Constant-acceleration space travel
Brachistochrone Trajectory Calculator
Calculate flip-and-burn travel time, midpoint velocity, and ship time from distance and acceleration.
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Ready
Journey result
Total observer time
2 days 1 hour
Earth → Mars at 1g
Orbital-radius estimate
Accelerate halfway, flip, then decelerate.
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Trajectory profile
The calculation model
How the flip-and-burn trajectory works
The calculator models a straight-line journey that starts and ends at rest in the external observer frame.
Accelerate
The spacecraft maintains the selected proper acceleration for the first half of the distance.
Flip
At the midpoint, the spacecraft rotates and reaches the highest velocity of the journey.
Decelerate
Equal acceleration in the opposite direction brings the spacecraft to rest at the destination.
Relativistic calculation
Formula and output
The calculator uses constant proper acceleration and automatically compares the relativistic result with the Newtonian approximation.
For total distance d, proper acceleration a, and light speed c:
q = ad / 2c²
t = 2(c/a)√((1 + q)² − 1)
τ = 2(c/a) acosh(1 + q)
t is observer time and τ is elapsed ship time. The midpoint peak velocity remains below the speed of light.
Practical output
Use the result in a complete journey plan
Science-fiction writing
Build consistent travel timelines for custom star systems and compare ships with different acceleration limits.
Games and simulations
Estimate the midpoint flip and braking phase for idealized continuous-thrust routes.
Physics learning
Compare Newtonian travel time with observer time, ship time, and relativistic peak velocity.
Questions
Brachistochrone trajectory FAQ
What is a brachistochrone trajectory in space travel?
In this calculator, it describes a continuous-thrust trip that accelerates for half the distance, flips at the midpoint, and decelerates for the remaining half.
What distance do the planet presets use?
Planet presets use the difference between average orbital radii. The custom-distance mode lets you enter a route distance from another source.
What is the difference between observer time and ship time?
Observer time is measured in the departure and destination reference frame. Ship time is the proper time experienced onboard during the journey.
Does the calculator include a coasting phase?
The selected profile applies acceleration throughout the journey: positive acceleration for the first half and equal deceleration for the second half.