4
$\begingroup$

when an aircraft is flying from let us say Paris to New York on a fixed altitude it is not flying on a straight line but on an orbital / circular trajectory around the earth. Assuming, the aircraft is flying Mach 5 or even faster, does this significantly reduce the amount of required lift, since the centrifugal force is acting on the aircraft?

$\endgroup$

1 Answer 1

7
$\begingroup$

We can calculate this by comparing the centrifugal force to the force of gravity:

$$ F_\text{centrifugal} = m \omega^2 r $$ $$ F_\text{gravity} = G \frac{m M}{r^2} $$

The ratio is then given by

$$ \frac{F_\text{centrifugal}}{F_\text{gravity}} = \frac{m \omega^2 r}{G m M / r^2} = \frac{\omega^2 r^3}{G M} $$

Let us assume we fly at FL600 (where Concorde flew), so $ h \approx 18 \, 288 \, \text{m} $ and then $ r = R + h$ (with Earth radius $R$ and mass $M$). The speed of sound at FL600 is $ c \approx 295.1 \, \text{m/s} $, which gives a speed at Mach 5 of $ v = 5c \approx 1476 \, \text{m/s} $. On a circular "orbit", we have $\omega = v/r $. Plugging in the numbers, I get

$$ \frac{F_\text{centrifugal}}{F_\text{gravity}} \approx 3.48 \, \% $$

So flying at Mach 5 would reduce the required lift by about 3.5%.

However, as Ralph pointed out in the comments, you need to take the Earth's own rotation into account. If you are on the equator, the above effect is already ~0.3% without moving at all (w.r.t. the Earth's surface). You need to add or subtract the Earth's motion from the velocity above. On the equator, that gives a difference of about 463 m/s (add when moving East, subtract when moving West). Repeating the calculation above then gives:

$$ \text{East:} \; \frac{F_\text{centrifugal}}{F_\text{gravity}} \approx 6.02 \, \% \qquad \text{West:} \; \frac{F_\text{centrifugal}}{F_\text{gravity}} \approx 1.64 \, \% $$

$\endgroup$
5
  • 4
    $\begingroup$ Is this a case where the rotational velocity of the Earth would actually affect the numbers, i.e. adding it to an eastbound jet, subtracting it for a westbound, and (roughly) no change for a north or southbound jet? $\endgroup$
    – Ralph J
    Jan 16, 2023 at 12:13
  • 2
    $\begingroup$ @RalphJ Good point! In space, there is no difference between retrograde and prograde orbits (Space.SE). But in the atmosphere, it can make quite a difference. I added a paragraph with some more details. $\endgroup$
    – Bianfable
    Jan 16, 2023 at 13:45
  • 1
    $\begingroup$ So the tl;dr might be something like "Yes, but not much, and mostly if you're flying towards the East. The accountants will be pleased at a bit of fuel saved." $\endgroup$ Jan 16, 2023 at 20:34
  • 2
    $\begingroup$ @WayneConrad: The extra lift scales with speed squared, so Mach 10 would have 4x the effect, about 14% +- east / west difference. But yeah I guess that's still "not much", and much higher speeds Mach 20 is more like re-entry from actual orbit. Even Mach 10 is much less realistic than Mach 5. $\endgroup$ Jan 16, 2023 at 23:34
  • 1
    $\begingroup$ @WayneConrad I think the accountants will be pretty mad at you for flying at Mach 5 because that really increases fuel consumption :D $\endgroup$
    – Bianfable
    Jan 17, 2023 at 7:15

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .