Low-Thrust and Multi-Target Missions

Ion propulsion, spirals, Edelbaum's approximation, phasing a low-thrust transfer, and Dawn at Vesta and Ceres

Vesta, 5 September 2012

For fourteen months Dawn had orbited Vesta, the second-largest body in the asteroid belt, spiralling down from 2,750 km to 210 km and back up as it mapped it. Now it was leaving — not with a burn, but by thrusting gently for weeks, its orbit widening turn by turn until Vesta let go. Ahead lay two and a half years of thrust to reach Ceres, the largest body in the belt, where it would enter orbit in March 2015 and stay until its hydrazine ran out in 2018. No spacecraft before had orbited two bodies beyond the Earth–Moon system. Dawn's three ion engines each pushed with about 90 millinewtons — the weight of a sheet of paper — and together over the mission they changed its velocity by about 11.5 km/s, more than any spacecraft had ever achieved with its own propulsion.

💡Why chemistry could not do it

The rocket equation,

m_f/m_0 = e^{-\Delta v/(I_{sp} g_0)}

, sets the propellant for a given Δv. At Dawn's Δv of 11.5 km/s:

a bipropellant engine at $I_{sp} = 320$ s needs 97.4% of the spacecraft to be propellant — a mass ratio of 39, impossible for a spacecraft that must also carry instruments, power and structure;
an ion engine at $I_{sp} = 3{,}100$ s needs 31.5%.

Dawn launched at 1,218 kg with 425 kg of xenon. The first lab shows that this tank, at 3,100 s, is worth 13.1 km/s.

T = \frac{2\eta P}{g_0 I_{sp}}

Jet power is ½ ṁ v_e² = ½ T v_e. With P = 2.3 kW and efficiency η = 0.6, T ≈ 91 mN at 3,100 s. At fixed power, higher Isp means less thrust: ion propulsion buys propellant efficiency with time.

With 91 mN on a 1,000 kg spacecraft the acceleration is 0.091 mm/s²: it would take 3.5 days to reach 100 km/h. Delivering 11.5 km/s takes about four years of thrusting. Dawn's engines in fact ran for about 5.9 years in total — about 54% of the time it spent in space.

A low-thrust trajectory is therefore not a sequence of impulses joined by conics. It is a continuous spiral, and it needs different tools.

Slow and steady wins the race.

– The moral of Aesop's fable The Tortoise and the Hare, and, it turns out, of ion propulsion

💡Spirals between circular orbits

With thrust along the velocity and an acceleration much smaller than local gravity, the orbit stays nearly circular as it grows or shrinks. Then the energy of a circular orbit,

-v^2/2

, changes at the rate

v\,a_T

, and

\dot v = -a_T

: the speed falls exactly as fast as the thrust acceleration. The spiral's cost between circular orbits is simply

\Delta v_{\text{spiral}} = |v_0 - v_1| = \left|\sqrt{\mu/r_0} - \sqrt{\mu/r_1}\right|

A spiral from low orbit to escape costs the whole circular speed — about 2.4 times an impulsive escape burn. Low thrust spends Δv freely and saves propellant through Isp.

The second lab integrates Dawn's transfers at Vesta (91 mN, 1,000 kg, point-mass gravity). From HAMO (680 km) to LAMO (210 km): 55.8 m/s, exactly $|v_0 - v_1|$ , in 7.1 days and 25 revolutions; a Hohmann transfer would cost 54.2. From the survey orbit to HAMO the thrust is no longer small compared with Vesta's weak gravity at 3,000 km, and the spiral ends slightly eccentric. Escaping from LAMO costs 191 m/s by spiral against 79 m/s impulsively.

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💡Edelbaum's approximation

Between two circular orbits of speeds

v_0

and

v_1

whose planes differ by

\Delta i

, the minimum low-thrust Δv — with the out-of-plane thrust spread optimally around every revolution — was found by Theodore Edelbaum in 1961:

\Delta v = \sqrt{v_0^2 + v_1^2 - 2 v_0 v_1 \cos\!\left(\frac{\pi}{2}\Delta i\right)}

With Δi = 0 it reduces to |v₀ − v₁|. With constant acceleration a, the speed evolves as v(t) = √(v₀² − 2v₀at cos β₀ + a²t²) and the transfer takes Δv/a.

