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.
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.