### Mathematical models and methods for the analysis of interplanetary tracking data

I am involved in three research projects whose common denominator is the “radio science”. Radio science means the achievement of scientific goals, such as determining the gravity field of a planet or a minor planet, constraining its rotation status or testing gravitational theories, using tracking data of an interplanetary probe.

The three research projects are related to three distinct space missions: BepiColombo (launched in 2018), Juno (now @Jupiter) and Hera (launch 2024).

**BepiColombo ESA mission**BepiColombo is an European Space Agency mission launched in 2018, with the goal of an in-depth exploration of the planet Mercury; it has been identified as one of the most challenging long-term planetary projects. Only two NASA missions had Mercury as target in the past, the Mariner 10, which flew by three times in 1974-5 and Messenger, which carried out its flybys on January and October 2008, September 2009 and started its year-long orbiter phase in March 2011. The BepiColombo mission is composed by two spacecraft to be put in orbit around Mercury. The**Radio Science Experiment (RSE)**is one of the on board experiments, which would coordinate a gravimetry, a rotation and a relativity experiment, using a very accurate range and range rate tracking. These measurements will be performed by a full 5-way link to the Mercury orbiter; by exploiting the frequency dependence of the refraction index, the differences between the Doppler measurements (done in Ka and X band) and the delay give information on the plasma content along the radiowave path. In this way most of the measurements errors introduced can be removed, improving of about two orders of magnitude with respect to the past technologies.

My group developed the orbit determination code Orbit14 that will be used for**MORE (Mercury Orbiter Radioscience Experiment)**, and is running a lot of simulations to prove that the mission could satisfy the scientific goals, that are a better knowledge of the gravity field of Mercury and of its rotation state, the testing of general relativity and the improving of planetary ephemerides.**Juno NASA mission**JUNO is a NASA mission launched on 5 August 2011 and inserted in orbit around Jupiter on 4th July, 2016. JUNO’s goal is to understand the origin and evolution of Jupiter. Using a spinning, solar powered spacecraft JUNO will produce global maps of the gravity, magnetic fields, and atmospheric composition from a unique polar orbit with a close perijove. My group developed the orbit determination code Orbit14 that we are using to analyze the Juno tracking data.**Hera mission**Hera is an approved ESA mission, the evolution of the Asteroid Impact Mission (AIM) that has been a candidate mission arrived to phase B1 (a group of the University of Bologna, lead by Prof. Paolo Tortora, and my research group had a contract to study the possible scientific results performing a radio science experiment with AIM). Hera is mission of planetary protection and technological demonstration, but an important by-product will be to improve the scientific knowledge of small asteroids, in particular that relating to a binary system, never reached before by a space mission. Hera’s main scientific objectives are: to characterize the mass, dimensions, morphology and density of Dimorphos, the natural satellite of the primary Didymos; to determine the dynamic properties of the binary system; to determine the surface, sub-surface, thermophysical properties and internal structure of Dimorphos. This last goal, if achieved, will give us the opportunity to study the dynamic evolution driven by the thermal effects Yarkovsky, YORP (Yarkovsky – O’Keefe– Radzievskii – Paddack) and BYORP (Binary YORP). Furthermore, Hera will provide useful elements for the so-called “planetary protection” because it will determine the transfer of momentum from the impact of the NASA DART (Double Asteroid Redirection Test) mission by measuring the mass of Dimorphos and the resulting effects on its surface (for example , the size of the crater). Many of these objectives will be achieved thanks to a radio science experiment that will use, in addition to the ground-based Doppler tracking of the spacecraft, also the Inter Satellite Link (ISL) with the two CubeSats (Juventas and Milani) that are part of the mission.

See publications [12], [14],[15],[17],[18],[20],[21],[22],[23],[24],[25],[27],[28],[29],[32],[33],[35],[37],[39],[40],[42],[45],[46],[50]

### Dynamics and Orbit Determination of NEOs (Near-Earth Objects)

During my M.Sc. thesis and my Ph.D period I studied the dynamics of Near-Earth Objects (NEOs). The NEOs population includes both asteroids and active/extinct comets having perihelion distances q<=1.3 AU

and aphelion distances Q >=0.983 AU. In particular my research focused on the * Impact Monitoring (IM)*, that is a”not so easy” consequence of the process of orbit determination of asteroids.

When a celestial body (asteroid, comet) has been observed only over a short time, its orbits is strongly undetermined and it could be anywhere in a *confidence region* (a subset of the 6-dimensional space of the orbital elements) where the astrometric residuals are acceptable. This region can be sampled by a set of *Virtual Asteroids* (VAs): one of them is real, but we do not know which one. The goal of IM is to establish whether the confidence region contains some *Virtual Impactor* (VI), a small subset of initial conditions leading to a collision with the Earth. The collision subset for a given epoch may be disconnected, when the same collision can be reached through different dynamics ways (for example

resonant returns in different resonances): in this case a VI is a connected component of the collision subset. The *Impact Probability* (IP) of a VI is proportional to the volume of the VI in the orbital elements space; finding an initial condition belonging to the VI, the *VI representative*, when the IP is small, would require a very dense sampling. The problem is how to do this sampling to guarantee completeness to VI search, taking into account the computational costs. One class of methods (it includes Monte Carlo and Statistical Ranging methods, uses random sampling of the confidence region to study the probabilistic distributions of the orbits through the swarm of VAs. When a large catalogs of asteroids has to be handled and we need to detect small probabilities computing a small number of VAs orbits, it

is more efficient to sample the confidence region with a geometrical object, such as a smooth manifold. In the last years we have developed 1-dimensional sampling methods based upon the *Line Of Variation *(LOV), a differentiable curve, which can represent, in some cases, the spine of the confidence region. The LOV is sampled uniformly, thus it is possible to interpolate between consecutive VAs. This is the basis for the current algorithms of impact monitoring, used by the automatic systems CLOMON2, that I contributed to develop in the first decade of 2000s. The LOV

method is also used in orbit determination for recovery of lost asteroids and for identification of independent discoveries of the same object.

