SPH Methods in the Modelling of Compact Objects
Abstract
We review the current status of compact object simulations that are based on the smooth particle hydrodynamics (SPH) method. The first main part of this review is dedicated to SPH as a numerical method. We begin by discussing relevant kernel approximation techniques and discuss the performance of different kernel functions. Subsequently, we review a number of different SPH formulations of Newtonian, special- and general relativistic ideal fluid dynamics. We particularly point out recent developments that increase the accuracy of SPH with respect to commonly used techniques. The second main part of the review is dedicated to the application of SPH in compact object simulations. We discuss encounters between two white dwarfs, between two neutron stars and between a neutron star and a stellar-mass black hole. For each type of system, the main focus is on the more common, gravitational wave-driven binary mergers, but we also discuss dynamical collisions as they occur in dense stellar systems such as cores of globular clusters.
Keywords
Hydrodynamics Smoothed particle hydrodynamics Binaries White dwarfs Neutron stars Black holes1 Introduction
1.1 Relevance of compact object encounters
The vast majority of stars in the Universe will finally become a compact stellar object, either a white dwarf, a neutron star or a black hole. Our Galaxy therefore harbors large numbers of them, probably ∼ 10^{10} white dwarfs, several 10^{8} neutron stars and tens of millions of stellar-mass black holes. These objects stretch the physics that is known from terrestrial laboratories to extreme limits. For example, the structure of white dwarfs is governed by electron degeneracy pressure, they are therefore Earth-sized manifestations of the quantum mechanical Pauli-principle. Neutron stars, on the other hand, reach in their cores multiples of the nuclear saturation density (2.6 × 10^{14} g cm^{−3}), which makes them excellent probes for nuclear matter theories. The dimensionless compactness parameter \(\mathcal{C} = (G/{c^2})(M/R) = {R_{\rm{S}}}/(2R)\), where M, R and R_{S} are mass, radius and Schwarzschild radius of the object, can be considered as a measure of the strength of a gravitational field. It is proportional to the Newtonian gravitational potential and directly related to the gravitational redshift. For black holes, the parameter takes values of 0.5 at the Schwarzschild radius and for neutron stars it is only moderately smaller, \(\mathcal{C} \approx 0.2\), both are gigantic in comparison to the solar value of ∼ 10−^{6}. General relativity has so far passed all tests to high accuracy (Will, 2014), but most of them have been performed in the limit of weak gravitational fields. Neutron stars and black holes, in contrast, offer the possibility to test gravity in the strong field regime (Psaltis, 2008).
Compact objects that had time to settle into an equilibrium state possess a high degree of symmetry and are essentially perfectly spherically symmetric. Moreover, they are cold enough to be excellently described in a T = 0 approximation (since for all involved species i the thermal energies are much smaller than the involved Fermi-energies, kT_{i} ≪ E_{F,i}) and they are in chemical equilibrium. For such systems a number of interesting results can be obtained by (semi-) analytical methods. It is precisely because they are in a “minimum energy state” that such systems are hardly detectable in isolation, and certainly not out to large distances.
Compact objects, however, still possess — at least in principle — very large energy reservoirs and in cases where these reservoirs can be tapped, they can produce electromagnetic emission that is so bright that it can serve as cosmic beacons. For example, each nucleon inside of a carbon-oxygen white dwarf (CO WD) can potentially still release ≈ 0.9 MeV via nuclear burning to the most stable elements, or approximately 1.7 × 10^{51} erg per solar mass. The gravitational binding energy of a neutron star or black hole is even larger, E_{grav} ∼ GM^{2}/R = CMc^{2} = 3.6 × 10^{53} erg (C/0.2)(M/M_{⊙}). Tapping these gigantic energy reservoirs, however, usually requires special, often catastrophic circumstances, such as the collision or merger with yet another compact object. For example, the merger of a neutron star with another neutron star or with a black hole likely powers the sub-class of short gamma-ray bursts (GRBs) and mergers of two white dwarfs are thought to trigger type Ia supernovae. Such events are highly dynamic and do not possess enough symmetries to be accurately described by (semi-) analytical methods. They require a careful, three-dimensional numerical modelling of gravity, the hydrodynamics and the relevant “microphysical” ingredients such as neutrino processes, nuclear burning or a nuclear equation of state.
1.2 When/why SPH?
Many of such modelling efforts have involved the smoothed particle hydrodynamics method (SPH), originally suggested by Gingold and Monaghan (1977) and by Lucy (1977). There are a number of excellent numerical methods to deal with problems of ideal fluid dynamics, but each numerical method has its particular merits and challenges, and it is usually pivotal in terms of work efficiency to choose the best method for the problem at hand. For this reason, we want to collect here the major strengths of, but also point out challenges for the SPH method.
The probably most outstanding feature of SPH is that it allows in a straight forward way to exactly conserve mass, energy, momentum and angular momentum by construction, i.e., independent of the numerical resolution. This property is ensured by appropriate symmetries in the SPH particle indices in the evolution equations, see Section 3, together with gradient estimates (usually kernel gradients, but see Section 3.2.4) that are anti-symmetric with respect to the exchange of two particle indices.^{1} For example, as will be illustrated in Section 4.1, the mass transfer between two stars and the resulting orbital dynamics are extremely sensitive to the accurate conservation of angular momentum. Eulerian methods are usually seriously challenged in accurately simulating the orbital dynamics in the presence of mass transfer, but this can be achieved when special measures are taken, see, for example, D’Souza et al. (2006) and Motl et al. (2007).
Another benefit that comes essentially for free is the natural adaptivity of SPH. Since the particles move with the local fluid velocity, they naturally trace the flow motion. As a corollary, simulations are not bound, like usually in Eulerian simulations, to a pre-defined “simulation volume”, but instead they can follow whatever the astrophysical system decides to do, be it a collapse or a rapid expansions or both in different parts of the flow. This automatic “refinement on density” is also closely related to the fact that vacuum does not pose a problem: such regions are simply devoid of SPH particles and no computational resources are wasted on regions without matter. In Eulerian methods, in contrast, vacuum is usually treated as a “background fluid” in which the “fluid of interest” moves and special attention needs to be payed to avoid artifacts caused by the interaction of these two fluids. For example, the neutron star surface of a binary neutron star system close to merger moves with an orbital velocity of ∼ 0.1c against the “vacuum” background medium. This can cause strong shocks and it may become challenging to disentangle, say, physical neutrino emission from the one that is entirely due to the artificial interaction with the background medium. There are, however, also hydrodynamic formulations that would in principle allow to avoid such an artificial “vacuum” (Duez et al., 2003).
On the other hand, SPH’s “natural tendency to ignore vacuum” may also become a disadvantage in cases where the major interest of the investigation is a tenuous medium close to a dense one, say, for gas that is accreted onto the surface of a compact star. Such cases are probably more efficiently handled with an adaptive Eulerian method that can refine on physical quantities that are different from density. Several examples of hybrid approaches between SPH and Eulerian methods are discussed in Section 4.1.4.
SPH is Galilean invariant and thus independent of the chosen computing frame. The lack of this property can cause serious artefact’s for Eulerian schemes if the simulation is performed in an inappropriate reference frame, see Springel (2010b) for a number of examples. Particular examples related to binary mergers have been discussed in New and Tohline (1997) and Swesty et al. (2000): simulating the orbital motion of a binary system in a space-fixed frame can lead to an entirely spurious inspiral and merger, while simulations in a corotating frame may yield accurate results. For SPH, in contrast, it does not matter in which frame the calculation is performed.
Another strong asset of SPH is its exact advection of fluid properties: an attribute attached to an SPH particle is simply carried along as the particle moves. This makes it particularly easy to, say, post-process the nucleosynthesis from a particle trajectory, without any need for additional “tracer particles”. In Eulerian methods high velocities with respect to the computational grid can seriously compromise the accuracy of a simulation. For SPH, this is essentially a “free lunch”, see for example Figure 9, where a high-density wedge is advected with a velocity of 0.9999 c through the computational domain.
The particle nature of SPH also allows for a natural transition to n-body methods. For example, if ejected material from a stellar encounter becomes ballistic so that hydrodynamic forces are negligible, one may decide to simply follow the long-term evolution of point particles in a gravitational potential instead of a fluid. Such a treatment can make time scales accessible that cannot be reached within a hydrodynamic approach (Faber et al., 2006a; Rosswog, 2007a; Lee and Ramirez-Ruiz, 2007; Ramirez-Ruiz and Rosswog, 2009; Lee et al., 2010).
SPH can straight forwardly be combined with highly flexible and accurate gravity solvers such as tree methods. Such approaches are extremely powerful for problems in which a fragmentation with complicated geometry due to the interplay of gas dynamics and self-gravity occurs, say in star or planet formation. Many successful examples of couplings of SPH with trees exist in the literature. Maybe the first one was the use the Barnes-Hut oct-tree algorithm (Barnes and Hut, 1986) within the TreeSPH code (Hernquist and Katz, 1989), closely followed by the implementation of a mutual nearest neighbor tree due to Press for the simulation of white-dwarf encounters (Benz et al., 1990). By now, a number of very fast and flexible tree variants are routinely used in SPH (e.g., Dave et al., 1997; Carraro et al., 1998; Springel et al., 2001; Wadsley et al., 2004; Springel, 2005; Wetzstein et al., 2009; Nelson et al., 2009) and for a long time SPH-tree combinations were at the leading edge ahead of Eulerian approaches that only have caught up later, see for example Kravtsov (1999); Teyssier (2002). More recently, also ideas from the fast multipole method have been borrowed (Dehnen, 2000, 2002; Gafton and Rosswog, 2011; Dehnen, 2014) that allow for a scaling with the particle number N that is close to O(N) or even below, rather than the O(N log N) scaling that is obtained for traditional tree algorithms.
But like any other numerical method, SPH also has to face some challenges. One particular example that had received a lot of attention in recent years, was the struggle of standard SPH formulations to properly deal with some fluid instabilities (Thacker et al., 2000; Agertz et al., 2007; Springel, 2010a; Read et al., 2010). As will be discussed in Section 3.1, the problem is caused by surface tension forces that can emerge near contact discontinuities. This insight triggered a flurry of suggestions on how to improve this aspect of SPH (Price, 2008; Cha et al., 2010; Read et al., 2010; Heß and Springel, 2010; Valcke et al., 2010; Junk et al., 2010; Abel, 2011; Gaburov and Nitadori, 2011; Murante et al., 2011; Read and Hayfield, 2012; Hopkins, 2013; Saitoh and Makino, 2013). Arguably the most robust way to deal with this is the reformulation of SPH in terms of different volume elements as discussed in Section 3.1, an example is shown in Section 3.7.3.
Another notoriously difficult field is the inclusion of magnetic fields into SPH. This is closely related to preserving the \(\nabla \cdot \vec B = 0\) constraint during the MHD evolution, but also here there has been substantial progress in recent years (Børve et al., 2001, 2004; Price and Monaghan, 2004a, b, 2005; Price and Rosswog, 2006; Børve et al., 2006; Rosswog, 2007a; Dolag and Stasyszyn, 2009; Bürzle et al., 2011a, b; Tricco and Price, 2012, 2013). Moreover, magnetic fields may be very important in regions of low density and due to SPH’s “tendency to ignore vacuum”, such regions are poorly sampled. For a detailed discussion of the current state of SPH and magnetic fields we refer to the recent review of Price (2012).
Artificial dissipation is also often considered as a major drawback. However, if dissipation is steered properly, see Section 3.2.5, the performance should be very similar to the one of approximate Riemann solver approaches. A Riemann solver approach may from an aesthetic point of view be more appealing, though, and a number of such approaches have been successfully implemented and tested (Inutsuka, 2002; Cha and Whitworth, 2003; Cha et al., 2010; Murante et al., 2011; Puri and Ramachandran, 2014).
Contrary to what was often claimed in the early literature, however, SPH is not necessarily a very efficient method. It is true that if only the bulk matter distribution is of interest, one can often obtain robust results already with an astonishingly small number of SPH particles. To obtain accurate results for the thermodynamics of the flow, however, still usually requires large particle numbers. In large SPH simulations, it becomes a serious challenge to maintain cache-coherence since particles that were initially close in space and computer memory can later on become completely scattered throughout different (mostly slow) parts of the memory. This can be improved by using cache-friendly variables (aggregate data that are frequently used together into common variables, even if they have not much of a physical connection) and/or by various sorting techniques to re-order particles in memory according to their spatial location. This can be done — in the simplest case — by using a uniform hash grid, but in many astrophysical applications hierarchical structures such as trees are highly preferred for sorting the particles, see e.g., Warren and Salmon (1995); Springel (2005); Nelson et al. (2009); Gafton and Rosswog (2011). While such measures can easily improve the performance by factors of a few, they come with some bookkeeping overhead, which, together with, say, individual time steps and particle sinks or sources, can make codes rather unreadable and error-prone.
Finally, we will briefly discuss in Section 3.6 the construction of accurate initial conditions where the particles are in a true (and stable) numerical equilibrium. This is yet another a non-trivial SPH issue.
1.3 Roadmap through this review
In Section 2 we discuss those kernel approximation techniques that are needed for the discussed SPH discretizations. We begin with the basics and then focus on issues that are needed to appreciate some of the recent improvements of the SPH method. Some of these issues are by their very nature rather technical. Readers that are familiar with basic kernel interpolation techniques could skip this section at first reading.
In Section 3 we discuss SPH discretizations of ideal fluid dynamics for both the Newtonian and the relativistic case. Since several reviews have appeared in the last years, we only concisely summarize the more traditional formulations and focus on recent improvements of the method.
The last section, Section 4, is dedicated to astrophysical applications. We begin with double white-dwarf encounters (Section 4.1) and then turn to encounters between two neutron stars and between a neutron star and a stellar-mass black hole (Section 4.2). In each of these cases, our emphasis is on the more common, gravitational wave-driven binary systems, but we also discuss dynamical collisions as they may occur, for example, in a globular cluster. Sections 2 and 3 provide the basis for an understanding of SPH as a numerical method and they should pave the way to the most recent developments. The less numerically inclined reader may, however, just catch the most basic SPH ideas from Section 3.2.1 and then jump to his/her preferred astrophysical topic in Section 4. The modular structure of Sections 2 and 3 should allow, whenever needed, for a selective consultation on the more technical issues of SPH.
2 Kernel Approximation
2.1 Function interpolation
2.2 Function derivatives
We restrict the discussion here to the first-order derivatives that we will need in Section 3, for higher-order derivatives that are required for some applications we refer the interested reader to the literature (Español and Revenga, 2003; Monaghan, 2005; Price, 2012). There are various ways to calculate gradients for function values that are known at given particle positions. Obviously, the accuracy of the gradient estimate is of concern, but also the symmetry in the particle indices since it can allow to enforce exact numerical conservation of physically conserved quantities. An accurate gradient estimate without built-in conservation can be less useful in practice than a less accurate estimate that exactly obeys Nature’s conservation laws, see Section 5 in Price (2012) for a striking example of how a seemingly good gradient without built-in conservation can lead to pathological particle distributions. The challenge is to combine exact conservation with an accurate gradient estimate.
2.2.1 Direct gradient of the approximant
2.2.2 Constant-exact gradients
2.2.3 Linear-exact gradients
2.2.4 Integral-based gradients
While being slightly less accurate than Eqs. (21)–(23), see Section 2.2.5, the approximation Eq. (26) has the major advantage that \(\vec G\) is anti-symmetric with respect to the exchange of \(\vec r\) and \({\vec r_b}\) just as the direct gradient of the radial kernel, see Eq. (3). Therefore, it allows in a straightforward way to enforce exact momentum conservation,^{3} though with a substantially more accurate gradient estimate. This type of gradient has in a large number of benchmark tests turned out to be superior to the traditional SPH approach.
