On encounter rates in star clusters

The most common method of calculating encounter times is based on the fundamental assumptions that a star is travelling through an effectively infinite, uniform density medium at a constant speed (see e.g. the derivation in Binney & Tremaine, 2008, note that these assumptions are perfectly adequate if one is interested in e.g. the Galactic disc).

The encounter rate for any individual star is typically given by

where $n$ is the number density of the stars, $\sigma$ is the velocity dispersion, $r_{\rm e}$ is the closest distance during the encounter, $G$ is the gravitational constant, and $m$ is the typical mass of the stars (Binney & Tremaine, 2008). The second term in the brackets is associated with the gravitational focusing effect which deflects the trajectories and decreases the distance of closest approach for slow encounters or encounters between more massive stars.

For an ensemble of $N$ stars (e.g. a cluster), it seems reasonable to assume that the total encounter rate (total number of encounters per unit time, $\mathcal{E}$) scales with the total number of stars. Since each encounter involves two stars, the encounter rate should scale with $N/2$, i.e. $\mathcal{E}\simeq N\varepsilon/2$.

In convenient units where $r_{\rm e}$ is in au, $\sigma$ in km s^-1, $n$ in pc^-3, and $m$ in M_☉, the encounter rate in a cluster of $N$ stars is then

Therefore, it would seem that to calculate the rate of encounters, $\mathcal{E}$, at a particular distance of interest, $r_{\rm e}$, in an ensemble of $N$ stars, the correct values of (a) number density, $n$, and (b) velocity dispersion, $\sigma$, are required. If there is a distribution of stellar masses, an appropriate value for $m$ must be taken.

While various assumptions that go into this simple calculation are clearly wrong for star clusters (e.g. moving through an effectively infinite uniform density medium at a constant speed) one might think that some simple variation on this approach might work (e.g. taking some appropriate average speed and density). However, we will show that this approach in star clusters gives an often wrong, and an always misleading, ‘answer’.

It is important to clarify what we want to know about encounters in a cluster. In most cases, what we would like is an estimate of what fraction of stars have had a close encounter as this tells us the relative levels of disc/planetary system/multiple system perturbation/destruction. It is important to remember that this is not what an estimate of an encounter rate gives without a further assumption of how the encounters are distributed between stars.

As an example let us take a cluster that we shall examine in detail later: an $N=300$, $M=300$ M_☉, equal-mass (so $m=1$ M_☉) virialised Plummer sphere cluster with half-mass radius $r_{\rm h}=0.5$ pc. If we want to know the number of stars that have had an encounter within, e.g. $r_{\rm e}=1000$ au, we can calculate that $\mathcal{E}\sim 25$ Myr^-1 by taking the values for the velocity dispersion and half-mass density of this cluster and putting them into equation (2).

If we assume this encounter rate estimate is correct, to calculate how many stars have had an encounter after some time we need to make a further assumption that encounters are random so that after $2$ Myr there will be 50 stars that have had an encounter, and after $10$ Myr $250$ stars (ie. $>80$ per cent) will have had an encounter. (One could be somewhat more sophisticated and estimate as the encounter fraction starts to approach unity how many stars have had zero, one, two etc. encounters.)

In assuming encounters are random, this calculation ignores that encounters are much more likely to occur in the core, and that after a few crossing times some stars in the core are likely to have had multiple encounters, while those in the halo may have had none. Indeed, when stated like this, this approach does seem extremely naive and it would be surprising if it gave the correct answer.

It would seem reasonable to assume that encounter rates and fractions should depend in some way on both number density and velocity dispersion. Below we go into some detail on how the number density and velocity dispersion of a cluster might be quantified. This is important to show that the encounter rates we measure in simulations disagree with simple calculations because the assumptions that underlie them are wrong for clusters, rather than that we are using the ‘wrong’ values for number density or velocity dispersion, or not accounting for how they change with time. A reader happy to take our word for this can skip the details below.

We start by comparing encounter rates at various distances in two clusters with $N=300$ and $N=600$ equal-mass stars. Both are initially virialised Plummer Spheres with $r_{\rm h}=0.5$ pc, with stars each of mass $1$ M_☉.

Figure 1 shows the cumulative number of encounters over $10$ Myr in the $N=300$ (top panel) and $N=600$ (bottom panel) clusters. In each panel the lines from bottom-to-top are the cumulative numbers of encounters at $r_{\rm e}<50$ (orange), $100$ (green), $500$ (magenta), and $1000$ (blue) au respectively. At the top left of each sub-figure are three numbers for each of the encounter distances: the first is the actual encounter rate (Myr^-1) as found in the simulation, the next two are time-averaged estimates that we will return to later, but note for now that all three numbers are often quite different. The two simulations shown are N3SMR050-A (top) and N6SMR050-A (bottom).

