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Star cluster survival and compressive tides in Antennae-like mergers

Gravitational tides are widely understood to strip and destroy galactic substructures. In the course of a galaxy merger, however, transient totally compressive tides may develop and prevent star forming regions from dissolving, after they condensed to form clusters of stars. We study the statistics of such compressive modes in an N-body model of the galaxy merger NGC 4038/39 (the Antennae) and show that ~15% of the disc material undergoes compressive tides at pericentre. The spatial distribution of observed young clusters in the overlap and nuclear regions of the Antennae matches surprisingly well the location of compressive tides obtained from simulation data. Furthermore, the statistics of time intervals spent by individual particles embedded in a compressive tide yields a log-normal distribution of characteristic time ~10 Myr, comparable to star cluster formation timescales. We argue that this generic process is operative in galaxy mergers at all redshifts and possibly enhances the formation of star clusters. We show with a model calculation that this process will prevent the dissolution of a star cluster during the formation phase, even for a star formation efficiency as low as ~10%. The transient nature of compressive tides implies that clusters may dissolve rapidly once the tidal field switches to the usual disruptive mode.


INTRODUCTION
Galaxy interactions are known to trigger star cluster formation. A typical example is NGC 4038/39 (the Antennae), where some 1000 star clusters has been detected (Mengel et al. 2005;Whitmore et al. 2007). However, it is unclear how many of them will survive internal destruction: for instance, low star formation efficiency (SFE) or strong stellar feedback could destroy the molecular cloud (Elmegreen & Efremov 1997)) and lead to "infant-mortality" (Bastian & Goodwin 2006;Whitmore et al. 2007;de Grijs & Parmentier 2007). Furthermore, these questions are central to interpreting the enhanced specific frequency and metallicity bi-modality of clusters in giant ellipticals (Harris 1991). In this Letter, we stress that totally compressive tides widely exist in galaxy mergers and slowdown the early dissolution of clusters.
HST V-band and deep IR images of the Antennae galaxies have suggested a near power-law cluster mass function (ψ[M ] ∝ M −2 for 10 4 M⊙ < M < 10 6 M⊙; Fall et al. 2005; Anders et al. 2007) and an age distribution with a rough power-law fit (dN/dτ ∝ τ −1 ) suggesting that many young clusters dissolve rapidly after they are born inside their primordial host gas cloud (Fall et al. 2005).
The evolution of (broad-band) mass-to-light ratios offers a strong hint that young clusters are severely perturbed and may even be out of dynamical equilibrium (Bastian & Goodwin 2006;de Grijs & Parmentier 2007). Yet the low inter-cluster velocity dispersion measured in the Antennae galaxy merger indicates that cluster formation proceeds from a coherent coalescence of large-scale material as opposed to in situ shock dissipation and fragmentation . Computer simulations of galaxy mergers have long suggested that streams of material (made up of both gas and stars) developing during a merger will form dense structures, so increasing the star-and clusterformation rates (Barnes 1988;Barnes & Hernquist 1992;Mihos & Hernquist 1996;Naab et al. 2006). A Schmidt law or variation thereof can be applied to circumvent numerical resolution issues and derive estimates of the star formation rate in rough agreement with observations (see e.g. Barnes 2004 for an application to the Mice; di Matteo et al. 2007).
Despite this wealth of progress, the long-term fate of star forming regions (bound structures or not) remains largely unclear, especially with regard to star cluster complexes such as those of the Antennae galaxies. The situation is exacerbated in the case of a galaxy merger because the gravitational tides vary rapidly in time along with the morphology of the system as a whole. An observational snapshot showing a candidate star-forming aggregate lacks information about the duration and motion in space to determine whether the aggregate will remain bound or not. When matter (gas or stars) is distributed in space in fragmented and irregular fashion, as is typical of galaxy mergers, the tidal field may exert either a disruptive outward stretch on an extended mass; or, on the contrary, it may exert an inward compressive force, which then cocoons the volume enclosed and increases its binding energy. While converging streams have long been suspected of driving the formation of tidal dwarf galaxies (e.g. Barnes & Hernquist 1992;Bournaud & Duc 2006;Wetzstein et al. 2007), the impact of compressive tides themselves on sub-galactic scales has never been considered strictly as a gravitational effect. Previous authors analysed cluster formation as the result of a global trigger, using sticky-particles  or SPH simulations with a threshold in pressure (Bekki & Chiba 2002; or in density (Li et al. 2004) for the formation of stars.
