comparison (e.g. ∆e50 = e50,eωρ − e 50,umb ). We show ∆e k = {0.05, 0.1, 0.15} in grey, as well as the ideal 1-to-1 line shown in black. These comparisons generally lie close to the 1-to-1 line, implying that the results of the two models are approximately equivalent. We see no trends in the residuals of these comparisons.

Both methods return accurate results
We have demonstrated that our alternative transit modeling approach produces equivalently accurate results relative to our baseline model, but we have not yet considered if these models yield the correct results (relative to the injected parameters). It is known in the field of exoplanet characterization that photometric eccentricity constraints (and ω constraints) tend to have large uncertainties for individual planets (see, e.g. Van Eylen et al. 2019). Here, we qualitatively assess these uncertainties across our set of injection-recovery tests.
Since our sample is not representative of the observed planet population, we describe the observed trends among our e measurements according to different quadrants of e − b parameter space. We split up our tests into four broad scenarios based on their injected transit properties: (1) low e and low b, (2) low e and high b, (3) high e and low b, (4) high e and high b. We show demonstrative examples of of these four scenarios in Figure 4 with several ω values, all at SNR = 20. In all four quadrants, the posterior distributions of e and b are broad, non-Gaussian, and display a range of outcomes, but we describe the general trends that we observe below. We also offer some additional discussion regarding how ω can affect these posterior constraints. We limit our discussion to only the posterior distributions of the T 14 + umb modeling approach since the two approaches produce nearly equivalent results.
In scenario 1 (low e and low b), transit models accurately measure low values for both e and b with little posterior mass at higher values (Figure 4, top left), regardless of ω. In scenario 2 (low e and high b), models tends to significantly overestimate e but produce more accurate measurements of b (Figure 4, bottom left), regardless of ω. The opposite is true in scenario 3 (high e and low b) where b tends to be overestimated while e is measured more accurately (Figure 4, top right), except near apastron where both are measured fairly accurately. In scenario 4 (high e and high b), transit models tend to accurately measure high values for both parameters with little posterior mass at lower values ( Figure 4, bottom right), except near apastron where neither is measured well. We avoid providing a quantitative description of these observed trends because the non-Gaussian posterior distributions are not well-represented by simple summary statistics.
When e is high (e.g. scenarios 3 and 4), the value of ω true can significantly impact the posterior constraints on e and b due to the degenerate influence that these parameters can have on the observed transit duration, particularly near apastron. On the other hand, we do not observe any noteworthy trends in model accuracy as a function of SNR. For a typical Kepler planet which has low e, non-grazing b, and ω closer to periastron, we would generally expect to measure e and b posterior distributions that are somewhat consistent with the true underlying orbital geometry of the planet based on the trends that we observe in Figure 4. In Appendix B, we briefly explore whether using a different sampler (i.e. might yield even more accurate posterior constraints, but our findings there are inconclusive. 5.3. Our T 14 + umb method is more efficient than the e-ω-ρ method We have shown that the T 14 +umb basis can be used as an alternative to the e-ω-ρ basis, achieving equivalent results while also reducing the number of parameters by two. This parameter reduction should increase the efficiency of the T 14 + umb model, but this approach also requires three separate sampling runs -one for each of the three umbrellas. To evaluate the overall model efficiencies, we compared the number of effective samples per second (η) achieved by each method for all injectionrecovery tests.
For the e-ω-ρ method, we measure the number of effective samples from the r posterior distribution for each test using Geyer's initial monotone sequence criterion via arviz (Geyer 1992;Gelman et al. 2013;Kumar et al. 2019). We select r because it is a common output between our models and is less affected by complicated parameter degeneracies. We then divide N eff by Figure 5. Ratio of sampling efficiencies η T 14 +umb /ηeωρ as a function of duration ratio T14/T 14,ref and SNR. We bin the data across every 10 th percentile of the duration ratio distribution, showing a single point per bin per SNR (bins are separated by vertical grey lines). Each point shows the 15 th , 50 th , and 85 th percentiles of a given bin. The increased efficiency of the T14 +umb method relative to the e-ω-ρ method depends on the SNR of the modeled lightcurve. At higher SNR the T14 + umb method is significantly more efficient, but at moderate-to-low SNR the two methods have more similar efficiencies. At shorter transit durations, the T14 + umb method is always more efficient, but this behavior changes around T14/T 14,ref ≈ 0.8. The large spread in some uncertainties reflects the heterogeneity of our injected lightcurve parameters.
the total run-time for this model to achieve the e-ω-ρ sampling efficiency: η eωρ . For the T 14 + umb method, we average N eff of the r posteriors from each umbrella model, weighted by their respective umbrella weights. We divide this weighted average by the sum of the runtimes for the three umbrella models (e.g. the CPU runtime) to achieve the overall T 14 + umb sampling efficiency: η T14+umb .
