I’ve been following aviation closely since the mid 1960s when I got old enough to have some understanding of the words alongside the pix. There’s never been a loop in an airliner that I heard about.
IMO/IME
You’re not going to succeed at performing a loop in an airliner. And definitely not in the hopelessly under-powered and high drag 727.
The problem is the comparatively low G-limits multiplied by the comparatively low engine power.
Even lightly loaded with no pax, low fuel load, etc. It doesn’t work. You can certainly build up speed in a shallow dive at full power to right up near or even a bit past redline maximum speed.
Then you start to pull the nose up. It’ takes 1G to offset gravity, and 2.5 is the limit, but for a one time experiment where you’re going to throw the airplane away afterwards you can probably push that to 3-ish. Which means you only have 2 G available for turning in teh vertical dimension. one of your 3 usable G’s is just fighting gravity. This concept is called “radial G”.
A net 2G pull means that at somewhere just before or just after getting vertical (= nose pointed straight up), you run out of airspeed and fall or stall out of the maneuver. ANd remember you can only sustain the full 3G pull wihle you’re fast. As soon as your speed starts decaying you need to start reducing G to avoid a high speed stall. You might start at 350 knots in a typical airliner, but by the time you’re down to ~200, you can’t pull much more than 1G without stalling.
With massively more power you could have thrusted your way all the way to the inverted top, then used gravity plus thrust to gain speed down the backside. With massively greater G available you could have turned more tightly = less time spent, and gotten at least well around to onto your back before the speed ran out and you sort of mushed your way over the top then once pointed downhill inverted you could wait a few seconds to regain enough speed to resume the pull.
If you did somehow succeed at the first half of a loop and now find yourself slow and inverted, but not stalled while pulling gently at low G. the good news is now gravity is pulling the same way you are and so now that 1G is adding to your turning, not subtracting from it. The bad news is that since you’re very, very slow, you may only be able to pull 1/2G without stalling. So you’re still only netting 1-1/2G turning force.
And now you get to the next problem. Avoiding getting supersonic and shedding parts on the way back downhill. Again it comes down to the hardest you can pull doesn’t get you past vertical straight down and into the dive recovery quickly enough to avoid having the airspeed get out of range. On the way up you ran out. On the way down, your cup most surely runneth over. That’s bad.
And all of that assumes you have a G-meter in your airliner so you can optimize your performance up until the airspeed exceedance low or high. Real airliners don’t have them. Guesswork on the pilots’ part can only detract from the theoretical optimum path with the best (but still hopeless) chance of success.
The effects of altitude on controlability are complicated, but by and large by flying based on indicated airspeed, many of those issues wash out.
There are two real problems with high altitude applicable to attempted loops
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Engine power declines precipitously with altitude. At high altitude cruise total thrust is maybe 10% of sea level thrust. Such that even getting unwitting slow in level flight can provoke a situation where ever full throttle won’t accelerate you even going dead horizontally. That situation rapidly deterioates and the only remedy is a descent (not quite a dive) to rebuild speed by trading altitude. Engines that can barely keep you at constant speed level are certainly not going to have the excess oomph to thrust you upwards and around in a loop.
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Mach. I said by flying based on indicated airspeed, controlability is about the same as at low altitude. E.g. seeing 250 knots on the airspeed indicator means the control surfaces are feeling the same poundage of air going by per second as they would at 250 actual knots at sea level. The gotcha is the air is actually going by at say, 450 knots, so you’re processing twice as much space of air per unit time, with each volume of air holding half as much molecules and pressure.
And this is where Mach gets into it. Your maximum speed in a jet is twofold. If you exceed either the maximum indicated airspeed or the maximum Mach bad things happen. Excess airspeed leads to loss of fragile moving parts. Excess Mach leads to Mach tuck and loss of control. Down low indicated airspeed is the limiting factor. Up high it’s Mach.
Trying to start a loop up high means your limiting airspeed, instead of being, say, the 350 it is down low, is now 250. Which leaves you even less headroom to slow down before you run out of indicated airspeed and stall.
More later. Gotta run.