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If a supersonic fighter jet wanted to reach the highest possible apogee it could in a parabola, what's the optimal pitch angle to do so? A too high pitch may result in insufficient thrust and a sooner stall while a too low pitch may result in lower altitudes than could be reached. But I don't think 45 degrees would be the optimal angle because some rocket planes ascend at a higher pitch, right (my question is on jet planes, however)? Or does it mostly depend on the plane in question?

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    $\begingroup$ A recent analysis [The symmetry of the Paris gun: when theory is foiled by atmospheric effects Physics Education, July 2019] of the WW1 'Paris Gun' points to an optimal firing angle for maximum range of 52,7º. Of course, this is not for maximum shell altitude, for which the best angle would have been 90º... $\endgroup$
    – xxavier
    Oct 26 at 9:53
  • $\begingroup$ @xxavier Interesting. $\endgroup$
    – Giovanni
    Oct 26 at 9:57
  • $\begingroup$ This is a high school physics or trigonometry problem; it's not really even about aviation per se. I think the core question, however, is intended to be about zoom climbs rather than ballistics. If this is true, please edit the question to clarify that's what you're talking about. $\endgroup$
    – Zeiss Ikon
    Oct 26 at 11:09
  • $\begingroup$ @ZeissIkon I rephrased the question a bit, but I made clear it is concerning air-breathing jets before already, hence it belongs to aviation imho. $\endgroup$
    – Giovanni
    Oct 26 at 11:33
  • $\begingroup$ Much clearer what you're after, and it no longer looks like a high school problem. $\endgroup$
    – Zeiss Ikon
    Oct 26 at 12:05
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It depends on the plane in question, more specifically the weight, wing, excess thrust, G loading limits, and its Vnever exceed speed.

Looking at the flight profile of the modified F-15 "Streak Eagle" of the 1970s, one can see the technique is to build up as much speed as possible, then try to climb at as high an angle as possible.

The Streak Eagle, stripped of all unnecessary equipment (including its paint), and carrying limited fuel, was able to achieve a thrust to weight ratio of 1.4! This airbreathing rocket could accelerate in vertical flight!

But the drag of turning to vertical cost the jet precious airspeed (kinetic energy), so the optimum climb angle was around 60 degrees, with the transition from building speed to climbing done at around 30,000 feet, which is above the denser layers of the atmosphere but still allows for aerodynamic manuvering. As the aircraft climbs into thinner air, less oxygen means less thrust available. The highest altitude comes from the straightest path$^1$ up.

For more typical subsonic aircraft with a thrust to weight ratio closer to 0.25, speed can be built in a shallow dive, but will be rapidly lost in a steep climb. Their "parabola" will not be much different than the max service ceiling of the aircraft.

In summary, the optimal profile for a given aircraft for a maximum altitude parabolic flight is to climb as steeply as possible as fast as possible from the highest altitude possible with the least amount of drag. Essentially, at that point, the aircraft is a projectile.

$^1$ this cost General C. Yeager one last bit of glory when the NF- 104 was "pulled up" to increase angle of climb during the ballistic phase of flight, resulting in a lower apogee (from increased drag), and a stall.

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In general, the highest altitude for a ballistic parabola flight in an air-breathing jet (which will be flamed out for the highest part of the flight, assuming it's the kind of high performance craft like a MiG 31 or F-104 that's usually used for this kind of maneuver) will be a trajectory that maximizes the vertical component of velocity.

This is done with what's called a "zoom climb" in which maximum horizontal speed is traded off for maximum vertical velocity with a pull-up. Although a simple-seeming maneuver by appearance (and one that dates back to the First World War), this is a complex maneuver to optimize, and is highly dependent on the aircraft type used, because different combinations of airframe and engine have different high altitude speeds and remaining thrust at a given altitude. The common factors, however, are a near-terminal horizontal sprint, a pull up at a G level that wastes the least energy, and a final angle that gets the most acceleration or least deceleration during the remaining engine run time before flameout due to lack of air pressure to keep the engine running.

As noted, the end goal to maximize altitude (hence zero-G time for the paying passenger, as that's pretty much the only use of this kind of maneuver since the early days of almost-spaceflight) is to maximize the total of vertically oriented kinetic energy and gravitational potential energy at the point where power is lost. Since the altitude of engine flameout due to lack of air pressure is fairly constant for a given engine, this reduces to maximum upward velocity at flameout.

For the few aircraft types capable of this maneuver (zero-G parabola above flameout altitude), there will be an optimum pull-up rate for each type that gives the highest final vertical velocity, and since altitude is the goal rather than distance, the final angle will be significantly closer to vertical than the 45° for no-drag maximum range, but lower than the 90° for maximum altitude of a purely ballistic trajectory (because the longer T/W can be kept above 1, the more velocity can be kept or built up).

As far as I'm aware, the only aircraft type currently in use for this kind of flight is the MiG 31, a high altitude interceptor originally intended to shoot down the SR-71 (some claim it never came close, some records indicate otherwise). In additional to high altitude and high speed capability (around Mach 3 at 20 km, give or take), it has a reaction control system that permits maintaining control when the air is too thin for control surfaces to be effective. The F-104 was capable of this maneuver, but due to lack of RCS as built it was a very dangerous stunt rather than something that could be done routinely.

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  • $\begingroup$ So the pull-up shouldn't be too fast, instead the pitch should slowly get to more than 45°? $\endgroup$
    – Giovanni
    Oct 26 at 12:33
  • $\begingroup$ Excessive G increases drag too much relative to lift, wasting energy. Worse, at high enough G and high altitude, there's a risk of accelerated stall, even at high speed, which (in supersonic flight and extreme altitude) might lead to an unrecoverable condition (flat spin, airframe breakup, etc.). $\endgroup$
    – Zeiss Ikon
    Oct 26 at 12:36
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    $\begingroup$ A good historical reference is the F-15 Streak Eagle flight. The last flight to 30km included a zoom between 30k ft to 40k ft to Mach 2.2 followed by a 4g pull to a 60 deg climb. The engines flamed out around 60k ft. Above that it was a ballistic flight where the goal was to maintain near zero angle of attack for minimum drag and to ensure the plane descends in a controllable attitude. There were a lot of simulations run to determine the best flight profile. $\endgroup$
    – Gerry
    Oct 26 at 12:57
  • $\begingroup$ There are at least two records of MIG-31 having being in a position to shoot SR-71 down at will. It was, in fact, a successful repellent for SR-71 missions in Russian airspace. $\endgroup$
    – Jpe61
    Oct 26 at 14:04

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