Figure 1 shows a number of features one might expect.

1. 1.

   The total number of encounters grows roughly linearly with time (in these two clusters at least).

2. 2.

   The number of encounters at different distances scales very roughly with $r_{\rm e}^{2}$ (e.g. for $N=300$ after $10$ Myr there have been $317$ encounters at $r_{\rm e}<500$ au, and $27$ at $<50$ au).

3. 3.

   Increasing both $N$ (and therefore also $n$) by a factor of two results in about $4$ times more encounters (e.g. $317$ when $N=300$, and $1165$ when $N=600$ at $r_{\rm e}<500$ au).

The second and third numbers in the top left are the estimates of encounter rate as calculated from equation (2) using the time-averaged values of the half-mass number density ($n_{\rm h}$) and the mean number density ($n_{\rm m}$) respectively.

In both cases using $n_{\rm h}$ under-estimates the number of encounters by a factor of $\sim 2$. Using $n_{\rm m}$ seems better, often giving a reasonable estimate (but sometimes being off by a factor of $\sim 2$). So at a first glance at just these two simulations one might consider that using $n_{\rm m}$ often provides a reasonable estimate of the encounter rate in a cluster.

However, we have only compared two simulations which just happened to be those labelled A in our ensembles. As we show below, when we look at the whole ensemble the picture becomes much more complicated, and this emphasises the importance of looking at ensembles of simulations when dealing with $N$-body systems.

We now consider all ten clusters in our $N=300$ equal-mass stars, and a half-mass radius of $r_{\rm h}=0.5$ pc ensemble. The only difference between these clusters is the random number seed used to generate the initial positions and velocities. Therefore one would expect that the encounter rates in each would be similar - and ideally be close to an analytic estimate.

The top panel of Fig. 2 shows the final encounter rates ($\mathcal{E}_{\rm sim}$) for each of our identical clusters measured after $10$ Myr from $r_{\rm e}=0$ to $1000$ au.¹¹1The exact values of encounter rates at separations of less than a few au may be affected by our softening, but a close encounter did happen, we just might not be able to trust the distance of closest approach too precisely. The simulation shown in the top panel of Fig. 1 is A which is the black line towards the middle-bottom of all the lines.

The most important thing to note about the top panel of Fig. 2 is the total encounter rates after 10 Myr varies very significantly between clusters with a difference of almost an order of magnitude. There appears to be no ‘typical’ clusters and some outliers – just a seemingly random spread in encounter rates between clusters that are initially statistically identical.

Interestingly, all of the curves in the top panel of Fig. 2 follow a distribution that goes roughly as $r_{\rm e}^{2}$ suggesting that the distribution of encounter distances is what would be expected for unbound encounters. However, the distribution can slightly deviate from $r_{\rm e}^{2}$, often due to three-body encounters between a single star and a binary (which has formed during the simulation). These encounters can cause an increase in the encounter rate at $r_{\rm e}\sim 1000$ au, as can be most obviously seen in cluster I (dark red line at the top of the figure).

The other panels of Fig. 2 show the difference between the analytically estimated encounter rates and the actual encounter rates for each cluster. In the middle panel the half-mass density ($n_{\rm h}$) is used for the estimate, and in the bottom panel it is the mean number density ($n_{\rm m}$). A good match to the analytic estimate is the black dashed line at a ratio of unity.

Using the half-mass density never gives a good estimate, and can be wrong by an order of magnitude. Using the mean density is slightly better - three or four clusters stay reasonably close to unity, but most clusters are always wrong by factors of several.

One obvious explanation would be that different clusters have changed significantly (some expanding and some contracting?).

In Table 2 we give the initial, time averaged and final values of $\sigma$, $n_{\rm h}$, and $n_{\rm m}$ for all ten clusters, as well as the final cumulative encounter rate. The averages and variance of each quantity over all the clusters are given in the bottom line.

The velocity dispersions ($\sigma$) are very similar between all clusters, but measures of density can change significantly over time with quite different time averaged and final values, and between different measures (half mass or mean).

However, these variations do not seem to correlate with the vastly different encounter rates. The last three columns show the encounter rates estimated with $n_{\rm h}$ and $n_{\rm m}$ and then the actual encounter rates from the simulations. Only once (cluster F) does $n_{\rm h}$ get close to predicting the actual encounter rate. Estimates using $n_{\rm m}$ are reasonable for 5/10 of the clusters, but very wrong for 5/10. We think this was seen by Craig & Krumholz (2013) who note that their simulations had far more encounters than one might expect, but substructure complicated their analysis.

