On the reliability of N-body simulations

The gravitational N-body problem aims to solve Newton’s equations of motion under gravity for N stars (Newton 1687). A popular integrator to perform this task is the fourth-order Hermite predictor-corrector scheme (Makino and Aarseth 1992), using double-precision arithmetic. The experiments we discuss in Section 4 will use this integrator as a benchmark test. We adopt a shared, adaptive time-stepping scheme with the following criterion: Here η is the time-step parameter and \(\Delta r_{ij}\) and \(\Delta a_{ij}\) are the relative distance and acceleration for the pair of particles i and j. We implement no further constraints on the time-step size.

To test how inaccurate we are allowed to integrate while still obtaining accurate statistics (Smith 1979; Quinlan and Tremaine 1992) we vary the time-step parameter η, to obtain statistics from conventional simulations with different precision.

The results obtained with the benchmark integrator are compared to those obtained with Brutus, which uses an arbitrary-precision library.¹ With this library we can specify the number of bits, \(L_{\mathrm{w}}\), used to store the mantissa, while the exponent has a fixed word-length. The length of the mantissa can be specified and increased, with the aim of controlling the round-off error.

The integration of the equations of motion is realised using the Verlet-leapfrog scheme (Verlet 1967). The time-step is shared among all particles, but varies for every step according to equation (1).

To control the discretisation error, we implemented the Bulirsch-Stoer (BS) method, which uses iterative integration and polynomial extrapolation to infinitesimal time-step size (Bulirsch and Stoer 1964; Gragg 1965). An integration step is accepted, when two subsequent BS iterations have converged to below the BS tolerance level, ϵ.

The time-step parameter η and the BS tolerance ϵ, both influence the performance. If η is too big, convergence may not be achieved for any tolerance. If η is too small, the many integration steps will render the integration too expensive. There is an optimal value for η as a function of ϵ. We measured this relation empirically, which results in: For \(\epsilon< 10^{-50}\) the powerlaw converges to \(A=0.029\) and \(B=0.45\). Extrapolating this relation to \(\epsilon> 10^{-50}\) will cause the time-step size to become larger than the time scale for the closest encounter in the system. Therefore this relation saturates to a flatter powerlaw for \(\epsilon> 10^{-50}\) with \(A=0.012\) and \(B=-0.40\). Compared to a fixed value for η, this relation speeds up the iterative procedure by about a factor three or more. The code is implemented as a community code in the AMUSE framework (Portegies Zwart et al. 2012) under the name Brutus.

For every simulation we have to define the BS tolerance parameter ϵ and the word-length \(L_{\mathrm{w}}\). In an iterative procedure we vary both parameters systematically, each time carrying out a simulation until \(t=t_{\mathrm{end}}\). We subsequently calculate the phase space distance, \(\delta^{2}_{\mathrm{A},\mathrm{B}}\), between two solutions A and B: The first summation is over all particles and the second summation is over the six phase-space coordinates denoted by q (Miller 1964). We normalise by 6N, so that δ represents the average difference per phase-space coordinate between two solutions A and B. In our experiments we adopt Hénon units² (Hénon 1971; Heggie and Mathieu 1986), in which the typical values for the distance and velocity are of the same order. We will also use the distance in just position or just velocity space as they might behave differently.

We consider the solutions A and B to be converged when \(\delta _{\mathrm{A},\mathrm{B}}<10^{-p}\) at all times during the simulation. Note that converged in this case means convergence of the total solution, contrary to convergence per integration step as in the previous section. This criterion for convergence is roughly equivalent to comparing the first p decimal places of the positions and velocities for all N stars, in two subsequent calculations A, B. In most of our experiments we adopt \(p=3\), i.e. all coordinates have to converge to about three decimal places or more. We perform a subset of simulations with \(p=15\) to investigate the effect of small errors (see Section 5.4.3).

Each simulation starts by specifying the initial positions and velocities of N stars in double-precision (see Section 4). The simulation is carried out with the parameter set \((\epsilon, L_{\mathrm{w}})\). We start each simulation with \(\epsilon=10^{-6} \) and \(L_{\mathrm{w}}=56\mbox{ bits}\). This corresponds to a level of accuracy similar to what we reach with the conventional Hermite integrator. After this simulation, we increase \(L_{\mathrm{w}}\), for example to 72 bits (∼22 decimal places), redo the simulation and calculate \(\delta^{2}\). We repeat this procedure until \(\delta< 10^{-p}\). When this is achieved, we have obtained a solution in which the round-off error is below a specified number of decimal places for this particular value of ϵ.