2.2.5 Accuracy assessment of different gradient prescriptions
We perform a numerical experiment to measure the accuracy of different gradient prescriptions for a regular particle distribution. Since, as outlined above, kernel function and particle distribution are not independent entities, such tests at fixed particle distribution are a useful accuracy indicator, but they should be backed up by tests in which the particles can evolve dynamically. We place SPH particles in a 2D hexagonal lattice configuration in [−1, 1] × [−1, 1]. Each particle is assigned a pressure value according P(x, y) = 2 + x and we use the different prescriptions discussed in Sections 2.2.1–2.2.4 to numerically measure the pressure gradient. The relative average errors, \(\epsilon = {N^{ - 1}}\sum\nolimits_{b = 1}^N {{\epsilon _b}}\) with \({\epsilon _b} = \vert{({\partial _x}P)_b} - 1\vert\), as a function of the kernel support are shown in Figure 3. The quantity η determines the smoothing length via h_{b} = η (m_{b}/ρ_{b})^{1/D}, while D denotes the number of spatial dimensions. For this numerical experiment the standard cubic spline kernel M_{4}, see Section 2.3, has been used. We apply the “standard” SPH-gradient, Eq. (7), the approximate integral-based gradient (“IA gradient”), Eq. (26), the full integral-based gradient (“fIA gradient”), Eq. (21), and the linearly exact-gradient (“LE gradient”), Eq. (14). Note that for this regular particle distribution, the constant-exact gradient, Eq. (10), is practically indistinguishable from the standard prescription, since it differs only by a term that is proportional to the first “gradient quality indicator”, Eq. (9), which vanishes here to a very good approximation. Clearly, the more sophisticated gradient prescriptions yield vast improvements of the gradient accuracy. Both the LE and fIA gradients reproduce the exact value to within machine precision, the IA gradient which has the desired anti-symmetry property, improves the gradient accuracy of the standard SPH estimate by approximately 10 orders of magnitude.
2.3 Which kernel function to choose?
2.3.1 Kernels with vanishing central derivatives
“Bell-shaped” kernels with their vanishing derivatives at the origin are rather insensitive to the exact positions of nearby particles and, therefore, they are good density estimators (Monaghan, 1992). For this reason they have been widely used in SPH simulations. The kernels that we discuss here and their derivatives are plotted in Figure 4. More recently, kernels with non-vanishing central derivatives have been (re-)suggested, some of them are discussed in Section 2.3.2.
2.3.1.1 B-spline functions: M_{4} and M_{6} kernels
2.3.1.2 A parametrized family of kernels
Parameters of W_{H,n} and QCM_{6} kernels. Table adapted from Rosswog (2015).
Normalization σ_{H,n} of W_{H,n} kernels | |||||||
---|---|---|---|---|---|---|---|
n = 3 | n = 4 | n = 5 | n = 6 | n = 7 | n = 8 | n = 9 | |
1D | 0.66020338 | 0.75221501 | 0.83435371 | 0.90920480 | 0.97840221 | 1.04305235 | 1.10394401 |
2D | 0.45073324 | 0.58031218 | 0.71037946 | 0.84070999 | 0.97119717 | 1.10178466 | 1.23244006 |
3D | 0.31787809 | 0.45891752 | 0.61701265 | 0.79044959 | 0.97794935 | 1.17851074 | 1.39132215 |
Parameters of the QCM_{6} kernel
q_{c} | 0.75929848 |
A | 11.01753798 |
B | −38.11192354 |
C | −16.61958320 |
D | 69.78576728 |
σ_{1D} | 8.24554795 × 10^{−3} |
σ_{2D} | 4.64964683 × 10^{−3} |
σ_{3D} | 2.65083908 × 10^{−3} |
2.3.1.3 Wendland kernels
An interesting class of kernels with compact support and positive Fourier transforms are the so-called Wendland functions (Wendland, 1995). In several areas of applied mathematics Wendland functions have been well appreciated for their good interpolation properties, but they have not received much attention as SPH kernels and have only recently been explored in more detail (Dehnen and Aly, 2012; Hu et al., 2014; Rosswog, 2015). Dehnen and Aly (2012) have pointed out in particular that these kernels are not prone to the pairing instability, see Section 2.3.4, despite having a vanishing central derivative. These kernels have been explored in a large number of benchmark tests (Rosswog, 2015) and they are high appreciated for their “cold fluid properties”: the particle distribution remains highly ordered even in dynamical simulations (e.g., Figure 6, upper right panel) and only allows for very little noise.
2.3.2 Kernels with non-vanishing central derivatives
A number of kernels have been suggested in the literature whose derivatives remain finite in the center so that the repulsive forces between particles never vanish. The major motivation behind such kernels is to achieve a very regular particle distribution and in particular to avoid the pairing instability, see Section 2.3.4. However, as recently pointed out by Dehnen and Aly (2012), the pairing instability is not necessarily related to a vanishing central derivative, instead, non-negative Fourier transforms have been found as a necessary condition for stability against pairing. Also kernels with vanishing central derivatives can possess this properties. We explore here only two peaked kernel functions, one, the linear quartic core (LIQ) kernel, that has been suggested as a cure to improve SPH’s performance in Kelvin-Helmholtz instabilities and another one, the quartic core M_{6} (QCM_{6}) kernel, mainly for pedagogical reasons to illustrate how an even very subtle change in the central part of the kernel can seriously deteriorate the approximation quality. For a more extensive account on kernels with non-vanishing central derivatives we refer to the literature (Thomas and Couchman, 1992; Fulk and Quinn, 1996; Valcke et al., 2010; Read et al., 2010).
2.3.2.1 Linear quartic kernel
2.3.2.2 Quartic core M_{6} kernel
The different kernels and their derivatives are summarized in Figure 4. As mentioned above, the M_{6}, QCM_{6} and W_{3,3} kernels have been rescaled to a support of Q = 2 to allow for an easy comparison. Note how the kernels become more centrally peaked with increasing order, for example, the W_{H,9} kernel only deviates noticeably from zero inside of q = 1, so that it is very insensitive to particles entering or leaving its support near q = 2.
2.3.3 Accuracy assessment of different kernels
2.3.3.1 Density estimates
We assess the density estimation accuracy of the different kernels in a numerical experiment. The particles are placed in a 2D hexagonal lattice configuration in [−1, 1] × [−1, 1]. This configuration corresponds to the closest packing of spheres with radius r_{s} where each particle possesses an effective area of \({A_{{\rm{eff}}}} = 2\sqrt 3 r_{\rm{s}}^2\). Each particle is now assigned the same mass m_{b} = ρ_{0}A_{eff} to obtain the uniform density ρ_{0}. Subsequently, the densities at the particle locations, ρ_{b}, are calculated via the standard SPH expression for the density, Eq. (56), and the average error \(\epsilon = {N^{ - 1}}\sum\nolimits_{b = 1}^N {{\epsilon _b}} \) with \({\epsilon _b} = \vert{\rho _b} - {\rho _0}\vert/{\rho _0}\) is determined. Figure 5, upper panel, shows the error as a function of η, which parametrizes the kernel support size via h_{b} = η (m_{b}/σ_{b})^{1/D}, D being again the number of spatial dimensions.
Interestingly, the “standard” cubic spline kernel (“CS”, solid black line in the figure) actually does not perform well. At typical values of η near 1.3 the relative density error is a few times 10^{−3}. Just replacing it by the quintic spline kernel (“M_{6}”, solid red line in the figure) improves the density estimate for similar values of η already by two orders of magnitude.^{5} The Wendland kernel (“W_{3,3}”, dashed blue line in the figure) continuously decreases the error with increasing η and therefore does not show the pairing instability at any value of η (Dehnen and Aly, 2012). It maintains a very regular particle distribution with very little noise (Rosswog, 2015), see also the Gresho-Chan vortex test that is shown in Section 3.7.4. In these fixed particle distribution tests the W_{H,n}-kernels perform particularly well. At large smoothing length they deliver exquisite density estimates. For example, the W_{H,9} kernel is at η > 1.9 more than two orders of magnitude more accurate than M_{6}.
2.3.3.2 Gradient estimates
Again, the “standard” cubic spline kernel (solid black) does not perform particularly well, only for η > 1.8 reaches the gradient estimate an accuracy better than 1%. In a dynamical situation, however, the particle distribution would at such a large kernel support already fall prey to the pairing instability. For moderately small supports, say η < 1.6, the M_{6} kernel is substantially more accurate. Again, the accuracy of the Wendland kernel increases monotonically with increasing η, and the three W_{H,n}-kernels perform best in this static test. As in the case of the density estimates, the peaked kernels perform rather poorly and only achieve a 1% accuracy for extremely large values of η.
2.3.4 Kernel choice and pairing instability
Figure 5 suggests to increase the kernel support to achieve greater accuracy in density and gradient estimates (though at the price of reduced spatial resolution). For many bell-shaped kernels, however, the support cannot be increased arbitrarily since for large smoothing lengths the so-called “pairing instability” sets in where particles start to form pairs. In the most extreme case, two particles can merge into effectively one. Nevertheless, this is a rather benign instability. An example is shown in Figure 6, where for the interaction of a blast wave with a low-density bubble^{6}, we have once chosen (left panels) a kernel-smoothing length combination (M_{4} kernel with η = 2.2) that leads to strong particle pairing, and once (right panels) a combination (Wendland kernel with η= 2. 2) that produces a very regular particle distribution. Note that despite the strong pairing in the left column, the continuum properties (the lower row shows the density as an example) are still reasonably well reproduced. The pairing simply leads to a loss of resolution, though at the original computational cost.
2.4 Summary: Kernel approximation
The improvement of the involved kernel approximation techniques is one obvious way how to further enhance the accuracy of SPH. One should strive, however, to preserve SPH’s most important asset, its exact numerical conservation. The simplest possible, yet effective, improvement is to just replace the kernel function, see Section 2.3. We have briefly discussed a number of kernels and assessed their accuracy in estimating a uniform density and the gradient of a linear function for the case where the SPH particles are placed on a hexagonal lattice. The most widely used kernel function, the cubic spline M_{4}, does actually not perform particularly well, neither for the density nor the gradient estimate. At moderate extra cost, however, one can use substantially more accurate kernels, for example the quintic spline kernel, M, or the higher-order members of the W_{H,n} family. Another, very promising kernel family are the Wendland functions. They are not prone to the pairing instability and therefore show much better convergence properties than kernels that start forming pairs beyond a critical support size. Moreover, the Wendland kernel that we explored in detail (Rosswog, 2015) is very reluctant to allow for particle motion on a sub-resolution scale and it maintains a very regular particle distribution, even in highly dynamical tests. The explored peaked kernels, in contrast, performed rather poorly in both estimating densities and gradients.
3 SPH Formulations of Ideal Fluid Dynamics
Smoothed particle hydrodynamics (SPH) is a completely mesh-free, Lagrangian method that was originally suggested in an astrophysical context (Gingold and Monaghan, 1977; Lucy, 1977), but by now it has also found many applications in the engineering world, see Monaghan (2012) for a starting point. Since a number of detailed reviews exists, from the “classics” (Benz, 1990; Monaghan, 1992) to more recent ones (Monaghan, 2005; Rosswog, 2009; Springel, 2010a; Price, 2012), we want to avoid becoming too repetitive about SPH basics and therefore put the emphasis here on recent developments. Many of them have very good potential, but have not yet fully made their way into practical simulations. Our emphasis here is also meant as a motivation for computational astrophysicists to keep their simulation tools up-to-date in terms of methodology. A very explicit account on the derivation of various SPH aspects has been provided in Rosswog (2009), therefore we will sometimes refer the interested reader to this text for more technical details.
The basic idea of SPH is to represent a fluid by freely moving interpolation points — the particles — whose evolution is governed Nature’s conservation laws. These particles move with the local fluid velocity and their densities and gradients are determined by the kernel approximation techniques discussed in Section 2, see Figure 7. The corresponding evolution equations can be formulated in such a way that mass, energy, momentum and angular momentum are conserved by construction, i.e., they are fulfilled independent of the numerical resolution.
In the following, we use the convention that the considered particle is labeled with “a”, its neighbors with “b” and a general particle with “k”, see also the sketch in Figure 1. Moreover, the difference between two vectors is denoted as \({\vec A_{ab}} = {\vec A_a} - {\vec A_b}\) and the symbol B_{ab} refers to the arithmetic average B_{ab} = (B_{a} + B_{b})/2 of two scalar functions.
3.1 Choice of the SPH volume element
The problem can be alleviated if also the internal energy is smoothed by applying some artificial thermal conductivity and this has been shown to work well for Kelvin-Helmholtz instabilities (Price, 2008). But, it is actually a non-trivial problem to design appropriate triggers that supply conductivity exclusively where needed and not elsewhere. Artificial conductivity applied where it is undesired can have catastrophic consequences, for example by removing physically required energy/pressure gradients for a star in hydrostatic equilibrium.
An alternative cure comes from using different volume elements in the SPH discretization process. The first step in this direction was probably taken by Ritchie and Thomas (2001) who realized that by using the internal energy as a weight in the SPH density estimate a much sharper density transition could be achieved than by the standard SPH density sum where each particle is, apart from the kernel, only weighted by its mass. In Saitoh and Makino (2013) it was pointed out that SPH formulations that do not include density explicitly in the equations of motion do avoid the pressure becoming multi-valued at contact discontinuities. Since the density usually enters the equation of motion via the choice of the volume element, a different choice can possibly avoid the problem altogether. This observation is consistent with the findings of Heß and Springel (2010) who used a particle hydrodynamics method, but calculated the volumes via a Voronoi tessellation rather than via smooth density sum. In their approach no spurious surface tension effects have been observed. Closer to the original SPH spirit is the class of kernel-based particle volume estimates that have recently been suggested by Hopkins (2013) as a generalization of the approach from Saitoh and Makino (2013). Recently, such volume elements have been generalized for the use in special-relativistic studies (Rosswog, 2015).
3.2 Newtonian SPH
3.2.1 “Vanilla ice” SPH
3.2.2 SPH from a variational principle
More elegantly, a set of SPH equations can be obtained by starting from the (discretized) Lagrangian of an ideal fluid (Gingold and Monaghan, 1982; Speith, 1998; Monaghan and Price, 2001; Springel and Hernquist, 2002; Rosswog, 2009; Price, 2012). This ensures that conservation of mass, energy, momentum and angular momentum is, by construction, built into the discretized form of the resulting fluid equations. Below, we use the generalized volume element Eq. (45) without specifying the choice of the weight X so that a whole class of SPH equations is produced (Hopkins, 2013). In this derivation the volume of an SPH particle takes over the fundamental role that is usually played by the density sum. Therefore we will generally express the SPH equations in terms of volumes rather than densities, we only make use of the latter for comparison with known equation sets.
Explicit forms of the equations for special choices of the weight X for Newtonian and special-relativistic SPH.
