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Sunday, August 7, 2011

To the Stars!

Chapter 7E

To the Stars!

Note: Although a little technical in a couple of places, I hope the reader will bear with me, for this is one of those cases where I need to bring up the technicalities for discussion. As I hope you will see, it is easier to see the point by doing a little bit of elementary algebra or chemistry.

Human Expansion throughout the Galaxy

With present technology a trip to the nearest stars would take tens of thousands of years. Perhaps with an extension of our present capabilities we may be able to cut the journey to only a few centuries. Unless a truly fantastic technology for suspended animation is discovered, the trip would have to be completed by the descendants of the astronauts that begin it. Under those conditions the best way to travel to the stars might be to turn one of O'Neill's colonies into a vehicle and set it to depart from our solar system. But surely, some critics might say, people in their right minds would not wish to take their space colony into interstellar space for journeys that would last thousands of years -- although how preposterous the idea is will have to be determined by a level of technology and an abundance of resources in interstellar space that we are in no position to predict now. At any rate, even people in their right minds may consider precisely such a journey if they knew of some unavoidable catastrophe that was to befall the solar system, or for other reasons that we may not fathom at this time.

Nevertheless, in journeys so long that only the descendants of the original travelers can complete them, a successful outcome may seem remote at best. Accordingly, some feel that the next great barrier to space exploration is the development of technology that would permit interstellar travel during a human lifetime. Since the closest stars are at least four light years away, and our galaxy is about one hundred thousand light years across, we would need starships that achieve velocities close to that of light.

Many scientists, however, believe that Einstein's special theory of relativity does not permit to accelerate spaceships close to that of light. And accelerating spaceships beyond the speed of light is simply forbidden by that theory. Nevertheless as we will see below, Einstein's physics does not in principle preclude either of these options.

Let us consider the first option. Although Alpha Centauri is only four light years away, the majority of stars of interest in the galaxy are tens, hundreds, or thousands of light years away. It may seem then that even if we could achieve relativistic velocities, traveling to the stars may take as long as the astronauts’ life spans, or longer. Fortunately, distance and time are relative to the inertial frame of reference in which they are measured (in an inertial frame of reference the velocity is uniform). In a ship that travels at great velocity with respect to us, time slows down and distances shorten, even though the astronauts themselves detect no abnormality. At velocities close to that of light, 300,000 Km/s, distances are so short (or alternatively, the dilation of time is so large that apparently unbelievably long journey became feasible. According to calculations by Carl Sagan, we could go to many interesting stars and come back in a decade or two, ship time[1]. Six years of ship time would go by in a round trip to Alpha Centaury (eight years Earth time), 22 to the Pleiades (800 years Earth time), and to the Galaxy of Andromeda, which is over a million light years away, the round trip would only take about five decades!

In the meantime nearly three million years would have gone by on the Earth, and so the return may offer more of a shock than what we might find in Andromeda. Most of us would not want to go on such a journey, but I imagine that the project would suffer no dearth of volunteers. The main problem, however, would be the energy required. At a constant acceleration of 1 g our spaceship would reach 99% of the speed of light in one year. But in reaching a velocity that high we would need to spend, according to some calculations, energy equal to the entire consumption in United States during a period of a million years! Enthusiasts like to point out that the first spaceship that went to the Moon spent an amount of energy tens of thousands of times larger than what many societies used only a century earlier. Bernard Oliver, who frowned on the idea of interstellar travel, thought that the requirements would be of this order of magnitude[2]. Even if such calculations are off by an order of magnitude or two, we are talking about staggering amounts of energy.

Some skeptical theorists have thought that the project is impossible, anyway, because as the velocity increases, so does the mass (also according to the special theory of relativity). But a larger mass requires larger energies to increase the velocity, which then increases the mass, and so on. This continuous increase in the mass of the spaceship eventually defeats the attempt to increase its velocity: we never reach a velocity close to that of light.

I do not believe that this objection works, however, for it does not take into account that from the point of view of the ship itself the mass has not increased. On the contrary, as most ships are conceived, it necessarily decreases as the engine burns fuel.

