From cavitation to astrophysics: Explicit solution of the spherical collapse equation

Area of Science:

• Physics
• Applied Mathematics
• Probability Theory

Background:

• Differential equations of the form R'' = -kR^γ are crucial for modeling physical phenomena.
• Existing solutions are often specific to integer values of γ or limited to certain fields like hydrostatics and hydrodynamics.

Purpose of the Study:

• To present a universal explicit solution for the differential equation R'' = -kR^γ for all real γ.
• To reveal a connection between collapse dynamics and probability theory.
• To unify conceptually distinct scientific fields and enhance understanding of these equations.

Main Methods:

• Application of the beta distribution to derive a universal explicit solution.
• Analysis of the derived solution for insights into symmetry properties.

Main Results:

• A novel, explicit, universal solution for R'' = -kR^γ applicable to all real γ.
• Demonstration of a link between collapse motions and probability theory.
• Conceptual unification of phenomena like gravitational collapse, bubble implosion, and orbital decay.

Conclusions:

• The beta distribution provides a powerful tool for solving a fundamental class of differential equations.
• This unified approach offers new avenues for analytical manipulation and deeper understanding of physical collapse phenomena.