arXiv subjects

Giovanni F. Gronchi

Publications and source records attributed to Giovanni F. Gronchi.

15 recordsLinked to original sources

arXiv2024

Linking tracklets over the years in large datasets

We present a new procedure to identify observations of known objects in large data sets of unlinked detections. It begins with a Keplerian integrals method that allows us to link two tracklets, computing preliminary orbits, even when the tracklets are separated in time by a few years. In the second step, we represent the results in a `graph' where the tracklets are the nodes and the preliminary orbits are the edges. Then, acceptable `3-cycles' are identified and a least squares orbit is computed for each of them. Finally, we construct sequences of $n \geq 4$ tracklets by searching through the orbits of nearby 3-cycles and attempting to attribute the remaining tracklets. We calculate the technique's efficiency at identifying unknown objects using real detections that attempt to mimic key parameters of the Minor Planet Center's Isolated Tracklet File (ITF) and then apply the procedure to the ITF to identify tens of thousands of new objects.

astro-ph.IM↗

arXiv2024

Numerical behavior of the Keplerian Integral methods for initial orbit determination

We investigate the behaviour of two recent methods for the computation of preliminary orbits. These methods are based on the conservation laws of Kepler's problem, and enable the linkage of very short arcs of optical observations even when they are separated in time by a few years. Our analysis is performed using both synthetic and real data of 822 main belt asteroids. The differences between computed and true orbital elements have been analysed for the true linkages, as well as the occurrence of alternative solutions. Some metrics have been introduced to quantify the results, with the aim of discarding as many of the false linkages as possible and keeping the vast majority of true ones. These numerical experiments provide thresholds for the metrics which take advantage of the knowledge of the \emph{ground truth}: the values of these thresholds can be used in normal operation mode, when we do not know the correct values of the orbital elements and whether the linkages are true or false.

astro-ph.EP↗

arXiv2023

Orbit determination from one position vector and a very short arc of optical observations

In this paper we address the problem of computing a preliminary orbit of a celestial body from one topocentric position vector and a very short arc (VSA) of optical observations. Using the conservation laws of the two-body dynamics, we write the problem as a system of 8 polynomial equations in 6 unknowns. We prove that this system is generically consistent, namely it admits solutions at least in the complex field. From this system we derive a univariate polynomial $\mathfrak{v}$ of degree 8 in the unknown topocentric distance at the mean epoch of the VSA. Through Gröbner bases theory, we show that the degree of $\mathfrak{v}$ is minimum among the degrees of all the univariate polynomials solving this problem. The proposed method is relevant for different purposes, e.g. the computation of a preliminary orbit of an Earth satellite with radar and optical observations, the detection of maneuvres of an Earth satellite, and the recovery of asteroids which are lost due to a planetary close encounter. We also show some numerical tests in the case of asteroids undergoing a close encounter with the Earth.

math-ph↗

arXiv2023

Revisiting the computation of the critical points of the Keplerian distance

We consider the Keplerian distance $d$ in the case of two elliptic orbits, i.e. the distance between one point on the first ellipse and one point on the second one, assuming they have a common focus. The absolute minimum $d_{\rm min}$ of this function, called MOID or orbit distance in the literature, is relevant to detect possible impacts between two objects following approximately these elliptic trajectories. We revisit and compare two different approaches to compute the critical points of $d^2$, where we squared the distance $d$ to include crossing points among the critical ones. One approach uses trigonometric polynomials, the other uses ordinary polynomials. A new way to test the reliability of the computation of $d_{\rm min}$ is introduced, based on optimal estimates that can be found in the literature. The planar case is also discussed: in this case we present an estimate for the maximal number of critical points of $d^2$, together with a conjecture supported by numerical tests.

math-ph↗

arXiv2022

A dynamical definition of the sphere of influence of the Earth

The concept of sphere of influence of a planet is useful in both the context of impact monitoring of asteroids with the Earth and of the design of interplanetary trajectories for spacecrafts. After reviewing the classical results, we propose a new definition for this sphere that depends on the position and velocity of the small body for given values of the Jacobi constant $C$. Here we compare the orbit of the small body obtained in the framework of the circular restricted three-body problem, with orbits obtained by patching two-body solutions. Our definition is based on an optimisation process, minimizing a suitable target function with respect to the assumed radius of the sphere of influence. For different values of $C$ we represent the results in the planar case: we show the values of the selected radius as a function of two angles characterising the orbit. In this case, we also produce a database of radii of the sphere of influence for several initial conditions, allowing an interpolation.

