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Kilonova Light CurvesCatie Ball

I. INTRODUCTION

LIGO’s recent observations of NS-NS and BH-BH mergershave ushered in an era of decreasing ambiguity when studyingcompact object mergers. Though the gravitational signaturesof these mergers are valuable objects for study in their ownright, the characteristics of the electromagnetic counterpartsof these events are still largely an open question in obser-vational astronomy. Kilonovae are proposed electromagneticcounterparts to NS-NS mergers that could illuminate thecreation of the heaviest elements in the universe. Kilonovaeare thermal transients observable in visible and infrared bands,powered by the radioactive decay of r-process elements andtheorized to peak on timescales between days and weeks long.The wavelength- and time-dependence of kilonovae are fieldsof active study today, as understanding how kilonovae areproduced and radiated is closely tied to open questions suchas the equation of state of neutron stars and sites of productionfor r-process elements.

II. EJECTION MECHANISMS

Hydrodynamical simulations suggest that there are multiplemethods for loss of mass in a neutron star merger event.In one case, material can be ejected dynamically during ormilliseconds after the merger, whether through tidal ejectionor through shock heating at the interface of the collision. Itcan also be ejected on timescales > 1s through outflow winds,whether caused by advection or neutrino-driven winds. Therelative importance of these types of ejecta, and the full natureof r-process elements that are created in each regime, are areasunder current investigation.

Initially proposed as an effect of black-hole-neutron starcollisions [13], tidal ejecta of compact object collisions pro-duce dense, neutron-rich material, making them a possible sitefor the creation of r-process elements. Creating simulationsthat collide both symmetrical mass neutron stars of 1.35M�and neutron stars with slightly different masses, Goriely etal. [7] find that the decompressed material tidally ejectedfrom the crust and outer regions of the neutron stars im-mediately following their collision is a site for creation ofheavy A > 140 r-process elements, and can replicate thesolar abundance of such elements with a NS-NS merger rateof ∼ (3 − 5) × 10−5M�, which is possible given currentobservational constraints [7].

However, this mechanism does not produce the expectedabundances for A < 140 [7]. Light r-process elements are alsotheorized to be created in material ejected from the merger onlonger timescales, after which neutron irradiation can createa higher electron fraction Ye in outflowing material [11]. Inthis case, the ejecta is expected to come from outflows fromtorus-like accretion disks [17]. These outflows are driven by

both neutrinos and viscous heating and, in hydrodynamicalsimulations, have been found to account for the production ofthe missing A < 140 r-process elements that are not createdin the tidal ejecta discussed above [8].

Other studies have suggested that light r-process elementscan also be produced in the dynamical ejecta released in theearly stages of the kilonova, and that heavy r-process elementscan be produced in the accretion torus, depending on theconditions of the remaining remnant object [1]. In these cases,light r-process material can be created in the shock heatingof material dynamically ejected from the system - weakinteractions will increase the material’s Ye, and material witha low lanthanide fraction (comprised preferentially of light r-process elements) can be created and ejected on timescalessimultaneous with the merger [3]. Beyond this, if the twoneutron stars coalesce quickly into a black hole remnant,there is not as much time for neutrino irradiation of theaccretion torus, meaning that this can also be a site of heavyr-process element production [11]. In these cases, we see thatneutron star mergers have potential to produce a variety ofcompositions of r-process elements - constraining the fractionof r-process elements produced in these events will requireobservational constraints on the composition of merger ejecta.Hydrodynamical simulations suggest a dependence on theequation of state of neutron stars and the composition of theirr-process ejecta ([3],[7]), so having observational constraintson the amount of r-process material produced in different massbins, and through different processes, can help us understandthe equation of state of neutron stars. One way to differentiatebetween the two types of ejecta is to consider their velocities- intuitively, material that is ejected on dynamical timescalesis unbound by the collision, and will have higher velocities(∼ .2 − .3c) than material that is ejected through outflows(with typical velocities ≤ 0.1c [11]).

III. RADIOACTIVE DECAY AND THERMALIZATION

Radioactive decay of unstable r-process elements powers thelight curves of kilonovae. Because of this, understanding theintrinsic signature of these events necessitates that astronomersconsider the decay of these elements. Metzger et al. [14] findthat a material composed of a broad variety of radioactivelydecaying elements will generate energy according to a powerlaw: E ∝ t−7/5.

