Analytical solutions for the dynamical clock A+ indicator in a toy model of pure dynamical friction
Mario Pasquato
astro-ph.GA
























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1INAF, Osservatorio Astronomico di Padova, vicolo dell'Osservatorio 5, I–35122 Padova, Italy.
2INFN, Sezione di Padova, Via Marzolo 8, I–35131 Padova, Italy.


Pasquato

Analytical A+ indicator solutions






  * M. Pasquato, INAF, Osservatorio Astronomico di Padova, vicolo dell'Osservatorio 5, I–35122 Padova, Italy.




M. Pasquato


Pasquato, M.


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.







Galaxy: globular clusters: general
Methods: analytical
Stars: blue stragglers






Analytical solutions for the dynamical clock A+ indicator in a toy model of pure dynamical friction
    
  M. Pasquato,1,2
    December 30, 2023
===================================================================================================






§ INTRODUCTION


Blue straggler stars are found in all globular clusters observed to date in the Milky way <cit.>. They are heavier than the average star in their host clusters, as they originate from stellar mergers either through direct collision <cit.> or close-binary mass transfer <cit.> or both <cit.>.
Since the first observations of a bimodality in the radial distribution of blue straggler stars when normalized to a reference population <cit.>, attempts at understanding its origin and evolution have been made based on simulations run with different software and various levels of realism <cit.>.

One of the main goals of these efforts was to reproduce the formation of a minimum in the normalized blue straggler star radial distribution, or zone of avoidance, starting from initial conditions where the blue straggler progenitors (i.e. close binary stars if for the moment we exclude the direct collision channel) are not mass segregated, sharing the same distribution as the other stars in the cluster. A secondary goal was to predict the temporal evolution of the position of the minimum and other characteristics of their distribution.
In particular <cit.> interpreted the position of the minimum as the hand of a dynamical clock, powered essentially by mass segregation of  blue straggler stars, revealing the state of dynamical relaxation of the host cluster. Further work based on direct N-body simulations <cit.> and on Montecarlo simulations <cit.> found the formation and motion of the blue straggler minimum somewhat hard to reproduce reliably due to a combination of statistical noise (as blue straggler stars are already low in number in real clusters, simulations necessarily contain an even lower number of particles that a real cluster, and the blue straggler number is even lower -by definition- in the minimum of their distribution) and possibly intrinsic dynamical noise. To solve the issue of statistical noise <cit.> introduced a technique named artificial oversampling where they simulate a large number of blue stragglers in a Montecarlo simulation while treating them as tracers, i.e. without letting them affect the dynamics of regular stars. Still, while  <cit.> manage to reliably observe the formation of a minimum in the blue straggler radial distribution, their simulations encounter difficultes in reproducing the motion of the minimum in absolute physical units <cit.> even though the radial position of their minimum increases with respect to the core radius of the cluster, which is shrinking in physical units while the cluster moves toward core collapse.

In a previous paper <cit.> showed that the physical ingredients underlying the formation and motion of the minimum 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, <cit.> 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 to 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 <cit.> 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 <cit.> 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, <cit.> 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.




§ CALCULATIONS



 §.§ A toy model of dynamical friction

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

    ṙ = - r/τ(r) = - v(r),

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

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

    ∫_r_0^rτ (x) dx/x = -t,

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:

    r(r_0, t) = I^-1(I(r_0) - t),

where the primitive

    I(r) = ∫τ (x) dx/x

is an invertible function because τ(x)/x always is positive. It is easy to see that r(r_0, t) is a monotonically decreasing function of t for every t > 0 and for every r_0, i.e. that orbit radii keep shrinking over time.
Similarly

    r_0(r, t) = I^-1(I(r) + t),

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) = 0 and lim_r →∞ N(r, t) = 1 for all t. If for any two particles at time t=0 the condition r_0_1 < r_0_2 held, then at any subsequent t, r_1(t) < r_2(t) will also hold. Therefore

    N(r, t) = N(r_0(r, t), 0)

as 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 

    N(r, t) = N(I^-1(I(r) + t), 0)

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).



