1.Introduction

recently introduced a new parameter, intended as a quantitative measurement of the
dynamical age of old star clusters. This indicator, dubbed A+ for short, is based on
measuring the radial distribution of massive tracers, such as blue straggler stars ,
which are undergoing mass segregation by virtue of being heavier than the average
stars in the host cluster. The A+ indicator, in particular, involves taking the
integral of the cumulative distribution of blue straggler stars as a function of log
radius, minus the same for a reference population, such as horizontal branch stars.
The A+ indicator is easier to measure compared to other attempts at building a
dynamical clock out of the blue straggler star distribution as it does not depend on
finding the exact position of the minimum of said distribution. While it was
originally introduced in the context of N-body simulations, the A+ indicator was
quickly shown to be able to capture the dynamical evolutionary stage of observed
globular clusters, displaying a strong anticorrelation with their half-mass
relaxation time . Throughout their paper, argue that the A+ indicator strictly
increases with time, and indeed show that this is the case in the large set of
simulations they run, spanning a wide range of initial conditions and dynamical
ingredients. This holds of course except for short term fluctuations due to
numerical, or possibly dynamical, noise. Monotonicity with time is a crucial
property for the A+ as well as for any other indicator of star-cluster dynamical
aging, which is an intrinsically irreversible process. Another property of A+ that
is apparent in the simulations run by , e.g. in their Figure 5, is that the
derivative of A+ also increases with time, i.e. that A+(t) is a convex function. In
a companion paper (^{Pasquato 2019}, referred as
P19 for short in the following), I worked out an analytical formula for the (three
dimensional) A+ indicator in a simplified model of mass segregation under pure
dynamical friction, and showed that it evolves monotonically as a function of time
for any initial blue straggler star distribution, and any radially increasing
dynamical friction timescale law. Here I build on these results to obtain a few
interesting properties of the A+ indicator. In particular I show that its derivative
is a non-negative function, which is non-decreasing as a function of time if the
dynamical friction timescale is non-decreasing with radius, making the A+ indicator
a convex function of time. I provide a simple formula for the value of the A+
indicator derivative at the beginning of dynamical evolution, which directly
connects it to the dynamical friction timescale. Finally, I suggest that in the late
stages of dynamical evolution the A+ indicator derivative should approach a
constant, making the A+ indicator asymptotically linear.

2.Calculations

In the context of the simplified model of evolution under dynamical friction introduced in P19, the A+ indicator is calculated as

Where N(r,t) is the cumulative distribution of blue straggler stars as a function of
radius and time. The function

Where τ(r) is the dynamical friction timescale as a function of radius.

2.1.Time Derivative of the A+ Indicator

The second term on the right hand side of equation 1 does not depend on time, so its derivative vanishes. Regarding the first term, I redefine for simplicity

so

which, by using the inverse function derivative rule, becomes

that is

where

by further substituting, for a given t, r_{0} = r_{0}(r,t), I
obtain

Where

This is a general formula for the derivative of the A+ indicator at time t. It is
always non-negative because the integrand is non-negative as long as the
derivative of the initial cumulative distribution function is. This condition is
equivalent to requesting that density is always non-negative. In other words,
the monotonicity of the A+ indicator derives from the condition that the number
density of blue straggler stars is everywhere non-negative. Note how in equation
9 the time dependence is all enclosed within the τ(r) term (not
τ(r_{0})!) at the denominator. If τ(r) is constant I recover the result
of P19 where the A+ indicator is linear with time, as its derivative equals
1/τ(0) and its initial value is always zero by construction.

