# Space Travel Calculator

Created by Davide Borchia
Last updated: Sep 09, 2022

Humans are barely a spacefaring civilization, as we only entered our spatial neighborhood: our space travel calculator will answer the question "what if..."

• What if I board a ship that travels in space at constant acceleration?
• What if I can ignore the speed of light in calculating the travel time in space?
• What if Einstein was right (he is) and space travel is relativistic?

And much more.

## Traveling in space: an introduction

Traveling in space is a whole different kettle of fish. No air means no friction, the ideal rocket equation rules undisputed, and usually, your destination is not exactly behind the corner.

Spaceflight is hard: humanity ventured as far as the Moon (slightly beyond if you consider the orbits around it) and did so only six times between 1969 and 1972. Since then, we have only ventured into Earth's orbit. However, the push for exploration didn't make vane; we are limited by technology and physics!

In this tool, we will consider what would happen to a spaceship that travels in space at constant acceleration. The good news is that since there is no friction up there, we don't have to burn fuel to maintain a constant speed. If our engine is on, we are accelerating (in fact, most of the time spent in space by a craft consists of coasting, engines off, and patiently waiting to reach the time for a correction in the trajectory).

Input the spacecraft mass, your destination (trust us on the directions), and what you want to do precisely: a fly-by or a full stop (in this case, we will calculate your space travel in two parts, the latter at a constant deceleration that would bring you at destination with zero speed, à la Expanse).

🙋 Feel free to input a destination of your choice by inserting any distance in the proper variable's field.

The last choice before the departure: is your universe following the rules of Newton or Einstein? We'll see the differences in a second. Board the spaceship Calculator, buckle up and wait for the countdown.

🔎 To explain our space travel calculator, we will assume a constant $1g$ acceleration (the most comfortable for a human) and an empty spacecraft mass of $1.000\ \text{t}$. The destination we chose for our spaceship calculator is the center of our galaxy, a supermassive black hole $27,900$ light years away.

## Before Einstein: non-relativistic space travel

Gravity rules Newton's universe alone. There is no speed limit and no one of the weird relativistic effects we will meet shortly. We calculate your space travel using the equation for motion in a purely classic framework.

If you choose to arrive at your destination at the maximum speed possible, then we input your acceleration in space in the formula:

$v_{\text{f}} = a\cdot t$

Where:

• $a$ — The acceleration;
• $t$ — The time of flight; and
• $v_{\text{f}}$ — The final speed.

To calculate the time, we use the distance $d$:

$t = \sqrt{\frac{2\cdot d}{a}}$

If you plan on visiting Sagittarius A, then you need to decelerate. In this case, the final speed is $$v_{\text{ f}} = 0$$, obviously, and the time of flight changes accordingly:

$t = \sqrt{\frac{4\cdot d}{a}}$

The time required to travel such a distance is... astronomical. As you can see in our constant acceleration space travel calculator:

• For a maximum speed flyby, the time is $232.5\ \text{y}$; and
• To stop at destination, $328.8\ \text{y}$.

The maximum velocity in the first case is $240$ times the speed of light. If Einstein could hear this, he would be utterly disappointed. To right this wrong, we will calculate the travel time if the speed of light genuinely represent an impenetrable barrier.

## How to calculate the travel time: speed of light as ultimate speed limit

We enter the territory of relativistic effects. Relativistic space travel calculations are a bit more complicated. In layman's terms, the faster you go, the slower time passes for you, and the perceived length for you, the traveler, also reduces. These two effects, described by the theory of special relativity, are coded in two equations:

$T = \gamma \cdot t$

And:

$L = \frac{l}{\gamma}$

$\gamma$ is the Lorentz factor:

$\gamma = \frac{1}{\sqrt{1-\beta^2}} = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Where $\beta$ is the ratio, always smaller than $1$, between the spacecraft's speed and the light's speed.

## Travel in a relativistic spaceship: calculations for time and speed

To find the time required to reach a given destination in a universe ruled by Einstein's relativity theory, with constant acceleration in space, the formula we've seen before must be changed and split: time is relative, and because of this, the trip will have two durations.

For a maximum speed fly-by from the perspective of a stationary observer:

\begin{align*} t &= \frac{c}{a} \cdot \sinh{\left(\frac{a\cdot T}{c}\right)} \\ \\ &= \sqrt{\left(\frac{d}{c}\right)^2+\frac{2\cdot d}{a}} \end{align*}

The duration of the journey as experienced by our astronauts is:

\begin{align*} T& = \frac{c}{a} \cdot \sinh{}^{-1}\left(\frac{a\cdot t}{c}\right)\\ \\ &=\frac{c}{a} \cdot \cosh{}^{-1}\left(\frac{a\cdot d}{c^2} + 1\right) \end{align*}