We now reduce the tolerance parameter ϵ, for example by a factor of 100, and repeat the procedure of increasing \(L_{\mathrm{w}}\). This series will again lead to a converged solution, but this time it is obtained using a smaller ϵ, and is likely to be different than the previous converged solution. We continue decreasing the value of ϵ by factors of 100 and repeat the procedure, until two subsequent iterations in ϵ lead to a converged solution with a value of \(\delta< 10^{-p}\). By this time we are assured of having a solution to the gravitational N-body problem, that is accurate up to at least p decimal places.

In practice, we speed up the procedure by writing the word-length as a function of BS tolerance. Consider for example a BS tolerance of 10⁻²⁰. To reach convergence up to this level, we need at least 20 decimal places. Adding an extra buffer of 10 digits gives a total of 30 digits, or equivalently a word-length of about 112 bits. For this example, 112 bits turns out to be a good minimum word-length. For a first estimate of the word-length, we use: With this relation, we will only have to specify a single parameter ϵ, which directly controls the discretisation error and indirectly controls the round-off error. For most of the systems in our experiment the discretisation error turns out to be the dominant source of error and as a consequence ϵ has to decrease quite drastically. When ϵ decreases, \(L_{\mathrm{w}}\) increases, even up to the point that there are many more digits available than really needed to control the round-off error. In the case when the discretisation error dominates, the above defined minimum word-length for a given BS tolerance will result in the converged solution. When the round-off error dominates the word-length should be varied independently.

To show that our method works, we adopt the Pythagorean 3-body system (Burrau 1913). Previous numerical studies have shown that this system dissolves into a binary and an escaper (Szebehely and Peters 1967; Aarseth et al. 1994). After many complex, close encounters the dissolution happens at about \(t=60\) time units (Dejonghe and Hut 1986), or about 16 crossing times.

We adopt the initial conditions for the Pythagorean problem and integrate up to \(t=100\). To illustrate how the method works we start with a high tolerance and short word-length (\(\epsilon=10^{-2}\), \(L_{\mathrm{w}}=40\mbox{ bits}\)), which is less precise than double-precision. In Figure 1, this calculation is compared to a simulation with (\(\epsilon=10^{-4}\), \(L_{\mathrm{w}}=48\mbox{ bits}\)), through the yellow (upper) curves in the first three panels. After the first BS integration step, δ obtains a value of the order of the BS tolerance, and continues to increase due to exponential divergence, to eventually exceed \(\delta \sim 10^{-1}\), after which the errors become on the order of the typical distance and speed in the system.

In the following step, we repeat the calculation with a precision of (\(\epsilon=10^{-6}\), \(L_{\mathrm{w}}=56\mbox{ bits}\)), and compare the result with the calculation using (\(\epsilon=10^{-4}\), \(L_{\mathrm{w}}=48\mbox{ bits}\)). The comparison is represented by the orange curves (second from above) in Figure 1. The overall behaviour of δ is similar, but the system diverges at a later time due to a higher initial precision.

We continue the iterative procedure until a converged solution has been obtained. In the first three panels of Figure 1, it can be seen that subsequent simulations with higher precision shift the curve to lower values of δ. Superposed on the steady growth of the error are sharp spikes, where the error grows by several orders of magnitude, after which the error restores again (Miller 1964). These spikes are dominated by errors in the velocity, as can be deduced by comparing the magnitude of the spikes in position and velocity-space. Eccentric binaries which are out of phase when comparing two solutions cause large, periodic errors in the velocity. We finish the procedure when a solution is obtained for which the criterion for convergence is fulfilled, considering the magnitude of the error between the sharp spikes (bottom, black curves).

