Landing on Moons: Powered Descent and Atmospheric Entry

Descent budgets on airless worlds, gravity turns and gravity loss, thrust-to-weight trades, and entry and parachute descent at Titan

Titan, 14 January 2005

Huygens had ridden on Cassini for seven years, then flown alone for three weeks. It hit the top of Titan's atmosphere at about 6 km/s. In three minutes its heat shield slowed it from about 18,000 to 1,400 km/h; at about 180 km altitude, now down to around 400 m/s, it fired out a 2.6 m pilot chute, which pulled out the main parachute and released the heat shield. Fifteen minutes later it cut the main parachute loose and fell on a smaller drogue for the rest of the way. It broke through the haze only about 30 km up and saw, for the first time, river channels running down to a dark plain. It landed after a descent of about two and a half hours, at around 5 m/s, on ground with the consistency of damp sand, and kept transmitting from the surface for over an hour until Cassini sank below its horizon. It is still the most distant landing ever made.

Every landing has the same problem: the spacecraft arrives with orbital speed — 1.4 km/s at Europa, 1.7 at the Moon, 0.18 at Enceladus — and must reach the surface with essentially none. On an airless world all of it is removed by the engine. On a world with a thick atmosphere almost all of it is removed by drag, and the engine (if any) does only the last few metres.

💡The ideal descent budget

The cheapest two-impulse descent from a circular orbit of radius

r_1

is a small deorbit burn onto an ellipse whose periapsis grazes the surface, then a braking burn at the surface that cancels the periapsis speed:

\Delta v_{\text{ideal}} = \underbrace{\sqrt{\frac{\mu}{r_1}} - \sqrt{\mu\Big(\frac{2}{r_1} - \frac{1}{a}\Big)}}_{\text{deorbit}} + \underbrace{\sqrt{\mu\Big(\frac{2}{R} - \frac{1}{a}\Big)}}_{\text{braking}},\qquad a = \frac{r_1 + R}{2}

Nearly all the cost is the braking burn: from 100 km, Europa 22 + 1,455 m/s; Enceladus 12 + 183 m/s.

World	Surface g (m/s²)	Escape speed	Ideal descent Δv	Each minute of hover
Mercury	3.70	4.25 km/s	3,065 m/s	222 m/s
Moon	1.62	2.38 km/s	1,726 m/s	97 m/s
Ganymede	1.43	2.74 km/s	1,974 m/s	86 m/s
Europa	1.32	2.03 km/s	1,476 m/s	79 m/s
Callisto	1.24	2.44 km/s	1,761 m/s	74 m/s
Triton	0.78	1.45 km/s	1,063 m/s	47 m/s
Ceres	0.28	0.52 km/s	399 m/s	17 m/s
Vesta	0.25	0.36 km/s	294 m/s	15 m/s
Enceladus	0.11	0.24 km/s	195 m/s	7 m/s

A real engine is not an impulse. A lander with thrust-to-weight ratio

T/W

(in local gravity) takes time to brake, and during that time gravity is pulling it down. The usual steering law for the main braking phase is the gravity turn: thrust exactly opposite the velocity, so the trajectory bends naturally from horizontal to vertical as the speed falls. The equations, in polar coordinates with

\gamma

the flight-path angle:

\dot v = -\frac{T}{m} - g\sin\gamma,\qquad v\dot\gamma = -\Big(g - \frac{v^2}{r}\Big)\cos\gamma,\qquad \Delta v_{\text{grav}} = \int g\sin\gamma\,dt

The gravity loss is the integral of the component of gravity along the flight path — it grows with burn time.

📐Worked Example: thrust-to-weight on Europa

The second lab simulates gravity-turn descents on Europa (Isp 320 s), choosing the starting orbit so the lander comes to rest exactly at the surface:

T/W 1.5: start at 58 km, 650 s burn, 434 km downrange, 1,648 m/s — gravity loss 190 m/s (13%);
T/W 2: 31 km, 465 s, 1,549 m/s — 102 m/s (7.1%);
T/W 3: 13.6 km, 300 s, 1,482 m/s — 44 m/s (3.0%);
T/W 5: 4.8 km, 177 s, 1,449 m/s — 15 m/s (1.0%);
T/W 10: 1.2 km, 88 s, 1,436 m/s — 3 m/s (0.2%).

