Minimizing energy cost on the interplanetary express

kw: analysis, energy, space travel, economics

It is frustrating. Space fiction is filled with 35^th Century, or 135^th Century folks flitting about space in their interstellar runabouts, going to Mars or Neptune like we might go to Omaha or Yokohama, and catching some kind of hyperspace express to cruise out to Aldebaran or some other locale a few hundred parsecs distant, for a rather modest cost.

The fact is, space travel requires a lot of energy, and energy costs something. At the moment, though, it costs more than it should because a space vehicle has to carry the fuel to make its entire journey, and we take advantage of tricks like using the atmosphere of Earth to slow the return module to parachute speed (or landing speed, for a shuttle-type vehicle, not that any currently exist).

A number of new technologies have been proposed to get a vehicle off the Earth without using any on-board fuel, such as laser propulsion. I don't propose to get into such a discussion here. Rather, given that some kind of remote assist is developed, what is the lowest cost of getting something from point A to point B?

For comparison, we might consider that it costs a few dollars ($20 or less) to ship a kilogram of any legal substance via public carriers or even the US Postal Service, say from western Pennsylvania to Massachusetts, a distance of about 800 km. If I were to personally deliver the package by driving both ways, it would cost more. My car gets 30 miles per gallon, or 48 km/gal, on the highway. That's also about 12.7 km/l. Gas (petrol) cost alone for the 1,600 km trip comes to 33.3 gallons at $4, or $133. But that's partly because the material being moved now weighs a metric ton, not just one kg. On a per kilo basis, the cost is thirteen cents. So the USPS or other carrier is only a few percent efficient, compared to my own costs, if I were carrying lots of packages in my one-ton car (and if I worked for free).

In actuality, the energy costs to the Postal Service or FedEx or whoever, are still a minor portion of total costs. But let's consider that energy-only cost a baseline: $0.133/kg to go 1,600 km, or about 8 cents per 1,000 km. Now let's consider moving a more modest 200 km, but straight up. That'll get us in the neighborhood of the ISS. USPS might charge only $5, but I doubt it. We'll consider achieving orbital velocity separately.

What's the gravitational potential difference between Earth's surface and an altitude of 200 km? Considering the Earth as a point object, which is mathematically valid from its surface outward, potential V = -GM/r. At the surface, V_s = -6.64×10^-11×5.97×10²⁴/6.37×10⁶ = -6.255×10⁷ J/kg. Add 200 to the 6,370 km radius of the earth and recalculate, and we get V_orbit = -6.065×10⁷ J/kg. Subtracting these two, we get 1.90×10⁶ J/kg. So what does that amount of energy cost?

In the US, gasoline costs $4 per gallon, and has an energy content of 3.2×10⁷ J/l or 1.2×10⁸ J/gal. The most efficient methods of using gasoline are only 30% efficient, however, so the usable energy cost is about ten cents per megajoule, or 10^-7 $/J. Liquid hydrogen can be bought for about $0.40/l, and running the figures I find it costs about 20% more than gasoline for a joule of energy obtained from hydrogen. We can use the 10¢/MJ figure for our calculations. Thus, lifting a kilogram to orbital altitude costs nineteen cents.

Keeping it there requires moving it at orbital velocity, however, which is 7,910 m/s. E_k = ½MV² = 3.13×10⁷ J/kg. This comes to $3.13/kg, more than sixteen times the cost of achieving altitude. That's an important fact about getting around in space: δv (delta vee), or change in velocity, can be a larger factor than the gravitational potential. However, at this point, let's consider that, if we truly could achieve costs as low as $3/kg to get an object into orbit, it would be revolutionary: Attaining orbit presently costs about $10,000/kg. With such a reduced cost we could think about visiting the outer planets.

