Suborbital Flight Mechanics
The effects of the earth’s curvature are quite negligible on the airplane modes of interest to the stability and control engineer under ordinary flight conditions. However, some significant effects are expected for the suborbital case. A number of investigators have extended the flat-earth equations to spherical or oblate models in order to examine these effects.
Linearized airplane motions have been examined in perturbations from great-circle and minor-circle trajectories about a spherical earth (Myers, Klyde, McRuer, and Larson, 1993). In principle, this is the same procedure followed by Bairstow (1914) in his extension of the Bryan equations of motion to perturbations from steady turning flight. An extra longitudinal mode of motion is found, in addition to the usual short-period and phugoid modes. This is a first-order density mode, also referred to as an altitude mode. Aside from this extra complexity, with a typical hypersonic configuration at Mach numbers from 3 to 20 the density mode occasionally couples with real phugoid poles.
There is also an extra lateral-directional real mode, in addition to the usual Dutch roll, spiral, and roll modes. This is called a kinematic mode, generally of very long time constant. At some high Mach numbers, the kinematic mode couples into the spiral mode, producing a very low-frequency stable oscillation.