# Estimating outflow masses and velocities in merger simulations: impact of
r-process heating and neutrino cooling

Francois Foucart  Department of Physics & Astronomy, University of New
Hampshire, 9 Library Way, Durham NH 03824, USA  Philipp Mösta  GRAPPA, Anton
Pannekoek Institute for Astronomy and Institute of High-Energy Physics,
University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
Teresita Ramirez  Department of Physics & Astronomy, University of New
Hampshire, 9 Library Way, Durham NH 03824, USA  Gravitational Wave Physics and
Astronomy Center, California State University Fullerton, Fullerton, California
92834, USA  Center for Interdisciplinary Exploration & Research in
Astrophysics (CIERA), Physics & Astronomy, Northwestern University, Evanston,
IL 60202, USA  Alex James Wright  ORCID: 0000-0002-5953-4221  Siva Darbha
Department of Physics, University of California, Berkeley, Berkeley, CA 94720,
USA  Daniel Kasen  Department of Physics, University of California, Berkeley,
Berkeley, CA 94720, USA  Department of Astronomy and Theoretical Astrophysics
Center, University of California, Berkeley, Berkeley, CA 94720, USA  Nuclear
Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA

###### Abstract

The determination of the mass, composition, and geometry of matter outflows in
black hole-neutron star and neutron star-neutron star binaries is crucial to
current efforts to model kilonovae, and to understand the role of neutron star
merger in r-process nucleosynthesis. In this manuscript, we review the simple
criteria currently used in merger simulations to determine whether matter is
unbound and what the asymptotic velocity of ejected material will be. We then
show that properly accounting for both heating and cooling during r-process
nucleosynthesis is important to accurately predict the mass and kinetic energy
of the outflows. We also derive a model accounting for both of these effects
that can be easily implemented in merger simulations. We show, however, that
the detailed velocity distribution and geometry of the outflows can currently
only be captured by full 3D fluid simulations of the outflows, as non-local
effect ignored by the simple criteria used in merger simulations cannot be
safely neglected when modeling these effects. Finally, we propose the
introduction of simple source terms in the fluid equations to approximately
account for heating/cooling from r-process nucleosynthesis in future seconds-
long 3D simulations of merger remnants, without the explicit inclusion of out-
of-nuclear statistical equilibrium reactions in the simulations.

##  I Introduction

Multimessenger observations of neutron star-neutron star (NSNS) and black
hole-neutron star (BHNS) binaries provide us with information about the
properties of dense matter [1, 2, 3], the origin of heavy elements [4, 5, 6],
the population of compact objects [7], their formation mechanism [8], and even
the expansion rate of the Universe [9, 10]. So far, one NSNS merger (GW170817)
has been observed through both gravitational and electromagnetic waves [11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. A few additional
mergers that likely involved at least one neutron star have been observed in
gravitational waves only, most notably the likely NSNS merger GW190425 [26]
and the likely BHNS mergers GW200105 and GW200115 [27]. A much larger number
of likely binary mergers have also been observed for decades as short gamma-
ray bursts [28, 29, 30, 31, 32], and a small number of these gamma-ray bursts
were followed by optical/infrared transients (kilonovae) that may have been
powered by radioactive decays in the matter ejected by BHNS and/or NSNS
mergers [33, 34, 35, 36].

While observing mergers through gravitational waves, gamma-ray bursts, or
kilonovae alone is certainly valuable, the outsize scientific impact of
GW170817 has clearly demonstrated the value of multimessenger observations of
these systems. However, the interpretation of electromagnetic signals powered
by NSNS and BHNS mergers remains difficult. Gamma-ray burst observations
provide very limited information about the properties of the merging compact
objects beyond the likely presence of a neutron star, in large part because
the exact process powering these bursts remain poorly understood today.
Kilonovae, which are powered by radioactive decay of the elements produced
through r-process nucleosynthesis in neutron rich outflows [4, 37, 38], are in
theory easier to connect to the masses, sizes, and spins of merging compact
objects: the magnitude, color, and duration of a kilonova are strongly
impacted by the mass, velocity, and composition of the matter ejected by a
merger [39]; and the properties of these outflows can themselves be connected
to the properties of the merging compact objects. Nevertheless, uncertainties
in the numerical simulations used to model these outflows, in the properties
of the elements produced during the r-process, and in the heating rate of the
ejecta due to radioactive decays create significant difficulties when
attempting to interpret kilonovae signals, with highly uncertain error bars in
any inference made about the parameters of the merging compact objects (see
e.g. [40, 41, 42]).

Predicting the properties of matter outflows in NSNS and BHNS mergers is
crucial to our ability to perform detailed analysis of kilonovae. Such
predictions are typically made by developing analytical fits to the results of
merger and post-merger simulations of these systems [43, 44, 45, 46, 47, 42].
A reliable determination of the properties of unbound material in merger
simulations is accordingly important to kilonova modeling efforts.
Unfortunately, the finite size of computational grids, finite length of
numerical simulations, and approximate treatment of nuclear physics in these
simulations can make it difficult to robustly determine whether a given region
of a simulation contains bound or unbound material. Historically, most general
relativistic simulations have determined whether matter is bound or unbound by
assuming that all matter beyond a certain radius follows geodesics of a time-
independent metric [48]. This ignores any impact of cooling (e.g. neutrino
emission), heating (e.g. r-process), pressure forces, or time variations in
the metric. An alternative method, commonly used today, is to rely on the
general relativistic Bernoulli criterion, which assumes a steady-state flow in
a time-independent metric. This criterion partially accounts for the
decompression of the fluid, and the conversion of internal and/or thermal
energy into kinetic energy, but ignores neutrino cooling and the evolution of
the metric in time.

