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Register a new userA Method for Accurate and Efficient Propagation of Satellite Orbits: A Case Study for a Molniya Orbit
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Fast and precise propagation of satellite orbits is required for mission design, orbit determination and payload data analysis. We present a method to improve the computational performance of numerical propagators and simultaneously maintain the accuracy level required by any particular application. This is achieved by determining the positional accuracy needed and the corresponding acceptable error in acceleration on the basis of the mission requirements, removing those perturbation forces whose effect is negligible compared to the accuracy requirement, implementing an efficient and precise algorithm for the harmonic synthesis of the geopotential gradient (i.e., the gravitational acceleration) and adjusting the tolerance of the numerical propagator to achieve the prescribed accuracy level with minimum cost. In particular, to achieve the optimum balance between accuracy and computational performance, the number of geopotential spherical harmonics to retain is adjusted during the integration on the basis of the accuracy requirement. The contribution of high-order harmonics decays rapidly with altitude, so the minimum expansion degree meeting the target accuracy decreases with height. The optimum degree for each altitude is determined by making the truncation error of the harmonic synthesis equal to the admissible acceleration error. This paper presents a detailed description of the technique and test cases highlighting its accuracy and efficiency.
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Fast and precise propagation of satellite orbits is required for mission design, orbit determination in support of operations and payload data analysis. This demand must also comply with the different accuracy requirements set by a growing variety of scientific and service missions. This contribution proposes a method to improve the computational performance of orbit propagators through an efficient numerical integration that meets the accuracy requirements set by the specific application. This is achieved by appropriately tuning the parameters of the numerical propagator (relative tolerance and maximum time step), establishing a threshold for the perturbing accelerations (Earths gravitational potential, atmospheric drag, solar radiation pressure, third-body perturbations, relativistic correction to gravity) below which they can be neglected without altering the quality of the results and implementing an efficient and precise algorithm for the harmonic synthesis of the geo-potential and its first-order gradient. In particular, when performing the harmonic synthesis, the number of spherical harmonics to retain (i.e., the expansion degree) is determined by the accuracy requirement. Given that higher-order harmonics decay rapidly with altitude, the expansion degree necessary to meet the target accuracy decreases with height. To improve the computational efficiency, the number of degrees to retain is determined dynamically while the trajectory is being computed. The optimum expansion degree for each altitude is determined by ensuring that the truncation error of the harmonic synthesis is below the threshold acceleration. The work is a generalization to arbitrary orbits of a previous study that focused on communication satellites in geosynchronous inclined orbits. The method is presented and a set of test cases is analysed and discussed.
Numerical integration of orbit trajectories for a large number of initial conditions and for long time spans is computationally expensive. Semi-analytical methods were developed to reduce the computational burden. An elegant and widely used method of semi-analytically integrating trajectories of objects subject to atmospheric drag was proposed by King-Hele (KH). However, the analytical KH contraction method relies on the assumption that the atmosphere density decays strictly exponentially with altitude. If the actual density profile does not satisfy the assumption of a fixed scale height, as is the case for Earths atmosphere, the KH method introduces potentially large errors for non-circular orbit configurations. In this work, the KH method is extended to account for such errors by using a newly introduced atmosphere model derivative. By superimposing exponentially decaying partial atmospheres, the superimposed KH method can be applied accurately while considering more complex density profiles. The KH method is further refined by deriving higher order terms during the series expansion. A variable boundary condition to choose the appropriate eccentricity regime, based on the series truncation errors, is introduced. The accuracy of the extended analytical contraction method is shown to be comparable to numerical Gauss-Legendre quadrature. Propagation using the proposed method compares well against non-averaged integration of the dynamics, while the computational load remains very low.
A method for classifying orbits near asteroids under a polyhedral gravitational field is presented, and may serve as a valuable reference for spacecraft orbit design for asteroid exploration. The orbital dynamics near asteroids are very complex. According to the variation in orbit characteristics after being affected by gravitational perturbation during the periapsis passage, orbits near an asteroid can be classified into 9 categories: (1) surroundingto-surrounding, (2) surrounding-to-surface, (3) surroundingto-infinity, (4) infinity-to-infinity, (5) infinity-to-surface, (6) infinity-to-surrounding, (7) surface-to-surface, (8) surfaceto-surrounding, and (9) surface-to- infinity. Assume that the orbital elements are constant near the periapsis, the gravitation potential is expanded into a harmonic series. Then, the influence of the gravitational perturbation on the orbit is studied analytically. The styles of orbits are dependent on the argument of periapsis, the periapsis radius, and the periapsis velocity. Given the argument of periapsis, the orbital energy before and after perturbation can be derived according to the periapsis radius and the periapsis velocity. Simulations have been performed for orbits in the gravitational field of 216 Kleopatra. The numerical results are well consistent with analytic predictions.
To date, only 18 exoplanets with radial velocity (RV) semi-amplitudes $<2$ m/s have had their masses directly constrained. The biggest obstacle to RV detection of such exoplanets is variability intrinsic to stars themselves, e.g. nuisance signals arising from surface magnetic activity such as rotating spots and plages, which can drown out or even mimic planetary RV signals. We use Kepler-37 - known to host three transiting planets, one of which, Kepler-37d, should be on the cusp of RV detectability with modern spectrographs - as a case study in disentangling planetary and stellar activity signals. We show how two different statistical techniques - one seeking to identify activity signals in stellar spectra, and another to model activity signals in extracted RVs and activity indicators - can enable detection of the hitherto elusive Kepler-37d. Moreover, we show that these two approaches can be complementary, and in combination, facilitate a definitive detection and precise characterisation of Kepler-37d. Its RV semi-amplitude of $1.22pm0.31$ m/s (mass $5.4pm1.4$ $M_oplus$) is formally consistent with TOI-178bs $1.05^{+0.25}_{-0.30}$ m/s, the latter being the smallest detected RV signal of any transiting planet to date, though dynamical simulations suggest Kepler-37ds mass may be on the lower end of our $1sigma$ credible interval. Its consequent density is consistent with either a water-world or that of a gaseous envelope ($sim0.4%$ by mass) surrounding a rocky core. Based on RV modelling and a re-analysis of Kepler-37 TTVs, we also argue that the putative (non-transiting) planet Kepler-37e should probably be stripped of its confirmed status.
We discuss the feasibility of and present initial designs and approximate cost estimates for a large ($Nsim2000$) network of small photometric telescopes that is purpose-built to monitor $V lesssim 15$ Gaia Mission program stars for occultations by minor solar system bodies. The implementation of this network would permit measurement of the solar systems tidal gravity field to high precision, thereby revealing the existence of distant trans-Neptunian objects such as the proposed Planet Nine. As a detailed example of the network capabilities, we investigate how occultations by Jovian Trojans can be monitored to track the accumulation of gravitational perturbations, thereby constraining the presence of undetected massive solar system bodies. We also show that the tidal influence of Planet Nine can be discerned from that of smaller, nearer objects in the Kuiper belt. Moreover, ephemerides for all small solar system bodies observed in occultation could be significantly improved using this network, thereby improving spacecraft navigation and refining Solar System modeling. Finally, occultation monitoring would generate direct measurements of size distributions for asteroid populations, permitting a better understanding of their origins.
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