MNRAS 000, 000–000 (0000) Preprint 5 October 2021 Compiled using MNRAS LATEX style file v3.0

Dynamical ejecta of neutron star mergers with nucleonicweak processes II: Kilonova emission

O. Just1,2, I. Kullmann3, S. Goriely3, A. Bauswein1, H.-T. Janka4,C. E. Collins11GSI Helmholtzzentrum für Schwerionenforschung, Planckstrasse 1, 64291 Darmstadt, Germany2Astrophysical Big Bang Laboratory, RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan3Institut d’Astronomie et d’Astrophysique, CP-226, Université Libre de Bruxelles, 1050 Brussels, Belgium4Max-Planck-Institut für Astrophysik, Postfach 1317, 85741 Garching, Germany

Released 2021 Xxxxx XX

ABSTRACTThe majority of existing results for the kilonova (or macronova) emission from ma-terial ejected during a neutron-star (NS) merger is based on (quasi-)one-zone modelsor manually constructed toy-model ejecta configurations. In this study we present akilonova analysis of the material ejected during the first ∼ 10ms of a NS merger,called dynamical ejecta, using directly the outflow trajectories from general relativis-tic smoothed-particle hydrodynamics simulations including a sophisticated neutrinotreatment and the corresponding nucleosynthesis results, which have been presented inPart I of this study. We employ a multi-dimensional two-moment radiation transportscheme with approximate M1 closure to evolve the photon field and use a heuristicprescription for the opacities found by calibration with atomic-physics based refer-ence results. We find that the photosphere is generically ellipsoidal but augmentedwith small-scale structure and produces emission that is about 1.5-3 times strongertowards the pole than the equator. The kilonova typically peaks after 0.7−1.5 days inthe near-infrared frequency regime with luminosities between 3− 7× 1040 erg s−1 andat photospheric temperatures of 2.2− 2.8× 103 K. A softer equation of state or higherbinary-mass asymmetry leads to a longer and brighter signal. Significant variationsof the light curve are also obtained for models with artificially modified electron frac-tions, emphasizing the importance of a reliable neutrino-transport modeling. None ofthe models investigated here, which only consider dynamical ejecta, produces a tran-sient as bright as AT2017gfo. The near-infrared peak of our models is incompatiblewith the early blue component of AT2017gfo.

Key words: nuclear reactions, nucleosynthesis, abundances – radiative transfer –stars: neutron – gravitational waves – methods:numerical – transients: neutron starmergers

1 INTRODUCTION

The recent gravitational-wave observation of a binaryneutron-star (NS) merger, GW170817 (e.g. Abbott et al.2017), together with its optical and near-infrared (IR) elec-tromagnetic counterparts, AT2017gfo (e.g. Kasen et al.2017; Nicholl et al. 2017; Smartt et al. 2017; Tanaka et al.2017; Tanvir et al. 2017; Villar et al. 2017), provideslong-sought observational support to the idea (Lattimer &Schramm 1976; Eichler et al. 1989) that a substantial frac-tion of material expelled during NS mergers undergoes therapid neutron-capture (or r-) process (see, e.g., Frebel &Beers 2018; Kajino et al. 2019; Arnould & Goriely 2020;Cowan et al. 2021, for recent reviews), which is responsi-

ble for the nucleosynthesis of about half of the trans-ironelements in the Universe. Major evidence for this interpre-tation, hence for the interpretation of AT2017gfo as a so-called kilo- or macronova (e.g. Li & Paczyński 1998; Kulka-rni 2005; Metzger et al. 2010; Metzger 2019), comes fromthe fact that the light curve of AT2017gfo declined at earlytimes, t, roughly as t−1.3 characteristic of the energy releaserate connected to the consecutive radioactive decay of newlyforged r-process elements (Metzger et al. 2010; Roberts et al.2011; Goriely et al. 2011). Additional support is provided byspectroscopic features of AT2017gfo, which point to the ex-istence of strontium (Watson et al. 2019) and lanthanidesin the ejecta (Kasen et al. 2017; Tanaka et al. 2017). More-over, the appearance of (at least) two different, red and blue,

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components in AT2017gfo suggests the existence of multiplechannels of matter ejection. Yet, no uniform consensus isreached as to what ejecta component was (mainly) responsi-ble for which kilonova component. The most dominant typesof ejecta are the dynamical ejecta (e.g. Ruffert et al. 1996;Rosswog et al. 1999; Goriely et al. 2011; Korobkin et al. 2012;Bauswein et al. 2013; Sekiguchi et al. 2015; Palenzuela et al.2015; Foucart et al. 2016; Radice et al. 2018b), neutrino-driven winds (e.g. Metzger & Fernández 2014; Perego et al.2014; Just et al. 2015a; Fujibayashi et al. 2018; Foucart et al.2020), and ejecta driven by turbulent viscosity (e.g. Fernán-dez & Metzger 2013; Just et al. 2015b; Siegel & Metzger2017; Fujibayashi et al. 2018; Miller et al. 2019).

The light curve of a kilonova delivers unique informationabout the outflow mass, velocity, and composition to the ob-server. These properties are of paramount importance for,among others, chemical evolution models that attempt tounravel the dominant sites of heavy-element nucleosynthesis(e.g. Shen et al. 2015; Vangioni et al. 2016; Côté et al. 2018;van de Voort et al. 2020), in order to understand the evolu-tion and lifetime of the NS remnant and the nuclear equationof state (EoS) (e.g. Bauswein et al. 2013; Hotokezaka et al.2013; Lippuner et al. 2017; Bauswein et al. 2017; Margalit& Metzger 2017; Radice et al. 2018a; Rezzolla et al. 2018;Coughlin et al. 2018; Shibata et al. 2019), or to constrainthe processes that launch and collimate ultrarelativistic jetsof short gamma-ray bursts (e.g. Just et al. 2016; Ito et al.2021; Hamidani et al. 2020; Gottlieb & Nakar 2021). How-ever, deciphering an observed kilonova light curve remainsan ambitious endeavor because of the complexity of involvedphysics ingredients. Profound theoretical understanding isrequired of the neutrino-magneto-hydrodynamical matterejection processes that shape the ejecta structure and set theneutron density (cf. aforementioned references for differentejecta components), of nuclear reaction networks that de-termine the elemental abundances and the radioactive heat-ing rates powering the emission (e.g. Goriely et al. 2013;Mumpower et al. 2015; Lippuner & Roberts 2017; Barneset al. 2021), and of atomic physics and radiative processesthat define the opacities and observer-angle dependent emis-sion rates in various frequency bands (e.g. Kasen et al. 2013;Tanaka et al. 2020; Fontes et al. 2020).

While tremendous progress has been made in the re-cent years, each of these aspects still faces considerable chal-lenges and uncertainties. As for the dynamical ejecta, a long-standing uncertainty is connected to their lanthanide frac-tion and the question whether they would produce rather ared or a blue kilonova, or could even produce both. Thelanthanide fraction resulting after the r-process is deter-mined mainly by the electron fraction, or proton-to-baryonratio, which, in turn, is set during the matter ejection pro-cess as a result of the competing emission and absorptionprocesses of electron neutrinos and antineutrinos. Early NSmerger simulations, which neglected neutrinos (Bausweinet al. 2013) or included neutrino emission but ignored ab-sorption (Freiburghaus et al. 1999; Korobkin et al. 2012),found a solar-like abundance pattern for all elements withmass numbers above A ∼ 130 − 140 and thus a high lan-thanide fraction and correspondingly high opacity. These re-sults implied a kilonova that peaks on rather long timescalesand in red spectral colors (Tanaka & Hotokezaka 2013;Grossman et al. 2014). Subsequent models using various

approximate prescriptions for neutrino absorption revealedthat the composition is not as robust as previously thoughtand may sensitively depend on neutrino-transport effects(Wanajo et al. 2014; Goriely et al. 2015; Martin et al. 2018;Radice et al. 2018b). However, since models including neu-trino absorption are still rare and all of them still employsome kind of approximations, and since self-consistent kilo-nova calculations based on these neutrino-hydrodynamicsmodels are even rarer, the question of the kilonova signa-ture of the dynamical ejecta remains open.

In this study, we take a step towards resolving thisquestion by investigating the kilonova that results fromdynamical ejecta modeled by using a state-of-the-art neu-trino scheme, namely the Improved Leakage-Equilibration-Absorption Scheme (ILEAS; Ardevol-Pulpillo et al. 2019),which in particular includes neutrino-absorption effects. Thepresent study follows up on the nucleosynthesis analysis pre-sented in Kullmann et al. (2021) (called “Part I” hereafter)and discusses the kilonova emission for exactly the samemodels.

Many studies, in fact most of the discovery studies con-nected to GW170817, are based on a simple but powerfulone-zone approximation (e.g. Arnett 1982), which modelsthe ejecta as an expanding homogeneous sphere with a con-stant, grey (i.e. frequency-independent) opacity. This ap-proach, or similarly simple considerations (e.g. Grossmanet al. 2014; Metzger 2019; Hotokezaka & Nakar 2020), areoften employed in a post-processing step to estimate the kilo-nova emission based on the bulk properties (total mass, aver-age velocity, average electron fraction) of ejecta obtained inhydrodynamical simulations. Alternatively, schemes are em-ployed that solve the radiative-transfer equations very ac-curately using state-of-the-art opacity descriptions (Kasenet al. 2013; Tanaka & Hotokezaka 2013; Wollaeger et al.2018; Bulla 2019), but which often assume manually con-structed toy-model distributions of the ejecta and that arenumerically involved and computationally expensive. In thispaper we apply a scheme, based on the M1 approximationof radiative transfer, that in terms of accuracy and com-plexity fills the gap between the two aforementioned ap-proaches. Due to its close relationship to hydrodynamics, itcan be implemented more readily in existing hydrodynam-ics codes than schemes solving the full Boltzmann equation.As opposed to most existing studies, we adopt the outflowdistributions directly from neutrino-hydrodynamics simula-tions, and we take the composition and the radioactive heat-ing rate from the consistently post-processed nucleosynthesistracers.

This paper is organized as follows: In Sect. 2 we brieflyreview the hydrodynamic models and corresponding nucle-osynthesis results from Part I, describe the employed map-ping from hydrodynamic tracers to the velocity space, andwe outline the governing equations and additional assump-tions that enter the kilonova evolution scheme. In Sect. 3,we present and compare the resulting kilonova light curves.Subsequently, in Sect. 4 we briefly speculate about the im-plications of our models to the interpretation of AT2017gfo,point out the advantages of a tracer-based scheme comparedto one-zone models, and contrast our results with some pub-lished works. Finally, we summarize in Sect. 5.

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Dynamical ejecta with weak processes II: Kilonova 3

2 METHODOLOGY

2.1 Investigated models and adopted quantities

We study the kilonova emission based on the four neutrino-hydrodynamical models of binary NS mergers that havebeen introduced in Part I. These models have been obtainedfrom smoothed-particle-hydrodynamics (SPH) simulationsthat were performed in general relativity using the conformalflatness approximation and that employed the recently de-veloped method ILEAS (Ardevol-Pulpillo et al. 2019) for thetreatment of neutrinos. In optically thick regions, ILEAS isconsistent with the diffusion law and ensures lepton-numberconservation, while in optically thin regions it describes neu-trino absorption using an approximation to ray-tracing. InPart I, the neutrino-hydrodynamics simulations were post-processed by extracting the hydrodynamic properties (den-sity, entropy, and electron fraction) along the trajectories ofall ejected SPH particles and computing the nucleosynthe-sis yields along each of these trajectories using a state-of-the-art nuclear network (see, e.g., Lemaître et al. 2021, andreferences therein). In the four hydrodynamical models thenuclear EoS is varied between SFHo (Steiner et al. 2013)and DD2 (Typel et al. 2010), and the two binary mass con-figurations 1.35M-1.35Mand 1.25M-1.45Mare used.We additionally consider three variations of the symmet-ric DD2 model, which were already introduced in Part I:In model DD2-135135-noneu we ignore all neutrino interac-tions that take place after the time of the merger, which wedefine by the time when the general relativistic lapse func-tion reaches a first minimum. This model was denoted as the“no neutrino” case in Part I. Moreover, to test the impact ofuncertainties in the electron fraction, Ye, predicted by theneutrino-hydrodynamics simulations the two models DD2-135135-Ye−01 and DD2-135135-Ye+01 consider the caseswhere along all trajectories Ye was artificially decreased orincreased by 0.1, respectively. Table 1 summarizes the ejectamasses, velocities, average electron fractions, and averagelanthanide mass fractions for all models. See Part I for moredetails regarding the underlying neutrino-hydrodynamicssimulations, method of ejecta extraction, employed nuclearnetwork, and the nucleosynthesis results of all models.