In practice, the Bernoulli criterion tends to overestimate the mass of unbound
material while the geodesic criterion tends to underestimate it; and
differences between the predictions of the two methods can be large. For
example, the Bernoulli criterion can predict more than twice as much unbound
matter than the geodesic criterion for hot outflows close to NSNS merger
remnants [49]. As the material expands and cools down, however, the two
criteria will begin to agree. As the Bernoulli criteria accounts for the
thermal energy of the outflows, one would expect it to perform better for hot
outflows close to the remnant; the Bernoulli criteria has thus been
particularly useful to study polar outflows in NSNS binaries (see e.g. [50]).
For cold ejecta, on the other hand, it is not as clear which criterion
performs best: some simulations have shown that the Bernoulli criterion flags
as unbound material that in fact never expands enough to make use of its
internal energy before falling back onto the remnant [51].

As long as one neglects nuclear reactions in the outflows, these two criteria
will both converge to the correct answer as matter moves away from the
remnant. Accordingly, they do a good job of predicting matter outflows in
simulations using simple, composition independent equations of state with
limited microphysics. In a realistic merger, however, the energy released
through r-process nucleosynthesis will be partially thermalized in the ejecta,
and partially lost to neutrino emission. The $\sim 3\,{\rm MeV}$ per nucleon
deposited in the ejecta by r-process nucleosynthesis for neutron-rich matter
($Y_{e}\lesssim 0.1$) can have a significant impact on the dynamics of the
outflows. It is for example enough to bring marginally bound matter to an
asymptotic velocity $v\sim 0.08c$. Including r-process heating thus leads to a
correction to the asymptotic velocity of unbound matter [52, 53], and to
predictions for the mass of matter ejected by a merger. Whether r-process
heating is sufficient to unbind marginally bound material is however a complex
question that depends on how the orbital timescale and heating timescale
compare [54].

Heating due to r-process nucleosynthesis is generally not directly included in
general relativistic merger simulations: the timescale for the r-process is
$\sim 1\,{\rm s}$, while merger simulations are typically continued for
$\lesssim 0.1\,{\rm s}$ post-mergers. Only a few long-term simulations of the
ejecta have been performed so far, including continuations of the smoothed
particle hydrodynamics merger simulations of [55, 56] in [57, 58], a post-
processing of the general relativistic, grid-based BHNS merger simulations of
[52] in [59], and post-processing the evolution of the outflows observed in
simulations of post-merger remnants [60] in [53]. Using a careful
implementation of the Bernoulli criterion, it is possible to partially correct
for energy deposition from the r-process in merger simulations; e.g. [49, 61]
use versions of the Bernoulli criteria that effectively thermalize $\sim
100\%$ of the energy released by the r-process in the ejecta. Accounting for
both r-process heating and neutrino losses, however, requires more care in the
definition of unbound matter. In this manuscript, we investigate a few
potential methods to predict whether a given region of a simulation contains
unbound material or not, including new criteria that attempt to account for
neutrino cooling and thermalization efficiency in the outflows.

In Sec. II, we first review the geodesic criterion, as well as various
implementations of the Bernoulli criterion that effectively account in
different ways for r-process heating and thermalization. Based on this
discussion, we then propose an improved version of the Bernoulli criterion
that explicitly accounts for both r-process heating and thermalization
efficiency, for any initial composition of the outflows. We note that we use a
formalism that makes explicit its assumptions about the energy released by the
r-process and the losses due to neutrino emission. This will allow us to
modify the algorithm as needed when improved estimates for these quantities
become available. We additionally rely on the study of r-process heating in
initially bound outflows performed by Desai et al [62] to estimate what part
of the initially bound matter is heated fast enough by the r-process to become
unbound before reaching apoastron. Finally, we discuss how our simple model
can be adapted to approximately include both r-process heating and neutrino
cooling in future seconds-long general relativistic simulations performed
using standard finite volume methods, without the inclusion of complex out-of-
nuclear statistical equilibrium nuclear reaction network.

In Sec. III, we then compare the predictions of these different models for
BHNS and NSNS mergers, and additionally compare these results to the outflows
measured in a 3D smoothed-particle hydrodynamics simulation of BHNS merger
outflows that does follow the outflows for multiple seconds and includes an
explicit heating term [59]. We argue that in relatively short merger
simulations, including both future cooling and future heating terms is crucial
to properly account for the energy budget of the outflows. On the other hand,
we note that none of these simple models properly predict the detailed
distribution of outflow velocities observed in hydrodynamics simulations,
emphasizing the need to post-process simulations results to properly determine
the geometry of merger outflows.

