Transport and Mixing of r-process elements in neutron star binary merger blast waves
The -process nuclei are robustly synthesized in the material ejected during a neutron star binary merger (NSBM), as tidal torques transport angular momentum and energy through the outer Lagrange point in the form of a vast tidal tail. If NSBM are indeed solely responsible for the solar system -process abundances, a galaxy like our own would require to host a few NSBM per million years, with each event ejecting, on average, about of -process material. Because the ejecta velocities in the tidal tail are significantly larger than in ordinary supernovae, NSBM deposit a comparable amount of energy into the interstellar medium (ISM). In contrast to extensive efforts studying spherical models for supernova remnant evolution, calculations quantifying the impact of NSBM ejecta in the ISM have been lacking. To better understand their evolution in a cosmological context, we perform a suite of three-dimensional hydrodynamic simulations with optically-thin radiative cooling of isolated NSBM ejecta expanding in environments with conditions adopted from Milky Way-like galaxy simulations. Although the remnant morphology is highly complex at early times, the subsequent radiative evolution that results from thermal instability in atomic gas is remarkably similar to that of a standard supernova blast wave. This implies that sub-resolution supernova feedback models can be used in galaxy-scale simulations that are unable to resolve the key evolutionary phases of NSBM blast waves. Among other quantities, we examine the radius, time, mass and kinetic energy content of the NSBM remnant at shell formation as well as the momentum injected to the ISM. We find that the shell formation epoch is attained when the swept-up mass is about , at this point the mass fraction of -process material is drastically enhanced up to two orders of magnitude in relation to a solar metallicity ISM.
Subject headings:hydrodynamics—nucleosynthesis, abundances—shock waves—stars: neutron—galaxies: ISM—ISM: supernova remnants
The specific physical conditions and nuclear physics pathways required for the -process were originally identified in the pioneering work by Burbidge et al. (1957). But the particular astrophysical site remains open to more than one interpretation (Sneden et al., 2008). Early work on the subject identified both type II supernovae (Woosley et al., 1994; Takahashi et al., 1994) and neutron-star binary mergers (NSBMs; Lattimer et al., 1977; Freiburghaus et al., 1999) as likely candidate events to hold r-process. NSBMs are significantly much rarer than type II supernovae (e.g. Argast et al., 2004) and take place far from their birth places, reaching distance up to a few Mpc from their host halo (e.g. Kelley et al., 2010). The two mechanisms synthesize different quantities of -process material, about and for each type II supernovae and NSBM, respectively (e.g. Cowan & Thielemann, 2004). These differences should give rise to clear signatures in the enrichment pattern of -process elements in galaxies and may ultimately help constrain the dominant production mechanism (Shen et al., 2015).
With many difficulties getting the necessary conditions to produce the -process in type II supernova winds (Qian & Woosley, 1996; Takahashi et al., 1994) , the NSBM model has recently been extensively studied and shown to be a viable alternative. The -process nuclei have been found to be robustly synthesized (Metzger et al., 2010; Roberts et al., 2011; Korobkin et al., 2012; Bauswein et al., 2013; Grossman et al., 2014; Ramirez-Ruiz et al., 2015; Goriely et al., 2015; Mendoza-Temis et al., 2015) and the predicted galactic enrichment history of NSBMs is consistent with the abundance patterns observed in halo stars (Shen et al., 2015; Tsujimoto & Shigeyama, 2014; van de Voort et al., 2015; Vangioni et al., 2016). This suggests that the injection of -process material by NSBMs has been operating in a fairly robust manner over long periods of time in galactic history, while the resultant chemical abundance dispersions in -process elements such as Eu suggests an early, chemically unmixed and inhomogeneous early Milky Way galaxy. At later times, these localized inhomogeneities would fade out as more events happen and -process products are given more time to be transported and mixed throughout the galaxy.
An accurate treatment of the evolution of the ejecta in NSBMs is thus crucial not only for models of electromagnetic transients (Nakar & Piran, 2011; Rosswog et al., 2014; Kelley et al., 2013; Metzger & Berger, 2012) following coalescence but also for models of -process enrichment in galaxies. In contrast to the extensive efforts developing models for supernova remnant evolution, (Cox & Smith, 1974; McKee & Ostriker, 1977; Blondin et al., 1998; Joung & Mac Low, 2006; Martizzi et al., 2015; Kim & Ostriker, 2015) studies quantifying the impact of NSBM remnants in a cosmological context have not been carried out. The efficacy of transport and mixing of -process in NSBM remnants has not been properly quantified mainly due to the highly inhomogeneous initial conditions and the excessive radiative losses expected during the shock propagation.
