The ECH capacities for the rotating Kepler problem

TL;DR

This paper computes the embedded contact homology (ECH) capacities of the rotating Kepler problem at energies below the critical value, introducing a new method for calculating weights via a specialized tree structure.

Contribution

It introduces a novel approach using a new tree structure to compute weights for ECH-capacities in the rotating Kepler problem at sub-critical energies.

Findings

Computed ECH-capacities for energies below the critical value.

Proved the first weight is the largest for all energies below the critical.

Provided a numerical example at the critical energy level.

Abstract

In this paper, I am going to compute the ECH-capacities of the rotating Kepler problem when the energy is less than or equal to the critical energy value $-\dfrac{3}{2}$. To compute the ECH-capacities, I will use the special concave toric domain of the rotating Kepler problem, that is explained in [arXiv:2108.04581] and obtain the weights of the special concave toric domain. I will use a new method to compute the weights via the new tree [arXiv:2108.04581] and then give an order to them that is necessary to do the computation. The weights are continuous function belongs to the energy parameter. So we can use the computation for all energy level below the critical energy. Finally, we will prove that the first weight is the biggest weight for all energy $c \leq -\dfrac{3}{2}$ and will see a numerical example of the ECH-capacities computation for the critical energy.