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NSRC 2023

HGS-1 illustration
More sophisticated versions of the lunar free return trajectory used to salvage AsiaSat 3/HGS-1 could be used to send spacecraft to the Moon with far less propellant than conventional techniques. (credit: Hughes Space and Communications/file)

From chaos, a new order

Last Tuesday night a Proton M rocket lifted off from the Baikonur Cosmodrome in Kazakhstan, carrying the Arabsat 4A commercial communications satellite. The launch initially seemed to proceed normally, but several hours after liftoff International Launch Services (ILS) declared a “launch anomaly”: for reasons yet to be determined, the Proton’s Breeze M upper stage shut down prematurely and released the Arabsat satellite while still in a low transfer orbit. Without the upper stage, the satellite lacks the ability to reach its final geostationary orbit (GEO), and its operators will likely declare it a total loss, if they have not done so already.

Last week’s failure was eerily similar to one in December 1997 when another Proton suffered a launch failure and stranded its payload, AsiaSat 3, in an elliptical transfer orbit. (That failure involved a different upper stage, the Block DM.) While AsiaSat declared the satellite a total loss, the spacecraft’s consortium of insurers, along with manufacturer Hughes Space and Communications (now part of Boeing), were able to salvage the spacecraft. Although the spacecraft lacked the propellant needed to reach GEO, controllers were able to perform an innovative series of maneuvers that sent the spacecraft around the Moon twice, allowing it to reach GEO. The satellite, renamed HGS-1 and later PAS-22 after being acquired by PanAmSat, was able to operate in a limited capacity for several years. According to at least one report, there is a discussion underway now to see if there is a way to use a similar technique to salvage Arabsat 4A.

The successful salvage of AsiaSat 3 raised awareness of the use of unconventional trajectories in spaceflight, not just for rescuing stranded communications satellites but also for a whole host of other applications. Some experts foresee using trajectories far more complex than the one used for AsiaSat 3, taking advantage of the chaotic physics of multibody dynamics to send spacecraft to the Moon and beyond more efficiently than conventional trajectories. Such trajectories could also explain natural phenomena like the formation of the Moon and even, perhaps, why there is life on Earth.

Vespa versus Ferrari

Chaotic dynamics has its roots in one of the most famous conundrums of classical physics: the three-body problem. In the late 19th century the French mathematician Henri Poincaré, trying to prove that the solar system is stable, found instead that the three-body problem could not be solved analytically. A slight change in initial conditions can result in a dramatically different outcome, making the system look unpredictable if that initial change is too small to be measured.

The classic Hohmann transfer and capture maneuver is “a fuel hog,” says Belbruno. “There’s got to be a better way to do it.”

This sensitivity to initial conditions may be a challenge to theoreticians, but it provides an opportunity for those who want to leverage it in spaceflight applications. One of the first to explore the potential of chaotic dynamics in spaceflight is Ed Belbruno, a visiting researcher at Princeton University. Working as an orbit analyst at JPL twenty years ago—new to the field of astrodynamics but with a doctorate in mathematics—he looked to alternatives to the traditional method to sending spacecraft to the Moon and planets. That approach, a Hohmann transfer orbit from the Earth and a “capture maneuver” at the destination, works well, but requires a high change in velocity, or delta-v, to perform the capture maneuver. Barring the availability of alternative deceleration techniques, like aerobreaking, this delta-v requirement translates into substantial propellant, and thus a heavier spacecraft.

“I view this, along with some of my colleagues, as a fuel hog,” said Belbruno during a session on chaos and astrodynamics at the American Association for the Advancement of Science (AAAS) annual meeting in St. Louis last month. “There’s got to be a better way to do it.”

Belbruno asked if there was a way for a spacecraft to perform a “ballistic capture” maneuver: to arrive at the Moon, for example, on a specific trajectory that would allow it to enter orbit without any delta-v at all. While his JPL colleagues at the time were convinced that it wasn’t possible, Belbruno studied the problem and found there was a way, as he put it, “to slowly creep up” on the Moon, arriving such that all the forces were balanced, allowing the spacecraft to go into orbit rather than escaping from or crashing into the Moon.