2.2 Data mapping

Ideally, we would conduct the hydrodynamical simulationsof the ejecta until they reach homology, i.e. until the thermalpressure becomes dynamically irrelevant, mixing ceases, andthe velocities of Lagrangian fluid elements freeze out. Forthis study, however, we avoid the significant computationalefforts required to follow the long-term expansion and simplyassume that homology is already reached at the times thyd

when the hydrodynamical simulations have been stopped,where thyd ≈ 10 − 20ms after the NSs fall into each other.While this assumption is likely to be justified for the high-velocity component of the dynamical ejecta, it may be lessappropriate for the slower outflow particles. At t = thyd

roughly 10 − 20 % of the total ejecta energy still resides inthermal energy, suggesting that the average velocities couldstill increase by about 5− 10 %. We note that our approachof fixing the velocities early after the merger also impliesthat we cannot describe fallback of gravitationally bound

material (e.g. Fernández et al. 2015; Ishizaki et al. 2021) orthe dynamical impact of radioactive r-process heating (e.g.Rosswog et al. 2014; Klion et al. 2021).

Our method of computing the kilonova adopts as inputthe hydrodynamic properties and nucleosynthesis data of ex-actly the same outflow trajectories as discussed in Part I. Ho-mologous expansion means that the radial coordinate, r, canbe replaced by the radial velocity, v, because v(r, t) = r/tholds at any given time t. Since our kilonova solver operateson an axisymmetric finite-volume grid, we first need to mapall required data from particle trajectories, along which thenucleosynthesis calculations have been performed, onto a 2Dspherical polar grid that is spanned in velocity space by thenormalized radial coordinate

x ≡ r

ct(1)

(with the speed of light c) and polar angle, θ. The coordi-nates of each trajectory particle in this velocity space aregiven by its radial velocity and the polar angle1 along whichit is ejected, both measured at t = thyd. For our calcula-tions we ignore ejecta in the southern hemisphere and as-sume equatorial symmetry, which is found to be rather wellfulfilled as shown in Part I. Having identified the positionof each ejecta particle in axisymmetric velocity space, wecan now interpolate the quantities needed for the upcom-ing kilonova evolution to all cells of the 2D velocity grid.We map altogether five quantities, namely the mass den-sity, ρ, the electron fraction at the onset of network calcu-lations, Ye(ρnuc), the specific heating rate, Qheat(t) (definedin Sect. 2.4), the lanthanide mass fraction2, XLA(t), as wellas the average baryon mass, 〈Anuc〉(t). Note that Ye(ρnuc)is extracted merely for diagnostic purposes (e.g. Fig. 2) butotherwise not needed by our kilonova scheme, because thedependence of the kilonova on the electron fraction is alreadyincluded in the quantities Qheat, XLA, and 〈Anuc〉 providedby the nucleosynthesis calculations. While ρ and Ye(ρnuc)only need to be mapped once3 the remaining quantities aretime dependent and therefore need to be mapped for a suffi-ciently large number (∼ 1000 in our case) of discrete times,between which linear interpolation will be used in the up-coming evolution scheme.

For the mapping we adopt interpolation methods thatare well known from SPH schemes (see, e.g., Price & Mon-aghan 2007). We stress, however, that our mapping proce-dure is completely independent of the SPH code that wasused to perform the hydrodynamical simulations. Thus, thekilonova scheme presented here can equally well be appliedin cases where the outflow trajectories have been extractedfrom grid-based simulations. For a set of N outflow particleswith masses mj and position vectors xj (where x ≡ xer(θ)with radial unit vector er and j = 1, . . . , N), we obtain the

1 The center and orientation of the axisymmetric spherical polarcoordinate system is uniquely defined by the vector of the totalorbital angular momentum of the stellar binary and the centerof mass. As usual, the polar angle, θ, is the angle between thecoordinate vector and the north pole.2 As in Part I, we subsume both lanthanides and actinides underwhat we call lanthanide fraction here.3 For homologous expansion, the density at any given time canbe obtained using ρ ∝ t−3.

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4 O. Just et al.

Table 1. Global properties of our models as measured in the entire volume (total), measured only in the polar regions defined by thecones of opening angle π/4 around the two poles (polar), and measured in the remaining equatorial sub-volume (equat.), namely theejecta mass, average velocity, average electron fraction, average lanthanide mass fraction, average initial opacity, as well as the time,bolometric luminosity, surface-averaged photospheric temperature, and surface-averaged photospheric velocity of the resulting kilonovaat peak epoch. We define the peak time as the first time when the total luminosity equals the total heating rate, and all other peakquantities are measured at that time. Note that a spherically symmetric mass distribution would correspond to Mpolar

ej /Mtotalej = 0.292.

model region Mej 〈v/c〉 〈Ye,ρnet 〉 〈XLA〉 〈κLA〉 tpeak Lpeak 〈Tph〉peak 〈vph/c〉peak

[M(Mtotalej )] [cm2 g−1] [d] [1040erg s−1] [103 K]

DD2-135135 total 1.99e-3(1) 0.255 0.27 1.07e-1 26.5 0.77 4.21 2.75 0.36polar 3.52e-4(0.18) 0.252 0.34 4.42e-2 18.3 0.66 6.30 3.24 0.30equat. 1.62e-3(0.82) 0.256 0.26 1.20e-1 28.3 0.85 3.32 2.50 0.38

DD2-125145 total 3.20e-3(1) 0.248 0.22 1.52e-1 29.1 1.04 4.12 2.39 0.34polar 5.01e-4(0.16) 0.255 0.29 6.60e-2 22.3 0.88 5.98 2.98 0.28equat. 2.66e-3(0.84) 0.247 0.21 1.68e-1 30.4 1.15 3.35 2.19 0.35

SFHo-135135 total 3.29e-3(1) 0.302 0.26 1.15e-1 27.2 0.91 5.34 2.49 0.43polar 5.49e-4(0.17) 0.241 0.30 8.15e-2 24.6 0.78 7.53 2.96 0.33equat. 2.70e-3(0.83) 0.314 0.25 1.22e-1 27.8 0.99 4.24 2.26 0.46

SFHo-125145 total 8.67e-3(1) 0.241 0.24 1.14e-1 27.4 1.49 7.51 2.20 0.37polar 1.77e-3(0.20) 0.225 0.28 6.62e-2 23.2 1.33 9.64 2.52 0.33equat. 6.75e-3(0.80) 0.244 0.23 1.27e-1 28.5 1.57 6.72 2.09 0.38

DD2-135135-noneu total 1.99e-3(1) 0.255 0.13 2.46e-1 32.3 0.86 2.79 2.35 0.35polar 3.53e-4(0.18) 0.252 0.17 2.22e-1 32.0 0.73 3.80 2.67 0.30equat. 1.62e-3(0.82) 0.256 0.12 2.51e-1 32.4 0.96 2.30 2.20 0.36

DD2-135135-Ye−01 total 1.99e-3(1) 0.255 0.17 2.02e-1 30.7 0.85 3.18 2.47 0.35polar 3.53e-4(0.18) 0.252 0.24 1.37e-1 27.6 0.70 4.69 3.04 0.29equat. 1.62e-3(0.82) 0.256 0.16 2.16e-1 31.4 9.48 2.55 2.25 0.36

DD2-135135-Ye+01 total 1.99e-3(1) 0.255 0.37 2.31e-2 12.2 0.63 5.09 3.14 0.38polar 3.53e-4(0.18) 0.252 0.44 1.04e-2 9.42 0.53 7.36 3.23 0.34equat. 1.62e-3(0.82) 0.256 0.36 2.59e-2 12.9 0.70 3.88 2.96 0.39

density at an arbitrary position x using

ρ(x) =1

2πR

∑j

mjW2D(x− xj , hj) , (2)

while all other quantities, represented by A, are interpolatedas

A(x) =

∑Nj

mjρ2D,j

AjW2D(x− xj , hj)∑Nj

mjρ2D,j

W2D(x− xj , hj), (3)

from the corresponding values at the particle positions, Aj .In the above equations, W2D is the two-dimensional cubicspline kernel (see, e.g., Monaghan 1992), hj is the smoothinglength,

ρ2D,j =

N∑i

miW2D(xj − xi, hi) (4)

is the particle representation of the 2D density, ρ2D ≡ 2πRρ(e.g. García-Senz et al. 2009), R is the cylindrical radius innormalized velocity space, and R = maxR, h/2 with h in-terpolated from hj using Eq. (3). As commonly done in SPHschemes, we fix the smoothing length, hj , by the conditionthat any sphere of radius hj around particle j should con-tain roughly the same number of particles (∼ 50 in our case;see, e.g., Price & Monaghan 2007, for explicit equations). Inorder to reduce numerical artefacts close to the polar axis,namely at cylindrical radii R h that are unresolved bythe particle data, we limit the conversion factor (2πR)−1

between ρ2D and ρ from below by replacing R with R inEq. (2).

2.3 Evolution equations

After mapping the particle data onto the grid, we com-pute the kilonova light curves in a simplified fashion using atruncated two-moment scheme with analytic closure (or of-ten called M1-scheme; e.g. Minerbo 1978; Levermore 1984;Audit et al. 2002; Just et al. 2015b). Instead of evolvingthe specific intensity Iν(x,n) as function of the photon fre-quency ν and photon momentum unit vector n, as is donein full-fledged radiative transfer schemes (e.g. Kasen et al.2006; Tanaka & Hotokezaka 2013), our approximate schemeevolves the 0th and 1st angular moments of Iν as measuredin the comoving (or fluid) frame, namely the monochromaticenergy density,

Eν(x) ≡ 1

c

∫IνdΩ , (5)

and the monochromatic energy-flux density vector,

F iν(x) ≡∫Iν n

idΩ , (6)

where∫

dΩ denotes angular integration over the full spherein photon momentum space. The O(v/c) evolution equa-tions for these quantities as function of the velocity coordi-nate vector x can be derived from the ordinary two-momentequations (e.g. Mihalas & Mihalas 1984; Just et al. 2015b)by making use of the homology condition, v(r, t) = r/t, andthey are given by (see, e.g., Pinto & Eastman 2000; Ross-wog et al. 2018 for an analog derivation of the 0th-moment

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Dynamical ejecta with weak processes II: Kilonova 5

equation):

dEνdt

+1

ct∇x · Fν +

4Eνt− 1

t

∂

∂ν(νEν) = cκ(Eeq

ν − Eν) ,

(7a)dFνdt

+c

t∇x ·Pν +

4Fνt− 1

t

∂

∂ν(νFν) = −cκFν , (7b)

where κ is the absorption opacity (see Sect. 2.4 for thecomputation), Eeq

ν is the equilibrium (Bose-Einstein) energydensity, and Pν/Eν is the normalized 2nd moment (Edding-ton) tensor with the components

P ijνEν≡ 1

cEν

∫Iν n

injdΩ . (8)

The M1 approximation consists of assuming that the Ed-dington tensor is given in terms of a local closure relationas function of Eν and Fν . We employ the closure relationby Minerbo (1978). In Eqs. (7) the time derivative d/dt istaken at constant velocity, x, and the spatial derivatives,∇x, are taken with respect to x. The individual terms en-tering the time derivative in Eqs. (7) describe, from left toright, the propagation of radiation fluxes, losses due to ex-pansion, Doppler-shift, and (emission and absorption) inter-actions with ions and electrons.

The energy equation of photons (cf. Eq. (7a)), is coupledto the energy equation for the remaining particles – whichwill collectively be denoted as gas or fluid in this work –via the first law of thermodynamics for Lagrangian fluidelements moving with velocity v = r/t,

de

dt+

5e

t= ρQheat −

∫ ∞0

cκ(Eeqν − Eν)dν , (9)

where e is the thermal energy density of the gas. Equa-tion (9) takes into account pdV expansion work, energyinput from radioactive decay of freshly synthesized ele-ments that powers the kilonova (see Sect. 2.4 for its com-putation), and exchange of energy due to emission andabsorption of radiation. Since by the time t > tKN,where 0.01 d <∼ tKN <∼ 100 d are typical kilonova emissiontimescales, most electrons are recombined and free neutrons,protons, and positrons have disappeared, the EoS of the gasis dominated by heavy ions and given by

e =3ρkBT

2〈Anuc〉mu, (10)

where T , kB , and mu are the fluid temperature, Boltzmannconstant, and atomic mass unit, respectively, and the aver-age mass number of ions, 〈Anuc〉, is provided by the nucle-osynthesis calculations.

The evolution during the intermediate phase betweenthe merger and the kilonova emission, namely during thyd <t < tKN, is less important, because in this adiabatic phasethe total (photon plus gas) energy quickly converges towardsa time-dependent quasi-equilibrium that is determined bythe balance between adiabatic expansion and radioactiveheating. Hence, as long as the time of initialization, t0,is chosen to be early enough for the system to reach thequasi-equilibrium well before t ≈ tKN, the resulting lightcurve should be insensitive to the particular choice of ini-tial conditions. Motivated by sensitivity tests using differentvalues of t0 and different initial gas temperatures (see Ap-pendix B) we initiate our simulations at t = 100 s after the

merger using as initial conditions a homogeneous tempera-ture of T = 100K and negligibly small radiation energies.The duration of our kilonova simulations is constrained bythe fact that we assume local thermodynamic equilibrium(LTE), which implies that our scheme is not applicable atlate times, typically beyond t ∼ 5− 20d (depending on theejecta properties) when non-LTE effects become dominant(e.g. Waxman et al. 2019). We furthermore note that theneutron precursor (Metzger et al. 2015), which is poweredby the decay of free neutrons in the fastest (v/c >∼ 0.5) layersof the ejecta, is not discussed in this study, mainly becauseit cannot properly be described by our O(v/c) scheme.