##  II Outflow models: unbound matter and asymptotic velocity

###  II.1 Geodesic criterion

The simplest method often used in numerical simulations to estimate whether a
fluid element is unbound, and what its asymptotic velocity might be, is the
geodesic criterion. When using that criterion, we make the assumption that the
ejecta follows spacetime geodesics in a time-independent, asymptotically flat
spacetime [48]. Then, the time component of the 4-velocity one-form ($u_{t}$)
is a conserved quantity. A particle following a geodesic is unbound if
$u_{t}<-1$, and its Lorentz factor at infinity ($\Gamma_{\infty}$) is given by

$\Gamma_{\infty}=-u_{t}.$ (1)

A few milliseconds after merger, and when $GM/(rc^{2})\ll 1$ (with $M$ the
mass of the remnant and $r$ the distance between the remnant and the ejecta),
the assumption of a time-independent, asymptotically flat spacetime is usually
well justified. However, assuming that particles follow spacetime geodesics is
less accurate. Both thermal energy and nuclear binding energy may be partially
converted into kinetic energy as the fluid expands in the surrounding
interstellar medium, and that acceleration of the ejecta is entirely ignored
by the ‘$u_{t}$’ criterion. As a result, this simple method typically seems to
work well when (a) the ejecta is cold; and (b) simple equations of state that
ignore nuclear binding energies are used.

We note that while the first condition may be physically correct for some
outflows (most notably the dynamical ejecta of black hole-neutron star
mergers), the second always introduces an error in the inferred value of
$\Gamma_{\infty}$. The $u_{t}$ condition may appear to work well for cold
ejecta in numerical simulations, in so far that $u_{t}$ appears to remain
constant as the ejecta moves away from the remnant, but this is somewhat
misleading. Merger simulations currently assume that the matter is in nuclear
statistical equilibrium (NSE), while the r-process is an out-of-equilibrium
process. Even if merger simulations were able to follow the ejecta for the
$\sim 1\,{\rm s}$ timescale over which r-process heating occurs, they would
simply predict that the outflows expand at constant composition (possibly up
to minor composition changes due to neutrino-matter interactions) if using a
composition-dependent equation of state, or at some predetermined
’equilibrium’ composition for composition-independent equations of state (e.g.
polytropes). Those simulations ignore any impact of an out-of-NSE evolution of
the composition of the outflows, and of the energy lost to neutrino emission
during that out-of-NSE evolution.

###  II.2 Bernoulli criterion version 1: constant composition

A first improvement to the geodesics criterion is the Bernoulli criterion,
used in many merger simulations today. The Bernoulli criterion is often
written as the assumption that a particle is unbound if $hu_{t}<-1$; and that,
if it is unbound, its asymptotic Lorentz factor is

$\Gamma_{\infty}=-hu_{t}.$ (2)

Here, $h=1+\epsilon+P/\rho$ is the enthalpy, $\epsilon$ is the specific
internal energy, and $P$ is the pressure. In composition-dependent equations
of state, $\epsilon$ includes not only the thermal energy, but also the
nuclear binding energy of the nuclei when in NSE at a given density,
temperature, and composition.

The Bernoulli criterion relies on the fact that $hu_{t}$ is constant along a
given streamline of a steady-state flow. This is not formally applicable to
merger outflows, as these are not steady-state flows, yet the Bernoulli
criterion appears to do a reasonable job of capturing the transformation of
thermal energy and nuclear binding energy into kinetic energy during the
expansion of the outflows. As a result, it is probably the most broadly used
criterion in current simulations. It does however have important limitations,
besides the (technically wrong but apparently reasonably accurate) assumption
of a steady-state flow.

The first issue is that, to use Bernoulli properly, we need to know the
asymptotic enthalpy of the fluid $h_{\infty}$. In fact, in the form stated
above, we implicitly assumed that $h_{\infty}=1$: if $hu_{t}$ is constant, the
correct criteria for matter to be unbound is [61]

$h(-u_{t})>h_{\infty}$ (3)

and the asymptotic Lorentz factor is

$\Gamma_{\infty}=-\frac{hu_{t}}{h_{\infty}}.$ (4)

Most of the simpler equations of state used in merger simulations (polytropes,
piecewise polytropes, spectral) are composition independent, and satisfy
$h_{\infty}=1$ by construction (asymptotically, $\rho\rightarrow 0$,
$\epsilon\rightarrow 0$, $P/\rho\rightarrow 0$). For composition-dependent
equations of state, on the other hand, the asymptotic enthalpy of the flow
depends on the asymptotic composition of the flow, i.e.

$\lim_{\rho\rightarrow 0}h=h_{\infty}(Y_{e,\infty})$ (5)

with $Y_{e,\infty}$ the asymptotic electron fraction (the temperature of the
outflows is small at late times, and can be safely ignored in this
expression). Typically, $h(\rho=0,T=0,Y_{e})$ can vary by as much as $1\%$
with $Y_{e}$, due to the difference between the rest mass energy of free
neutrons ($Y_{e}\sim 0$) and the rest mass energy per nucleon of the most
strongly bound nuclei (e.g. ${}^{56}{\rm Ni}$). As merger simulations ignore
out-of-equilibrium nuclear reactions, and thus have outflows with nearly
constant $Y_{e}$ once $GM/(rc^{2})\ll 1$, the criteria

$-hu_{t}>h(\rho=0,T=0,Y_{e})$ (6)

with $Y_{e}$ the current electron fraction of the outflows in the simulation
will perform very well at predicting the amount of matter that will be unbound
by the merger within the limited set of physical processes included in the
simulations. The formula

$\Gamma_{\infty}=-\frac{hu_{t}}{h(\rho=0,T=0,Y_{e})}$ (7)

thus provides good predictions for the asymptotic Lorentz factor under the
same assumptions.

