### 3. THEORY OF THE TIDAL TAIL FORMATION IN INTERACTING GALAXIES

After having reproduced numerically some of the extragalactic tidal structures observed in the Universe, several physical and mathematical descriptions of the phenomenon have been proposed to better understand the tides at galactic scale. The complexity of the task comes from the diversity of possible configurations, which translates into a large number of parameters. In this section, we review the role of the first order parameters and illustrate their respective effects thanks to numerical simulations of interacting galaxies. A mathematical description of the tidal field is also presented.

By definition, the tides are a differential effect of the gravitation. Let's consider a galaxy, immersed in a given gravitational field. At the position of a point within the galaxy, the net acceleration can be split into the effect from the rest of the galaxy aint, and the acceleration due to external sources aext. The latter can itself be seen as a part common to the entire galaxy (usually the acceleration of the center of mass), and the differencial acceleration, that differs from point to point within the galaxy. In other terms, the net acceleration at the position rP, in the reference frame of the center of mass of the galaxy (which lies at the position rg), is given by

 (1)

For small = rP - rg with respect to rg, one can develop at first order and get

 (2)

 (3)

or simpler

 (4)

when using Einstein's summation convention. The effect of the external sources on the galaxy are described by the term

 (5)

which is the j,i term of the 3 × 3 tensor T called tidal tensor (Renaud et al. 2008). Such a tensor encloses all the information about the differential acceleration within the galaxy. Therefore, the (linearized) tidal field at a given point in space is described by the tensor evaluated at this point.

Note that the tidal tensor is a static representation of the tidal field: the net effect on the galaxy also depends on its orbit in the external potential, or in other words, on the variations of intensity and orientation of the tidal field. This can be accounted for by writing to pseudo-accelerations (centrifugal, Coriolis and Euler) in the co-rotating (i.e. non-inertial) reference frame, or by the means of a time-dependent effective tidal tensor in the inertial reference frame. For simplicity, in the following we focus on static, purely gravitational tides and refer the reader to Renaud et al. (2011) for more details.

Because the acceleration aext derives from a gravitational potential ext, one can write

 (6)

(Several examples of tidal tensors of analytical density profiles are given in Renaud et al. 2009, see also the Appendix B of Renaud 2010.) It is important to note that these considerations are scale-free and applies to any spatially extended object, such as galaxy clusters, galaxies, star clusters, stars, planets, etc.

For example, let's consider the Earth-Moon system and compute the tidal field with the Moon as source of gravitation. It can been seen from the Earth as a point-mass, and yields a potential of the form

 (7)

with r = (xi + xj2 + xk2)1/2. The components of the tidal tensor are

 (8)

where ij = 1 if i = j and 0 otherwise. When computed at the distance d along the i-axis (i.e. for r = d and xj = xk = 0), the tidal tensor becomes

 (9)

The signs of the diagonal terms (which are, in this case, the eigenvalues because the tensor is writen in its proper base) denotes differential forces pointing inward along the i-axis, and outward along the other two axes. A rapid study of the differential forces around the Earth (see Figure 5) shows indeed, that they point toward the Earth along the axes perpendicular to the direction of Moon. One speaks of a compressive effect. Along the Earth-Moon axis however, the differential forces point away from the planet: the effect is extensive.

 Figure 5. Gravitational attraction (black, dotted line) of the Moon on the Earth, and the differential forces (grey). The tidal effect appears to be extensive in A and C, while it is compressive in B and D.

Back to the general case, it follows from Equation 6 that any tidal tensor is symmetric. Because it is also real-valued, it can be set in diagonal form, by switching to its proper base. In this case, three eigenvalues {i} denote the strength of the tides along the associated eigenvectors. The trace of the tensor (which is base-invariant) reads

 (10)

which can be connected to the local density ρ thanks to Poisson's equation:

 (11)

The condition on the sign of the trace implies that it is impossible to compute simultaneously three strictly positive eigenvalues. Remains the cases of two, one or no positive eigenvalues, as mentioned by Dekel et al. (2003). For two or one positive 's, the tidal field is called (partially) extensive, like e.g. in our Earth-Moon example. When all three eigenvalues are negative, the tides are (fully) compressive. By noticing that T is minus the Hessian matrix of the potential, one can show that a change of curvature of the potential implies a change of sign for T. Therefore, compressive tides are located in the cored regions of potentials only, and never in cusps.

Note that a compressive mode (three negative 's) implies that the local density due to the source of gravitation is non-zero. Although such a situation does not exist with point-masses, it can occurs when considering extended mass distributions, like e.g. for galaxies embedded in a dark matter halo.