In the bottom right panel of Figure 1, we compare solutions obtained by the Hermite integrator to the converged solution. The different curves belong to different time-step parameters; η = 2⁻³, 2⁻⁵, 2⁻⁷ up to 2⁻¹³. Note that for a time-step parameter \(\eta< 2^{-9}\), the curve is not shifted to lower values of δ, but even increases again. At this point round-off error becomes important, making the solution less accurate. The final close encounter in the Pythagorean problem occurs around 60 time units, after which a permanent binary and an escaper are formed. The Hermite integrator is able to accurately reproduce the evolution up to this point, but not subsequently, because δ has increased to values of order unity or higher. This can be explained by a small error in the final close encounter between all three stars, such that the direction of the escaper is slightly different.

To obtain the converged solution up to the first three decimal places, a tolerance of 10⁻¹⁴ and a word-length of 88 bits were needed. The simulation was about twice as slow compared to the Hermite simulation with \(\eta=2^{-9}\). The Hermite simulation, however, had a slightly different solution and a final, relative energy conservation of 10⁻⁸. Decreasing the value of η will improve the level of energy conservation, but due to round-off error δ will not decrease.

As a second test case, we adopt the 3-body equilateral triangle as an initial condition (Lagrange 1772). In the exact solution this configuration remains intact, but small perturbations, such as produced by numerical errors, quickly cause the triangle to fall apart. For this problem, we also have a source of error in the initial conditions. Whereas the Pythagorean problem can be set up using integers, the initial condition for the equilateral triangle contains irrational numbers. To control the error in the initial condition, we calculate the initial coordinates with the same word-length as used for the simulation.

In the left panel of Figure 2, a similar diagram is shown as for the Pythagorean problem in the lower left panel of Figure 1. The starting precision is \(\epsilon=10^{-10}\) and the word-length is a function of ϵ as in equation (4). Subsequent simulations are performed with a 10 orders of magnitude higher precision. For a short initial phase of 5 time units, the rate of divergence follows a power law. At later time, the solutions start to diverge exponentially with a characteristic rate independent of the tolerance and word-length. To investigate this transition, we redo the simulations with a large, fixed word-length of 512 bits (green dotted curves). This way, we reduce the amount of round-off error. As a consequence the rate of divergence is first dominated by the accumulation of discretisation errors and this phase lasts for a longer time, until the transition in the behaviour of the divergence, is reached, but now at ∼45 time units. The time of the transition depends on word-length. Why the exponential divergence starts once the round-off error has kicked in, is a question that is still under investigation.

The red dashed curves in the same diagram in Figure 2 give the results of the fourth-order Hermite, which are compared with the most precise Brutus simulation (with \(\epsilon=10^{-80}\), \(L_{\mathrm{w}}=352\mbox{ bits}\)). The time-step parameter η = 10⁻¹, 10⁻², 10⁻³ and 10⁻⁴ for subsequent curves. The Hermite integrations show a similar behaviour as the Brutus results, which could imply that the rate of divergence is a physical property of the configuration, rather than a property of the integrator.

In the right panel of Figure 2 we show the duration for which the triangular configuration remains intact as a function of BS tolerance. For this experiment we halt the simulation when the distance between any two particles has increased or decreased by 10%, after which the triangle falls apart quickly. This diagram also illustrates the linear relation between accuracy and time in this system, which is caused by the constant number of digits being lost during every unit of time. The small scatter is due to the discrete times at which we check the triangular configuration. The solid, blue line is a fit to the data and its slope is \(-0.52(3)\), which is equivalent to a loss of \(1.9(1)\) digits per cycle.

As a third test we simulate the dynamical formation of the first hard binary in a small star cluster. We select a moderate number of sixteen equal mass stars and draw them randomly from a Plummer distribution (Plummer 1911). We integrate this system for about ten crossing times and apply the method of convergence. In Figure 3 we present how two solutions with the same initial conditions, but different precisions, diverge as a function of time. The rate of exponential divergence, on average, starts rather constant, with a loss of ∼2/3 digits per time unit. This is equivalent to an e-folding time of \(t_{\mathrm{e}}=0.65\), which is consistent with the results of Goodman et al. (1993) (see their Figure 8). From \(t=20\) onwards, the rate of divergence experiences systematic changes, in particular a steep rise of the error of about 10 orders of magnitude between \(t=26\) and \(t=29\). Such rises are a signature for the presence of a hard binary interacting with surrounding stars.

