Physics of Luminous Transient Light Curves: A New Relation Between Peak Time and Luminosity
DD RAFT VERSION O CTOBER
2, 2019Typeset using L A TEX twocolumn style in AASTeX62
Physics of Luminous Transient Light Curves: A New Relation Between Peak Time and Luminosity D AVID
K. K
HATAMI AND D ANIEL
N. K
ASEN
1, 21
Department of Astronomy, University of California, Berkeley, CA, 94720 Nuclear Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720
ABSTRACTSimplified analytic methods are frequently used to model the light curves of supernovae and other energetictransients and to extract physical quantities, such as the ejecta mass and amount of radioactive heating. Theapplicability and quantitative accuracy of these models, however, have not been clearly delineated. Here wecarry out a systematic study comparing certain analytic models to numerical radiation transport calculations. Weshow that the neglect of time-dependent diffusion limits the accuracy of common Arnett-like analytic models,and that the widely-applied Arnett’s rule for inferring radioactive mass does not hold in general, with an errorthat increases for models with longer diffusion times or more centralized heating. We present new analyticrelations that accurately relate the peak time and luminosity of an observed light curve to the physical ejecta andheating parameters. We further show that recombination and the spatial distribution of heating modify the peakof the light curve and that these effects can be accounted for by varying a single dimensionless parameter in thenew relations. The results presented should be useful for estimating the physical properties of a wide variety oftransient phenomena.
Keywords: radiative transfer – supernovae: general INTRODUCTIONWide-field surveys are gathering data on an increasingnumber of common supernovae (SNe) and related transientssuch as tidal disruption events, fast-evolving luminous tran-sients, superluminous supernovae, and kilonovae. The gen-eral physics controlling the light curves of these events issimilar – energy deposited either by a propagating shock(e.g. Type II-P SNe) or a heating source (e.g. radioactivityor a central engine) radiatively diffuses through the opticallythick and expanding ejecta, undergoing adiabatic loses untilthe radiation reaches the surface and escapes. Analysis of theobserved light curves can provide information on the ejectaproperties and the nature of the powering source.With the increasing number of observed transients, thereis increased need for fast, empirical techniques to infer theirphysical properties and discriminate between competing the-oretical explanations. Simplified analytic models are com-monly used to analyze observations and make theoretical pre-dictions (e.g. Li & Paczy´nski (1998); Chatzopoulos et al.(2012); Inserra et al. (2013); Villar et al. (2017); Nichollet al. (2017); Guillochon et al. (2018)). Most common amongthese are the Arnett models (Arnett 1980, 1982), in whichbolometric light curves are calculated through a simple nu-merical integral. These models also provide several “rulesof thumb” for estimating physical properties from the lightcurve brightness and duration. In particular, “Arnett’s rule” states that the instantaneous heating rate at peak is equal tothe peak luminosity (Arnett 1982). For Type I supernovae,this rule in principle allows one to extract the mass of ra-dioactive Ni (Stritzinger et al. 2006; Valenti et al. 2007;Drout et al. 2011; Scalzo et al. 2014; Prentice et al. 2016).Despite the frequent application of these analytic modelsand rules, a systematic study of their accuracy and applica-bility has not been carried out. Previous numerical modelsof Type Ia SNe have noted that Arnett’s rule is usually ac-curate to ∼
20% (Blondin et al. 2013; Hoeflich et al. 2017;Khokhlov et al. 1993). For Type Ib/c SNe Arnett’s rule is typ-ically off by ∼
50% (Dessart et al. 2015, 2016), and for TypeII SNe like SN1987A the error is a factor of ∼ a r X i v : . [ a s t r o - ph . H E ] S e p K HATAMI AND K ASEN
We derive a new relation between the peak time and peakluminosity of transient light curves which accurately capturesthe results of numerical models. We study how the relationdepends on the spatial distribution of heating as well as theeffects of a non-constant opacity due to recombination. Thenew relation is parameterized by a dimensionless constant β ,and works for a variety of assumed heating sources and ejectacharacteristics.In Section 2, we describe the assumptions and limitationsof the Arnett light curve models and Arnett’s rule. In Sec-tion 3, we derive the new peak-time luminosity relation andcompare it to numerical simulations. In Section 4, we in-vestigate the relation between the peak and diffusion time.In Section 5, we examine the effects of concentration of theheating source. In Section 6, we look into the effects of anon-constant opacity due to recombination on the light curveand the new relation. Finally, in Section 7, we apply the re-sults and relation to radioactive Ni-powered transients. InAppendix A, we provide a table of the new relation for a va-riety of luminous transients. LIMITATIONS OF ARNETT-LIKE MODELSThe analytic light curve modeling approach of Arnett(1980, 1982) (hereafter A80,A82) is widely used to analyzeluminous transients. A closely related “one-zone” model-ing approach (Arnett 1979; Kasen & Bildsten 2010; Villaret al. 2017) differs in its mathematical details but results in asimilar expression for calculating the light curve.2.1.