The duality of compressive/extensive tidal modes plays a role in the formation, early evolution and dissolution rates of star clusters. It has been noted that observed young clusters were preferentially found in the regions of compressive tides (see Renaud et al. 2008 in the case of the Antennae galaxies), and a compressive mode would slow down the dissolution of young globulars (Renaud et al. 2011).

In isolation, a galaxy keeps its material, which is made of dark matter, stars, gas and dust, bound thanks to the gravitation. However, when it moves in an external potential, created for instance by neighbor galaxies, it can experience gravitational forces which are different from one side of the galaxy to the other. In other words, the galaxy is plunged in a tidal field. As a result, its material undergoes deforming effects that re-arrange the individual components of the galaxy. On the one hand, when this material was initially distributed in an (almost) random way in phase-space (as opposed to e.g. sharing a common velocity pattern), the net tidal effect does not translate into a clear global change for an entire region of the galaxy. Therefore, such tides are difficult to detect. On the other hand, when large scale, regular patterns exists in the distribution of the galactic material in phase-space (e.g. a disk), the tides have a similar impact on stars that already lied in the same region of phase-space. All these stars are affected the same way and thus, the effect is much more visible. In the end, a given tidal field is easier to detect when it affects a regular, organized distribution of matter, than when it applies to isotropic structures. This is the reason why tidal features like tails and bridges are well visible around disk galaxies where the motion is well-organized, and merely inexistent in ellipticals, which yield much more isotropic distributions of positions and velocities. This last point can be extended to all structures with a high degree of symmetry (halos, bulges, and so on), as opposed to axisymmetric components like disks.

As a consequence, the tidal structures gather the matter that occupy a well-defined region in phase-space. Figure 6 (top row) shows the N-body toy-simulation of an encounter between a composite galaxy (disk+bulge+dark matter halo) and a point mass. Particles being part of one of the tails are tagged so that it is possible to track them back in time to their initial position in the disk. As mentioned above, these particles are distributed in a more or less confined region of phase-space at the time of the pericenter passage of the intruder, so that their individual motions are re-organized in a similar way. It is interesting to note that they cover a wide range of radii in the disk and thus, because of the differential rotation, the zone they occupy before the interaction is far from being symmetrical.

 Figure 6. Top: morphology of a disk galaxy, seen face-on, during its coplanar interaction with a point-mass (mass ratio = 1), before the interaction (left), at pericenter (middle) and after (right). The dashed line indicates the trajectory of the point-mass (from top to bottom). The black dots tag a subset of particles that are situated in one of the tidal tails at t = 500 Myr. Bottom: same but for an elliptical galaxy. No tidal structures are visible.

When the same experiment is repeated with an elliptical galaxy (Figure 6, bottom row), the velocities are distributed almost isotropically and thus, no structure is created by the tidal field. As a conclusion, strong galactic tidal bridges and tails are formed from the material of disks galaxies. Note that the experiment we conducted above applies to any mass element, and thus can be, in principle, extended to both the gaseous and stellar components of a galaxy.

In the case of a flyby, the galaxies do not penetrate in the densest regions of their counterpart, do no loose enough orbital energy to become bound to each other, and thus they escape without merging. However, when the exchange of orbital angular momentum (through dynamical friction) is too high, the mean distance between the progenitors rapidly decreases (as a damped oscillation) before they finally merge, forming a unique massive galaxy. On the external regions of the merger, the tidal tails (if they exist) expand in the intergalactic medium and slowly dissolve. Because the tails are generally long-lived, they can indicate past interactions, as discussed in Struck (1999). As a result, tidal features can point to interacting events, even when what has caused their creation (i.e. a counterpart progenitor) has disappeared in a merger or has flown away.

The response of the gas to a galactic interaction can be seen as either an outflow or an inflow. For distant, non-violent encounters, a large fraction of the hot gas (T > 103 K) can be tidally ejected into the intergalactic medium, thus forming broad gaseous tails and/or halos around galaxies (see e.g. Kim et al. 2009). It has been noted that while the least bound material would expand widely, more bound structures could easily fall back into the central region of the galaxies within less than ~ 1 Gyr (Hibbard & Mihos 1995, Hibbard & van Gorkom 1996).

During a first, distant passage, some galactic material is stripped off thanks to the transformation of the orbital energy of the progenitor galaxies. As a result, and because of dynamical friction, the interacting pair becomes more and more concentrated and can, under precise conditions (see e.g. White 1978), experience other passage(s) and finally end as a merger (Barnes & Hernquist 1992a). During such a second, closer interaction, tidal forces can induce shocks covering a large fraction of the galactic disk, which gives the gas a significantly different behavior than that of the stars (Negroponte & White 1983). In particular, when stellar and gaseous bars form, the symmetry of the galaxy is broken: gravitational torques remove the angular momentum of this gaseous structure (Combes & Gerin 1985) and make it fall onto the nucleus of the merger (< 1 kpc, see e.g. Noguchi 1988, Barnes & Hernquist 1991, Hernquist & Mihos 1995, Mihos & Hernquist 1996). Such an inflow fuels the central region of the merger and participates in the nuclear starburst (Springel 2000, Barnes 2002, Naab et al. 2006) often observed as an excess of infrared light or a strong nuclear activity (Larson & Tinsley 1978, Lonsdale et al. 1984, Soifer et al. 1984, Genzel et al. 2001, Younger et al. 2010). At large radii in a disk, one gets the opposite effect: the gravitational torques push the material to the outer regions. This outflow enhances the formation of the tails already formed by the tidal field itself (Bournaud 2010).

