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Register a new userAccurate and Efficient Propagation of Satellite Orbits in the Terrestrial Gravity Field
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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.
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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.
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.
The climate and circulation of a terrestrial planet are governed by, among other things, the distance to its host star, its size, rotation rate, obliquity, atmospheric composition and gravity. Here we explore the effects of the last of these, the Newtonian gravitational acceleration, on its atmosphere and climate. We first demonstrate that if the atmosphere obeys the hydrostatic primitive equations, which are a very good approximation for most terrestrial atmospheres, and if the radiative forcing is unaltered, changes in gravity have no effect at all on the circulation except for a vertical rescaling. That is to say, the effects of gravity may be completely scaled away and the circulation is unaltered. However, if the atmosphere contains a dilute condensible that is radiatively active, such as water or methane, then an increase in gravity will generally lead to a cooling of the planet because the total path length of the condensible will be reduced as gravity increases, leading to a reduction in the greenhouse effect. Furthermore, the specific humidity will decrease, leading to changes in the moist adiabatic lapse rate, in the equator-to-pole heat transport, and in the surface energy balance because of changes in the sensible and latent fluxes. These effects are all demonstrated both by theoretical arguments and by numerical simulations with moist and dry general circulation models.
The formation and the evolution of protoplanetary disks are important stages in the lifetime of stars. The processes of disk evolution and planet formation are intrinsically linked. We spatially resolve with GRAVITY/VLTI in the K-band the sub au-scale region of 27 stars to gain statistical understanding of their properties. We look for correlations with stellar parameters, such as luminosity, mass, temperature and age. Our sample also cover a range of various properties in terms of reprocessed flux, flared or flat morphology, and gaps. We developed semi-physical geometrical models to fit our interferometric data. Our best models correspond to smooth and wide rings, implying that wedge-shaped rims at the dust sublimation edge are favored, as found in the H-band. The closure phases are generally non-null with a median value of ~10 deg, indicating spatial asymmetries of the intensity distributions. Multi-size grain populations could explain the closure phase ranges below 20-25 deg but other scenarios should be invoked to explain the largest ones. Our measurements extend the Radius-Luminosity relation to ~1e4 Lsun and confirm the significant spread around the mean relation observed in the H-band. Gapped sources exhibit a large N-to-K band size ratio and large values of this ratio are only observed for the members of our sample that would be older than 1 Ma, less massive, and with lower luminosity. In the 2 Ms mass range, we observe a correlation in the increase of the relative age with the transition from group II to group I, and an increase of the N-to-K size ratio. However, the size of the current sample does not yet permit us to invoke a clear universal evolution mechanism across the HAeBe mass range. The measured locations of the K-band emission suggest that these disks might be structured by forming young planets, rather than by depletion due to EUV, FUV, and X-ray photo-evaporation.
During its mission in the Saturn system, Cassini performed five close flybys of Dione. During three of them, radio tracking data were collected during the closest approach, allowing estimation of the full degree-2 gravity field by precise spacecraft orbit determination. The gravity field of Dione is dominated by $J_{2}$ and $C_{22}$, for which our best estimates are $J_{2} times 10^6 = 1496 pm 11$ and $C_{22} times 10^6 = 364.8 pm 1.8$ (unnormalized coefficients, 1-$sigma$ uncertainty). Their ratio is $J_{2}/C_{22} = 4.102 pm 0.044$, showing a significative departure (about 17-$sigma$) from the theoretical value of $10/3$, predicted for a relaxed body in slow, synchronous rotation around a planet. Therefore, it is not possible to retrieve the moment of inertia directly from the measured gravitational field. The interior structure of Dione is investigated by a combined analysis of its gravity and topography, which exhibits an even larger deviation from hydrostatic equilibrium, suggesting some degree of compensation. The gravity of Dione is far from the expectation for an undifferentiated hydrostatic body, so we built a series of three-layer models, and considered both Airy and Pratt compensation mechanisms. The interpretation is non-unique, but Diones excess topography may suggest some degree of Airy-type isostasy, meaning that the outer ice shell is underlain by a higher density, lower viscosity layer, such as a subsurface liquid water ocean. The data permit a broad range of possibilities, but the best fitting models tend towards large shell thicknesses and small ocean thicknesses.
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