We calculate the ratio of these two efficiencies for all injection-recovery tests and find that η T14+umb /η eωρ > 1 for ∼73% of tests, suggesting that the T 14 + umb approach is generally more efficient across our set of injected planet parameters. The median value of η T14+umb /η eωρ across our sample is 2.0, implying that the T 14 + umb approach is typically 2× more efficient than the e-ω-ρ method, although the range of this efficiency ratio is broad. When we consider η T14+umb /η eωρ as a function of SNR, however, we measure a median efficiency increase of 5.7× at SNR = 80, 1.2× at SNR = 40, and 1.1× at SNR = 20 (see Figure 5). We also find that the T 14 + umb method is only more efficient than the e-ω-ρ method in ∼52% of low-SNR tests. These findings suggest that the T 14 + umb method tends to be less efficient when the transit signal is weaker.
From Figure 5, we also see that the efficiency ratio changes with respect to the duration ratio T 14 /T 14,ref .
For tests with SNR = 20, the median efficiency ratio η T14+umb /η eωρ decreases significantly as the duration ratio increases, dropping from 2.1× at T 14 /T 14,ref ≤ 0.8 to 0.6× at T 14 /T 14,ref > 0.8. This trend is likely due to differences in how the two methods explore the high-b grazing regime. As the duration ratio approaches unity or higher, high b values are significantly less likely, but the T 14 + umb approach continues to carefully explore the high-b regime via three umbrella models even when it is not necessary. On the other hand, injection-recovery tests with higher b values (and generally shorter transit durations) are more efficiently sampled by the T 14 +umb approach. This behavior is consistent with what we would expect, given that umbrella sampling is specifically intended to ensure accurate measurements of the high-b parameter space.
Our set of injected transit properties, however, is not completely representative of observed planet demographics. To make a more representative comparison, we estimate the efficiency ratio for a typical Kepler planet based on both SNR and duration ratio T 14 /T 14,ref . We use the latter metric because it reflects the combined effects of e, b, and ω in a single variable. For a typical confirmed Kepler planet with SNR ≈ 20 − 40 and T 14 /T 14,ref ≈ 0.6 − 1.1, we estimate an efficiency ratio of ∼0.9×. Based on these findings, we assert that the two methods generally have similar sampling efficiencies for real planetary transit signals, with the T 14 + umb approach excelling for signals with higher SNR or lower duration ratio.
The efficiency increase from the T 14 +umb approach is more significant when we consider wall-clock time rather than CPU time. Since the three umbrella models can be run in parallel, we can reduce the apparent run-time of the T 14 + umb approach by up to a factor of a few. In this parallelized case, the apparent sampling efficiency of the T 14 + umb method is ∼1.2× faster than the e-ω-ρ method for a typical Kepler planet. As another added benefit, the posteriors of the T 14 + umb approach can be importance sampled for updated values of ρ ⋆ (as they become available) without re-running the NUTS sampling process (see §4.2), which is a major advantage in the long-term efficiency of the T 14 + umb parameterization.

CONCLUSIONS
In this work, we presented an updated photoeccentric transit modeling method using a durationbased parameterization {P, t 0 , r, b, T 14 } (with umbrella sampling) and post-hoc importance sampling which efficiently achieves accurate constraints on e, ω, and b. Through a suite of synthetic injection-and-recovery tests, we demonstrated that our approach produces equivalent eccentricity constraints relative to the more common eccentricity-explicit transit model parameterization {P, t 0 , r, b, e, ω, ρ}. We find that our modeling method generally has a higher sampling efficiency than the e-ω-ρ method when the true e or b value is high or a similar efficiency otherwise. Our approach can also be parallelized to increase its relative sampling efficiency several-fold more.
A key advantage of our modeling method is that posthoc importance sampling allows us to successfully derive accurate e and ω posterior distributions (relative to the e-ω-ρ method) without including e, ω, or ρ as explicit model parameters. Our importance sampling routine is fast and flexible enough to easily incorporate an updated prior on e and/or ω, which is critical for hi-erarchical modeling approaches at the population level. Our method also allows us to update parameter posterior distributions according to updated values of ρ ⋆ (e.g. from new Gaia data releases) without any loss of generality. In the modern era of high-precision stellar characterization, this sort of "future-proofing" will be invaluable as the number of transit candidates around well-characterized stars continues to grow.
