What is the theoretical fuel cost to launch 1 kg of payload to orbit on an ideal rocket (rocket with 0 kg dry mass)?
We can use the rocket equation to get a rough idea of the fuel required.
$$\delta V = v_e ln \frac{m_0}{m_f}$$
- $\delta V$ required to reach LEO is 9.4 km/s
- $v_e$ is the exhaust velocity of the rocket, 3 km/s is pretty good for a chemical rocket
- $m_0$ is the initial total mass, including fuel.
- $m_f$ is the final mass, 1 kg.
We need to solve for $m_0$.
$$m_0 = m_f e^{\frac{\delta V}{v_e}}$$
Plugging in the numbers...
$$m_0 = 1 \text{ kg} \times e^{\frac{9.4 km/s}{3 km/s}}$$
$$m_0 = e^{3.13} \text{kg}$$
$$m_0 = 23 \text{ kg}$$
An initial mass of 23 kg means 22 kg of fuel to get a 1 kg of payload on a zero mass rocket to LEO.
According to this answer a Falcon 9 uses 2:1 LOX to RP-1 so that's about 14 kg LOX and 7 kg RP-1. And they say LOX is about \$0.20/kg while RP-1 is \$1.20/kg.
About \$11. Though so little probably won't get you SpaceX's bulk discount.
However chemical rockets are used for lift off because they have the necessary oomph to lift the many tons of rocket, fuel, and payload against the force of gravity. With just 1 kg you might be able to get away with a more efficient, but less powerful, method of propulsion.
1 kg in Earth gravity exerts only 10 N of force. Our most efficient engines are ion thrusters. There's a whole raft of reasons it's a bad idea to use these inside an atmosphere, but let's say they do work. The problem remains that existing ion thrusters have thrusts measured in micro Newtons. Some Magnetoplasmadynamic thrusters (MPDT) on the drawing board can, theoretically, provide the necessary thrust.
Let's assume we have a zero mass MPDT with enough thrust to lift 1 kg. How much fuel would it need? These have exhaust velocities up to 60 km/s.
$$m_0 = 1 \text{ kg} \times e^{\frac{9.4 km/s}{60 km/s}}$$
$$m_0 = e^{0.157} \text{kg}$$
$$m_0 = 1.17 \text{ kg}$$
1.17 kg initial mass means 0.17 kg of fuel to lift 1kg of mass into orbit. Our hypothetical zero-mass MPDT would need about 12 N of trust to lift its payload an fuel. That is inside what we believe to be achievable with an MPDT (though zero mass and operating inside an atmosphere is not).
However, this is 0.17 kg of a noble gas. Traditional ion thrusters use Xenon propellant. At roughly \$850/kg we're looking at about \$150. However, MPDTs could use much cheaper propellant such as helium, hydrogen, or lithium.
Unlike chemical rockets, ion thrusters use electricity to accelerate ions. They need a power source. Typically these are solar panels, but an MPDT requires far more power such as a small nuclear reactor or power beamed via ground-based lasers. We would also have to assume the power source is zero mass.
Let's put this to the limit. What if the exhaust velocity was the speed of light, a photon rocket! Let's be clear, this is like trying to move your car with a flashlight. There is no way it has enough thrust to launch 1 kg, this is just an exercise.
$$m_0 = 1 \text{ kg} \times e^{\frac{9.4 km/s}{300,000 km/s}}$$
$$m_0 = e^{0.0000313} \text{kg}$$
$$m_0 = 1.00003 \text{ kg}$$
A photon rocket needs 0.03 grams of fuel to lift 1kg of payload to LEO. That is the hypothetical best we could do assuming we can build a zero mass rocket.
