where D is the straight distance to Proxima Centauri, about 4.3 light years or 4 x 1016m, and Mf is the final mass of the ship after the trip is over. The ratio F/Mf is an acceleration, precisely that at the trip end (not during the trip!) and for the present purpose can be assumed to be a constant (for instance, 1 g). F is kept constant.
The inverse dependence of trip time on Isp on equation (8.15) is striking, but it was also found in a different form in Section 7.18, where time to accelerate, tacc, was found to be proportional to Isp. The dependence on F is tempered by the square root. For Isp in the upper range enabled by fusion strategies (106 to 107 s, see Figure 8.4), the first term is much smaller than the second and can be neglected.
Actual numbers using equation (8.15) indicate that reaching Proxima Centauri and back takes 508 and 51 years at Isp = 106 and 107 s, respectively. Average speed,
av tES 4
With the approximation made, this average velocity depends only on Isp, and is of order 106 or 107 m/s, respectively. This means Newtonian mechanics can still be used if Isp is in the low range, while a small relativistic correction could be made in the high range (where Vav/c, about 8%, is not completely negligible). These mission times are substantial; since in relativistic physics Isp has an absolute maximum, the speed of light c = 3 x 108m/s), the conclusion is that a mission at constant thrust might still not be the best strategy over stellar distances.
To reduce trip time, it appears the trip should be made at a speed as close to c as practicable. Neglecting relativistic effects (therefore violating the self-imposed rule of Section 8.3), at the speed of light the round trip would take of course 8.4 years for the crew. Traveling at average V = 0.5c would double the trip to nearly 17 years, not accounting for the acceleration and deceleration periods. This strategy means that thrust should have a history in which acceleration ramps rapidly, followed by a period in which it stays constant until V reaches a significant fraction of c. Finally, a deceleration period should slow the spacecraft down, to enable orbit capture near the star or planet. For a given final mass, Mf, this means the power demand must be very high, since thrust power P = F1sp, but only during the acceleration period, when F is increasing or constant. Once the ship moves at the planned fraction of c, power can be turned off and F = 0, the ship coasting at high speed.
A crude example may help in understanding the terms of the problem. If the time-averaged ship mass is of order 100 metric tons, and a = 3g (a modest increase over 1g calculations made before, but barely tolerable by a human crew), F would be 3 x 106 N, and P, at an optimistic 7sp = 107 m/s, would reach above 104 GW. Fusion energy release is of order 3 x 1014J/kg, and about 100 g/s of D-T fuel (see Section 8.6, below) would have to be fused. However, the mass conversion ratio in fusion is only about 0.3%, meaning the actual fuel flow-rate injected inside a fusion chamber would have to be 1/0.003 times higher, or 33kg/s. During only one day, the total mass of fuel injected would be of order 2,850 tons, two orders of magnitude greater than the assumed mass of the ship. Working close to the theoretical Isp, say, 108 m/s, the fuel consumption would reduce to 285 tons/day, still an astonishing figure. More encouragingly, fusing a proton, and an antiproton, p~ (mass annihilation, 100% mass conversion) yields J = c 2/kg = 9 x 1016 J/kg; so in the same example the mass consumption would drastically reduce to 9.6 kg/day; see also [Borowski, 1995].
As of now, no nuclear process exists with yield in between that of fusion and that of annihilation. Percent mass conversion is either in the few parts per thousand (using D, T or H fuels and kinetics) or 100% (annihilation). The reason is the binding energy of Figure 8.6, that is no higher than about 8MeV per nucleon. Until annihilation becomes a practical process, and provided relativistic effects can be dealt with, practical QI and stellar travel with technologies within our grasp and ship masses below 0(103) ton will depend essentially on distance, and will be limited by how long acceleration (thrust) can be maintained to reach a substantial fraction of the speed of light.
Before examining the details of high energy density propulsion based on fusion, an important aspect of practical QI and stellar missions is that the length of a mission, calculated in this section from the viewpoint of a spacecraft crew, may be different for the mission support team left on Earth. Effects due to missions performed at constant acceleration and reaching relativistic speeds, together with their consequence on mass ratio have already been mentioned in Section 8.3, but differences in times have not, and are found in Chapter 9.
The considerations made about travel times and mass consumption in this and in the previous section should warn about presuming too much from propulsion as we know it, that is, based on Newton's Third Principle. Power and mass consumption, together with distances to cross and mission times are formidable hurdles, although mastering mass annihilation may overcome the first two. Notwithstanding all this, because of its energy density, fusion is the only power source viable for future QI, if not interstellar, space travel, and is a source that has been studied at least for half a century.
What follows deals with how fusion energy can actually be harnessed and work in a space propulsion system, with emphasis on the different technologies proposed, their drawbacks and their advantages (see [Leifer, 1999] for a brief summary).
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