Reference · Mission Constants

Mission Constants

The single source of truth for every physical constant and unit used throughout the Mars mission simulator.

Orbital mechanics relies on well-defined physical constants. By storing these values in a single shared location, every calculation, from orbital velocity to Hohmann transfer trajectories, uses consistent and scientifically accurate data.

Solar Constantsi

μ☉m³/s²

Sun Gravitational Parameter

1.327 × 10²⁰m³/s²

Determines the strength of the Sun’s gravity. Used when calculating orbital velocity and transfer trajectories.

Earthi

r⊕m

Earth Orbit Radius

1.496 × 10¹¹m

149,597,871 km

The average distance between Earth and the Sun, the starting orbit of the mission (1 AU).

Marsi

r♂m

Mars Orbit Radius

2.279 × 10¹¹m

227,939,200 km

The average distance between Mars and the Sun, the destination orbit of the mission (≈ 1.524 AU).

Unitsi

AUm

Astronomical Unit

1.496 × 10¹¹m

149,597,871 km

The standard unit of distance used throughout the Solar System, the average Earth–Sun distance.

Interactive · Orbital Mechanics

Hohmann Transfer Calculator

A step-by-step Earth-to-Mars transfer, built from the shared constants. Adjust the orbit radii to see how the semi-major axis, transfer time, and Delta-v respond.

STEP 1

Mission Inputs

The transfer depends on the Sun's gravity and the two orbit radii. The Sun's gravitational parameter is fixed; the radii are adjustable so you can experiment with different destinations.

μ☉fixed

Sun grav. parameter

1.327 × 10²⁰ m³/s²

r₁1.000 AU

Departure orbit · Earth

149,597,871 km

r₂1.524 AU

Arrival orbit · Mars

227,987,155 km

Spacecraft assumptions

Dry masskg

Fuel masskg

Exhaust velocitym/s

Presets

Live response

How the mission changes with your assumptions

Δv needed · transfer5.60 km/s baseline

Time taken · transfer259 days baseline

Departure velocity32.7 km/s baseline

Arrival velocity21.5 km/s baseline

Available Δv · rocket8.06 km/s baseline

Δv margin · spare+2.47 km/s enough

Fuel required · propellant12,340 kg baseline

Δv needed, time and velocities are set by the orbits. Available Δv, margin and propellant mass depend on the spacecraft.

Fuel efficiency

Why this route uses the least fuel

Total Delta-v is the fuel metric. The chart plots every faster route to the same Mars orbit, and each one costs more Delta-v. The Hohmann transfer between these two radii sits right at the minimum.

Chosen route · Hohmann

5.596 km/s

✓Lowest-fuel option

Drag the radii above to watch the curve and its minimum shift.

STEP 2

The Transfer Orbit

A Hohmann transferis the most fuel-efficient two-impulse manoeuvre between two circular orbits. The spacecraft rides an elliptical orbit that touches Earth's orbit at perihelion and Mars's orbit at aphelion.

Just two burns are needed: one to leave the inner orbit, and one to settle into the outer orbit on arrival.

STEP 3

Orbital Calculations

Each calculation is shown in three steps: the general formula, your current values substituted in, and the final answer. They update live as you change the inputs above. Tap a card to collapse it.

Here r₁is Earth's orbital radius and r₂is Mars's. Averaging them gives aₜ, the semi-major axis of the transfer orbit.

Formula

a_t = \dfrac{r_1 + r_2}{2}

Substituted

a_t = \dfrac{1.496\times10^{8} + 2.280\times10^{8}}{2}\ \text{km}

Answer

a_t = 1.888\times10^{8}\ \text{km} \;=\; 1.262\ \text{AU}

Using the semi-major axis aₜand the Sun's gravitational parameter μ, this gives half of the transfer orbit's period, giving the time to coast from Earth to Mars.

Formula

t = \pi\sqrt{\dfrac{a_t^{3}}{\mu}}

Substituted

t = \pi\sqrt{\dfrac{(1.888\times10^{11})^{3}}{1.327\times10^{20}}}

Answer

t = 2.237\times10^{7}\ \text{s} \;=\; 258.9\ \text{days}

The velocity change needed to leave Earth's circular orbit and enter the transfer orbit.

Formula

\Delta v_1 = \sqrt{\dfrac{\mu}{r_1}}\left(\sqrt{\dfrac{2r_2}{r_1+r_2}}-1\right)

Substituted

\Delta v_1 = \sqrt{\dfrac{1.327\times10^{20}}{1.496\times10^{11}}}\left(\sqrt{\dfrac{2(2.280\times10^{11})}{1.496\times10^{11}+2.280\times10^{11}}}-1\right)

Answer

\Delta v_1 = 2.946\ \text{km/s}

The velocity change needed to match Mars's circular orbital speed once the craft reaches its orbit.

Formula

\Delta v_2 = \sqrt{\dfrac{\mu}{r_2}}\left(1-\sqrt{\dfrac{2r_1}{r_1+r_2}}\right)

Substituted

\Delta v_2 = \sqrt{\dfrac{1.327\times10^{20}}{2.280\times10^{11}}}\left(1-\sqrt{\dfrac{2(1.496\times10^{11})}{1.496\times10^{11}+2.280\times10^{11}}}\right)

Answer

\Delta v_2 = 2.650\ \text{km/s}

Formula

\Delta v_{\text{total}} = |\Delta v_1| + |\Delta v_2|

Substituted

\Delta v_{\text{total}} = |2.946| + |2.650|

Answer

\Delta v_{\text{total}} = 5.596\ \text{km/s}

This is a simplified heliocentric estimate. It does not include launch from Earth's surface, atmospheric drag, course corrections, or landing on Mars. A real mission needs more than this.

