Fundamental cost of putting stuff in orbit: theoretical minimum vs RLV

What is the minimum energy of orbit, and how does that compare to the energy in a chemical rocket’s propellant?

Accessing a 150km LEO orbit requires first the energy to get to 150km. That’s roughly (in Energy/mass, or J/kg, aka m^2/s^2, the unit I’ll mostly use here): 150km*9.8m/s^2.

Orbital velocity at 150 km altitude is just v=sqrt(mu/a), where the distance from the center of the Earth a = r_Earth + 150km. Mu is the “standard gravitational parameter” of Earth, or ~3.986*10^14 m^3/s^2.

(BTW, I’ll write numbers like 3.986*10^14 in a more compact notation: 3.986E14.)

So v= sqrt(3.986E14m^3/s^2/(r_Earth+150km)) = 7814m/s ( here is the google calculation: https://www.google.com/webhp?q=sqrt(3.986E14m^3/s^2/(r_Earth%2B150km)) ).

But we can minus the speed from the rotation of the Earth: v= sqrt(3.986E14m^3/s^2/(r_Earth+150km)) – 2*pi*r_Earth/day

Now we need to make this in terms of energy in order to add that potential energy from being 150km high:
E_specific (energy/mass) = .5*(sqrt(3.986E14m^3/s^2/(r_Earth+150km)) – 2*pi*r_Earth/day) + 150km*9.8m/s^2

Which is roughly: 28,480,000 m^2/s^2 or 28.5MJ/kg. That’s 7.9kWh/kg or just under $1 per kg to LEO at typical 10-12 cents per kWh.
And in terms of delta-v, it’s: v = sqrt(2*E) = 7550m/s or so.

That’s zero aero or gravity drag, launching due East on the equator. Imagine a 150km tall tower with a 100% efficient electromagnetic launch mechanism on the top, including the energy required to lift stuff up that tower and assuming no energy loss from the sled, no mass for the encapsulating of the payload, and 100% efficiency for electromagnetic launch. None of these are realistic assumptions.

Let’s compare with chemical launch. Assume a hypothetical stoichiometric methane/oxygen rocket engine operating at 3.7km/s exhaust velocity. This is very aggressive (especially at sea level), would probably melt the engine due to operating stoichiometrically, but it may actually be possible.

A stoich methane/oxygen mix, with methane having 55.5MJ/kg specific energy and the mix having 11.1MJ/kg, would have a theoretical exhaust velocity, if you totally convert chemical energy to jet energy, of 4.712km/s, so 3.7km/s isn’t physically impossible in the least (would be feasible in vacuum, but would require incredibly high pressures at sea level).
Anyway, let’s assume a mass ratio of, say, 25 for each stage. Let’s assume a 100 ton payload. The first stage weighs 120 tons dry (25 times that wet), and the next stage 10 tons dry (etc). That gets us 9km/s delta-v, which we’ll say is good enough, launching on the equator due East to 150km altitude.

Work: 3.7*ln((25*120+(25*10+100))/((25*10+100)+120)+3.7*ln((25*10+100)/(100+10)

We assume the dry mass magically can be recovered at no mass penalty (I will address this in another post…).

Mass of the propellant is: 120*24 + 10*24 = 3120 tons. Or 31.2 kg of propellant per kg to orbit. At 11.1MJ/kg, that’s 346MJ/kg of chemical energy in the form of methane. Natural gas is about $0.30 per therm in bulk. A therm is about 105MJ. So the cost of chemical energy to put stuff in orbit via chemical rocket like I described is actually ALSO $1/kg, and with arguably more realistic (though also aggressive) assumptions.

Moral of the story: It’s not, and never ever has been, about the cost of energy to get to orbit. Such arguments are flawed.

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7 Responses to Fundamental cost of putting stuff in orbit: theoretical minimum vs RLV

  1. Andrew_W says:

    Or 31.2 kg of propellant per kg to orbit. At 11.1MJ/kg, that’s 346MJ/kg of chemical energy in the form of methane. Natural gas is about $0.30 per therm in bulk. A therm is about 105MJ. So the cost of chemical energy to put stuff in orbit via chemical rocket like I described is actually ALSO $1/kg, and with arguably more realistic (though also aggressive) assumptions.