In this paper, we present the results of a series of controlled three-dimensional hydrodynamic simulations of NSBM remnants. To calculate the ejected mass and initial structure of the tidal tails, we make use of three-dimensional smoothed particle hydrodynamics simulations (Roberts et al., 2011) of NSBMs (Section 2.1). The resultant homologous structure of the tidal tail is then mapped into an adaptive mesh refinement (AMR) simulation with optically-thin radiative cooling (Section 2.2). Our goal is to understand the evolution of single NSBM remnants, which is quantified in Section 3, and how it might depend on the properties of the surrounding medium, which are derived using a cosmological simulation of the formation of the Milky Way (Section 2.3). Our simulations of isolated NSBMs are used to construct sub-resolution prescriptions for galaxy-scale simulations with inadequate resolution to properly define the the cooling radius of NSBM blast waves. Discussion of the results as well as detail comparisons with studies from spherical models are presented in Section 4.
2.1. Initial Conditions
Tidal tails are a common feature formed during mergers and collisions between compact objects. These are typically a few thousand kilometers in size by the end of the merger event in the case of neutron star disruptions by black holes and NSBMs (Lee & Ramirez-Ruiz, 2007; Faber & Rasio, 2012). Some small fraction of the material () in the tails is actually unbound and will escape to the surrounding medium. The exact mass and structure (density and velocity distribution) of the ejected material depends on the mass ratio and details of the equation of state (EoS; e.g. Roberts et al., 2011)
To calculate the density and velocity structure of the unbound tidal tails we use a similar method to the one described by Roberts et al. (2011). We use three-dimensional smoothed particle hydrodynamics (SPH) simulations (Lee & Ramirez-Ruiz, 2007; Lee et al., 2010) to follow the merger of two neutron stars during which the geometry, densities, and timescales change violently (Rasio & Shapiro, 1994; Lee, 2000; Rosswog et al., 2003). As a representative example, we study the dynamics of the material ejected during a NSBM with mass ratio (Roberts et al., 2011). A hybrid EoS, similar to that implemented by Shibata et al. (2005), is used which combines the cold Friedman-Pandharipande-Skyrme nuclear EoS with an ideal gas component. Once the initial dynamical interaction is realized, the fluid elements in the unbound tails are verging on ballistic trajectories, moving primarily under the influence of the central mass potential. The hydrodynamical calculations are stopped only after the expansion of the unbound material becomes homologous. Figure 1 in Roberts et al. (2011) depicts the structure of the tidal tails at homology for a wide range of NSBMs. A clear progression is observed from equal mass tails formed in the case to almost no secondary tail produced in the case studied here.
2.2. Hydrodynamical Evolution of NSBM Remnants
Here we follow the expansion of the tidal tail produced by a NSBM with in three dimensions with the parallel, adaptive-mesh, hydrodynamical code FLASH (Fryxell et al., 2000). We evolve the ideal fluid in three dimensional cartesian coordinates using a metallicity-dependent cooling function, which is constructed using the ion-by-ion cooling efficiencies for low-density gas derived by Gnat & Ferland (2012) for gas temperatures between and K. As it is generally adopted, we do not include metal fine structure transitions or molecular line cooling and so the cooling function is effectively truncated below K.
Figure 1 shows the cooling function for an optically thin thermal plasma with solar abundances assuming collisional equilibrium (blue curve). The full radiative spectra of -process material at the relevant densities and temperatures is not well known as complete atomic line lists for these heavy species are not available (Kasen et al., 2013; Tanaka et al., 2014; Lippuner & Roberts, 2015). We therefore simply assume that the energy integral of the full radiative spectra of -process material is similar to that of Fe. The computed cooling curves with enhanced Fe abundances are plotted in Figure 1. This assumption is, however, of no consequence at all as radiative losses begin to influence the NSBM remnant evolution only when the metal content is almost exclusively dominated by the swept up gas (see Section 3 for specifics). As such, the resulting evolution is quantitatively similar to that computed using solar abundances, but note that we have implemented metal enhanced cooling rates here for completeness.
2.3. Properties of the Ambient Medium
|Tidal Tail||NSBM with|
The evolution of a NSBM remnant depends on the character of the ambient medium. In Figure 2 we show the expected properties of the gas in and around the NSBM injection sites, derived using the cosmological zoom-in simulation Eris, which at redshift is a close analog of the Milky Way (Guedes et al., 2011). The merger rate is inferred by convolving the star formation history with a standard delay-time distribution of mergers modeled by a power-law (Piran, 1992; Kalogera et al., 2001; Shen et al., 2015). The resulting NSBM history in the simulation is shown in Figure 1 of Shen et al. (2015). The spatial distribution of NSBM mergers is then assumed to roughly follow the stellar distribution. Using these two key model ingredients, the number and location of NSBMs can then be estimated, which in turn can be used to infer the density, temperature and metallicity of the surrounding ambient gas (Figure 2). Motivated by this, we run three different types of simulations:
isolated NSBM in a homogeneous ISM with , K and ;
isolated NSBM in a homogeneous, dense ISM with , K and ;
isolated NSBM in a homogeneous, rarefied ISM with , K and .
