A vehicle arriving from Earth meets the top of Mars's atmosphere, about 125 km up, at between 5.5 and 7.5 km/s. The 2026 transfer of the Lambert lesson arrives with a hyperbolic excess of 2.57 km/s, and Mars's gravity adds the rest: at the entry interface the speed is $v_{\infty}^{2} + 2 μ / r = 5.57$ km/s. That is 15.5 MJ of kinetic energy in every kilogram of the vehicle. Seven minutes or so later it has to be on the ground, at a chosen spot, moving at less than a couple of metres a second, and nobody on Earth can help: the signal takes between three and twenty-two minutes to arrive.

The problem, precisely: from a given entry state — speed, flight-path angle, position — remove essentially all of the vehicle's energy, keep the peak deceleration and heating within what the structure and the heat shield can bear, and arrive at the surface slowly enough to land, using as little propellant as possible for the part the air cannot do.

On the Moon every metre per second of that is paid for with propellant; on Earth the atmosphere does nearly all of it, and a parachute finishes the job at a few metres a second. Mars is the awkward middle. Its atmosphere is thick enough to heat a vehicle to thousands of degrees and to take off more than nine tenths of the entry speed — and so thin that, for anything heavy, it runs out of height before it has finished. At the datum the simulator's Mars air has a density of 0.014 kg/m³, 1.1 % of Earth's at sea level, with a scale height of about 11 km.

One number for the vehicle

Drag decelerates a vehicle at

\frac{D}{m} = \frac{\frac{1}{2} ρ v ^{2} C _{D} A}{m} = \frac{q}{β}, β = \frac{m}{C _{D} A}

where $ρ$ is the air density, $v$ the speed, $q = \frac{1}{2} ρ v^{2}$ the dynamic pressure, $C_{D}$ the drag coefficient, $A$ the frontal area and $m$ the mass. The ballistic coefficient $β$ is everything about the vehicle that matters to a ballistic entry, in one number: mass per unit of effective drag area. A vehicle with low $β$ is a feather and decelerates high up, where the air is thin and the heating mild; one with high $β$ is a stone and falls deep before the air takes hold.

The difficulty at Mars is how $β$ scales. Mass grows with the volume of what is being landed; frontal area grows only with the square of the diameter, and the diameter is capped by the launch vehicle's fairing. Viking's 3.5 m aeroshell carried about a tonne at $β \approx 64$ kg/m². Curiosity's 4.5 m shell carried 3.3 t at 146. Doubling the diameter again is not possible inside any fairing that has flown to Mars.

The equations of an entry

Treat the planet as a non-rotating sphere of radius $R$ , the vehicle as a point of constant $β$ with no lift, and measure the flight-path angle $γ$ below the local horizontal. At altitude $h$ , radius $r = R + h$ and local gravity $g = μ / r^{2}$ ,

\overset{v}{˙} = - \frac{ρ v ^{2}}{2 β} + g sin γ, \overset{γ}{˙} = (\frac{g}{v} - \frac{v}{r}) cos γ, \dot{h} = - v sin γ

The second equation contains the physics the simple theory will leave out. The path bends down under gravity and up under its own curvature; above the local circular speed — 3.49 km/s at 125 km on Mars — the second wins, and an entry that starts shallow gets shallower before the air grabs it.

The straight-line solution

Allen and Eggers, working on ballistic missile warheads in the 1950s, found the solution that still organises the subject. Assume the drag is so much larger than gravity that $g$ can be dropped, so the path is a straight line at constant $γ$ ; and assume an exponential atmosphere, $ρ = ρ_{s} e^{- h / H}$ , with surface density $ρ_{s}$ and scale height $H$ . Dividing the first equation by the third turns time into altitude:

\frac{d v}{d h} = \frac{ρ v}{2 β sin γ} ⟹ v = v_{e} exp (- \frac{ρ H}{2 β sin γ})

where $v_{e}$ is the entry speed, reached where $ρ \to 0$ . Every result that follows comes from this one line.