2.2.Convexity

The evolution of a star’s radius given its initial value 𝑟 0 and the elapsed time t is given by equation 3 of P19:

For any

and, defining

so for a given r_{0}, the function r(r_{0},t) is monotonically
decreasing with 𝑡. Stars can only sink towards smaller radii over time. Now, if
τ(r) is non-decreasing, _{1} > r_{2}. Then the denominator τ(r)
in equation [alessdernicenicenice] cannot increase with time because 𝑟
decreases as shown in equation [breaku], and

proving that A+ is a convex function of time. This is a quite interesting result in the light of Figure 5 of where the (two dimensional) A+ indicators calculated for different simulations appear to evolve with increasing slope for increasing times, modulo the numeric error. The physical interpretation for this result is that the evolution becomes faster as the blue straggler stars drop to smaller radii, where the dynamical friction timescale is shorter.

Equation 7 can also be used to calculate the slope of the A+ indicator at the start of the dynamical evolution, e.g. in a simulation, by setting t = 0.

To further carachterize this I can write

So the derivative of the A+ indicator at the beginning of the dynamical evolution (e.g. at the start of a simulation) is the mean of the reciprocal of the dynamical friction timescale considered as a function of the initial cumulative number distribution of blue stragglers.

2.3.Limiting Behaviour at Large Times

If I set

by substitution into equation 9, as τ(r) is increasing with radius.

In general, since a constant τ(r) = τ (0) holds for the approximately constant-density cores of globular clusters, I expect the final stages of the evolution, when most blue stragglers have sunk to the core, to have a value of the A+ indicator’s derivative that approaches the constant value 1/τ(0). As I have shown, any value attained later in evolution should be larger than previous values of the derivative. This is another point where comparison with Figure 5 of is reassuring, as all the slopes in that figure are quite smaller than one, even though their values of the A+ indicator are referred to the half-mass relaxation timescale, which is longer than the central dynamical friction timescale. Further caveats for a comparison are that:

at some point during the evolution of a cluster, my simplified picture of dynamical friction will no longer represent reality, as diffusion effects will start to dominate right in the central regions, so the limiting value of dA

^{+}/dt = 1/τ(0) is not necessarily reached;blue stragglers on very eccentric orbits already depart from my simplified picture, even at the beginning of their evolution;

use a projected, two-dimensional version of the A+ indicator;

the reference stars (horizontal branch stars) they use are also evolving over time, as they are likely heavier than the average star in the simulations they run.

In subsequent papers I will address the third point, obtaining a better basis for comparison with simulations. The fourth point is addressed in the following.

2.4.Moving Reference Stars

Until now I assumed that the reference stars for the A+ indicator are fixed, i.e. that they keep the initial cumulative distribution over time. If I relax this assumption, while still assuming that the initial cumulative distribution of blue straggler and reference stars coincides, equation 1 becomes

where

and

Where m_{BSS} is the mass of a blue straggler star and m_{HB} the
mass of a reference star. This is in line with e.g. , who assume that dynamical
friction scales inversely with mass. With this in mind it is easy to see that
the time derivative of A+ calculated in equation 9 becomes

which, by substituting in equation [bastatelefonateperfavore], leads to the final result

which shows that the slope of the time evolution of A+ is merely rescaled by a constant factor. Since both the A+ indicator referred to a static population and the one referred to a population that also suffers dynamical friction are zero at the beginning of evolution, then A+ itself is merely rescaled by a constant factor. Thus all the results I obtained above still hold.

3.Conclusions

Working within the same pure dynamical friction picture I introduced in the companion paper P19, I have calculated the time derivative of the A+ indicator and found that it is not decreasing in time if the dynamical friction timescale τ(r) is non-decreasing with radius. Thus the A+ indicator is a convex function of time. P19 results show that in a constant dynamical friction timescale scenario the A+ indicator is linear with time, so convexity implies that over time the A+ indicator evolves with increasing slope becoming asymptotically linear, as the late stages of evolution take place in the uniform density cores of globular clusters, where the dynamical friction timescale becomes constant. Finally, I have shown that the choice of a reference population that is also affected by mass-segregation (as is likely the case in an observational setting) merely leads to a constant rescaling of the A+ indicator.