Getting done for next release

Version 0.13 is getting closer

Version 0.13 is expected to be tagged in about a month. As any other release, it includes new features such as a new method in the Orbit class called change_attractor or the trail parameter in StaticOrbitPlotter. However, this release will also include really important bug fixes such as:

• Minor issues related to Lambert's problem.
• Propagator's are supposed not to hang out anymore due to robust solutions and propagation methods.

Now that my final exams ended I am completely free to work in poliastro and direct all my efforts to the software. I want to close issues #495 and #475 in the next days since they are previously commented bugs.

Propagator methods in the twobody problem

Kepler's equation (KE) allows us to solve either for the time at a given position or the position for a known time. Although it is a very short equation, there have been lots of numerical methods along with history since it has not an analytical solution.

The most famous numerical method related to root finding is probably the so called Newton-Raphson method. By making use of the function derivative it is possible to reach with great accuracy and speed the solution to the equation. Other methods also use higher order derivatives to achieve a faster connvergence. Although the Newton-Raphson methods works really well most of the cases, it is possible that sometimes the derivative at some point of the evaluated function becomes zero, making to diverge the numerical method.

Along the years, different approaches have been developed to solve this famous equation: conversion to a third order polynomial, higher order derivatives iterative methods, series expansion... Most of the different papers available claim convergence regarding the speed of computation, somehting that is not fair if we consider that CPUs computational power evolve as years pass. Numerical methods should be compared fixing a relative error and number of iterations. But remember: there is not a perfect numerical methods. While for examples the bisection method always converges it does really slowly, on the other hand, Newton-Rapshon based methods have a great rate convergence but are subjected to zero derivative singularities.

After a huge research on the topic, the most interesting algorithms from my point of view are the following ones:

Both are universal, meaning that no matter the geometrical nature of the orbit we can solve the KE. They claim low interations with huge numerical precission onli limited by mantissa errors (floating point precission).

Implementation of previous methods

As said before, there is no such a perfect numerical solver for the KE. I would like to implement not only the previous cited ones, but also more of them and compare their performance on the near-parabolic regime.

Fukushima's work also includes some Fortran77 codes and his research on the KE topic is huge. It looks really promising. On the other hand, the SDG group at UPM may be a really good contact source if I got stucked with the implementation of the algorithm.

Last week I already added a new one that was called by me as kepler_improved and although it is a non-iterative method it has a great accuracy. But after some tests, I realized that it was diverging for eccentricities near 0.999. I will contact with the Julia Astro people to compare the results.

Stay tuned!🚀