Space Travel Calculator
Enter a distance and spacecraft acceleration to see how long a journey takes from both the crew's perspective (ship time) and Earth's perspective (Earth time). Einstein's special relativity means the two diverge dramatically at high speeds: a trip to Andromeda that takes millions of years on Earth might last only decades aboard ship. The calculator covers both arrive-at-speed and arrive-and-stop modes, shows max velocity as a fraction of light speed, computes the relativistic Lorentz factor, and walks through every step of the math.
Formula
Worked example
Journey to Proxima Centauri (4.24 ly) at 1 g acceleration, arrive and stop. alpha_half = (9.807 x 4.24 x 9.461e15) / (2 x (3e8)^2) = 2.177. Ship time per half = (c/a) x arccosh(3.177) = 3.579 years, total 7.16 years. Earth time per half = (c/a) x sinh(arccosh(3.177)) = 5.75 years, total 11.5 years. Peak velocity = c x tanh(arccosh(3.177)) = 0.948 c (94.8 % of light speed). Lorentz factor at peak = cosh(arccosh(3.177)) = 3.177.
What is the space travel calculator?
This calculator applies Einstein's special relativity to a spacecraft travelling under constant proper acceleration. It answers two distinct questions: how long does the journey feel to the crew, and how long does it take from Earth's perspective. Because nothing with mass can reach the speed of light, the relationship between these two times becomes increasingly non-linear as you push the acceleration longer.
The calculator covers two travel modes: arrive and stop, where the ship accelerates to the midpoint then flips to decelerate, and flyby, where it accelerates the whole way and passes the destination at peak speed. Arrive-and-stop requires roughly twice the proper-time but produces far higher fuel demands because the ship must carry propellant for the deceleration phase as well.
The relativistic rocket equations explained
Classical Newtonian physics uses distance = 0.5 x a x t^2 and velocity = a x t. These break down once velocity becomes a significant fraction of the speed of light (roughly above 10 % of c). The correct relativistic equations replace the naive linear growth with hyperbolic functions that enforce the speed-of-light ceiling.
For a constant-acceleration flyby over distance d:
- Ship time: T = (c/a) x arccosh(a*d/c^2 + 1)
- Earth time: t = (c/a) x sinh(a*T/c)
- Peak velocity: v = c x tanh(a*T/c)
For arrive-and-stop, replace d with d/2 in each half, then double. The hyperbolic arccosh and sinh ensure velocity never exceeds c regardless of how long or hard you accelerate.
The Lorentz factor gamma = 1 / sqrt(1 - v^2/c^2) tells you how much clocks slow down aboard the ship relative to Earth. At 90 % of c, gamma is about 2.3, meaning Earth ages 2.3 years for every 1 year the crew experiences. At 99.9 % of c, gamma is roughly 22.
Fuel requirements and why interstellar travel is hard
The fuel mass calculation here assumes a perfect photon rocket: 100 % conversion of propellant mass to directed photon thrust. This is the theoretical best case, far beyond any actual technology. Even with this assumption, the Tsiolkovsky-Lorentz equation shows that fuel requirements grow exponentially with desired velocity:
M_fuel / m = exp(a*T/c) - 1
For a 1 g trip to Proxima Centauri (arrive and stop), this ratio is about 37: 37 kg of antimatter fuel per kg of dry ship mass. The entire International Space Station masses about 420,000 kg, so its equivalent trip would need roughly 15 million tonnes of antimatter, compared with Earth's current total production of a few nanograms per year.
Longer journeys become astronomically demanding. A 1 g trip to the Galactic Centre (26,000 ly) requires a fuel ratio of roughly 10^11. This is why most serious interstellar-travel concepts focus on lower accelerations, laser sails that carry no propellant at all (Breakthrough Starshot), or multi-generation generation ships that simply accept long Earth-frame travel times.
The twin paradox and what these numbers mean
These results are a worked instance of the famous "twin paradox." If one twin boards a spacecraft at 1 g to Andromeda and returns, they age about 56 years. The twin left on Earth ages roughly 5 million years - and civilization, as we know it today, would almost certainly no longer exist upon the traveller's return.
There is no paradox in the physics: the travelling twin's path through spacetime is genuinely shorter (in proper time) because their worldline curves through the velocity dimension. The asymmetry arises because only the ship-twin accelerates; acceleration breaks the symmetry between the two frames.
Length contraction accompanies time dilation. At 94.8 % of c (Proxima Centauri trip peak), the 4.24 light-year distance appears Lorentz-contracted to only 1.37 ly from the crew's perspective, which is why their journey time is shorter than 4.24 years even when travelling at nearly c.
Practical caveats: this calculator ignores gravitational fields (which add general-relativistic corrections), interstellar medium collisions (a serious concern at relativistic speeds), and the assumption of continuous thrust. Real mission designs would need detailed trajectory optimization.
