By Bryan Laubscher
Introduction
There is an old adage: “Getting there isn’t half the fun, it is all the fun!” I want to adapt it to spaceflight: “Getting there isn’t half the effort, it is all of the effort!” The reason for this is the physics of the rocket equation and the depth of Earth’s gravity well. In this missive I will perform a back of the envelope calculation for the cost per kilogram (kg) of transporting mass to the vicinity of Mars. This calculation is rough and certainly not the last word. It does not include the cost of rocket research, development or engineering. Moreover, the overhead of rocket structure, tanks and staging are not included. Thus the rocket equation represents the best case scenario – true rocket performance will be less. However, the resulting cost is stunning and creates a lot of questions around how much exploration, especially manned exploration, can we afford to do with rocket technology!
Note that in this discussion, the vicinity of Mars means that it is not in orbit nor on the surface – those two maneuvers cost more fuel and hence money. I chose the vicinity of Mars because at Mars aerobraking can be used to get into orbit or to the surface without fuel expenditure. Another reason for the choice is that the ΔV, (the amount of velocity change) to reach the vicinity of Mars is less that that required to “soft” land on the surface of the Moon. Thus all costs that I derive are greater for a lunar surface mission
ΔV Considerations
Earth’s gravity well is deep. It takes a ΔV of 9.7 kilometers per second (km/sec) to reach low Earth orbit (LEO) from Earth’s surface. Now, LEO is actually a range of altitudes each requiring slightly different energies but this is a rough calculation so I’m not sweating the details. This ΔV is achievable with chemical energy, albeit just barely. As an illustration, remember that the mighty Saturn V rocket configured for the Apollo mission expended 95% of its total mass to place the remaining 5% into LEO. As an astronomer, I wonder about our civilization had we evolved on a more massive world such that chemical energy was impractical to achieve orbit. In that case, we might have waited until nuclear propulsion was perfected. Where would we be at the present time in such a world?
For comparison, note that the ΔV from LEO to the vicinity of Mars is 3.8 km/sec and from LEO to the lunar surface is 5.5 km/sec. This is the origin of Heinlein’s declaration (paraphrased): “When you get to LEO, you are halfway to anywhere!”
The Rocket Equation
Consider the forms of the rocket equation:
ΔV = Vp ln(Mi/Mf)
And
Mi/Mf = exp(ΔV/Vp)
Where ΔV is the velocity change, Vp is the propellant velocity of the rocket, Mi is the initial rocket mass and Mf is the final rocket mass (payload). Note that the mass ratio, Mi/Mf depends exponentially on the ratio of velocities, ΔV/Vp. This means that if the required ΔV doubles, the initial mass must increase by a factor of around 7.8. If the ΔV increases by a factor of 5, the initial mass must increase by a factor of about 172. This is the reality of rocket technology.
One way to think about the reason for this “inefficiency” is that a rocket carries a tremendous amount of fuel to burn at higher altitudes. Lifting all that fuel takes additional fuel that ends up dominating the mass carried at most points in the trajectory. One can imagine getting around the rocket equation by burning all the fuel on the launch pad – an artillery shell. It is hard to imagine, however, containing an explosion of the enormity needed to throw a communications satellite to geosynchronous orbit. Also a satellite robust enough to survive the tremendous acceleration (thousands of times the acceleration of gravity) would be heavily built and quite massive requiring an even larger explosion and greater acceleration. Moreover, the air resistance encountered by a large payload achieving its maximum velocity at a hundred meters (or so) altitude after launch would be prodigious and require an even greater amount of initial energy and speed to achieve orbit above the atmosphere. Indeed, to minimize air resistance, modern rockets fly an “efficient trajectory” by throttling up only high in the atmosphere, purposefully carrying fuel to high altitudes.
As an aside, nuclear rockets, by virtue of relying on propellant are also subject to the rocket equation. The advantage of nuclear fission rockets is that Vp is 2 to 3 times higher than for chemical reactions. Fusion and anti-matter rockets are predicted to have much higher propellent velocities than fission rockets. Of course, some drawbacks exist to nuclear rockets although the negative impacts decrease outside the atmosphere.
I already stated that the Saturn V, the most powerful rocket ever flown, placed 5% of is mass into LEO. Therefore, let me start with that information. This fact means that for every kilogram of payload lifted to LEO, 20 kilograms of fuel was required. Using 20 as the mass ratio corresponding to the ΔV to LEO I will use the rocket equation to scale the mass ratio from LEO to Mars vicinity. I get the result that 2.39 kg of fuel are required to transport each kg of payload in LEO to the vicinity of Mars. Thus the total mass lifted to LEO is 3.39 kilograms, 1.0 kg of payload and 2.39 kg of fuel to take that payload to the vicinity of Mars. Because this is a back of the envelope calculation, I’ll simplify this to 3.4 kilograms.