astro-ph.EP↗

arXiv2021

Symmetric constellations of satellites moving around a central body of large mass

We consider a $(1+N)$-body problem in which one particle has mass $m_0 \gg 1$ and the remaining $N$ have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the $N$ bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form $U \sim \frac{1}{r^α},$ where $α\in [1,2)$ and $r$ is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use $Γ$-convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional $Γ$-converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the $Γ$-limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of $m_0$. We discuss some examples, where the symmetry is defined by an action of the groups $Z_4$ , $Z_2 \times Z_2$ and the rotation groups of Platonic polyhedra on the set of loops.

math-ph↗

arXiv2021

Generalization of a method by Mossotti for initial orbit determination

Here we revisit an initial orbit determination method introduced by O. F. Mossotti employing four geocentric sky-plane observations and a linear equation to compute the angular momentum of the observed body. We then extend the method to topocentric observations, yielding a quadratic equation for the angular momentum. The performance of the two versions are compared through numerical tests with synthetic asteroid data using different time intervals between consecutive observations and different astrometric errors. We also show a comparison test with Gauss's method using simulated observations with the expected cadence of the VRO-LSST telescope.

astro-ph.EP↗

arXiv2018

Long term dynamics for the restricted N-body problem with mean motion resonances and crossing singularities

We consider the long term dynamics of the restricted N-body problem, modeling in a statistical sense the motion of an asteroid in the gravitational field of the Sun and the solar system planets. We deal with the case of a mean motion resonance with one planet and assume that the osculating trajectory of the asteroid crosses the one of some planet, possibly different from the resonant one, during the evolution. Such crossings produce singularities in the differential equations for the motion of the asteroid, obtained by standard perturbation theory. In this work we prove that the vector field of these equations can be extended to two locally Lipschitz-continuous vector fields on both sides of a set of crossing conditions. This allows us to define generalized solutions, continuous but not differentiable, going beyond these singularities. Moreover, we prove that the long term evolution of the 'signed' orbit distance (Gronchi and Tommei 2007) between the asteroid and the planet is differentiable in a neighborhood of the crossing times. In case of crossings with the resonant planet we recover the known dynamical protection mechanism against collisions. We conclude with a numerical comparison between the long term and the full evolutions in the case of asteroids belonging to the 'Alinda' and 'Toro' classes (Milani et al. 1989). This work extends the results in (Gronchi and Tardioli 2013) to the relevant case of asteroids in mean motion resonance with a planet.

math-ph↗

arXiv2018

Existence of periodic orbits near heteroclinic connections

We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1) \end{equation} has a family of $T$-periodic solutions $u^T$ which, along a sequence $T_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $a_-$ to $a_+$. We then focus on the elliptic system \begin{equation} Δu=W_u(u),\;\; u:R^2\rightarrow R^m, \hskip 2cm (2) \end{equation} that we interpret as an infinite dimensional analogous of (1), where $x$ plays the role of time and $W$ is replaced by the action functional \[J_R(u)=\int_R\Bigl(\frac{1}{2}\vert u_y\vert^2+W(u)\Bigr)dy.\] We assume that $J_R$ has two different global minimizers $\bar{u}_-, \bar{u}_+:R\rightarrow R^m$ in the set of maps that connect $a_-$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions $u^L:R^2\rightarrow R^m$, which is $L$-periodic in $x$, converges to $a_\pm$ as $y\rightarrow\pm\infty$ and, along a sequence $L_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $\bar{u}_-$ to $\bar{u}_+$.

math.DS↗

arXiv2017

Preliminary orbits with line-of-sight correction for LEO satellites observed with radar

We propose a method to account for the Earth oblateness effect in preliminary orbit determination of satellites in low orbits with radar observations. This method is an improvement of the one described in (Gronchi et al 2015), which uses a pure Keplerian dynamical model. Since the effect of the Earth oblateness is strong at low altitudes, its inclusion in the model can sensibly improve the initial orbit, giving a better starting guess for differential corrections and increasing the chances to obtain their convergence. The input set consists of two tracks of radar observations, each one composed of at least 4 observations taken during the same pass of the satellite. A single observation gives the topocentric position of the satellite, where the range is very accurate, while the line of sight direction is poorly determined. From these data we can compute by a polynomial fit the values of the range and range rate at the mean epochs of the two tracks. In order to obtain a preliminary orbit we wish to compute the angular velocities, that is the rate of change of the line of sight. In the same spirit of (Gronchi et al 2015), we also wish to correct the values of the angular measurements, so that they fit the selected dynamical model if the same holds for the radial distance and velocity. The selected model is a perturbed Keplerian dynamics, where the only perturbation included is the secular effect of the $J_2$ term of the geopotential. The proposed algorithm models this problem with 8 equations in 8 unknowns.

