Air breathing saves wet-mass, but increases dry mass. So far, the dry mass increase has been enough to make it not worthwhile. Especially since the change in trajectory required results in much higher drag losses, and more worse thermal performance (rocket gets hotter)

With the notable exception of anti satellite weapons which have often been fired from military aircraft. It’s cheaper because an individual flight on an existing high altitude high velocity aircraft is cheap. Unfortunately, it would require an enormous number of space flights for this first stage to be worthwhile otherwise.

The most advanced project of this nature is Skylon and its Sabre engine (https://en.wikipedia.org/wiki/Skylon_(spacecraft)).

> There have been some experiments in this direction, notably Richard Branson's.

No? Richard Branson invested in SpaceShipTwo which uses a hybrid rocket motor with liquid oxidizer and solid propellant. It is NOT air-breathing.

It's air-launched but the carrier aircraft is powered by traditional jet engines (not rockets).

> but the carrier aircraft is powered by traditional jet engines (not rockets).

Hum... That's the name we give to air breathing engines.

But we're talking about air-breathing rockets, not merely engines. Crucially, a rocket supplies its own oxidiser, and an air-breathing rocket supplements this with intake air. An example of such would be Sabre: https://en.wikipedia.org/wiki/SABRE_%28rocket_engine%29

Ok, you are reacting to the exact term the GP used, but it doesn't look like it was used as a term of trade, just as a general characteristic.

Using an airplane as a first stage of a rocket gives the exact same air-breathing advantages as using a rocket (or better, because the planes uses air as reaction mass too). That makes both equivalent on practice. Yes, an hypersonic engine makes for a larger stage, and an dual one that can operate with and without air makes for an even larger stage, but they are of the same broad category.

I am not an expert on this, but I have read that the problem with that sort of rocket is you have to go horizontally through the atmosphere for a long ways to get up to speed. That produces a tremendous amount of drag that requires a corresponding amount of fuel to overcome. With a regular rocket you go vertical from the start and get out of the atmosphere in a minute or two, and then turn horizontal to get into orbit, and that takes a whole lot less fuel.

Or non-chemical propulsion.