Rocket Launch Costs
The energy required to launch one kilogram to LEO is well known – it is 17.2 kiloWatt hours (kWh). Home electricity prices are less than $0.10/kWh so using that value the corresponding cost for the energy to LEO is $1.72/kg!
Rockets are expensive and complicated. They are mechanical machines that operate at the limits of materials science and chemistry. They also occasionally fail. Decades of experience with rockets points to more than $10,000 per kilogram to LEO. Attempts to radically reduce this number have failed and the claims for the greatest reduction hover around a factor of 2. (See When Physics, Economics and Reality Collide: The Challenge of Cheap Orbital Access by Jurist, Dinkins and Livingston, 2005, for a well thought out treatise.)
This is our current reality and the rocket equation, coupled with our deep gravity well, combine to ensure that there will not be much movement in this number. Rocket technology is surprisingly mature. The issues have been thoroughly understood for many decades. As an example, the Saturn V first stage possessed a mass ratio of 94%. That means that 6% of the first stage’s mass was structure, tanks, engines and controls – the remainder was fuel.
Cost per Kilogram to Mars Vicinity
Using the value of $10,000/kg, the cost of moving one kilogram of payload from the surface of Earth to the vicinity of Mars is $34,000. The cost to the lunar surface is greater. I claim that this is good to at least a factor of two barring a glacial bureaucracy that would drive the cost higher than $68,000/kg. I do not foresee the lower limit of $17,000 per kilogram being realized but most people would agree it is a lower limit.
NASA’s Space Exploration Initiative called for a Mars rocket with a mass of 1000 metric tons which corresponds to $340 billion launch cost to Mars! The innovative Mars Direct plans called for a Mars spacecraft of 87 tons implying $2.9 billion launch cost to Mars! These costs do not include research, development, fabrication, construction or test flights. Also, these costs are not rigorous since these ships were to be constructed and launched from LEO so the mass may include their fuel to Mars – my source did not break down the mass. If the fuel to Mars is included, then the launch costs change to $100 billion and $853 million, respectively. In either case, the magnitude of these numbers is useful to realize what we are looking at in terms of launch costs.
A manned outpost or colony would require many, many tons of shelter, equipment, food, water etc. to be sent to Mars over a long period of time. If the plan is to “live off the land”, initial missions will still require tremendous amounts of logistical support. The moon requires even more in-situ support since it lacks the inherent resources and advantages of Mars. Of course many fewer resources are required for the 3-day trip to the moon versus the many months travel time to Mars.
Conclusion
My question is: How much exploration, especially manned exploration, of the moon and Mars will we be doing at $34,000 per kilogram? My guess is that we’ll do pretty much what we’ve done over the last 35 years since the last Apollo mission.
If you agree, then consider that the Space Elevator is the answer. The cost per kilogram for the first Space Elevator launch is expected to be around $1000 - $3000/kg and it can launch the payload to Mars with only a small rocket engine for a plane change maneuver. The potential for savings is stunning! How much exploration could we do with a launch infrastructure that cost $1000 per kilogram to LEO or GEO or to Mars? Moreover, Space Elevator technology is subject to the economy of scale. As an infrastructure of high-capacity Space Elevators is constructed, the cost will fall dramatically from $1000/kg to $50/kg. No matter how difficult it may seem to develop Space Elevator technology, it is the missing piece to open space as a place to solve problems here on Earth.
I want to hear thoughtful responses from the community. What do you think? What is your estimate of the cost? What is a realistic estimate for human missions to the moon and Mars? What conclusion do you draw from these numbers?
Although the numbers are discouraging for rocket launches, the "rocket crowd" has one advantage over the space elevator community: their product exists in reality, and not just on paper.
Rockets are expensive, but there is currently no other alternative at replacing them, although some companies like SpaceX are trying to reduce the rocket costs by cutting out bureaucracy.
I'm currently exploring the option of Maglifters (whether they are possible or not) but until a space elevator is built, we may not be able to migrate the masses off world unto other moons and planets.
I have no problems with the numbers on rocket launch costs, they are in the order of US/European launch services.
I do have doubts with the numbers you mentioned for a space elevator. All numbers I have seen so far are very arbitrary, from ridiculously low (10$/kg) to even more than rocket launch costs. Until a full life cycle cost breakdown is made for a space elevator, I don't take any number for granted.