The skeptics’ suggestion is, again, that as the traveler approaches the speed of light, the mass increases so that it takes more and more energy to keep accelerating at the same rate. Many physicists, using this line of reasoning, conclude that it is impossible to travel at the speed of light, let alone faster. At least it seems that such is the reasoning that leads physicists like Smolin to conclude that, “…her mass increases as she approaches the speed of light. Were her speed to match that of light, her mass would become infinite. But one cannot accelerate an object that has infinite mass, hence one cannot accelerate an object to the speed of light and beyond.”[3] Similar remarks are made by Brian Greene[4] and even by Stephen Hawking[5].

I think this line of reasoning is misleading in two ways. First, once again, as far as the special theory of relativity is concerned, the mass and the corresponding energy requirements increase only from the point of view of the observers left behind on Earth. But from the point of view of the star travelers, who are at rest with respect to the ship, the mass of the ship does not increase at all, and therefore accelerating the ship is not particularly more daunting than it was at lower velocities. If anything, it is easier because the longer the ship accelerates the more fuel it uses, and therefore the more mass it loses, as I pointed out above. At speeds close to that of light, its rest mass should be considerably less than at the beginning of its journey, as long as you have the standard means of propulsion, i.e., shooting something out the back. In practice, or course, the faster a starship travels, the greater the resistance from the interstellar medium, which could become significant depending on the ship’s design and other factors. But this is a different type of concern altogether.

Second, the reason why the ship cannot match the speed of light has nothing to do with the mass becoming infinite. What physicists like Smolin, Greene, and Hawking have in mind is Einstein’s equation:

m= m0/(1- v2/c2 )1/2

where m is the mass of the ship from the point of view of the observer, m0 is the rest mass, v is the velocity of the ship with respect to the observer, and c is the speed of light.

As v gets closer to c, the term v2/c2 approaches 1. This means that the denominator approaches 0, which makes m approach infinity.

But m can never reach infinity for the simple reason that, if the velocity of the ship reached that of light, the denominator would become 0 and the function would be undefined. The problem is not that an infinite mass is physically inconceivable, but that the mathematical expression makes no sense.

The main insight is flawed, in any event. Infinite mass has nothing to do with the relativistic speed limit. The reason why a ship cannot accelerate to the speed of light is that Einstein’s formula for addition of velocities (based in part on the postulate that the speed of light is a constant) will always yield final velocities less than c.

If I am traveling in a ship at .5 c, the speed of a ray of light with respect to me, whether it goes towards me or away from me, still is 300,000 km/sec. If I fire a probe that travels at .5 c with respect to me, the speed of that ray of light would still be 300,000 km/sec with respect to the probe.

The result is that, according to Einstein[6], in the special theory of relativity I cannot just add the velocities of my ship with respect to the ground (vs) and of the probe with respect to me (vp).

That addition must be divided by the term 1+ vs.vp/c2. When I add the velocities (my ship’s plus the probe’s) I do not get c, therefore, but only .8 c.

This corrected interpretation of the situation (from the point of view of the astronaut, and the appropriate equations from the special theory of relativity) still seems to forbid travel at or faster than the speed of light. It leaves open the question of building a spaceship that comes very close to the speed of light, though.

There have been other attempts to prove that near-light speed travel is impossible, and there have been many refutations of such attempts as well. Of the presently available starship technologies (available in theory, that is) some form of controlled fusion may offer the best hope to achieve relativistic speeds (though just barely about 1/10 of the velocity of light). The ideal apparently would be a matter- antimatter engine, for it would convert all of the fuel's mass into energy as the particles and antiparticles annihilate each other. A serious problem is how to produce the necessary amounts of antimatter without spending more energy than that required to propel the starship. And if you do produce it, you then have to worry about how to channel it so it goes out the nozzle only, otherwise it will radiate in all directions. And there is also the already familiar difficulty that if you include all the fuel you need to keep accelerating the starship, then you need to build a much larger starship, which then needs even more fuel, and thereby an even larger starship.