Unfortunately, physical outflows do not simply expand at constant $Y_{e}$. In
reality, they undergo r-process nucleosynthesis, a process that (a) changes
the electron fraction; (b) puts the outflows out of nuclear statistical
equilibrium; and (c) leads to the loss of a significant amount of energy
through emission of electron antineutrinos. Accordingly, no simulation
explicitly uses this constant-$Y_{e}$ model to determine $\Gamma_{\infty}$.
The energy released by the r-process is $\sim 7\,{\rm MeV}$ per nucleons (for
neutron rich outflows), and about half of that energy goes into the escaping
neutrinos. Both r-process heating and neutrino cooling are thus worth taking
into account: an energy difference of $7\,{\rm MeV}$ per nucleon can be the
difference between matter being marginally bound, and an ejecta with
asymptotic velocity $v\sim 0.13c$!

###  II.3 Bernoulli criterion version 2: r-process heating

To modify the constant-$Y_{e}$ Bernoulli criterion, let us first attempt to
tackle r-process heating, ignoring losses due to neutrino emission during the
r-process. This is reasonably simple to do, if we know the average binding
energy of the nuclei created by the r-process. Indeed, asymptotically
$P/\rho=0$, and thus if we know the asymptotic $\epsilon_{\infty}$, we know
$h_{\infty}$ and can use

$\Gamma_{\infty}=-\frac{hu_{t}}{h_{\infty}}$ (8)

As $h_{\infty}$ does not strongly depend on the initial properties of the
ejecta, this criteria is actually simpler than Bernoulli without heating: we
only need to know the current values of $h$ and $u_{t}$ in the ejecta, and the
value of $h_{\infty}$. This is a heating term because, for neutron rich
matter, $h(\rho=0,T=0,Y_{e})>h_{\infty}$. This is practically very close to
the criterion proposed by Fujibayashi et al [61]111In that work, $h_{\infty}$
is replaced by the minimum value of the function $h(\rho=0,T=0,Y_{e})$, which
differs from $h_{\infty}$ at the level of the difference between the binding
energy of the ashes of the r-process and the binding energy of the most bound
nuclei., as well as to the effective meaning of the condition
$\Gamma_{\infty}=-hu_{t}$ in composition-independent equations of state222As
composition-independent equations of state assume NSE and neutrinoless beta-
equilibrium, leading to a value of $h(\rho=0,T=0)$ close to the minimum of
$h(\rho=0,T=0,Y_{e})$, and thus tp results similar to Fujibayashi et al’s
method [61].

We note that, in simulations, there is a subtlety in the calculation of
$h_{\infty}=1+\epsilon_{\infty}$. The specific internal energy
$\epsilon_{\infty}$ is just the average binding energy of the nuclei formed by
the r-process; but the binding energy has to be defined with respect to a
reference value $m_{*}$ for the mass of a nucleon. Different equations of
state commonly used in numerical simulations make different choices for
$m_{*}$. For example, the DD2 and SFHo equations of state from [63] use
$m_{*}=1u$, with $u$ the atomic mass unit; but the LS220 equation of state
[64] uses a mass closer to the mass of a neutron. The result of simulations is
clearly independent of that choice; $m_{*}$ only provides us with an arbitrary
separation between what we call the ‘baryon rest mass energy density’ of the
fluid ($\rho=nm_{*}c^{2}$, with $n$ the baryon number density) and the
internal energy density of the fluid ($u=\rho\epsilon$). The baryon number (in
general) and total energy (in time-independent spacetimes) have to satisfy
conservation laws, but the separation between baryon mass and internal binding
energy is somewhat arbitrary. What is sometimes called the ‘rest mass
conservation’ equation in relativistic simulations is in fact the equation for
baryon number conservation, rescaled by $m_{*}$.

As a result, with the DD2 or SFHo equation of state we have $h_{\infty}\sim
1$, but with the LS220 equation of state we have $h_{\infty}\sim 0.992$. Using
$\Gamma_{\infty}=-hu_{t}$ with the LS220 equation of state leads to an
underestimate of the unbound mass and of the velocity of the ejecta, unless
the reference mass has been appropriately modified. We note that an error of
$\sim 0.008$ in $h_{\infty}$ results in an error of $\sim 7\,{\rm MeV}$ per
nucleon in the final kinetic energy of the ejecta, a non-negligible
difference.

