# Further properties of the dynamical clock A+ indicator in a toy model of pure dynamical friction

## TL;DR

This paper explores the properties and time evolution of the A$+$ indicator, a measure of star cluster dynamical evolution based on blue straggler star distribution, revealing its relation to dynamical friction timescales.

## Contribution

It provides a detailed analysis of the A$+$ indicator's properties and its time derivative within a simplified dynamical friction model, offering new insights into its physical interpretation.

## Key findings

- The time derivative of A$+$ equals the density-weighted reciprocal dynamical friction timescale.
- $dA^+/dt$ is non-negative and non-decreasing if the dynamical friction timescale is non-decreasing with radius.
- A$+$ is a convex function over time under certain conditions.

## Abstract

The Alessandrini A$+$ indicator is a quantitative measure of star cluster dynamical evolution based on the mass-segregation of blue straggler stars. A$+$ is defined as the integral of the cumulative distribution of blue straggler stars with the radius measured in logarithmic scale, minus a term related to the reference population used. In a companion paper I introduced a simplified model of dynamical friction and calculated the A$+$ indicator analytically. Here I show further properties of the time evolution of A$+$, focusing in particular on the physical interpretation of its time derivative $d{A^+}/dt$. I find that $d{A^+}/dt$ is the mean of the reciprocal dyamical friction timescale, weighted by the density of blue straggler stars. I show that it is non-negative (as expected based on monotonicity) due to the density of blue-straggler stars being non-negative and that, for a dynamical friction timescale that is non-decreasing with radius, $d{A^+}/dt$ is also non-decreasing with time, making the A$+$ indicator a convex function.