In the last years (2015-2017) we concentrated on short-arc orbit determination, that is crucial when an asteroid is first discovered. It often happens that in these cases the observations are too few to compute an orbit. We have developed an initial orbit computation method, based on the systematic ranging, which also makes use of the theory of the Admissible Region. We obtained a fully rigorous computation of the probability for the asteroid to impact the Earth within few days from the discovery, without any a priori assumption.

See publications [1], [2],[3],[4],[5],[6],[8],[9],[36],[41],[49],[52]

### Orbit Determination of Space Debris

In the early phase of my Post-Doc period I worked in the field of orbit determination of space debris, developing new OD techniques for the correlation problem.

The near-Earth space, filled by more than 300000 artificial debris particles with diameter larger than 1 cm, can be divided into three main regions: the Low Earth Orbit (LEO), below about 2000 km, the Medium Earth Orbit (MEO), above 2000 km and below 36000 km, and the Geosynchronous Earth Orbit (GEO) at about 36000 km of altitude.

Currently the orbits of more than 12000 objects larger than about 10 cm are listed in the so called Two Line Elements (TLE) catalogue. To produce and maintain such a catalogue a large number of optical and radar observations are routinely performed by the United States Space Surveillance Network. Nowadays also Europe has launched its Space

Situational Awareness (SSA) initiative aimed to increase the knowledge of the circumterrestrial environment. In this context the availability of efficient methods and algorithms for accurate orbit determination is extremely important.

Given two or more sets of observations, the main problem is how to identify which separate sets of data belong to the same physical object (the so-called ** correlation problem**). Thus the OD problem needs to be solved in two stages: first different sets of observations need to be correlated, then an orbit can be determined

See publications [7], [10],[11],[13],[16]

### Dynamics of transneptunian objects

The dynamical structure of the transneptunian region is still far from being fully understood, especially concerning high-perihelion objects and the link toward the Oort Cloud. For these objects, the orbital perturbations are very weak, both from inside (the planets) and from outside (passing stars and galactic tides). Before thinking of exotic theories, an exhaustive survey has to be conducted on the different mechanisms that could produce such trajectories involving only what we take for granted about the Solar System dynamics, that is the orbital perturbations by the known planets and/or by galactic tides. In the first paper of a series of three two semi-analytical one-degree-of-freedom secular models has been presented for the motion of small bodies beyond Neptune. A special attention has been given to trajectories entirely exterior to the planetary orbits. In the second paper we used a secular representation to describe the long-term dynamics of transneptunian objects in mean-motion resonance with Neptune. The parameter space was systematically explored, showing that the secular trajectories depend little on the resonance order. In the thrid paper we use a secular model to describe the non-resonant dynamics of trans-Neptunian objects in the presence of an external ten-Earth-mass perturber. The secular dynamics resulted analogous to an “eccentric Kozai mechanism” but with both an inner component (the four giant planets) and an outer one (the eccentric distant perturber). By the means of Poincaré sections, the cases of a non-inclined or inclined outer planet were successively studied, making the connection with previous works.

See publications [26], [30], [34]

### Reentry of space assets

Reentry trajectories to the Earth have been recently considered as a valuable end-of-life option also for Libration Point Orbits (LPO) missions. In a series of two papers, we investigated in detail the case corresponding to SOHO. On the one hand, we showed how the main uncertainties associated with the problem affect the probability of reentry and the corresponding point at the interface with the atmosphere. Monte Carlo propagations were applied to different cases of uncertainties. They correspond to the orbit determination, the efficiency of the maneuver required to target the Earth, and the characteristics of the spacecraft determining the solar radiation pressure effect. On the other hand, we provided a comparison between a classical reentry from a LEO and a hypervelocity reentry from a LPO, in terms of ground casualty area and demise percentage.

See publications [31], [38], [44]

### Dynamics of Molnyia satellites

On April 23, 1965 the first Molniya-1 spacecraft was launched by former Soviet Union and after that many others were launched until 2004. These satellites were initially designed for Russian communication networks and their orbits form a class of special orbits around Earth, the Molniya orbits: period of approximatively 12hours, eccentricity e≥0.7, inclination i=63.43◦, argument of perigee ω= 270◦. These orbits are particularly interesting from a dynamical point of view. The orbital period of a Molniya satellite is commensurable with the Earth’s rotation period: each day a Molniya satellite revolves around the Earth two times. The consequence is a 2 : 1 tesseral resonance whose effects couple with the critical inclination resonance effects. Moreover, a Molniya satellite undergoes several perturbations. The low value of the altitude of the perigee gives a non-negligible atmospheric drag, which deeply affects the evolution of the semi-major axis. Besides, the satellite spends the most of the time at high altitudes, thus the lunisolar effects play a fundamental role on timescale larger than a satellite orbital period. Some Molniya satellites launched before 1974 experienced a quick decay, but the satellites launched after 1974 did not. Molniya orbits are considered quite chaotic, that is, the dynamical evolution strongly depends on the initial conditions.

See publications [43], [47], [48],[51]