Equations of motion of ball lightning in a rocket airstream

Abstract

In the literature we often encounter descriptions of the motion of ball lightning of large radius in aircraft airstreams. There are well-known cases of the motion of ball lightning in the airstream created by the engines of a multiengined aircraft, when the lightning, whose dimensions are comparable with those of the aircraft, is captured by the engines and pursues the aircraft persistently, traveling at speeds of 150--200 m/sec while preserving its spherical shape and a constant distance from the tail [1, 2]. I f lightning with a radius o f 5 m were replaced by a rigid sphere o f the same radius, then for it to pursue the aircraft a force of the order o f several tons would have to be applied [3]. Obviously, in this case there can be no question o f the pursuit of the aircraft under the influence of hydrodynamic forces. However, numerous observations o f the motion of ball lightning of large radius in aircraft airstreams give reason to suppose that because of its special internal structure it behaves in a viscous air medium like a rigid sphere in an ideal incompressible fluid [4]. Without concerning ourselves with the physical causes of this behavior, we will accept this property as a finding of numerous experiments [5, 6]. In this case it is possible to investigate the laws of motion of ball lightning in the airstreams created by various flying bodies. We will consider the hydrodynamic interaction between ball lightning and a rocket traveling along a certain trajectory under its own engine power, which can be simulated in the first approximation by a moving point source of variable intensity. Since the lightning undergoes hydrodynamic interaction with a source moving relative to the stationary air medium, it is possible to assume that the stationary source and the lightning moving relative to it are located in a translational airstream variable in magnitude and direction, whose velocity components v x, Vy, and v z are determined by the kinematic equation of motion of the rocket. Bearing this in mind, we will describe the position of the center M of ball lightning of radius a by means of the spherical coordinates s, or, /3 of a fixed system, at whose center lies a point source of intensity 3' per unit solid angle. We make the center of the moving coordinate system xyz coincide with the point M, directing the z axis towards the source, .the y axis in the direction of the unit vector et3, and the x axis in the direction opposite to that of the unit vector ecc The motion of the airstream is described by the equation [7] 27 5L= ~i(r-s),5(cos 0-1) (1) $2