1.Introduction

Blue straggler stars are found in all globular clusters observed to date in the Milky
way . They are heavier than the average star in their host clusters, as they
originate from stellar mergers either through direct collision or by close-binary
mass transfer , or both . Since the first observations of a bimodality in the radial
distribution of blue straggler stars when normalized to a reference population were
done , attempts at understanding its origin and evolution have been made based on
simulations run with different software and various levels of realism . (^{Mapelli et al. 2004}, ^{2006}; ^{Ferraro et al.
2012}; ^{Hypki & Giersz 2013}; ^{Miocchi et al. 2015}; ^{Hypki & Giersz 2017}; ^{Sollima
& Ferraro 2019}).

In a previous paper showed that the physical ingredients underlying the formation and motion of the minimum of the distribution are dynamical friction and diffusion respectively. While the two are connected as they ultimately arise from the same phenomenon, i.e. scatter with lighter background stellar particles, varied the diffusion coefficient and dynamical friction independently, showing that when diffusion is too strong a minimum does not reliably form, whereas if diffusion is too weak a clear-cut minimum forms but does not move outwards over time. This suggests that simulation schemes should be carefully assessed regarding their ability to correctly model the dynamical friction and diffusion phenomena in order to reproduce the observed evolution of the blue straggler star distribution minimum with increasing dynamical age.

In this context introduced a new *dynamical clock* indicator which did
not require a measurement of the position of the minimum of the normalized blue
straggler star distribution, as it is based on the cumulative radial distribution of
blue straggler stars compared to the cumulative distribution of some other class of
reference stars. The indicator (or A+ for short) was introduced in the context of
direct N-body simulations, where it was shown that it increases with the dynamical
age of simulated clusters, acting as a mass-segregation powered dynamical clock.
Later, measured a slightly modified version of the A+ indicator on a sample of 25
Galactic globular clusters, showing that it correlates with the cluster dynamical
age measured in terms of a cluster’s current relaxation time.

The A+ indicator is defined as the difference between the integral of the cumulative distribution of the blue straggler stars, expressed as a function of the logarithm of the cluster-centric radius, and that of a reference distribution. In the following I will obtain some of its properties analytically under simplifying assumptions.

2.Calculations

2.1.A Toy Model of Dynamical Friction

I model blue straggler stars as a population of equal mass particles in circular orbits in a spherically symmetric fixed gravitational potential. The radius r of each orbit evolves due to dynamical friction, as

where r is the distance from the center and τ(r) is a positive, monotonically increasing function of r, representing the dynamical friction timescale at radius r.

Equation 1 shows that orbital radii contract with an instantaneous velocity v (r) > 0 that depends only on r. It can be integrated, obtaining

where r_{0} is the initial value of the radius at time t = 0 and r is its
current value at time 𝑡. In general r_{0} > r because the radii
contract over time. If the function τ(x) is known, the integral can be
calculated and r can be obtained as a function of r_{0} and t:

where the primitive

is a monotonically increasing function because τ(x)/x always is positive.
Consequently, it is invertible. Note also that r(r_{0},t) is a
monotonically decreasing function of t for every t > 0 and for every
r_{0}, i.e. the orbit radii keep shrinking over time. This can be
shown by writing

which holds for every t > 0, and applying I^{-1}, which is also
monotonic, to both sides, yields

Similarly to equation [rexplicit],

also holds.

I now denote with N(r,t) the cumulative distribution of particles at a given time
as a function of radius. This is by construction such that N(0,t) and _{0}(r,t) at time 0, as can be seen by placing an imaginary
particle exactly at r and observing that no other particle ever crosses its
path. In other words, Lagrangian radii behave exactly like particle radii.
So

the number of particles that had a radius less than a given r_{0} at the
beginning still have a radius less than r(r_{0},t) at time t. This can
be rewritten as

which, given knowledge of the function I is a general solution for N(r,t). Thus τ(r) fully determines N(r,t) given an initial N(r,0).

*2.2.Recovering the A+ Indicator*

In the following I will assume that the reference population of stars to which the blue stragglers are compared to build the A+ indicator initially shares the same distribution as the blue stragglers and does not evolve.

Under this assumption it is trivial to obtain the evolution of the (three-dimensional) A+ indicator from equation [cum]. I will write 𝑠=log𝑟, so that

and the A+ indicator becomes

2.3.Monotonicity

Note that at time

and the integrand

is positive for every s, because _{2} and t_{1} respectively: a particle that
took more time (

2.4.A+ Linear Dependence in Globular Cluster Cores

While equation 7 can be solved numerically for any

where σ is the velocity dispersion of background stars at radius r and p is their number density. For a Plummer model this works out as

where α is the model scale radius and τ(0) the scale time for dynamical friction at the center. Equation [integral] is solved exactly, for this dependence, by

where

which unfortunately cannot be inverted in terms of simple functions. However for small radii equation [tau] reduces to a constant, so equation [integral] becomes trivially

and, with reference to equation 7

so

As the central regions of a Plummer model have approximately constant density p(0), I can take at time t = 0.

with a radial cutoff at

after which

Therefore

which simplifies to

This result actually generalizes to any non-constant initial density as long as equation [expodep] holds, because of the interplay between the logarithm in the definition of the A+ indicator and the exponential dependence of equation [expodep], which leads to the first and the third term in equation [aless_const] canceling out. Thus the A+ indicator should evolve linearly with time if the dynamical friction timescale is constant with radius.

3.Conclusions

Working within a pure dynamical friction picture, under a set of simplifying assumptions, I have shown that the A+ indicator evolves monotonically in time and I have found an analytical solution for its time dependence. I worked out the case of a dynamical friction timescale that is constant with radius, which results in the A+ indicator increasing linearly with time. Monotonicity is an interesting result, as it proves that the A+ indicator is effectively a dynamical clock, as previously claimed by based on the results of a set of direct N-body simulations. As my simple model neglects diffusion, which was instead treated numerically by , I showed that the A+ indicator still works as a dynamical clock even in the absence of diffusion.