The parameters of the different simulations are outlined in Table 1. For comparison, standard blast wave simulations initiated by injecting the same total energy in a spherical region are computed in order to determine how the evolution of a NSBM is initially altered by its non-uniform original structure. In all cases, we consider computational boxes filled with an ISM initially in pressure equilibrium and adopt a refinement scheme based on pressure and density gradients which refines around the expanding shock.
3. Hydrodynamical Evolution of NSBM Remnants
The evolution of an isolated NSBM remnant expanding into a uniform medium can be broadly characterized by the well-known evolutionary stages of a supernova remnant (SNR):
the free expansion phase, during which the mass of the tidal ejecta is larger than the mass of the swept up ISM;
the energy conserving phase, during which radiative losses are not important;
the cooling-modified pressure-driven snowplow phase, during which shell formation occurs;
and the final momentum-conserving expansion phase.
However, differences in ejected mass and initial structure modifies the evolution of a NSBM remnant relative to that of a SNR, as illustrated in Figure 3 for our fiducial run (labelled ISM in Table 1). The top panel shows the evolution at time kyr, long after the free expansion phase and during the energy conserving phase. The kinetic energy of the NSBM tidal tail is erg. The ensuing strong blast wave converts much of this energy into thermal energy, which leads to lateral expansion111Similar to the case of remnants arising from jet-driven supernova explosions (Ramirez-Ruiz & MacFadyen, 2010; González-Casanova et al., 2014) or from the unbound debris of stars disrupted by massive black holes (Kasen & Ramirez-Ruiz, 2010; Guillochon et al., 2015).. As the NSBM remnant ages, the thermal energy is lost to radiative cooling, the shock slows down, and shell formation is established. The bottom panel in Figure 3 shows the effect of the blast wave propagation at time Myr, before the onset of the remnant’s cooling-modified snowplow phase.
The top panel of Figure 4 shows the evolution of the forward shock radius with time for the NSBM remnant case shown in Figure 3. For comparison, the evolution of a spherical blast wave with the same initial mass and energy is also shown together with the analytical Sedov-Taylor solution. For spherical simulations, is identified by measuring spherically averaged profiles, while for the NSBM simulations we approximate as the radius of the sphere enclosing 90% of the total energy (no significant difference in the measured radio was obtained from slightly different percentages). The radial temperature and metal mass fraction profiles are then calculated by measuring the spherically averaged values of such quantities at .
The initial configuration of the ejected tidal tail material is highly inhomogeneous, as can be seen in Figure 3, with shock expansion velocities being larger along the original orbital plane of the NS binary. As a result, the rate of ambient material swept up by the blast wave is reduced relative to the spherical case until lateral expansion becomes important. The Sedov-Taylor solution is an attractor and the NSBM remnant evolution slowly adjust to match this spherical, energy conserving solution. As shown in Figure 4, the NSBM remnant becomes radiative before matching the Sedov-Taylor solution, which increases the local cooling time and leads to a decrease in radiative losses when compared to the spherical case. Similar behavior is also seen in simulations of a NSBM remnant expanding into a dense ambient medium (Figure 5) and into a rarefied, hot external environment (Figure 6).
In a uniform medium, the shell formation epoch, which occurs when a SNR becomes radiative, is usually well characterized by the time the mass of swept up material attains . Previous studies of SNRs expanding in a homogeneous ambient medium (Cioffi et al., 1988; Thornton et al., 1998) approximate the cooling radius as
Here we evaluate the radius , swept up mass , and kinetic energy of the NSBM remnant at shell formation. The results for all simulations are given in Table 2. The radius, total remnant mass, and outward radial momentum at shell formation are similar to those obtain using spherical simulations and as such are close to the analytic estimates. Despite the initial differences in ejecta mass and geometry, our conclusions regarding shell formation and momentum injection in NSBM remnants are quite similar to those obtain for SNRs.
At the time of shell formation, the mass fraction arising from metals ejected during the NSBM is about . We thus only expect metal cooling from -process material to influence the dynamics of NSBM remnants if they expand into an ambient medium with , under the assumption that the metal cooling function of -process material is similar to that of Fe (Section 2.2). The -process enrichment of the gas depends on how efficiently the metals are mixed with the ambient material swept up by the blast wave. At the time of shell formation, we find the mass fraction of -process material in the NSBM remnant to be drastically enhanced in relation to solar metallicity to about .