Newtonian SPH | |
---|---|
WeightX = m | |
density | \({\rho _a} = \sum\nolimits_b {{m_b}} {W_{ab}}({h_a})\) |
momentum | \({\textstyle{{d{{\vec v}_a}} \over {dt}}} = - \sum\nolimits_b {{m_b}} \left\{ {{\textstyle{{{P_a}} \over {{\Omega _a}\rho _a^2}}}{\nabla _a}{W_{ab}}({h_a}) + {\textstyle{{{P_b}} \over {{\Omega _b}\rho _b^2}}}{\nabla _a}{W_{ab}}({h_b})} \right\}\) |
energy | \({\textstyle{{d{u_a}} \over {dt}}} = {\textstyle{{{P_a}} \over {{\Omega _a}\rho _a^2}}}\sum\nolimits_b {{m_b}} {{\vec v}_{ab}} \cdot {\nabla _a}{W_{ab}}({h_a})\) |
WeightX = 1 | |
density | \({\rho _a} = {m_b}\sum\nolimits_b {{W_{ab}}} ({h_a})\) |
momentum | \({\textstyle{{d{{\vec v}_a}} \over {dt}}} = - \sum\nolimits_b {{m_b}} \left\{ {{\textstyle{{{m_a}} \over {{m_b}}}}{\textstyle{{{P_a}} \over {{\Omega _a}\rho _a^2}}}{\nabla _a}{W_{ab}}({h_a}) + {\textstyle{{{m_a}} \over {{m_b}}}}{\textstyle{{{P_b}} \over {{\Omega _b}\rho _b^2}}}{\nabla _a}{W_{ab}}({h_b})} \right\}\) |
energy | \({\textstyle{{d{u_a}} \over {dt}}} = {\textstyle{{{P_a}{m_a}} \over {{\Omega _a}\rho _a^2}}}\sum\nolimits_b {{{\vec v}_{ab}}} \cdot {\nabla _a}{W_{ab}}({h_a})\) |
WeightX = P^{k} | |
density | \({\rho _a} = {m_b}\sum\nolimits_b {{{\left({{\textstyle{{{P_b}} \over {{P_a}}}}} \right)}^k}{W_{ab}}} ({h_a})\) |
momentum | \({\textstyle{{d{{\vec v}_a}} \over {dt}}} = - \sum\nolimits_b {{m_b}} \left\{ {{\textstyle{{{m_a}} \over {{m_b}}}}{\textstyle{{P_a^{1 - k}P_b^k} \over {\Omega _a^2\rho _a^2}}}{\nabla _a}{W_{ab}}({h_a}) + {\textstyle{{{m_a}} \over {{m_b}}}}{\textstyle{{P_a^kP_b^{1 - k}} \over {{\Omega _b}\rho _a^2}}}{\nabla _a}{W_{ab}}({h_b})} \right\}\) |
energy | \({\textstyle{{d{u_a}} \over {dt}}} = {\textstyle{{{P_a}{m_a}} \over {{\Omega _a}\rho _a^2}}}\sum\nolimits_b {{{\left({{\textstyle{{{P_b}} \over {{P_a}}}}} \right)}^k}{{\vec v}_{ab}}} \cdot {\nabla _a}{W_{ab}}({h_a})\) |
Special-relativistic SPH | |
WeightX = ν | |
CF bar. num. density | \({\rho _a} = \sum\nolimits_b {{\nu _b}{W_{ab}}} ({h_a})\) |
spec. canon. momentum | \({\textstyle{{d{{\vec S}_a}} \over {dt}}} = - \sum\nolimits_b {{\nu _b}} \left\{ {{\textstyle{{{P_a}} \over {{\Omega _a}N_a^2}}}{\nabla _a}{W_{ab}}({h_a}) + {\textstyle{{{P_b}} \over {{\Omega _b}N_a^2}}}{\nabla _a}{W_{ab}}({h_b})} \right\}\) |
spec. canon. energy | \({\textstyle{{d{\epsilon _a}} \over {dt}}} = - \sum\nolimits_b {{\nu _b}} \left\{ {{\textstyle{{{P_a}} \over {{{\tilde \Omega }_a}N_a^2}}}{{\vec v}_b} \cdot {\nabla _a}{W_{ab}}({h_a}) + {\textstyle{{{P_b}} \over {{{\tilde \Omega }_b}}}}{{\vec v}_a} \cdot {\nabla _a}{W_{ab}}({h_b})} \right\}\) |
WeightX = 1 | |
CF bar. num. density | \({N_a} = {\nu _a}\sum\nolimits_b {{W_{ab}}} ({h_a})\) |
spec. canon. momentum | \({\textstyle{{d{{\vec S}_a}} \over {dt}}} = - \sum\nolimits_b {{\nu _b}} \left\{ {{\textstyle{{{\nu _a}} \over {{\nu _b}}}}{\textstyle{{{P_a}} \over {{\Omega _a}N_a^2}}}{\nabla _a}{W_{ab}}({h_a}) + {\textstyle{{{\nu _b}} \over {{\nu _a}}}}{\textstyle{{{P_b}} \over {{\Omega _b}N_a^2}}}{\nabla _a}{W_{ab}}({h_b})} \right\}\) |
spec. canon. energy | \({\textstyle{{d{\epsilon _a}} \over {dt}}} = - \sum\nolimits_b {{\nu _b}} \left\{ {{\textstyle{{{\nu _a}} \over {{\nu _b}}}}{\textstyle{{{P_a}} \over {{{\tilde \Omega }_a}N_a^2}}}{{\vec v}_b} \cdot {\nabla _a}{W_{ab}}({h_a}) + {\textstyle{{{\nu _b}} \over {{\nu _a}}}}{\textstyle{{{P_b}} \over {{{\tilde \Omega }_b}}}}{{\vec v}_a} \cdot {\nabla _a}{W_{ab}}({h_b})} \right\}\) |
WeightX = P^{k} | |
CF bar. num. density | \({N_a} = {\nu _a}\sum\nolimits_b {{{\left({{\textstyle{{{P_b}} \over {{P_a}}}}} \right)}^k}{W_{ab}}} ({h_a})\) |
spec. canon. momentum | \({\textstyle{{d{{\vec S}_a}} \over {dt}}} = - \sum\nolimits_b {{\nu _b}} \left\{ {{\textstyle{{{\nu _a}} \over {{\nu _b}}}}{\textstyle{{P_a^{1 - k}P_b^k} \over {\Omega _a^2N_a^2}}}{\nabla _a}{W_{ab}}({h_a}) + {\textstyle{{{\nu _b}} \over {{\nu _a}}}}{\textstyle{{P_a^kP_b^{1 - k}} \over {{\Omega _b}N_a^2}}}{\nabla _a}{W_{ab}}({h_b})} \right\}\) |
spec. canon. energy | \({\textstyle{{d{\epsilon _a}} \over {dt}}} = - \sum\nolimits_b {{\nu _b}} \left\{ {{\textstyle{{{\nu _a}} \over {{\nu _b}}}}{\textstyle{{P_a^{1 - k}P_b^k} \over {{{\tilde \Omega }_a}N_a^2}}}{{\vec v}_b} \cdot {\nabla _a}{W_{ab}}({h_a}) + {\textstyle{{{\nu _a}} \over {{\nu _b}}}}{\textstyle{{P_a^kP_b^{1 - k}} \over {{{\tilde \Omega }_b}}}}{{\vec v}_a} \cdot {\nabla _a}{W_{ab}}({h_b})} \right\}\) |
3.2.2.1 The Gadget equations
3.2.3 Self-regularization in SPH
SPH possesses a built-in “self-regularization” mechanism, i.e. SPH particles feel, in addition to pressure gradients, a force that aims at driving them towards an optimal particle distribution. This corresponds to (usually ad hoc introduced) “re-meshing” steps that are used in Lagrangian mesh methods. The ability to automatically re-mesh is closely related the lack of zeroth order consistency of SPH that was briefly described in Section 2.2.1: the particles “realize” that their distribution is imperfect and they have to adjust accordingly. Particle methods without such a re-meshing mechanism can quickly evolve into rather pathological particle configurations that yield, in long-term, very poor results, see Price (2012) for an explicit numerical example.
3.2.4 SPH with integral-based derivatives
3.2.5 Treatment of shocks
The equations of gas dynamics allow for discontinuities to emerge even from perfectly smooth initial conditions (Landau and Lifshitz, 1959). At discontinuities, the differential form of the fluid equations is no longer valid, and their integral form needs to be used, which, at shocks, translates into the Rankine-Hugoniot conditions, which relate the upstream and downstream properties of the flow. They show in particular, that the entropy increases in shocks, i.e., that dissipation occurs inside the shock front. For this reason the inviscid SPH equations need to be augmented by further measures that become active near shocks.
Another line of reasoning that suggests using artificial viscosity is the following. One can think of the SPH particles as (macroscopic) fluid elements that follow streamlines, so in this sense SPH is a method of characteristics. Problems can occur when particle trajectories cross since in such a case fluid properties at the crossing point can become multi-valued. The term linear in the velocities in Eq. (82) was originally also introduced as a measure to avoid particle interpenetration where it should not occur (Hernquist and Katz, 1989) and to damp particle noise. Read and Hayfield(2012) designed special dissipation switches to avoid particle properties becoming multi-valued at trajectory crossings.
3.2.5.1 Common form of the dissipative equations
3.2.5.2 Time dependent dissipation parameters
3.2.5.3 New shock triggers
3.2.5.4 Noise triggers
Note that in all these triggers (Cullen and Dehnen, 2010; Read and Hayfield, 2012; Rosswog, 2015) the gradients can straight forwardly be calculated from accurate expressions such as Eq. (21) or (14) rather than from kernel gradients.
3.2.5.5 Limiters
3.3 Special-relativistic SPH
The special-relativistic SPH equations can — like the Newtonian ones — be elegantly derived from a variational principle (Monaghan and Price, 2001; Rosswog, 2009, 2010a, b). We discuss here a formulation (Rosswog, 2015) that uses generalized volume elements, see Eq. (45). It is assumed that space-time is flat, that the metric, η_{μν}, has the signature (-,+,+,+) and units in which the speed of light is equal to unity, c=1, are adopted. We use the Einstein summation convention and reserve Greek letters for space-time indices from 0…3 with 0 being the temporal component, while i and j refer to spatial components and SPH particles are labeled by a, b and k.
Note that by choosing the canonical energy and momentum as numerical variables, one avoids complications such as time derivatives of Lorentz factors, that have plagued earlier SPH formulations (Laguna et al., 1993a). The price one has to pay is that the physical variables (such as υ and \(\vec v\)) need to be recovered at every time step from N, e and \(\vec S\) by solving a non-linear equation. For this task approaches very similar to what is used in Eulerian relativistic hydrodynamics can be applied (Chow and Monaghan, 1997; Rosswog, 2010b).
3.3.1 Integral approximation-based special-relativistic SPH
3.4 General-relativistic SPH
For binaries that contain a neutron or a black hole general-relativistic effects are important. The first relativistic SPH formulations were developed by Kheyfets et al. (1990) and Mann (1991, 1993). Shortly after, Laguna et al. (1993a) developed a 3D, general-relativistic SPH code that was subsequently applied to the tidal disruption of stars by massive black holes (Laguna et al., 1993b). Their SPH formulation is complicated by several issues: the continuity equation contains a gravitational source term that requires SPH kernels for curved space-times. Moreover, owing to their choice of variables, the equations contain time derivatives of Lorentz factors that are treated by finite difference approximations and restrict the ability to handle shocks to only moderate Lorentz factors. The Laguna et al. formulation has been extended by Rantsiou et al. (2008) and applied to neutron star black hole binaries, see Section 4.2.6. We focus here on SPH in a fixed background metric, approximate GR treatments are briefly discussed in Section 4.2.
3.4.1 Limiting cases
It is a straight forward exercise to show that, in the limit of vanishing hydrodynamic terms (i.e., υ and P = 0), the evolution equations (141) and (144) reduce to the equation of geodesic motion (Tejeda, 2012). If, in the opposite limit, we are neglecting the gravitational terms in Eqs. (141) and (144) and assume flat space-time with Cartesian coordinates one has \(\sqrt { - g} \rightarrow 1\) and Θ → γ, and Eq. (134) becomes N = γn. The momentum and energy equations reduce in this limit to the previous equations (108) and (113) (for X = ν).
3.5 Frequently used SPH codes
Frequently used Newtonian SPH codes
reference | name/group | SPH equations | self-gravity | AV steering | EOS, burning |
---|---|---|---|---|---|
Gadget | entropy equation, Eq. (71) | oct-tree | fixed parameters | polytrope | |
Faber et al. (2010) | StarCrash | “vanilla ice” | FFT on grid | fixed parameters | polytrope |
Rosswog and Davies (2002) | Leicester | “vanilla ice” | binary tree (Benz et al., 1990) | time-dep., Balsara-switch | Shen |
Rosswog et al. (2008) | Bremen | “vanilla ice” | binary tree (Benz et al., 1990) | time-dep., Balsara-switch | Helmholtz + network |
Rosswog and Price (2007) | Magma | hydro & self-gravity from Lagrangian | binary tree | time-dep., Balsara-switch, conductivity | Shen |
Guerrero et al. (2004) | Barcelona | “vanilla ice” | oct-tree | fixed parameters, Balsara-switch | ions, electrons, photons + network |
Fryer et al. (2006) | SNSPH | “vanilla ice” | oct-tree | fixed parameters | polytrope, Helmholtz + network |
Wadsley et al. (2004) | Gasoline | “vanilla ice” | K-D tree | fixed parameters, Balsara-switch | polytrope |
3.6 Importance of initial conditions
In experiments (Rosswog, 2015) different kernels show a very different noise behavior and this seems to be uncorrelated with the accuracy properties of the kernels: kernels that are rather poor density and gradient estimators may be excellent in producing very little noise (see, for example, the QCM_{6} kernel described in Rosswog, 2015). On the other hand, very accurate kernels (like W_{H,9}) may be still allow for a fair amount of noise in a dynamical simulation. The family of Wendland kernels (Wendland, 1995) has a number of interesting properties, among them the stability against particle pairing despite having a vanishing central derivative (Dehnen and Aly, 2012) and their reluctance to tolerate sub-resolution particle motion (Rosswog, 2015), in other words noise. In those experiments where particle noise is relevant for the overall accuracy, for example in the Gresho-Chan vortex test, see Section 3.7.4, the Wendland kernel, Eq. (34), performs substantially better than any other of the explored kernels.
A heuristic approach to construct good, low-noise initial conditions for the actually used kernel function is to apply a velocity-dependent damping term, \({\vec f_{{\rm{damp}}}} \propto - {\vec v_{{\rm{damp}},{\rm{a}}}}/{\tau _a},\), with a suitably chosen damping time scale τ_{a} to the momentum equation. This “relaxation process” can be applied until some convergence criterion is met (say, the kinetic energy or some density oscillation amplitude has dropped below some threshold). This procedure becomes, however, difficult to apply for more complicated initial conditions and it can become as time consuming as the subsequent production simulation. For an interesting recent suggestion on how to construct more general initial conditions see Diehl et al. (2012).
3.7 The performance of SPH
Here we cannot give an exhaustive overview over the performance of SPH in general. We do want to address, however, a number of issues that are of particular relevance in astrophysics. Thereby we put particular emphasis on the new improvements of SPH that were discussed in the previous sections. Most astrophysical simulations, however, are still carried out with older methodologies. This is natural since, on the one hand, SPH is still a relatively young numerical method and improvements are constantly being suggested and, on the other hand, writing a well-tested and robust production code is usually a rather laborious endeavor. Nevertheless, efforts should be taken to ensure that latest developments find their way into production codes. In this sense the below shown examples are also meant as a motivation for computational astrophysicists to keep their simulation tools up-to-date.
higher-order kernels, for example, the W_{H,n} or the Wendland kernel W_{3,3}, see Section 2.3
different volume elements, see Section 3.1
more accurate integral-based derivatives, see Section 2.2.4
modern dissipation triggers, see Section 3.2.5. In the examples shown here, we use a special-relativistic SPH code (“SPHINCS_SR”, Rosswog, 2015) to demonstrate SPH’s performance in a few astrophysically relevant examples. Below, we refer several times to the “\({\mathcal{F}_1}\) formulation”: it consists of baryon number density calculated via Eq. (104) with volume weight X = P^{0}^{05}, the integral approximation-based form of the SPH equations, see Eqs. (121) and (122), and shock trigger Eq. (88) and noise triggers, see Rosswog (2015) for more details.