###  II.4 Bernoulli criterion version 3: r-process heating and neutrino losses

The last important effect that we would like to take into account is energy
losses due to neutrino emission during r-process nucleosynthesis. We note that
during the first few seconds of evolution, when most of the r-process energy
is released, we can safely assume that only neutrinos escape the ejecta, while
other products of nucleosynthesis are thermalized. If a fraction $f_{\rm
loss}$ of the fluid’s rest mass energy is lost to neutrinos during the
r-process, we can use the condition

$-hu_{t}(1-f_{\rm loss})>h_{\infty}$ (9)

to flag unbound material, and

$\Gamma_{\infty}=-\frac{hu_{t}}{h_{\infty}}(1-f_{\rm loss})$ (10)

for the asymptotic Lorentz factor. The value of $f_{\rm loss}$ is uncertain.
Desai et al [62], using the output of the nuclear reaction code SkyNet for
black hole-neutron star merger outflows [65, 66] and assuming that $45\%$ of
the energy released by the r-process is lost to neutrinos, find a quasi linear
relation between $Y_{e}$ and the heating rate (for $Y_{e}<0.2$) which, in our
notation, would be

$f_{\rm loss}\approx 0.0032-0.0085Y_{e}.$ (11)

Overall, we thus get the corrected expression

$\Gamma_{\infty}=-\frac{hu_{t}}{h_{\infty}}\left(0.9968+0.0085Y_{e}\right).$
(12)

For higher $Y_{e}$, r-process heating will become less important. We thus use
this expression is $f_{\rm loss}>0$ ($Y_{e}<0.376$), and set $f_{\rm loss}=0$
otherwise. This expression still neglects non-local effects, but should at
least perform better when it comes to calculating the total kinetic energy of
the outflows.

###  II.5 Unbound criterion: Impact of the heating timescale

Whether the expression derived in the previous section does well at predicting
the mass of the outflows is uncertain, in part because we assume that there is
enough time to heat marginally bound material before it starts falling back to
the post-merger remnant, interrupting the r-process [54]. Desai et al [62]
find that there is enough time to heat the ejecta for massive remnants
($M\gtrsim 12M_{\odot}$, with some dependence on the heating rate). For lower
mass systems, it is possible to estimate which material will receive enough
r-process energy to be unbound if we assume that the heating is roughly
constant in time and distributed over a time $t_{\rm heat}$. If the initial
binding energy per nucleon of the fluid is $E_{0}$ and its binding energy per
nucleon at apoastron (after heating, for bound matter) is $E_{f}$, we should
have

$E_{0}-E_{f}=Q\frac{t_{\rm orb}(E_{f})}{2t_{\rm heat}}$ (13)

with $Q$ the total r-process heating per nucleon and $t_{\rm orb}(E)$ the
orbital timescale of fluid with binding energy $E$. This is [62]

$t_{\rm orb}(E)=1.6\,{\rm s}\left(\frac{E}{1\,{\rm
MeV}}\right)^{-1.5}\left(\frac{M_{\rm rem}}{5M_{\odot}}\right),$ (14)

with $M_{\rm rem}$ the gravitational mass of the post-merger remnant. Our
equation for $E_{f}$ only has a solution for

$E_{0}>E_{0,\rm lim}=1.96\left(Q\frac{0.8\,{\rm s}}{t_{\rm heat}}\frac{M_{\rm
rem}}{5M_{\odot}}\right)^{0.4}\,{\rm MeV}.$ (15)

Material with a lower initial binding energy is unbound due to r-process
heating. The condition for material to be unbound can then be approximated as

$\frac{-hu_{t}}{h(\rho=0,T=0,Y_{e})}>1-\frac{E_{0,\rm lim}}{m_{p}c^{2}}$ (16)

with $m_{p}$ the mass of a proton. Note the use of $h(\rho=0,T=0,Y_{e})$ and
not $h_{\infty}$ here, as heating/cooling during the r-process is accounted
for in the right hand side. For our choice of $f_{\rm loss}$ above, and
assuming $45\%$ of the r-process energy is lost to neutrinos, we get

$Q=(3.669-9.745Y_{e})\,{\rm MeV}$ (17)

for $Y_{e}<0.2$333While we include 3 significant digits in this formula, we
note that the uncertainty in $Q$ is much larger than $10^{-3}$.. A reasonable
choice for the heating timescale is $t_{\rm heat}\approx 1\,{\rm s}$. We again
assume that this expression is correct as long as it predicts $Q>0$, and we
set $Q=0$ otherwise. As long as we have a good estimate of $M_{\rm rem}$,
these expressions are relatively easy to use in numerical simulations.

We now have at our disposal a model that approximately includes r-process
heating, neutrino cooling, thermal expansion, but still neglects non local
effects. The asymptotic Lorentz factor in this model is given by Eq. 12, and
the condition for matter to be unbound by Eq. 16. These equations rely on the
choice of a uniform r-process heating timescale $t_{\rm heat}$ and a total
r-process heating $Q(Y_{e})$ (excluding the energy lost to neutrino emission)
which are easy to modify when our understanding of their correct physical
values improves.

###  II.6 Toy model for numerical simulations

Going further than these estimates will require numerical simulations of the
outflows. We note that it is possible to emulate the heating due to the
r-process in an approximate yet straightforward manner in simulations that
evolve the general relativistic equations of fluid dynamics in conservative
form. To do so, we note that our formulae predict that $Q=0$ for $Y_{e}\approx
0.38$. We also note that, looking at the output of the nuclear reaction
network Skynet for r-process nucleosynthesis in neutron rich ejecta [65, 66],
the final electron fraction of the ashes of the r-process is $Y_{e,f}\approx
0.38$. We will thus assume that, for $Y_{e}<0.38$, the r-process drives
$Y_{e}$ to that final value on a timescale $t_{rp}$. This can be done by the
addition of a source term to the equation of lepton number conservation