Relativistic travel times to famous destinations at 1 g constant acceleration (arrive and stop)
| Destination | Distance (ly) | Ship time | Earth time | Peak velocity |
|---|---|---|---|---|
| Moon | 4.06e-8 | 3.5 hours | 3.5 hours | 0.02 % c |
| Mars (avg) | 2.38e-5 | 3.5 days | 3.5 days | 0.50 % c |
| Jupiter (avg) | 8.22e-5 | 6.5 days | 6.5 days | 0.92 % c |
| Pluto (avg) | 6.24e-4 | 18.0 days | 18.0 days | 2.54 % c |
| Proxima Centauri | 4.24 | 3.54 years | 5.87 years | 94.95 % c |
| Alpha Centauri A | 4.37 | 3.58 years | 6.00 years | 95.17 % c |
| Barnard's Star | 5.96 | 4.04 years | 7.66 years | 96.94 % c |
| Sirius | 8.60 | 4.61 years | 10.36 years | 98.30 % c |
| Vega | 25.05 | 6.44 years | 26.92 years | 99.74 % c |
| Pleiades | 444.00 | 11.88 years | 445.95 years | 100.00 % c |
| Galactic Centre | 26000.00 | 19.76 years | 26003 years | 100.00 % c |
| Andromeda Galaxy | 2537000.00 | 28.63 years | 2.54 million years | 100.00 % c |
Computed using the relativistic rocket equations. Ship time experienced by the crew; Earth time elapsed at origin. All values assume 100% fuel efficiency.
Frequently asked questions
How long would it take to reach Proxima Centauri at 1 g?
At 1 g constant acceleration with arrive-and-stop mode, the crew would experience about 7.2 years of ship time while about 11.5 years pass on Earth. The spacecraft would reach a peak velocity of roughly 94.8 % of the speed of light at the midpoint (2.12 ly), then decelerate symmetrically. These numbers come directly from the relativistic hyperbolic-motion equations, not the classical d = 0.5at^2.
What is the difference between ship time and Earth time?
Ship time (proper time) is the time measured by a clock aboard the spacecraft, experienced by the crew. Earth time (coordinate time) is the time elapsed on Earth or in the original rest frame during the same journey. Special relativity predicts that a moving clock runs slow relative to a stationary one; this is time dilation. The faster the ship travels, the slower its clock runs, so the crew ages less than observers back home.
Why do I need so much fuel for interstellar travel?
The relativistic version of the rocket equation (Tsiolkovsky-Lorentz) shows that fuel requirements grow exponentially with the final velocity. Even with a 100 % efficient antimatter engine - the theoretical maximum - reaching 95 % of light speed requires roughly 37 kg of fuel per kg of dry ship mass. For a 10-tonne spacecraft, that is 370 tonnes of antimatter; for a crewed mission with life support, the numbers become staggering. Alternative concepts like solar sails or laser propulsion avoid carrying fuel by pushing the craft with external energy.
What does the Lorentz factor (gamma) mean?
The Lorentz factor gamma = 1 / sqrt(1 - v^2/c^2) is the ratio by which time, length, and relativistic mass scale at velocity v. A gamma of 3 means clocks aboard the ship tick three times slower than Earth clocks, and the ship appears three times shorter in the direction of motion to an outside observer. At everyday speeds gamma is essentially 1; at 90 % of c it is about 2.3; at 99.9 % of c it is roughly 22; at 99.9999 % of c it exceeds 700.
What is the difference between arrive-and-stop and flyby mode?
In arrive-and-stop mode the spacecraft accelerates to the midpoint of the journey, then reverses thrust to decelerate and arrive at rest at the destination. This allows orbiting or landing but requires much more fuel because the ship must carry propellant for both the deceleration phase and, if it carries that fuel from the start, for the whole first half as well. In flyby mode the ship accelerates the entire way and arrives at maximum velocity. Flyby is faster (shorter ship time for the same distance) and less fuel-hungry, but the ship blasts past the destination at near-light speed with no way to stop.
Could you really travel to another galaxy in a human lifetime?
Mathematically, yes. At 1 g constant acceleration, a crew would reach the Andromeda Galaxy (2.537 million ly) in about 28 years of ship time, while 2.5 million years pass on Earth. The barrier is engineering, not physics: carrying enough fuel is currently impossible, and interstellar dust at near-light speed would deliver enormous radiation doses to the hull. But the relativistic time compression is real - it is exactly what the equations predict.
Does this calculator account for general relativity?
No. The equations used here are from special relativity and assume flat spacetime with no significant gravitational fields. General relativistic corrections become important near massive objects (stars, black holes) or over cosmological distances where spacetime curvature and cosmic expansion matter. For interstellar travel at moderate distances the special-relativistic treatment is an excellent approximation.