arXiv subjects

Zachary Murray

Publications and source records attributed to Zachary Murray.

9 recordsLinked to original sources

arXiv2025

Closed Form Expressions for the Potentials and Accelerations of Generalized Ring Models

We present several closed-form expressions of useful mass distributions. These include the potentials and accelerations of circular rings and arcs, the potentials of uniform density rings and arcs at arbitrary eccentricities, and the potentials and accelerations of rings and arcs when the mass is time-averaged over a Kepler orbit. We show that these expressions can be expressed, often simply, in terms of elliptic functions of complex arguments. We show that in a few limiting cases, the expressions are entirely real. We expect that these expressions will allow for more rapid modeling in many areas of celestial mechanics.

astro-ph.EP↗

arXiv2025

Numerical Determination of the Gravitational Cross Sections of an Accreting Binary

A significant amount of work has been devoted to the study of small binary solar system objects. The majority of these binaries, especially among the near-earth or main belt asteroids have small radius ratios, implying a large difference in size between the primary and its companion. Farther from the sun, the binary fraction increases, with the Kuiper Belt having many known binaries with radius ratios of order unity. In this paper, we consider the runaway growth of a binary system in an accretionary stream of small particles. We perform brute-force integrations, each with 10 million test particles and numerically compute the gravitational cross sections for each member of the binary as a function of the system's separation and mass ratio. We show that the behavior of the cross section is complex, and it can be either diminished or enhanced depending on the orbital configuration. In the regime where gravitational focusing dominates the accretion process, we show that binaries grow towards smaller mass ratios than would be expected given single-body cross sections. Finally, we provide a grid of these cross sections for use in the future study of such systems.

astro-ph.EP↗

arXiv2025

Detecting False Positives With Derived Planetary Parameters: Experimenting with the KEPLER Dataset

Recent developments in computational power and machine learning techniques motivate their use in many different astrophysical research areas. Consequently, many machine learning models have been trained to classify exoplanet transit signals - typically done by using time series light curves. In this work, we attempt a different approach and try to improve the efficiency of these algorithms by fitting only derived planetary parameters, instead of full time-series light curves. We investigate and evaluate 4 models (Logistic Regression, Random Forest, Support Vector Machines, and Convolutional Neural Networks) on the KEPLER dataset, using precision-recall trade-off and accuracy metrics. We show that this approach can identify up to about 90% of false positives, implying the planetary parameters encompass most of the relevant information contained in a light curve. Random Forest and Convolutional Neural Networks produce the highest accuracy and the best precision-recall trade-off. We also note that the accuracies as a function of the stellar eclipse flag SS have the best performance.

astro-ph.EP↗

arXiv2025

Boosting decision trees for Main Belt Asteroid selection in planetary ephemerides: an alternative model

One of the main bottleneck in assessing the accuracy of Mars orbit is the unknown value of the asteroids in the Main Asteroid Belt. Nowadays a modeling with 343 asteroids as point masses is used, with the relative masses fitted to observational data. In the current work we propose an innovative methodology to reduce the number of asteroids implemented as point masses, thus reducing the number of parameters to be fitted, without a significant degradation of the postfit residuals.

astro-ph.EP↗

arXiv2023

The Challenge of Measuring Asteroid Masses with Gaia DR2 astrometry

The Gaia second data release contains high-accuracy astrometric measurements of thousands of solar system bodies. These measurements raise the possibility of determining asteroid masses by modeling scattering events between massive objects observed by Gaia. In this paper, we identify promising encounters between small asteroids that occur during the second data release and quantify the various errors involved in mass determination. We argue that in the best case, Gaia astrometry can provide constraints as tight as 1 km on the positions of asteroids. Further, we find that even with general relativistic corrections, integrations of the solar system accumulate 1 km errors after 700 days. While not a problem for modeling DR2 astrometry, future Gaia data releases may require models accounting for additional effects such as gravitational harmonics of the sun and planets. Additionally, due to sub-optimal astrometric uncertainty, the geometry of the observations, and the Gaia observing pattern result in much looser constraints in most cases, with constraints being several orders of magnitude weaker in some cases. This suggests that accurate mass determination for the smallest asteroids will require additional observations, either from future Gaia data releases or from other sources. We provide a list of encounters that are most promising for further investigation.