Peak deceleration. The deceleration is $ρ v^{2} /2 β$ . Written in terms of $ρ$ and set stationary, it peaks where $ρ = β sin γ / H$ , with the speed down to $v_{e} e^{- 1/2} = 0.61 v_{e}$ , and its value there is

a_{max} = \frac{v _{e}^{2} sin γ}{2 eH}

— independent of $β$ . A heavier vehicle does not feel a harder peak; it feels the same peak lower down, at $h^{*} = H ln (ρ_{s} H / β sin γ)$ . The steepness of the entry sets the load; the ballistic coefficient sets the altitude where it happens.

Peak heating. Convective heating at the stagnation point follows Sutton and Graves, $\overset{q}{˙} = k ρ / r_{n} v^{3}$ , with $r_{n}$ the nose radius and $k = 1.9027 \times 1 0^{- 4}$ in SI units for a CO₂ atmosphere. Maximising $ρ v^{3}$ along the same solution puts the heating peak at $ρ = β sin γ /3 H$ — a third of the density, so higher up — at $v_{e} e^{- 1/6} = 0.85 v_{e}$ . The vehicle is hottest before it is most heavily loaded.

The height at which it slows down. The speed of sound near the ground on Mars is about 235 m/s. Solving the velocity law for the density at which $v$ has fallen to some speed $v_{1}$ ,

ρ_{1} = \frac{2 β sin γ}{H} ln \frac{v _{e}}{v _{1}}

and the vehicle reaches $v_{1}$ above the ground only if $ρ_{1} < ρ_{s}$ , that is, only if

β < \frac{ρ _{s} H}{2 sin γ ln ( v _{e} / v _{1} )}

For a Curiosity-like entry — 5.85 km/s at 15.5° — with $ρ_{s} = 0.0139$ kg/m³ and $H = 11.1$ km, the vehicle gets below Mach 2 before the ground only if $β < 114$ kg/m², and below Mach 1 only if $β < 90$ . Curiosity's 146 is past both. This is the Mars entry problem in one inequality: the air can take the speed off, but for a heavy vehicle it cannot finish before the ground arrives.

The corridor

The flight-path angle has limits on both sides. Too steep, and the peak load — which grows with $sin γ$ — and the heating grow past what the vehicle survives. Too shallow, and a vehicle arriving faster than escape speed passes through the upper atmosphere without shedding enough energy to be captured, and leaves again on a hyperbola. Between them is the entry corridor. For a ballistic vehicle at 5.57 km/s it runs from about 10° at the interface, below which it skips out, to about 18°, where a Curiosity-class vehicle passes 15 g. In terms of the approach hyperbola, the shallow edge is a periapsis roughly 40–45 km above the datum; the corridor is what the approach navigation has to hit, from hundreds of millions of kilometres away.

Figure · a ballistic entry at Mars

VEHICLE

BALLISTIC COEFFICIENT β146 kg/m²

ENTRY ANGLE15.5 °

ENTRY SPEED5.85 km/s

OUTCOME: Reaches the datum
PEAK DECELERATION: 13.2 g · 18.9 km
STRAIGHT-LINE THEORY: 15.5 g
PEAK HEATING, 1 m NOSE: 80 W/cm² · 29.0 km
MACH 2: not above the datum
SUBSONIC: not above the datum
AT THE DATUM: 577 m/s · Mach 2.5

From the 125 km entry interface to the datum, no lift, through the simulator's mean Mars atmosphere. Left: speed against altitude; the grey lines are Mach 1 and Mach 2. The filled dot is peak deceleration, the ring peak heating. Right: deceleration in Earth g, against the 15 g structural limit. The presets are roughly the published entry states of those vehicles, flown here without the lift the later ones used. Move β and the deceleration pulse slides down the altitude axis; at steep angles its peak barely changes, at shallow ones it falls. Move the angle and the peak grows with it.

The figure integrates the full equations, gravity and curvature included, through the simulator's own Mars atmosphere, and puts the straight-line prediction beside the result. For Viking they agree to within a tenth: 11.9 g against a predicted 10.9. For Curiosity they agree less well, 13.2 against 15.5, because its faster entry flattens more on the way in. Move the Viking preset's $β$ from 64 up to 500 and the deceleration pulse slides down from 24 km to 2.4 km while its peak barely changes, from 11.9 g to 9.7, just as the theory says; Mach 2 is already below the datum by $β = 150$ . Now pick Odyssey Mars, at 11°. The theory predicts 10 g; the integration finds 3.5. The entry is shallow and faster than escape, the path flattens as it descends — from 11° at the interface to about 2.5° near 20 km — and the vehicle spends a long time in thin air, bleeding speed gently. The straight-line assumption has failed, and in a direction that helps.

