There’s a certain misunderstanding common in aerospace that rockets are horribly inefficient and that long term we need air breathing ramjets or scramjets to efficiently launch things, with the idea that we can thus avoid accelerating oxygen to flight speed, which is considered wasted energy. “Airbreathing hypersonics are five times as efficient as rockets” they say. This, however, is not so.
The misunderstanding comes in part by considering oxygen as just as costly as fuel. Oxygen is not. It can be condensed out of the atmosphere with little energy and is available by the truckload at $100/ton or less. A dedicated production plant can produce it for as low as $10/ton. That compares to $1200 to $3500 per ton for industrial liquid hydrogen which is often the fuel being compared to.
A stoichiometric rocket burns 8 times as much oxygen as it does hydrogen. So if an airbreather consumes a factor of 5 times less propellant than a rocket, that means it consumes nearly twice the hydrogen!
Hydrogen requires the vast bulk of the energy to produce compared to oxygen, a couple orders of magnitude more energy. So for our purposes we can ignore the energy needed to produce liquid ...