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

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.

💡 Research Summary

The paper presents a concise yet rigorous nonlinear analysis of the temperature evolution of a small satellite that is assumed to be isothermal. The thermal model incorporates external heat inputs—solar radiation, planetary albedo, and planetary infrared emission—as well as internal dissipation (e.g., electronics, attitude‑control actuators). Heat loss is modeled by black‑body radiation from the satellite’s surface, giving rise to a fourth‑power temperature term. The governing equation is therefore a first‑order ordinary differential equation (ODE) of the form

  C dT/dt = Q_in(t) – εσA T⁴,

where C is the satellite’s heat capacity, ε its emissivity, σ the Stefan‑Boltzmann constant, A the radiating area, and Q_in(t) a periodic function with the orbital period τ.

The authors first perform a qualitative phase‑space analysis. By bounding Q_in(t) between its minimum and maximum values they show that all trajectories remain within a physically admissible temperature interval. Using the Poincaré‑Bendixson theorem they prove that the system possesses a unique attracting limit cycle that is synchronized with the orbital period. Consequently, irrespective of the initial temperature, the satellite’s temperature converges to a periodic steady‑state.

Next, a perturbation (averaging) approach is applied. Q_in(t) is decomposed into a constant mean component Q̄ and a small periodic fluctuation ε q(t). The temperature is expanded as T = T₀ + ε θ(t). The averaged equation

  C dT₀/dt = Q̄ – εσA T₀⁴

has an exact steady‑state solution T_eq = (Q̄/(εσA))¹⁄⁴. Linearizing around T_eq yields

  dθ/dt = –λ θ + q(t)/C, with λ = 4εσA T_eq³/C.

The decay rate λ determines how quickly the temperature perturbations are damped. Two distinct regimes emerge when λ is compared with the orbital frequency ω = 2π/τ:

1. Slow‑decay regime (λτ ≪ 1). The damping is weak; the temperature approaches the limit cycle over many orbital periods. In this case the averaged equation alone provides an accurate description of the long‑term behavior, and the satellite experiences relatively gentle temperature swings.

2. Fast‑decay regime (λτ ≫ 1). The damping is strong; the temperature settles onto the limit cycle within a fraction of an orbit. Here the transient dynamics are brief, and the system’s response is dominated by the forced periodic component.

Numerical integration (fourth‑order Runge‑Kutta) is used to validate the analytical predictions across a range of parameter sets (varying C, ε, Q̄, and the amplitude of q(t)). The simulations confirm the existence of the attracting limit cycle and illustrate the transition between the two regimes. In particular, increasing internal power dissipation raises λ, pushing the system into the fast‑decay regime, whereas a larger thermal mass (higher C) reduces λ, leading to slow convergence.

The discussion translates these findings into practical satellite design guidance. If rapid temperature stabilization is desired (e.g., to protect temperature‑sensitive payloads), designers can increase emissivity or internal dissipation to raise λ, ensuring that the satellite quickly reaches its periodic thermal state. Conversely, if smoother temperature variations are preferred, a higher heat capacity or lower emissivity can be employed to keep λ low, allowing the temperature to evolve more gradually. The paper also notes that the simple isothermal model, despite its abstraction, captures the essential physics of satellite thermal dynamics and can serve as a baseline for more sophisticated multi‑node thermal network models that incorporate conduction, convection (in rarefied atmospheres), and variable radiative properties.

In conclusion, the study demonstrates that a satellite’s temperature dynamics under periodic heating and radiative cooling are governed by a nonlinear ODE that inevitably settles onto a stable, periodic limit cycle. The rate of convergence to this cycle is controlled by the ratio of the radiative damping coefficient λ to the orbital frequency, leading to either slow or fast thermal stabilization. These insights provide a clear, quantitative framework for thermal‑design trade‑offs in small satellite missions.