The right panel in Figure 3 shows the energy conservation (black bullets, solid line) and the normalized phase space distance (red triangles, dashed line) versus ϵ. Energy conservation is proportional to ϵ, but the solutions only start to converge for \(\epsilon<10^{-34}\). More generally, even if conserved quantities like total energy are conserved to machine-precision or better, it is not guaranteed that the solution itself has converged.

The highest precision Brutus simulation in this example (\(\epsilon=10^{-50}\), \(L_{\mathrm{w}}=232\mbox{ bits}\)), took about a day of wall-clock time, which is about 7,000 times slower than a simulation with Hermite using \(\eta=2^{-9}\).

The use of arbitrary-precision arithmetic dramatically increases the CPU time of N-body simulations. Also the BS method, which performs integration steps iteratively, makes an integration scheme more expensive by at least a factor two or more. To investigate for example how feasible it would be to run a converged N-body simulation for 10³ stars through core collapse, we perform a scaling test in which we vary the number of particles and the precision, ϵ and \(L_{\mathrm{w}}\).

We randomly select positions and velocities for N equal mass stars from the virialised Plummer distribution (Plummer 1911), for \(N=2, 4, 8\), …, up to 1,024. The BS tolerance is fixed at a level of 10⁻⁶ and the word-length at 64 bits. We integrate the systems for one Hénon time unit and measure the wall-clock time. In the top left panel in Figure 4 we show the wall-clock time as a function of N, which fit the relation \(t_{\mathrm{CPU}} \propto N^{2.6}\).

For \(N>32\), it becomes efficient to parallellise the code. Our version implements i-parallellisation (Portegies Zwart et al. 2008) in the calculation of the accelerations. In the top right panel of Figure 4, we plot the speed-up, S, against the number of cores. For \(N = 1\mbox{,}024\), we obtain a speed up of a factor 30 using 64 cores.

In the lower panels of Figure 4 we present the scaling of the wall-clock time with BS tolerance and word-length. To measure the dependence on BS tolerance, we simulated a 16-body cluster for 1 Hénon time unit. We varied the BS tolerance while keeping the word-length fixed at \(L_{\mathrm{w}} = 1\mbox{,}024\mbox{ bits}\). The relation obtained converges to \(t_{\mathrm{CPU}} \propto\epsilon^{-0.032}\). A similar experiment was performed to measure the dependence on word-length. This time we fixed the BS tolerance at \(\epsilon=10^{-10}\) and varied the word-length. For \(L_{\mathrm{w}} < 1\mbox{,}024\), the relation can be estimated as \(t_{\mathrm{CPU}} \propto L_{\mathrm{w}}^{0.33}\), while for \(L_{\mathrm{w}} > 1\mbox{,}024\), \(t_{\mathrm{CPU}} \propto L_{\mathrm{w}}\). This transition depends on the internal workings of the arbitrary-precision library which we will not discuss here.

Using a very long word-length of 4,096 bits, i.e. ∼10³ digits, results in a slowdown of a factor \(f_{\mathrm{s}} \sim16\) compared to 64 bits. But for some simulations a BS tolerance smaller than 10⁻⁵⁰ can easily be required to reach convergence, and this will result in a slowdown of a factor \(f_{\mathrm{s}} > 100\). The very small BS tolerance is often the main cause for the slowdown of the simulations, instead of the increased word-length.

Using the above results, we can construct the following model to estimate the wall-clock time for integrating 1 Hénon time unit with \(L_{\mathrm{w}} < 1\mbox{,}024\mbox{ bits}\): Integrating \(N=1\mbox{,}024\) with standard precision (\(\epsilon =10^{-6}\), \(L_{\mathrm{w}}=64\mbox{ bits}\)), up to core collapse at ∼300 time units, and taking into account a speed up of a factor 30 due to parallellisation, we estimate a total wall-clock time of a week. Increasing the precision to (\(\epsilon=10^{-20}\), \(L_{\mathrm{w}}=112\mbox{ bits}\)), will take about a month. A precision of (\(\epsilon=10^{-50}\), \(L_{\mathrm{w}}=232\mbox{ bits}\)) will take roughly a year. To estimate how much precision is needed, we will assume that the rate of exponential divergence before the formation of the first hard binary is approximately constant. In the left panel of Figure 3, the initial slopes correspond to a loss of ∼2/3 digits per time unit. We construct the following approximate model for the initial BS tolerance needed to end up with a converged solution: Here ϵ is the BS tolerance parameter, \(\delta _{\mathrm{final}}\) is the final precision of all the coordinates in the system, \(R_{\mathrm{div}}\) is the approximately constant rate of divergence, e.g. the number of accurate digits lost per unit of time, and \(t_{\mathrm{cc}}\) is the core collapse time. We set the final precision to 10⁻⁶, i.e. convergence to the first 6 decimal places, and we set the core collapse time to ∼300 as before. If we adopt \(R_{\mathrm{div}} = 2/3\), we estimate that we need an \(\epsilon\sim10^{-206}\). This would take about 10⁵ years to finish. It would be more practical to simulate a 256-body cluster. If we set the core collapse to 100 time units we estimate \(\epsilon\sim 10^{-73}\), which would take about a month on a cluster of 64 Intel^® Xeon^® E5530 cores.