We are grateful to Dan Foreman-Mackey for helpful conversations about this work. This study made use of computational resources provided by the University of California, Los Angeles and the California Planet Search.  For each injection-recovery test, we measure the values e k,T14+dyn from the T 14 + dyn eccentricity posterior at the k = 15 th , 50 th , and 85 th percentiles of the distribution and compare to the results of the T 14 + umb method like in §5.1 (Figure 6). We find that the T 14 + umb and T 14 + dyn methods yield eccentricity results that are broadly in agreement. However, there appears to be more differences between samplers (T 14 +umb versus T 14 +dyn) than between parameterizations (T 14 + umb versus e-ω-ρ ). The comparison between parameterizations yielded no test results that consistently differed by |∆e k | ≥ 0.05, but the comparison between samplers yields 42 of such discrepancies. Among these, there are three tests that differ by |∆e k | ≥ 0.15 and yield entirely different posterior topologies for e. The discrepant measurements of ∆e k are most common at the k = 15 th percentile, implying that the two sampling methods differ most at sampling the low-e tail of the eccentricity distribution. We observe that the T 14 + dyn method produces e posterior distributions with much less posterior weight in the low-e tail as compared to the results of the T 14 + umb approach. We also see a similar divergence of the two methods in the upper tail of the b posterior distributions. This is consistent with our additional observation that the majority of the discrepancies occur in tests with shorter duration ratios (T 14 /T 14,ref ≲ 0.5). Most discrepancies also occur at higher SNR levels, counter to expectations. Together, these criteria for discrepant results only match with ∼1% of observed Kepler transit signals, implying that real systems are highly unlikely to fall into this subset. Figure 6. Comparison of e values measured from the T14 + umb and T14 + dyn modeling methods at the 15 th , 50 th , and 85 th percentiles of their distributions, along with the residuals ∆e k for each comparison (e.g. ∆e50 = e 50,dyn − e 50,umb ). One outlier residual lies beyond the bounds of our residual plots, as indicated by the arrow pointing towards the outlier at ∆e50 = 0.417. We show the ∆e k = {0.05, 0.1, 0.15} in grey, as well as the ideal 1-to-1 line shown in black. These comparisons generally lie close to the 1-to-1 line, implying that the results of the two models are approximately equivalent. However, we do observe 42 tests where the T14 + umb and T14 + dyn model results are discrepant by more than |∆e k | ≲ 0.05. In these instances, we find that the T14 + dyn model tends to overestimate e relative to the T14 + umb model, as seen in the residuals and discussed in §B.1.

B.2. Accuracy
We compare the true underlying eccentricity of each injection-recovery test with the measured posterior distribution of e from the T 14 + dyn modeling method. Overall, we find that the qualitative trends in e and b measured via the T 14 + dyn method are roughly equivalent to those measured from the T 14 + umb method (see §5.2). We do, however, find a significant difference between the accuracies of the two modeling methods among the three most discrepant injection-recovery tests, where |∆e 50 | ≥ 0.15. For these discrepant tests, seen as outliers in Figure 6, the T 14 + dyn method achieves more accurate posterior constraints on both e and b. This may suggest that differences between samplers can, in some cases, lead to significant differences in the accuracy of modeled parameters. While all three of these tests have e true = 0.8, we unfortunately do not find any discernible rules by which to distinguish when sampler differences will lead to substantial differences in the accuracy of posterior results.

B.3. Efficiency
We also compare these two modeling approaches according their sampling efficiencies. We calculate the efficiency η T14+dyn of the T 14 + dyn approach for each injection-recovery test, based on the number of effective samples measured via the Kish 1965 approach using dynesty. Similar to §5.3, we compute the efficiency ratio between the T 14 +dyn model and our T 14 + umb approach (η T14+umb /η T14+dyn ) and show these results in Figure 7. The distribution of efficiency ratios among our sample is broad but suggests that the two methods generally have similar sampling efficiencies, with a median efficiency ratio of η T14+umb /η T14+dyn ≈ 1.1. At lower duration ratios, the T 14 + umb approach is ∼1.4× more efficient, which is to be expected since this part of parameter space includes higher b values -the specialty of umbrella sampling as implemented by Gilbert (2022).
For a typical Kepler planet, however, we estimate that the T 14 + dyn method is ∼1.6× faster than the T 14 + umb approach. This observation, along with an occasional improvement in accuracy, leans in favor of dynamic nested sampling compared to NUTS sampling + umbrella sampling for our tests, but there are many other compounding factors that are beyond the scope of our experiment. Overall, both sampling methods offer their own benefits with neither winning out 100% of the time, but it is clear that the duration-based parameterization performs well regardless of the underlying sampling method. Figure 7. Ratio of sampling efficiencies η T 14 +umb /η T 14 +dyn as a function of duration ratio T14/T 14,ref and SNR. We bin the data across every 10 th percentile of the duration ratio distribution, showing a single point per bin per SNR (bins are separated by vertical grey lines). Each point shows the 15 th , 50 th , and 85 th percentiles of a given bin. The efficiency of the T14 + umb method relative to the T14 + dyn method depends partially on the SNR of the modeled lightcurve. At shorter transit durations, the T14 + umb method is typically more efficient, but the opposite is true at longer transit durations. The large spread in some uncertainties reflects the heterogeneity of our injected lightcurve parameters.