orbital insights

The Global Trajectory Optimization Competition (GTOC) brings together astrodynamics, optimization, and creative problem-solving: it is the  "America's Cup of rocket science" . This year I took part in the competition for the first time, and managed to enter the leaderboard with a respectable final score.  In this post, the first part gives a high-level overview of how I approached the challenge, while the (more technical) second part will discuss the implementation details of my solution.   This year's edition revolved around the exploration of the exoplanetary system of the fictional star Altaira, whose planets sport a variety of pop-culture-inspired names. Assuming the interstellar leg is already taken care of, the task was to design a trajectory that maximizes the number and scientific value of reconnaissance close approaches with Altaira’s planets, asteroids, and comets. The dynamical model includes unpowered gravity assists with the major planets (minor bo...

West Sands Beach near St. Andrews, Scotland, where the tidal range (the difference between high tide and low tide) can be up to 4 meters.    Few natural phenomena capture the imagination and, at the same time, affect daily experience as much as the oceanic tides. With their daily rhythm, tides may have even shaped the evolution of life itself: it has been suggested that tides played a role in the transition from marine to land animals at the end of the Devonian period [1].   Understanding tides allows us to fully appreciate a hidden clockwork that connects celestial mechanics with the ebb and flow of the sea, and its impact on life on Earth.  Tides follow highly regular patterns, as is immediately obvious to anyone who lives near a coastal area with a tidal range of the order of meters (as opposed to the puny tens of cm in my Mediterranean hometown!). In this post I will discuss how the most readily observable periodicities in the rise and fall o...

A tool often employed in the early stages of space mission design is the patched conics approximation. In this formalism, a trajectory is approximated by a sequence of two-body orbits, defined with respect to an appropriately changing primary body, according to the updated position of the spacecraft. The validity of this approach rests on the assumption that in each portion of the trajectory the acceleration of the spacecraft mostly arises from the gravitational attraction by a single body, which dominates over all the other perturbations. For instance, to an excellent degree of approximation, the motion of an interplanetary probe is mostly governed by solar gravity, except when in the immediate vicinity of a planet. The "Sphere of Influence" (SOI) of a celestial body with respect to a distant, major primary is formally defined below. At the lowest order of approximation, the SOI is a sphere centered on the secondary body, and of radius $R_S = D \, \left(\frac{m}{M}\right)^{2...

A spacecraft on a free-return trajectory departs a primary body and is sent back to its vicinity by a gravity assist from a secondary body; in other words, after orbit injection, no additional propulsive maneuver is required for the return trip. Free-return trajectories between the Earth and the Moon, in particular, have been of great practical importance for both unmanned and manned missions since the beginning of the Space Age. In this post I discuss the construction of a circumlunar free-return trajectory by a patched-conics approach. For the Earth-Moon case, this type of analysis must account for the fact that a significant portion of the trajectory is spent inside the Sphere of Influence (SOI) of the Moon, whose extension cannot be neglected as in the case of heliocentric interplanetary missions. I will focus in particular on the method described in Section 9.3 of Battin (1999), which was originally published in a technical report from the early Apollo era [2]. Besides its hi...

Many systems commonly encountered in celestial mechanics and astrodynamics can be effectively modeled as a two-body problem subject to relatively small perturbations. Under these circumstances, the deviations from the two-body orbit due to the perturbative acceleration can be described in terms of time-variable orbital elements .   In this post I will follow the elegant geometric treatment of the perturbation of the orbital elements given in Chapter VIII of Moulton (1902), from which I borrowed the key figures. This approach is complementary to the analytical derivation, usually leading to Lagrange's planetary equations. Having its roots in the first-ever treatment of orbital perturbations, which appeared in Newton's Principia , the geometric picture emphasizes intuitive, qualitative understanding over quantitative results.    We shall focus on elliptic (i.e., bound) orbits, and restrict the discussion to the five classical orbital elements that describe the shape and o...

In some cases, unexpectedly deep connections allow far-reaching conclusions to be drawn from commonplace observations. This post provides an example of such an occurrence. A readily observable fact about the orbits of bound celestial objects is that they regularly repeat. Closed orbits are not only prevalent in the solar system, but can also be directly observed, for instance, in visual binary stars, as shown in the figure below. This simple observation places strong constraints on the nature of the gravitational force.  Observed orbit projected in the plane of the sky of the secondary component of the Sirius system relative to the primary (open and filled circles represent data obtained with different techniques, as detailed in the legend; the black filled circle at the origin marks the position of the primary). The best-fitting orbit is shown by the black line (data and orbital solution from Bond et al. 2017, ApJ 840, 70).   A somewhat obscure theorem of classical mecha...

I owe my first encounters with celestial mechanics to a popular science magazine and a university-level supplementary textbook. The unlikely pair consisted of a special issue of Le Scienze (the Italian edition of Scientific American), devoted to the scientific biography of Sir Isaac Newton, and of the Italian translation of Schaum's Outline of Theoretical Mechanics, which, from its dusty appearance, had spent quite some time forgotten on a shelf of a surprisingly well-stocked high school library.     Still barely halfway through high school, and equipped only with a shaky understanding of limits and derivatives, I marveled at the genius of the Man who invented Calculus, while the dusty textbook inspired, intrigued, and (more often than not!) frustrated me. These two oddly complementary readings cemented my choice of Physics as university major. More than twenty years later, the better part of which I have spent pursuing a PhD in Astrophysics and later working as a researcher,...