Assumptions and Light Curve Solution
The Arnett-like models begin with the first law of thermo-dynamics ˙ E = − P ˙ V + ε − ∂ L ∂ m (1)where E is the specific (i.e. per unit mass) energy density, P the pressure, V = 1 /ρ is the specific volume, ε is the specificheating rate, and L the emergent luminosity. Several simpli-fying assumptions are then made: (1) the ejecta is expandinghomologously and so the radius evolves as R ej ( t ) = v ej t (2)where v ej is the maximum ejecta velocity; (2) Radiation pres-sure dominates over the gas pressure and so we can expressthe specific energy density as E = 3 P /ρ = aT /ρ (3)where T is the temperature; (3) The luminosity is describedby the spherical diffusion equation L ( r ) = − π r c κρ ∂ e ∂ r (4) where e = ρ E is the energy density (per unit volume) and κ the opacity; and (4) the ejecta is characterized by a constantopacity.The Arnett models make an additional consequential, butoften overlooked, assumption: (5) The energy density profileis self-similar, i.e., the spatial dependence is fixed and onlythe overall normalization changes with time e ( x , t ) = E int ( t ) V ( t ) ψ ( x ) (5)where E int is the total internal energy of the ejecta and x = r / R ej ( t ) is the (comoving) dimensionless coordinate. The di-mensionless function ψ ( x ) describes the spatial dependenceof the radiation energy density, which by assumption doesnot change with time. Substituting Eq. 5 into the diffusionequation (Eq. 4) gives the emergent luminosity at r = R ej L = tE int ( t ) τ d (6)where τ d = (cid:20) π κ M ej v ej c ξ (cid:21) / (7)is the characteristic diffusion time through the ejecta withmass M ej . The quantity ξ = d ψ/ dx | x =1 specifies the energydensity gradient at the ejecta surface and is a constant whenself-similarity is assumed. The one-zone models make the ansatz ξ = 1. Arnett (1982) uses a more sophisticated sepa-ration of variables method to derive a self-consistent solutionfor e ( x , t ). This requires making a final assumption: (6) Thespatial distribution of the heating is proportional to the en-ergy density. Eq. 1 can then be solved to find e ( x , t ) = E int ( t ) V ( t ) (cid:20) π π x ) x (cid:21) (8)which gives ξ = π / dE int ( t ) dt = − E int ( t ) t + L heat ( t ) − L ( t ) (9)where L heat ( t ) is the total input heating rate. Using Eq. 6 toreplace E int = L τ d / t and rearranging gives τ d t dLdt = L heat ( t ) − L ( t ) (10)Arnett’s rule follows, since the condition of an extremum dL / dt = 0 implies L = L heat . UMINOUS T RANSIENT L IGHT C URVES AT P EAK E int ( t ) and hence the emergent luminosity L ( t ) = 2 τ d e − t / τ d (cid:90) t t (cid:48) L heat ( t (cid:48) ) e t (cid:48) / τ d d t (cid:48) (11)Both the one-zone and the separation of variables approachesresult in the same expression for the light curve; the only dif-ference is the value of ξ , which reflects different assumptionsabout the shape of the self-similar energy density profiles.The diffusion time for the one-zone models is a factor of π/ √ ≈ t d = (cid:115) κ M ej v ej c (12)and so τ d = [3 / πξ ] / t d . Physically, the characteris-tic diffusion timescale gives the time at which the ex-pansion timescale t exp = R ej / v ej equals the diffusion time t diff = κρ R / c . The peak time scales with the diffusiontimescale t peak ∝ t d , but the numerical coefficient relatingthem depends on the distribution of heating, nature of theopacity, and other effects.The self-similarity assumption will be shown below