Space Travel Calculator | Relativistic Rocket Equation

The space travel calculator is a comprehensive tool that allows you to estimate many essential parameters in theoretical interstellar space travel. Have you ever wondered how fast can we travel in space, how much time will it take to get to the nearest star or galaxy, or how much fuel does it require? In the following article, we'll try to answer questions is interstellar travel possible? and can humans travel at the speed of light? using a relativistic rocket equation. Explore the world of light speed travel of (hopefully) future spaceships with our relativistic space travel calculator!

If you're interested in astrophysics, check out our other calculators. Find out the speed required to leave the surface of any planet with the escape velocity calculator or estimate the parameters of the orbital motion of planets using the orbital velocity calculator.

Interstellar space is a rather empty place. Its temperature is not much more than the coldest possible temperature, i.e., an absolute zero. It equals about 3 Kelvins - minus 270 C or minus 455 F. You can't find air there, and therefore there is no drag or friction. On one hand, humans can't survive in such a hostile place without expensive equipment like a spacesuit or a spaceship, but on the other hand, we can make use of space conditions and its emptiness.

The main advantage of future spaceships is that, since they are moving through a vacuum, they can theoretically accelerate to infinite speeds! However, this is only possible in the classical world of relatively low speeds where Newtonian physics can be applied. Even if it's true, let's imagine, just for a moment, that we live in a world where any speed is allowed. How long will it take to visit the Andromeda Galaxy, the nearest galaxy to the Milky Way?

We will begin our intergalactic travel with a constant acceleration of 1 g (9.81 m/s or 32.17 ft/s) because it ensures that crew experiences the same comfortable gravitational field as the one on Earth. By using this space travel calculator in Newton's universe mode, you can find out that you need about 2200 years to arrive at the nearest galaxy! And, if you want to stop there, you need an additional 1000 years. Nobody lives for 3000 years! Is intergalactic travel impossible for us, then? Luckily, we have good news. We live in a world of relativistic effects where unusual phenomena readily occur.

In the previous example, where we traveled to Andromeda Galaxy, the maximum velocity was almost 3000 times greater than the speed of light c = 299,792,458 m/s, or about c = 3 * 10 m/s using scientific notation. You can always use our speed converter to find its value in any other speed units.

However, as velocity increases, relativistic effects start to play an essential role. According to special relativity proposed by Albert Einstein, nothing can exceed the speed of light. How can it help us with interstellar space travel? Doesn't it mean we will travel at a much lower speed? Yes, it does, but there are also few new relativistic phenomena, including time dilation and length contraction to name a few. The former is crucial in relativistic space travel. Time dilation is a difference of time measured by two observers, one being in motion and second at rest (relative to each other). It is something we are not used to on Earth. Clocks in a moving spaceship tick slower than the same clocks on Earth! Time passing in a moving spaceship T and equivalent time observed on Earth t are related by the following formula:

T = * t,

where is the Lorentz factor that comprises the speed of the spaceship v and speed of light c:

= 1/(1 - ) = 1/(1 - v/c),

where = v/c.

For example, if = 10 (v = 0.995c), then every second passing on Earth corresponds to ten seconds passing in the spaceship. Inside the spaceship, events take place 90 percent slower; the difference can be even greater for higher velocities. Note that both observers can be in motion, too. In that case, to calculate the relative relativistic velocity, you can use our velocity addition calculator.

Let's go back to our example again, but this time we're in Einstein's universe of relativistic effects trying to reach Andromeda. The time needed to get there measured by the crew of the spaceship equals only 15 years! Well, this is still a long time, but is more achievable in a practical sense. If you would like to stop at the destination, you should start decelerating halfway through. In this situation, the time passed in the spaceship will be extended by about 13 additional years.

Unfortunately, this is only a one-way journey. You can, of course, go back to Earth but nothing will be the same. During your interstellar space travel to the Andromeda Galaxy, about 2,500,000 years have passed on Earth. It would be a completely different planet, and nobody can foresee the fate of our civilization. A similar problem was considered in the first Planet of the Apes movie, where astronauts crash landed back on Earth. While these astronauts had only aged by 18 months, 2000 years had passed on Earth (sorry for the spoilers, but the film is over 50 years old at this point, you should have seen it by now). How about you? Would you be able to leave everything you know and love about our galaxy forever, and begin a life of space exploration?

Now that you know whether interstellar is travel possible and how fast can we travel in space, it's time for some formulas. In this section, you can find the "classical" and relativistic rocket equations that are included in the relativistic space travel calculator. There could be four combinations since we want to estimate how long it takes to arrive at the destination point at full speed as well as arrive at the destination point and stop. Every set contains distance, time passing on Earth and in the spaceship (only relativity approach), expected maximum velocity and corresponding kinetic energy (if you turn on the advanced mode), and the required fuel mass (see Intergalactic travel - fuel problem section for more information). The notation is:

Relativistic space travel calculator is dedicated to very long journeys, interstellar or even intergalactic, in which we can neglect the influence of the gravitational field, e.g., from Earth. We didn't include in destination list our closest celestial bodies like Moon or Mars, because it would be pointless. For them, we need different equations that also take into consideration gravitational force.

