Nonlinear analysis of a simple model of temperature evolution in a satellite

TL;DR

This paper models the temperature evolution of a satellite considering heat transfer, internal dissipation, and black-body radiation, revealing conditions for stable periodic temperature patterns through nonlinear analysis.

Contribution

It introduces a nonlinear differential equation model for satellite temperature dynamics and analyzes the conditions leading to stable periodic temperature cycles.

Findings

Temperature approaches a periodic limit cycle.

Decay towards the limit cycle can be slow or fast depending on parameters.

An average equation describes slow decay scenarios.

Abstract

We analyse a simple model of the heat transfer to and from a small satellite orbiting round a solar system planet. Our approach considers the satellite isothermal, with external heat input from the environment and from internal energy dissipation, and output to the environment as black-body radiation. The resulting nonlinear ordinary differential equation for the satellite's temperature is analysed by qualitative, perturbation and numerical methods, which show that the temperature approaches a periodic pattern (attracting limit cycle). This approach can occur in two ways, according to the values of the parameters: (i) a slow decay towards the limit cycle over a time longer than the period, or (ii) a fast decay towards the limit cycle over a time shorter than the period. In the first case, an exactly soluble average equation is valid. We discuss the consequences of our model for the thermal stability of satellites.