Why parachutes only get you so far

Every Mars lander so far has opened a supersonic parachute — a disk-gap-band canopy, qualified to open at up to about Mach 2 — after the aeroshell has done its work. A parachute's job is to take a vehicle from Mach 2 to a slow, steady descent, and it can only do the second part as well as the air allows. At its terminal speed, drag balances weight:

v_{t} = \frac{2 m g}{ρ C _{D} A _{p}}

with $A_{p}$ the canopy's area. Take a canopy 21.5 m across, with a drag coefficient of about 0.6, carrying 2.5 t: through Earth's sea-level air it settles at 13.6 m/s, and through the simulator's Mars air at the datum at 75 m/s. That is a crash, not a landing, so every Mars parachute is followed by something else: retrorockets, airbags, or a sky crane.

Scaling makes it worse. $v_{t}$ grows as $m / A_{p}$ , so the same canopy under 25 t falls at 238 m/s, and bringing it back to 75 m/s needs a canopy of 68 m. And the parachute also needs time and height: it has to open at Mach 2 or below, and a vehicle of high $β$ reaches Mach 2 lower down, leaving less air between the deployment and the ground. At $β \approx 150$ the margin is already thin; well above it, there is none.

Supersonic retropropulsion

The way past the limit is to light the engines while the vehicle is still supersonic, firing into the oncoming flow. Supersonic retropropulsion takes over the job the parachute cannot do, and for a vehicle of tens of tonnes it is the only known way to land propulsively on Mars — the alternative being a much larger decelerator, such as an inflatable heat shield, to bring $β$ back down.

It is not a landing burn started early. The exhaust plume pushes the shock wave in front of the vehicle outwards and fills the region behind it, and most of the aeroshell's drag disappears when the engines light. The rockets must then supply most of the deceleration themselves, while the vehicle is still fast, which costs propellant; the flow round the plume is unsteady, which the control system has to fight; and the engines have to start and run in a supersonic headwind. No Mars mission has done it. The nearest flight experience is on Earth: the entry burns of returning Falcon 9 boosters, which fire into supersonic flow high in Earth's atmosphere and which NASA has studied as an analogue.

The sequence, Viking to Perseverance

Every landing so far has used the same three-part solution: an aeroshell for the hypersonic part, a parachute for the supersonic part, and something propulsive or cushioned for the last tens of metres per second.

Mission	Landed	Entry mass	Aeroshell	β, kg/m²	Entry	The last part
Viking 1 and 2	1976	about 1 t	3.5 m	64	from orbit, about 4.7 km/s	parachute, then throttled retrorockets on legs
Mars Pathfinder	1997	584 kg	2.65 m	63	direct, about 7.3 km/s	parachute, solid rockets, airbags
Spirit, Opportunity	2004	about 830 kg	2.65 m	94	direct, about 5.5 km/s	parachute, solid rockets, airbags
Phoenix	2008	about 600 kg	2.65 m	about 65	direct, about 5.6 km/s	parachute, pulsed rockets on legs
Curiosity	2012	about 3.3 t	4.5 m	146	direct, about 5.9 km/s, guided	parachute, sky crane
Perseverance	2021	about 3.4 t	4.5 m	about 150	direct, about 5.4 km/s, guided	parachute, sky crane

Read down the table and the pressure is visible. The aeroshell went from 2.65 m to 4.5 m, about as wide as a 5 m launch fairing allows; $β$ still more than doubled; and the solutions at both ends grew more elaborate to compensate. Curiosity and Perseverance fly a lifting entry: the capsule's centre of mass is offset so it trims at an angle of attack and develops a little lift, and the flight computer rolls that lift vector left and right to control range and to hold the vehicle higher, where it has time to open its parachute. That, with the landing sites chosen low — Curiosity's in Gale crater, about 4.5 km below the datum, where there is more air overhead — is how a $β$ of 146 lands at all. Perseverance added a range trigger, which opens the parachute on the distance to the target rather than on a fixed speed, and terrain-relative navigation, which compares the ground below with an onboard map and picks a safe spot. The landing ellipse shrank from Curiosity's 20 by 7 km to Perseverance's 7.7 by 6.6 km.

