Relativistic Rocket Calculator

How long would it take to travel to cosmic destinations using a constantly accelerating rocket? Compute ship time, Earth time, mass ratio, and fuel requirements with real special-relativity physics.

Journey Parameters

Destination

Distance (light-years)

Flight Mode

Acceleration: 1g (9.81 m/s²)

Coast Phase: 0% of distance

Exhaust Velocity (v_e/c)

100% = photon drive (e.g. antimatter annihilation)

Ship Dry Mass

100.00 tonnes

Ship Time

3 years, 7 months

Earth Time

6 years

You'd age 3 years, 7 months while 6 years pass on Earth — a time dilation factor of 1.7×.

Journey Details

Distance:4.37 ly

Acceleration:1g (9.81 m/s²)

Mode:Brachistochrone

Peak Velocity:95.17% c

Peak Lorentz Factor (γ):3.26

Time Dilation Ratio:1.68×

Rocket Mass Ratio

Mass ratio40 : 1

Total launch mass4037 tonnes

Fuel required3937 tonnes

Energy (mc²)353.9 ZJ

610× global annual energy use

Feasibility

Launch mass = 10× the mass of the ISS

If water density: 9.9 m sphere radius

If tungsten density: 3.7 m sphere radius

All Destinations at 1g (Brachistochrone)

Destination	Distance	Ship Time	Earth Time	Peak Speed	γ
Proxima Centauri	4.25 ly	3 years, 7 months	5 years, 11 months	94.97%	3.2
Alpha Centauri	4.37 ly	3 years, 7 months	6 years	95.17%	3.3
Tau Ceti	11.9 ly	5 years, 2 months	13 years, 8 months	99.02%	7.1
40 Eridani	16.3 ly	5 years, 8 months	18 years, 2 months	99.43%	9.4
Vega	25 ly	6 years, 5 months	26 years, 10 months	99.74%	13.9
Betelgeuse	500 ly	12 years, 1 month	501 years, 11 months	100.00%	259
Orion Nebula	1,344 ly	14 years	1,346 years	100.00%	695
Sagittarius A* (Galactic Center)	26,000 ly	19 years, 9 months	26,003 years	100.00%	13,421
Far side of Milky Way	100,000 ly	22 years, 4 months	100,005 years	100.00%	51,617
Large Magellanic Cloud	160,000 ly	23 years, 3 months	160,006 years	100.00%	82,587
Andromeda Galaxy	2,500,000 ly	28 years, 7 months	2.5 million years	100.00%	1,290,555
Virgo Cluster	53,800,000 ly	34 years, 7 months	53.8 million years	100.00%	30,011,996
Edge of Observable Universe	46,500,000,000 ly	47 years, 8 months	46.5 billion years	100.00%	∞

About Relativistic Rocket Travel

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.

Key Equations

γ = cosh(aτ/c) — Lorentz factor from proper time
β = v/c = tanh(aτ/c) — velocity as fraction of c
Earth time (half-trip): t = √( (d/c)² + 2d/a )
Ship proper time (half-trip): τ = (c/a) · acosh(1 + ad/c²)
One-way mass ratio: R = γ(1 + β)
Brachistochrone mass ratio: R² (accel + decel)
Adjusted for exhaust velocity η: R^1/η
Fuel energy: E = m_fuel · c²

Flight Modes

Brachistochrone: Accelerate for the first half, flip the ship, then decelerate to arrive at rest. This is the minimum-time trajectory that stops at the destination.
Fly-by: Accelerate the entire way — arriving at enormous velocity. Useful for probes or if you have a separate braking mechanism.

Coast Phase

The coast phase allows you to cruise at peak velocity for a portion of the journey without burning fuel. This dramatically reduces fuel requirements at the cost of longer travel time. A 0% coast means constant thrust throughout.

Based on the Relativistic Rocket FAQ by John Baez.