The desire to reach for the sky runs deep in our human psyche.
- Cesar Pelli
Interstellar space travel refers to unmanned or manned travel to the stars and it is vastly more difficult that interplanetary space travel as the distances involved are many orders of magnitude greater, even for the nearest stars. The distances to the stars are so immense that a light-year is employed as the unit of measurement, where one light-year is the distance a beam of light travels in one year and it has a value of 9.46 trillion kilometers.
Alpha Centauri is one of the closest stars and it is already located at a distance of 4.37 light-years or 41.34 trillion kilometers away from us. To put this impressive distance into perspective, 41.34 trillion kilometers is over a billion times the circumference of the Earth, or over 100 million times the distance of the Moon from the Earth. Even traveling at a velocity of 100 kilometers per second, it will take over 13000 years to traverse that distance! Hence, in order to reach the nearest stars within a reasonable amount of time, a spacecraft will have to be accelerated to much larger velocities and this is where the immense difficulty of interstellar space travel arises.
If the total worldwide energy consumption in 2009 were used to accelerate a 10 ton spacecraft, it will only accelerate the spacecraft to a velocity of only 10 percent the speed of light and that spacecraft will still have to take over 4 decades to reach Alpha Centauri. Furthermore, upon reaching Alpha Centauri, the spacecraft will not be able to spend any meaningful amount of time at its destination since it will simply speed past Alpha Centauri at 10 percent the speed of light unless a similar amount of energy is employed to decelerate the spacecraft.
In this article, I will assume that the immense scientific and technological barriers of interstellar space travel have been crossed and the capability to accelerate to near the speed of light is possible. This possibility is enabled by having a propulsion system that can generate exhaust velocities at close to the speed of light and some hypothetical form of antimatter-based propulsion system can be a possible candidate. It should be noted that the speed of light in a vacuum is exactly 299792458 meters per second since one meter is officially defined as the distance traveled by light in a vacuum in 1/299792458 of a second.
To begin, I shall describe a set of equations that I developed not long ago which extends the classical rocket equations into the relativistic regime. In other words, the relativistic rocket equations that I have derived account for the effects of relativity as the rocket’s velocity approaches a significant fraction of the speed of light and such relativistic effects include time dilation and length contraction. Additionally, I have also written a program which employs the equations to compute the characteristic of various mission scenarios.
In almost all other literature that I have reviewed, a constant acceleration is assumed for the relativistic rocket equations. However, in the equations that I have derived, a constant proper thrust is assumed rather than a constant acceleration because in practice, it is more realistic for a rocket to maintain a constant thrust rather than having a varying thrust to maintain a constant acceleration. It should be noted that the thrust is constant from the perspective of an observer traveling together with the rocket. This observer will also experience a gradual increase in acceleration as the total proper mass of the rocket decreases due to the burning of propellant, while the thrust remains constant throughout.
Using the set of equations and the computer program that I have developed, I will start with an unmanned spacecraft that has a total initial mass of 1 million kilograms (one thousand metric tons). This spacecraft is in orbit around the Earth and it is poised for a one-way journey to the stars. Which destination should the spacecraft visit? Alpha Centauri? Tau Ceti? Sirius? In this mission, I shall choose the red dwarf star Gliese 581 as the interstellar destination for the spacecraft.
… to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no one has gone before.
- Gene Roddenberry
Why Gliese 581? The reason is that Gliese 581 has a total of six known planets in orbit around it and in my previous post, I wrote about one of the planets which is the most Earth-like one discovered so far. This planet is designated Gliese 581 g and it orbits Gliese 581 at a comfortable distance where the temperatures are estimated to be just right to support Earth-like conditions. The star Gliese 581 is located 20.3 light years or 192 trillion kilometers away from us and the spacecraft will need to accelerate to close to the speed of light to get there within a reasonable period of time. Upon reaching Gliese 581, the spacecraft will also have to decelerate itself from its incredibly huge velocity so that it will not merely zip pass Gliese 581.
As stated previously, the spacecraft has a total initial mass of 1 million kilograms and most of which is in the form of fuel. The spacecraft also has a propulsion system which can generate an exhaust velocity that is 80 percent the speed of light. Furthermore, the spacecraft’s propulsion system is able to generate a constant 24 million Newton of thrust and this force is equivalent to approximately 7 times the weight of a Boeing 747 airliner. It is important to note that the mass of the spacecraft and the generated thrust is measured from the perspective of an ‘observer’ traveling with the spacecraft since the effect of relativity will give a different measured reading for an observer at rest.
To generate a constant 24 million Newton of thrust, the spacecraft will have to burn its propellant at a rate of 0.1 kilograms per second and direct the high energy exhaust out at 80 percent the speed of light. From rest, the spacecraft will accelerate at a constant thrust for a total duration of 7.5 million seconds (86.8 days) as measured from onboard the spacecraft. However, due to the effect of relativistic time dilation, 8.6 million seconds (99.6 days) would have already elapsed on Earth during the entire acceleration phase!
Initially, the spacecraft will experience an acceleration of 2.40 g’s which gradually increases to 9.59 g’s at the end of the acceleration phase because the proper mass of the spacecraft decreases while the thrust remains constant. At the end of the acceleration phase, the spacecraft will attain a final velocity of 240949550 meters per second which is slightly over 80 percent the speed of light. During the acceleration phase, the spacecraft would have already traveled over a trillion kilometers, or approximately one-tenth of a light year.
The spacecraft then shuts off its engine and begins its high speed cruise across the vast expanses of interstellar space, towards the direction of Gliese 581. It should be noted that the total mass of the spacecraft is now 250 thousand kilograms. Cruising at an incredible velocity of 240949550 meters per second, the spacecraft still has to take 25 years to get to Gliese 581! Additionally, the effect of relativistic time dilation means that only 15 years would have elapsed for a hypothetical observer onboard the spacecraft.
Upon reaching Gliese 581, the spacecraft will turn on its engine and commence its deceleration phase with the same constant thrust of 24 million Newton. The spacecraft will take 1.875 million seconds (21.7 days) to decelerate from 80 percent the speed of light so that it will be slow enough to enter orbit around Gliese 581. However, due to relativistic time dilation, 2.151 million seconds (24.9 days) would have elapsed back on Earth during the entire deceleration phase. Initially, the spacecraft will experience a deceleration of 9.59 g’s which gradually increases to 38.4 g’s at the end of the deceleration phase because the proper mass of the spacecraft decreases while the thrust remains constant. The spacecraft would have traveled another quarter of a trillion kilometers during the deceleration phase.
Orbiting around Gliese 581, the spacecraft now has a total mass of just 62.5 thousand kilograms as 93.75 percent of its initial mass is basically the propellant required for the journey. It is up to you to imagine the various kinds of payloads that can makeup the 62.5 metric tons of the spacecraft’s final mass. Due to the finite speed of light, the Earth will only receive the first signals from the spacecraft 20.3 years after the spacecraft has reached Gliese 581…
Now when we think that each of these stars is probably the centre of a solar system grander than our own, we cannot seriously take ourselves to be the only minds in it all.
- Percival Lowell