Space travel calculator

Do you want to travel to another planet? Or perhaps even another star system?

Then you can use this calculator to work out how long it will take you, how much energy your spacecraft needs and what your maximum velocity will be. If you travel close to the speed of light, you can also see how much time it will take from your point of view and from the point of view of the people on earth. You can also see how the length of your spacecraft will shorten for observers watching it from earth, if only they had powerful enough telescopes.

This is the simplest way to use the space travel calculator:

Enter a distance to a planet or star. Don't know any? Then type Pr and press the down arrow. The distance to Proxima Centauri appears. Select it and the distance will be filled in. Try other places in space.
Click Calculate. The calculator determines the remaining unfilled values.
Click Run. Watch the space rocket travel from earth to your destination. Also watch the clocks of the observer and the traveler.

Known problems

The animation spacecraft is at a different scale to the distance between the observer and destination. Even for the shortest space travel distances, for example the earth to the moon, the spacecraft would occupy less than a pixel. This problem will not be fixed.
As an object moves further into the distance it appears smaller to an observer. This change in perspective distance is not represented in the animation. The reduction in the spacecraft length from the observer's framework at velocities approaching the speed of light is an entirely different concept to perspective distance.
If you set the iterations on the animation to a low number, e.g. less than 20, the animation's spaceship time will not be calculated accurately if the observer and traveler times diverge substantially.
The code is old and the user interface needs to be refreshed. (Also the PHP component is overkill and was only used for learning purposes.) You're encouraged to improve the code and place the travel calculator on your own website; it's FLOSS.

A bug fix was made in June 2016. The calculation for the fuel needed for the trip did not take into account conservation of momentum. These two webpages helped me correct the error and I am grateful to the various people contributed the notes that helped me fix this (Physics Stack Exchange users user2096078, Qmechanic and udrv, Don Koks for the Relativistic Rocket, and John F who emailed me) :

This is the distance from earth to your destination. Either enter a value or search the database for a distance to a space object by typing the first few letters of its name. All objects in the database matching that start with the letters you have typed will appear. Select the one you want. Distances are approximate because the planets' positions change continuosly relative to the earth. If you leave distance blank, it will be calculated --if you enter the observer time elapsed and the traveler's maximum velocity-- using this equation:

\frac{cvt}{c + \sqrt{(c^{2} - v^{2})}}

   where
    c = the speed of light,
    v = maximum velocity,
    t = time elapsed in observer timeframe.

Source: Most Direct Derivation of Relativistic Constant Acceleration Distance Formula

This is the constant acceleration of the traveler's spacecraft. Half way through the journey, the spacecraft starts decelerating at the same rate.

If you leave the acceleration blank, it will be calculated using Newton's laws of motion (depending on which fields have values):

a = \frac{s}{t^{2} / 4}

a = \frac{v^{2}}{s}

a = \frac{2v}{t}

   where
    s = distance,
    v = maximum velocity and
    t = time elapsed in observer timeframe

This is increasingly inaccurate as you approach the speed of light, so for large distances, such as to the nearest stars, it is better to enter the acceleration manually.

If a spacecraft accelerates constantly at 1g --or 9.8m/s-- the travelers on board can experience earth-like gravity. Unfortunately interstellar travel at this acceleration will likely never be achieved because of the huge amount of energy required. It is not possible to travel to the nearest stars at this acceleration if the fuel must be carried onboard the spacecraft, no matter what kind of fuel is used.

This is the maximum velocity the spacecraft will reach, from the perspective of an observer on earth. This occurs when the spacecraft is half way to its destination. This is calculated using this equation:

\frac{c}{\sqrt{1 + \frac{c^{2}}{a^{2} {(\frac{T}{2})}^{2}}}}

   where
    c = speed of light,
    a = acceleration and
    t = time elapsed to end of journey in observer timeframe.

Source: The Sky This Week.

This is the time elapsed for the whole journey from the observer on earth's time frame. This is calculated using this equation:

t = 2 \sqrt{\frac{c^{2}}{a^{2}} [{(\frac{d}{2 c^{2} / a} + 1)}^{2} - 1]}

   where
    c = speed of light,
    d = distance of the journey and
    a = acceleration.

Source: The Sky This Week.

This is the time elapsed for the whole journey from the perspective of the spacecraft. This is calculated using this equation:

t = \frac{2c}{a} \cosh^{- 1} [\frac{d}{2 c^{2} / a} + 1]

   where
    c = speed of light,
    d = distance of the journey and
    a = acceleration.

Source: The Sky This Week.

This is the amount of energy your spacecraft's payload will need to constantly accelerate to half way to your destination and then decelerate at the same rate until you reach your destination. This is calculated using this equation:

e = {2mc}^{2} [\frac{1}{\sqrt{(1 - v^{2} / c^{2})}} - 1]

   where
    c = speed of light,
    v = maximum velocity and
    m = spacecraft mass.

Source: The Sky This Week.

This is the mass of the fuel needed to for your journey. This is calculated using this equation:

\frac{M}{m} = \frac{2 \frac{v}{c}}{1 - \frac{v}{c}}

v = maximum velocity and
c = speed of light.

Source: The Relativistic Rocket and Physics Stack Exchange. (Thanks to users user2096078, Qmechanic and udrv. Also thanks to John F for informing me of a bug that has now hopefully been corrected.)

This is the length of the spacecraft at the beginning of the journey. Note that the spacecraft length always stays the same for the people in it. This is calculated using this equation:

l_{0} = \frac{l}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}

   where
    l = length of traveler from observer's perspective,
    v = maximum velocity of traveler and
    c = speed of light.

Source: Hyperphysics.

This is the length of the spacecraft from the observer on earth's perspective. Of course spacecrafts are small, so it would be impossible to see a spacecraft from earth on an interstellar voyage. This is calculated using this equation:

l = l_{0} \sqrt{1 - \frac{v^{2}}{c^{2}}}

   where
    l₀ = original length of spacecraft on earth,
    v = maximum velocity of traveler and
    c = speed of light.

Source: Hyperphysics.