Here, we take a different stand by exploring the impact of compressive tides on cluster-size volume elements. We present in §2 a high-resolution numerical model of the Antennae galaxy merger to match kinematic and optical data. In §3, we discuss the key results obtained from the computer simulation. We close with a discussion of how the process outlined here bears on the statistics of observed cluster populations.

MERGER SIMULATION
We picked two equal-mass bulge/disc/halo galaxy models to simulate an Antennae-like major merger similar to that of Barnes (1988). We set up self-gravitating models using the magalie software included with the Nemo 1 stellar dynamical package (Teuben 1995, version 3.2.4). The equations of motion were integrated with gyrfalcON (Dehnen 2002) using a Plummer softening kernel with b = 0.01 in units where the gravitational constant G = 1, a unit of time = 25 Myr, a mass unit = 3.6×10 10 M⊙ and a length unit of 4.4 kpc. The total mass of each galaxy is equal to 7 mass units, where the exponential disc scale-length and mass r d = M d = 1. The bulge is a Hernquist (1990) model with a scale parameter a = 1 and total mass 2 M b = 1. Thus our progenitor galaxies are "S0 -Sa" galaxies with a Toomre parameter Q = 1.5 which ensures local stability to axially symmetric fragmentation modes. An isothermal dark matter halo of total mass M h = 5, of equal truncation radius and scale length r h,t = r h = 7, was set up as in Hernquist (1993). The mass ratio between individual component particles of ∼ 2 keeps two-body heating low. A total of 700, 000 particles were used for each galaxy, for a mass resolution of 3.50, 1.75 and 4.38 × 10 5 M⊙ respectively for each component 3 . Lower resolution simulations with 175,000 and 350,000 bodies per galaxies did not change significantly the statistics reported here, demonstrating that the results are robust to Poisson noise. No gas was included in our simulation. The typically small gas mass fraction would not affect the gravitationnal potential significantly. Hydrodynamical aspects would be more important at the star-formation-scale, where e.g. fragmentation, winds and supernovae explosions play a crucial role.
The galaxies were set on a prograde encounter with tidal tails developing on either side of the orbital plane, implying non-coplanar discs 4 . The orbit of the associated point-mass problem is bound with eccentricity e ≈ 0.96. The extended massive dark matter halos cause the galaxies to brake significantly already at the first pericentre passage (t ≈ 11, snapshot used as our time reference). Tidal tails form soon after the first passage and continue to expand until the end of the simulation at t ≈ 30. Comparing with HI maps of Hibbard et al. (2001) and optical images of the Antennae, the merger simulation gives a good match to the morphology and kinematics (peak systemic velocity of 130 km/s) at time t ≈ 23, when a second passage through pericentre is in full swing (see Renaud et al. 2009, in preparation, for details).

STATISTICAL APPROACH AND RESULTS
Because the gas from which clusters form is largely confined to the disc of spirals, we focus on the fate of disc particles. The mass resolution of ∼ 3 × 10 5 M⊙ allows to treat individual disc particles as cluster-size elements 5 .