Though radioactive decay is what powers kilonova lightcurves, we observe an electromagnetic signature when thedecay products of this radioactive material thermalize, ortransfer energy to the ejecta material such that it can radiatethermal emission. However, this process is not completelylossless - not all of the energy released in decay is reprocessedas thermal emission, which necessitates characterization of

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Fig. 1. Kasen & Barnes [12] analytic illustration of the relationship betweenradioactive power, heating, and the full luminosity of the light curve withsome simple diffusion timescale estimate.

the efficiency of this process given the current constraintson kilonova ejection material and its radioactive constituents.Figure 1 illustrates the different time dependences of lumi-nosity, input radioactive power, and radioactive heating. Manypapers characterize radioactive heating using thermalizationefficiency, which describes the relationship between the ra-dioactive heating and the power released in radioactive decay:

f(t) =Ethermal

Edecay(1)

Understanding compositions helps constrain the relativeimportance of different decay channels. As alpha particles,beta particles, fission, and neutrinos will all interact differentlywith the ejecta material, the relative abundance of each typeof particle - which is determined by the composition of theejecta and the preferential decay channels for these materials-helps to set the thermalization efficiency [2]. For instance,one way that the efficiency of thermalization processes isimmediately and intrinsically reduced is in the production ofneutrinos in beta decays. In this case, neutrinos are extremelyunlikely to interact with the intervening material of the ISM.As these neutrinos free stream out of the merger, there is animmediate loss of 35% of the decay energy that cannot gointo thermalization [2].

Barnes et al. [2] outlines different methods for thermaliza-tion for each type of particles:

• γ rays lose energy through compton scattering, photoion-ization

• β particles lose energy through coulomb interactions withthermal electrons and collisional excitation. relativisticβ particles may also lose energy to Bremsstrahlung andsynchrotron emission.

• α particles lose energy in scattering off electrons, bothfree and bound

In analyzing the way each type of suprathermal decay productcomes into thermal equilibrium with the material, Barnes etal.[2] characterize the timescales for decay products to come

Fig. 2. (top panel) Barnes et al. [2] model of thermalization efficiencydecomposed by decay channel. This depends on both the amount of energygenerated in each decay channel over time (bottom panel), but also theprocesses through which these decay products interact with the ejecta material.

into thermal equilibrium with the ejecta. This allows themto calculate the relative efficiency of energy in each type ofdecay product in a time-evolving sense, giving a completepicture of the heat transfer as a function of time and decayproducts in figure 2. Similar decay channel-dependent analysesof energy loss over time are leveraged in Kasen & Barnes[12], wherein the thermalization efficiency is calculated byconsidering the time evolution of energy input and output tothe ejecta material. In considering the thermalization efficiencyof electrons, Kasen et al. [12] find a semi-analytic solution forthe thermalization efficiency of beta decay:

fβ(τ) = pe(1 +t

te)−n + pγ(1− e−t

2γ/t

2

) (2)

where pe, pγ are the fractional amounts of the decay energyreleased as e− and γ rays. Such a model illustrates thecomplicated time dependence of ejecta thermal energy, whichdepends not only on the relative amount of energy releasedthrough each type of decay, but evolves according to differenttimescales when considering each decay product (te, tγ here).

Beyond the energy input from thermalization, the ejecta alsoloses energy to scattering and to adiabatic expansion of thematerial - though we expect kilonova ejecta to have someinitial velocity imparted by its ejection from the merger, mostmodern treatments assume homologous expansion of ejectamaterial, meaning that there may be some energy lost to inexpansion. Fernandez & Metzger [6] sketch the full evolutionof ejecta thermal energy:

dE

dt= Eheating −

E

tdiff− P dV

dt(3)

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where the first term is the input energy from radioactive decayf(t)Edecay , the second term is energy lost to scattering, andthe third term accounts for adiabatic expansion of the ejectamaterial.

IV. R-PROCESS MATERIAL OPACITY

Once there is thermalization of these decay products, thisthermal radiation needs to escape through the material of theejecta. This is complicated by the fact that the ejecta is denseand full of heavy elements, making it a non-trivial radiativetransfer problem to understand how this light diffuses fromthe merger to the outside observer.