As for the space elevator; the development, operational and maintenance costs will be the main financial drivers, not the energy costs.
Also bear in mind that the space elevator will be an even bigger (multi-national) project than rocket development, so leaving bureaucracy out of the equation would be a mistake.
Readers:
Bryan posits only an incomplete answer: “How much exploration…will we be doing at $34,000 per kilogram? My guess is that we’ll do pretty much what we’ve done over the last 35 years since the last Apollo mission.”
Bryan is right, but misses the larger and more important distinction.
The key word is Exploration. At these costs space travel can never be anything more than exploration, experimentation. No commercial development, no human occupation of space. And that would be a tragedy!
Right now the world is awash in progress; commerce, trade, dare we say, capitalism has captured the harts and minds of even the communists, (except for the Islamists, of course). In 20 years or so, what will 7 to 8 Billion people do when they realize that human progress has ended, that the world standard of living will, from then on, be spiraling downward?
Yes space is fascinating, yes science is fun, but it is no longer a game for we, the technologically literate, to dabble in at our leisure. It’s time to get on a stick(!) - the future of the human race depends on the full development of space and our collective fate in this regard will be decided in the first half of this century.
Eric Westling
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The Mars Exploration Rovers had a cost of 425M including design, construction, launch and operation (1 yr) and had a mass of 174 Kg. That's about 2,400,000 dollars per kilogram for a complete program. I haven't found the cost of just the launch component.
If an 87 ton craft were to be delivered under the same sort of (low cost) budget, then the estimate would be about 200 billion dollars for the entire project. (Ten times higher than "projected" for the Mars Direct Plan.) Since manned missions are more complicated, the costs would be considerably higher.
Launch cost is not the major expense.
A space elevator would provide a steady stream of materials into orbit. I think THAT would gradually result in many changes to space technology and utilization. THEN the costs of everything related to space travel would drop considerably.
The problem with using delta-V in this kind of discussion is that the delta-V for ever higher orbits goes down with counterintuitive consequences in this context.
It is certainly clear that not having to carry fuel reduces both inetia and the work required to lift unspent fuel as you climb. The problem is that the percentage of energy diverted to lift fuel overall is not as great as one might have thought. Its significant to be sure, but not stunning. As a quick example, suppose I have a rocket with liftoff mass of 100,000 kg of which 90% is fuel and I burn it at a steady rate such that all of it is gone in 200 secs. Assume that the exhaust velocity is 4500 m/sec (space shuttle main engine performance). The result is that my cutoff velocity going straight up (disregarding coriolis drift) is about 8470 m/sec at 450 km. The percentage of my total enegy invested in potential energy is only 10.8%. Looked at that way it does not seem like I wasted much by carrying fuel on the way up.
So, if you are going to contrast this with space elevator operations you need to stop and think about how really hard it is to compare things. In my example the fact is that the rocket had lots of kinetic energy at that point which, albeit carried only by the payload, will get dissipated by potential energy as it keeps going up. The space elevator cargo moves slowly (compared to a rocket) and so never picks up a significant amount of kinetic energy. But that's only part of the reconciliation. At some chosen reference point the elevator cargo just gets to hold on to the ribbon and sit there -- whereas the rocket does not quite have an equivalent state.
So if you are going to compare efficiencies you have to construct roughly equivalent scenarios for both systems. We might agree to equate energy required to propel a rocket payload that drifts up until it turns around (somewhere) with the energy the elevator needs to lift a payload of same mass to the same turnaround point. Or maybe you'd like to tweak that comparison some more.
Below GEO, payloads will need their own boosters to achieve stable orbits. Above GEO the elevator could, in principle, fling its cargo into space though maneuvering fuel and booster hardware would still be needed.
Its messy and so its also vulnerable to criticism. I for one would like to see a cost (energy and cost of energy considered) comparison that is convincingly rigorous. So would the cynics and the true believers. Somebody should work on this very seriously. And no, I am not volunteering -- but I will critique the result. Maybe these comments will help someone get started!
I want to thank the individuals that responded to my blog. I would like to comment on their comments:
Darnell Clayton
It is certainly true that the Space Elevator has the disadvantage of being only at the conceptual level now. My argument is that if you do a “requirements analysis” you can conceive of the kinds of launch costs you need to dramatically increase our access to space. Pursuing that type of system you are forced to eliminate chemical rockets. The Space Elevator is the kind of transportation system that will open space as a place to solve problems here on Earth.
I request that Darnell carry out a study of Maglifters and share his results here on this website.