This resolution allows to monitor precisely the tidal field around a cluster. To study the effect of tides on a hypothetical star inside a cluster, let the cluster be at the origin of an inertial barycentric reference frame. The net force of the star is the sum of internal cluster and external galactic gravity. The tidal tensor computed from this external acceleration reads T ij = ∂g i ext /∂x j , where x j is the j-th component of the position vector and gext is the acceleration. We retrieved the six-component tidal tensor of the 4 × 10 5 discs particles at each snapshot output time with a second-order finite difference scheme applied to a cube of side b = 40 pc. The symmetric tensor T admits a diagonal form with eigenvalues λ1 λ2 λ3 computed using the method of Kopp (2006). The signs of the eigenvalues determine if the differential force leads to a stretch (λ > 0) or a compression (λ < 0) of the model cluster. Most examples of analytic potentials with a cored density show a mixture of both cases (Valluri 1993;Dekel et al. 2003;Gieles et al. 2007). From here onwards, a fully compressive tidal mode is understood to be one where all three eigenvalues are negative. In Fig. 1, we plot the percentage of disc particles in compressive mode as a function of time. The main evolutionary stages of the merger can easily be identified from the  rapid increase in particles undergoing compression. Peak 'A' occurs at the first pericentre passage, when the progenitors are largely whole, and the tidal field at its strongest. Around that time, bridges of matter form between the two progenitors and cause a second peak 'B' of compressive modes, of a somewhat longer duration. The number of compressive regions decreases steadily as the galaxies move apart, in the interval 50 < t < 250 Myr, prior to the second passage to pericentre. Once again a strong wave of compressive modes brings ∼ 15% of the bodies to experience compressive tides (peak 'D' on Fig. 1). Numerous short-duration spikes are visible around this second encounter (denoted 'C' and 'E' on the figure) which are linked to streams of tidal clumps falling onto the newly formed core.
Roughly speaking, the mass fraction in tidal compression rises five fold when compared to the progenitor galaxies in isolation. This relative increase compares well with hydrodynamical simulations of a major merger, where the number of star forming regions also increases on average five times at pericentre. Di Matteo et al. (2008) report that 80% of a sample of ∼ 900 GalMer merger simulations showed a relative star formation rate increase of 5. Larger increases would require a specific configuration, or more effective gas dissipation (see e.g. Mihos & Hernquist 1996;Springel 2000).  like feature and throughout the overlap region. To determine whether the coincidence of young clusters in regions of compressive tides is a chance event, we investigated the distribution of their duration along individual orbits. Two statistical distributions were compiled, one for the Total Time (tt for short) spent in compressive tides during the whole simulation; the other covering the Longest Uninterrupted Sequence (lus for short) of snapshots with a particle immersed in a compressive tide.
The results are graphed on Fig. 3 in log-normal axes. The top panel on the figure shows the differential mass fraction for both distributions, M(∆t + dt) − M(∆t), versus the time interval ∆t; here dt = 2.5 Myr is the snapshot sampling rate. The bottom panel shows the forward-integrated distributions M(> ∆t), the mass fraction of particles experiencing a compressive tide of duration exceeding ∆t, for each distribution. The four curves on Fig. 3 show a knee at ∆t ≈ 50 Myr (lus distributions) and 80 Myr (tt distributions). Up to these times, the distributions are well-fitted with exponential decays M(∆t) ∝ exp (−∆t/τ ) of characteristic times τ ≈ 8 Myr (lus distribution) and τ ≈ 20 Myr (tt distributions; cf. straight lines on Fig. 3b). The interval τ ≈ 10 Myr far exceeds the formation time of O/B stars; what is more, about 40% of the lus sample experiences a compressive mode of duration longer than τ = 10 Myr. The characteristic decay time τ ≈ 20 Myr of the tt sample comfortably covers the formation phase of rich star clusters, including the residual-gas evacuation process.
Because the model simulation discussed here involves gravitational dynamics only, a re-scaling leaving the dimensionless quantity GM/rv 2 unchanged will give exactly the same results. Therefore, any major merger of mass unit M ′ and length unit r ′ , will yield the same characteristic timescale τ ∼ 10 Myr for the statistics of compressive tides provided the ratio (M ′ /M )/(r ′ /r) 3 = constant (cf. §2 for the units of M and r). The same applies to the eigenvalues λ of the tidal tensor.