The numerical methods employed to solve the M1 equa-tions, of which the canonical form can be recovered by rescal-ing the time coordinate as shown in Appendix B of Just et al.(2021), are detailed in Just et al. (2015b). We employ thesame code ALCAR that is described in Just et al. (2015b)and was previously used to evolve the M1 equations for neu-trino transport.

A few comments are in order regarding the advantagesand disadvantages of the M1 scheme. While the M1 schemehas been employed already in a large number of previousapplications in the context of photon and neutrino trans-port (e.g. Cernohorsky & Bludman 1994; Smit et al. 2000;Pons et al. 2000; Audit et al. 2002; McKinney et al. 2014;Just et al. 2015b; O’Connor 2015; Kuroda et al. 2016; Weihet al. 2020) the present scheme is, to our knowledge, thefirst application to the computation of kilonovae. A com-pelling advantage of the M1 scheme is its computationalefficiency and algorithmic simplicity compared to full scaleradiative transfer schemes that resolve the angular distribu-tion of the photon field. The hyperbolic nature of the M1equations allows to integrate spatial derivatives explicitlyin time and by that avoids inversions of large matrices dur-ing each integration step (e.g. Just et al. 2015b), keeping thecomputational expense comparable to that of hydrodynamicsolvers. The accuracy of the scheme is generally expectedto be superior to some widely employed, more approximatemethods. For instance, the leakage-like model by Grossmanet al. (2014) does not solve a conservation equation for thephoton energy, and as a consequence it systematically un-derestimates the luminosities around peak epoch4. In con-trast to (quasi-)one-zone approximations that do solve anenergy conservation equation for photons (e.g. Arnett 1982;Goriely et al. 2011; Villar et al. 2017; Metzger 2019; Ho-tokezaka & Nakar 2020), the M1 formulation does not de-pend on manually chosen estimates of the diffusion rate,because it self-consistently resolves the spatial propagationof radiation through the ejecta. Moreover, it is able to con-sistently describe heating due to reabsorption of photonstransported from one location to another, and it providesa genuinely multi-dimensional framework that can handlenon-radial fluxes.

On the other hand, the main disadvantage of the M1

4 This is because the luminosity estimated in the model by Gross-man et al. (2014) is always bound to be lower than or equal tothe current global heating rate, which is inconsistent with moredetailed calculations where the luminosity typically exceeds theheating rate at times close to the peak. We refer the reader to theAppendix of Rosswog et al. (2018) for a comparison of the modelby Grossman et al. (2014) with more accurate schemes.

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6 O. Just et al.

scheme is its poor ability to describe radiation in the opti-cally thin regime, particularly in the case of geometricallycomplex radiation sources (e.g. Audit et al. 2002; Just et al.2015b; Weih et al. 2020). While the underlying Boltzmannequation predicts linear superposition of radiation packetsin the optically thin regime, the non-linear closure relationof the M1 scheme causes radiation beams to interact witheach other even where collisional interaction rates vanish. Aparticularly noteworthy consequence of this shortcoming isthat radiation emitted in the optically thin phase of evolu-tion, after the photosphere has diappeared, is not isotropic,as it should be, but still retains a certain dependence on theobserver angle, θobs (cf. Sect. 3.2). In view of the approx-imate handling in M1 of the angular photon distributionfunction, we consider the obtained dependence of radiationfluxes on specific observer angles to be less reliable than cor-responding averages over finite angle intervals. This is whywe will restrict most of our discussion to considering lumi-nosities averaged over finite solid angle domains instead ofones measured at specific angles.

2.4 Heating rate and opacity

The quantity Qheat on the right-hand side of Eq. (9) is com-puted as

Qheat ≡ fth(Q+Qneu) (11)

and represents the effective heating rate per unit of mass,namely the fraction fth of the total radioactive energy re-lease rate, Q + Qneu, that is converted to thermal energyon timescales shorter than the evolution timescale. We fol-low the notation of Part I and use Q to denote the releaserate without neutrinos, while Qneu stands for the contribu-tion that is carried away by neutrinos (and not available forthermal heating in any case). We compute the thermaliza-tion efficiency, fth, exactly as in Rosswog et al. (2017) (whoadopted the formalism by Barnes et al. 2016), namely as

fth =Qβ[ζγfγ + ζefe

]+Qαfα +Qfisffis

Q+Qneu, (12)

where Qβ , Qα, and Qfis are the partial release rates5 from β-decay, α-decay, and fission, and the thermalization efficien-cies fi (i ∈ γ, e, α, fis) for photons, electrons, α particles,and fission products are functions of τi/t with the thermal-ization timescales τi; see Rosswog et al. (2017) for the ex-plicit expressions. Since in our notation Qβ does not containthe energy release going into neutrinos, the weighting fac-tors ζγ = 0.69 and ζe = 0.31 differ from the ones (ζγ = 0.45and ζe = 0.2) employed in Rosswog et al. (2017). Note thatwhile the timescales τi are functions only of the total massand average velocity of the ejecta, fth can be different alongeach trajectory because of different relative contributions ofQβ , Qα, and Qfis. The average thermalization efficiencies forall investigated models as functions of time are displayed inFig. 1. Almost perfect thermalization (Qheat ≈ Q) prevailsuntil about t ∼ 0.5− 1d, whereafter fth declines roughly as∝ t−1. Lower ejecta masses or faster expansion velocities ac-celerate the decay of fth. For completeness we also plot the

5 See Sect. 3.2 and Fig. 9 in Part I for a discussion of the indi-vidual contributions to Q.

fraction fneu ≡ Qneu/(Q + Qneu) of energy that is carriedaway by neutrinos, which amounts to about ≈ 50− 60 % inour models.

Finally, we need a reasonable prescription for the opac-ities, which are dominated mainly by thousands of possi-ble transition lines between energy levels of f -shell elementsnewly created by the r-process. Instead of using a detailedatomic-physics based opacity model (e.g. Kasen et al. 2013;Tanaka et al. 2020; Wollaeger et al. 2018) in our approxi-mate kilonova solver, we express the opacity in a parametricfashion as a function of the lanthanide mass fraction and gastemperature as

κ(XLA, T ) = κLA × κT (13)

where the XLA-dependent part is

κLA ≡

30 cm2 g−1(XLA/10−1)0.1 , XLA > 10−1 ,

3 cm2 g−1(XLA/10−3)0.5 , 10−3 < XLA < 10−1 ,

3 cm2 g−1(XLA/10−3)0.3 , 10−7 < XLA < 10−3 ,

0.2 cm2 g−1 , XLA < 10−7 ,

(14)

and the temperature-dependent part is

κT ≡

1 , T > 2000K(

T2000 K

)5 , T < 2000K .(15)

The above prescription was motivated by fits to bolometriclight curves from the atomic-physics based models of Kasenet al. (2017); see Appendix A for comparison plots. Oursimplified opacities are not able to reproduce all quantita-tive features in all models by Kasen et al. (2017), but theyproduce agreement in the bolometric luminosities typicallywithin a factor of two, and they rather reliably predict thecorrect frequency regime and epoch of maximum brightnessof individual broadband light curves. While our prescrip-tion is only a crude approximation to atomic-physics basedmodels, it is more advanced than many previous treatmentsusing (piecewise) constant opacities (e.g. Villar et al. 2017;Perego et al. 2017; Rosswog et al. 2018; Hotokezaka & Nakar2020) in two important respects: First, the functional depen-dence between XLA and κ is more consistent with sophisti-cated opacities. In particular, it ensures that variations inthe lanthanide fractions always result in variations of theopacity and therefore of the light curve, which is not guar-anteed for a piecewise constant prescription. Second, thetemperature-dependent reduction factor, κT , incorporatesthe effect that electron recombination and the concomitantdisappearance of atomic transition lines lead to a decline ofthe opacities at late times (e.g. Kasen et al. 2013; Tanakaet al. 2020; Zhu et al. 2021).

2.5 Grid setup

The hydrodynamical NS-NS merger simulations sample theoutflow in each model with N ∼ 800 − 1400 trajectories(see Part I). The kilonova evolution is simulated on a veloc-ity grid containing about 400 zones in radial direction, dis-tributed uniformly within v/c = 0.02−1.26, and 60 equidis-tant zones in θ direction from 0 to π/2. Frequency space is

6 The reason for locating the outer boundary at v > c is only toprevent numerical artifacts, which may be encountered at early

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Dynamical ejecta with weak processes II: Kilonova 7

10 1 100 101

time [d]

10 2

10 1

100

f

thermalization efficiency

DD2-135135DD2-125145SFHo-135135SFHo-125145

fth = Qheat/(Q + Qneu)fneu = Qneu/(Q + Qneu)

10 1 100 101

time [d]

10 2

10 1

100

f

thermalization efficiency

DD2-135135-noneuDD2-135135-Ye-0.1DD2-135135-Ye+0.1

Figure 1. Mass-averaged thermalization efficiencies as function of time for all of our models as computed with Eq. (12) (solid lines) aswell as the mass-averaged fraction of the radioactive energy release rate that is carried away by neutrinos (dashed lines).

discretized by 50 frequency bins, which are distributed loga-rithmically between ν = 0 and 2.42×1015 Hz. The midpointsof the outermost bins are located at wavelengths of 130 nmand 289µm.

2.6 Extraction of observer light curves

In contrast to a complete radiative transfer solver, the ap-proximate two-moment system with analytic closure thatwe solve, Eq. (7), does not evolve the angular distributionof photons in momentum space, but only the 0th- and 1st-angular moments thereof (cf. Eqs. (5) and (6)). Since themoments are defined in the frame locally comoving with theejecta the fluxes F iν measured at some velocity coordinate vmabove the photosphere are Doppler red-shifted and retardedin time compared to the fluxes measured by a Eulerian ob-server, who is assumed to be at rest with respect to thecenter of mass of the NS binary. Moreover, due to the prox-imity of vm to spatially extended cloud of ejecta a significantfraction of photons is still moving in lateral directions, i.e.the angular distribution of F rν is not the final one as mea-sured by an observer. We apply the following scheme basedon discretizing the radiation field into radiation packets inorder to reconstruct the approximate angular distribution ofthe radiation field and to correct for the frame-dependence(inspired by a related treatment in Lucy 2005).

The first step consists of recording the comoving-frameradiation moments, Eν , F rν , and F θν at the time-dependentvelocity coordinate

vm(t) = max0.5c,min0.99c, 1.2〈vph〉 , (16)

which is chosen to be 20 % larger than the average locationof the photosphere (see Sect. 3.2 and Eq. (29) for the defi-nition of the photosphere), but within c and c/2. The lowerlimit of 0.5c is applied in order to ensure that monitoring isperformed outside of regions of significant photon produc-tion also after the ejecta become optically thin.

In the next step we reconstruct the specific intensity, Iν ,

times when radiation crosses the boundary with low values of theflux factor, |Fν |/(cEν).

using explicitly the closure prescription of Minerbo (1978).This closure assumes Iν to be of the form

Iν(µ) ∝ eaν µ (17)

as function of the cosine

µ ≡ n · Fν|Fν |

(18)

of the angle between the momentum vector of photons, n,and the flux-density unit vector, Fν/|Fν |, and where theparameter aν = L−1(|Fν |/(cEν)) with the inverse of theLangevin function, L−1. The normalization constant of Iνis obtained from the definition of either of the evolved mo-ments (cf. Eqs. (5) or (6)). Next, we discretize the momen-tum space of photons, using as coordinates µ ≡ n · er ∈[−1, 1], the cosine of the angle between the photon mo-mentum and the radial direction, and the azimuthal angleΦ ∈ [0, 2π) between the projection of n onto the plane nor-mal to er and the unit vector in θ direction, eθ. The differen-tial energy of a radiation packet emitted at time t, velocityv = vm(t), polar angle θ, and frequency ν into the directionof n(µ,Φ) is then given by

Epack = Iν(vm, θ, µ, φ, t)µdνdµdΦ× 2π(vmt)2 sin θdθdt

(19)

and the number of photons carried by that packet is Npack =Epack/(hν) (with Planck constant h). While Npack is frameindependent, the energy, momentum vector, and arrival timeof each packet as measured by an observer at rest transformas

Epack,obs = EpackΓ(

1 + n · vc

), (20a)

nobs =Epack

Epack,obs

[n + Γ

v

c

(1 +

Γ

Γ + 1n · v

c

)], (20b)

tobs = t(

1− nobs ·v

c

), (20c)

where v ≡ vmer and Γ ≡ (1 − v2m/c

2)−1/2. Since the timeelapsed in the v = 0 frame is the same as the observer time,we will from now on drop the subscript “obs” from the ob-server time coordinate and, for the sake of a compact pre-sentation, use t to denote both ejecta properties and theobserver signal.