Jasper Bouwmeester
Jasper is correct about our cost estimates. In fact, the operational costs for the Space Elevator are one of the least understood aspects of the Space Elevator conceptual design. I would like to identify an industrial or process engineer to take this problem and give us a first cut at the operational scenario and costs. However, I used triple the cost estimates of my colleagues. Finally, I do feel confident that an infrastructure of Space Elevators will be subject to the economy of scale and will be much lower in cost/kg than the first elevator.
Eric Westling
Eric expresses concisely the fear many of us harbor concerning the development of the Space Elevator: By the time society at large realizes the imperative of building the Space Elevator to open space as a place to solve problems here on Earth, our societies will not have the resources to actually build the elevator. Energy use is currently a direct measure of the wealth of societies. If we delay until the world has only expensive energy, we may miss a step in our evolution and our society could be forfeit.
James Hughes
I think it is important to realize that no one would place a $100,000 satellite on a $100 million launch vehicle. Because space access is so expensive, the payloads must be of very high reliability including their ability to survive the violent rocket launch environment. In this way launch costs do set the stage for expensive payloads. A few of my colleagues at Los Alamos National Laboratory could build inexpensive space instrumentation in their garage if launch costs were low enough to allow for a few failures and had enough capacity to allow easy access. Imagine a world in which every high school in America hosts a “space fair” in which the winning entry is launched into space by holding a bake sale to pay for the launch. What kind of world would we live in then?
Steve Patamia
Steve goes through some analysis to try to compare “hanging” on the Space Elevator and a similar rocket launch. I actually meant that the Space Elevator will be placing the payloads into orbit.
Remember that to place a payload into LEO, the Space Elevator must take the payload very high (around 14,000 km) and release the payload into a highly elliptical orbit. A small rocket is used to circularize the orbit. The elevator has provided the vast majority of the energy for the orbit.
I agree with Steve in calling for more work on these aspects of Space Elevator placing payloads into orbit so direct comparisons to be made. In contrast to Steve, I would be happy with a series of cogent, back-of-the-envelope calculations to give us a general understanding. Such a series of simple calculations would match our current conceptual understanding of Space Elevator technology.
Please post more responses!
Hi Bryan,
Well,of course, you can choose orbital placement at a specified altitude to be the result to be compared. Your choice of a particular orbital goal -- say LEO versus GEO -- will no doubt alter the results. You can also decide that the comparable involves reaching and staying at some altitude with further exploits in mind. For example "hanging out" at GEO (where hanging on is not itself a problem though hanging around an oscillating ribbon might be) could be a valid goal for comparative methods since the objective would reasonably be to compare the cost of assembling a larger space ship or armada before firing off to more distant objectives. Lots of choices -- and no doubt distinctions in the outcomes. If you can do this using back of the envelope calculations then I certainly do not object. I've done some analysis (since my original post) myself and found that the penalty of carrying fuel to a selected altitude can be characterized succinctly -- but it remains only a piece of the picture.
Having said all that, your article started out talking about the inherent limit described by the rocket equation. Let's return to *that* for just a minute. I think you were correct to frame the issue in terms of delta V since, really, that is what the rocket equation is about even more than mass ratio effects -- i.e. its about exhaust velocity as a limit on interplanetary and interstellar travel times. My concern was that getting carried away trying to fit this into the problem of the cost to orbit was misplaced and much more difficult to quantify than you tried to achieve.
So, play around all you want putting things in orbit by rocket versus space elevator, though if you want to talk about the particular consequences of lifting off with the fuel you need you will be surprised to discover that the penalty is not as onerous as generally believed. I am not saying that the space elevator will therefore not save much -- my personal suspicion is that could indeed save lots of bucks -- but you will have to work much harder to show that comprehensively than anyone I have seen attempt so far. It would be great if someone could make a convincing succinct case with a little math -- but cost accounting is the single hardest kind of accounting that there is and I suspect that this problem will not be solved on the back of an envelope precisely because it is not so much a physics problem as it is an elusive economic one. It is, in fact, too hard for me to take on without time and lots of financial support. Offer me a grant and I will reconsider.
Hi all
Newbie here with a question on whether the 'circling line' concept has been considered for the space elevator requirement.
This idea is based on the idea that an aircraft at altitude can be banked and piloted into a turn to circle continually around any given spot on the earth's surface. If a line is dropped from the aircraft it can be tethered on the ground, at the centrepoint of the aircraft's orbit. This end of the line can be anchored and the aircraft can be put into a spiral climb or dive whilst circling, to keep the tether taut. The line will thus slowly trace-out a conic path in the sky, with the angle depending on the height and the radius of turn of the aircraft. A payload can thus be lifted up the line to the aircraft, and lowered from it.