Note that star formation in mergers is also considered to be triggered by energy dissipation through shocks (Barnes 2004). This is, however, quite sensible to the orbital parameters of the galaxies. Details about merger-induced starbursts are out of the scope of the present document. The reader can find a mine of information on this topic in Hopkins et al. (2006), Robertson et al. (2006), Di Matteo et al. (2007), Cox et al. (2008), Hopkins et al. (2009), Teyssier et al. (2010) and references therein.

Interestingly, Springel & Hernquist (2005) showed that the collision between two gas-dominated disks could form a spiral-like galaxy instead of an elliptical one, as one could expect. In this case, a significant fraction of the gas is not consumed by the burst of star formation induced by the merger. Through conservation of the angular momentum, dissipation transforms the gaseous structure into a star-forming disk (Hopkins et al. 2009). Owning that the gas fraction in galaxy increases with redshift (as suggested by Faber 2007, Lotz et al. 2010), this last point sheds light on the formation history of low-redshift spiral galaxies.

The details on the formation of tidal structures are adjusted by several parameters that mainly concern the orbit of the galaxies, i.e. the way one sees the gravitational potential of the other. Because an analytical study of the influence of these parameters is very involved, many authors conducted numerical surveys to highlight the trends obtained from several morphologies.

In their pioneer study, Toomre & Toomre (1972) already mentioned the influence of the spin-orbit coupling of the progenitors. For simplicity, let's consider two galaxies A and B separated by a distance rAB, and whose disks lie in the orbital plane. The norm of the velocity of an element of mass of the galaxy A situated at a radius r, relative to the galaxy B is rAB r , where denotes orbital rotational velocity and the (internal) rotation speed of the galaxy A (i.e. the spin). The sign of the second term depends on the alignment of with . For a prograde encounter, the spin () and the orbital motion () are coupled (i.e. aligned). Therefore, the relative velocity is lower (rAB - r ) than for a retrograde encounter (rAB + r ) and the net effect of the tides is seen for a longer period of time. As a result, the structures formed during prograde encounters are much more extended than those of retrograde passages.

Although this conclusion can be exported to inclined orbits, the strongest responses of the disks are seen for planar orbits, i.e. with a zero-inclination. The highly inclined configurations, called polar orbits, give generally birth to a single tail, as opposed to the bridge/tail pairs (Howard et al. 1993). In short, because an observed tidal effect does not only depend on the strength of the differential forces, but also on the duration of their existence, long tails are associated with prograde configurations.

Another key parameter is the mass ratio of the progenitors. In the hierarchical scenario, the galaxies form through the repeated accretion of small satellites (see e.g. Stewart et al. 2008 and references therein), and interactions between a main galaxy and number of smaller progenitors would occur more or less continuously. It is usual to distinguish the major mergers where the mass ratio is smaller than 3:1 (i.e. almost equal-mass galaxies), from the minor mergers involving a larger ratio (e.g. 10:1). In the last case, tidal tails are generally thin and small, while the same features are more extended and survive for a longer time in major mergers (Namboodiri & Kochhar 1985).

The dependence of the structure of the remnant of the interaction (disky or boxy elliptical, as opposed to more symmetric galaxies) on the mass ratio of the progenitors has been extensively debated but is not directly connected to the tidal activity, and thus is out of the scope of this review (see Schweizer 1982, Barnes & Hernquist 1991, Barnes & Hernquist 1992a, Hernquist 1992, Hernquist 1993, Naab & Burkert 2003, Bournaud et al. 2005, Bournaud et al. 2007 for much more details).

During the interaction, the impact parameter plays an indirect role: a close, penetrating encounter will drive one galaxy deep inside the high density regions of the other, which implies a strong dynamical friction (see e.g. Bertin et al. 2003). In this case, the separation of the progenitors after such a passage would be much smaller than for a more distant encounter.

Furthermore, a close passage generally corresponds to a significant tidal stripping. This situation occurs repeatedly for satellites orbiting within the halo of major galaxies (Read et al. 2006). Only the densest satellites can survive such a disruption (Seguin & Dupraz 1996), while more fragile object would be converted into stellar streams (Johnston et al. 1999, Mayer et al. 2002, Peñarrubia et al. 2009), as observed in the local Universe (Ibata et al. 2001).

