The Long and the Short of Flights

Two aircraft, a tanker and a bomber, depart the equator at the Greenwich meridian on a true heading of 045° with unlimited fuel and no wind.  The tanker maintains a constant true heading of 045°.  The bomber whose flight control system failed immediately after departure continues on its last orientation (straight and level pitch, roll and heading remain fixed) which it maintains for the entire flight.  Assume that the Earth is a perfect sphere and each aircraft completes one natural cycle of its flight path, describe each aircraft’s path and distance traveled.

Note:  Since there is no wind, the true heading equals the track flown as there is no drift and the true heading coincides with the track.  Ignore the effects of Coriolis force.

Hint:  A great circle is the path traced by the intersection of a plane passing through the center of a sphere and the surface of the sphere.  It is the shortest distance between two points on a sphere.  A rhumb line is the path traced on the surface of the Earth by a craft that maintains a constant track (crossing all meridians at the same angle).  It is the longer distance of the two paths discussed between two points on the Earth.  The distance (D) of a rhumb line is D= rsecaDf where r is the mean radius (nmi) of the Earth, a is the track angle and Df is the difference (in degrees) between the departure and terminal latitudes.  Assume the Earth to be a perfect sphere.

a.     Both the tanker and bomber fly great circle paths for a roundtrip distance of 21,614.5      nmi (mean circumference).

b.     Both the tanker and bomber fly rhumb line paths for a distance of 7,641.8.

c.     The bomber flies a rhumb line for a distance of 7,641.8 nmi and the tanker flies a great      circle for a distance of 21,614.5 nmi (mean circumference).

d.     The tanker flies a rhumb line for a distance of 7,641.8 nmi and the bomber flies a great       circle for a distance of 21,614.5 nmi (mean circumference).

The answer is:

Since both aircraft depart on the same heading, the reader is forced to differentiate their paths in the light of the flight control failure of the bomber.

The Tanker Path

The tanker maintains a constant course of 045°.  The path flown therefore is a rhumb line which intersects each successive meridian at the same angle resulting in a loxodromic path which ultimately spirals toward the North Pole.  The distance flown is:

D = r sec aDf

where    r    = mean radius [(2a+b)/3] of the Earth or 3,440.06 nmi
                    (WGS 84)

            a   = track angle or 045°

            Df = difference in latitude or 90°

Thus D = 3,440.06 nmi x 1.4142 x 90/360 x 2p = 7,641.8 nmi

Note the comparable great circle path approximation between the equator’s intersection with the Greenwich meridian (latitude 0° and longitude 0°) and the North Pole (latitude 90°, longitude undefined) is simply the latitude difference multiplied by 60nmi/° or 90° x 60 nmi/° = 5,400 nmi.  This illustrates the advantage of the great circle over the rhumb line in distance savings in the most extreme case ~2,242 nmi (about 3.74 hours savings in a jet flying at 600 knots).  The great circle path would be straight up the Greenwich meridian from the equator to the North Pole.  It is interesting to note that in the higher latitudes, the change in magnetic variation and the convergence of the meridians are a close match which enables a magnetic rhumb line to be closer to a great circle in distance (example flights between Gander and Shannon) than it is to a true rhumb line path.

The Bomber Path

The bomber maintains the same track with respect to inertial space.  At the instant of flight control failure (flying straight and level with no lateral acceleration—control surfaces fixed) the bomber continues its last space orientation and therefore remains in a plane that is always parallel to its moving tangent plane to the Earth immediately below.  Thus, the bomber flies a great circle path in a plane always perpendicular to a plane passing through the center of the Earth.

However, the bomber’s track with respect to the Earth’s coordinate system is constantly changing.  The natural cycle of the bomber’s flight is one revolution of the Earth with a distance equal to its circumference.  The distance flown would be:

Using the formula of

c = 2pr

and the mean radius of

3,440.06 nmi,

D = 21,614.5 nmi (mean circumference) or a close
approximation:

D = 360° x 60 nmi/° = 21,600 nmi.

It is important to recognize that in order to fly a rhumb line path, the aircraft must maintain a constant course with reference to the Earth’s coordinate system.  The aircraft must be continually turning for the aircraft track angle to remain constant as the meridians are continually converging.  This is the distinction between the flight paths flown by the two aircraft.

Lindbergh’s flight across the Atlantic between New York and Paris in 1927 illustrates the use of short segmented rhumb lines of 100 miles that were carefully plotted to approximate the great circle path (between the two cities) resulting in a savings of 140 nmi.  Lindbergh used a planning chart with a gnomonic projection where a great circle course is obtained by a straight line between departure and termination.  He divided the resulting straight line (between New York and Paris) into100 mile segments and transferred the coordinates of the extremities of these segments to a Mercator chart which he used for his historic flight.  Figure 38 illustrates the rhumb line and great circle comparison.

Note:  The Earth is an oblate spheroid with a polar radius slightly less than its equatorial radius.  Thus, the shortest distance between two points is a geodesic.  Very often in practical navigation, this distinction is ignored and the Earth is considered as a perfect sphere.  In an inertial navigation system the oblateness of the Earth is recognized.

 

Figure 38.  Rhumb Line and Great Circle Comparison