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Exoplanet transits contain substantial information about the architecture of a system. By fitting transit lightcurves we can extract dynamical parameters and place constraints on the properties of the planets and their host star. Having a well-defined probabilistic model plays a crucial role in making robust measurements of these parameters, and the ability to differentiate the model provides access to more robust inference tools. Gradient-based inference methods can allow for more rapid and accurate sampling of high-dimensional parameters spaces. We present a fully differentiable photodynamical model for multi-planet transit lightcurves that display transit-timing variations. We model time-integrated exposures, compute the dynamics of a system over the full length of observations, and provide analytic expressions for derivatives of the flux with respect to the dynamical and photometric model parameters. The model has been implemented in the Julia language and is available open-source on GitHub., Comment: 20 pages, 13 figures, 3 tables. Re-submitted to Monthly Notices of the Royal Astronomical Society in response to referee report. Software package at: https://github.com/langfzac/Photodynamics.jl Comments/corrections welcome

We present the first set of trans-Neptunian objects (TNOs) observed on multiple nights in data taken from the DECam Ecliptic Exploration Project (DEEP). Of these 110 TNOs, 105 do not coincide with previously known TNOs and appear to be new discoveries. Each individual detection for our objects resulted from a digital tracking search at TNO rates of motion, using two to four hour exposure sets, and the detections were subsequently linked across multiple observing seasons. This procedure allows us to find objects with magnitudes $m_{VR} \approx 26$. The object discovery processing also included a comprehensive population of objects injected into the images, with a recovery and linking rate of at least $94\%$. The final orbits were obtained using a specialized orbit fitting procedure that accounts for the positional errors derived from the digital tracking procedure. Our results include robust orbits and magnitudes for classical TNOs with absolute magnitudes $H \sim 10$, as well as a dynamically detached object found at 76 au (semi-major axis $a\approx 77 \, \mathrm{au}$). We find a disagreement between our population of classical TNOs and the CFEPS-L7 three component model for the Kuiper belt., Comment: Accepted to AJ, companion paper do DEEP III. Objects will be released in the journal version (or contacting the authors)

We present a detailed study of the observational biases of the DECam Ecliptic Exploration Project's (DEEP) B1 data release and survey simulation software that enables direct statistical comparisons between models and our data. We inject a synthetic population of objects into the images, and then subsequently recover them in the same processing as our real detections. This enables us to characterize the survey's completeness as a function of apparent magnitudes and on-sky rates of motion. We study the statistically optimal functional form for the magnitude, and develop a methodology that can estimate the magnitude and rate efficiencies for all survey's pointing groups simultaneously. We have determined that our peak completeness is on average 80\% in each pointing group, and our magnitude drops to $25\%$ of this value at $m_{25} = 26.22$. We describe the freely available survey simulation software and its methodology. We conclude by using it to infer that our effective search area for objects at 40 au is $14.8\deg^2$, and that our lack of dynamically cold distant objects means that there at most $8\times 10^3$ objects with $60 < a < 80$ au and absolute magnitudes $H \leq 8$., Comment: Accepted to AJ, companion paper to DEEP VI

When fitting N-body models to astronomical data - including transit times, radial velocity, and astrometric positions at observed times - the derivatives of the model outputs with respect to the initial conditions can help with model optimization and posterior sampling. Here we describe a general-purpose symplectic integrator for arbitrary orbital architectures, including those with close encounters, which we have recast to maintain numerical stability and precision for small step sizes. We compute the derivatives of the N-body coordinates and velocities as a function of time with respect to the initial conditions and masses by propagating the Jacobian along with the N-body integration. For the first time we obtain the derivatives of the transit times with respect to the initial conditions and masses using the chain rule, which is quicker and more accurate than using finite differences or automatic differentiation. We implement this algorithm in an open source package, NbodyGradient.jl, written in the Julia language, which has been used in the optimization and error analysis of transit-timing variations in the TRAPPIST-1 system. We present tests of the accuracy and precision of the code, and show that it compares favorably in speed to other integrators which are written in C., Comment: 24 pages; 7 figures. Revised for submission to MNRAS. Links in equations are provided to code at the repository https://github.com/ericagol/NbodyGradient.jl Some documentation and example scripts for running the code are provided in the repository

We have collected transit times for the TRAPPIST-1 system with the Spitzer Space Telescope over four years. We add to these ground-based, HST and K2 transit time measurements, and revisit an N-body dynamical analysis of the seven-planet system using our complete set of times from which we refine the mass ratios of the planets to the star. We next carry out a photodynamical analysis of the Spitzer light curves to derive the density of the host star and the planet densities. We find that all seven planets' densities may be described with a single rocky mass-radius relation which is depleted in iron relative to Earth, with Fe 21 wt% versus 32 wt% for Earth, and otherwise Earth-like in composition. Alternatively, the planets may have an Earth-like composition, but enhanced in light elements, such as a surface water layer or a core-free structure with oxidized iron in the mantle. We measure planet masses to a precision of 3-5%, equivalent to a radial-velocity (RV) precision of 2.5 cm/sec, or two orders of magnitude more precise than current RV capabilities. We find the eccentricities of the planets are very small; the orbits are extremely coplanar; and the system is stable on 10 Myr timescales. We find evidence of infrequent timing outliers which we cannot explain with an eighth planet; we instead account for the outliers using a robust likelihood function. We forecast JWST timing observations, and speculate on possible implications of the planet densities for the formation, migration and evolution of the planet system., Comment: Final version to be published in the Planetary Sciences Journal. 56 pages, 30 figures. Data from the paper and a complete table of forecast JWST times may be found at https://github.com/ericagol/TRAPPIST1_Spitzer/