The last part is the sky crane. After the parachute, the backshell is released a couple of kilometres up, still moving at 80–90 m/s, and a rocket-powered descent stage flies the rover down, slows to a hover about 20 m above the surface, lowers the rover on cables, sets it down at well under a metre per second, and flies away to crash at a distance. Perseverance, at 1,025 kg, is the heaviest thing yet landed on Mars.

A worked example: Odyssey Mars

The simulator's Mars stack ends in a lander of 25.36 t behind a 7 m aeroshell: $A = π (3.5)^{2} = 38.48$ m², and with the blunt-body drag coefficient of about 1.3 the simulator uses at entry Mach numbers,

β = \frac{25 360}{1.3 \times 38.48} = 507 kg/m^{2}

— more than three times Curiosity's. The same mass on Odyssey's 4.2 m lunar lander would be about 1,400 kg/m², which is why the Mars stack is a separate vehicle.

It arrives on the 2026 transfer at $v_{\infty} = 2.57$ km/s. The reference program steers the approach so that periapsis would be 20 km above the datum, which means an entry at 5.57 km/s and 10–11° at the 125 km interface — just inside the corridor, whose shallow edge for this vehicle is at about 10.2°. Set the figure to Odyssey Mars, at 11°, and read the result. Peak heating is 77 W/cm² for a 1 m nose, at 28 km. Peak deceleration is a gentle 3.5 g, at 21 km. At 10 km the vehicle is still moving at 1.9 km/s — Mach 8.3 — and it reaches the datum at 1,009 m/s, Mach 4.3, 301 s after entry. There is no parachute regime at all: the vehicle is never below Mach 2 above the ground.

Try the same vehicle steeper. At 14° the peak is 8.6 g at 9 km and the datum comes at 2.11 km/s: a steeper entry reaches denser air sooner but gives it less time to work. At 10.1° the vehicle skims through the upper atmosphere and leaves again at 4 km/s; at 9°, at 5.4. The shallow edge is where a heavy vehicle wants to be, and it is also the edge where it skips out.

So the engines have to take out at least the thousand metres a second the air leaves at the datum, and in practice more: they must light well above it, hypersonic, and while they burn most of the aeroshell's drag is gone. The simulator's own design estimate was that entry, descent and landing need about 1.2 km/s from the lander. Flown in the full aerodynamics, it is not enough — see below.

In Vivapse

Mars's air is marsAtmosphere(h) in src/sim/bodies.ts. Below 30 km it is a hydrostatic column: Seiff's mean temperature profile from the Viking era, integrated upwards from Viking Lander 1's two-year mean surface pressure, which puts 567 Pa at the datum. From 30 to 50 km it blends into the Mars Climate Sounder means; above that it is built from Mars Climate Sounder retrievals up to 75 km and Mars Global Surveyor's aerobraking accelerometer from 102.5 to 140 km, carried hydrostatically through a 110 K mesopause between them. The speed of sound uses a ratio of specific heats that follows the temperature, as CO₂'s does. It is a mean model: no dust storms, no daily cycle, no variation with latitude. Against the two entries that measured the air on the way down, it sits between them — 10 to 37 % denser than Pathfinder's night-time profile, 5 to 18 % thinner than Phoenix's afternoon one. (Until the third physics audit, the lowest 40 km were an engineering fit that ran up to 2.7 times too dense at 26–40 km, which flattered every entry in this lesson by 190–470 m/s at 10 km.) The real atmosphere scatters about any mean by a factor of up to 2.7, so an entry that relies on it should assume the density could be half or double. The physics model has its accuracy band by band, and the entry figure above uses the same table.

