Interstellar Traverse

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

Resembling Earth

For many planet hunters, though, the ultimate goal is still greater (or actually, smaller) prey: terrestrial planets, like Earth, circling a star like the Sun. Astronomers already know that three such planets orbit at least one pulsar. But planet hunters will not rest until they are in sight of a small blue world, warm and wet, in whose azure skies and upon whose wind-whipped oceans shines a bright yellow star like our own.

- Ken Croswell

Located 20 light-years or 190 trillion kilometers away is a humdrum red dwarf star called Gliese 581. As of October 2010, the star Gliese 581 has a total of six known planets in orbit around it and one of which is the most Earth-like planet discovered so far. This planet is designated Gliese 581 g and it is the fourth planet from its parent star. Gliese 581 g orbits its parents star at a distance of 22 million kilometers, taking 36.6 days to complete one orbit.

The orbit of the planet Gliese 581 g is located well within the habitable zone where the distance from its parent star is just right to support Earth-like surface temperatures. Thus, Gliese 581 g is located neither too close nor too far from its parent star. Since the star Gliese 581 is much less luminous that our Sun, the planet Gliese 581 g is able to support Earth-like surface temperatures even though it is located much closer to its parent star than our Earth is from the Sun.

In addition to orbiting its parent star within the “Goldilocks Zone”, the mass of Gliese 581 g is estimated to be between 3.1 to 4.3 times the mass of the Earth. If Gliese 581 g is a dense rocky planet like the Earth, its diameter will be somewhere between 1.3 to 1.5 times of the Earth’s diameter. The surface gravity of Gliese 581 g is also expected to be between 1.1 to 1.7 times the surface gravity of the Earth, making it not too different from the Earth. In fact, it will not be much of a problem for a human being to walk on the surface of Gliese 581 g.

With an Earth-like greenhouse effect, the average surface temperature of Gliese 581 g is estimated to be between 236 to 261 degrees Kelvin. In comparison, the average surface temperature of the Earth is 288 degrees Kelvin or 15 degrees Centigrade. However, because Gliese 581 g is more massive than the Earth, it is possible that the planet will have a more massive atmosphere which can create a larger greenhouse effect than Earth’s atmosphere. This can increase the average surface temperature of Gliese 581 g closer to the average surface temperature of the Earth.

One key difference between Gliese 581 g and the Earth is that Gliese 581 g is probably tidally locked whereby the same hemisphere of the planet perpetually faces its parent star. This is somewhat like the Earth-Moon system where the same side of the Moon always faces the Earth. In such a scenario, one side of Gliese 581 g will be in eternal daylight while the other side will experience eternal night. On such a world, temperatures can range from blazing hot at the sub stellar point on the day side to freezing cold on the night side. The sub stellar point on the surface of Gliese 581 g is where its parent star is forever directly overhead and it is the spot with maximum insolation. Between the two extremes, Earth-like temperatures can exist where a world that is not too different from ours is both easily conceivable and highly probable.

The mere fact that a potentially habitable planet has been discovered so soon around such a nearby star, suggests that habitable planets are far more common than previously believed. This means that potential of having many billions of Earth-like planets in our Milky Way galaxy alone is extremely probable. The paper detailing this discovery is by Steven S. Vogt, at al. (2010) and it is entitled “The Lick-Carnegie Exoplanet Survey: A 3.1 M Earth Planet in the Habitable Zone of the Nearby M3V Star Gliese 581”.

We live in a changing universe, and few things are changing faster than our conception of it.

- Timothy Ferris

Of the 500 or so known extrasolar planets, Gliese 581 g is probably the most interesting one discovered so far. Apart from the remarkable discovery of Gliese 581 g, a paper by Daniel Kubas, at al. (2010) entitled “A frozen super-Earth orbiting a star at the bottom of the Main Sequence” describes the discovery of a super-Earth which orbits a faint red dwarf star whose mass is close to the lower limit for a star.

This planet is designated MOA-2007-BLG-192Lb and its mass is 3.2 times the mass of the Earth. The planet orbits its parent star at a distance of about 100 million kilometers and this is about two-thirds the distance of the Earth from the Sun. However, because the parent star of MOA-2007-BLG-192Lb is a mere 8.4 percent the mass of the Sun, the planet receives over a thousand times less insolation that the Earth gets from the Sun even though it is located closer to its parent star than the Earth is from the Sun.

MOA-2007-BLG-192Lb was discovered using the gravitational microlensing technique as it provides a unique opportunity for the detection of low mass planets that are currently beyond the reach of most other methods. Gravitational microlensing occurs when a foreground star passes in front of a background star and the gravity of the foreground star acts as a lens and magnifies the apparent brightness of the background star. If the foreground star has a planet orbiting it, the gravity of the planet can induce a perturbation to the microlensing light curve. The duration of the perturbation depends on the mass of the planet, where a more massive planet will induce a perturbation with a longer duration. The discovery of the MOA-2007-BLG-192Lb shows that planet formation can occur down to the very low mass end of the stellar population.

The surface temperature of MOA-2007-BLG-192Lb is estimated to be around 55 degrees Kelvin and this is just below the melting temperature of pure nitrogen. However, internal heat generated from the decay of radioisotopes can raise the temperature on the surface of MOA-2007-BLG-192Lb to beyond the melting temperature of pure nitrogen. This can enable seas and oceans of liquid nitrogen to exist on the planet’s surface as long as the atmospheric pressure on the planet’s surface exceeds 0.1 bars.

The flow of internal heat onto the surface of a terrestrial planet can be strongly heterogeneous, making it highly probably that the surface temperatures on specific locations on MOA-2007-BLG-192Lb can exceed not just the melting point of nitrogen, but also the melting point of methane and even water. Therefore, lakes of liquid hydrocarbons like those on Saturn’s moon Titan can exist on MOA-2007-BLG-192Lb and open bodies of liquid water may even exist in volcanically active locales.