MNRAS 000, 000–000 (0000)

8 O. Just et al.

The differential luminosity, dLν , at time tobs, observerangle θobs, and frequency ν is then given by the sum of en-ergy packets Epack,obs arriving between tobs and tobs + dtinto the solid-angle interval dΩ = 2π sin θobsdθobs and withmean energy Epack,obs/Npack between hν and hν + hdν, di-vided by dt. The total luminosity going into a given solidangle is then just the sum of dLν over the appropriate an-gle interval. We compute the isotropic-equivalent luminosityinto a given observer angle as

Liso,ν(θobs) = 4πdLνdΩ

(21)

and the flux density measured by an observer at distance das

Fν,obs(θobs) =Liso,ν(θobs)

4πd2. (22)

Corresponding bolometric quantities, for which we will usethe same symbol but without the subscript ν, are obtainedby integration over frequency. Absolute AB magnitudes arecomputed from the above fluxes as7

MAB ≡ −2.5× log10Fν,obs[erg s−1 cm−2 Hz−1] − 48.6 ,(23)

while in this study we restrict ourselves to the three bands g,z, and H. The g band represents green optical frequencies,the z band the blue end of the near-IR domain, and theH band the red end of the near-IR domain. For computingthe magnitudes in these bands, we adopt the fluxes at themidpoint wavelength of each band, namely at 514, 902, and1630 nm, respectively, i.e. we do not apply a filter function.

Furthermore, we extract, for each observer angle, θobs,and time, t, a temperature, Tobs, and velocity, vobs, by fit-ting the observed isotropic-equivalent luminosities, Liso,ν , toa blackbody luminosity, LBB

iso,ν , that would result if all pho-tons were emitted with a blackbody spectrum of tempera-ture Tobs from a spherical surface of radius vobst, where

LBBiso,ν(Tobs, vobs) ≡

8π2h(vobst)2

c2ν3

exphν/(kBTobs) − 1(24)

with Boltzmann constant kB. The quantities Tobs and vobs

estimate the temperature and velocity of the photospherebased on observable fluxes only. In Sects. 3.1 and 3.2 we willadditionally introduce the actual temperature and velocityof the photosphere, Tph and vph. We will provide both typesof quantities for each model in order to illustrate the con-nection between properties of the expanding ejecta and theobservable signal.

3 RESULTS

In the following we consider the spatial structure of theejecta and the photosphere in Sect. 3.1, then we examinethe light curves as functions of time and observer angle fora fiducial model, and subsequently we investigate the modeldependence of the results.

We summarize global properties of the ejecta and the

7 By convention, absolute magnitudes assume the flux Fν,obs inEq. (23) to be measured at a distance of 10 pc.

corresponding kilonova in Table 1. We stress that through-out this paper our definition of the kilonova peak is notstrictly mathematical, i.e. we do not identify the peak as thepoint where the brightness reaches a global maximum. Maxi-mum brightness can appear already very early, in which caseit is produced only by a small amount of mass located at theouter edge of the ejecta (see, e.g., Banerjee et al. 2020, fora light-curve comparison between different choices of massdistributions in the outermost shells). Since we are not inter-ested in the emission coming from the fastest ejecta shells,but rather in that from the bulk ejecta, we measure the peakas the time when the bolometric luminosity, L, first startsto exceed the volume integral of the instantaneous heatingrate,

q ≡∫V

ρQheatdV , (25)

which signals the onset of the optically thin phase8.In Table 1, as well as in the following discussions, we

often distinguish between polar and equatorial properties.If not explicitly stated otherwise, “equatorial” always refersto the region within angles of −π/4 and +π/4 around theequator, while “polar” refers to the remaining volume of thesphere9.

3.1 Ejecta structure

Before considering the light curves, we first inspect the spa-tial distribution of the ejecta. To that end, we provide inFig. 2 a tableau of contour plots showing for all models thedensity, electron fraction, lanthanide fraction, opacity, andheating rate in the polar plane at a time of t = 1 d postmerger. Moreover, Fig. 3 provides for all models the angle-integrated mass distributions as function of the velocity co-ordinate.

As already discussed in Part I, and in qualitative agree-ment with results by other groups (Sekiguchi et al. 2015;Foucart et al. 2016; Radice et al. 2018b), all models show,at least for velocities v/c <∼ 0.5, a characteristic pole-to-equator asymmetry in the electron fraction, Ye, which in-creases from Ye ≈ 0.15 − 0.2 near the equator to about0.3− 0.35 near the poles (cf. panels in second row from leftin Fig. 2). High values of Ye beyond ∼ 0.25 are mainly aresult of neutrino absorption, which tends to dominate neu-trino emission near the poles and drives Ye towards ∼ 0.5(e.g. Goriely 2015; see also Part I). As a result of the re-duced neutron-richness, the mass-averaged lanthanide frac-tion, 〈XLA〉, in the polar cones is about a factor of twosmaller than in the equatorial region (cf. Table 1). As a

8 The crossing of q and L is a generic feature of radioactivelypowered transients (e.g. Arnett 1982; Kasen & Barnes 2019; Ho-tokezaka & Nakar 2020) and a result of the fact that radioac-tive heating effectively charges the thermal energy content of theejecta in the late optically thick phase. The subsequent rapid re-lease (called “diffusion wave”; see, e.g., Waxman et al. 2018) ofthis surplus of energy commencing once optically thin conditionsare reached creates luminosities in excess of the instantaneousheating rates.9 We caution the reader that the two characteristic regions usedhere are different from the three regions (polar, middle, equato-rial) used in Part I.

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Dynamical ejecta with weak processes II: Kilonova 9

Figure 2. Color maps of various quantities in velocity space at time t = 1 d post merger. All plots in a row belong to the model givenin the right panel. From left to right the logarithmic density, electron fraction at the time when the network calculation was initiated,logarithmic lanthanide mass fraction, opacity, and specific heating rate after thermalization are shown. The black line in each paneldenotes the photosphere (cf. Eq. (26)) at t = 1 d, while the grey lines in the left panels additionally show the photospheres at t = 0.3, 0.5,

and 2 d going from high to low velocities.

consequence, the mass-averaged opacities, 〈κLA〉, in the cor-responding regions differ by about 15 − 30 %. Besides thecomposition asymmetry, polar ejecta also exhibit a slightlyreduced mass loading compared to equatorial ejecta (cf. bot-tom panel of Fig. 3). The mass ratio Mpolar

ej /M totalej of polar

to total ejecta is in all models about a factor of ∼ 2 smallerthan the value of 0.292 that would result for an isotropicejecta distribution (cf. Table 1 and see Part I).

The black and grey lines in Fig. 2 denote the photo-sphere, which we define as the surface rph(θ) where the ra-dial optical depth, τ , equals unity, i.e.

τ(rph(θ), θ) =

∫ ∞rph

κρdr = 1 . (26)

Around times of t ∼ 1d, i.e. close to the epoch of peakemission, the photospheres of all models resemble, to firstorder, an oblate ellipsoid with semi-major axis ratio of about

MNRAS 000, 000–000 (0000)

10 O. Just et al.

10 6

10 5

10 4

10 3

10 2

Mej

(>v)

[M]

mass distribution

DD2-135135DD2-125145SFHo-135135SFHo-125145

10 1 100

v/c

10 6

10 5

10 4

10 3

10 2

Mej

(>v)

[M]

totalpolarequatorial

Figure 3. Mass distribution of all investigated models showingfor a given velocity v the mass of all ejected material expandingfaster than v. In the top panel the angle integration is performedover the entire sphere while in the bottom panel Mej(> v) isshown separately for polar (0 < θ < π/4 and 3π/4 < θ < π) andequatorial (π/4 < θ < 3π/4) regions.

1.2 − 2. However, the shape of the photosphere varies withtime and becomes more spherical or even prolate for somemodels at very early or late times. Moreover, apart fromthe global geometry, the ejecta also exhibit a significant de-gree of clumpiness and small-scale structure, i.e. structureson scales of ∆v ∼ 0.1 c and less. Such variations can be aresult of the case dependent merger dynamics, e.g. of massshedding during the plunge, or oscillations of the mergerremnant (e.g. Bauswein et al. 2013), or of other processessuch as spiral-waves (e.g. Nedora et al. 2019). As a conse-quence, the ejecta of all investigated models develop a pho-tosphere with wiggles and irregular shapes and a complextime dependence (see left panel of Fig. 2 for photospheres atdifferent times), clearly different from the idealized geome-tries that were assumed in previous studies (e.g. Kawaguchiet al. 2018; Darbha & Kasen 2020; Korobkin et al. 2021).The degree of clumpiness seen in our models may be exag-gerated by our particle-based interpolation method and thefact that the hydrodynamics are not followed all the wayuntil self-similar expansion, in particular because r-processheating on longer timescales may lead to smoother densityprofiles (Rosswog et al. 2014). On the other hand, our mod-els may also underestimate the level of anisotropy to someextent, because they are azimuthally averaged, do not in-clude magnetic fields (e.g. Ciolfi & Kalinani 2020), or pos-sibly under-resolve the small-scale structure due to limitednumerical resolution in the ejecta.

3.2 Light curves of fiducial model

We now discuss the light curve starting with the fidu-cial model DD2-135135. In Fig. 4 we plot, as functions of

time, the angle-averaged bolometric luminosity, the totaland mass-averaged specific heating rates, q and q/Mej, re-spectively, the surface-averaged absolute magnitudes in g, z,and H bands, the mass-averaged opacity,

〈κ〉 =

∫VρκdV

Mej, (27)

as well as the surface-averaged temperature and velocity atthe photosphere,

〈Tph〉 =1

2

∫ π

0

T (rph, θ) sin θdθ , (28)

and

〈vph〉 =1

2

∫ π

0

rph

tsin θdθ , (29)

respectively, together with the corresponding surface-averaged estimates 〈Tobs〉 and 〈vobs〉 of the photosphericconditions based on observable fluxes.

The bulk of the ejecta starts to become optically thinat about t ≈ 0.8 d, which is the time when the luminosityexceeds the heating rate. By the time of t ≈ 3 d the opti-cal depth has dropped below unity, i.e. the photosphere hasdisappeared, implying that re-absorption of thermal photonshas become irrelevant.

In Fig. 4, the solid (dashed) lines denote quantities forwhich only the polar (equatorial) solid angles have beentaken into account for the corresponding volume- or surface-integration. At early times of t <∼ 0.2− 0.3d, when the pho-tosphere is still located at high velocities of v >∼ 0.5 c, thecomposition is nearly independent of the polar angle andmaterial is lanthanide-rich in both polar and equatorial di-rections (see Fig. 2). Once the photosphere travels to slowerejecta mass shells, with v <∼ 0.5 c, the composition gradientalong the polar angle, mentioned in Sect. 3.1, starts to be-come relevant. The lower opacities (cf. bottom middle panelin Fig. 4) together with the reduced mass loading in thepolar region allow the photosphere to reach lower veloci-ties and higher temperatures in the polar compared to theequatorial ejecta for a given time. The fact that due to theangular composition gradient radiation is released from po-lar ejecta more readily than from equatorial ejecta, providesone reason for the excess of polar compared to equatorialvalues of Liso at times t >∼ 0.3 d. Another reason, which isalso observed for all considered models, is the oblate geom-etry of the photosphere at times t ∼ 1 d, which results ina larger projected surface area when viewed face-on com-pared to edge-on (Darbha & Kasen 2020; Wollaeger et al.2018). On the other hand, angular variations of the specificradioactive heating rate do not seem to play a significantrole for producing pole-to-equator emission asymmetries, asis suggested by the top middle panel in Fig. 4 (see Part Ifor an in-depth discussion of the heating rate as function ofpolar angle and velocity).