A strong engine almost eliminates the gravity loss — but it starts its burn barely a kilometre up, with no margin for a ridge or a navigation error. Engine mass also grows with thrust. Real descent designs settle around T/W of 2–3 for the main braking phase, then throttle down for a terminal descent and a hover over the chosen spot.

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⚠Light time rules out piloting

Light takes 33–53 minutes to travel one way between Earth and Jupiter, and 68–84 minutes for Saturn. Every landing at an outer moon must be fully autonomous: the lander measures its altitude and velocity with radar or lidar, recognises terrain from images it carries in memory (terrain-relative navigation, first used on Mars by Perseverance in 2021), finds a safe spot and lands, without help. The descent Δv budget therefore always includes hover and divert margins — at 79 m/s per minute on Europa, a minute of searching is not cheap.

Houston, Tranquility Base here. The Eagle has landed.

– Neil Armstrong, Apollo 11, 20 July 1969. He flew the last minutes of the descent by hand, with Houston 1.3 light-seconds away. At Europa the delay is over half an hour

💡Entry through an atmosphere

Where there is air, drag removes the orbital energy as heat. The deceleration is

D/m = \rho v^2/(2\beta)

, with ballistic coefficient

\beta = m/(C_D A)

. On a steep entry into an exponential atmosphere, the classical Allen–Eggers result is that peak deceleration does not depend on

\beta

at all:

a_{\max} = \frac{v_E^2 \sin|\gamma_E|}{2 e H},\qquad \rho_{\text{at peak}} = \frac{\beta\sin|\gamma_E|}{H}

Entry speed v_E, flight-path angle γ_E, scale height H. The ballistic coefficient sets where the peak happens, not how hard it is.

Titan is the easiest place in the Solar System to land on. Its surface pressure is 1.5 bar, its surface air is about four times as dense as Earth's, and its gravity is 1.35 m/s² — less than the Moon's. Terminal velocity under a parachute scales as

\sqrt{2mg/(\rho C_D A)}

: low

g

and high

\rho

make it small.

The third lab builds a hydrostatic atmosphere from Huygens's measured temperature profile and flies a Huygens-like probe (6 km/s, −65°, β ≈ 37 kg/m²):

peak deceleration 12.6 g at 218 km;
slowed to 400 m/s at 153 km (Huygens: about 180 km);
main parachute (8.3 m) for 15 minutes, then the 3.03 m drogue: descent 2.33 hours, touchdown at 5.1 m/s (Huygens: about two and a half hours, about 5 m/s);
the same probe and drogue on Earth would hit the ground at 28 m/s.

A planar model with a fixed drag coefficient lands within tens of kilometres and tens of minutes of the real descent — good enough to size parachutes.

🛰Dragonfly

NASA's Dragonfly is a rotorcraft lander for Titan, scheduled to launch in July 2028 on a Falcon Heavy and arrive in 2034. After entry and descent under parachute it will release itself and fly to a landing — Titan's dense air and low gravity make flight there far easier than on Earth — and then hop between sites tens of kilometres apart, recharging its batteries from a radioisotope power source between flights. Its target is the dune fields near Selk crater, where liquid water from the impact may once have mixed with Titan's organic chemistry.

💡Landing on a comet or an asteroid

At the other extreme, a 4 km comet or a 500 m asteroid has escape speeds of centimetres to a metre per second. The difficulty is not braking but staying: Philae, Rosetta's lander, bounced twice off comet 67P in November 2014 when its anchoring harpoons failed, and came to rest in a shadowed spot over a kilometre from its target. Missions to small bodies — Hayabusa, Hayabusa2, OSIRIS-REx — mostly touch and go, sampling in seconds and backing away.

↓But there is a problem

Every mission so far in this unit has had one destination. Galileo, Cassini and Juno stayed with one planet; Voyager and New Horizons flew past many worlds without stopping at any. In 2007 a spacecraft left Earth intending to do what none had done: enter orbit around one body in the asteroid belt, map it, leave, travel to a second, and enter orbit around that one too. The total Δv it needed was larger than any chemical rocket could carry. Its engines pushed with the weight of a sheet of paper.