# Improved Initial Guesses for Numerical Solutions of Kepler's Equation

**Authors:** Kevin J Napier  
**arXiv:** [2411.15374](https://arxiv.org/abs/2411.15374)  
**Date:** 2024-11-22  
**Summary page:** https://www.summarizepaper.com/en/paper/2411.15374v1/

## Key points

- Numerical solutions of Kepler's Equation are crucial in celestial mechanics software applications
- Efficiency of root-finding algorithms depends on quality of initial guess provided
- Paper focuses on improving initial guesses for elliptical and hyperbolic forms of Kepler's Equation
- Machine learning approach implemented for elliptical form in Section 2, extended to hyperbolic orbits in Section 4
- Traditional methods like analytic estimates or empirical formulas used for initial guesses have computational overhead compared to data-driven approaches
- Data-driven approaches involve pre-calculating solutions or training neural networks for more accurate initial guesses
- Study introduces new, easily implementable initial guesses that improve speed when solving Kepler's Equation numerically

## Summary

In the field of celestial mechanics, numerical solutions of Kepler's Equation play a crucial role in various software applications. These solutions are often computational hot spots due to the frequency with which they need to be solved. The efficiency of a numerical root-finding algorithm heavily relies on the quality of the initial guess provided, striking a balance between complexity and the number of iterations required for convergence. This paper delves into improving initial guesses for both elliptical and hyperbolic forms of Kepler's Equation. Section 2 introduces and implements a machine learning approach specifically tailored for the elliptical form, while Section 3 investigates the efficacy of this solution. Building upon this success, Section 4 extends the same methodology to address initial guesses for hyperbolic orbits. The findings and implications are summarized in Section 5. Traditionally, analysts have relied on analytic estimates or ad-hoc expressions to generate initial guesses for solving Kepler's Equation. For instance, series expansions like E = M + e sin M or empirical formulas such as E = M + 0.85e have been commonly used. While some analytical expressions offer high precision with minimal error rates across the solution space, their computational overhead often outweighs the benefits compared to iterative root-finding algorithms with simpler initial guesses. Alternatively, a data-driven approach can be employed to enhance the accuracy of initial guesses. This could involve pre-calculating a fixed grid of solutions and interpolating within it to derive initial guesses at arbitrary points in parameter space. Another avenue is training neural networks to closely approximate the underlying function, thereby providing more accurate starting points for iterative solvers. By leveraging symbolic regression and genetic learning algorithms, this study introduces new, easily implementable initial guesses that yield notable speed improvements for both elliptical and hyperbolic orbits when solving Kepler's Equation numerically. The results showcase promising advancements in computational efficiency within celestial mechanics software, underscoring the importance of continuously refining numerical methods in astronomical research and exploration efforts.