However, the mass captured by a more massive companion (mass ratio close to 1:1) seems to be higher for short pericenter distances, as noted by Wallin & Stuart (1992). The lost of material into the intergalactic medium is also higher under these circumstances.

In addition to the effect of orbital parameters, several authors noted the role played by the dark matter halo of the progenitor on the morphology of the merger, mainly the lenght of the tails. E.g. Dubinski et al. (1996) showed that long, massive tidal tails are associated with light halos, while the deep potential created by more massive ones would prevent the creation of extended structures. Note that, for a given mass, a dense halo appears to be more efficient in retaining the stellar component bound (Mihos et al. 1998). An important conclusion of this work was that galaxies exhibiting striking tails are likely to have relatively light halo (i.e. a dark to baryonic mass ratio smaller than ~ 10:1).

However, Springel & White (1999) qualified this by stating that the important parameter is in fact the ratio of escape velocity to circular velocity of the disk, at about solar radius (see also Dubinski et al. 1999). Therefore, even massive halos (e.g. mass ratio 40:1) can allow the growth of tails, provided the kinetic energy of the disk material is high enough to balance the depth of the gravitational potential of the massive dark matter halo. See Section 6.3 for more details.

Although they are the most visible structures formed during galactic interactions, the tidal tails and bridges are not the only signatures of encounters. Other mechanisms (not directly of tidal origin) lead to disrupted morphology. We briefly mentioned them here, for the sake of completeness.

• Shells or "ripples" describe the arcs and loops showing sharp edges in the envelope of galaxies. They originate from the collision between a massive galaxy and a small companion, 10 to 100 times lighter (Quinn 1984). The material of the satellite is spread by an extensive tidal field in the potential well of the primary, along a given orbit of low-angular momentum (see Athanassoula & Bosma 1985 and references therein). A sharp ridge forms near the turnaround points of the orbit. The multiplicity of the shells corresponds to an initial spread in energy, leading to several possible radii for the ridges.

• A ring galaxy forms from the head-on collision between a large disk and a compact, small perturber (Freeman & de Vaucouleurs 1974, Theys & Spiegel 1977). The density wave created by the collision empties the central region of the disk and forms a ring in radial expansion (Lynds & Toomre 1976, see also Appleton & Struck-Marcell 1996 for an observational and theoretical review).

• Warped disks can be created by gravitational torques due to an infalling satellite galaxy (mass ratio ~ 10:1, see Huang & Carlberg 1997, Revaz & Pfenniger 2001). Note that warps and bending instabilities can also form through the torques exerted by a misaligned dark halo, or via accretion of matter (Jiang & Binney 1999, Bailin & Steinmetz 2003, Revaz & Pfenniger 2004).

The galactic tides are a purely gravitational effect, which means that they rely on scale-free quantities like the relative mass of the galaxies, the inclination of the orbits, their relative velocities and so on. Therefore, the conclusions presented above can be applied to any scales, from planetary to cosmological. If true in principle, this statement must be qualified because the requirements of the galactic-type tides themselves do not exist at all scales.

In the case of planetary tides, for example in the Earth-Moon system, the source of gravity does not penetrate in the object experiencing tides, and is generally situated at a distance large enough that it can be approximated by a point-mass. Furthermore, the binding energy of a solid and/or dense body like a planet is much higher than those of the galaxy on its stars. That is, the planetary tidal effects are weaker than the galactic ones. Note however that both the planetary and the galactic tides can destroy an object, like the comet Shoemaker-Levy 9 pulled apart by Jupiter's tidal field, or dwarf galaxies that dissolves in the halo of a larger galaxy, generally forming streams.

Another major difference arises from the periodicity of the motion. While a binary star or a planet is orbiting in a regular, periodic way, the galaxies show more complex trajectories, highly asymmetric, and rarely closed (because of high velocity dispersion and/or orbital decay). As a consequence, the tides at stellar or planetary scales can be seen as a continuous, or at least periodic effect, while they are rather well-defined in time and never occur twice the same way at galactic scales.

Therefore, the tidal effects seen at planetary or stellar scales, like the deformation of the oceans, atmospheres or external stellar envelops strongly differ from their equivalent phenomena in galaxies. At intermediate scale, the star clusters share properties of both tidal regimes. When orbiting an isolated galaxy, they undergo rather regular tidal effects and can, by filling their Roche lobe, evacuate stars through the Lagrange points. As a results, some globular clusters exhibit tidal tails, as seen in observations and reproduced by simulations (see e.g. Belokurov et al. 2006, Fellhauer et al. 2007, Küpper et al. 2010 and references therein).