astro-ph.EP↗

arXiv2023

Using neural networks to model Main Belt Asteroid albedos as a function of their proper orbital elements

Asteroid diameters are traditionally difficult to estimate. When a direct measurement of the diameter cannot be made through either occultation or direct radar observation, the most common method is to approximate the diameter from infrared observations. Once the diameter is known, a comparison with visible light observations can be used to find the visible geometric albedo of the body. One of the largest datasets of asteroid albedos comes from the NEOWISE mission, which measured asteroid albedos both in the visible and infrared. We model these albedos as a function of proper elements available from the Asteroid Families Portal using an ensemble of neural networks. We find that both the visible and infrared geometric albedos are significantly correlated with asteroid position in the belt and occur in both asteroid families and in the background belt. We find that the ensemble's prediction reduces the average error in albedo by about 37% compared to a model that simply adopts an average albedo, with no regard for the dynamical state of the body. We then use this model to predict albedos for the half million main belt asteroids with proper elements available in the Asteroid Families Portal and provide the results in a catalog. Finally, we show that several presently categorized asteroid families exist within much larger groups of asteroids of similar albedos - this may suggest that further improvements in family identification can be made.

astro-ph.EP↗

arXiv2023

Can the gravitational effect of Planet X be detected in current-era tracking of the known major and minor planets?

Using Fisher information matrices, we forecast the uncertainties $σ_M$ on the measurement of a "Planet X" at heliocentric distance $d_X$ via its tidal gravitational field's action on the known planets. Using planetary measurements currently in hand, including ranging from the Juno, Cassini, and Mars-orbiting spacecraft, we forecast a median uncertainty (over all possible sky positions) of $σ_M=0.22M_\oplus (d_x/400\,\textrm{AU})^3.$ A definitive $(5σ)$ detection of a $5M_\oplus$ Planet X at $d_X=400$ AU should be possible over the full sky but over only 5% of the sky at $d_X=800$ AU. The gravity of an undiscovered Earth- or Mars-mass object should be detectable over 90% of the sky to a distance of 260 or 120 AU, respectively. Upcoming Mars ranging improves these limits only slightly. We also investigate the power of high-precision astrometry of $\approx8000$ Jovian Trojans over the 2023--2035 period from the upcoming Legacy Survey of Space and Time (LSST). We find that the dominant systematic errors in optical Trojan astrometry (photocenter motion, non-gravitational forces, and differential chromatic refraction) can be solved internally with minimal loss of information. The Trojan data allow useful cross-checks with Juno/Cassini/Mars ranging, but do not significantly improve the best-achievable $σ_M$ values until they are $\gtrsim10\times$ more accurate than expected from LSST. The ultimate limiting factor in searches for a Planet X tidal field is confusion with the tidal field created by the fluctuating quadrupole moment of the Kuiper Belt as its members orbit. This background will not, however, become the dominant source of Planet X uncertainty until the data get substantially better than they are today.

astro-ph.EP↗

arXiv2022

Constructive Analysis in the Agda Proof Assistant

Proof assistant software has recently been used to verify proofs of major theorems, yet even the libraries of some of the most prominent proof assistants lack much of undergraduate mathematics. In particular, the Agda proof assistant has no formalization of the real numbers and their arithmetic. In this thesis, I present my implementation of Errett Bishop's constructive real numbers in Agda, including their arithmetic, ordering, and fundamental results, such as uncountability and Cauchy completeness. We will also survey the basic concepts of constructive analysis and the Agda proof assistant.

cs.LO↗

arXiv2022

The Effects of Disk Induced Apsidal Precession on Planets Captured into Mean Motion Resonance

The process of migration into resonance capture has been well studied for planetary systems where the gravitational potential is generated exclusively by the star and planets. However, massive protoplanetary disks add a significant perturbation to these models. In this paper we consider two limiting cases of disk-induced precession on migrating planets and find that small amounts of precession significantly affect the equilibrium reached by migrating planets. We investigate these effects with a combination of semi-analytic models of the resonance and numerical integrations. We also consider the case of the disk's dispersal, which can excite significant libration amplitude and can cause ejection from resonance for large enough precession rates. Both of these effects have implications for interpreting the known exoplanet population and may prove to be important considerations as the population of well-characterized exoplanet systems continues to grow.

astro-ph.EP↗