3.7.1 Advection
3.7.2 Shocks
3.7.3 Fluid instabilities
3.7.4 The Gresho—Chan vortex
3.8 Summary: SPH
The most outstanding property of SPH is its exact numerical conservation. This can straight forwardly be achieved via symmetries in the particle indices of the SPH equations together with anti-symmetric gradient estimates. The most elegant and least arbitrary strategy to obtain a conservative SPH formulation is to start from a fluid Lagrangian and to derive the evolution equations via a variational principle. This approach can be applied in the Newtonian, special- and general relativistic case, see Sections 3.2, 3.3 and 3.4. Advection of fluid properties is essentially perfect in SPH and it is in particular not dependent on the coordinate frame in which the simulation is performed.
SPH robustly captures shocks, but they are, at a given resolution, not as sharp as those from state-of-the-art high-resolution shock-capturing schemes. Moreover, standard SPH has been criticized for its (in)ability to resolve fluid instabilities under certain circumstances. Another issue that requires attention when performing SPH simulations are initial conditions. When not prepared carefully, they can easily lead to noisy results, since the regularization force discussed in Section 3.2.3 leads for poor particle distribution to a fair amount of particle velocity fluctuations. This issue is particular severe when poor kernel functions and/or low neighbor numbers are used, see Section 2.
Recently, a number of improvements to SPH techniques have been suggested. These include a) more accurate gradient estimates, see Section 2.2, b) new volume elements which eliminate spurious surface tension effects, see Section 3.1, c) higher-order kernels, see Section 2.3 and d) more sophisticated dissipation switches, see Section 3.2.5. As illustrated, for example by the Gresho-Chan vortex test in Section 3.7.4, enhancing SPH with these improvements can substantially increase its accuracy with respect to older SPH formulations.
4 Astrophysical Applications
two white dwarfs (Section 4.1),
two neutron stars (Section 4.2.5) and
a neutron star with a black hole (Section 4.2.6). In each case the focus is on gravitational-wave-driven binary mergers. In locations with large stellar number densities, e.g., globular clusters, dynamical collisions between stars occur frequently and encounters between two neutron stars and a neutron star with a stellar-mass black hole may yield very interesting signatures. Therefore, such encounters are also briefly discussed.
In each of these fields, a wealth of important results have been achieved with a number of different methods. Naturally, since the scope of this review are SPH methods, we will focus our attention here to those studies that are at least partially based on SPH simulations. For further studies that are based on different methods we have to refer to the literature.
4.1 Double white-dwarf encounters
4.1.1 Relevance
White dwarfs (WDs) are the evolutionary end stages of most stars in the Universe, for every solar mass of stars that forms ∼ 0.22 WDs will be produced on average. As a result, the Milky Way contains ∼ 10^{10} WDs (Napiwotzki, 2009) in total and ∼ 10^{8} double WD systems (Nelemans et al., 2001b). About half of these systems have separations that are small enough (orbital periods < 10 hrs) so that gravitational wave emission will bring them into contact within a Hubble time, making them a major target for the eLISA mission (Amaro-Seoane et al., 2013). Once in contact, in almost all cases the binary system will merge, in the remaining small fraction of cases mass transfer may stabilize the orbital decay and lead to long-lived interacting binaries such as AM CVn systems (Paczyński, 1967; Warner, 1995; Nelemans et al., 2001a; Nelemans, 2005; Solheim, 2010).
Those systems that merge may have a manifold of interesting possible outcomes. The merger of two He WDs may produce a low-mass He star (Webbink, 1984; Iben Jr and Tutukov, 1986; Saio and Jeffery, 2000; Han et al., 2002), He-CO mergers may form hydrogen-deficient giant or R CrB stars (Webbink, 1984; Iben Jr et al., 1996; Clayton et al., 2007) and if two CO WDs merge, the outcome may be a more massive, possibly hot and high B-field WD (Bergeron et al., 1991; Barstow et al., 1995; Segretain et al., 1997). A good fraction of the CO-CO merger remnants probably transforms into ONeMg WDs which finally, due to electron captures on Ne and Mg, undergo an accretion-induced collapse (AIC) to a neutron star (Saio and Nomoto, 1985; Nomoto and Kondo, 1991; Saio and Nomoto, 1998). Given that the nuclear binding energy that can still be released by burning to iron group elements (1.6 MeV from He, 1.1 MeV from C and 0.8 MeV from O) is large, it is not too surprising that there are also various pathways to thermonuclear explosions. The ignition of helium on the surface of a WD may lead to weak thermonuclear explosions (Bildsten et al., 2007; Foley et al., 2009; Perets et al., 2010), sometimes called “.Ia” supernovae. The modern view is that WDWD mergers might also trigger type Ia supernovae (SN Ia) (Webbink, 1984; Iben Jr and Tutukov, 1984) and, in some cases, even particularly bright “super-Chandrasekhar” explosions, e.g., Howell et al. (2006); Hicken et al. (2007).
SN Ia are important as cosmological distance indicators, as factories for intermediate mass and iron-group nuclei, as cosmic ray accelerators, kinetic energy sources for galaxy evolution or simply in their own right as end points of binary stellar evolution. After having been the second-best option behind the “single degenerate” model for decades, it now seems entirely possible that double degenerate mergers are behind a sizable fraction of SN Ia. It seems that with the re-discovery of double degenerates as promising type Ia progenitors an interesting time for supernovae research has begun. See Howell (2011) and Maoz et al. (2013) for two excellent recent reviews on this topic.
Below, we will briefly summarize the challenges in a numerical simulation of a WDWD merger (Section 4.1.2) and then discuss recent results concerning mass transferring systems (Section 4.1.3) and, closely related, to the final merger of a WDWD binary and possibilities to trigger SN Ia (Section 4.1.4). We will also briefly discuss dynamical collisions of WDs (Section 4.1.5). For SPH studies that explore the gravitational wave signatures of WDWD mergers we refer to the literature (Lorén-Aguilar et al., 2005; Dan et al., 2011; van den Broek et al., 2012).
Note that in this section we explicitly include the constants G and c in the equations to allow for a simple link to the astrophysical literature.
4.1.2 Challenges
WDWD merger simulations are challenging for a number of reasons not the least of which are the onset of mass transfer and the self-consistent triggering of thermonuclear explosions.
The onset of mass transfer represents a juncture in the life of WDWD binary, since now the stability of mass transfer decides whether the binary can survive or will inevitably merge. The latter depends sensitively on the internal structure of the donor star, the binary mass ratio and the angular momentum transport mechanisms (e.g., Marsh et al., 2004; Gokhale et al., 2007). Due to the inverse mass-radius relationship of WDs, the secondary will expand on mass loss and, therefore, tendentially speed up the mass loss further. On the other hand, since the mass is transferred to the higher mass object, momentum/center of mass conservation will tend to widen the orbit and, therefore, tendentially reduce mass transfer. If the circularization radius of the transferred matter is smaller than the primary radius it will directly impact on the stellar surface and tend to spin up the accreting star. In this way, orbital angular momentum is lost to the spin of the primary which, in turn, decreases the orbital separation and accelerates the mass transfer. If, on the other hand, the circularization radius is larger than the primary radius and a disk can form, angular momentum can, via the large lever arm of the disk, be fed back into the orbital motion and stabilize the binary system (Iben Jr et al., 1998; Piro, 2011). To make things even more complicated, if tidal interaction substantially heats up the mass donating star it may impact on its internal structure and, therefore, change its response to mass loss.
To capture these complex angular momentum transfer mechanisms reliably in a simulation requires a very accurate numerical angular momentum conservation. We want to briefly illustrate this point with a small numerical experiment. A 0.3 + 0.6 M_{⊙} WD binary system is prepared in a Keplerian orbit, so that mass transfer is about to set in. To mimic the effect of numerical angular momentum loss in a controllable way, we add small artificial forces similar to those emerging from gravitational wave emission (Peters and Mathews, 1963; Peters, 1964; Davies et al., 1994) and adjust the overall value so that 4% or 0.5% of angular momentum per orbit are lost. These results are compared to a simulation without artificial loss terms where the angular momentum is conserved to better than 0.01% per orbit, see Figure 13. The effect on the mutual separation a (in 10^{9} cm) is shown in the upper panels and the gravitational wave amplitude, h_{+} (r is the distance to the observer) as calculated in the quadrupole formalism, are shown the lower panels.
Even the moderate loss of 0.5% angular momentum per orbit leads to a quick artificial merger and a mass transfer duration that is reduced by more than a factor of three. These conservation requirements make SPH a natural choice for WDWD merger simulations and it has indeed been the first method used for these type of problems.
There is also a huge disparity in terms of time scales. Whenever nuclear burning is important for the dynamics of the gas flow, the nuclear time scales are many orders of magnitude shorter than the admissible hydrodynamic time steps. Therefore, nuclear networks are usually implemented via operator splitting methods, see e.g., Benz et al. (1989); Rosswog et al. (2009b); Raskin et al. (2010); García-Senz et al. (2013). Because of the exact advection in SPH the post-processing of hydrodynamic trajectories with larger nuclear networks to obtain detailed abundance patterns is straight forward. For burning processes in tenuous surface layers, however, SPH is seriously challenged since here the resolution is poorest. For such problems, hybrid approaches that combine SPH with, say, AMR methods (Guillochon et al., 2010; Dan et al., 2015) seem to be the best strategies.
4.1.3 Dynamics and mass transfer in white-dwarf binaries systems
Three-dimensional simulations of WDWD mergers were pioneered by Benz et al. (1990). Their major motivation was to understand the merger dynamics and the possible role of double degenerate systems as SN Ia progenitors. They used an SPH formulation as described in Section 3.2.1 (“vanilla ice”) together with 7000 SPH particles, an equation of state for a non-degenerate ideal gas with a completely degenerate, fully relativistic electron component and they restricted themselves to the study of a 0.9–1.2 M_{⊙} system. No attempts were undertaken to include nuclear burning in this study (but see Benz et al., 1989). Each star was relaxed in isolation, see Section 3.6, and subsequently placed in a circular Keplerian orbit so that the secondary was overfilling its critical lobe by ∼ 8%. Under these conditions the secondary star was disrupted within slightly more than two orbital periods, forming a three-component system of a rather unperturbed primary, a hot pressure supported spherical envelope and a rotationally supported outer disk. About 0.6 % of a solar mass were able to escape, the remaining ∼ 1.7 M_{⊙}, supported mainly by pressure gradients, showed no sign of collapse.
Rasio and Shapiro (1995) were more interested in the equilibrium and the (secular, dynamical and mass transfer) stability properties of close binary systems. They studied systems both with stiff (Γ > 5/3; as models for neutron stars) and soft (Γ = 5/3) polytropic equations of state, as approximations for the EOS of (not too massive) WDs and low-mass main sequence star binaries. They put particular emphasis on constructing accurate, synchronized initial conditions (Rasio and Shapiro, 1994). These were obtained by relaxing the binary system in a corotating frame where, in equilibrium, all velocities should vanish. The resulting configurations satisfied the virial theorem to an accuracy of about one part in 10^{3}. With these initial conditions they found a more gradual increase in the mass transfer rate in comparison to Benz et al. (1990), but nevertheless the binary was disrupted after only a few orbital periods.
Segretain et al. (1997) focussed on the question whether particularly massive and hot WDs could be the result of binary mergers (Bergeron et al., 1991). They applied a simulation technology similar to Benz et al. (1990) and concentrated on a binary system with non-spinning WDs of 0.6 and 0.9 M_{⊙}. They showed, for example, that such a merger remnant would need to lose about 90% of its angular momentum in order to reproduce properties of the observed candidate WDs.
In their study, such carefully constructed initial conditions were compared to the previously commonly used approximate initial conditions. Apart from the inaccuracies inherent to the analytical Roche lobe estimates, approximate initial conditions also neglect the tidal deformations of the stars and, therefore, seriously underestimate the initial separation at the onset of mass transfer. For this reason, such initial conditions have up to 15% too little angular momentum and, as a result, binary systems with inaccurate initial conditions merge too violently on a much too short time scale. As a result, temperatures and densities in the final remnant are over- and the size of tidal tails are underestimated. The carefully prepared binary systems all showed dozens of orbits of numerically resolvable mass transfer. Given that, due to the finite resolution, the mass transfer is already highly super-Eddington when it starts being resolvable all the results on mass transfer duration have to be considered as strict lower limits. One particular example, a 0.2 M_{⊙} He-WD and a 0.8 CO-WD, merged with approximate initial conditions within two orbital periods (comparable to earlier SPH results), but only after painfully long 84 orbital periods when the initial conditions were prepared carefully. This particular example also illustrated the suitability of SPH for such investigations: during the orbital evolution, which corresponds to ≈ 17000 dynamical time scales, energy and angular momentum were conserved to better than 1%! All of the investigated (according to the Marsh et al. analysis unstable) binary systems merged in the end although only after several dozens of orbital periods. Some systems showed a systematic widening of the orbits after the onset of mass transfer. Although they were still disrupted in the end, this indicated that systems in the parameter space vicinity of this 0.2–0.8 M_{⊙} system may evolve into short-period AM CVn systems.
In two recent studies, Dan et al. (2012, 2014) systematically explored the parameter space by simulating 225 different binary systems with masses ranging from 0.2 to 1.2 M_{⊙}. All of the initial conditions were prepared carefully as Dan et al. (2011). Despite the only moderate resolution (40 K particles) that could be afforded in such a broad study, they found excellent agreement with the orbital evolution predicted by mass transfer stability analysis (Marsh et al., 2004; Gokhale et al., 2007).
4.1.4 Double white-dwarf mergers and possible pathways to thermonuclear super-novae
The Barcelona group were the first to explore the effect of nuclear burning during a WD merger event (Guerrero et al., 2004). They implemented the reduced 14-isotope α-network of Benz et al. (1989) with updated reaction rates into a “vanilla-ice” SPH code with artificial viscosity enhanced by the Balsara factor, see Sections 3.2.1 and 3.2.5. They typically used 40 K particles, approximate initial conditions as described above and explored six different combinations of masses/chemical compositions. They found an orbital dynamics similar to Benz et al. (1989); Segretain et al. (1997) and, although in the outer, partially degenerate layers of the central core temperatures around 10^{9} K were encountered, no dynamically important nuclear burning was observed. Whenever it set in, the remnant had time to quench it by expansion, both for the He and CO accreting systems. Therefore, they concluded that direct SN Ia explosions were unlikely, but some remnants could evolve into subdwarf B objects as suggested in Iben Jr (1990).