For direct N-body codes, the time for integrating up to core collapse usually scales as \(\mathcal{O}(N^{3})\). Using the analysis above, we estimate that the time for converged core collapse simulations scales approximately exponentially. This is effectively caused by the exponential divergence.

In Figure 5, the distributions obtained by converged solutions are given for the following quantities: velocity and kinetic energy of the escaper in the barycentric reference frame, and semimajor axis, binding energy and eccentricity of the binary. We start by looking at the eccentricity distributions (bottom panel in Figure 5). These distributions can be estimated analytically by assuming that the probability of a certain configuration is proportional to the associated volume in phase space (Monaghan 1976; Valtonen and Karttunen 2006) or by considering an equilibrium distribution of binary stars in a cluster (Heggie 1975). The resulting thermal distribution in the three-dimensional case is given by and in the two-dimensional case by The 3-body cold collapse problem is essentially a two-dimensional problem. We compare the empirical and theoretical distributions by means of the K-S test (see also next section). It turns out that the distributions in eccentricity are statistically distinguishable. By inspection by eye we observe that in the virialised case, there are slight deviations at high eccentricities. In the case of the equal-mass, cold systems, there are more low eccentricity binaries compared to the theoretical prediction. They coincide at an eccentricity of about 0.7, after which they deviate again. For the cold systems with unequal masses, this behaviour is the other way around. The analytical predictions are able to capture the empirical distributions only in a qualitative manner.

The velocity distribution of the single escaping star can be estimated analytically in a similar way as was done for the eccentricities. The resulting distribution is predicted to be a double powerlaw given by (Monaghan 1976; Valtonen and Karttunen 2006): We fit this model to the data (see Figure 5, first panel) and obtain values for α and β which are given in Table 1. The powerlaw indices vary with mass ratio and total angular momentum. To remove the dependence on mass ratio, we plot the kinetic energy of the escaper (see Figure 5, top right panel). Again, we fit a double powerlaw of a similar form as equation (9), and the powerlaw indices are given in Table 1. Both the escaper velocity and kinetic energy are consistent with a double powerlaw distribution.

The binary semimajor axis and binding energy are related quantities. We fit the binding energy distribution (see Figure 5, middle right panel) to a powerlaw (Heggie 1975; Monaghan 1976; Valtonen and Karttunen 2006): The fitted powerlaw indices are given in Table 1. The empirical distributions are consistent with a powerlaw, although somewhat steeper than predicted (Heggie 1975; Monaghan 1976; Valtonen and Karttunen 2006). The slopes do tend to vary somewhat as a function of angular momentum (Monaghan 1976; Valtonen and Karttunen 2006).

The empirical distributions obtained by Brutus are in qualitative agreement with the analytical estimates present in the literature (Heggie 1975; Monaghan 1976; Valtonen and Karttunen 2006). Slight variations are present due to the dependence on total angular momentum, a limited statistical sampling and assumptions made in the derivation of the analytical distributions. Nevertheless, a similar qualitative agreement has been obtained between the analytical distributions discussed above and empirical distributions from an ensemble of conventional numerical solutions, e.g. not converged (Valtonen and Karttunen 2006, Chapters 7-8 and references therein). The question remains to what extend conventional and converged solutions agree quantitatively.