What is particularly interesting is that there is no systematic change in encounters rates with any measure of density or how they evolve. Cluster F has the lowest final mean density (484 pc^-1) and the lowest encounter rate (30 Myr^-1), but cluster I has the second lowest final mean density (531 pc^-1), but the highest encounter rate (208 Myr^-1). Clusters B and E have almost the same encounter rate, but final mean densities that are different by a factor of over two (735 and 1813 pc^-1).

One might think that maybe there was some extreme deviation in density at some point in time that the time averaged densities do not properly include, however this is not the case. In Fig. 3 we show the numbers of encounters with $r_{\rm e}<1000$ au (top panel), half-mass density (middle panel), and mean density (lower panel) for clusters D and I (cf. Fig. 1).

Cluster D (black line) shows a relatively linear increase in encounter numbers to end with $\sim 1400$ encounters within 10 Myr. Cluster I (red line) is similar to cluster D until a period between 6–7 Myr when the encounter rate increases significantly.

There is no reason to think that the increased encounter rate in cluster I is due to density variations however. The middle panel shows that both cluster’s half-mass densities are very similar, and both fairly constant and declining slightly. The bottom panel shows more variation in the mean densities with short-lived fluctuations of factors of a few, but both clusters show this behaviour, and, if anything, cluster D has higher densities. There are fluctuations in the mean density of cluster I around when the encounter rate increases significantly, but there are others when it does not.

Examination of the data shows that the large numbers of encounters in cluster I at 6–7 Myr is due to a few pairs of stars having multiple self-encounters in weakly bound pairs (cf. Moeckel & Clarke, 2011).

In all the ensembles of statistically identical clusters we have run we find no systematic relationship between any measure of cluster density and encounter rates (apart from occasionally in just a few of the clusters, but these could be chance given that many fluctuations do not correlate).

We clearly see that the total number of encounters between stars at any particular distance can be different by maybe an order of magnitude in initially statistically identical clusters, in a way that cannot be explained by density fluctuations.

The encounter rates we have shown in Fig. 2 and Table 2 are the number of times two stars come closer together than a particular distance. However, this measure does not include information on if a particular star, or a particular pair of stars, have had multiple encounters.

Star clusters have a density distribution with a high density core and increasingly lower density as one moves outwards, and a Plummer profile is a reasonable approximation to young, bound clusters.

Within this density distribution stars can have a variety of orbits (which can change after encounters). Some stars will spend a significant amount of their time in the high density core, some will spend most of their time in the low density halo, and various combinations in between (orbits can be radial or circular etc. and can change over time).

This means that each individual star will have a unique encounter history. Those that spend a lot of time in the core may have multiple encounters, while those in the halo may have none. In addition (as seen above), some stars may get into loosely bound multiples (cf. Moeckel & Clarke, 2011) and potentially have numerous self-encounters which can inflate the encounter rate significantly (see above).

Despite the large variation in encounter rates, When we measure the encounter fraction in each of our ten clusters we find that this value is statistically the same. In Table 3 we show the number ($N_{\rm s}$) and fraction ($f_{\rm s}$) of stars in each of the ten statistically identical clusters from Fig. 2 and Table 2 that have had an encounter within 1000 au after 10 Myr. This number is between 143 and 170 ($157\pm 9$) – statistically consistent with being the same number, and a little over half the stars in the cluster at $f_{\rm s}=0.58\pm 0.03$.

This tells us that in all clusters in this ensemble the same fraction of stars are having very different numbers of encounters.

We can also examine other ensembles of statistically identical clusters and we find that the encounter fraction in different clusters in the same ensemble is statistically the same.

The mean and variances of encounter fractions for each ensemble are given in Table 4, but to summarise for $r_{\rm e}<1000$ au: for $N=300$ clusters with $r_{\rm h}=0.75$pc, the encounter fraction is $0.38\pm 0.02$; for $N=300$ clusters with $r_{\rm h}=1$pc, the encounter fraction is $0.28\pm 0.03$; and for $N=600$ clusters with $r_{\rm h}=0.5$pc, the encounter fraction is $0.60\pm 0.02$.

We do not present the data here in detail, but the same is true for different encounter distances: ie. the encounter fraction is lower when the distance is e.g. 500au, but the encounter fraction is statistically the same within each ensemble. (It is difficult to say anything about extremely close encounters as we are into small-$N$ statistics.)

That the encounter fraction is constant is extremely interesting, as the most useful measure of encounters is often how many stars have had at least one encounter closer than a particular distance over a particular timescale.