Yoon et al. (2007) challenged the “classical picture” of the cold WD accreting from a thick disk as an oversimplification. Instead, they suggested that the subsequent secular evolution of the remnant would be better studied by treating the central object as a differentially rotating CO star with a central, slowly rotating, cold core engulfed by a rapidly rotating hot envelope, which, in turn, is embedded and fed by a centrifugally supported Keplerian accretion disk. The further evolution of such a system is then governed by the thermal cooling of the hot envelope and the redistribution of angular momentum inside of the central remnant and the accretion of the matter from the disk into the envelope. They based their study of the secular remnant evolution on a dynamical merger calculation of two CO WDs with 0.6 and 0.9 M_{⊙}. To this end, they used an SPH code originally developed for neutron star merger calculations (Rosswog et al., 1999, 2000; Rosswog and Davies, 2002; Rosswog and Liebendörfer, 2003; Rosswog et al., 2008) extended by the Helmholtz EOS (Timmes, 1999) and a quasi-equilibrium reduced a-network (Hix et al., 1998). Particular care was taken to avoid artefacts from the artificial viscosity treatment and time-dependent viscosity parameters (Morris and Monaghan, 1997) and a Balsara-switch (Balsara, 1995), see Section 3.2.5, were used in the simulation. As suggested by the work of Segretain et al. (1997), they assumed non-synchronized stars and started the simulations from approximate initial conditions, see Section 4.1.3. Once a stationary remnant had been formed, the results were mapped into a 1D hydrodynamic stellar evolution code (Yoon and Langer, 2004) and its secular evolution was followed including the effects of rotation and angular momentum transport. They found that the growth of the stellar core is controlled by the neutrino cooling at the interface between the core and the envelope and that carbon ignition could be avoided provided that a) once the merger reaches a quasi-static equilibrium temperatures are below the carbon ignition threshold, b) the angular momentum loss occurs on a time scale longer than the neutrino cooling time scale and c) the mass accretion from the centrifugally supported disk is low enough (M ≤ 5 × 10^{−6}–10^{−5}M_{⊙} yr^{−1}). From such remnants an explosion may be triggered ∼ 10^{5} years after the merger. Such systems, however, may need unrealistically low viscosities.
A more recent study (Shen et al., 2012) started from two remnants of CO WD mergers (Dan et al., 2011) and followed their viscous longterm evolution. Their conclusion was more in line with earlier studies: they expected that the long-term result would be ONe or an accretion-induced collapse to a neutron star rather than a SN Ia.
In recent years, WD mergers have been extensively explored as possible pathways to SN Ia. Not too surprisingly, a number of pathways have been discovered that very likely lead to a thermonuclear explosion directly prior to or during the merger. Whether these explosions are responsible for (some fraction of) normal SNe Ia or for peculiar subtypes needs to be further explored in future work. Many of the recent studies used a number of different numerical tools to explore various aspects of WDWD mergers. We focus here on those studies where SPH simulations were involved.
4.1.4.1 Explosions prior to merger
Dan et al. (2011) had carefully studied the impact of mass transfer on the orbital dynamics. In these SPH simulations the feedback on the orbit is accurately modelled, but due to SPH’s automatic “refinement on density” the properties of the transferred matter are not well resolved. Therefore, a “best-of-both-worlds” approach was followed in Guillochon et al. (2010)/Dan et al. (2011) where the impact of the mass transfer on the orbital dynamics was simulated with SPH, while recording the orbital evolution and the mass transfer rate. This information was used in a second set of simulations that was performed with the Flash code (Fryxell et al., 2000). This second study focussed on the detailed hydrodynamic interaction of the transferred mass with the accretor star. For He-CO binary systems where helium directly impacts on a primary of a mass > 0.9 M_{⊙}, they found that helium surface explosions can be self-consistently triggered via Kelvin-Helmholtz (KH) instabilities. These instabilities occur at the interface between the incoming helium stream and an already formed helium torus around the primary. “Knots” produced by the KH instabilities can lead to local ignition points once the triple-alpha time scale becomes shorter than the local dynamical time scale. The resulting detonations travel around the primary surface and collide on the side opposite to the ignition point. Such helium surface detonations may resemble weak type Ia SNe (Bildsten et al., 2007; Foley et al., 2009; Perets et al., 2010) and they may drive shock waves into the CO core which concentrate in one or more focal points, similar to what was found in the 2D study of Fink et al. (2007). This could possibly lead to an explosion via a “double-detonation” mechanism. In a subsequent large-scale parameter study, Dan et al. (2012, 2014) found, based on a comparison between nuclear burning and hydrodynamical time scales, that a large fraction of the helium-accreting systems do produce explosions early on: all dynamically unstable systems with primary masses < 1.1 M_{⊙} together with secondary masses > 0.4 M_{⊙} triggered helium-detonations at surface contact. A good fraction of these systems could also produce in addition KH-instability-induced detonations as described in detail in Guillochon et al. (2010). There was no definitive evidence for explosions prior to contact for any of the studied CO-transferring systems.
4.1.4.2 Explosions during merger
Pakmor et al. (2010) studied double degenerate mergers, but — contrary to earlier studies — they focussed on very massive WDs with masses close to 0.9 M_{⊙}. They used the Gadget code (Springel, 2005) with some modifications (Pakmor et al., 2012a), for example, the Timmes EOS (Timmes, 1999) and a 13 isotope network were implemented for their study. In order to facilitate the network implementation, the energy equation (instead of, as usually in Gadget, the entropy equation) was evolved. No efforts were undertaken to reduce the constant, untriggered artificial viscosity. They placed the stars on orbit with the approximate initial conditions described above and found the secondary to be disrupted within two orbital periods. In a second step, several hot (> 2.9 × 10^{9} K) particles were identified and the remnant was artificially ignited in these hot spots. The explosion was followed with a grid-based hydrodynamics code (Fink et al., 2007; Röpke and Niemeyer, 2007) that had been used in earlier SN Ia studies. In a third step, the nucleosynthesis was post-processed and synthetic light curves were calculated (Kromer and Sim, 2009). The explosion resulted in a moderate amount of ^{56}Ni (0.1 M_{⊙}), large amounts (1.1 M_{⊙}) of intermediate mass elements and oxygen (0.5 M_{⊙}) and less than 0.1 M_{⊙} of unburnt carbon. The kinetic energy of the explosion (1.3 × 10^{51} erg) was typical for a SN Ia, but the resulting velocities were relatively small, so that the explosion resembled a sub-luminous 1991bg-like supernova. An important condition for reaching ignition is a mass ratio close to unity. Some variation in total mass is expected, but cases with less than 0.9 M_{⊙} of a primary mass would struggle to reach the ignition temperatures and — if successful — the lower densities would lead to even lower resulting ^{56}Ni masses and therefore lower luminosities. For substantially higher masses, in contrast, the burning would proceed at larger densities and, therefore, result in much larger amounts of ^{56}Ni. Based on population synthesis models (Ruiter et al., 2009) they estimated that mergers of this type of system could account for 2–11 % of the observed SN Ia rate. In a follow up study (Pakmor et al., 2011) the sensitivity of the proposed model to the mass ratio was studied. The authors concluded that binaries with a primary mass near 0.9 M_{⊙} ignite a detonation immediately at contact, provided that the mass ratio q exceeds 0.8. Both the abundance tomography and the lower-than-standard velocities provided support for the idea of this type of merger producing sub-luminous, 1991bg-type supernovae.
As a variation of the theme, Pakmor et al. (2012b) also explored the merger of a higher mass system with 0.9 and 1.1 M_{⊙}. Using the same assumption about the triggering of detonations as before, they found a substantially larger mass of ^{56}Ni (0.6 M_{⊙}), 0.5 M_{⊙} of intermediate mass elements, 0.5 M_{⊙} of Oxygen and about 0.15 M_{⊙} of unburnt carbon. Due to its higher density, only the primary is able to burn ^{56}Ni and, therefore, the brightness of the SN Ia would be closely related to the primary mass. The secondary is only incompletely burnt and thus provides the bulk of the intermediate mass elements. Overall, the authors concluded that such a merger reproduces the observational properties of normal SN Ia reasonably well. In Kromer et al. (2013) the results of a 0.9 and 0.76 M_{⊙} CO-CO merger were analyzed and unburned oxygen close to the center of the ejecta was found which produces narrow emission lines of [O1] in the late-time spectrum, similar to what is observed in the sub-luminous SN 2010lp (Leibundgut et al., 1993).
Fryer et al. (2010) applied a sequence of computational tools to study the spectra that can be expected from a supernova triggered by a double-degenerate merger. Motivated by population synthesis calculations, they simulated a CO-CO binary of 0.9 and 1.2 M_{⊙} with the SNSPH code (Fryer et al., 2006). They assumed that the remnant would explode at the Chandrasekhar mass limit, into a gas cloud consisting of the remaining merger debris. They found a density profile with ρ ∝ r^{−4}, inserted an explosion (Meakin et al., 2009) into such a matter distribution and calculated signatures with the radiation-hydrodynamics code Rage (Fryer et al., 2007, 2009). They found that in such “enshrouded” SNe Ia the debris extends and delays the X-ray flux from a shock breakout and produces a signal that is closer to a SN Ib/c. Also the V-band peak was extended and much broader than in a normal SN Ia with the early spectra being dominated by CO lines only. They concluded that, within their model, a CO-CO merger with a total mass > 1.5 M_{⊙} would not produce spectra and light curves that resemble normal type Ia supernovae.
One of the insights gained from the study of Pakmor et al. (2010) was that (massive) mergers with similar masses are more likely progenitors of SNe Ia than the mergers with larger mass differences that were studied earlier. This motivated Zhu et al. (2013) to perform a large parameter study where they systematically scanned the parameter space from 0.4–0.4 M_{⊙} up to 1.0–1.0 M_{⊙}. In their study, they used the Gasoline code (Wadsley et al., 2004) together with the Helmholtz EOS (Timmes, 1999; Timmes and Swesty, 2000), no nuclear reactions were included. All their simulations were performed with non-spinning stars and approximate initial conditions as outlined above. Mergers with “similar” masses produced a well-mixed, hot central core while “dissimilar” masses produced a rather unaffected cold core surrounded by a hot envelope and disk, consistent with earlier studies. They found that the central density ratio of the accreting and donating star of ρ_{a}/ρ_{d} > 0.6 is a good criterion for those systems that produce hot cores (i.e., to define “similar” masses).
Dan et al. (2014) also performed a very broad scan of the parameter space. They studied the temperature distribution inside the remnant for different stellar spins: tidally locked initial conditions produce hot spots (which are the most likely locations for detonations to be initiated) in the outer layers of the core, while irrotational systems produce them deep inside of the core, consistent with the results of Zhu et al. (2013). Thus, the spin state of the WDs may possibly be decisive for the question where the ignition is triggered which may have its bearings on the resulting supernova. Dan et al. found essentially no chemical mixing between the stars for mass ratios below q ≈ 0.45, but maximum mixing for a mass ratio of unity. Contrary to Zhu et al. (2013), nowhere complete mixing was found, but this difference can be convincingly attributed to the different stellar spins that were investigated (tidal locking in Dan et al., 2014, no spins in Zhu et al., 2013). In addition to the helium-accreting systems that likely undergo a detonation for a total mass beyond 1.1 M_{⊙}, they also found CO binaries with total masses beyond 2.1 M_{⊙} to be prone to a CO explosion. Such systems may be candidates for the so-called super-Chandrasekhar SN Ia explosions. They also discussed the possibility of “hybrid supernovae” where a ONeMg core with a significant helium layer collapses and forms a neutron star. While technically being a (probably weak) core-collapse supernova (Podsiadlowski et al., 2004; Kitaura et al., 2006), most of the explosion energy may come from helium burning. Such hybrid supernovae may be candidates for the class of “Ca-rich” SNe Ib (Perets et al., 2010) as the burning conditions seem to favor the production of intermediate mass elements.
Raskin et al. (2012) explored 10 merging WDWD binary systems with total masses between 1.28 and 2.12 M_{⊙} with the SNSPH code (Fryer et al., 2006) coupled to the Helmholtz EOS and a 13-isotope nuclear network. They used a heuristic procedure to construct tidally deformed, synchronized binaries by letting the stars fall towards each other in free fall, and repeatedly set the fluid velocities to zero. They had coated their CO WDs with atmospheres of helium (smaller than 2.5% of the total mass) and found that for all cases where the primary had a mass of 1.06 M_{⊙}, the helium detonated, no carbon detonation was encountered, though.
Given the difficulty to identify the central engine of a SNe Ia on purely theoretical grounds, Raskin and Kasen (2013) explored possible observational signatures stemming from the tidal tails of a WD merger. They followed the ejecta from a SNSPH simulation (Fryer et al., 2006) with n-body methods and — assuming spherical symmetry — they explored by means of a 1D Lagrangian code how a supernova would interact with such a medium. Provided the time lag between merger and supernova is short enough (< 100 s), detectable shock emission at radio, optical, and/or X-ray wavelengths is expected. For delay times between 10^{8} s and 100 years one expects broad NaID absorption features, and, since this has not been observed to date, they concluded that if (some) type Ia supernovae are indeed caused by WDWD mergers, the delay times need to be either short (< 100 s) or rather long (> 100 years). If the tails can expand and cool for ∼ 10^{4} years, they produce the observable narrow NaID and Ca II K & K lines which are seen in some fraction of type Ia supernovae.
An interesting study from a Santa Cruz-Berkeley collaboration (Moll et al., 2014; Raskin et al., 2014) combined again the strengths of different numerical methods. The merger process was calculated with the SNSPH code (Fryer et al., 2006), the subsequent explosion with the grid-based code Castro (Almgren et al., 2010; Zhang et al., 2011) and synthetic light curves and spectra were calculated with Sedona (Kasen et al., 2006). For some cases without immediate explosion the viscous remnant evolution was followed further with ZEUS-MP2 (Hayes et al., 2006). The first part of the study (Moll et al., 2014) focussed on prompt detonations, while the second part explored the properties of detonations that emerge in later phases, after the secondary has been completely disrupted. For the first part, three mergers (1.20–1.06, 1.06–1.06 and 0.96–0.81 M_{⊙}) were simulated with the SPH code and subsequently mapped into Castro, where soon after simulation start (< 0.1 s) detonations emerged. They found the best agreement with common SNe Ia (0.58 M_{⊙} of ^{56}Ni) for the binary with 0.96–0.81 M_{⊙}. More massive systems lead to more ^{56}Ni and therefore unusually bright SNe Ia. The remnant asymmetry at the moment of detonation leads to large asymmetries in the elemental distributions and therefore to strong viewing angle effects for the resulting supernova. Depending on viewing angle, the peak bolometric luminosity varied by a factor of two and the flux in the ultraviolet even varied by an order of magnitude. All of the three models approximately fulfilled the width-luminosity relation (“brighter means broader”; Phillips et al., 1999).
The companion study (Raskin et al., 2014) explored cases where the secondary has become completely disrupted before a detonation sets in, so that the primary explodes into a disk-like CO structure. To initiate detonations, the SPH simulations were stopped once a stationary structure had formed, all the material with ρ > 10^{6} g cm^{−3} was burnt in a separate simulation and, subsequently, the generated energy was deposited back as thermal energy into the merger remnant and the SPH simulation was resumed. As a double-check of the robustness of this approach, two simulations were also mapped into Castro and detonated there as well. Overall there was good agreement both in morphology and the nucleosynthetic yields. The explosion inside the disk produced an hourglass-shaped remnant geometry with strong viewing angle effects. The disk scale height, initially set by the mass ratio and the different merger dynamics and burning processes, turned out to be an important factor for the viewing angle dependence of the later supernova. The other crucial factor was the primary mass that determines the resulting amount of ^{56}Ni. Interestingly, the location of the detonation spot, whether at the surface or in the core of the primary, had a relatively small effect compared to the presence of an accretion disk. While qualitatively in agreement with the width-luminosity relation, the lightcurves lasted much longer than standard SNe Ia. The surrounding CO disk from the secondary remained essentially unburnt, but, impeding the expansion, it lead to relatively small intermediate mass element absorption velocities. The large asymmetries in the abundance distribution could lead to a large overestimate of the involved ^{56}Ni masses if spherical symmetry is assumed in interpreting observations. The lightcurves and spectra were peculiar with weak features from intermediate mass elements but relatively strong carbon absorption. The study also explored how longer-term viscous evolution before a detonation sets in would affect the supernova. Longer delay times were found to produce likely larger ^{56}Ni masses and more symmetrical remnants. Such systems might be candidates for super-Chandra SNe Ia.