Newton's universe - arrive at destination at full speed

It's the simplest case because here T equals t for any speed. To calculate distance covered, at constant acceleration during a certain time, you can use the following classical formula:

d = 1/2 * a * t.

Since acceleration is constant and we assume that the initial velocity equals zero, you can estimate the maximum velocity using this equation:

v = a * t,

and the corresponding kinetic energy:

KE = m * v / 2.

Newton's universe - arrive at destination and stop

In this situation, we're accelerating to the half-way point, reaching maximum velocity and then decelerating to stop at the destination point. Distance covered during the same time is, as you may expect, smaller than before:

d = 1/4 * a * t.

Acceleration remains positive until we're half-way there (then it is negative - deceleration), so the maximum velocity is:

v = a * t/2,

and the kinetic energy equation is the same as the previous one.

Einstein's universe - arrive at destination at full speed

The relativistic rocket equation has to consider the effects of light speed travel. These are not only speed limitations and time dilation, but also how every length becomes shorter for a moving observer which is a phenomenon of special relativity called length contraction. If l is the proper length observed in rest frame and L is length observed by a crew in a spaceship, then:

L = l / .

What does it mean? If spaceship moves with the velocity of v = 0.995c, then = 10 and the length observed by a moving object is ten times smaller than the real length. For example, the distance to the Andromeda Galaxy equals about 2,520,000 light years with Earth as the frame of reference. For a spaceship moving with v = 0.995c, it will be "only" 252,200 light years away. That's a 90 percentage decrease or 164 percentage difference!

Now you probably understand why special relativity allows us for intergalactic travel. Below you can find relativistic rocket equation for the case in which you want to arrive at destination point at full speed (without stopping). You can find its derivation in the book by Messrs Misner, Thorne (Co-Winner of the 2017 Nobel Prize in Physics) and Wheller titled Gravitation, section 6.2. Hyperbolic motion. More accessible formulas are in the mathematical physicist's, John Baez, article The Relativistic Rocket:

t = c/a * sh[a*T/c] = [(d/c) + 2*d/a],

T = c/a * sh[a*t/c] = c/a * ch[a*d/c + 1],

d = c/a * [ch(a*T/c) - 1] = c/a * [(1 + (a*t/c)) - 1],

v = c * th[a*T/c] = a*t / [1 + (a*t/c)],

EK = mc * ( - 1)

The symbols sh, ch and th are respectively sine, cosine, and tangent hyperbolic functions, which are analogs of the ordinary trigonometric functions. In turn, sh and ch are the inverse hyperbolic functions that can be expressed with natural logarithms and square roots according to the article Inverse hyperbolic functions on Wikipedia.

Einstein's universe - arrive at destination point and stop

Most websites with relativistic rocket equations consider only arriving at desired place at full speed. If you want to stop there, you should start decelerating at the halfway point. Here, you can find set of equation estimating interstellar space travel parameters in situation when you want to stop at destination point:

t = 2*c/a * sh[a*T/(2*c)] = [(d/c) + 4*d/a],

T = 2*c/a * sh[a*t/(2*c)] = 2*c/a * ch[a*d/(2*c) + 1],

d = 2*c/a * [ch(a*T/(2*c)) - 1] = 2*c/a * [(1 + (a*t/(2*c))) - 1],

v = c * th[a*T/(2*c)] = a*t / (2 * [1 + (a*t/(2*c))]),

EK = mc * ( - 1)

So after all of these considerations, can humans travel at the speed of light, or at least at a speed close to it? Jet-rocket engines, used e.g. by NASA, taught us that rockets need a lot of fuel per unit of weight of the rocket. You can use our rocket equation calculator to see how much fuel you need to obtain a certain velocity (e.g., with an effective exhaust velocity of 4500 m/s).

Hopefully future spaceships will be able to produce energy from the matter-antimatter annihilation. This process releases energy from two particles that have mass (e.g., electron and positron) into photons. These photons may be then shot out at the back of the spaceship, and accelerate the spaceship due to the conservation of momentum. If you want to know how much energy is contained in matter, check out our E = mc calculator which is about the famous Albert Einstein equation.

Now that you know the maximum amount of energy you can acquire from matter, it's time to estimate how much of it you need for intergalactic travel. Appropriate formulas are derived from conservation of momentum and energy principles. For the relativistic case:

M = m * (exp(a*T/c) - 1),

where exp(x) is an exponential function, and for classical case:

M = m*v / (2*c) + m*v / c.

Remember that it assumes 100% efficiency! One of the promising future spaceships sources of power is the fusion of hydrogen into helium which provides energy of 0.008 mc. As you can see, in this reaction, efficiency equals only 0.8%.

Let's check whether fuel mass amount is reasonable for sending a mass of 1 kg to the nearest galaxy. With space travel calculator you can find out that, even with 100% efficiency, you would need 5,200 tons of fuel to send only 1 kilogram of your spaceship. That's a lot! So can humans travel at the speed of light? Right now it seems impossible, but technology is still developing. For example, photonic laser thruster is a good candidate since it doesn't require any matter to work, only photons. Infinite and beyond is actually within our reach!

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Space Travel Calculator | Relativistic Rocket Equation