    You don’t seem to have allowed for the LOX.

  2. Andrew_W says:

    Though I suppose LOX and liquid CH4 probably cost about the same anyway.

  3. Paul D. says:

    Cost of liquefying natural gas is about $0.50/million BTU, which is maybe 20% of the current pipeline price in the US. I think LOX is considerably cheaper (in both energy and $ terms) than LNG.

  4. Chris Stelter says:

    The actual energy cost of liquefying methane is around 550kJ/kg. If we assume we’re only 55% efficient, let’s say it’s around 1MJ/kg.

    For oxygen, it takes like half that, so about 300kJ/kg, so 600kJ/kg after efficiency is taken into account. And you have 4 times as much oxygen as methane in a stoich mix, so we’re talking a total extra energy cost of around 3.4MJ per kilogram of fuel. That’s still small compared to our actual fuel energy. Whether this makes any difference depends on whether you’re using cheap fracked methane or you’re electrolyzing the methane. If you’re electrolyzing, then it isn’t significant. If you’re using fracked fuel, then it increases the price by about 30-80%. So you’re looking at like $1.5/kg instead of $1/kg. Not a big deal. The vast majority of the energy cost is in the fuel itself.

    By the way, at scales relevant to these calculations (i.e. ridiculously high flight rate), you’re better off with an on-site cryogenic liquefaction plant than trucking (or even piping) in liquid oxygen and liquid methane. Just transporting that much liquid cryogen probably makes up the majority of the cost, so you’re better off doing it on-site.

  5. Ian Woollard says:

    It sets a lower limit on the cost of achieving orbit.

    Some of the reusable launch systems I’ve modeled have a cost to orbit of around $100-$150/kg of payload. If the propellant costs ~$30/kg of payload that’s starting to be important; variation in the propellant costs can significantly affect the launch cost.

  6. Chris Stelter says:

    Good point. And interesting that Blue Origin, SpaceX, and Masten are all pursuing methane as the fuel for their RLVs. Methane (well, natural gas) is the cheapest energy source on the planet right now (and is actually pretty cheap to produce using electricity and air and water, just like on Mars… though not as cheap as fracking). This puts the propellant costs down through the floor. It even makes point-to-point transport via orbit (or near-orbit) look not-dumb.

  7. Allen Hall says:

    You guys are missing the real issues. Fuel is nothing and always has been and will be. I have been involved with the space program in the USA and Europe and I have over 25 years in the Industry. The cost is the rocket, the systems and the support. It takes a lot of people to put a rocket up without it exploding, which is a problem that ist likely to go away, as the the systems are designed to the ragged edge of performance. It is an unforgiving process and the costs come from the efforts in keeping a controlled explosion from becoming uncontrolled. Elon’s strategy, like everything is space exploration, is old. It will be a constant battle for them as reusing the rocket will require some overhaul… it will. As soon as the have a failure with a reuseable first stage, the costs will start increasing as more and more recovery procedures are put in place for reusing the hardware.

    Our current technology for space travel has not significantly change in over 70 years, it has only evolved a bit but its the same methodology with refinements and nothing is on the horizon that is going to change that. Anyone who believes we are making technological progress here is just deluding themselves. All we have learned how to do is blow our selves up less often, fundamentally it is still the same process that powered the V2 rocket in WW2. Ion propulsion isnt a new idea either. First proposed over 100 years ago and experimented with over 60 years ago, it isn’t a game changer as you cannot get into orbit with it.

    The truth is that we have gone backward the past 40 years. The pinnacle of our success was the Saturn 5. It could launch over 300,000 lbs into LEO and do it at a price competitive with SpaceX in adjusted 2016 dollars.. Since then the price to orbit has skyrocketed while at the same time our launch capabilities have shrunk to a fraction of what the Saturn 5 could have done. We could have put the ISS up in 5 launches, not the 100 that it took. . To top it off, America lost the ability for manned space flight 5 years ago. Now we are trying to pay private businesses to catch up with engineering we mastered over 50 years ago and they are struggling. There was never a launch failure in the entire Saturn family history of 3 rockets and 32 launches. In 2016 we are nowhere near that rate of success. We have come a long way baby … backwards.

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