Relativistic Rocket Calculator

How long would it take to travel to cosmic destinations using a constantly accelerating rocket? (Like in Project Hail Mary.)

Journey Parameters

g
%

100% = photon drive (eg. Spin Drive)

%
kg

100.0 tonnes

Journey Results

Ship Time

3 years, 11 months

You'd age 3 years, 11 months while 13 years, 2 months pass on Earth — a time dilation factor of 3.4×.

Earth Time

13 years, 2 months

You'd age 3 years, 11 months while 13 years, 2 months pass on Earth — a time dilation factor of 3.4×.

Journey Details

  • Distance:11.9 ly
  • Acceleration:1.5g (14.71 m/s²)
  • Mode:Brachistochrone
  • Peak Velocity:99.52% c
  • Peak Lorentz Factor (γ):10
  • Time Dilation Ratio:3.37×

Rocket Mass Ratio

  • Mass ratio:416 : 1
  • Total launch mass:41.59 thousand tonnes
  • Fuel required:41.49 thousand tonnes
  • Energy (mc²):3.7 YJ
  • 6,429× global annual energy use

Feasibility

Launch mass = 99× the mass of the ISS

If water density: 21.5 m — building-sized

If tungsten density: 8.0 m — building-sized

If neutronium density: 291.7 μm — marble-sized

Schwarzschild radius: 0.1 am — smaller than a proton

v0.5Source Code

This calculator uses the hyperbolic motion equations from special relativity to compute proper (ship) time and coordinate (Earth) time for a spacecraft maintaining constant proper acceleration.

The travel-time side of the calculator answers the pure relativity question: if a ship can maintain a chosen proper acceleration profile, how much ship time and Earth time pass over the journey? Those numbers come from special relativity directly.

The mass-ratio and fuel estimates below are a separate idealized step based on the relativistic rocket equation and a user-selected effective exhaust velocity:

γ = cosh(aτ)    // Lorentz factor

β = v/c = tanh(aτ)    // velocity as fraction of c

R = γ(1 + β)    // one-way mass ratio

Rtotal = R²    // brachistochrone (flip & decelerate)

Radj = R1/η    // adjusted for exhaust velocity efficiency η

Efuel = mfuel · c²    // fuel energy (perfect conversion)

That makes this a reasonable first-pass model for something like Project Hail Mary if you assume spent reaction mass is discarded, but it does not model tankage, leftover inert residue kept onboard, waste heat, thrust limits, or other engine-specific losses. At 1g, a human-comfortable acceleration, you can reach the Andromeda Galaxy in about 28 years of ship time — while 2.5 million years pass on Earth.

Based on the Relativistic Rocket FAQ by John Baez, and Ask a Mathematician. if you find any mistakes.