$\frac{d(\rho_{*}Y_{e})}{dt}=...-\rho_{*}\frac{Y_{e}-Y_{e,f}}{t_{rp}}$ (18)

with $\rho_{*}=\rho\sqrt{-g}u^{t}$, $u^{t}$ the time component of the fluid
4-velocity, and $g$ the determinant of the spacetime metric. As the electron
fraction evolves in time, the r-process releases energy into the fluid. This
is already partially taken into account by the equation of state, when using a
nuclear equation of state: the equation of state assumes nuclear statistical
equilibrium, and to first order the binding energy of the ashes of the
r-process is comparable to the binding energy of the nuclei formed in NSE (the
difference is typically $\sim 10\%$ of the energy released by the r-process).
What is not yet taken into account is the energy lost to neutrinos emitted
during the r-process. Accordingly, we need to include a cooling term in the
evolution equations of the fluid

$\nabla_{\mu}T^{\mu\nu}=-\dot{Q}_{r}u^{\nu}$ (19)

and not a heating term. Consistency with our earlier choices for $Q,f_{\rm
loss},t_{\rm heat}$ would lead to the formula

$\dot{Q}_{r}=0.0085\rho_{0}\frac{Y_{e}-Y_{e,f}}{t_{rp}}.$ (20)

and $t_{rp}\sim(0.5-1)\,{\rm s}$. If included in merger simulations continued
for a few seconds post merger, these additional source terms should provide a
reasonably good approximation to the impact of nuclear heating on the dynamics
of the outflows, including non-local effects – at least up to the $\sim 10\%$
difference between the binding energy of the ashes of the r-process and the
binding energy of nuclei in NSE, as well as up to the impact of a more complex
time-dependence of the r-process heating.

Adding these simple source terms to numerical simulations evolved for a few
seconds post-merger could thus significantly improve the accuracy of the
predicted outflow properties. We have verified using single-cell simulations
that these simple evolution equations provide heating comparable to the output
of the SkyNet nuclear reaction network for initially neutron rich ejecta. We
note however that this model does not account for heating on timescales $\gg
1\,{\rm s}$. Heating on longer timescales does not significantly impact the
dynamics of the outflows, but it is the main source of energy powering
kilonovae. At late times, the heating is expected to be more weakly dependent
on the initial $Y_{e}$, but on the other hand the thermalization efficiency of
particles produced through nuclear reactions is more complex to model [67].
Our model could thus be used to get outflows to their nearly homogeneous
expansion phase, but more advanced heating terms and thermalization models
will still be needed when studying the production of kilonovae and when
calculating lightcurves.

##  III Impact on measured simulation outflows

To illustrate the impact of these different assumptions, we now post-process
the outflows produced in a neutron star-neutron star merger, in the early
post-merger remnant of a neutron star-neutron star binary, and in a black
hole-neutron star merger, using different prediction methods. We also compare
the black hole-neutron star results with the final velocity observed in Darbha
et al [59], where one of our black hole-neutron star merger simulations is
continued using the smoothed particle hydrodynamics code PHANTOM [68], and an
approximate model for r-process heating is included in the simulation.

The models considered here are:

  * •

Model A: $\Gamma_{\infty}=-u_{t}$, with unbound criterion $u_{t}<-1$ (geodesic
criterion)

  * •

Model B: $\Gamma_{\infty}=-hu_{t}/h_{\infty}$, with unbound criterion
$hu_{t}<-h_{\infty}$ (Bernoulli without neutrino losses)

  * •

Model C:
$\Gamma_{\infty}=-\frac{hu_{t}}{h_{\infty}}\left(0.9968+0.0085Y_{e}\right)$,
with unbound criterion $\Gamma_{\infty}>1$ (Bernoulli with neutrino losses)

  * •

Model D:
$\Gamma_{\infty}=-\frac{hu_{t}}{h_{\infty}}\left(0.9968+0.0085Y_{e}\right)$,
with unbound criterion

$-hu_{t}>h(\rho=0,T=0,Y_{e})\left(1-\frac{E_{0,\rm lim}}{m_{p}c^{2}}\right).$
(21)

(Bernoulli without neutrino losses and with corrections for the finite
duration of an orbit)

For the black hole-neutron star merger, we also consider an additional Model E
that includes the same amount of heating as the PHANTOM simulation. As the
PHANTOM simulation only evolves fluid elements with $u_{t}<-1$ in the merger
simulation, we use Model E with the unbound criteria $u_{t}<-1$, for
consistency.

###  III.1 Neutron star-neutron star merger

Figure 1: Predicted mass of unbound matter as a function of $Y_{e}$ (Top) and
of the asymptotic speed $v_{\infty}$ (Bottom) in a neutron star-neutron star
simulation for our four outflow models.

We begin with the outflows observed in a neutron star-neutron star merger
simulation [69] performed with the SpEC merger code [70]. That simulation
includes Monte-Carlo radiation transport, but not magnetic fields. It uses
neutron stars of mass $m_{1}=1.58M_{\odot}$, $m_{2}=1.27M_{\odot}$, and the
DD2 equation of state. The simulation ends $5\,{\rm ms}$ after merger. The
total mass of material flagged as unbound is $M_{\rm
ej}=(0.0034,0.0083,0.0068,0.0066)M_{\odot}$ for models $(A,B,C,D)$
respectively, while the estimated asymptotic kinetic energy of the ejecta is
$K_{\rm ej}=(6.6,14.1,12.5,12.5)\times 10^{-5}M_{\odot}c^{2}$. Model $A$,
which ignores thermal energy and nuclear binding energy in the outflows,
vastly underestimates the ejected mass. Model B, which ignores cooling,
overestimates the ejected mass. The difference between models C and D is
smaller, which is probably for the best considering that the unbound criteria
used in model D is the most ad-hoc part of our estimates. From these results,
it would be reasonable to assume that $\sim(0.0065-0.0070)M_{\odot}$ is
ejected in our simulation (ignoring finite resolution errors for now…).