In essence it represents an instance of an elevator based on a type of geostationary orbit.
For a continually circling aircraft, the tether could be used to transmit fuel, and/or electrical power, up to the aircraft. So, an aircraft could consist of an airframe, suitable motors, and an automatic flight control system, but would not require a fuel-load.
The tether could thus be used to lift and lower payloads to and from the aircraft. The maintenance of the desired height with variation in payload would be compensated by adjusing the angle of wing incidence as normal for an aircraft.
Failure modes could include the aircraft being released to glide to a normal landing, with the payload descending via parachute, and so on.
Given that such an aircraft could be constructed to fly at an altitude of say 30 km plus, the question would be whether this system could be a useful part of a space-elevator. Whilst it might make for a shorter elevator cable and thus reduce the gravitational loading, as a concept it might be interesting to see it as a first-stage of a multistage system.
For example whilst the limit of the first stage of circa 30km is determined by aerodynamic considerations, we can pump huge continual amounts of energy up there, for example liquid or gaseous fuel (or even coal!) without the payload penalty. And so we could postulate a second-stage circling-body system with non-aerodynamic engines providing the thrust as well as the lift-forces.
At this point, we can conceive of progressing by added circling-body stages right up to the geostationary orbit, with the stage-lengths being determined by the strength of existing ordinary materials.
In conceptual terms, I suppose the idea is a dynamic version of Goddard's rocket-stage thinking, as it gets around the fuel-mass problem.
I'd be keen to hear what people reckon on the concept, especially if they can pick holes!
(Incidentally, one of the potential uses for a 'space-elevator' even limited to the aerodynamic limit of around 30km, is that at this height an aicraft with an LD ratio of 50 could glide 1500 kmn without using engine power. So if the stratosphere-elevator was used to hoist up airliners and let then glide to their destinations, wouldn't that make a really 'green' airline?)
While an interesting idea, I see a few problems. First, the higher the aircraft goes, the less atmosphere and thus the less lift it can generate, the less lift it can generate the less weight it can support, and the less weight it can support the less payload it would be able to bring up. Burt Rutan has found a "sweet spot", an aircraft capable of lifting to altitude a vehicle capable of launching from that altitude the rest of the way into space, but only barely... flights on SpaceShip One can only achieve a couple of minutes of weightlessness at the top of a very low (in space terms) suborbital flight. If the White Knight also had to support a tether that was over 50,000ft long, I doubt it would be able to lift a craft capable of reaching even the very edge of space. Then there is the problem of transferring a liquid fuel up this tether... Oil rigs can pump oil up from depths of 30,000ft or more, but this requires bringing it up in multiple "steps" through extremely strong (and very heavy) steel pipes. If you had a tether stretching 50,000ft into the air full of water, the water pressure at the bottom of the tether would be over 20,000 pounds per square inch!
However, as a concept, I like it, and I'd recommend one change... instead of using an aircraft at the top, use a solar powered helium-filled airship. It wouldn't have to move, just support the weight of the tether and the payload, and the tether wouldn't have to be able to lift fuel (you could replenish helium through the tether of course, but that would actually make the tether lighter, and it wouldn't have to be pressurized at all). The problem with this is that the airship would have to be MONSTROUS. Since air is so much less dense at that altitude, it would require a HUGE airship just to get that high, let alone to be able to support a significant amount of weight. And then you reach the real problem... Helium is expensive, and becoming more and more scarce. Hydrogen is cheap and plentiful, but it's bad habit of exploding generally rules it out. The idea of a "rockoom" (rocket launched from a balloon) isn't at all new, but with Lockheed Martin now developing a whole new generation of hybrid airship, perhaps it's time for a new look. For example, what about a modular design? Or airships that do fly a circular path, generating some lift from good old aerodynamics like Lockheed's new hybrids?
Hi, just a question? Why is everyone trying so hard to creat ONE tether or one carrier or One "MONSTROUS" airship? I would approach the task in an organic way...not so stiff. If I had the possibility to experiment, I would first try to send a thin light PIPE(common materials) in a sort of carrying Cocoon driven by helium 200-300 Km in space, and in case it works than we got it, because the Pipe could than transport materials for bigger better and more Hi-Tech systems. Like pipe-mail. I excuse my unqualified opinion. But I really think I could make it in just ca. one year. Its maybe too simple to understand ? I would love to see it before I go. mehrdad