A quantitative way to compare global distributions is by performing two-sample Kolmogorov-Smirnov tests (K-S tests) (Kolmogorov 1933; Smirnov 1948). The K-S test gives the likelihood that two samples are drawn from the same distribution, quantified by the value called p. When the p-value is below five percent, the distributions are considered to be significantly different.

In Figure 6 we plot the p-value obtained by comparing the Brutus distribution with the Hermite distribution versus time-step parameter η used for Hermite. In the panel showing the data for the binary semimajor axis, the distributions of the cold systems become significantly different for \(\eta> 2^{-6}\). The distributions from the initially virialised systems start to differ for \(\eta> 2^{-4}\). The cold systems are harder to model accurately, because of the close encounters that occur shortly after the start. The reason the distributions start to become significantly different at large time-steps is because at these large time-steps most simulations violate energy conservation by \(\vert \Delta E/E \vert > 0.1\). When this occurs, solutions might reach regions in 6N-dimensional phase-space, which theoretically are forbidden. The distribution then becomes biased by these outlier solutions.

In Figure 7, we present the fraction of triple systems which are undissolved, i.e. still interacting, as a function of time. The results by Brutus are represented by the data points: equal-mass Plummer (black bullets), Plummer with different masses (red triangles), equal-mass cold Plummer (blue squares) and cold Plummer with different masses (green stars). The results by Hermite for a time-step parameter \(\eta= 2^{-5}\) are represented by the curves appearing to go through the data points.

The initially cold systems dissolve faster than the initially virialised systems. This is somewhat expected due to the close triple encounter resulting from the initial cold collapse: the rate of energy exchange can be very high for these encounters (Johnstone and Rucinski 1991). After ∼180 crossing times, about 40% of the systems which started with an equal-mass Plummer initial configuration, are undissolved, compared to about 10% for the cold Plummer with different masses. Systems which include stars with different masses dissolve faster than their equal mass counterparts. Energy equipartition tends to cause the lightest particle to quickly reach the escape velocity.

In Figure 7, the grey curves through the data points represent the interpolated Hermite results. Even though Hermite and Brutus use different algorithms and precisions to solve the equations of motion, we find that the lifetime of an unstable triple is statistically indistinguishable between converged Brutus and non-converged Hermite solutions (but see also Section 6.3).

In Figure 8, we plot the maximum CPU time and minimum BS tolerance, both as a function of dissolution time. This is shown for the Brutus simulations, for the four different initial conditions. The longer it takes for a system to dissolve, the longer the CPU time and the higher the precision needed to reach a converged solution. To reach ∼180 crossing times, there are systems which require a BS tolerance of the order 10⁻¹⁰⁰, with the final converged run taking of the order a few days. The average CPU time as a function of time is about an order of magnitude smaller than the maximum CPU time. The average BS tolerance ranges from ∼10⁻²⁰ to 10⁻³⁰. For systems which dissolve within 100 crossing times, Brutus is on average about a factor 120 slower than Hermite.

We were able to obtain a complete ensemble of systems dissolving within ∼180 crossing times. Simulations which take longer than this are not taken into account in this experiment. The fraction of long-lived systems as obtained by Hermite and Brutus are consistent. For our purpose of comparing results from conventional integrators with the converged solution, integrating up to ∼180 crossing times is sufficient, in the sense that there is enough time for conventional solutions to diverge from the true solution (see Section 5.4.1). Including the long-lived triple systems may however influence the statistical distributions and biases on the long term.

For the individual comparison, we take a certain initial realisation and compare the solutions of Hermite and Brutus. In Figure 9 we show scatter plots of the Hermite solution (with time-step parameter \(\eta=2^{-5}\)) versus the converged Brutus solution for the equal-mass Plummer data set.

Data points on the diagonal represent accurate solutions, whereas the scatter around it represents inaccurate Hermite solutions. The diagonal is present in each panel and extends throughout the range of possible outcomes. The width of the diagonal is very narrow. When the normalized phase-space distance between the Hermite and Brutus solution \(\delta< 10^{-1}\), then the coordinates are accurate enough to produce derived quantities accurate to at least one decimal place and Hermite and Brutus will give similar results. Once \(\delta> 10^{-1}\), the solution has diverged to a different trajectory in phase-space leading to a different outcome. This outcome could in principle be any of the possible outcomes as can be derived from the amount of scatter in the Hermite solutions at a fixed Brutus solution.

