← INDEX — FIG. 00 — HOW TO GET FROM ONE PLANET TO ANOTHER HOHMANN, 1925

TRANSFER
EARTH TO MARS: DISTANCE, TIMING, FUEL AND RADIATION

Ten chapters on the trip every Mars mission has to make.

BEGIN — CHAPTER 01 →

CH.01 / 11 — WHERE MARS IS, TODAY AND ANY DAY

The distance to Mars changes by a factor of seven

Earth circles the Sun in one year at about 150 million km.

The model below solves Kepler's equation for both planets.

DATE (UTC)—

Distance over 2000–2045.

CH.02 / 11 — WHY YOU CAN'T LAUNCH WHENEVER YOU WANT

The door to Mars opens once every 26 months

A spacecraft leaving Earth is already moving with Earth.

The half-orbit takes about 259 days.

SYNODIC PERIOD — HOW OFTEN THE GEOMETRY REPEATS

1 / S = 1 / T_Earth − 1 / T_Mars → S = 1 / (1/365.26 − 1/686.98) ≈ 780 d

Earth laps Mars like a faster runner on an inner track.

Idealised window dates.

Real windows are found with porkchop plots.

CH.03 / 11 — THE HOHMANN TRANSFER

Aim at where Mars will be. Try it

In 1925 Walter Hohmann showed the least-fuel path.

Set how far ahead of Earth Mars is at launch.

MARS AHEAD OF EARTH AT LAUNCH20°

TRANSFER TIME — HALF THE ELLIPSE'S PERIOD

t = π √(a³ / μ_☉), a = (r_E + r_M) / 2

a = 1.262 AU gives t ≈ 259 days.

SPEED ANYWHERE ON THE ELLIPSE (VIS-VIVA)

v² = μ_☉ (2/r − 1/a)

At Earth: 32.73 km/s.

CH.04 / 11 — FASTER COSTS MORE — MUCH MORE

Trade days for fuel

Push harder at departure.

Move the slider.

FARTHEST POINT OF TRANSFER ORBIT—

TRIP TIMEΔv FROM LOW EARTH ORBIT

black: daysred: departure Δv

Model: circular, co-planar orbits.

CH.05 / 11 — THE FUEL BILL

Most of what leaves Earth orbit is propellant

Reaching low Earth orbit already costs about 9.4 km/s.

At Mars you either brake with the atmosphere or fire engines.

DEPARTURE BURN FROM A CIRCULAR PARKING ORBIT

Δv = √(v_∞² + 2μ_⊕/r) − √(μ_⊕/r)

v∞ = speed left over after escaping Earth.

ROCKET EQUATION

m₀ / m_f = e^{Δv / (I_sp · g₀)}

See IMPULSE for its derivation.

PARKING ORBIT ALTITUDE300 km

TRIP (SET IN CHAPTER 4)—

ARRIVAL:

ENGINE:

WHAT LEAVES EARTH ORBIT, PER TONNE ARRIVING—

payload + structure (1 t)propellant

Engine Isp values are approximate.

CH.06 / 11 — RADIATION ON THE WAY

Outside Earth's shield, the dose is measured, not guessed

On Earth, the magnetic field stops almost all space radiation.

Curiosity carried a detector, RAD.

TRIP OUT (DAYS)180

DAYS ON MARS500

TRIP BACK (DAYS)180

TOTAL DOSE vs NASA CAREER LIMIT (600 mSv)—

in cruiseon Mars600 mSv career limit

Honest limits.

CH.07 / 11 — RADIOACTIVITY ABOARD: THE NUCLEAR BATTERY

Curiosity and Perseverance run on decaying plutonium

NASA's two biggest rovers carry an MMRTG.

Plutonium-238 decays by emitting alpha particles.

YEARS AFTER FUELING0

HEAT FROM Pu-238 DECAY100%

0%HALF-LIFE 87.7 YEARS100%

RADIOACTIVE DECAY

P(t) = P₀ · 2^{−t / 87.7 y}

Heat only.

CH.08 / 11 — TALKING ACROSS THE GAP

Nobody can joystick a Mars rover

Radio travels at the speed of light.

About every two years the Sun passes between the planets.

CH.09 / 11 — ARRIVING AT A DIFFERENT WORLD

Hitting the top of the air at 5–6 km/s

Falling into Mars's gravity well adds speed.

YOUR MASS (kg):

Values from NASA's planetary fact sheets.

CH.10 / 11 — LEGACY: THE TRIPS THAT WERE MADE

Sixty years of seven-month journeys

Cruise times of real missions.

“Mars is there, waiting to be reached.”

BUZZ ALDRIN

CH.11 / 11 — REFERENCES

Sources cited throughout this guide

Bracketed numbers in the text above correspond to the sources below.

TRANSFEREARTH TO MARS: DISTANCE, TIMING, FUEL AND RADIATION