One way to characterize these kilonova is to estimate thetimescales where their emission peaks. Though the energyinput through radioactive decay decreases as Q ∝ t−α

[14], the fact that ejecta material has some opacity meanswe expect a finite diffusion timescale through the material.Including a geometric factor B, which accounts for the factthat ejecta material may not be isotropic [14], our materialhas a diffusion timescale td =

BκMej

cR for some radius R.Since our ejecta is expanding, our photons will only diffusethrough the expanding ejecta on timescales where td ≥ texp,where texp = R/v for homologous expansion, a standardassumption in contemporary kilonova models. Given this andthe constantly decreasing rate of power input into the kilonova,we expect the peak to occur when this condition is just satisfied- when R

v =BκMej

cR . This corresponds to

tpeak ≈

√BκMej

cvej(4)

We can use this timescale to estimate the effective temper-ature of our thermal emission, assuming some luminosity thatdepends on the thermal enery of the ejecta:

Lp ≈ f(t)Edecay (5)

using the Stefan-Boltzmann law (L = 4πr2σT 4) at the peakradius gives:

T ≈ (f(tp)Edecayσv2ejt

2p

)1/4 = (cf(tp)EdecayσvejBκMej

)1/4 (6)

[10]. We see, directly, how opacity affects the temperature ofthe light curve - higher opacities mean later kilonova peaksand lower - redder - temperatures.

As the ejected material expands away from the merger, thelight emitted in radioactive decay is constantly being redshiftedwith respect to the material it is moving through. To accountfor this, models of the opacity of this material follow theprescription described in Karp et al. [9], wherein the constantredshift of light in the ejecta rest frame allows for an increasedprobability of absorption of these photons. This effect is givenaccording to:

dν

dt= −ν

c(dv

dt) ≈ −ν

t(7)

[9], which gives a relation between the travel time through anexpanding medium and the final frequency of the photon:

νf = νie−(δx)/ct (8)

Fig. 3. Karp et al.[9] illustration of effective expansion opacity - comparisonbetween a line of finite width in a static medium (top panel) and an infinitelythin line in a medium with some expansion velocity (bottom panel)

- the longer a photon travels through an expanding medium,the greater the range of apparent frequencies of the photon inthe ejecta rest frame. The effect of this expansion on a singleline is illustrated in figure 3.This formalism allows for directcalculation of the effective expansion opacity of an r-processmaterial, given some composition and some knowledge of theavailable spectral transitions-

αexp =1

ct

∑z

Nz∑i

λi∆λ

(1− exp−τi(ρz)) (9)

[12], which poses the expansion opacity as a sum over zatomic species with i transitions, each with a distinct wave-length and optical depth (λi, τi). Understanding the energiesat which atoms ionize, or at which spectral transitions canbe excited, in ejecta material is a nontrivial consideration inthese calculations, as high Z elements have many possibletransitions. Early work on kilonova light curves made as-sumptions for the nature of these light curves based on therelatively well-studied opacity and spectral line informationof iron. However, detailed modeling of the atomic structure ofr-process elements suggests that, especially for heavy r-processelements like lanthanides, the opacity of these materials canbe orders of magnitude more extreme than these early models[10].

Detailed contemporary treatments of r-process material ex-pansion opacities compute temperature-dependent effectiveopacity spectra. The available transitions for different r-process materials do not necessarily have the same spacing inwavelength-space. Beyond this, the temperature of the ejectamaterial sets the ionization state of the r-process species,meaning that there is some dependence of mean expansionopacity on temperature [10]. In this, the degeneracy of anumber of different variables means spectroscopic detectionof the exact atomic constituents of kilonova is beyond thescope of current study. Instead, current observational analysesapply broad characteristics of material opacities that come

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Fig. 4. Tanaka et al. [16] model of the opacity of possible kilonova ejectacomponents. The orange component corresponds to material with the expectedcomposition of dynamical ejecta, and the blue and green lines are expectedto be representative of abundances more common in post-merger outflows.

Fig. 5. Further results from Tanaka et al.[16] with element-specific opacities,showing that the greatest contributors to the increased opacity of the orangeline in figure 4 are elements like those in the right panel of this image -lanthanides.

from detailed studies like Kasen et al. [10] and Tanaka etal. [16].

Tanaka et al. [16] use atomic structure codes to calculateand confirm the available spectral transitions for a number ofdifferent r-process elements with different open orbitals andmass numbers. These simulations account for mostly neutral,singly-, and doubly-ionized species of r-process elements witha variety of mass numbers, finding numbers of availabletransitions for these species ranging from a few hundred (419lines for Te III) to tens of millions (70,366,259 for Nd I). Theeffective opacities of these materials are complicated, but tendto be distinctly higher at all wavelengths for lanthanides withopen f-shell orbitals [16].