In the scatter plot of the dissolution time, we observe that for small times (\(t < 10\)), Hermite and Brutus agree on the solution in the sense that the data points lie on the diagonal. Systems which dissolve after a short time don’t have sufficient time to accumulate enough error to diverge to another trajectory in phase-space. Once however this level of divergence is reached, the scatter immediately covers the entire, available outcome space. This randomisation is also observed in the other panels.

In this section we compare the solutions obtained with Hermite and Brutus individually, by looking at which star eventually becomes the escaper and which form the binary. We define preservation if the Hermite and the Brutus solution both have the same star as the escaper. We define it as exchange if the escaping star is different. A further distinction can be made in the preservation category, if the Hermite simulation is also accurate. We can typify each Hermite simulation as follows:

In Figure 13 we present the fraction of each category as a function of time. As expected, systems which dissolve quickly, hardly have time to develop errors and are categorized as accurate simulations. In time however, because errors grow exponentially, the solutions become inaccurate. The fractions of preservation and exchange start to grow. For a small time-step parameter (\(\eta= 2^{-11}\), top row in Figure 13), this growth starts after ∼20 crossing times for the initially virialised systems. For the initially cold systems, the inaccurate fractions already start to grow after a single crossing time.

The cold collapse with equal-mass stars is the hardest problem to integrate as the accurate fraction is of comparable magnitude as the preservation and exchange fractions. The accurate fraction generally remains dominant, with a final fraction varying from about 0.4 for the equal-mass cold Plummer to about 0.7 for the Plummer with different masses. For the lesser precision (\(\eta= 2^{-3}\), bottom row in the figure), the accurate fractions decrease to below 0.2.

In the panels in Figure 13, which include the data for the systems with different masses, preservation is more common than exchange. This can be understood, because due to energy equipartition, the lightest particle will be more likely to escape and therefore the identity is more often correct than in the equal mass case. For the equal mass case, the fraction of preservation and exchange is comparable, except in the case of the equal-mass cold Plummer with the low precision (\(\eta= 2^{-3}\), the bottom row). If we regard the identity of the escaping star to be completely random once the solution has become inaccurate, we would expect the fraction of exchange to be twice the fraction of preservation. This is roughly what we observe in the equal mass cold collapse case with low precision. Because of the low precision and the initial close encounter, solutions will diverge very quickly. In the panel with the higher precision this trend is not observed because the solutions are less randomised. The preservation category includes solutions which slightly differ from the converged solution only in the escape angle of the escaper. Also the long-lived triples are not taken into account here, which will alter these fractions.

In every ensemble of Hermite solutions there are some that grossly violate conservation of energy \(\vert \Delta E/E \vert > 0.1\). This deformation of the energy hyper-surface in phase-space can allow solutions to reach parts of phase-space which are theoretically forbidden. This affects the global statistical distributions. In Figure 14, we replot the average error in the binary semimajor axis as a function of the time-step parameter. We produce similar diagrams as presented in Figure 11, but this time we introduce a maximum allowed error in the energy. If we filter out simulations with a relative energy conservation \(\vert \Delta E/E \vert > 1\), or \(\vert \Delta E/E \vert > 0.1\), we observe that the bias in the average error of the semimajor axis of the binaries vanishes. We conclude that this bias is caused by a few simulations which grossly violate energy conservation. A similar bias in the velocity of the escaping star is less pronounced.

In Section 5.4.3, we discussed an asymmetry at small errors. In Figure 15, we present similar diagrams as in Figure 12 for the positive and negative errors. This time we add the errors in the total energy and angular momentum of the system and the error in the velocity of the escaper.

We also vary the integration method because different methods produce different (biased) error distributions in energy and angular momentum. We use a standard Leapfrog integrator, a standard Hermite integrator and a Hermite integrator which uses the \(P(EC)^{n}\) method (we adopted \(n=3\)) (Kokubo et al. 1998). This last method adds an iterative procedure to the algorithm to improve the predictions and corrections, which improves the time-symmetry. For each method we implement a shared, adaptive time-step criterion as in equation (1), with a time-step parameter \(\eta= 2^{-7}\). As a consequence they will not be time-symmetric nor symplectic.

