Modelling the reflective thermal contribution to the acceleration of the Pioneer spacecraft

We present an improved method to compute the radiative momentum transfer in the Pioneer 10 & 11 spacecraft that takes into account both diffusive and specular reflection. The method allows for more reliable results regarding the thermal acceleration of the deep-space probes, confirming previous findings. A parametric analysis is performed in order to set an upper and lower-bound for the thermal acceleration and its evolution with time.

💡 Research Summary

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The paper addresses the long‑standing “Pioneer anomaly” – an unexplained, approximately constant sun‑ward acceleration of the Pioneer 10 and 11 spacecraft measured at about 8.74 × 10⁻¹⁰ m s⁻². While early analyses dismissed thermal recoil as too small, later work suggested that the spacecraft’s own heat could account for a substantial fraction of the effect. The authors build on their previous point‑source Lambertian model and introduce a more sophisticated treatment of surface reflection that explicitly separates diffuse and specular components using the Phong shading model, a technique borrowed from computer graphics.

Methodology

1. Geometric Simplification – The spacecraft is reduced to its most relevant structures: the high‑gain parabolic antenna, the main equipment compartment, and the two radio‑thermal generators (RTGs). Each of these is represented by a small set of Lambertian point sources (e.g., two sources on the RTG bases, four on the side walls of the compartment, six on the back wall, etc.). This discretisation captures the main heat‑emitting surfaces while keeping the model computationally light.

2. Radiative Momentum Transfer – The time‑averaged Poynting vector for a Lambertian source at position x₀ with power W is expressed as
   **S(x) = (W/π‖x−x₀‖²) ·