The correlation between higher opacity and compositionallows for a direct relation between observable quantities andthe composition of ejecta material - as discussed above, tpeakscales with opacity, so material that has a higher opacity -lanthanides, most distinctly, will result in a later, redder peakto a kilonova light curve. This is illustrated in figure 4, wheresample ejecta opacities are calculated and are distinguishedbased on the ejection scenario and initial Ye values. In this,we see that material with some low electron fraction materialhas significantly higher opacity, despite the possibility of somehigh electron fraction material in dynamical ejecta.

Fig. 6. Kasen et al. [11]’s two-component model for the observed GW170817light curve.

V. OBSERVATIONAL EVIDENCE

The era of multi-messenger, time-domain astronomy is aparticularly fruitful time for the study of kilonovae, as thecomplementary study of NS-NS mergers through gravitationalwaves and electromagnetic counterparts help us better under-stand both phenomena.

Prior to the LIGO observation of GW170817, IR luminosityexcesses coincident with GRBs were used as the first fewattempts to characterize kilonovae light curves. Berger et al.[4] observed a slight excess in IR luminosity in the afterglowof GRB130603B that faded on timescales more similar withthose of kilonovae than GRB afterglows. Though it is theorizedthat, because of the offset of GRBs from host galaxies, thesimilarity of the predicted GRB occurrence rates and theoccurrence rates of compact binary mergers, short GRBs aretheorized to be observational signatures of compact objectmergers [4]. However, as the source of short-hard GRBsis still unconfirmed, these results are not as unambiguousas the kilonova observed coincident with gravitational wavedetections of NS-NS merger GW170817.

GW170817s kilonova light curve is characterized by arapid decrease in blue light coincident with a longer-timescaleincrease in red and near-infrared light [11]. Such a timeevolution is fit by models that contain two distinct types ofejecta, a quickly-fading blue light curve primarily composed oflight r-process elements, and a redder one comprised of heavyr-process material [11]. As discussed in Section III, lanthanide-rich heavy r-process material is much more opaque to photonsthan light r-process materials, making the diffusion timescalesin these materials longer. These models are consistent withdiffering electron fractions and masses for the distinct ejecta,

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and include a component with low Ye ejected by wind anda dynamically ejected component [15]. Cowperthwaite etal. [5]’s models of the kilonova, which includes one twocomponent model and one three component model that arealmost indistinguishable, recover values for the ejecta massand velocity for the multiple components:

TABLE IEJECTA COMPONENTS

Component Mass (M�) Velocity (c)2 Component, Blue 0.014+0.002

−0.001 0.266+0.007−0.002

2 Component, Red 0.036+0.001−0.002 0.123+0.012

−0.014

3 Component, Blue 0.014+0.002−0.001 0.267+0.006

−0.011

3 Component, Purple 0.034+0.002−0.002 0.110+0.011

−0.010

3 Component, Red 0.010+0.002−0.001 0.160+0.030

−0.025

The colors denoted in this model signify different opacitiesfor the models - in the three-component models, fixed opacitiesof 0.5,3.0, and 10.0 cm2g−1 are adopted for the blue, purpleand red components. In the two-component model, the bluecomponent has an opacity set to 0.5 cm2g−1, where theopacity of the red component is left as a free parameter andfound to be 3.349+0.364

−0.337 cm2g−1. As these best-fit modelsfind a high-velocity blue component and lower velocity redcomponent, this suggests little neutrino irradiation of the torus,and a short lifetime for any remnant hypermassive neutronstar [5]. As a single data point, and due to the complicatedrelationship between ejecta composition and recieved spectra,GW170817 is not a totalizing constraint on the sites of r-process element production in the universe. However, the factthat this first observation suggests a diversity of r-processmaterials can be produced in these mergers lends credenceto the theory that NS-NS mergers are a major site of r-process element production [11], as both must occur in orderto produce solar r-process element abundances [8].

With unambiguous detections of NS-NS mergers fromLIGO and with the increased number of time-domain surveyslike LSST, the next decade will be particularly fruitful for thestudy of kilonovae and the characterization of their properties.As mentioned in Section I, the nature of material ejectedin NS-NS mergers is sensitive to a variety of parametersincluding the final state of the merging objects, the equationof state of neutron stars, and the mass ratio of the mergingobjects. Studying these events further will serve to illuminatethe nature of these compact objects, as well as their importancein creating heavy elements.

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