The orange lines in the right panels of Fig. 7 also show,for comparison to the corresponding quantities measured atthe photosphere, the spectral temperature, 〈Tobs〉, and ve-locity, 〈vobs〉, inferred from the fluxes assuming blackbodyemission. The agreement between both types of quantitiesis good, though not perfect, at least around peak emissiontimes of 0.7 d <∼ t <∼ 3 d. At earlier times the spectral tem-perature (velocity) based on the blackbody model systemati-cally overestimates (underestimates) the corresponding pho-

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Dynamical ejecta with weak processes II: Kilonova 11

1037

1038

1039

1040

1041

1042

L,q

[erg

/s]

bolometric luminosity

Liso, polarLiso, equatorialq

106

107

108

109

1010

1011

q/M

ej [e

rg/g

/s]

specific heating ratepolarequatorial

2

4

6

8

<T

> [K

]

1e3 photospheric temperaturepolarequatorial

< Tph >< Tobs >

10 1 100 101

time [d]

14

13

12

11

abso

lute

AB

mag

nitu

de

broadband magnitudesgzH

polarequatorial

10 1 100 101

time [d]

0

5

10

15

20

25

30<

> [c

m2 /g

]opacity

polarequatorial

10 1 100 101

time [d]

10 2

10 1

100

<v

>[c

]

photospheric velocity

polarequatorial

< vph >< vobs >

Figure 4. Global properties as functions of time characterizing the kilonova for the fiducial model DD2-135135. The left column showsin the top panel the bolometric isotropic-equivalent luminosity (cf. Eq. (21) together with the thermalized heating rate (cf. Eq. (25)) andin the bottom panel absolute AB magnitudes (cf. Eq. (23)), the middle column displays the mass-averaged specific heating rate (top)and mass-averaged opacity (cf. Eq. (27); bottom), and the right column provides in the top panel surface averages of the photospherictemperature (cf. Eq. (28); top) and photospheric velocity (cf. Eq. (29); bottom). Orange lines refer to the estimates of the photospherictemperature and velocity, Tobs and vobs, respectively, based on spectral fits to a blackbody spectrum (cf. Eq. (24). Solid (dashed) linesdenote volume averages (q, 〈κ〉) or surface averages (remaining quantities) over the polar (equatorial) domain.

0.5

1.0

1.5

2.0

2.5

3.0

L iso

(ob

s)/L i

so(9

0)

viewing angle dependenceDD2-135135DD2-125145SFHo-135135SFHo-125145

surface average

0 10 20 30 40 50 60 70 80 90obs [ ]

0.5

1.0

1.5

2.0

2.5

3.0

L iso

(ob

s)/L i

so(9

0) DD2-135135

DD2-135135-noneuDD2-135135-Ye-0.1DD2-135135-Ye+0.1

surface average

Figure 5. Isotropic-equivalent luminosity as function of the in-clination angle between rotation axis and observer line-of-sight,θobs, at t = 1 d normalized to the value at θobs = 90 (solid lines).Dashed lines show the corresponding surface averages.

tospheric quantity. At late times, t >∼ 3 d, 〈Tobs〉 and 〈vobs〉do not characterize the photospheric conditions anymore,since by then the photosphere has disappeared.

Looking at magnitudes in given frequency bands (cf.

103 104

wavelength [nm]

10 1

100

101

L iso

, [1

040er

g/s]

spectral distribution

polarequatorial

103 104

wavelength [nm]

spectral distribution

DD2-135135DD2-125145SFHo-135135SFHo-125145

Figure 6. Monochromatic, isotropic-equivalent luminosity timesfrequency, νLiso,ν , as function of wavelength for the four modelsmentioned in the right panel. Solid (dashed) lines denote surfaceaverages over the polar (equatorial) domain.

bottom left panel of Fig. 4), the kilonova of the dynamicalejecta peaks near the red end of the near-IR band, repre-sented by H in our case, and it does so in most of ourmodels. The H-band magnitude near peak epoch is, alsoroughly representative of all our models, ∼ −14mag andabout 0.5mag higher for polar observers than for equatorialones.

The dependence of Liso on the observer angle θobs isshown at t = 1 d in Fig. 5. The pole-to-equator ratio of Liso

is close to 2 for this model and similar to other models witha comparable semi-major axis ratio of the photophere (cp.black lines in Fig. 2). Model DD2-125145 exhibits a highervalue close to 3, which is likely the result of a significantlymore oblate ejecta geometry compared to the other models.

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12 O. Just et al.

Finally, the behavior of the spectral distribution of νLiso,ν ,plotted in Fig. 6, does not seem to exhibit a strong depen-dence on the observer angle.

3.3 Model dependence

In the following we examine how the previously identifiedproperties change when switching the nuclear EoS, decreas-ing the mass ratio of the NS binary, ignoring neutrino inter-actions, and artificially reducing or enhancing the electronfraction Ye by 0.1.

3.3.1 Equation of state and binary mass ratio

We show in Fig. 7 a similar selection of quantities as inFig. 4, but now for the four models DD2-135135, DD2-125145, SFHo-135135, SFHo-125145 and not distinguishingbetween polar and equatorial regions in the computation ofthe corresponding quantities. As shown in Part I, the abun-dance pattern of nucleosynthesis yields differs only by mod-erate amounts between these models and, hence, the initialopacities (cf. bottom middle panel of Fig. 7) as well as thespecific heating rates (see Fig. 8 in Part I) are rather similarfor all models. The observed differences in the kilonova emis-sion must therefore be connected to variations of the ejectamasses and velocities. Consistent with previous studies (e.g.Bauswein et al. 2013; Hotokezaka et al. 2013; Radice et al.2018b), the ejecta mass and average velocity increase for thesofter SFHo EoS compared to the DD2 EoS, and the ejectamass increases as well when reducing the mass ratio of thestellar binary. Indeed, the bolometric luminosities of the fourmodels are roughly consistent with the order correspondingto the ejecta masses, at least in the late, optically thin phaseof evolution, during which the luminosity is essentially givenby the thermal heating power,

L ≈ Lthin ≡ 〈Qheat〉Mej = q . (30)

Deviations from an exactly linear proportionality betweenLthin and Mej at a given time t are mainly connected tothe thermalization efficiency, fth, which grows withMej anddecreases with the ejecta velocity, 〈v〉 (cf. Fig. 1). In contrastto the behavior at late times, the luminosity differences rightaround the peak epochs are significantly smaller. The reasonis that more massive ejecta take longer to become opticallythin than less massive ejecta, namely roughly until

tpeak ∝M12

ej 〈v〉− 1

2 〈κ〉12 (31)

as estimated based on the simplified case of a uniform dis-tribution of density and opacity (e.g. Arnett 1982; Metzgeret al. 2010; Grossman et al. 2014). Hence, since the radioac-tive heating rate declines approximately as

〈Qheat〉 ≈ 〈fth〉 ×Q0 ×(t

1 d

)−α(32)

(where Q0 ≈ 2 × 1010 erg g−1 s−1 and α ≈ 1.3 reproducewell the total radioactive energy release rate, Q+Qneu, re-sulting from our nucleosynthesis calculations; see Part I10),

10 Note that the analytic fit presented in Part I with a scalingfactor of 1 × 1010 erg g−1 s−1 was for the heating rate withoutneutrino contributions, Q, whereas here we consider Q+Qneu.

the luminosity at peak epoch scales only weakly with theejecta mass (e.g. Grossman et al. 2014; Kasen et al. 2017;Wollaeger et al. 2018):

Lpeak ≈ Lthin(tpeak)

∝ Q0〈fth〉peakM0.35ej 〈v〉0.65〈κ〉−0.65 . (33)

Assuming that the implicit dependence of the thermalizationefficiency at peak, 〈fth〉peak ≡ 〈fth〉(tpeak), on Mej and 〈v〉is subdominant11, the behavior of tpeak and Lpeak in ourmodels (cf. Table 1) can be understood reasonably well bymeans of Eqs. (31) and (33), respectively.

In Table 1 we also provide the peak photospheric tem-peratures, 〈Tph〉peak, and find that they all lie rather closeto each other within 2.2 . . . 2.7×103 K, i.e. neither very sen-sitive to the nuclear EoS nor to the binary mass ratio. Theweak dependence on Mej and 〈v〉, is in agreement with ex-pectations from an analytical estimate of the peak photo-spheric temperature, which can be derived using the Stefan-Boltzmann law under the assumption that the ejecta emitblack-body radiation from their surface at about r ≈ 〈v〉t(e.g. Grossman et al. 2014):

〈Tph〉peak ≈[

Lpeak

σSB4π(〈v〉tpeak)2

] 14

∝ Q0.250 〈fth〉0.25

peakM−0.16ej 〈v〉−0.09〈κ〉−0.41 (34)

(where σSB is the Stefan-Boltzmann constant). Measured interms of the power by which they determine 〈Tph〉peak inthis simplified analytical estimate, the mass and velocity ofthe ejecta are less important than the opacity and heatingrate. Not surprisingly, considering the rather similar valuesof 〈Tph〉peak, all models peak in the same (H) frequencyband.

In summary, in our kilonova models the luminosity(spectral temperature) is rather sensitive (insensitive) tovariations of the nuclear EoS and the binary mass ratio.Since the composition is fairly independent of the EoS andmass ratio, the bolometric luminosity is therefore mainly de-termined by the ejecta mass and velocity. We note that thesimplified grey opacities used here are not suited for an in-depth analysis of spectral features. More sophisticated cal-culations using opacities based on atomic lines may revealsensitivities that are obscured by our scheme.

3.3.2 Ignoring neutrino interactions

The impact of neutrino interactions on the composition andthe kilonova of dynamical ejecta (as well as other ejectacomponents) represents one of the most uncertain and chal-lenging aspects in theoretical multi-messenger modeling ofNS mergers given the difficulties of solving the Boltzmannequation for neutrinos (e.g. Wanajo et al. 2014; Gorielyet al. 2015; Martin et al. 2018; Radice et al. 2018b; Fou-cart et al. 2018; Sumiyoshi et al. 2021). Until a few yearsago most merger simulations therefore either completely ne-glected neutrino interactions or implemented them in a waythat neglects net neutrino absorption and/or the advection

11 We stress that 〈fth〉(tpeak) varies much less between individualmodels than 〈fth〉(t) at some fixed time, t, as can be seen fromFig. 1 using the peak times provided in Table 1.

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Figure 7. Same as Fig. 4 but comparing the four hydrodynamic models with different nuclear EoSs and binary mass ratios and notdistinguishing between polar and equatorial regions, i.e. only spherically averaged or integrated quantities are shown. Note that the topmiddle panel now shows q instead of q/Mej.

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Figure 8. Same as Fig. 7 but comparing the fiducial model DD2-135135 with its counterpart DD2-135135-noneu, in which neutrinoreactions have been neglected after the plunge.

and equilibration of trapped neutrinos. The advection oftrapped neutrinos can be accounted for without significantefforts and even without computing any neutrino interactionrates – basically by replacing the electron fraction with thetotal lepton fraction in optically thick regions (e.g. Gorielyet al. 2015; Ardevol-Pulpillo et al. 2019). However, the mainchallenge lies in the reliable description of the emission andabsorption rates in the semi-transparent regions surroundingthe hot merger remnant once it is formed. In order to quan-

tify the impact of (not) including any neutrino interactionsafter the first touch of the two stars, we introduced in Part Ia variation of model DD2-135135 (called DD2-135135-noneuhere), in which Ye is held constant after the point when thetwo stars plunge into each other. In this model the abun-dances of elements with mass numbers 90 <∼ A <∼ 140 arefound to be reduced, while the mass fractions of lanthanidesand heavier elements are enhanced by a factor of 2−3. More-over, the pole-to-equator composition gradient basically dis-

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14 O. Just et al.

appeared and high values of XLA >∼ 0.1 are now found at alllatitudes (see fifth row of Fig. 2). For the heating rate thelack of low-mass elements causes a reduction at early times(0.1 d <∼ t <∼ 1 d), while the excess of heavy elements pro-duces an increase at late times (1 d <∼ t <∼ 100 d) comparedto the full neutrino model (cf. Sect. 3.2 and Fig. 8 of PartI).

Figure 8 provides insight about the extent to which thekilonova light curve and related quantities are affected by ne-glecting neutrino interactions during (most of) the mergerprocess. Initially, when the temperatures are above the re-combination temperatures, the enhancement factor of ∼ 2 inthe lanthanide fraction translates into an opacity increase of≈ 20 %. The weaker radioactive heating under more opaqueconditions at t < 1 d explains why the emission peak isfainter by about 30 − 50 % and is delayed by ∼ 10 % (cf.top left panel of Fig. 8 and Table 1). At late times, whenopacity effects have ceased and heavier elements dominatethe heating rate, the situation is reversed because of thehigher abundance of heavier elements in the “noneu” model.Considering broadband light curves, the kilonova peak re-mains in the red regime of the near-IR band, representedhere by theH-band, while its magnitude is reduced by about0.3mag. The photospheric temperatures near peak epochare reduced, though only mildly.

3.3.3 Sensitivity to Ye

An additional sensitivity test to bracket the uncertaintiesin the treatment of weak interactions can be conducted bymanually varying the electron fraction at the onset of the nu-clear network calculations, Ye(ρnet). To this end, we set uptwo models that are identical to the fiducial model but whereYe of all trajectories is increased and reduced, respectively,by 0.1 (with a lower limit of 0.005 imposed). The nucleosyn-thesis yields of these models have already been discussed inPart I, see Sect. 3.3 and Fig. 11 therein.