Belbruno first proposed taking advantage of “weak stability boundary theory” in 1986 for a proposed small lunar orbiter that could be launched from a Get Away Special canister in the shuttle’s cargo bay. The spacecraft’s thrusters were too weak to perform a conventional capture maneuver, so Belbruno proposed an alternative trajectory, using the spacecraft’s thrusters to slowly spiral out from Earth and coast to the weak stability boundary, where the spacecraft would be captured into the Moon’s orbit, then use the thrusters to spiral down to the final orbit. The drawback was that it took the spacecraft two years to reach the Moon. The reaction the concept got within JPL, Belbruno recounted, was along the lines of “very interesting, but who cares? It takes three days to get to the Moon—we’ve been there already—why take two years?”

That comment encapsulates the tradeoffs involved with the application of chaotic dynamics to spacecraft trajectories: while techniques like weak stability boundary transfer are far more fuel-efficient than traditional approaches, they take far longer to execute. It’s akin to the difference between a Vespa scooter and a Ferrari sports car: the Vespa gets far better gas mileage, but the Ferrari will get you to your destination much faster.

Belbruno did get a chance to put this concept into action in 1990 when the Japanese were trying to salvage its first lunar mission. Hiten was in a highly elliptical Earth orbit; a companion spacecraft, Hagoromo, was deployed from Hiten to go into lunar orbit, but suffered a communications failure. Belbruno developed a series of maneuvers that, over the course of three months, put the spacecraft temporarily into lunar orbit in October 1991. Later, ESA used a version of Belbruno’s lunar Get Away Special trajectory for its SMART-1 lunar orbiter mission.

Future applications

The success of Hiten and SMART-1 shows that chaotic dynamics does offer a viable alternative to traditional trajectories. The challenge in using these techniques lies not in the theory, which is based on work over a century old, but in the details implementing them in practice. “This is nothing new in terms of anyone thinking of it, but it is new in terms of operationally implementing it,” said David Folta of the Flight Dynamics Analysis Branch of NASA’s Goddard Space Flight Center, speaking in the same AAAS conference session as Belbruno.

Traditional versus chaotic dynamics is akin to the difference between a Vespa scooter and a Ferrari sports car: the Vespa gets far better gas mileage, but the Ferrari will get you to your destination much faster.

Chaotic dynamics’ sensitivity to initial conditions poses a problem for those who try to take advantage of it. Minor effects that are often ignored in conventional trajectory design, such as solar wind and atmospheric models, must be taken into account when using chaotic dynamics or else the trajectory can quickly diverge. “This stuff gets—gee, annoying isn’t the word,” Folta said, “but after running many, many simulations, trying to come up with the right trajectory, it does become annoying.”

To grapple with all those effects, Folta and his Goddard colleagues have developed models that take all those possible perturbations into account in trajectory analysis. “Our models are the best we can possibly get to at this point,” he said. Those models include high-precision gravitational models for the Earth and Moon, solar radiation pressure, and the solar wind. “It’s even to the point where the software includes relativistic effects.”

Folta is looking at ways that chaotic dynamics can be used to support NASA’s Vision for Space Exploration by developing trajectories for lunar spacecraft. One example he presented compared the differences between a conventional, or “direct transfer” trajectory, versus a weak stability transfer approach for putting a spacecraft into a 100-kilometer circular lunar orbit. The weak stability transfer approach requires nearly 20 percent less delta-v than the direct approach, which in turn can result in mass savings for the spacecraft. The tradeoff, again, is time: the direct approach gets the spacecraft to the Moon in four days, while the weak stability transfer trajectory, which sends the spacecraft out to a distance of 1.5 million kilometers from the Earth, takes 98 days to reach lunar orbit.