 §.§ 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 Eq. <ref>. I will write s = log r, so that

    N(r, t) = N(I^-1(I(e^s) + t), 0)

so the A+ indicator becomes
 
    A^+(t) = ∫_-∞^+∞ N(I^-1(I(e^s) + t), 0) ds - ∫_-∞^+∞ N(e^s, 0) ds




 §.§ Monotonicity

Note that at time t_2 > t_1
 
    A^+(t_2) -  A^+(t_1) = ∫_-∞^+∞[ N(I^-1(I(e^s) + t_2), 0) - N(I^-1(I(e^s) + t_1), 0) ] ds

and the integrand
 
    N(I^-1(I(e^s) + t_2), 0) - N(I^-1(I(e^s) + t_1), 0)

is positive for every s, because I^-1(I(e^s) + t_2) > I^-1(I(e^s) + t_1) as the two terms represent, per Eq. <ref>, the initial radius of a particle that is at r = e^s at t_2 and t_1 respectively: a particle that took more (t_2 > t_1) to fall to r was further away at the beginning.
This implies that A^+(t) is a monotonically increasing function of time, i.e. a working dynamical clock.



 §.§ A+ linear dependence in globular cluser cores

While Eq. <ref> can be solved numerically for any τ(r), some choices of τ(r) will lead to a simple analytical solution.
 For example, following Eq. 1 of <cit.> I take
 
    τ(r) ∝σ^3(r)/ρ(r)

 where σ is the velocity dispersion of background stars at radius r and ρ is their number density.
 For a Plummer model this works out as
 
    τ(r) = τ_0 ( 1 + r^2/a^2)^7/4

where a is the model scale radius and τ_0 the scale time for dynamical friction at the center. From Eq. <ref>, setting τ_0 = 1 and a=1 I obtain

    I(u) = 1/2[ log(u-1) - log(u + 1) ] + arctan(u) + 2/7 u^7 + 2/3 u^3

where

    u = ( 1 + r^2)^1/4 > 1

which unfortunately cannot be inverted in terms of simple functions. 
However for small radii Eq. <ref> reduces to a constant, so Eq. <ref> becomes trivially

    I(r) = τ_0 log(r/a)

and

    r = r_0 e^-t/τ_0

so

    N(r, t) = N(a e^log(r/a) + t/τ_0, 0) = N(r e^t/τ_0, 0)

As the central regions of a Plummer model have approximately constant density ρ_0, I can take

    N_c(r, 0) = 4 πρ_0 r^3

with a radial cutoff at

    r_c_0 = ( 4 πρ_0 )^-1/3

after which N_c(r, 0) becomes identically 1.
At time t the radius at which N_c(r, t) becomes identically 1 is

    r_c = r_c_0 e^-t/τ_0

Therefore

    A^+(t) = ∫_-∞^logr_c N_c(e^s + t/τ_0, 0) ds + ∫_logr_c^logr_c_0 1 ds - ∫_-∞^logr_c_0 N_c(e^s, 0) ds

which simplifies to

    A^+(t) =  logr_c_0/r_c = t/τ_0.

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



§ CONCLUSIONS

Working within a pure dynamical friction picture, under a set of simplifying assumptions, I have shown that the <cit.> 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 <cit.> based on the results of a set of direct N-body simulations. As my simple model neglects diffusion, which was instead treated numerically by <cit.>, I showed that the A+ indicator still works as a dynamical clock even in the absence of diffusion.



§ ACKNOWLEDGMENTS

This project has received funding from the European Union's Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 664931.
I wish to thank Dr. Pierfrancesco di Cintio, Dr. Paolo Miocchi, Dr. Alessandro Cobbe, and Dr. Stefano Pugnetti for helpful discussion on this subject.

rmaa