In Fig. 9 we compare both models with the fiducialmodel. Artificially reducing Ye by 0.1 yields results that arevery similar to the case of completely neglecting neutrinointeractions (see previous section and cf. Fig. 8), namely adelayed and fainter peak. This is not surprising consider-ing that the average electron fractions of the correspondingmodels are comparable (〈Ye〉 ≈ 0.17 and 0.13, respectively;cf. Table 1). In the opposite case of increasing Ye by 0.1, themass fraction of lanthanides and the opacity are reducedroughly by factors of 4 and 2, respectively. The lower opac-ities result in a significantly shorter duration of the kilo-nova and a slightly more luminous peak. The luminosityenhancement due to the reduced opacity is, however, par-tially compensated by the weaker radioactive heating ratesin the Ye+0.1 model. At t = 1d, for instance, the heat-ing rate is about q/(1040 erg s−1) = 1.66 compared to 2.32in the reference model. The reduced opacities also shift thespectral peak to the blue end of the near-IR domain, as canbe seen by the enhanced z-band magnitudes in the lowerleft panel of Fig. 9. Yet, despite the considerable shift of Yeand resulting reduction of the lanthanide fraction down to〈XLA〉 ≈ 0.02, the emission in the visible bands is still sub-dominant compared to that in the near-IR domain, i.e. thetransient would classify as a red kilonova.

Summarizing the results of Sects. 3.3.2 and 3.3.3: Dif-

ferences in the treatment of neutrino effects can induce vari-ations in brightness of up to a factor of two and in durationof ∼ 20 %, i.e. an amount comparable to that resulting fromchanging the binary mass ratios or the nuclear EoSs.

4 DISCUSSION

4.1 Role of dynamical ejecta inAT2017gfo/GW170817

A comprehensive understanding of the origin of each compo-nent in the kilonova accompanying GW170817, AT2017gfo,is still elusive. Although the lack of the secular post-mergerejecta in our study and the small set of considered EoSsand NS merger configurations defies a conclusive compari-son, we can nevertheless briefly speculate whether our mod-els, which in contrast to most previous models are based onconsistent nucleosynthesis trajectories from hydrodynamicalsimulations, would, or not, support the possibility that thedynamical ejecta alone are responsible for either of the two,red or the blue, kilonova components. The observed bolo-metric luminosity of AT2017gfo, as constructed by Waxmanet al. (2018), is shown by filled circles in Fig. 7 as well asestimates of the spectral temperature and photospheric ve-locity, which can be compared directly to 〈Tobs〉 and 〈vobs〉(orange lines in the right panels of Fig. 7). As can be seen,none of our models appears to be compatible with eithercomponent of AT2017gfo.

The early, blue component of AT2017gfo, i.e. the emis-sion observed until t ∼ 1 − 1.5d (e.g. Nicholl et al. 2017),exhibited a bolometric luminosity of almost one order ofmagnitude higher than the surface-averaged values predictedby our models, and its spectral temperatures were roughlytwice as high as in our models. The combination of both dis-crepancies, too faint luminosities and too red colors, couldhardly be resolved by just increasing the ejecta mass, be-cause more massive ejecta would rather reduce the peak tem-peratures, as suggested by Eq. (34) neglecting uncertaintiesconnected to the thermalization efficiency. The comparisonsuggests that, in agreement with previous studies based onmore simplified one-zone models (Villar et al. 2017), signif-icantly lower lanthanide mass fractions, and correspondingopacities, may be required to reproduce the early compo-nent. Even though the average electron fraction is raisedsignificantly as a result of a sophisticated inclusion of neu-trinos in our models (e.g. from 〈Ye〉 = 0.13 to 0.27 for modelDD2-135135; cf. Table 1), the amount of lanthanides stillseems to be too high to produce an optical transient like theone in AT2017gfo. However, our set of investigated mod-els is rather small, and more exhaustive explorations usingother EoSs and NS binary configurations compatible withGW170817 are needed to clarify whether cases exist in whichthe lanthanide mass fraction becomes significantly smallerthan observed for our models.

The relatively red spectral colors seen in the kilonovasignals of our models suggests a better agreement of ourmodels with the red component of AT2017gfo (e.g. Chornocket al. 2017)12, especially keeping in mind that (cf. Fig. 5)

12 We remark that post-merger disk outflows are often consid-ered more suitable to explain the red component (e.g. Kasen et al.

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Figure 9. Same as Fig. 7 but comparing the fiducial model DD2-135135 with its counterparts DD2-135135-Ye-0.1 and DD2-135135-Ye+0.1, in which the electron fraction, Ye, was shifted by −0.1 (+0.1) for all trajectories.

isotropic luminosities observed from polar viewing anglescan be enhanced by some factor compared to the surface-averaged isotropic luminosities plotted in Fig. 7. However,the ejected mass would presumably need to be significantlylarger (by a factor of a few) than the 0.009M of our mostmassive model in order for the peak to occur sufficientlylate and luminous (cp. Fig. 7 and Eqs. (31) and (33)). Thiscould possibly be achieved by very asymmetric cases and asoft equation of state (e.g. Bauswein et al. 2013; Sekiguchiet al. 2016; Radice et al. 2018b). Moreover, a crucial role maybe played by the uncertainties associated with the thermal-ization efficiency, fth (Barnes et al. 2016; Kasen & Barnes2019; Wu et al. 2019). The fact that fth increases with theejecta mass could improve the prospects in this regard.

In any case, our analysis remains incomplete and incon-clusive, because we only consider a small set of EoSs and bi-nary configurations and we only include a single ejecta com-ponent. A more realistic model of AT2017gfo would need toaccount for genuine multi-component effects, such as pho-ton transport from one ejecta component to another (e.g.Kawaguchi et al. 2018), or photon blocking of one compo-nent by another (often called “lanthanide curtain”; see, e.g.,Kasen et al. 2015; Korobkin et al. 2021).

4.2 Importance of consistent ejecta sampling

An often employed approximation in modeling ejecta andkilonovae from NS mergers is to assume representative bulkvalues of the electron fraction, expansion velocity, and en-tropy for the entire ejecta, or for sub-domains within the

2017), because they are typically more massive than the dynam-ical ejecta while carrying comparable lanthanide fractions (e.g.Just et al. 2015a; Wu et al. 2016; Siegel & Metzger 2017; Justet al. 2021). See, e.g., Tanaka et al. (2017); Perego et al. (2017);Kawaguchi et al. (2018) for the discussion of alternative scenarios.

ejecta, and to derive from those the composition, heatingrate, and opacity using a single nucleosynthesis calculation.Obviously, such single-trajectory models carry some uncer-tainty, because in reality the ejecta exhibit a broad spectrumof hydrodynamic and thermodynamic properties, which mayresult in large variations of heavy-element yields among dif-ferent fluid elements in the ejecta. The yields obtained by av-eraging over a collection of nucleosynthesis trajectories maypredict results that cannot be accounted for by a single nu-cleosynthesis calculation along a trajectory of averaged fluidproperties. Our method of detailed ejecta sampling by post-processing individual outflow trajectories using consistenthydrodynamic data allows to test the uncertainty of single-trajectory models by comparing the average opacities andheating rates predicted by either approach.

We first consider the opacity. It is well known (e.g. Ruf-fert et al. 1997; Hoffman et al. 1997; Lippuner & Roberts2015; Lemaître et al. 2021) that the production of lan-thanides along a typical outflow trajectory from NS mergersbecomes inefficient as soon as the electron fraction exceedsa threshold value, Y crit

e , which, for typical merger outflowsand mainly depending on the entropy per baryon, lies withinY crite = 0.25 − 0.3. As can be seen by the dots in the mid-

dle panel of Fig. 10, this behavior is reproduced well bynucleosynthesis yields for individual ejecta trajectories ofour model DD2-135135: All trajectories with Ye >∼ 0.28 ex-hibit extremely low lanthanide fractions of XLA < 10−4.This step-like behavior is often used to characterize alsothe bulk ejecta or sub-regions of the ejecta by assigningvery low values of XLA to regions with average Ye > Y crit

e .However, if instead of individual trajectories we considerfinite sub-domains of the ejecta, for example obtained bydividing the northern hemisphere into 30 angular bins (seeblue crosses in Fig. 10), we receive quite a different rela-tion between Ye and XLA. Due to mixing of low-XLA fluidelements with high-XLA fluid elements, the sharp cut-off

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16 O. Just et al.

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Figure 10. Example illustrating the discrepancy between correlations obtained for individual trajectories and correlations betweenaverages over finite domains. All plotted data is taken from model DD2-135135. Three left plots: Mass-averages of the electron fraction,Ye, lanthanide mass fraction, XLA, and (temperature-independent part of the) opacity, κLA, over the volumes defined by 30 equidistantpolar-angle bins dividing the northern hemisphere. Three right plots: Pair-wise correlation plots between the same data as in the leftplots. For comparison, orange dots denote the corresponding data measured for individual outflow trajectories. A sharp cut-off of XLA

around Ye ∼ 0.28 is only visible for single-trajectory data, while much higer values of XLA(Ye), and therefore of κLA(Ye), can be reachedwhen considering collections of trajectories in finite regions.

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DD2-135135 (average)DD2-125145 (average)SFHo-135135 (average)SFHo-125145 (average)DD2-135135-noneu (average)DD2-135135-Ye-0.1 (average)DD2-135135-Ye+0.1 (average)

Figure 11. Comparison of total radioactive energy-release rates,Q + Qneu, along individual outflow trajectories of model DD2-135135 and the corresponding rates mass-averaged over all tra-jectories of a given model. Orange lines denote the subset of par-ticles of model DD2-135135 with Ye close to the average value of〈Ye〉 = 0.27 and with velocities v/c < 0.4. Values of Q + Qneu

differ more strongly between individual trajectories than betweenensemble averages of different models, even for models with ratherdiverse values of 0.13 < 〈Ye〉 < 0.37.

is replaced by a much more shallow decline, such that re-gions with average Ye > 0.4 can still contain a high fractionof lanthanides XLA > 0.01 and posess a correspondinglyhigh opacity. If, for instance, we had estimated the opacityof angular bins based on the κLA(Ye) behavior of individ-ual trajectories, e.g. by using κLA(Ye > 0.28) = 0.2 andκLA(Ye < 0.28) = 30 cm−2g−1, the average opacity of thepolar (θ < π/4) ejecta would be 0.2 cm−2g−1, i.e. grosslyunderestimated compared to the value of ≈ 6.2 cm2g−1 thatwould result if we had used the same κLA(Ye) function forindividual particles instead. Hence, approximating XLA orκLA based on the average Ye of an ensemble of trajectoriescan lead to a significant underestimation of XLA or κLA,

which tends to be more serious for a higher degree of in-homogeneity of the ejecta. While here we consider only thedynamical ejecta, this aspect is likely to be relevant alsofor turbulence-driven secular ejecta, which typically exhibita broad spectrum of Ye (e.g. Fernández & Metzger 2013;Siegel & Metzger 2017; Just et al. 2021), while it might beless relevant for neutrino-driven winds, which often have arather smooth structure (e.g. Perego et al. 2014; Fujibayashiet al. 2018).

Next we take a look at the radioactive heating rates. Asis already clear from the analytic scaling laws for Lthin andLpeak (cf. Eqs. (30) and (33), respectively) a reliable predic-tion of the heating rate is mandatory for any kilonova modelto properly infer the ejecta mass from observations duringthe peak or the optically thin phase (e.g. Wanajo 2018; Ross-wog et al. 2018; Wu et al. 2019; Barnes et al. 2021). For sim-plicity, we now ignore the additional complexity connectedto the thermalization efficiency and assume fth = 1. Forthe heating rates the analysis is not as straightforward asfor the opacities, where the strong non-linearity of XLA(Ye)represent the main problem, because the heating rates aredetermined by decay chains involving a large number of iso-topes, of which the abundances are quite sensitive also tothe entropy and expansion timescale. Nevertheless, we canget a basic idea of the uncertainty of single-trajectory mod-els by comparing the heating rates for individual trajectorieswith the average heating rates. To that end, in Fig. 11 weplot, apart from the global heating rates for each model,the heating rates of all particles of model DD2-135135 us-ing black, slightly transparent lines, while orange lines areused to denote only the subset of these particles that haveYe close to the average value of 〈Ye〉 ≈ 0.27 and velocitiesof v/c < 0.4. Even for this subset of presumably represen-tative trajectories (in the sense that Ye and v are in theballpark of expected bulk values), the range of variation ofthe heating rates is substantial and amounts to factors of2-4 during the entire period of 0.1 d< t < 20d that is rele-vant for the kilonova emission. On the other hand, the sub-stantially smaller model-by-model variations of the averageheating rates (thick lines in Fig. 11) indicate that the aver-

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Figure 12. Exemplary comparison of the surface-integrated bolo-metric luminosity (top panel) and the AB magnitudes in selectedbands (bottom panel) of model B2 (low-Ye version) of Korobkinet al. (2021) with our model SFHo-125145. The good agreementwith Korobkin et al. (2021), who employ a Monte-Carlo radiative-transfer scheme with atomic-physics based opacities supports thevalidity of our results obtained with an approximate radiativetransfer solver with parametrized opacity treatment.

age heating rate of ensembles of trajectories are subject to amuch reduced level of randomness than the heating rate ofindividual particles. This finding is particularly remarkable,because we even consider models with artificially changedvalues of Ye that span a large range of values for 〈Ye〉 be-tween 0.13 and 0.37 (cf. Table 1).

