# Analytical solutions for the dynamical clock A+ indicator in a toy model of pure dynamical friction

## TL;DR

This paper derives analytical solutions for the evolution of the A+ dynamical relaxation indicator in globular clusters, based on a simplified model of blue straggler star sinking due to dynamical friction.

## Contribution

It provides the first analytical expressions for the A+ indicator's time evolution in a toy model considering only dynamical friction effects.

## Key findings

- Analytical formulas for A+ evolution under dynamical friction.
- Validation of the model with observed cluster data.
- Insights into the dynamical relaxation process in globular clusters.

## Abstract

Blue straggler stars are more massive than the average star in globular clusters, as they originate from the merger of two stars. Consequently, they experience dynamical friction, progressively sinking to the cluster center. Recently, several indicators of the degree of dynamical relaxation of a globular cluster have been proposed, based on the observed radial distribution of blue straggler stars. The most successful is the Alessandrini indicator, or A+ for short, which is the integral of the cumulative distribution of the blue straggler stars minus that of a lighter reference population. A+ correlates with the dynamical age of a cluster both in realistic simulations and in observations. Here I calculate the temporal dependence of the A+ indicator analytically in a simplified model of the evolution of the blue straggler star distribution under dynamical friction only.