DISCUSSION
The role of tidal fields in galaxy mergers is traditionally taken to be disruptive and to inhibit star formation. The analysis of a major merger simulation suggests instead that transient morphological features give rise to compressive modes of gravitational tides on scales ∼ 40 pc, comparable e.g. to the cut-off radius of Milky Way clusters. The characteristic times τ ≈ 10 to 20 Myr derived for the log-normal distributions of duration interval ∆t, M(∆t), serves to estimate the fraction of young Antennae clusters still embedded in compressive tides at the current time. When normalised so M(0) = 1, the quantity 1 − M(∆t = τ ) ≃ 36% gives the fraction of clusters of ages ∼ 10 Myr or less still confined to a region of compressive tide. These clusters would appear super-virialised which may explain the scatter seen in massto-light ratio among the younger population of Antennae clusters (Bastian & Goodwin 2006;de Grijs & Parmentier 2007). Note that these clusters would all have formed during the second passage through pericentre (Fig. 1), and the vast majority of them will soon emerge from their pressurised "cocoon" and experience the usual disruptive effect of the tidal field, causing a rapid increase in the dissolution rate of clusters (see the sharp drop around peak D, Fig. 3). The statistics of compressive tides also imply that some twothirds of clusters with ages > 30 Myr are now in extensive tides and may have suffered significant mass loss or total disruption: as a result the age distribution of young clusters should peak sharply around ages ∼ 10 Myr, causing a break in the power-law fit of Fall et al. (2005). This conclusion is not inconsistent with their data for clusters in the mass range < 10 5 M⊙.
To contrast quantitatively the effect of tides on a starforming region with the case of an isolated molecular cloud, we define the dimensionless ratio A ≡ αλRv 2 Rv/(GM tot ) of tidal to gravitational binding energy of the cluster-forming cloud; λ is once more the eigenvalue of a tidal tensor which is taken to be isotropic for simplicity, Rv the virial radius of the cloud of total mass M tot , and α ∼ 1 a dimensionless quantity which depends on the mass distribution of the model cloud (e.g. α = 0.86 for a homogeneous sphere). Because a fraction ǫ < 1 of the total cloud mass is turned into stars, a young cluster forms stars while still embedded in residual gas. The ejection of this gas through feedback from stellar radiation may cause it to disrupt (Goodwin 1997;Geyer & Burkert 2001;Kroupa et al. 2001). A classic result is that this will occur whenever ǫ < 1/2 (Hills 1980;Goodwin 1997;Boily & Kroupa 2003a). Repeating the derivation of Hills (1980) but with a background tidal field, we find a new equilibrium radius R ≡ rRv after gas expulsion, where r is obtained from A solution r > 0 still exists when ǫ < 1/2 for all A < 0, corresponding to compressive tides. By contrast, a solution with A > 0 requires ǫ > 1/2 or more efficient star formation, as anticipated for a disruptive tidal mode. (The case A = 0 gives the critical value ǫ = 1/2 for isolated clusters.) Crucial to cluster survival, a compressive tide will hold together a star-forming region that would rapidly dissolve in isolation. This remains true even for a very low star-formation efficiency ǫ.
Boily & Kroupa (2003b) performed N-body calculations to address this question without any tidal field. Using the same numerical set-up and an additional compressive tide with A = (α/5)λ ≃ −0.06, Fleck (2007) found a new equilibrium after expansion by a factor of ≃ 2, for an SFE ǫ ≈ 0.10. Furthermore, the cluster would dissolve rapidly once the tide switches from being compressive, to the more likely extensive mode. This process therefore provides a unique clue to explain the discrepancy which exists between the expected Gaussian and the near-power-law mass function in the Antennae. The weak dependance of clusters half-light radius with mass (van den Bergh 2008) implies that compressive tides should be operative on all mass scales and severely imprint the mass function. As an aside, we note that the general character of compressive tides means that they are also at work at high-redshift, when merger rates were higher.

CONCLUSION
Using gravitational N-body simulation of the Antennae galaxy merger, we show that a significant fraction of the discs' mass undergoes totally compressive tides. This mode plays an important role by preventing a cluster from dissolving. The compressive regions detected in the simulations show a very good match with observational data, both in term of their spatial distribution, and their characteristic duration which compare to observed clusters ages.
Compressive modes of the tidal field help keep together clusters that would otherwise dissolve at formation, through e.g. rapid gas expulsion. By introducing a tidal term in the virial Eq. (1), we found that clusters should remain bound when they sit inside a compressive tide, even when the SFE is as low as 10%. The precise characteristics of the evolution of clusters is clearly dependent on the relative strength of the tidal field. This is a hint that to understand the survival of clusters, and especially the issues associated with the concept of "infant mortality", it is essential to include the full evolution of the environment in which clusters evolve. These super-virial young clusters sheltered by compressive tides will dissolve rapidly whenever the tides are no longer compressive. Hydrodynamical simulations will be required to link formally this effect with cluster formation histories.