Fig. 1 provides more information about the initial electron fraction and
asymptotic velocity of the outflows according to these four models. Model A
underestimates the outflows across the entire parameter space, but model B,C,D
are in good agreement for less neutron-rich ejecta, when nuclear heating plays
a smaller role in the dynamics of the outflows. Models C and D mostly disagree
for low-$Y_{e}$, low-velocity outflows: these outflows have the largest amount
of r-process heating, and may end up bound or unbound depending on when / how
fast the r-process energy is deposited in the fluid.

###  III.2 Neutron star-neutron star merger remnant

Figure 2: Predicted mass of unbound matter as a function of $Y_{e}$ (Top) and
of the asymptotic speed $v_{\infty}$ (Bottom) for the remnant of a neutron
star-neutron star merger $32\,{\rm ms}$ post-merger. We show all outflow
models except model D, which is expected to behave very similarly to model C.

Next we analyze the outflows observed in a neutron star- neutron star merger
remnant simulation presented in [71] and performed with the open-source
Einstein Toolkit [72] module GRHydro [73]. The simulation included a magnetic
field and employed the $K_{0}=220\,\mathrm{MeV}$ variant of the equation of
state of [64] and the neutrino leakage/heating approximations described in
[74] and [75]. The simulation was carried out until the remnant hypermassive
neutron star collapses to a black hole (approximately 42ms after merger). The
data analyzed in this manuscript considers all matter flagged as unbound by a
model $32\,{\rm ms}$ after merger. We only consider models A,B,C. As in other
examples, we expect that the results of model D would be very similar to those
of model C.

Models A,B and C predict an unbound mass on the grid of, respectively, $M_{\rm
ej}=(1.7,6.3,5.8)\times 10^{-4}M_{\odot}$; they also predict total asymptotic
kinetic energy $K_{\rm ej}=(2.1,11.9,11.5)\times 10^{-6}M_{\odot}c^{2}$. We
see that, in this post-merger remnant, model A still vastly underestimates the
amount of matter unbound by the system. This is not a surprising results:
post-merger outflows typically have higher enthalpy than the cold, neutron-
rich dynamical ejecta; model A is thus very inaccurate when used on these
outflows. On the other hand, models B and C are in much better agreement. We
can improve our understanding of the reasons behind this agreement by looking
at Fig. 2. Most of the ejecta has $Y_{e}\gtrsim 0.25$, and thus releases a lot
less energy during the early stages of r-process nucleosynthesis than the very
neutron-rich tidal ejecta studied in the previous section.

###  III.3 Black hole-neutron star binary

Figure 3: Same as Fig. 1, but for a black hole-neutron star binary.

Finally, we consider a black hole-neutron star binary merger. We use
simulation M14-M5-S9-I60 from [52], the merger of a $1.4M_{\odot}$ neutron
star and a $5M_{\odot}$ black hole with dimensionless spin $\chi_{\rm BH}=0.9$
initially inclined by $60^{\circ}$ with respect to the orbital angular
momentum of the binary. The properties of the outflows are measured $5\,{\rm
ms}$ after merger. In this simulation, neutrinos are modeled using a simple
leakage scheme, while magnetic fields are ignored. For this system, we predict
an ejected mass $M_{\rm ej}=(0.0140,0.0200,0.0176,0.0175)M_{\odot}$ for models
$(A,B,C,D)$. There is better agreement on the ejected mass than in the neutron
star binary because there is nearly no marginally bound ejecta, and the ejecta
is relatively cold. We find larger variations in the estimated kinetic
energies of the ejecta, $K_{\rm ej}=(20.7,33.9,28.4,28.4)\times
10^{-5}M_{\odot}c^{2}$. Again, models C and D are in very good agreement. The
predicted $Y_{e}$ and $v_{\infty}$ distributions of the ejecta for each model
are shown in Fig. 3. The $Y_{e}$ distribution is always narrow and centered on
$Y_{e}\sim 0.05$. The main impact of including r-process heating/cooling is to
shift the tail of the velocity distribution, and to increase the mass of the
outflows. Models C and D are indistinguishable on the figure.

Figure 4: Comparison of predictions for the asymptotic velocity of the ejecta
between simple models post-processing merger simulations without ($A$) and
with ($E$) heating, and the result of a hydrodynamics simulation with heating
included ($H4$), and without heating ($H0$).