tolimit the accuracy of the Arnett models. However, this is nota necessary assumption as Eq.8 is only the first eigenfunc-tion of the separated spatial equation. The full solution of theenergy density can be expressed as an infinite sum of higherorder eigenfunctions whose normalization will be set by thespatial distribution of heating and boundary conditions. Pinto& Eastman (2000) show how such an approach can be used torelax the assumptions (5) and (6) and produce more accuratelight curves. However, due to the more complicated nature ofthe solution, the full solution with higher-order eigenmodesis rarely used in practice.2.2. Comparison to Numerical Simulations
To assess the accuracy of the Arnett solutions, we comparethem to numerical monte-carlo radiation transport calcula-tions run with Sedona (Kasen et al. 2006). We adopt similarassumptions as A82: homologous expansion, uniform den-sity, and a constant opacity. Non-constant opacity will beconsidered in Section 6. In this section, the ejecta has a dif-fusion timescale t d = 100 days and the heating source is atthe center and follows L heat ( t ) = L e − t / t s , where the timescale t s = 10 days.Fig. (1) compares the numerical light curve to the Ar-nett analytic solution. The numerical models have an ini-tial “dark period” until t ∼ . t d , before which the photonshave not had sufficient time to diffuse from the center of theejecta (Piro & Nakar 2013). In contrast, the analytic solu-tions predict a steeper rise beginning at t = 0, a consequence t / t d L / L Monte-Carlo RTArnettOne-zoneInput
Figure 1.
Light curves from the Arnett solution Eq.(11) with differ-ent choice in the diffusion timescale factor ξ (red and blue lines),compared with a numerical monte-carlo radiation transport solu-tion using Sedona (teal line with points). The input heating (dashedblack line) consists of a centrally-located exponential source withluminosity L in ( t ) = L exp (cid:2) − t / t s (cid:3) with a timescale t s = 10 days and acharacteristic diffusion timescale t d = 100 days. of the assumption that radiation energy is immediately dis-tributed throughout the ejecta. The A82 solution predicts apeak time a factor of 2 shorter than the numerical result, butgives roughly the correct peak luminosity. The peak time ofthe one-zone model is closer to the numerical simulation, butunder-predicts the peak luminosity and is overall too broad.There is no choice of ξ such that the analytic solution closelymatches the numerical light curve.This inaccuracy of the analytic models is more pronouncedfor more centrally concentrated heating sources. Fig. (2)shows numerical models where the heating source has beenuniformly mixed out to dimensionless radius x s , with x s = 0corresponding to a central source. The Arnett analytic so-lution most closely resembles a well-mixed numerical modelwith x s ≈ .
8. We can define a heating-weighted radius wherethe bulk of heating occurs as (cid:104) x s (cid:105) = (cid:32) (cid:82) x ˙ e heat ( x ) dx (cid:82) ˙ e heat ( x ) dx (cid:33) / (13)where ˙ e heat ( x ) is the energy density heating rate at x . Forconstant heating out to radius x s , we have the relation (cid:104) x s (cid:105) = x s / √
3. In the Arnett solution, ˙ e heat ( x ) ∝ e ( x ) and using Eq.(8)we find that (cid:104) x s (cid:105) ≈ .
4, which indeed corresponds to x s ≈ . HATAMI AND K ASEN
Time (Days) B o l o m e t r i c L u m i n o s i t y ( e r g s ) x s = 0.90.80.70.5 0.30.0 Exponential Source t s = 10d Figure 2.