From the above discussions we conclude that predic-tions for the nucleosynthesis yields and the correspondingkilonova signal that are based on one-zone models or single-trajectory modeling may carry substantial systematic un-certainties in cases where the thermodynamic properties arenot homogeneous throughout the ejecta but given by a broaddistribution.

4.3 Comparison with previous studies

We will now briefly compare some of our results with pre-vious studies. The level of sophistication of kilonova mod-els is growing quickly, and many studies employ their ownways of dealing with the large number of required physicsingredients. Moreover, given the large parameter space it isdifficult to find studies that discuss models with a match ofthe basic characteristic parameters (geometry, mass, veloc-ity, composition, etc.), which would be required for a mean-ingful quantitative comparison. Hence, we will only comparea few elementary features with selected studies.

While the number of kilonova studies directly based onthe outputs from hydrodynamical simulations is still verysmall, quite a few studies exist by now that investigate multi-dimensional kilonovae based on manually constructed ejectaconfigurations with homogeneous distributions of the heat-ing rate and opacity (e.g. Wollaeger et al. 2018; Barbieriet al. 2019; Bulla 2019; Darbha & Kasen 2020; Korobkinet al. 2021; Kawaguchi et al. 2020; Heinzel et al. 2021).Darbha & Kasen (2020) studied the emission characteris-

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Figure 13. Comparison of mass-averaged total radioactiveenergy-release rates resulting from our nucleosynthesis calcula-tions for model DD2-135135 with heating rates employed in otherkilonova studies (not accounting for thermalization efficiencies).The red curve (extracted from Fig. 5 of Wanajo et al. 2014) de-notes the heating rate obtained from a nucleosynthesis analysis ofa neutrino-hydrodynamical merger simulation and was employedfor dynamical ejecta in, e.g., Kawaguchi et al. (2018, 2020). Thegreen line (extracted from Fig. 4 of Wollaeger et al. 2018) is basedon a network calculation along an exemplary outflow trajectoryof model B of Rosswog et al. (2014) and was employed for dynam-ical ejecta in, e.g., Wollaeger et al. (2018); Korobkin et al. (2021);Wollaeger et al. (2021). Blue lines (cf. Eq. 2 and case BF of Peregoet al. 2017) represent the Ye-dependent prescription employed byPerego et al. (2017); Radice et al. (2018b), which is not based ona nuclear network calculation but deduced from a light curve fitto AT2017gfo.

tics as function of the observer angle based on parametrizedellipsoids, tori, and conical sections with a grey and con-stant opacity and fitted the results with an analytic func-tion for the ratio Liso(0)/Liso(90) in dependence of theprojected surface area of the emitting object. For an ellip-soid with a semi-major axis ratio of ∼ 1.5 − 2, which tofirst order resembles the shape of the photospheres that weobserve in our models (cf. Fig. 2), they predict values ofLiso(0)/Liso(90) ≈ 1.5 − 3 that are in good agreementwith our results (see Fig. 5).

Various types of parametrized, axisymmetric ejecta con-figurations have been employed in a series of papers by theLos Alamos group (Wollaeger et al. 2018; Korobkin et al.2021; Wollaeger et al. 2021, e.g.) using atomic-physics basedopacities (Fontes et al. 2020) and two different choices forthe composition (Even et al. 2020). The ellipsoidal geom-etry of photospheres found here for the dynamical ejectais roughly reproduced by the biconcave (“B”) configurationintroduced by Korobkin et al. (2021), which however wasdropped again in the set of models byWollaeger et al. (2021).Out of the single-component models considered by Korobkinet al. (2021), their model B2, with an average velocity of 0.2 cand a mass of 0.01M, posesses parameters with the closestagreement to one of our models, namely to SFHo-125145. Acaveat for our comparison is, however, that their “low-Ye”case exhibits a lower lanthanide fraction (XLA = 0.032; seeKorobkin et al. 2021 and Even et al. 2020) compared to our

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model (0.114). In Fig. 12 we compare13 the angle-averagedbolometric light curves and magnitudes for both models andfind rather good agreement, both quantitatively and quali-tatively regarding the property that the emission peaks inthe same near-IR frequency regime.

A similar approach as in the aforementioned workswas pursued in Kawaguchi et al. (2018, 2020), however, al-most exclusively considering multi-component (dynamicalplus secular) ejecta models. Broadband magnitudes for twosingle-component models of dynamical ejecta, with massesof 0.003 and 0.01M and average velocity of 0.25 c are pre-sented in Appendix A of Kawaguchi et al. (2020). Despitethe fact that their setup agrees fairly well with ours – themain difference is probably the more toroidally shaped ejectadistribution in their case – we find only modest agreement.Independent of the observer angle their kilonova is brighterby about 1mag in the near-IR bands and, at early times andpolar viewing angles, they obtain a considerably strongeroptical component than us. The origin of these differencesremains unclear but could be connected to the more ad-vanced set of opacities (which are based on Tanaka et al.2020) employed in these studies and to the different ejectageometry.

Among the few available kilonova models that adoptejecta properties directly from hydrodynamical simulationsare Kawaguchi et al. (2021) and Radice et al. (2018b).A direct comparison with the light curves of Kawaguchiet al. (2021) is not possible, however, because they addition-ally follow the long-term evolution of the NS-disk remnant,which produces massive viscously driven outflows that dom-inate the dynamical ejecta. Yet, a noteworthy difference tothe models in Kawaguchi et al. (2021) is the prolate geom-etry of their total ejecta, which is in contrast to the ratheroblate structures that we find for just the dynamical ejectaat peak epoch.

Radice et al. (2018b) reported kilonova light curves thatare obtained by a superposition of quasi-spherically sym-metric models adopting the formalism by Grossman et al.(2014); Martin et al. (2015); Perego et al. (2017). Radiceet al. (2018b) used data provided by binary NS merger sim-ulations for the emission from the dynamical ejecta, whilethey employed a parametric wind model to estimate the con-tribution from neutrino- plus viscously-driven post-mergerejecta. Although no light curves just for the dynamical ejectacomponents are shown in Radice et al. (2018b), one can in-directly infer from their results that the dynamical ejectaproduce significant emission in the g and z bands, with ab-solute AB magnitudes of about -15mag (corresponding toapparent magnitudes of ≈ 18mag when observed at a dis-tance of 40Mpc) for dynamical ejecta masses of ≈ 0.003M.Our models, which seem to broadly agree in the mass- andYe-distributions of the ejecta, instead produce significantlyfainter emission which peaks in bands with lower frequency.One reason for the higher luminosities may be that Radiceet al. (2018b) assigned the opacities as function of average Yevalues along angular bins (see Sect. 4.2 for a discussion). An-other, more likely explanation may come from the fact thatRadice et al. (2018b) employed the extremely high heating

13 The data for model B2 was downloaded fromhttps://ccsweb.lanl.gov/astro/transient/transients_astro.html.

rates by Perego et al. (2017). These rates, which are shownin Fig. 13, were not adopted from nucleosynthesis calcula-tions but were calibrated such that the semi-analytic kilo-nova models by Perego et al. (2017) reproduce the observedkilonova AT2017gfo14. Switching to such a high ad-hoc heat-ing rate enhances the peak luminosities substantially, be-cause the heating rate enters Lpeak linearly, whereas theejecta mass, velocity, and opacity only enter with a lowerpower (cf. Eq. (33)). Hence, we expect the heating rate tobe the main reason for the observed discrepancies betweentheir kilonova results and ours.

Although the high heating rates assumed in Peregoet al. (2017) cannot be excluded entirely, to our knowledgeno state-of-the-art network solver currently exists that pro-duces heating rates of that magnitude for the thermody-namic conditions under consideration. A systematic discus-sion of the sensitivity of the heating rate with respect tonuclear-physics uncertainties is, however, deferred to a fu-ture study. We point out that Barnes et al. (2021) and Zhuet al. (2021) have recently taken a step in that directionand report large variations between Q profiles with differentnuclear physics inputs. However, the range of uncertaintymay not be as dramatic as suggested by these studies, be-cause, first, some of their most extreme cases are producedby outdated nuclear models, which are already known to beinconsistent with currently known experimental data15, andsecond, their analysis is based on single outflow trajectoriesinstead of trajectory ensembles, which may artificially en-hance the range of scatter as indicated by the comparisondiscussed in Sect. 4.2 (Fig. 11 there).

5 SUMMARY AND CONCLUSIONS

In this study we applied a new methodical treatment topredict the kilonova emission of dynamical ejecta adopt-ing tracer particles from general relativistic neutrino-hydrodynamics simulations. In contrast to many previ-ous kilonova models using only the bulk properties of theejecta, the tracer-particle approach samples the local, time-dependent properties of the ejected material as well as thecorresponding nucleosynthesis results for the radioactiveheating rate, lanthanide mass fraction, and mean baryonnumber. The kilonova emission is computed by means of atwo-moment transport scheme employing the approximateM1 closure and assuming a simplified, heuristic expressionof the photon opacity as function of the lanthanide frac-tion and the temperature. We applied our approach to thedynamical ejecta of four different NS-merger configurationswith different nuclear equations of state and different binarymass ratio. Moreover, in order to quantify the sensitivitywith regard to uncertainties of weak interaction effects tak-ing place during the dynamical merger phase, we comparedwith three additional cases with artificially modified electronfractions. Our results are summarized as follows:

14 A similar approach involving even higher heating rates wasrecently adopted by Breschi et al. (2021).15 For example, Barnes et al. (2021) and Zhu et al. (2021) employthe SLy4 HFB masses, which are known to predict all experimen-tal masses of even-even nuclei with a root-mean-square deviationas large as 5 MeV (Stoitsov et al. 2003).

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Dynamical ejecta with weak processes II: Kilonova 19

• At times close to peak emission, all models exhibit pho-tospheres that globally resemble ellipsoids with semi-majoraxis ratio of 1.2 − 2. The globally oblate geometry of thephotosphere is a result of less efficient matter ejection to-wards the poles as well as systematically higher electronfractions, and therefore reduced opacities, in the polar com-pared to equatorial regions. Due to the violent, highly dy-namic matter-ejection processes during the merger, the pho-tosphere is not smooth but rather corrugated and charac-terized by irregular structures that change with time as thephotosphere travels deeper into the ejecta.• For the four dynamical ejecta models with regular treat-

ment of weak interactions, masses between 0.002−0.009M,average velocities of ≈ 0.25 c, and lanthanide mass fractionsof ≈ 0.1 the ejecta reach optically thin conditions at timesbetween 0.7 − 1.5d, while during that epoch they producebolometric luminosites of ∼ 3− 7× 1040 erg s−1, which peakin near-IR frequency bands, and they exhibit photospherictemperatures within 2.2−2.8×103 K. Since the compositionis rather similar for the four regular models, the observeddifferences of peak times and luminosities between thesemodels are a result mainly of the different ejecta masses.A softer equation of state as well as a more asymmetric bi-nary produce slightly more luminous transients with longerdurations.• The fluxes received by observers increase by a factor of

1.5− 3 when going from equatorial to polar viewing angles,while for different configurations this factor increases withthe level of oblateness of the ejecta. The spectra differ onlymarginally for different observer angles.• Neglecting all neutrino interactions after both NSs first

touched for a symmetric model with DD2 EoS reduces theaverage electron fraction from 0.27 to 0.13, increases the lan-thanide fractions by a factor of about 2, and moves radioac-tive heating power effectively from times t < 1 d to t > 1 d.The main consequences for the kilonova are an extendedpeak (by ∼ 15 %) and a luminosity reduction (by ∼ 40 %),while the spectrum is only mildly red-shifted. These changesare comparable to the changes obtained when reducing Ye by0.1 globally for all trajectories. Conversely, increasing Ye by0.1 reduces the lanthanide fractions by a factor of 4, but atthe same time reduces the heating rate at all times by about20 − 40 %. Hence, the emission is only marginally more lu-minous, but its duration is ∼ 20 % shorter and the spectrumshifted towards the blue end of the near-IR band. The factthat the impact of these Ye modifications is of comparablemagnitude as the variations due to different EoSs or binarymass ratios implies that neutrino interactions must be prop-erly accounted for in models that shall be used to reliablyinfer the ejecta properties.• None of our investigated models produces a transient

compatible with the blue or red component of AT2017gfothat accompanied GW170817. Agreement with the highspectral temperatures and high luminosities of the early,blue component of AT2017gfo would require significantlysmaller lanthanide mass fractions, and therefore opacities,than found for our models. While the low spectral temper-atures of our models are roughly compatible with the late,red component of AT2017gfo, the short duration and lowluminosity of our kilonova signals would call for a signifi-cantly higher ejecta mass than the 0.009M of our mostmassive case. However, our results are not conclusive regard-

ing the role of dynamical ejecta in AT2017gfo as they coveronly a small set of EoSs and binary parameters allowed byGW170817 and do not include secular post-merger ejecta.• We found that one-zone kilonova models, which assume

that the nucleosynthesis results for a single trajectory arerepresentative for the bulk of the ejecta, tend to system-atically underestimate the lanthanide fractions, and there-fore the opacities, compared to models adopting an ensem-ble of trajectories that samples a range of thermodynamicconditions. Moreover, the heating rates predicted by one-zone models may depend rather sensitively on the partic-ular choice of thermodynamic conditions along the trajec-tory and on the employed nuclear-physics input, in contrastto which the average heating rate for an ensemble of tra-jectories was found to be more robust with respect to thevariation of global parameters.