STEP 4

Mission Results

✓

Mission engineering status

Feasible

Available Δv

8.063 km/s

Required Δv

5.596 km/s

Surplus

+2.47 km/s · +44%

The spacecraft has more available delta-v than the transfer requires, including a useful safety margin. Based on this simplified calculation, the mission is feasible.

Available 8.06 km/s vs required 5.60 km/s, a surplus of 2.47 km/s (+44% margin).

akm

Semi-major axis

188,792,513

Average radius of the transfer ellipse.

tdays

Transfer time

258.9

Time to travel from Earth's orbit to Mars's orbit.

Δv₁km/s

Departure Delta-v

2.946

Velocity increase to leave Earth's orbit.

Δv₂km/s

Arrival Delta-v

2.650

Velocity adjustment when reaching Mars.

◆ Primary mission metric

Total Delta-v

Overall propulsion requirement for the transfer.

5.596 km/s

m₀kg

Initial mass

30,000

Fuelled spacecraft at launch (dry + fuel).

m_fkg

Final mass

5,000

Dry mass once all fuel is spent.

Δvkm/s

Available Δv · Tsiolkovsky

8.063

Δv = vₑ · ln(m₀ / m_f), the rocket's own capability.

1 · Formula

\Delta v = v_e \cdot \ln\!\left(\dfrac{m_0}{m_f}\right)

2 · Substituted

\Delta v = 4,500\ \cdot \ln\!\left(\dfrac{30,000}{5,000}\right)

3 · Answer

\Delta v = 4,500 \cdot \ln(6.000) = 8,063\ \text{m/s} \;=\; 8.063\ \text{km/s}

Adjust the values

m₀ · Initial mass30,000 kg

m_f · Dry mass5,000 kg

vₑ · Exhaust velocity4,500 m/s

✓

Mission Feasible

8.063 available vs 5.596 required km/s · +2.47 km/s (+44%)

v_eExhaust velocity: how fast the engine throws its propellant. A higher value means a more efficient engine.

m₀Initial mass: the fuelled spacecraft at launch (dry mass plus fuel mass).

m_fFinal mass: the spacecraft once all the fuel is spent (the dry mass).

lnThe natural logarithm of the mass ratio. Doubling the fuel does not double the Δv, which is why big missions need so much propellant.

Required vs available delta-v

Surplus: +2.47 km/s

Required · Hohmann transfer5.596 km/s

Available · this spacecraft8.063 km/s

✓

Sufficient delta-v

Margin of 2.47 km/s against the 5.60 km/s required.

In plain English

The rocket equation estimates how much the spacecraft can change its speed using the fuel it carries. A spacecraft with more fuel or a more efficient engine can usually produce more delta-v. For the mission to be feasible, the available delta-v must be equal to or greater than the Hohmann transfer requirement.

The spacecraft carries a healthy propulsion margin. Any surplus can be reserved for course corrections and orbit insertion.

STEP 5

Validation

Expected · Earth → Mars

Transfer time≈ 259 days

Total Delta-v≈ 5–6 km/s

This calculator

Transfer time258.9 days

Total Delta-v5.596 km/s

✓Matches expected range

Slight differences occur because the model treats orbits as circular and coplanar and ignores planetary gravity.

Engineering Trade-offs

Every choice is a balance

Designing a Mars mission is not one correct answer. It is a set of compromises. These cards read from the values above, so they update as you change the mission.

Fuel Mass

More fuel means more delta-v, but a heavier spacecraft. The benefit shrinks each time, because the rocket must also accelerate the extra fuel.

Fuel mass

25,000 kg

Fuel : dry

5.00 : 1

Wet mass

30,000 kg

Available Δv

8.06 km/s

More fuel raises Δv, but also makes the spacecraft heavier.

Engine Efficiency

Exhaust velocity sets how well fuel becomes delta-v. A more efficient engine does the same job with less fuel, but often gives less thrust.

Exhaust vel.

4,500 m/s

Specific impulse

459 s

Typical chemical

A more efficient engine cuts fuel, but may give less thrust.

Transfer Time

A Hohmann transfer saves fuel but is not the fastest route. A quicker trip needs more delta-v; a slower one exposes the crew to radiation and wear for longer.

Transfer time

259 days

Required Δv

5.60 km/s

A fuel-efficient transfer takes longer, raising crew exposure.

✓

Mission Risk

Fuel margin, transfer time, engine capability and mass all feed into overall risk. A mission can be possible yet still risky if the margin is thin.

Feasible+2.47 km/s (+44%)

A comfortable Δv margin absorbs navigation errors and unplanned burns, so the mission carries acceptable risk.

Engineering note

These constants live in a dedicated constants.ts module. Every constant is exported individually with a descriptive name and documented with JSDoc comments including its units.

Orbital calculations, transfer-orbit algorithms, Delta-v maths and UI components all import from this one file, so the simulator and this page never drift out of sync.

constants.ts

/** Sun gravity μ☉  @unit m³/s² */
export const SUN_GRAVITATIONAL_PARAMETER = 1.327e20;

/** Earth orbit radius  @unit m */
export const EARTH_ORBIT_RADIUS_M = 1.496e11;

/** Mars orbit radius  @unit m */
export const MARS_ORBIT_RADIUS_M = 2.279e11;

/** Astronomical Unit  @unit m */
export const ASTRONOMICAL_UNIT_M = 1.496e11;