Effects of varying concentration, in terms of the dimen-sionless radius x s . The heating is uniformly mixed out to x s . Thesolution of A82 is shown for comparison (red line), as well as theinput heating rate (dashed black line), here an exponential sourcewith timescale t s = 10 days. Arnett’s rule, which predicts that theinput heating rate should intersect the observed light curve exactlyat peak holds only for well mixed sources, x s ≈ . timescale ∼ t d . In Fig. (3), we show the evolution of the en-ergy density profile defined in Eq.(8) for a central exponentialheating source. At early times, a diffusion wave propagatesoutwards, and the self-similar assumption fails. By neglect-ing this diffusion wave, the Arnett-like models overestimatethe luminosity at early times. For a more uniformly mixedsource, self similarity is established earlier, and so the Arnettmodels are more applicable.Thus, it is a common misconception that the Arnett modelsassume a centrally-located heating source – while the energydensity increases towards the center, the heating luminosity peaks close to the surface, producing the faster rise and ear-lier peak compared to the central-heating numerical solution.This likely explains why the A82 solution more closelypredicts Type Ia rather than core collapse SN light curves– Type Ia SN typically have a much larger degree of mixing(Blondin et al. 2013) while core collapse SNe have more cen-trally concentrated Ni. We explore the effects of the spatialdistribution of the heating source in more detail in Section 5.2.3.
Arnett’s Rule
A specific prediction of the Arnett models is that the peakluminosity is equal to the heating rate at peak, i.e. L peak = L heat ( t peak ) (14)This is commonly referred to as Arnett’s rule (or Arnett’slaw) and is widely used to infer e.g. the nickel mass in ra-dioactive SNe (Stritzinger et al. 2006; Valenti et al. 2007). x ( x ) t = 0.1 t p Figure 3.
Evolution of the energy density profile Eq. (5) for acentral exponential heating source. Shown are profiles at differenttimes relative to peak, t p . The self-similar assumption breaks downat early times as the diffusion wave propagates outward. At times t > t p , the profile settles into a self-similar shape. Also shown forcomparison is the solution of Arnett in Eq.(8) (dashed red line). We see from Fig.(2) that Arnett’s rule does not hold ingeneral and that its accuracy depends on the heating sourceconcentration. For centrally concentrated sources ( x s (cid:46) . under-estimate of the true peak lumi-nosity, with the error being systematically worse for morecentralized heating. For nearly fully mixed heating sources( x s (cid:38) .
8) Arnett’s law is an over-estimate.The failure of Arnett’s rule again stems from the assump-tion of self-similarity, which implies a proportionality be-tween the luminosity and the total ejecta internal energy, L ∝ Et . Under this assumption, the light curve has to peaksimultaneously with E ( t ) – i.e., at the time when the rate ofenergy loss, L ( t ) equals the rate of energy gain, L heat ( t ). Inreality, the time-dependent propagation of a diffusion wavemeans that the luminosity does not strictly track the inter-nal energy. For central sources, L ( t ) generally lags E ( t ) andthe light curve peaks at a time when L > L heat . For fullymixed cases, L ( t ) leads E ( t ) and the light curve peaks when L < L heat . Arnett’s law holds only for the case of a specificconcentration ( x s ≈ .