4. Discussion and Conclusions
The importance of SNR evolution for detailed models of the ISM but also for models of galaxy evolution and chemical enrichment has been appreciated for decades. In recent years, it has become increasingly clear that an accurate treatment of NSBM remnant evolution is crucial not just for detailed models of electromagnetic transients (Nakar & Piran, 2011; Rosswog et al., 2014; Kelley et al., 2013; Metzger & Berger, 2012) but also for models of heavy element enrichment (Shen et al., 2015; Tsujimoto & Shigeyama, 2014; van de Voort et al., 2015). NSBMs are thought to play a predominant role in creating (Metzger et al., 2010; Roberts et al., 2011; Korobkin et al., 2012; Bauswein et al., 2013; Grossman et al., 2014; Ramirez-Ruiz et al., 2015; Goriely et al., 2015; Vangioni et al., 2016) and dispersing -process elements, yet modern cosmological simulations are still not able to resolve the expansion of SNRs. Motivated by this, we have perform a series of three-dimensional hydrodynamic simulations of isolated NSBM remnants expanding in environments with thermodynamical properties similar to those found in cosmological simulations (Section 2.3). Particularly important is the use of realistic merger simulations (Roberts et al., 2011) for constructing the initial conditions characterizing the structure of the ejected tidal debris (Section 2.2). We have centered our attention in particular on the evolution of NSBM remnants during the well-known stages that are commonly used to describe the evolution of SNRs.
First, a free expansion phase is observed, during which the mass of the ejecta is larger than the mass of the swept up gas. This evolutionary phase is significantly shorter than that of a typical SNR due mainly to the low mass content of the ejecta () and, to a lesser degree, the initial complex geometry.
Secondly, an energy conserving phase takes place, during which radiative losses are negligible (Figure 3). During this phase the rate of mass swept up by the blast wave is reduced when compared to a standard SNR but is then subsequently increased as the NSBM remnant becomes progressively more spherical. However, radiative losses begin to influence the NSBM remnant evolution before it is accurately described by a standard spherical solution (Figure 4). In addition, the overall evolution takes slightly longer, with the cooling time reaching its critical value only when the shock has travelled a distance that is a few times larger than in the spherical case. This is observed in all simulations listed in Table 1.
Thirdly, a pressure-driven snowplow phase sets in, during which radiative losses begin to influence the evolution of the NSBM remnant and shell formation is found to set in at a time when the swept up mass attains a value of , as it is commonly found in SNRs (Table 2). The radius, total swept-up mass, kinetic energy and outward radial momentum at shell formation for NSBM remnants, we conclude, are close to the standard estimates given for SNRs (Cox & Smith, 1974; McKee & Ostriker, 1977; Blondin et al., 1998; Joung & Mac Low, 2006).
Finally, during the momentum conserving phase, the evolution of a NSBM remnant is remarkably similar to that of a SNR. This implies that sub-resolution supernova feedback models (e.g. Martizzi et al., 2015; Kim & Ostriker, 2015) can be accurately used in galaxy-scale simulations that are unable to resolve the early evolutionary stages of a NSBM remnant.
While our simulations of NSBM remnants have confirmed the similarities and highlighted the differences of the well-known evolutionary stages of SNRs, one of the key distinct processes affecting their structure is the contribution of -process material to the cooling of the swept-up material. As complete atomic line lists for these heavy species are not available (Kasen et al., 2013), we have thus simply assumed that the energy integral of the full radiative spectra of -process material is similar to that of Fe. Under these conditions, we find that metal cooling from -process material is not expected to influence the dynamics of NSBM remnants expanding into an ambient medium with . The resulting NSBM remnant evolution in these environments should solely be determined by the metal cooling of the swept-up material. Yet, the mass fraction of -process material at shell formation within the remnant is expected to be severely enhanced to about , assuming a solar abundance ratio. In contrast, with a total -process mass per supernova of (Shen et al., 2015), we expect a local enhancement of only about when a Type II SNR attains shell formation.
Of particular relevance in this case are the heavy element composition of galactic halo stars with metallicity regime , which have been found to have pure -process products with a distribution that is characteristic of solar system matter but with a large star-to-star scatter in their -process concentrations (Sneden et al., 2008). In some stars with , the -process enrichment can be as large as , assuming a solar abundance ratio. The presence of -process material in these stars with a distribution that is characteristic of solar abundance ratios illustrates that -process enrichment has operated in a fairly robust manner, while their abundance dispersions further suggests that -process production sites must be rare and locally very enhanced, as expected from NSBM enrichment (e.g. Shen et al., 2015). These localized inhomogeneities would then be smoothed out as more NSBMs take place and -process material is given more time to be transported and mixed throughout the early Milky Way. Metal feedback from NSBM remnants is thus essential to understand the formation of -process enhanced stars in galaxies. Despite the initial differences, our conclusions regarding the efficacy of the transport and mixing of -process material by NSBM remnants in a cosmological context should be similar to that expected from single SNRs.
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