📐Worked Example: Dawn's budget

The third lab adds up the whole mission with JPL's orbital elements:

Earth → Vesta: relative inclination 7.14°, Edelbaum 11.41 km/s. The Delta II launcher supplied about 3.38 km/s (C₃ ≈ 11.4 km²/s²) and the Mars flyby in February 2009 about 2.6 km/s, leaving 5.63 km/s for the ion engines including the spiral down: 206 kg of xenon.
Vesta → Ceres: relative inclination only 4.93°, yet the plane change doubles the cost — 2.91 km/s instead of 1.47 coplanar. With the spirals out of Vesta and into Ceres: 3.38 km/s, 106 kg of xenon. Dawn's mission plan allocated 112 kg to this leg.
Total: 312 kg of the 425 kg carried. The margin paid for orbit changes at both bodies and an extended mission at Ceres.

The same Vesta–Ceres leg with a 320 s chemical stage would need 667 kg of propellant from a 1,012 kg spacecraft — and a heavier spacecraft would have needed a larger launcher to reach Vesta in the first place.

When can you leave? Vesta and Ceres line up only every 17.2 years (their synodic period). In the guided assignment for this unit you combine Edelbaum's time history with both asteroids' positions to find the departure dates from Vesta for which a constant-acceleration transfer arrives exactly where Ceres is. With 0.04 mm/s² the model finds 23 November 2012 → 17 March 2015; Dawn actually left on 5 September 2012 and entered orbit on 6 March 2015. Miss that opportunity, and the next is in 2029. Dawn's stay at Vesta had a hard deadline, set by orbital mechanics.

🕰From [Deep Space 1](https://en.wikipedia.org/wiki/Deep_Space_1) to Psyche

Dawn's engines descended from those of Deep Space 1 (1998), which first demonstrated ion propulsion as a spacecraft's main engine on a deep-space mission. JAXA's Hayabusa and Hayabusa2 used ion engines to reach and return from asteroids Itokawa and Ryugu; ESA–JAXA's BepiColombo is using them on its way into orbit around Mercury. NASA's Psyche, launched in October 2023, uses Hall-effect thrusters and a Mars flyby to reach the metal-rich asteroid 16 Psyche, with arrival planned for 2029. Solar-electric propulsion is also a leading option in recent studies for reaching Uranus faster.

💡A recipe for going anywhere

Every mission in this course, from a satellite in low Earth orbit to a lander on Titan, is assembled from the same steps:

When — launch windows from real ephemerides and Lambert's problem, with the synodic periods of every pair of bodies involved.
Which way — direct, or via gravity assists chosen with Tisserand graphs; Jupiter for the outer planets, Venus–Earth–Earth when the launcher is small.
Arrive — B-plane targeting, low and loose capture, a moon flyby or aerocapture if available.
Get close — a resonant moon tour and an endgame to lower v∞; or a low-thrust spiral.
Stay — orbits that the three-body dynamics and perturbations allow: frozen orbits, distant moons, halo orbits.
Land — powered descent or entry, autonomously.
Close the budget — Δv to propellant through the rocket equation with margins; and check the design in a full force model before trusting it.

↓But there is a problem

This is where the course ends. Go back to the beginning. A twenty-three-year-old in a plague year, in a garden, asking whether the force that drops an apple might be the same force that holds the Moon. From that one question you have derived the two-body equation of motion and the proof that its orbits are conics; the six elements and the frames they are written in; Kepler's equation; the rocket equation and optimal transfers; the perturbations that break every one of those results; Lambert's problem and the solvers that make it fast; launch windows; the patched-conic machinery that turns them into a mission to another planet; and, in this last unit, the gravity assists, moon tours, three-body orbits, landings and low-thrust spirals that reach everywhere else. Every equation, you derived. Every algorithm, you implemented and tested. And you know where it stops. The patched conic is a design model, wrong by a million kilometres at Mars until a full force model takes over. The three-body problem has no general closed-form solution — Poincaré showed in 1890 that its motion can be chaotic — and the low-energy routes between moons, the long-term fate of orbits around them, and the optimal steering of a low-thrust spacecraft are still being worked out. That is not a gap in this course. That is where the field is still alive: low-energy and low-thrust trajectory design, autonomous navigation and landing, missions to Uranus and Neptune, Enceladus and Europa. All of it is built on exactly what you now know. The capstone is where you start.