8) for which the light curve coinciden-tally peaks at the same time as does the internal energy. A NEW RELATION BETWEEN PEAK TIME ANDLUMINOSITYGiven the limits of Arnett’s rule, we look for a more robustrelationship between the peak time and peak luminosity of atransient light curve. We proceed by considering the evolu-
UMINOUS T RANSIENT L IGHT C URVES AT P EAK E , and rewrite Eq. 9 as d ( tE ) dt = t [ L heat ( t ) − L ( t )] (15)which integrates to tE ( t ) = (cid:90) t t (cid:48) L heat ( t (cid:48) ) dt (cid:48) − (cid:90) t t (cid:48) L ( t (cid:48) ) dt (cid:48) (16)Eq. (16) is similar to the analysis presented in (Katz et al.2013), which considers times t (cid:29) t peak when E ( t ) = 0. Herewe instead consider times around peak t ∼ t peak . Furthermore,we assume the initial energy content in the ejecta is zero andignore the initial stellar radius in our assumption of homol-ogy. Thus, the analysis presented here does not necessarilyapply to Type IIP/L SNe, whose light curves are dominatedby the initial shock-deposited energy.We rewrite Eq. (16) as t L peak = (cid:90) t t (cid:48) L heat ( t (cid:48) ) dt (cid:48) + (cid:15) ( t ) (17)where (cid:15) ( t ) = (cid:20) t L peak − (cid:90) t t (cid:48) L ( t (cid:48) ) dt (cid:48) (cid:21) − tE ( t ) (18)The first term in brackets can be shown to be positive (since L ( t ) ≤ L peak ) and monotonically increasing (see Appendix B).The second term tE ( t ) is also positive and is a decreasingfunction when L > L heat , which is typically obtained for t (cid:38) t peak . We therefore anticipate there may be a time when thetwo functions cross and cancel to give (cid:15) ( t ) = 0.We express this time as t = β t peak and rearrange Eq.(17) toget L peak = 2 β t (cid:90) β t peak t (cid:48) L heat ( t (cid:48) ) dt (cid:48) (19)which is our desired expression for L peak . In Appendix Bwe show that for common heating functions there is indeed atime when (cid:15) ( t ) = 0 for a value of β ∼ L heat ( t ) = L e − t / t s (20)the peak time-luminosity relation can be evaluated to get L peak = 2 L t s β t (cid:104) − (1 + β t peak / t s ) e − β t peak / t s (cid:105) (21)This can be contrasted with Arnett’s rule, which predicts L peak = L e − t peak / t s . The two expressions make similar predic-tions when t peak (cid:28) t s but increasingly diverges for t peak (cid:29) t s . t peak (Days) L p e a k / L Eq. (21)Arnetts RuleSedona
Figure 4.
Relation between peak time and peak luminosity for acentral exponential source with timescale t s = 10 days. Numericalradiation transport simulations with various peak times are shown(circles) compared to Arnett’s rule (black dashed line) and the newrelation Eq.(19) with β = 4 / In Fig. (4) we compare our expression for L peak to those ofnumerical light curve calculations for a central exponentialheating source with t s = 10 days. The numerical models spana wide range of ejecta masses, velocities, and opacities, andhence result in a range of peak times. Eq. 21 with β = 4 / L peak and be-comes progressively worse for larger values of t peak / t s .The peak time-luminosity relation Eq. 19 with β = 4 / L heat ( t ) = L (1 + t / t s ) (22)which can also be analytically evaluated (see Appendix A).Fig. 5 shows that the Eq. 19 with β = 4 / β does change if the source heating is spa-tially mixed or the opacity is non-constant due to recombi-nation, but that β remains largely independent of the heatingfunction, source timescale t s , or ejecta diffusion timescale t d . RELATION BETWEEN PEAK TIME ANDDIFFUSION TIMEAnalyses of observed light curves often attempt to con-strain the ejecta mass and velocity by setting the observed K
HATAMI AND K ASEN t peak (Days) L p e a k / L t s = d d d d Exp. Central SourceConstant Opacity0 20 40 60 80 100 120 t peak (Days) L p e a k / L t s = d d d d n = 2 Central SourceConstant Opacity Figure 5.