To get around these problems we might employ starships that do not carry their fuel on board but take it from their environment. The first such "design" was for an interstellar fusion ram-jet that would scoop hydrogen ions from space, the Bussard Ramjet[7]. Bussard’s interesting idea was marred by several difficulties, especially that it would require a scoop 160 Km in diameter and that it would use a proton-proton fusion reaction that may work only in temperatures as hot as the interior of stars[8]. A modified version, the Whitmire catalytic nuclear ramjet,[9] apparently solves some of the main theoretical problems (it works by scooping up the hydrogen ions and running them through a catalytic nuclear reaction cycle, i.e., a nuclear reaction that repeats itself again and again, and that returns to the starting point of the reaction extremely fast so a new batch of protons can be used to propel the starship).

One possible sequence or reactions would be, for example, that of the catalytic cycle of carbon-nitrogen-oxygen (CNO), which occurs in the thermonuclear reactions of very hot starts:

12C + 1Hà13N + γ

13N + 1Hà14O + γ

14Oà14N + e+ + ν

14N + 1Hà15O + γ

15Oà15N + e+ + ν

15N + 1Hà12C + 4He

As we can see, the hydrogen ions (protons) react with the carbon isotope (12C) to begin the cycle, which, after utilizing a total of four protons, ends again in 12C plus a helium nucleus that is expelled out the nozzle, thus propelling the ship forward. The positrons (e­+) react with the electrons that remain from the ionization process and liberate additional energy in the form of gamma rays[10].

Whitmire and others[11] have worked on the possibility of either electromagnetic or electrostatic scoops of dimensions in the hundreds of meters, rather than kilometers, to reduce the immense proton and electron drag expected to affect a ship moving at relativistic speed (and that would be more efficient in the collection of protons). There are several practical problems with Whitmire’s design, the most nagging of which is that the temperatures in his reactor might reach temperatures of one billion degrees Kelvin!

There are also problems raised by the gravitational impact of a ship that moves through a medium with respect to which its mass increases extraordinarily (even if it does not change for the astronauts). It is possible that the ship may affect the structure of spacetime in its path. There would be additional problems to describe mathematically the interactions between the ship and the environment, using the general theory of relativity, for the ship will exchange energy with the particles closest to it as it accelerates, which would then cause all sorts of difficulties for the calculation of the relevant masses.

Whether these problems can be solved eventually, I do not know. Nevertheless the interesting result is that, in principle, interstellar hydrogen can be used to accelerate a starship at 1g to achieve speeds arbitrarily close to that of light.

[1] C. Sagan, “Direct Contact Among Galactic Civilizations by Relativistic Spaceflight,” Planetary and Space Science 11 (1963): 485-498.

[2] B.M. Oliver, “Efficient Interstellar Rocketry,” Paper IAA-87-606, presented at 38th I. A. F. Congress, Brighton, UK., 10-17 October 1987.

[3] Smolin…

[4] Brian Greene, The Elegant Universe, Vintage Books (2003): 52.

[5] Stephen Hawking, The Universe in a Nutshell, Bantam Books (2001):12.

[6] Albert Einstein, “On the Electrodynamics of Moving Bodies,” reprinted in The Principle of Relativity, with H.A. Lorenz, H. Minkowski and H. Weyl, Dover Publications, 1952 (republication of translation published by Methuen and Company, 1923). His equation as it appears in Section 5 of the article (“Composition of Velocities”), when the direction of motion of v and w is along the X axis, is:

V= v + w/1+ vw/c2

[7] R.W. Bussard, “Galactic Matter and Interstellar Spaceflight,” Astronautica Acta 6 (1960): 179-194.

[8] For an interesting discussion of this and other possible starships, please see E. Mallowe and G. Matloff, The Starflight Handbook: A Pioneer’s Guide to Interstellar Travel, John Wiley and Sons, Inc (1989): 89-149.

[9] D. P. Whitmire, “Relativistic Spaceflight and the Catalytic Nuclear Ramjet,” Acta Astronautica 2 (1975): 497-509.

[10] Otros ciclos catalíticos como el del 20Ne también serían posibles. El tope de la energía generada por el reactor de Whitmire sería unos 1011 megawatts, cerca de 10.000 veces lo producido por el mundo entero hoy en día (Mallowe y Matloff, p. 114).

[11] See again Mallove and Matloff, op. cit., 124-133.


tonyon said...
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tonyon said...
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tonyon said...
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