Finally, we compare our results to PHANTOM simulations of the same system
[59]. The PHANTOM simulations are initialized from the snapshot of the SpEC
merger simulation analyzed in this section, but only regions with $u_{t}<-1$
are evolved. In [59], multiple heating models are considered; here we compare
results to the H0 model (no heating) and the H4 model (r-process heating of
$5.9\,{\rm MeV}$ per nucleon, i.e. order-of-magnitude accurate but a bit
larger than the most likely heating rates). We compare these two results to
model A (no heating) and model E ($\Gamma_{\infty}=-u_{t}(1+5.9\,{\rm
MeV}/(m_{p}c^{2}))$). All models agree on the mass of the ejecta, by
construction (they all use the $u_{t}<-1$ criteria). The predicted kinetic
energy of the ejecta is $K_{\rm ej}=(21.3,29.8)\times 10^{-5}M_{\odot}c^{2}$
for models (H0,H4) of the PHANTOM simulations and $K_{\rm
ej}=(20.0,29.6)\times 10^{-5}M_{\odot}c^{2}$ for the post-processing models
$(A,E)$. We see that given the same total heating, the full hydrodynamics
simulation and the post-processing agree on the total kinetic energy of the
ejecta. However, non-local effects do impact the distribution of velocities.
Fig. 4 shows $v_{\infty}$ for models $(A,E,H0,H4)$. We see that while the
average velocities of models $E$ and $H4$ are comparable, hydrodynamical
effects cause the production of a much broader velocity distribution in the
$H4$ model. The outflows also become more spherical, although this only
becomes obvious when looking at their spatial distribution. These changes were
already noted in [59], and similar effects on the geometry of the outflows in
the presence of r-process heating were observed in [57]. The sharp cutoff in
velocity at $v_{\infty}\sim 0.1c$ for $E$ is an artifact of the unbound
condition $u_{t}<-1$, taken to match the assumptions of the H4 simulation
rather than for its realism. The simulation without heating ($H0$) nearly
exactly agrees with the predictions of the model without heating ($A$),
indicating that the spread in the velocity observed in model $H4$ is truly due
to non-local redistribution of the energy released by heating, rather than
simply to the use of a hydrodynamical simulation.

##  IV Conclusions

In this manuscript, we reviewed the various prescriptions used in numerical
relativity simulations of neutron star mergers to determine whether a given
fluid element is unbound and, if unbound, how fast that fluid element will
move when far away from the post-merger remnant. We emphasize the need to
properly take into account heating due to r-process nucleosynthesis and
cooling due to neutrino emission on timescales typically not simulated by
merger codes ($\sim 1\,{\rm s}$), as well as the thermal energy of hot
outflows. We also note that to account for r-process nucleosynthesis, it is
crucial to understand the assumptions that go into the construction of an
equation of state for the composition of the fluid, including the definition
of the mass of a baryon.

Existing prescriptions for the asymptotic properties of matter outflows
typically ignore energy losses due to neutrino emission, and sometimes neglect
heating due to r-process nucleosynthesis. In this manuscript, we provide an
easy-to-use prescription to account for these two effects (Eq. 9 and Eq. 12)
[Model C in the figures], as well as a slight modification to the unbound
criterion obtained for that model that accounts for the possibility that
ejected material does not have enough time to be heated by the r-process to be
unbound before it starts falling back onto the black hole (Eq. 16) [Model D in
figures].

Applying our results to existing numerical simulations of BHNS and NSNS
mergers, we find that both heating and cooling effects as well as the fluid’s
internal energy are important to take into account in order to properly
predict the mass and kinetic energy of matter outflows. The impact of
r-process heating/cooling is particularly important for neutron-rich outflows,
i.e. for the matter ejected by the tidal disruption of a neutron star in
either neutron star-neutron star or black hole-neutron star mergers. On the
other hand, the impact of the heating timescale is found to be relatively
small. While model D (Eq. 16 and Eq. 12) is in theory the most accurate model
presented here, model C (Eq. 9 and Eq. 12) provides similar results with a
much simpler method.

Finally, we compare our results to 3D smooth particle hydrodynamics
simulations of merger outflows. We find that the simple models presented in
this manuscript provide good predictions for the total kinetic energy of the
unbound matter, given a heating/cooling model, but that 3D hydrodynamics
simulations are required to properly capture the geometry of the outflows.
Indeed, energy transfer between fluid elements during the expansion of the
outflows lead to a wider spread of velocities (and more spherical outflows) in
3D simulations than what would be predicted when blindly applying the simpler
models that can be used in merger simulations. While we recommend the use of a
more advanced criteria for the unbound material and its velocity (models C or
D) in simulations, we thus note that these models are no substitutes for more
costly yet more accurate 3D simulations of the outflows.

###### Acknowledgements.

The authors thank Stephan Rosswog for useful comments on an earlier version of
this manuscript. This research was carried out in part during the 2019 Summer
School at the Center for Computational Astrophysics, Flatiron Institute. The
Flatiron Institute is supported by the Simons Foundation. FF acknowledges
support from the DOE through grant DE-SC0020435, from NASA through grant
80NSSC18K0565. FF and TR acknowledge support from the NSF through grant
PHY-1806278. This research was supported in part by the U.S. Department of
Energy, Office of Science, Office of Nuclear Physics, under contract number
DE-AC02-05CH11231 and DESC0017616, by a SciDAC award DE-SC0018297, and by the
Gordon and Betty Moore Foundation through Grant GBMF5076. DK acknowledges
support from the Simons Foundation Investigator program under award number
622817. This collaborative work was supported in part by the NSF Physics
Frontier Center N3AS under cooperative agreement #2020275.

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