physics.space-ph↗

arXiv2017

On the existence of connecting orbits for critical values of the energy

We consider an open connected set $Ω$ and a smooth potential $U$ which is positive in $Ω$ and vanishes on $\partialΩ$. We study the existence of orbits of the mechanical system \[ \ddot{u}=U_x(u), \] that connect different components of $\partialΩ$ and lie on the zero level of the energy. We allow that $\partialΩ$ contains a finite number of critical points of $U$. The case of symmetric potential is also considered.

math.DS↗

arXiv2015

On the computation of preliminary orbits for space debris with radar observations

We introduce a new method to perform preliminary orbit determination for space debris on low Earth orbits (LEO). This method works with tracks of radar observations: each track is composed by $n\ge 4$ topocentric position vectors per pass of the satellite, taken at very short time intervals. We assume very accurate values for the range $ρ$, while the angular positions (i.e. the line of sight, given by the pointing of the antenna) are less accurate. We wish to correct the errors in the angular positions already in the computation of a preliminary orbit. With the information contained in a pair of radar tracks, using the laws of the two-body dynamics, we can write 8 equations in 8 unknowns. The unknowns are the components of the topocentric velocity orthogonal to the line of sight at the two mean epochs of the tracks, and the corrections $Δ$ to be applied to the angular positions. We take advantage of the fact that the components of $Δ$ are typically small. We show the results of some tests, performed with simulated observations, and compare this algorithm with Gibbs' method and the Keplerian integrals method.

math-ph↗

arXiv2015

Orbit Determination with the two-body Integrals. III

We present the results of our investigation on the use of the two-body integrals to compute preliminary orbits by linking too short arcs of observations of celestial bodies. This work introduces a significant improvement with respect to the previous papers on the same subject (see Gronchi et al. 2010, 2011). Here we find a univariate polynomial equation of degree 9 in the radial distance $ρ$ of the orbit at the mean epoch of one of the two arcs. This is obtained by a combination of the algebraic integrals of the two-body problem. Moreover, the elimination step, which in Gronchi et al. 2010, 2011 was done by resultant theory coupled with the discrete Fourier transform, is here obtained by elementary calculations. We also show some numerical tests to illustrate the performance of the new algorithm.

math-ph↗

arXiv2011

Orbit Determination with the two-body Integrals. II

The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or radar observations. We write polynomial equations for this problem, that we can solve using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. In this paper we continue the research started in [Gronchi, Dimare, Milani, 'Orbit determination with the two-body intergrals', CMDA (2010) 107/3, 299-318], where the angular momentum and the energy integrals were used. A suitable component of the Laplace-Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.

math-ph↗

arXiv2007

Orbit Determination with Topocentric Correction: Algorithms for the Next Generation Surveys

Given a set of astrometric observations of the same object, the problem of orbit determination is to compute the orbit and to assess its uncertainty and reliability. For the next generation surveys, with much larger number density of observed objects, new algorithms or substantial revisions of the classical ones are needed. The problem has three main steps, preliminary orbit, least squares orbit, and quality control. The classical theory of preliminary orbits was incomplete: the consequences of the topocentric correction had not been fully studied. We show that it is possible to account for this correction, possibly with an increase in the number of preliminary solutions, without impairing the overall orbit determination performance. We have developed modified least squares orbit determination algorithms that can be used to improve the reliability of the procedure. We have tested the complete procedure on two simulations with number densities comparable to that expected from the next generation surveys such as Pan-STARRS and LSST. To control the problem of false identifications we have introduced a quality control on the fit residuals based on an array of metrics and a procedure to remove duplications and contradictions in the output. The results confirm that large sets of discoveries can be obtained with good quality orbits and very high success rate losing only 0.6 to 1.3% of objects and a false identification rate in the range 0.02 to 0.06%.

astro-ph↗