The fact that our kilonova models based on single-component dynamical-ejecta configurations are unable toreproduce the blue or the red component of AT2017gfo isin line with previous analyses, such as Siegel (2019); Ne-dora et al. (2021). We note, however, that these works arebased on a comparison with the Arnett-type (i.e. essentiallyspherically symmetric one-zone) model of AT2017gfo by Vil-lar et al. (2017). Since radiative transfer models can predictquite different peak properties for specific observer direc-tions than one-zone models – particularly in the case ofmulti-component ejecta (Korobkin et al. 2021) – the masses,bulk velocities, and bulk opacities found in Villar et al.(2017) for the different components of AT2017gfo shouldbe taken with care.

The role of the dynamical ejecta in AT2017gfo may notbe constrained to the minor effects suggested by the single-component models considered in our study. First, our setof models considers only very few EoSs and binary massratios, while more extreme configurations may lead to sig-nificantly more massive and/or lanthanide-poor ejecta andcorrespondingly brighter and blue-shifted kilonova signals.Another, more general argument may be connected to thetendency16 that dynamical ejecta are expelled in all direc-tions, albeit with a greater mass loading in equatorial thanpolar directions (Bauswein et al. 2013; Sekiguchi et al. 2015;Radice et al. 2018b; Kullmann et al. 2021). The dynami-cal ejecta should therefore overcast all non-relativistic post-merger ejecta to a substantial extent, such that a reprocess-ing of the emission from interior ejecta by the envelopingdynamical ejecta can be naturally expected, in particular atearly times (t <∼ 1−1.5 d) when most of the dynamical ejectaare still optically thick. The model by Kawaguchi et al.(2018), which is based on an analytic distribution of multi-component ejecta, supports this notion, and it demonstratedthat the blue component of AT2017gfo can be produced evenby a lanthanide-rich dynamical-ejecta component, if the lat-

16 We note that kilonova studies (e.g. Wollaeger et al. 2018) in-formed by Newtonian merger models (e.g. Rosswog et al. 2014)often consider purely toroidal geometries to represent the dynam-ical ejecta. This is because matter ejection into polar directions asa result of collision shocks and corresponding heating is typicallyless pronounced in Newtonian simulations compared to the gen-eral relativistic case, mainly because of effectively weaker effectsby the gravitational interaction of the colliding bodies.

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20 O. Just et al.

ter is subject to heating by photons coming from an en-shrouded cloud of lanthanide-poor post-merger ejecta. Eventhough this model did not adopt a self-consistent ejecta con-figuration from a hydrodynamic simulation, the results high-light the difficulty of connecting specific kilonova (e.g. blueor red) components to individual (e.g. dynamical or viscousor neutrino-driven) ejecta components, because a kilonovafrom multiple ejecta components may not just be the sumof kilonovae from single-component ejecta models.

Keeping the aforementioned aspects in mind, we alsomention some alternative scenarios that could explain theblue component of AT2017gfo. The partial disruption of thedynamical ejecta due to a relativistic jet and its correspond-ing energy injection have been considered as a possible wayto make the ejecta shine brighter and at higher temperatures(Nativi et al. 2020; Klion et al. 2021b). Moreover, neutrinooscillations could change the nucleosynthesis signature of theejecta and reduce their lanthanide content (e.g. Zhu et al.2016; George et al. 2020). Apart from the dynamical ejectaand subsequent neutrino-driven winds from the surface ofthe hyper-massive NS, additional fast and lanthanide-poormaterial could be ejected in connection to magnetic-field ef-fects (e.g. Metzger et al. 2018; Mösta et al. 2020; Shibataet al. 2021) or in the form of a spiral-wave wind powered byan m = 1 mode (Nedora et al. 2019). Future, more refinedneutrino-magneto-hydrodynamic models of the merger andits remnant as well as self-consistent kilonova calculationsbased on these models will have to reduce the large uncer-tainties that, as of yet, prohibit a conclusive assessment ofthe constituents that shaped the kilonova in GW170817.

Finally, we summarize the shortcomings of our studythat should be improved upon in future work: Since the M1approximation is less reliable in the optically thin regime,our models do not have the same predictive power as kilo-nova calculations based on Boltzmann solvers, particularlyconcerning the emission into specific directions. Also, weonly include the early, dynamical ejecta here, but additionalejecta components produced by the merger remnant shouldbe included as well, ideally accounting for the dynamical andradiative interactions between components. Moreover, thetransition between the phase of matter ejection and homol-ogous expansion is not included but should be accounted forto ensure a correct mass distribution as input for the kilo-nova calculation. Finally, our phenomenological treatmentof the opacities is not based on atomic properties, whichthemselves depend on the local thermodynamic conditions.Our scheme could readily be improved by using an opacityframework based on atomic-physics models.

ACKNOWLEDGMENTS

OJ, AB, and CC acknowledge support by the European Re-search Council (ERC) under the European Union’s Horizon2020 research and innovation programme under grant agree-ment No. 759253 and by the - Project-ID 279384907 - Son-derforschungsbereich SFB 1245 by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation). OJ waspartially supported by JSPS Grants-in-Aid for ScientificResearch KAKENHI (A) 19H00693 and by the Interdis-ciplinary Theoretical and Mathematical Sciences Program(iTHEMS) of RIKEN. SG acknowledges financial support

from F.R.S.-FNRS (Belgium). This work has been sup-ported by the Fonds de la Recherche Scientifique (FNRS,Belgium) and the Research Foundation Flanders (FWO,Belgium) under the EoS Project nr O022818F. AB wasadditionally supported by - Project-ID 138713538 - SFB881 (“The Milky Way System”, subproject A10) and bythe State of Hesse within the Cluster Project ELEMENTS.At Garching, funding by the European Research Councilthrough Grant ERC-AdG No. 341157-COCO2CASA and bythe DFG through SFB-1258 “Neutrinos and Dark Matter inAstro- and Particle Physics (NDM)” and under Germany’sExcellence Strategy through Cluster of Excellence ORIGINS(EXC-2094)—390783311 is acknowledged. OJ acknowledgescomputational support by the HOKUSAI supercomputer atRIKEN/Japan and by the Max-Planck Computing and DataFacility (MPCDF) at Garching/Germany. The nucleosyn-thesis calculations benefited from computational resourcesmade available on the Tier-1 supercomputer of the Fédéra-tion Wallonie-Bruxelles, infrastructure funded by the Wal-loon Region under the grant agreement n1117545 and theConsortium des Équipements de Calcul Intensif (CÉCI),funded by the Fonds de la Recherche Scientifique de Bel-gique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by theWalloon Region.

Data availability: The data underlying this article willbe shared on reasonable request to the corresponding au-thor.

APPENDIX A: OPACITY CALIBRATION

Our kilonova treatment avoids the complexity connected toa microphysically consistent opacity, κ, which would needto account for thousands of transition lines between energylevels – most of them poorly known so far – of many iso-topes in dependence of the detailed composition, density,temperature, and degree of ionization. Instead, we employa simplified, grey version that parametrizes the opacity interms of the lanthanide mass fraction, XLA, and tempera-ture, T ; see Eqs. (13-15). The form of κ was calibrated17

in a way to reproduce, as accurately as possible, a set ofbolometric light curves by Kasen et al. (2017) using a state-of-the-art opacity treatment, which differ by the containedmass of lanthanides. In Fig. A1, left panel, we provide thelight curves based on which we calibrated. The ejecta massand bulk velocity for this reference case are 0.02M and0.1 c, respectively, while we refer to Kasen et al. (2017)for the precise form of the analytic mass distribution. Forthis test we use a radioactive heating rate that is pre-sumably similar, though possibly not exactly identical, tothe one used in Kasen et al. (2017). It is taken from Lip-puner & Roberts (2015) and given by qrad [erg g−1 s−1]=(1.0763 × 1010t−1.518

day + 9.5483 × 109e−tday/4.947) with tday

17 In order to prevent confusion, we note that a similar calibra-tion was already presented in the Appendix of Just et al. (2021),which suggested a slightly different form of κ(XLA, T ). The rea-son is that Just et al. (2021) did not correct their luminosities forframe-dependent effects as we do here (cf. Sect. 2.6) and, hence,needed different opacities in order to reproduce the same refer-ence results. We recommend using the newer version presentedhere.

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Dynamical ejecta with weak processes II: Kilonova 21

100 101

time [d]

1039

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1041

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1043

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rg/s

]

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bolometric luminosityXLA = 1e-1

our schemeKasen+17

14

12

10

MAB

Mej = 0.02M , vk = 0.1c, XLA = 0.1

broadband magnitudesgzH

10 1 100 101

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14

12

10

MAB

Mej = 0.0025M , vk = 0.2c, XLA = 10 5

our schemeKasen+17

Figure A1. Left panel: Results of our opacity calibration using a set of bolometric light curves from Kasen et al. (2017). Solid linesrefer to results obtained with our scheme, while dashed lines are the reference results from Kasen et al. (2017). Model parameters arelisted in the insets. The blue dotted line denotes the effective heating rate after thermalization. Right panel: Comparison of broadbandmagnitudes in three exemplary frequency bands for two qualitatively different models. The top panel shows results for a model leading toa near-IR kilonova, while the model in the bottom panel produces a kilonova peaking rather in the optical regime. Despite quantitativedifferences, our scheme reproduces the peaks in the correct bands.

being the time in units of days. The homogeneous thermal-ization efficiency is computed, as in Kasen et al. (2017),using the analytic fit formula as function of the ejecta massand velocity provided by Barnes et al. (2016).

A comparison of the peak behavior of broadband lightcurves provides another consistency check for our treatment.In Fig. A1 we compare the broadband light curves in theg, z, and H bands for two models. The model in the toppanel of Fig. A1 assumes rather massive, relatively slow,and lanthanide-rich ejecta, and the peak is in the near-IR(H) band. The model in the bottom panel assumes less mas-sive ejecta with higher velocities and a low lanthanide frac-tion, and the peak is therefore at higher spectral temper-atures, namely in the g and z bands. Albeit quantitativedifferences are noticeable, our scheme can reproduce the ba-sic peak behavior quite well. This result lends credibility tothe broadband behavior obtained in the dynamical ejectamodels investigated in this paper.

APPENDIX B: SENSITIVITY TO INITIALCONDITIONS

As argued in Sect. 2.3, the light curve at t >∼ 0.1 d is fairlyindependent of the choice of the initial conditions – namelythe initialization time, t0, as well as the initial distributionsof radiation energy and gas temperature – as long as t0 ischosen sufficiently early for the system to relax well beforet ∼ 0.1 d into the correct state of balance between radioac-tive heating and adiabatic expansion. In this Appendix weback this argument by showing in Fig. B1 light curves forvariations of the fiducial model, DD2-135135, using differ-ent initial conditions. All models in the main part of thisstudy are initialized at t0 = 100 s with a constant tempera-ture of T0 = 100K. The excellent agreement of the fiducialmodel with one model using instead t0 = 1 s and with an-other model using T0 = 10000K supports the robustness ofour results with respect to the initial conditions. Figure B1

1037

1038

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]

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T0 = 100 K, t0 = 100 sT0 = 100 K, t0 = 1 sT0 = 100 K, t0 = 1000 sT0 = 10000 K, t0 = 100 s

10 1 100 101

time [d]

14.013.513.012.512.011.5

abso

lute

AB

mag

nitu

de

broadband magnitudesgzH

Figure B1. Comparison of bolometric luminosities (top panel)and broadband magnitudes (bottom panel) for kilonova calcula-tions of model DD2-135135 initialized with different gas temper-atues, T0, and at different times, t0. Solid lines correspond to thestandard case used for all models in the main part of this study.The lines corresponding to t0 = 100 s and T0 = 10000K lie on topof the solid lines. Small differences at early times are only visiblefor the model initialized at t0 = 1000 s.

also reveals that a later initialization time of t0 = 1000 swould have led to a slight underestimation of the luminosi-ties, though only at early times t <∼ 0.5d.

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22 O. Just et al.

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