Such approaches can be used for a wide range of lunar missions, including polar and elliptical orbits to provide constant communications coverage over the lunar poles. The most fascinating, though, is a way that—theoretically—could allow future servicing of the James Webb Space Telescope (JWST). Unlike Hubble, which is in low Earth orbit, JWST will be located at the Earth-Sun L2 Lagrange point, about 1.5 million kilometers from the Earth. Whereas Hubble was designed to be regularly repaired and upgraded by shuttle missions, there are no plans to make JWST servicable because of its location. However, Folta said there is a way around this by taking advantage of the intersections between Sun-Earth and Earth-Moon dynamics that would allow JWST to maneuver back closer to the Earth. “Because of this intersection we could actually bring the JWST back into the Earth-Moon system. Someone could go out into the Earth-Moon system in three or four days and repair what they needed do, and then we could send JWST back out.” The cost of doing that, in terms of propellant for JWST? Two kilograms, according to Folta.

The Big Splat explained

The applications of chaotic dynamics go far beyond spacecraft trajectories, however. Belbruno belies that it can provide further evidence for the formation of the Moon itself. The current leading model for the formation of the Earth-Moon system is the collision of a Mars-sized body with the proto-Earth, the so-called “Big Splat” hypothesis. The model does a good job of explaining many of the attributes of the Earth-Moon system, including the differences in composition of the two worlds, but one major outstanding question is where the impactor came from: was this a chance encounter or an inevitable outcome of the dynamics of solar system formation?

Chaotic dynamics requires taking into account many factors that would otherwise be ignored. “This stuff gets—gee, annoying isn’t the word,” Folta said, “but after running many, many simulations, trying to come up with the right trajectory, it does become annoying.”

Belbruno examined a hypothesis by Richard Gott, a Princeton astrophysicist, that protoplanetary material accumulated at one of the two stable Earth-Sun Lagrange points, L4 and L5. As the debris accumulated, it would start to oscillate around the Lagrange point more and more, to the point where it would eventually collide with the Earth. The problem, Belbruno said, was that classical dynamics methods suggested that such a collision would be highly improbable.

Belbruno instead looked at the problem using chaotic dynamics. He found that when the oscillation reaches a certain point the object “breaks out” of the Lagrange point into solar orbit, making repeated close approaches to the Earth. “When the object ‘spins around’, the chance of a collision is very high,” he said: about 75 percent, according to a paper he published in the Astronomical Journal in 2005. In addition, the velocity of such an impact, according to this approach, is close to what is predicted by the Big Splat model.

Chaotic dynamics might also explain the propagation of life from solar system to solar system through a process called “lithopanspermia”, where rock fragments bearing primitive life are blown off the surface of one world, such as in an asteroid impact, and transported to another. Classical dynamics suggests that the odds of a rock making it from one solar system to another are very low, given the distances and volumes of space involved. However, Belbruno argues that weak stability boundary transfer makes this more likely this can work, at least in fairly dense star clusters. “This shows that, maybe, life did not evolve independently on different solar systems, but propagated” from one to another, he said.

Lithopanspermia might seem outlandish, but chaotic dynamics seems less so now than when Belbruno was alone in advocating it at JPL two decades ago. Besides designing the trajectory used by Hiten to reach the Moon and laying the groundwork for the SMART-1 trajectory, another of his accomplishments was proposing the rescue of AsiaSat 3. He and Rex Ridenoure first suggested to Hughes in January 1998 that the spacecraft could be sent around the Moon and brought back to GEO using only the propellant onboard the spacecraft. However, while they proposed a weak stability boundary transfer approach, Hughes elected to go with a more conventional free return trajectory around the Moon; the free return approach cost more propellant but allowed controllers to keep the spacecraft in tracking range. However, given the advances chaotic dynamics has demonstrated, one wonders if the owners of Arabsat 4A will be as conservative today when contemplating the options for their stranded satellite.