We first look at the error distributions in the total energy and angular momentum. We observe that none of them are symmetric, in the sense that the positive and negative errors have identical distributions, except for the angular momentum in the Leapfrog simulations. The Leapfrog solutions tend to gain energy, whereas the standard Hermite loses energy. The Hermite with the \(P(EC)^{n}\) method produces both positive and negative errors in the energy, but not in a symmetric manner.

To investigate whether the bias in energy and angular momentum conservation propagates to a bias in the binary and escaper properties, we estimate what the errors should be if we regard the error in the energy and angular momentum as a small perturbation to the converged solution. For the error in the velocity of the escaper, using the derivative of the kinetic energy with respect to velocity, we obtain the following expression: Here m is the mass of a star, v the velocity as obtained by Brutus, δE the energy error and δv the error in the velocity due to this energy error. For the binary semimajor axis we obtain: Here a is the semimajor axis from the Brutus solution. For the eccentricity we obtain: Here μ is the total mass of the binary, ϵ and l the specific energy and specific angular momentum of the binary as obtained by Brutus. The error in the eccentricity δe has contributions from errors in the energy δϵ and angular momentum δl.

If we compare the resulting error distributions to the actual error distributions, we find that the approximated error distribution is positioned at the asymmetry in the empirical error distribution. This is most clearly seen for the semimajor axis and eccentricity (see Figure 15).

The reason why the approximated error distribution overestimates the excess, is because not all errors are solely due to an error in the energy and angular momentum. In time, the numerical solution diverges from the true solution and this error due to divergence will become more dominant. With this in mind, we can approximate the error in a statistic as follows: Here S is a statistic that is related to energy and/or angular momentum, \(\delta S_{\mathrm{conservation}}\) is the error due to a small perturbation in the energy and/or angular momentum and \(\delta S_{\mathrm{divergence}}\) is the error due to divergence of the solution. When the solution has not diverged appreciably yet, the first type of error will dominate and possible biases can be observed. When the second type of error dominates, we observe that the symmetry is restored to within the statistical error.

Valtonen et al. (2004) state that the final statistical distributions forget the specific initial conditions and only depend on globally conserved quantities. This assumption makes predictions which are verified by our experiment. The results show that for a time-step parameter \(\eta< 2^{-5}\), the distributions are statistically indistinguishable, even though at least half of the solutions diverged from the converged solution. If however, energy conservation is grossly violated, biases are introduced in the statistics. In our experiment, a maximum level of relative energy conservation of \(\vert \Delta E/E \vert = 0.1\) was sufficient to remove the biases. This is a much milder constraint than the \(\vert \Delta E/E \vert \sim10^{-6}\) usually adopted in collisional simulations. Whether 0.1 is also sufficient for systems with more stars, should be verified experimentally. Heggie (1991) for example, finds that the energy of escaping stars in higher-N systems, depends sensitively on integration accuracy. The maximum required level of energy conservation should be such that it is below the energy taken away from the cluster by the escaping stars.

The chaoticity of the 3-body problem is illustrated by the scatter diagrams in Figure 9. For a certain value of a statistic obtained by Brutus, any other value in the allowed outcome space is reachable for the Hermite integrator. For example, if the converged solution gives an eccentricity for the binary of 0.6, a diverged solution can produce any eccentricity between 0 and 1. Once the solution has diverged from the true solution, it will start a random walk through or near the allowed phase-space until the 3-body system has dissolved. We observed that this randomisation happens in such a way that the available outcome space is still completely sampled and that it preserves global statistical distributions.

In Section 4, we discussed that the lifetime of an unstable triple does not depend on the integrator used nor on the accuracy of that integrator. This last point should be interpreted in the sense that when more effort is put into performing simulations with higher precision, that this does not change the global statistics, even though individual solutions will change with precision (see for example the Hermite results in Figure 1). If instead we continue to decrease the precision, there will be a point where biases start to appear. Urminsky (Urminsky 2008) analysed the 3-body Sitnikov problem and showed that the precision of the integration influences the average lifetime of triple systems, contrary to our results. The integration times in our experiment however, are much shorter. Obtaining a converged solution for a resonant 3-body system for longer than 200 crossing times, is still computationally challenging. Therefore any statistical difference on the long term will not be visible in our experiment.