A hydrodynamic mechanism of meteor ablation

II. Approximate analytical solution

Odesa National Maritime University, Mechnikova str., 34, 65029 Odesa, Ukraine
e-mail: club21@ukr.net

Received: 22 August 2016
Accepted: 24 July 2018

Abstract

Aims. We aim to obtain an approximate analytical solution to the equations of meteoroid dynamics within the frames of the suggested mechanism of ablation due to meteoroid melt spraying.

Methods. We have described the droplet breakaway of the melt in terms of hydrodynamic instability theory for the case of a shallow angle of meteoroid entry. The differential equation of meteoroid spraying was derived, together with the equation for the distribution function of sprayed particles by sizes. The latter was obtained considering two different frameworks for meteoroid motion law: empirical and theoretical.

Results. The trinal problem of an ablating meteoroid is solved analytically. The approximate solution is found for the system of equations of motion, ablation and number of sprayed droplets. The latter yields the equation for the distribution function of breakaway droplets by sizes, for which the approximate solution is obtained by providing the intermediate and final number distributions. The meteoroid ablation–deceleration balance remains in the regime of dynamic ablation, when the meteoroid velocity deficit due to aerodynamic drag is negligible until the instant of almost total meteoroid destruction. The responsible “h”-criterion is found, which depends only on the physical properties of the media, so that for melts of materials with lower viscosity such as iron the balance is closer to dynamic ablation than for the stony melt. The differential equation for meteoroid spraying is derived and integrated, showing a near-linear decrease of the meteoroid radius with time.

Conclusions. The proposed spraying model allows us to obtain approximate laws of meteoroid dynamics, which can serve as a base of a comprehensive study of meteor formation and evolution. The model provides intermediate and final number distributions of the breakaway droplets by sizes through direct numerical integration of the governing equations, or, alternatively, in the form of approximate relationships. The theory can be extended to the general case of a molten meteoroid entering atmosphere at an arbitrary angle.