4.1.5 Simulations of white dwarf—white dwarf collisions
This result is interesting for a number of reasons. First, the detonation mechanism is parameterfree and extremely robust: the free-fall velocity between WDs naturally produces relative velocities in excess of the WD sound speeds, therefore strong shocks are inevitable. Moreover, the most likely involved WD masses are near the peak of the mass distribution, ∼ 0.6 M_{⊙}, or, due to mass segregation effects, possibly slightly larger, but well below the Chandrasekhar mass. This has the benefit that the nuclear burning occurs at moderate densities (ρ ∼ 10^{7} g cm^{−3}) and thus produces naturally the observed mix of ∼ 0.5 M_{⊙}^{56}Ni and intermediate-mass elements, without any fine-tuning such as the deflagration-detonation transition that is required in the single-degenerate scenario (Hillebrandt and Niemeyer, 2000). However, based on simple order of magnitude estimates, the original studies (Raskin et al., 2009; Rosswog et al., 2009a) concluded that, while being very interesting explosions, the rates would likely be too low to make a substantial contribution to the observed supernova sample. More recently, however, there have been claims (Thompson, 2011; Kushnir et al., 2013) that the Kozai-Lidov mechanism in triple stellar systems may substantially boost the rates of WDWD collisions so that they could constitute a sizeable fraction of the SN Ia rate. Contrary to these claims, a recent study by Hamers et al. (2013) finds that the contribution from the triple-induced channels to SN Ia is small. Here further studies are needed to quantify how relevant collisions really are for explaining normal SNe Ia.
4.1.6 Summary: Double white-dwarf encounters
SPH has very often been used to model mergers of, and later on also collisions between, two WDs. This is mainly since SPH is not restricted by any predefined geometry and has excellent conservation properties. As illustrated in the numerical experiment shown in Figure 13, even small (artificial) losses of angular momentum can lead to very large errors in the prediction of the mass transfer duration and the gravitational-wave signal. SPH’s tendency to “follow the density” makes it ideal to predict, for example, the gravitational-wave signatures of WD mergers. But it is exactly this tendency which makes it very difficult for SPH to study thermonuclear ignition processes in low-density regions that are very important, for example, for the double-detonation scenario. This suggests to apply in such cases a combination of numerical tools: SPH for bulk motion and orbital dynamics and Eulerian (Adaptive Mesh) hydrodynamics for low-density regions that need high resolution. As outlined above, there have recently been a number of successful studies that have followed such strategies.
SPH simulations have played a pivotal role in “re-discovering” the importance of white-dwarf mergers (and possibly, to some extent, collisions) as progenitor systems of SNe Ia. In the last few years, a number of new possible pathways to thermonuclear explosions prior to or during a WD merger have been discovered. There is, however, not yet a clear consensus whether they produce just “peculiar” SN Ia-like events or whether they may be even responsible for the bulk of “standard” SN Ia. Here, a lot of progress can be expected within the next few years, both from the modelling and the observational side.
4.2 Encounters between neutron stars and black holes
4.2.1 Relevance
In addition, CBMs have long been suspected to be the engines of short gamma-ray bursts (sGRBs) (Paczyński, 1986; Goodman, 1986; Eichler et al., 1989; Narayan et al., 1992). Although already their projected distribution on the sky and their fluence distribution pointed to a cosmological source (Goodman, 1986; Paczyński, 1986; Schmidt et al., 1988; Meegan et al., 1992; Piran, 1992; Schmidt, 2001; Guetta and Piran, 2005), their cosmological origin was only firmly established by the first afterglow observations for short bursts in 2005 (Hjorth et al., 2005; Bloom et al., 2006). This established the scale for both distance and energy and proved that sGRBs occur in both early- and late-type galaxies. CBMs are natural candidates for sGRBs since accreting compact objects are very efficient converters of gravitational energy into electromagnetic radiation, they occur at rates that are consistent with those of sGRBs (Guetta and Piran, 2005; Nakar et al., 2006; Guetta and Piran, 2006) and CBMs are expected to occur in both early and late-type galaxies. Kicks imparted at birth provide a natural explanation for the observed projected offsets of ∼ 5 kpc from their host galaxy (Fong et al., 2013), and, with dynamical time scales of ∼ 1 ms (either orbital at the ISCO or the neutron star oscillation time scales), they naturally provide the observed short-time fluctuations. Moreover, for cases where an accretion torus forms, the expected viscous lifetime is roughly comparable with a typical sGRB duration (∼ 0.2 s). While this picture is certainly not without open questions, see Piran (2004); Lee and Ramirez-Ruiz (2007); Nakar (2007); Gehrels et al. (2009); Berger (2011, 2013) for recent reviews, it has survived the confrontation with three decades of observational results and — while competitors have emerged — it is still the most commonly accepted model for the engine of short GRBs.
Lattimer and Schramm (1974, 1976) and Lattimer et al. (1977) suggested that the decompression of initially cold neutron star matter could lead to rapid neutron capture, or “r-process”, nucleosynthesis so that CBMs may actually also be an important source of heavy elements. Although discussed convincingly in a number of subsequent publications (Symbalisty and Schramm, 1982; Eichler et al., 1989), this idea kept the status of an exotic alternative to the prevailing idea that the heaviest elements are formed in core-collapse supernovae, see Arcones and Thielemann (2013) for a recent review. Early SPH simulations of NSNS mergers (Rosswog et al., 1998, 1999) showed that of order ∼ 1% of neutron-rich material is ejected per merger event, enough to be a substantial or even the major source of r-process. A nucleosynthesis post-processing of these SPH results (Freiburghaus et al., 1999) confirmed that this material is indeed a natural candidate for the robust, heavy r-process (Sneden et al., 2008). Initially it was doubted (Qian, 2000; Argast et al., 2004) that CMBs as main r-process source are consistent with galactic chemical evolution, but a recent study based on a detailed chemical evolution model (Matteucci et al., 2014) finds room for a substantial contribution of neutron star mergers and recent hydrodynamic galaxy formation studies (Shen et al., 2014; van de Voort et al., 2015) even come to the conclusion that neutron star mergers are consistent with being the dominant r-process source in the Universe. Moreover, state-of-the-art supernova models seem unable to provide the conditions that are needed to produce heavy elements with nucleon numbers in excess of A = 90 (Arcones et al., 2007; Fischer et al., 2010; Hüdepohl et al., 2010; Roberts et al., 2010).^{12} On the other hand, essentially all recent studies agree that CBMs eject enough mass to be at least a major r-process source (Oechslin et al., 2007; Bauswein et al., 2013b; Rosswog et al., 2014; Hotokezaka et al., 2013b; Kyutoku et al., 2013) and that the resulting abundance pattern beyond A ≈ 130 resembles the solar-system distribution (Metzger et al., 2010b; Roberts et al., 2011; Korobkin et al., 2012; Goriely et al., 2011a,b; Eichler et al., 2015). Recent studies (Wanajo et al., 2014; Just et al., 2014) suggest that compact binary mergers with their different ejection channels for neutron-rich matter could actually even be responsible for the whole range of r-process.
In June 2013, the SWIFT satellite detected a relatively nearby (z = 0.356) sGRB, GRB 130603B, (Melandri et al., 2013) for which the HST (Tanvir et al., 2013; Berger et al., 2013) detected 9 days after the burst a nIR point source with properties close to what had been predicted (Kasen et al., 2013; Barnes and Kasen, 2013; Grossman et al., 2014; Tanaka and Hotokezaka, 2013; Tanaka et al., 2014) by models for “macro-” or “kilonovae” (Li and Paczyński, 1998; Kulkarni, 2005; Rosswog, 2005; Metzger et al., 2010a,b; Roberts et al., 2011), radioactively powered transients from the decay of freshly produced r-process elements. If this interpretation is correct, GRB 130603B would provide the first observational confirmation of the long-suspected link between CBMs, nucleosynthesis and gamma-ray bursts.
4.2.2 Differences between double neutron and neutron star black hole mergers
Both NSNS and NSBH systems share the same three stages of the merger: a) the secular inspiral where the mutual separation is much larger than the object radii and the orbital evolution can be very accurately described by post-Newtonian methods (Blanchet, 2014), b) the merger phase where relativistic hydrodynamics is important and c) the subsequent ringdown phase. Although similar in their formation paths and in their relevance, there are a number of differences between NSNS and NSBH systems. For example, when the surfaces of two neutron stars come into contact the neutron star matter can heat up and radiate neutrinos. Closely related, if matter from the interaction region is dynamically ejected, it may — due to the large temperatures — have the chance to change its electron fraction due to weak interactions. Such material may have a different nucleosynthetic signature than the matter that is ejected during a NSBH merger. Moreover, the interface between two neutron stars is prone to hydrodynamic instabilities that can amplify existing neutron star magnetic fields (Price and Rosswog, 2006; Anderson et al., 2008; Rezzolla et al., 2011; Zrake and MacFadyen, 2013). The arguably largest differences between the two types of mergers, however, are the total binary mass and its mass ratio q. For neutron stars, the deviation from unity ∣ 1 − q ∣ is small — for masses that are known to better than 0.01 M_{⊙}, J1807−2500B has the largest deviation with a mass ratio of q = 0.88 (Lattimer, 2012) — while for NSBH systems a broad range of total masses is expected. Since the merger dynamics is very sensitive to the mass ratio, a much larger diversity is expected in the dynamical behavior of NSBH systems. The larger bh mass has also as a consequence that the plunging phase of the orbit sets in at larger separations and therefore lower GW frequencies, see Eq. (154). Moreover, for black holes the dimensionless spin parameter can be close to unity, a_{BH} = cJ/GM^{2} ≈ 1, while for neutron stars it is restricted by the mass shedding limit to somewhat lower values a_{NS} < 0.7 (Lo and Lin, 2011), therefore spin-orbit coupling effects could potentially be larger for NSBH systems and lead to observable effects, e.g., Buonanno et al. (2003); Grandclément et al. (2004).
4.2.3 Challenges
Compact binary mergers are challenging to model since in principle each of the fundamental interactions enters: the strong (equation of state), weak (composition, neutrino emission, nucleosynthesis), electromagnetic (transients, neutron star crust) and of course (strong) gravity. Huge progress has been achieved during the last decade, but so far none of the approaches “has it all” and, depending on the focus of the study, different compromises have to be accepted.
Of comparable importance is the neutron star equation of state (EOS). Together with gravity it determines the compactness of the neutron stars which in turn impacts on peak GW frequencies. It also influences the torus masses, the amount of ejected matter, and, since it sets the β-equilibrium value of the electron fraction Y_{e} in neutron star matter, also the resulting nucleosynthesis and the possibly emerging electromagnetic transients. Unfortunately, the EOS is not well known beyond ∼ 3 times nuclear density. On the other hand, since the EOS seriously impacts on a number of observable properties from a CBM, one can be optimistic and hope to conversely constrain the high-density EOS via astronomical observations.
Since the bulk of a CBM remnant consists of high density matter (ρ > 10^{10} g cm^{−3}), photons are very efficiently trapped and the only cooling agents are neutrinos. Moreover, with temperatures of order MeV, weak interactions become so fast that they change the electron fractions Y_{e} substantially on a dynamical time scale (∼ 1 ms). While such effects can be safely ignored when gravitational-wave emission is the main focus, these processes are crucial for the neutrino signature, for the “engine physics” of GRBs (e.g., via \(\nu \bar \nu\) annihilation), and for nucleosynthesis, since the nuclear reactions are very sensitive to the neutron-to-proton ratio which is set by the weak interactions. Fortunately, the treatment of neutrino interactions is not as delicate as in the core-collapse supernova case where changes on the percent level can decide between a successful and a failed explosion (Janka et al., 2007). Thus, depending on the exact focus, leakage schemes or simple transport approximations may be admissible in the case of compact binary mergers. But also approximate treatments face the challenge that the remnant is intrinsically three-dimensional, that the optical depths change from τ ∼ 10^{4} in the hypermassive neutron star (e.g., Rosswog and Liebendörfer, 2003), to essentially zero in the outer regions of the disk and that the neutrino-nucleon interactions are highly energy-dependent.
Neutron stars are naturally endowed with strong magnetic fields and a compact binary merger offers a wealth of possibilities to amplify them. They may be decisive for the fundamental mechanism to produce a GRB in the first place, but they may also determine — via transport of angular momentum — when the central object produced in a NSNS merger collapses into a black hole or how accretion disks evolve further under the transport mediated via the magneto-rotational instability (MRI) (Balbus and Hawley, 1998).
4.2.4 The current status of SPH- vs grid-based simulations
A lot of progress has been achieved in recent years in simulations of CBMs. This includes many different microphysical aspects as well as the dynamic solution of the Einstein equations. The main focus of this review are SPH methods and therefore we will restrict the detailed discussion to work that makes use of SPH. Nevertheless, it is worth briefly comparing the current status of SPH-based simulations with those that have been obtained with grid-based methods. As will be explained in more detail below, SPH had from the beginning a very good track record with respect to the implementation of various microphysics ingredients. On the relativistic gravity side, however, it is lagging behind in terms of implementations that dynamically solve the Einstein equations, a task that has been achieved with Eulerian methods already more than a decade ago (Shibata, 1999; Shibata and Uryū, 2000). In SPH, apart from Newtonian gravity, post-Newtonian and conformal flatness approaches (CFA) exist, but up to now no coupling between SPH-hydrodynamics and a dynamic spacetime solver has been achieved.
Naturally, this has implications for the types of problems that have been addressed and there are interesting questions related to NSNS and NSBH mergers that have so far not yet (or only approximately) been tackled with SPH approaches. One example with far-reaching astrophysical consequences is the collapse of a hypermassive neutron star (HMNS) that temporarily forms after a binary neutron star merger. Observations now indicate a lower limit on the maximum neutronstar mass of around 2.0 M_{⊙} (1.97 ± 0.04 M_{⊙} for PSR J1614+2230, see Demorest et al., 2010, and 2.01 ± 0.04 M_{⊙} for PSR J0348+0432, see Antoniadis et al., 2013) and, therefore, it is very likely that the hot and rapidly differentially rotating central remnant of a NSNS merger is at least temporarily stabilized against a gravitational collapse to a black hole, see e.g., Hotokezaka et al. (2013a). It appears actually entirely plausible that the low-mass end of the NSNS distribution may actually leave behind a massive but stable neutron star rather than a black hole. In SPH, the question when a collapse sets in, can so far only be addressed within the conformal flatness approximation (CFA), see below. Being exact for spherical symmetry, the CFA should be fairly accurate in describing the collapse itself. It is, however, less clear how accurate the CFA is during the last inspiral stages where the deviations from spherical symmetry are substantial. Therefore quantitative statements about the HMNS lifetimes need to be interpreted with care. On the other hand, differential rotation has a major impact in the stabilization and therefore hydrodynamic resolution is also crucial for the question of the collapse time. Here, SPH with its natural tendency to refine on density should perform very well once coupled to a dynamic spacetime solver.
