an aircraft is tracking a great circle track, there is a large bearing change between departure and arrival points. If the aircraft were to fly the rhumb line heading of the track at the departure point, would it end up arriving overhead the destination airport?
The answer is
it depends on wheter the equator is crossed, and how
The reason for this is as follows:
an aircraft is tracking a great circle track, there is a large bearing change between departure and arrival points.
This indicates we are dealing with a great circle which is not along the equator, nor along a meridian. (This can be proven mathematically; since the difference in a great circle track and a rhumb line track in either end is +- 0,5*dlong*sin mlat, either dlong or sin mlat have to be zero for the function to be zero (ie for there to be no "bearing change"); dlong=0 occurs on a track exactly north/south (along a meridian), and sin mlat=0 occurs where the mean latitude is zero. This is along the equator (on a east-west track along the equator), but also at any other track crossing the equator at the semispan)
If the aircraft were to fly the rhumb line heading of the track at the departure point, would it end up arriving overhead the destination airport?
I assume this to mean "if the aircraft were to continously fly the GCTTinitial track, to make a rhumb line".
Initally, the answer would then be no: because we know there's a difference between the rhumb line track and the great circle track, then logically if one leads to the destination then the other will not, if they are diverging tracks (they start out on the same heading, one goes straight and the other one curves away, they are diverging.)
However, there is again one possibility; a track that has an east-west componet, and a north-south component, where the arrival and departure are in different hemispheres: is it possible to come up with two places that have both the same true rhumb line track and great circle true track initial?
Let's how about 4500S 00000E to 5800N 13554E -- the great circle true track starts at 48,07 degrees and ends at 96,29 degrees -- however the rhumb line track between them is
also 48,07 degrees! We see they are
not on the equator, they are not equidistant from the equator, and they are not diametrically opposite, and they are not on the same meridians, or antimeridians.
Two other points however, may or may not have the same features. However, you will not find such a track where both arrival and departure points are in the same hemisphere.
Edit: the precise tracks and thus the precice points I've chosen vary slightly from calculator to calculator, so you may get slightly different results when trying it out, but move the destination around a little bit and you'll see the principle still remains.