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NOTE: This is not the place to question the fundamental veracity of nuclear propulsion in aircraft and spacecraft.

For context, in a spaceplane, the higher speeds you can attain under air-breathing power the better, as they are inherently more efficient than rocket engines. In both air-breathing and rocket propulsion, using a nuclear heat source promises much higher efficiency than a chemical one, with almost infinite and ~1200 seconds specific impulse potentially being attainable, respectively. However, for any turbojet engine, principal limits in speed are the compressor inlet temperature and the maximum temperature of the hot section. In chemical jet engines, the energy density of the fuel puts a hard cap on the latter. Even notional nuclear turbojets, though, also have those same limitations due to the material constraints of constructing them.

So, in order to increase the speed of a turbojet, the compressor must be actively cooled and the temperature of the hot section boosted beyond what is normally possible. These problems could be potentially solved by using an air-breathing analogue of a LANTR (Liquid oxygen-Augmented Nuclear Thermal Rocket) cycle with elements of SABRE. First, the liquid hydrogen later used in rocket propulsion would be pumped through the compressor section, lowering its temperature while also recovering its heat energy. This would also aid in nuclear reactor thermal efficiency. Second, this hot hydrogen gas would be pumped through the nozzle, cooling it down while recovering more heat. This would then be injected at the nozzle throat. The very hot hydrogen and extremely hot nuclear-preheated air would combust at astronomical temperatures, potentially allowing higher speeds to be reached before the spaceplane would have to switch to pure rocket mode.

So, the question can be expanded and rephrased as this: Would such a "LHANTTJ" (Liquid Hydrogen-Augmented Nuclear Thermal TurboJet) engine actually produce a higher specific impulse than the subsequent pure-rocket stage and a higher exhaust velocity than reachable through the previous pure-nuclear turbojet stage? Also, would the systems be worth the extra weight?

Edit: If the answer to the first element of the above question is "yes", by about how much would the exhaust velocity and specific impulse be greater?

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No.

LANTR didn't increase ISP either. Its advantage was that it had increased thrust (relative to both weight and cost). @rokomijic's answer gives the reason why it doesn't:

Exhaust velocity ($v$) is proportional to the square root of temperature over particle mass. As:

$\frac {1}{2}{\overline {mv^{2}}}=\frac {3}{2}k_{B}T.$

Chemically combining to different molecules can increase $T$ but will also increase $m$ (by a large fraction if you start with H2). Given that nuclear already gets things very hot, you'd need a really big increase in temperature to get exhaust velocity back up to the 'pure' nuclear rocket. Way beyond what combustion can achieve.

But that doesn't make it a bad idea, the benefit is that for a given thrust requirement, you get away with a lighter, cheaper engine.

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I'm assuming that the goal is to get to orbit with a hybrid between a nuclear turbojet and a nuclear thermal rocket.

Be aware that H2O is a significantly heavier molecule than H2, so mixing your hydrogen with oxygen is unlikely to improve V_exh, even if it increases the temperature somewhat.

On that basis I'm going to guess the answer is no, and that the best solution for your hybrid is to get to the fastest possible steady state as a turbojet, then pitch up and escape the atmosphere as a nuclear rocket.

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