In these equations, $d$ is the distance. In this relativistic framework, we calculate it with the formula:

\begin{align*} d& = \frac{c^2}{a} \cdot \left(\cosh{\left(\frac{a\cdot T}{c}\right)} -1\right)\\ \\ &=\frac{c^2}{a} \cdot \left(\sqrt{1+\left(\frac{a\cdot t}{c}\right)^2}-1\right) \end{align*}

Lastly, we can calculate the maximum velocity in relativistic space travel without deceleration:

\begin{align*} v& = c \cdot \tanh{\left(\frac{a\cdot T}{c}\right)}\\ \\ &=\frac{a\cdot t}{\sqrt{1+ \left(\frac{a\cdot t}{c}\right)^2}} \end{align*}

For a visit to Sagittarius A*, the times required for relativistic travel at constant $1g$ acceleration would be:

\begin{align*} T& = 10.62\ \text{y}\\ t&=27,\!901\ \text{y} \end{align*}

The difference is noticeable, to say the least. The maximum speed would be $0.4$ parts per billion smaller than the speed of light: the dilation effects would be extreme.

The formulas would change slightly if we wanted to stop at our destination. From the observer's point of view, the time passed is:

\begin{align*} t &=2\cdot \frac{c}{a} \cdot \sinh{\left(\frac{a\cdot T}{2\cdot c}\right)} \\ \\ &= \sqrt{\left(\frac{d}{c}\right)^2+\frac{4\cdot d}{a}} \end{align*}

In our example, $t=27,902\ \text{y}$. From the perspective of the travelers, the time is:

\begin{align*} T& = 2\cdot\frac{c}{a} \cdot \sinh{}^{-1}\left(\frac{a\cdot t}{2\cdot c}\right)\\ \\ &=2\cdot\frac{c}{a} \cdot \cosh{}^{-1}\left(\frac{a\cdot d}{2\cdot c^2} - 1\right) \end{align*}

Corresponding to $20\ \text{y}$. The perceived time is much longer than before: almost two times. This is because the astronauts would not "enjoy" a noticeable time dilation during the initial and final parts of the journey.

For distance and maximum velocity, we apply the following formulas:

\begin{align*} d& = \frac{c2\cdot ^2}{a} \cdot \left(\cosh{\left(\frac{a\cdot T}{2\cdot c}\right)} -1\right)\\ \\ &=\frac{2\cdot c^2}{a} \cdot \left(\sqrt{1+\left(\frac{a\cdot t}{2\cdot c}\right)^2}-1\right)\\ \\ v& = c \cdot \tanh{\left(\frac{a\cdot T}{2\cdot c}\right)}\\ \\ &=\frac{a\cdot t}{2\cdot \sqrt{1+ \left(\frac{a\cdot t}{2\cdot c}\right)^2}} \end{align*}

You can use our space travel calculator also to find the kinetic energy of an object moving at such speeds. You won't be surprised to learn that the kinetic energy of an object moving almost at the speed of light is astronomical.

## Fuel calculator for space travel: astronomical pit-stop

Rocketry is another word for mastery in fuel economy: you can learn everything about it with our rocket thrust calculator. Imagining an interstellar journey using chemical, ionic, or nuclear rockets is wishful thinking. To even have a shot to the stars, we need to learn how to control the mass to energy conversion. The annihilation reaction between matter and antimatter would have a perfect yield, converting all the mass involved into energy.

Assuming this $100\%$ efficiency, we can compute the required mass for our journey both in the classic and relativistic case:

\begin{align*} M_{\text{clas}} &= \frac{m\cdot v^2}{2\cdot c^2}+\frac{m\cdot v}{c}\\ \\ M_{\text{rel}} &= m\cdot \left(e^{\frac{a\cdot T}{c}}-1\right) \end{align*}

The results of these equations are disheartening: to send our ship to the center of our galaxy and stop there, the required fuel in the relativistic case is almost $830$ billion tons.

Will humans ever reach the star? Will Enterprises and Millenium Falcons cross the darkness between other Suns? With the technology of today, it's unlikely. But things change quickly, and what looks impossible today may be tomorrow's science. Be hopeful and keep dreaming about touching the sky.

Davide Borchia
Dreaming of traveling into space? 🌌 Plan your interstellar travel (even to a Star Trek destination) using this calculator 👨‍🚀! Estimate how fast you can reach your destination and how much fuel you would need 🚀.
Prepare for travel 👩‍🚀👨‍🚀
Spaceship acceleration
g
Spaceship mass
US ton
Destination
Star or galaxy
Which star/galaxy?
Select...
Distance
ly
Aim
Arrive at destination and stop
Calculations mode
Einstein's universe
5-4-3-2-1-0 booster ignition and liftoff! 🚀
Time passed in spaceship
yrs
Time passed on Earth
yrs
Required fuel mass
US ton
Maximum velocity
c
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