Another question of high astrophysical significance is the amount of matter that becomes ejected into space during a NSNS or NSBH merger. It is likely one of the major sources of r-process in the cosmos and thought to cause electromagnetic transients similar to the recently observed “macronova” event in the aftermath of GRB 130603B (Tanvir et al., 2013; Berger et al., 2013). In terms of mass ejection, one could expect large differences between the results of fully relativistic, grid-based hydrodynamics results and Newtonian or approximate GR SPH approaches. The expected differences are twofold. On the one hand, Newtonian/approximate GR treatments may yield stars of a different compactness which in turn would influence the dynamics and torques and therefore the ejecta amount. But apart from gravity, SPH has a clear edge in dealing with ejecta: mass conservation is exact, advection is exact (i.e., a composition only changes due to nuclear reactions but not due to numerical effects), angular momentum conservation is exact and vacuum really corresponds to the absence of matter. Eulerian schemes usually face here the challenges that conservation of mass, angular momentum and the accuracy of advection are resolution-dependent and that vacuum most often is modelled as background fluid of lower density. Given this rather long list of challenges, it is actually encouraging that the results of different groups with very different methods agree reasonably well these days. For NSNS mergers, Newtonian SPH simulations (Rosswog, 2013) find a range from 8 × 10^{−3}–4 × 10^{−2}M_{⊙}, approximate GR SPH calculations (Bauswein et al., 2013b) find a range from 10^{−3}–2 × 10^{−2}M_{⊙}, and full GR calculations (Hotokezaka et al., 2013b) find 10^{−4}–10^{−2}M_{⊙}.^{13} Even the results from Newtonian NSBH calculations agree quite well with the GR results (compare Table 1 in Rosswog, 2013 and the results of Kyutoku et al., 2013).
Another advantage of SPH is that one can also decide to just focus on the ejected matter. For example, a recent SPH-based study (Rosswog et al., 2014) has followed the evolution of the dynamic merger ejecta for as long as 100 years, while Eulerian methods are usually restricted to very few tens of milliseconds. During this expansion the density was followed from supra-nuclear densities (> 2 × 10^{14} g/ccm) down to values that are below the interstellar matter density (< 10^{−25} g/ccm).
For many of the topics that will be addressed below, there have been parallel efforts on the Eulerian side and within the scope of this review we will not be able to do justice to all these parallel developments. As a starting point, we want to point to a number of excellent textbooks (Alcubierre, 2008; Baumgarte and Shapiro, 2010; Rezzolla and Zanotti, 2013) that deal with relativistic (mostly Eulerian) fluid dynamics and to various recent review articles (Duez, 2010; Shibata and Taniguchi, 2011; Faber and Rasio, 2012; Pfeiffer, 2012; Lehner and Pretorius, 2014).
4.2.5 Neutron star—neutron star mergers
4.2.5.1 Early Newtonian calculations with polytropic equation of state
The earliest NSNS merger calculations (Rasio and Shapiro, 1992; Davies et al., 1994; Zhuge et al., 1994; Rasio and Shapiro, 1994, 1995; Zhuge et al., 1996) were performed with Newtonian gravity and a polytropic equation of state, sometimes a simple gravitational-wave backreaction force was added. While these initial studies were, of course, rather simple, they set the stage for future efforts and settled questions about the qualitative merger dynamics and some of the emerging phenomena. For example, they established the emergence of a Kelvin-Helmholtz unstable vortex sheet at the interface between the two stars, which, due to the larger shear, is more pronounced for initially non-rotating stars. Moreover, they confirmed the expectation that a relatively baryon-free funnel would form along the binary rotation axis (Davies et al., 1994) (although this conclusion may need to be revisited in the light of emerging, neutrino-driven winds, see e.g., Dessart et al., 2009; Perego et al., 2014; Martin et al., 2015). These early simulations also established the basic morphological differences between tidally locked and irrotational binaries and between binaries of different mass ratios. In addition, these studies also drove technical developments that became very useful in later studies such as the relaxation techniques to construct synchronized binary systems (Rasio and Shapiro, 1994) or the procedures to analyze the GW energy spectrum in the frequency band (Zhuge et al., 1994, 1996). See also Rasio and Shapiro (1999) for a review on earlier research.
4.2.5.2 Studies with focus on microphysics
Studies with a focus on microphysics (in a Newtonian framework) were pioneered by Ruffert et al. (1996) who implemented a nuclear equation of state and a neutrino-leakage scheme into their Eulerian (PPM) hydrodynamics code. In SPH, the effects of a nuclear equation of state (EOS) were first explored in Rosswog et al. (1999). The authors implemented the Lattimer-Swesty EOS (Lattimer and Swesty, 1991) and neutrino cooling in the simple free-streaming limit to bracket the effects that neutrino emission may possibly have. To avoid artefacts from excessive artificial dissipation, they also included the time-dependent viscosity parameters suggested by Morris and Monaghan (1997), see Section 3.2.5. They found torus masses between 0.1 and 0.3 M_{⊙}, and, maybe most importantly, that between 4 × 10^{−3} and 4 × 10^{−2}M_{⊙} of neutron star matter becomes ejected. A companion paper (Freiburghaus et al., 1999) post-processed trajectories from this study and found that all the matter undergoes r-process and yields an abundance pattern close to the one observed in the solar system, provided that the initial electron fraction is Y_{e} ≈ 0.1. A subsequent study (Rosswog et al., 2000) explored the effects of different initial stellar spins and mass ratios q ≠ 1 on the ejecta masses.
The simulation ingredients were further refined in Rosswog and Davies (2002); Rosswog and Liebendörfer (2003); Rosswog et al. (2003). Here, the Shen et al. (1998a, b) EOS, extended down to very low densities, was used and a detailed multi-flavor neutrino leakage scheme was developed (Rosswog and Liebendörfer, 2003) that takes particular care to avoid using average neutrino energies. These studies were also performed at a substantially higher resolution (up to 10^{6} particles) than previous SPH studies of the topic. The typical neutrino luminosities turned out to be ∼ 2 × 10^{53} erg/s with typical average neutrino energies of 8/15/20 MeV for ν_{e}, \({\bar \nu _e}\) and the heavy lepton neutrinos. Since GRBs were a major focus of the studies, neutrino annihilation was calculated in a post-processing step and, barring possible complications from baryonic pollution, it was concluded that \(\nu \bar \nu \) annihilation should lead to relativistic ouflows and could produce moderately energetic sGRBs. Simple estimates indicated, however, that strong neutrino-driven winds are likely to occur, that could, depending on the wind geometry, pose a possible threat for the emergence of ultra-relativistic outflow/a sGRB. A more recent, 2D study of the neutrino emission from a merger remnant (Dessart et al., 2009) found indeed strong baryonic winds with mass loss rates Ṁ ∼ 10^{−3}M_{⊙}/s emerging along the binary rotation axis. Recently, studies of neutrino-driven winds have been extended to 3D (Perego et al., 2014) and the properties of the blown-off material have been studied in detail. This complex of topics, ν-driven winds, baryonic pollution, collapse of the central merger remnant will for sure receive more attention in the future. Based on simple arguments, it was also argued that any initial seed magnetic fields should be amplified to values near equipartition, making magnetically launched/aided outflow likely. Subsequent studies (Price and Rosswog, 2006; Anderson et al., 2008; Rezzolla et al., 2011; Zrake and MacFadyen, 2013) found indeed a strong amplification of initial seed magnetic fields.
In a recent set of studies, the neutron star mass parameter space was scanned by exploring systematically mass combinations from 1.0 to 2.0 M_{⊙} (Korobkin et al., 2012; Rosswog et al., 2013; Rosswog, 2013). The main focus here was the dynamically ejected mass and its possible observational signatures. One interesting result (Korobkin et al., 2012) was that the nucleosynthetic abundance pattern is essentially identical for the dynamic ejecta of all mass combinations and even NSBH systems yield practically an identical pattern. While extremely robust to a variation of the astrophysical parameters, the pattern showed some sensitivity to the involved nuclear physics, for example to a change of the mass formula or the distribution of the fission fragments. The authors concluded that the dynamic ejecta of neutron star mergers are excellent candidates for the source of the heavy, so-called “strong r-process” that is observed in a variety of metal-poor stars and shows each time the same relative abundance pattern for nuclei beyond barium (Sneden et al., 2008). Based on these results, predictions were made for the resulting “macronovae” (or sometimes called “kilonovae”) (Li and Paczyński, 1998; Kulkarni, 2005; Rosswog, 2005; Metzger et al., 2010a,b; Roberts et al., 2011). The first set of models assumed spherical symmetry (Piran et al., 2013; Rosswog et al., 2013), but subsequent studies Grossman et al. (2014) were based on the 3D remnant structure obtained by hydrodynamic simulations of the expanding ejecta. This study included the nuclear energy release during the hydrodynamic evolution (Rosswog et al., 2014). The study substantially profited from SPH’s geometric flexibility and its treatment of vacuum as just empty (i.e., SPH particle-free) space. The ejecta expansion was followed for as many as 40 orders of magnitude in density, from nuclear matter down to the densities of interstellar matter. Since from the latter calculations the 3D remnant structure was known, also viewing angle effects for macronovae could be explored (Grossman et al., 2014). Accounting for the very large opacities of the r-process ejecta (Barnes and Kasen, 2013; Kasen et al., 2013), Grossman et al. (2014) predicted that the resulting macronova should peak, depending on the binary system, between 3 and 5 days after the merger in the nIR, roughly consistent with what has been observed in the aftermath of GRB 130603B (Tanvir et al., 2013; Berger et al., 2013).
4.2.5.3 Studies with approximate GR gravity
A natural next step beyond Newtonian gravity is the application of post-Newtonian expansions. Blanchet et al. (1990) developed an approximate formalism for moderately relativistic, self-gravitating fluids which allows to write all the equations in a quasi-Newtonian form and casts all relativistic non-localities in terms of Poisson equations with compactly supported sources. The 1PN equations require the solution of eight Poisson equations and to account for the lowest order radiation reaction terms requires the solution of yet another Poisson equation. While — with nine Poisson equations — computationally already quite heavy, the efforts to implement the scheme into SPH by two groups (Faber and Rasio, 2000; Ayal et al., 2001; Faber et al., 2001; Faber and Rasio, 2002) turned out to be not particularly useful, mainly since for realistic neutron stars with compactness C ≈ 0.17 the corrective 1PN terms are comparable to the Newtonian ones, which can lead to instabilities. As a result, one of the groups (Ayal et al., 2001) decided to study “neutron stars” of small compactness (M < 1 M_{⊙}, R ≈ 30 km), while the other (Faber and Rasio, 2000; Faber et al., 2001; Faber and Rasio, 2002) artificially downsized the 1PN effects by choosing a different speed of light for the corresponding terms. While both approaches represented admissible first steps, the corresponding results are astrophysically difficult to interpret.
A second, more successful approach, was the resort to the so-called conformal flatness approximation (CFA) (Isenberg, 2008; Wilson and Mathews, 1995; Wilson et al., 1996; Mathews and Wilson, 1997; Mathews et al., 1998). Here the basic assumption is that the spatial part of the metric is conformally flat, i.e., it can be written as a multiple (the prefactor depends on space and time and absorbs the overall scale of the metric) of the Kronecker symbol γ_{ij} = Ψ^{4}δ_{ij}, and that it remains so during the subsequent evolution. The latter, however, is an assumption and by no means guaranteed. Physically this corresponds to gravitational-wave-free space time. Consequently, the inspiral of a binary system has to be achieved by adding an ad hoc radiation reaction force. The CFA also cannot handle frame dragging effects. On the positive side, for spherically symmetric space times the CFA coincides exactly with GR and for small deviation from spherical symmetry, say for rapidly rotating neutron stars, it has been shown (Cook et al., 1996) to deliver very accurate results. For more general cases such as a binary merger, the accuracy is difficult to assess. Nevertheless, given how complicated the overall problem is, the CFA is certainly a very useful approximation to full GR, in particular since it is computationally much more efficient than solving Einstein’s field equations.
The CFA was implemented into SPH by Oechslin et al. (2002) and slightly later by Faber et al. (2004). The major difference between the two approaches was that Oechslin et al. solved the set of six coupled, non-linear elliptic field equations by means of a multi-grid solver (Press et al., 1992), while Faber et al. used spectral methods from the Lorene library on two spherically symmetric grids around the stars. Both studies used polytropic equations of state (Oechslin et al., 2002 used Γ = 2.0, 2.6 and 3.0; Faber et al., 2004 used Γ = 2.0) and approximative radiation reaction terms based on the Burke-Thorne potential (Burke, 1971; Thorne, 1969). Oechslin et al. used a combination of a bulk and a von Neumann-Richtmyer artificial viscosity steered similarly as in the Morris and Monaghan (1997) approach, while Faber et al. argued that shocks would not be important and artificial dissipation would not be needed.
In a subsequent study, Oechslin et al. (2004) explored how the presence of quark matter in neutron stars would impact on a NSNS merger and its gravitational-wave signal. They combined a relativistic mean field model (above ρ = 2 × 10^{14} g cm^{−3}) with a stiff polytrope as a model for the hadronic EOS and added in an MIT bag model so that quark matter would appear at 5 × 10^{14} g cm^{−3} and would completely dominate the EOS for high densities (> 10^{15} g cm^{−3}). While the impact on the GW frequencies at the ISCO remained moderate (< 10%), the post-merger GW signal was substantially influenced in those cases where the central object did not collapse immediately into a BH. In a subsequent study, (Oechslin and Janka, 2006) implemented the Shen et al. (1998a, b) EOS and a range of NS mass ratios was explored, mainly with respect to the question how large resulting torus masses would be and whether such merger remnants could likely power bursts similar to GRBs 050509b, 050709, 050724, 050813. The found range of disk masses from 1–9% of the baryonic mass of the NSNS binary was considered promising and broadly consistent with CBM being the central engines of sGRB.
In Oechslin et al. (2007), the same group also explored the Lattimer-Swesty EOS (Lattimer and Swesty, 1991), the cold EOS of Akmal et al. (1998) and ideal gas equations of state with parameters fitted to nuclear EOSs. The merger outcome was rather sensitive to the nuclear matter EOS: the remnant collapsed either immediately or very soon after merger for the soft Lattimer-Swesty EOS and for all other cases it did not show signs of collapse for tens of dynamical time scales. Both ejecta and disk masses were found to increase with an increasing deviation of the mass ratio from unity. The ejecta masses were in a range between 10^{−3} and 10^{−2}M_{⊙}, comparable, but slightly lower than the earlier, Newtonian estimates (Rosswog et al., 1999). In terms of their GW signature, it turned out that the peak in the GW wave energy spectrum that is related to the formation of the hypermassive merger remnant has a frequency that is sensitive to the nuclear EOS (Oechslin and Janka, 2007). In comparison, the mass ratio and neutron star spin only had a weak impact.
In subsequent work, a very large number of microphysical EOSs was explored (Bauswein and Janka, 2012; Bauswein et al., 2012, 2013b, a). Here, the authors systematically explored which imprint the nuclear EOS would have on the GW signal. They found that the peak frequency of the post-merger signal correlates well with the radii of the non-rotating neutron stars (Bauswein and Janka, 2012; Bauswein et al., 2012) and concluded that a GW detection would allow to constrain the ns radius within a few hundred meters. In a follow-up study, Bauswein et al. (2013a) explored the threshold mass beyond which a prompt collapse to a black hole occurs. The study also showed that the ratio between this threshold mass and the maximum mass is tightly correlated with the compactness of the non-rotating maximum mass configuration.
