# Will a jet flown on a parabolic arc reach its highest altitude on a warm day or a cold one?

In level flight, on cold days jet-powered planes can fly higher because the air is denser. However, at the same time the true airspeed is lower because of the denser air.

At warmer temperatures a plane accelerates to a higher airspeed, and as a result the higher speed could allow for enough initial speed to reach higher peak altitudes when flown in a steep parabolic arc.

Can a (military jet) plane fly higher in a parabolic flight when the air is warmer?

• Interesting question! I reworded it to make it clearer, let me know if this still captures your intent. BTW, is it true that planes accelerate more quickly when the air is thinner? I would have thought the opposite, since there is less oxygen. And did you mean higher delta-v, or higher kinetic energy? Commented Oct 25, 2021 at 21:25
• @KennSebesta You're right as technically the plane's delta-v is always the same, I rather mean kinetic energy, I clarified that. Rather than accelerating more quickly the plane reaches a higher true airspeed than at colder temperatures (but this goes hand in hand), and I think it really works, higher apogees can be achieved. The reason the plane goes faster is the warm air which is less dense so there is less air resistance I guess. Commented Oct 26, 2021 at 5:22
• This is a very interesting questìon, and not a trivial one to answer. Yes, in hotter air TAS is higher, but how about engine efficiency? And how about lift, which is also imperative when considering flying parabolic, (but not ballistic) trajectories. Commented Oct 26, 2021 at 6:28
• @Jpe61 Um, wait, what's the difference between parabolic and ballistic trajectories? My question is concerning those that fly as high as possible, with a high pitch, to altitudes 110~130,000 ft where air is near vacuum. Commented Oct 26, 2021 at 7:11
• Strictly speakin ballistic trajectory is "the path of an unpowered object, as a missile". Commented Oct 26, 2021 at 7:22

## 1 Answer

The maximum airspeed of any aircraft is determined by thrust and drag$$^1$$.

At first glance it would seem lower density of warm air would favor faster true airspeed (more kinetic energy to climb), but (for airbreathing engines) one can consider the effects of warmer, less dense air on thrust production as well.

Jet engines rely on delta temperature by combustion of fuel to produce propulsive thrust through the expansion of gasses, just like their piston cousins. Therefor, colder air gives them a double advantage of more oxygen to burn and greater delta T.

This is why airliners like to fly in the Tropopause, a region in our atmosphere that is always very cold.

Very luckily, for our parabolic endeavors, the atmosphere warms (and thins) above the tropopause, favoring maximum altitude during the ballistic phase of flight.

Analysis of the flight profiles of the NF-104 and the F-15A "Streak Eagle" show a very similar approach, using maximum thrust to go supersonic in the tropopause, around 6-8 miles high, before angling upwards.

But we must remember to think vertically and have temperature data from all levels of the planned flight.

One other thing about a "cold day" on the ground is that pressure actually drops faster with altitude, enough to cause significant altimeter errors in extreme cases.

So, with the Streak Eagle, they did it on a very cold day.

$$^1$$ and structural strength to handle that speed

• Pressure drops faster with altitude on colder days? Commented Oct 29, 2021 at 13:53
• Yes, and if it got much colder air would be a puddle on the ground. It's "pressure drops faster with altitude". Commented Oct 29, 2021 at 15:17
• In other words, e.g. FL300 above the poles is at a completely different absolute altitude than above the Sahara desert? Commented Oct 29, 2021 at 15:56
• I have heard the stratosphere can be as low as 25,000 feet at the poles. Commented Oct 29, 2021 at 17:04
• Yes that's true (I even mean to have heard 20,000 ft) and at the equator the tropopause can be as high as 70,000 ft, but these values are about where the temperature stops decreasing, and where it starts rising again with altitude. This doesn't have to do with pressure directly, but with temperature and how high cumulonimbus clouds can go. Commented Oct 29, 2021 at 18:02