The major factor going from planet to planet is the gravitational potential relative to the Sun. At Earth, this comes to -8.85×10⁸ J/kg; at Neptune, it is much smaller: -2.95×⁷ J/kg. Subtracting these yields 8.55×10⁸ J/kg, which costs $88.50/kg. Getting out of Earth's gravity well is a fraction of this (about $6/kg, similar to the cost of going to the Moon). But now there is a time factor to consider. It takes fifteen to twenty years to get to Neptune on a ballistic orbit. In other words, if some kind of energy deposition mechanism gives our one kilogram package an initial velocity of about 40 km/s, it will coast out to Neptune, and have nearly no kinetic energy left, but it might take twenty years or more.

If we increase that to Solar escape velocity, measured from Earth vicinity, or 42 km/s, it'll arrive with velocity comparable to Neptune's orbital velocity of 5.4 km/s. However, it will have required 15-16 years to travel some five billion km. To get there in one year requires a lot more initial velocity, and almost as much δv at the other end to slow down. Initial velocity needs to be of the order of 158 km/s. Kinetic energy comes to 1.25×10¹⁰, which costs $1,250. So, take your choice. A decade and a half for $88.50 or a one year delivery time for $1,250, plus another thousand-dollar slowdown fee.

These costs assume we are not accelerating fuel, just the kilogram we want to deliver. Perhaps there will one day be installations, set up by earlier generations (plural, to be sure!), that use something like laser boosting to push a projectile to these velocities, or to push against an incoming package to slow it back down. These costs are just the incremental energy costs for moving a package about. I am ignoring amortization of sunk costs (you know, the odd quadrillion or quintillion dollars to get the laser boosters into Earth orbit—or onto the Moon—, Neptune orbit, and sundry places between).

If getting to Earth orbit drops to some $3/kg, then there is some hope for a 100 kg guy like me to afford an orbital vacation. I'd gladly pay $300 each way for tickets to visit a space station, particularly if a more comfortable one than the ISS is assembled. Of course, I suspect the daily room cost will be more than at your average hotel! Going to Neptune would be more costly. Since the express trip takes a year each way (I don't have thirty years for the slower round trip!), I need some support systems, including plenty of water, air and food. Call it a couple tons. At $1,250/kg to start, $1,200 to stop, and then the same amounts for the return trip, the energy costs alone will come to nearly ten million dollars.

I don't have even one million dollars, nor much prospect of obtaining it. Vacationing in the outer solar system will probably always remain available only to the rich. What about going farther out? Stellar travel has huge time requirements, and to make it practical, the energy has to be balanced against that time.

For a number of reasons, various researchers have settled on a tradeoff velocity of 0.13c, or 39,000 km/s. That'll get you to Proxima Centauri in 33 years and Barnard's Star in 46 years. What is the energy cost? You really need laser boosting, at least at the near end, to make it practical. The relativistic kinetic energy is 7.69×10¹⁴ J/kg, at a cost of $76.9 million/kg. How many kg will a vehicle weigh, that can keep a few people alive for decades? 10,000 tons? Assuming that would do it, the energy cost is now $769 billion, or about what each of the "stimulus" packages of 2008 and 2009 cost the US government.

That is the bottom line. Sending people to a star is going to cost trillions. It may be that bombing around the inner solar system will become affordable for many of us, but even visiting the outer solar system will never be within reach to folks like me. Just getting a useful-sized spacecraft up to 0.13c is a project for a nation or a consortium of nations. Getting a kg or two in the form of a Von Neumann self-replicating robot up to such a speed is no cheap undertaking, and sending along fuel enough to allow it to slow down is another huge cost, but much less than the cost of sending people.

This doesn't mean I don't think it will be done. I expect it to take a lot more time, ingenuity, and fortitude. We especially need the planetary will to invest in technologies that enable getting off Earth, into orbit, and off to the planets, at the very least, at greatly reduced incremental cost. In today's dollars, we spent a pretty good chunk of a trillion dollars going to the Moon a few times. With any luck at all, we ought to be able to return to the Moon for one percent of that cost. The Moon is a good base for big lasers to accelerate packages once they are outside the atmosphere; an Earth-based laser facility ought to be able to get them that far. That is step one, and further steps are up to future generations of dreamers.