In a separate study (Bauswein et al., 2013b), they used their large range of equations of state and several mass ratios to systematically explore dynamic mass ejection. According to their study, softer equations of state with correspondingly smaller radii eject a larger amount of mass. In the explored cases they found a range from 10^{−3} to 2 × 10^{−2}M_{⊙} to be dynamically ejected. For the arguably most likely case with 1.35 and 1.35 M_{⊙} they found a range of ejecta masses of about one order of magnitude, determined by the equation of state. Moreover, consistent with other studies, they found a robust r-process that produces a close-to-solar abundance pattern beyond nucleon number of A = 130 and they discussed the implications for “macronovae” and possibly emerging radio remnants due to the ejecta.
4.2.6 Neutron star—black hole mergers
A number of issues that have complicated the merger dynamics in the WDWD case, such as stability of mass transfer or the formation of a disk, see Section 4.1.3, are also very important for NSBH mergers.^{14} Here, however, they are further complicated by a poorly known high-density equation of state which determines the mass-radius relationship and therefore the reaction of the neutron star on mass loss, general-relativistic effects such as the appearance of an innermost stable circular orbit or effects from the bh spin and the fact that now the GW radiation-reaction time scale can become comparable to the dynamical time scale, see Eq. (156).
4.2.6.1 Early Newtonian calculations with polytropic equations of state
In a second set of calculations (≈ 80 K SPH particles), they explored non-rotating neutron stars that were modelled as compressible triaxial ellipsoids according to the semi-analytic work of Lai et al. (1993a, b, 1994), both with stiff (Γ = 2.5 and 3) (Lee, 2000) and soft (Γ = 5/3 and 2) (Lee, 2001) polytropic equations of state. They used the same simulation technology, but also applied a Balsara-limiter, see Eq. (95), in their artificial viscosity treatment and only purely Newtonian interaction between NS and BH was considered. For the Γ = 3 case, the neutron star survived again until the end of the simulation, with Γ = 2.5 it survived the first mass transfer episode but was subsequently completely disrupted and formed a disk of nearly 0.2 M_{⊙}, about 0.03 M_{⊙} were dynamically ejected.
Lee et al. (2001) also simulated mergers between a black hole and a strange star which was modelled with a simple quark-matter EOS. The dynamical evolution for such systems was quite different from the polytropic case: the strange star was stretched into thin matter stream that wound around the black hole and was finally swallowed. Although “starlets” of ≈ 0.03 M_{⊙} formed during the disruption process, all of them were in the end swallowed by the hole within milliseconds, no mass loss could be resolved.
4.2.6.2 Studies with focus on microphysics
The first NSBH studies based on Newtonian gravity, but including detailed microphysics were performed by Ruffert and Janka (1999) and Janka et al. (1999) using a Eulerian PPM code on a Cartesian grid.^{15} The first Newtonian-gravity-plus-microphysics SPH simulations of NSBH mergers were discussed in Rosswog et al. (2004); Rosswog (2005). Here the black hole was simulated as a Newtonian point mass with an absorbing boundary and a simple GW backreaction force was applied. For the neutron star the Shen et al. (1998a, b) temperature-dependent nuclear EOS was used and the star was modelled with 3 × 10^{5} − 10^{6} SPH particles. In addition, neutrino cooling and electron/positron captures were followed with a detailed multi-flavor leakage scheme (Rosswog and Liebendörfer, 2003). The initial study focussed on systems with low mass black holes (q = 0.5−0.1) since this way there are greater chances to disrupt the neutron star outside of the ISCO, see above. Moreover, both (carefully constructed) corotating and irrotational neutron stars were studied. In all cases the core of the neutron star (0.15−0.68 M_{⊙}) survived the initial mass transfer episodes until the end of the simulations (22 −64 ms). If disks formed at all during the simulated time, they had only moderate masses (∼ 0.005 M_{⊙}). One of the NSBH binary (M_{NS} = 1.4 M_{⊙}, M_{BH} = 3 M_{⊙}) systems was followed throughout the whole mass transfer episode (Rosswog, 2007b) which lasted for 220 ms or 47 orbital revolutions and only ended when the neutron star finally became disrupted and resulted the formation of a disk of 0.05 M_{⊙}. A set of test calculations with a stiff (Γ = 3) and a softer polytropic EOS (Γ = 2) indicated that such episodic mass transfer is related to the stiffness of the ns EOS and only occurs for stiff cases, consistent with the results of Lee (2000). Subsequent studies with better approximations to relativistic gravity, e.g., Faber et al. (2006b), have seen qualitatively similar effects for stiffer EOSs, but after a few orbital periods the neutron was always disrupted. Shibata and Taniguchi (2011) discussed such episodic, long-lived mass transfer in a GR context and concluded that while possible, it has so far never been seen in fully relativistic studies.
A study (Rosswog, 2005) with simulation tools similar to Rosswog et al. (2004) focussed on higher mass, non-spinning black holes (M_{BH} = 14… 20 M_{⊙}) that were approximated by pseudo-relativistic potentials (Paczyński and Wiita, 1980). While being very simple to implement, this approach mimics some GR effects quite well and in particular it has an innermost stable circular particle orbit at the correct location (6GM_{BH}/c^{2}), see Tejeda and Rosswog (2013) for quantitative assessment of various properties. In none of these high black hole mass cases was episodic mass transfer observed, the neutron star was always completely disrupted shortly after the onset of mass transfer. Although disks formed for systems below 18 M_{⊙}, a large part of them was inside the ISCO and was falling practically radially into the hole on a dynamical time scale. As a result, they were thin and cold and not considered promising GRB engines. It was suggested, however, that even black holes at the high end of the mass distribution could possibly be GRB engines, provided they spin rapidly enough, since then both ISCO and horizon move closer to the bh. The investigated systems ejected between 0.01 and 0.2 M_{⊙} at large velocities (∼ 0.5 c) and analytical estimates suggested that such systems should produce bright optical/near-infrared transients (“macronovae”) powered by the radioactive decay of the freshly produced r-process elements within the ejecta, as originally suggested by Li and Paczyński (1998).
4.2.6.3 Studies with approximate GR gravity around a non-spinning black hole
4.2.6.4 Studies in the fixed metric of a spinning black hole
Rantsiou et al. (2008) explored how the outcome of a neutron star black hole merger depends on the spin of the black hole and on the inclination angle of the binary orbit with respect to the equatorial plane of the black hole. They used the relativistic SPH code originally developed by Laguna et al. (1993a, b) to study the tidal disruption of a main sequence star by a massive black hole. The new code version employed Kerr-Schild coordinates to avoid coordinate singularities at the horizon as they appear in the frequently used Boyer-Lindquist coordinates. Since the spacetime was kept fixed, they focused on a small mass ratio q = 0.1, where the impact of the neutron star on the spacetime is sub-dominant. Both the black hole mass and spin were frozen at their initial values during the simulation and the GW backreaction was implemented via the quadrupole approximation in the point mass limit, similar to the one used by Lee and Kluźniak (1999b). The neutron star itself was modelled as a Γ = 2.0 polytrope with Newtonian self-gravity, the artificial dissipation parameters were fixed to 0.2 (instead of values near unity which are needed to properly deal with shocks). Note that Γ = 2 is a special choice, since a Newtonian star does not change its radius if mass is added or lost, see Eq. (161). The bulk of the simulations was calculated with 10^{4} SPH particles, in one case 10^{5} particles were used to confirm the robustness of the results.
4.2.7 Collisions between two neutron stars and between a neutron star and a black hole
Traditionally, the focus of compact object encounters have been GW-driven binary systems such as the Hulse-Taylor pulsar (Taylor and Weisberg, 1989; Weisberg et al., 2010). More recently, however, also dynamical collisions/high-eccentricity encounters between two compact objects have attracted a fair amount of interest (Kocsis et al., 2006; O’Leary et al., 2009; Lee et al., 2010; East et al., 2012; Kocsis and Levin, 2012; Gold and Brügmann, 2013). Unfortunately, their rates are at least as difficult to estimate as those of GW-driven mergers.
Collisions differ from gravitational-wave-driven mergers in a number of ways. For example, since gravitational-wave emission of eccentric binaries efficiently removes angular momentum in comparison to energy, primordial binaries will have radiated away their eccentricity and will finally merge from a nearly circular orbit. On the contrary, binaries that have formed dynamically, say in a globular cluster, start from a small orbital separation, but with large eccentricities and may not have had the time to circularize until merger. This leads to pronouncedly different gravitational-wave signatures, “chirping” signals of increasing frequency and amplitude for mergers and initially well-separated, repeated GW bursts that continue from minutes to days in the case of collisions. Moreover, compact binaries are strongly gravitationally bound at the onset of the dynamical merger phase while collisions, in contrast, have total orbital energies close to zero and need to get rid of energy and angular momentum via GW emission and/or through mass shedding episodes in order to form a single remnant. Due to the strong dependence on the impact parameter and the lack of strong constraints on it, one expects a much larger variety of dynamical behavior for collisions than for mergers.
Lee et al. (2010) provided detailed rate estimates of compact object collisions and concluded that such encounters could possibly produce an interesting contribution to the observed GRB rate. They also performed the first SPH simulations of such encounters. Using the SPH code from their earlier studies (Lee and Kluźniak, 1999a, b; Lee, 2000, 2001), they explored the dynamics and remnant structure of encounters with different strengths between all types of compact stellar objects (WD/NS/BH; typically with 100 K SPH particles). Here polytropic equations of state were used and black holes were treated as Newtonian point masses with absorbing boundaries at their Schwarzschild radii. Their calculations indicated in particular that such encounters would produce interesting GRB engines with massive disks and additional external reservoirs (one tidal tail for each close encounter) where a large amounts of matter (> 0.1 M_{⊙}) could be stored to possibly prolong the central engine activity, as observed in some bursts. In addition, a substantial amount of mass was dynamically ejected (0.03 M_{⊙} for NSNS and up to 0.2 M_{⊙} for NSBH systems).
In Rosswog et al. (2013), various signatures of gravitational-wave-driven mergers and dynamical collisions were compared, both for NSNS and NSBH encounters. The study applied Newtonian SPH (with up to 8 × 10^{6} particles) together with a nuclear equation of state (Shen et al., 1998a, b) and a detailed neutrino leakage scheme (Rosswog and Liebendörfer, 2003). As above, black holes were modelled as Newtonian point masses with absorbing boundaries at the Rs. A simulation result of a strong encounter between a 1.3 and a 1.4 M_{⊙} neutron star (pericenter distance equal to the average of the two neutron star radii) is shown in Figure 21. Due to the strong shear at their interface, a string of Kelvin-Helmholtz vortices forms in each of the close encounters before a final central remnant is formed. Such conditions offer plenty of opportunity for magnetic field amplification (Price and Rosswog, 2006; Anderson et al., 2008; Obergaulinger et al., 2010; Rezzolla et al., 2011; Zrake and MacFadyen, 2013). In all explored cases, the neutrino luminosity was at least comparable to the merger case, L_{v} ≈ 10^{53} erg/s, but for the more extreme cases exceeded this value by an order of magnitude. Thus, if neutrino annihilation should be the main agent driving a GRB outflow, chances for collisions should be at least as good as in the merger case. But both scenarios also share the same caveat: neutrino-driven, baryonic pollution could prevent in at least a fraction of cases the emergence of relativistic outflows. In NSBH collisions the neutron star took usually several encounters before being completely disrupted. In some cases its core survived several encounters and was finally ejected with a mass of ∼ 0.1 M_{⊙}. Of course, this offers a number of interesting possibilities (production of low-mass neutron stars, explosion of the NS core at the minimal mass etc.). But first of all, such events may be very rare and it needs to be seen whether such behavior can occur at all in the general-relativistic case.
4.2.8 Post-merger disk evolution
SPH simulations were also applied to study the long-term evolution of accretion disks that have formed during a CBM. Lee and collaborators (Lee and Ramirez-Ruiz, 2002; Lee et al., 2004, 2005, 2009) started from their NSBH merger simulations, see Section 4.2.6, and followed the fate of the resulting disks. Since the viscous disk time scale, see Eq. (157), by far exceeds the numerical time step allowed by the CFL condition, Eq. (158), the previous results were mapped into a 2D version of their code and they followed the evolution, driven by a Shakura-Sunyaev “α-viscosity” prescription (Shakura and Sunyaev, 1973), for hundreds of milliseconds. Consistent with their NSBH merger simulations, the black hole was treated as a Newtonian point mass with an absorbing boundary at R_{S} = 2 GM_{BH}/c^{2}, the disk self-gravity was neglected. In the first study (Lee and Ramirez-Ruiz, 2002) the disk matter was modelled with a polytropic EOS (Γ = 4/3) and locally dissipated energy was assumed to be emitted via neutrinos. Subsequent studies (Lee et al., 2004, 2005) applied increasingly more sophisticated microphysics, the latter study accounted for pressure contributions from relativistic electrons with an arbitrary degree of degeneracy and an ideal gas of nucleons and alpha particles. These latter studies accounted for opacity-dependent neutrino cooling and also considered trapped neutrinos as a source of pressure, but no distinction between different neutrino flavors was made. It turned out that at the transition between the inner, neutrino-opaque and the outer, transparent regions an inversion of the lepton number gradient builds up, with minimum values Y_{e} ≈ 0.05 close to the transition radius (∼ 10^{7} cm), values close to 0.1 near the BH and proton-rich conditions (Y_{e} > 0.5) at large radii. Such lepton number gradients drive strong convective motions that shape the inner disk regions. Overall, neutrino luminosities around ≈ 10^{53} erg/s were found and around 10^{52} erg were emitted in neutrinos over the lifetime of the disk (∼ 0.4 s).
4.2.9 Summary: Encounters between neutron stars and black holes
SPH has played a major role in achieving our current understanding of the astrophysics of compact binary mergers. The main reasons for its use were its geometrical flexibility, its excellent numerical conservation properties, and the ease with which new physics ingredients can be implemented. A broad range of physical ingredients has been implemented into the simulations that exist to date. These include a large number of different equations of state, weak interactions/neutrino emission and magnetic fields. In terms of gravitational physics, mergers have been simulated in Newtonian, post-Newtonian and in conformal flatness approximations to GR. An important milestone, however, that at the time of writing (June 2014) still has to be achieved, is an SPH simulation where the spacetime is self-consistently evolved in dynamic GR.
In the following we will drop the distinction between a function and its approximation to alleviate the notation.
We use the convention that W refers to the full normalized kernel while w is the un-normalized shape of the kernel.
Keep in mind that M_{6} has been rescaled everywhere to a support of 2h to ensure a fair comparison.
Note that we write the equations here as in the original paper, i.e., with a smoothing kernel that vanishes at 1h rather than at 2h.
For a discussion of artificial dissipation effects in an astrophysical context (gravitationally unstable protoplanetary disks), see, for example, Mayer et al.(2004)
A possible exception may be magnetically driven explosion of rapidly rotating stars (Winteler, 2012).
The fair comparison of these numbers is complicated by the fact that also different equations of state and mass ratios have been used.
If the neutron star mass distribution allows for small enough mass ratios, this applies as well to NSNS binaries.
Acknowledgements
This work has benefited from the discussions with many teachers, students, colleagues and friends and I want to collectively thank all of them. Particular thanks goes to W. Benz, M. Dan, M. B. Davies, W. Dehnen, O. Korobkin, W. H. Lee, J.J. Monaghan, D. J. Price, E. Ramirez-Ruiz, E. Tejeda and F. K. Thielemann. This work has been supported by the Swedish Research Council (VR) under grant 621-2012-4870, the CompStar network, COST Action MP1304, and in part by the National Science Foundation Grant No. PHYS-1066293. The hospitality of the Aspen Center for Physics is gratefully acknowledged. Some of the figures in this article have been produced by the visualization software Splash (Price, 2007).