The vehicle meets it with the same six-degree-of-freedom aerodynamics as on Earth. Flown base-first, the axial force coefficient runs from 0.85 subsonic to 1.45 at Mach 2 and 1.28 hypersonic — the source of the 1.3 in the $β$ above — and under thrust it collapses as $C_{A} / (1 + \frac{1}{2} C_{T})$ , with $C_{T}$ the thrust over $q$ times the reference area: the plume effect of the previous section, as a correlation. Heating is Sutton–Graves at an effective nose radius of 1.2 times the diameter for a blunt base, cut to 0.3 of its value while the engines fire into the flow, and absorbed by a heat shield modelled as a single lumped mass with a 2,400 K limit and no ablation. The Sutton–Graves constant follows the gas: $1.7415 \times 1 0^{- 4}$ for air, $1.9027 \times 1 0^{- 4}$ for Mars's CO₂ (src/sim/aero.ts). There is no parachute model: the simulator's Mars landing is aeroshell and then engines. The structure fails above 15 g, and above 250 kPa·° of bending load.

The program is mars-landing.js, for the Odyssey Mars stack. After the cruise it walks the approach's periapsis to within 30 km of a point 20 km above the datum — an entry at about 10° at the interface — drops the injection stage, and flies the entry base-first on fc.airRetrograde, at zero angle of attack with nothing to trim, letting the air do the work. Then comes guided retropropulsion, and the same terminal hoverslam as the Moon, aimed at 1.5 m/s, with the legs out at 1,200 m.

It does not yet land, and the third physics audit found out why. The flight used to arrive with most of its $Δ v$ gone: the parking-orbit burn ran on to a 185 × 10,300 km orbit, the program's own trajectory predictor was 449 km wrong at Mars, and the entry test lit the engines some 2,000 km out. With those fixed, the lander reaches the corridor with its full 1.96 km/s — and the aeroshell is what fails. Flown in the full aerodynamics with the engines off, it is still at 2.6 km/s at 30 km and 1.1 km/s at 1 km; once the plume's drag collapse is modelled, no ignition point stops it. The analytic estimate above was optimistic. A 9 m aeroshell, which lowers $β$ to about 390, brought a test lander to 40 m/s at 118 m with 2.7 t to spare; whether the reference vehicle should grow one is an open decision. Fidelity and its limits keeps the list of what is open.

Try it

The simulator does fly supersonic retropropulsion routinely — on Earth, in every booster return. Fly the full mission with the Aster preset from Cape Canaveral, with the weather set to Custom and no wind, gusts or turbulence, and add a log to the booster's program. At the top level of the Full mission — orbit & return from anywhere example, beside the other state:

let srpLogAt = 0;

and as the first lines inside function booster(fc) {:

if (fc.altitude < 60e3 && fc.t >= srpLogAt) {
  srpLogAt = fc.t + 1;
  fc.log(fc.mem.phase, 'Mach', fc.mach.toFixed(2), 'q', (fc.dynamicPressure / 1e3).toFixed(1), 'kPa',
    'drag', fc.aeroAccel.drag.toFixed(2), 'm/s²', 'thrust', (fc.thrust / 1e3).toFixed(0), 'kN');
}

Watch the console as the booster comes back down. Coasting at Mach 6.8 through 55 km, it reads 1.2 kPa of dynamic pressure and 0.41 m/s² of drag. Three engines light for the entry burn and push 2,796 kN into the oncoming flow; the drag drops at once to 0.13 m/s², though the dynamic pressure keeps rising. Through the burn it climbs to 4.3 kPa while the drag stays below 0.6 m/s². The burn cuts at 974 m/s, Mach 3.2, 35 km up — and half a second later, at 4.0 kPa, the drag reads 2.27 m/s². Allowing for the propellant burned, the engines were cancelling about three quarters of the drag the stage would otherwise have felt. That is the plume at work, and the reason a Mars lander that relies on retropropulsion cannot also count on its aeroshell while the engines are lit. The numbers are from a calm flight without hardware dispersions; yours will differ a little.

For scale: Odyssey Mars, flown ballistically in the figure, is at Mach 8.3 and over 10 kPa at 10 km. Retropropulsion there would begin at dynamic pressures two or three times anything the booster's entry burn meets.

Where this leaves you

The ladder started with a ratio of masses and ends here, at the bottom of a descent that needs every tool in it at once: the rocket equation for the propellant that remains, Lambert for the arrival speed, the conic for the corridor, the drag and heating of the flight lessons, and a powered descent at the end that cannot hover and cannot wait. The simulator implements all of it, and it does not yet land on Mars. That is not a gap in the lessons. It is the state of the problem, and a program that closes it is one worth writing.