Same as Fig.(4), but for different values of t s , and for anexponential (top) and magnetar (bottom) source. Numerical simula-tions are shown as points. Lines of constant diffusion timescales t d are indicated (dotted black lines). Eq.(19) with β = 4 / t s . time of peak, t peak , equal to the diffusion timescale τ d (e.g.,Drout et al. 2011; Prentice et al. 2018). Here we study thatrelation for constant opacity models, and show that t peak de-pends not only on t d , but also on the heating timescale t s .In Fig.(6), we show the dependence of t peak on the ratio t s / t d , for a large number of numerical models with uniformdensity ejecta and two different central heating sources. Themodels have a range of masses, velocities, and constant opac-ities, although only the combination t d is relevant for the lightcurve behavior. For t s / t d (cid:28)
1, the peak time asymptotes to t peak ≈ . t d independent of t s . In this limit, the source canthus be approximated as an instantaneous “pulse” of energydeposited at t s . The energy from such a pulse diffuses out and t s / t d t p e a k / t d Eq. (23)ArnettExponentialMagnetar
Figure 6.
Relation between the source timescale t s and the peaktime t peak , relative to the diffusion timescale t d . Shown are an expo-nential (teal circles) and magnetar (orange squares) central heatingsource. The best-fit Eq.(19) is also shown (black dashed line). Forcomparison, the Arnett t peak = t d relation is shown. peaks at around ∼ . t d . In comparison, the Arnett modelspredict t peak ≈ . t d (see Fig. (1)).For t s / t d (cid:38) .
1, the continuing source deposition beginsto lengthen the peak time. The dependence is fairly weak – t peak only increases by a factor of ∼ t s changes over threeorders of magnitude, implying that for the sources consideredthe light curve peak is mostly powered by heating depositedat early times.An equation that captures the peak time of numerical mod-els with constant opacity and central heating is t peak t d = 0 .
11 ln (cid:18) + t s t d (cid:19) + .
36 (23)In the limit that t s (cid:28) t d , Eq.(23) goes to t peak ≈ . t d , while for t s (cid:29) t d it grows logarithmically with t s . The relation is rela-tively insensitive to the functional form of the heating source(e.g. exponential vs. power-law) as long as the function issmoothly and gradually declining. SPATIAL DISTRIBUTION OF HEATINGAnother important effect in shaping the light curve is thespatial distribution of heating within the ejecta, e.g. differentamounts of “mixing” of Ni in Type I SNe or assumed dis-tribution of magnetar heating (see Dessart (2018)). Indeed,In Fig. (2), we showed how the spatial distribution impactsboth the peak time and luminosity of the light curve, and inparticular found that the Arnett solution and Arnett’s rule aremost appropriate for less concentrated/more uniform heating.To account for the spatial distribution of heating, we takethe heating rate to be uniform out to a (scaled) radius x s . UMINOUS T RANSIENT L IGHT C URVES AT P EAK t s / t d t p e a k / t d x s = 0.330.670.800.90 Figure 7.
Relation between source timescale to the peak time, rel-ative to the diffusion timescale. Different colors indicate differentlevels of concentration, parameterized by the concentration radius x s = 0 . , . , . , and 0 .
90. For comparison, the case of a cen-trally concentrated source ( x s = 0) is shown (grey squares). Peak Time (Days) L p e a k / L = / . . . x s = 0.330.670.800.90 Figure 8.
Peak time-luminosity relation for different spatial distri-butions of heating.
Fig.(7) shows how the concentration affects the time of peak.The overall effect is to systematically drop the relation, i.e.for a given t s and t d , concentration causes the light curve topeak earlier. This was shown for the case of t s = 10 daysin Fig.(2). Interestingly, the peak time does not differ muchunless the concentration radius is greater than x s > / t s (cid:28) t d . For the most mixed case x s = 0 .
9, therelation flattens as t peak ≈ . t d . This lends further caution to Time (Days) B o l o m e t r i c L u m i n o s i t y ( e r g s )
20 10 8 6 4 2 0 = T ion /1000 K Figure 9.
Light curves of a central exponential heating source with t s = 10 days and fixed ejecta properties, but varying the recombina-tion temperature T ion . The input heating rate is shown (dashed blackline). The case of T ion = 0K is identical to assuming a constant greyopacity. using t d as a proxy for t peak ; in addition to depending on t s ,there is also another dependence on x s .In Fig.(8), we show the peak time-luminosity relation forthe different spatial distributions of heating, for an exponen-tial source with t s = 10 days. Interestingly, for different con-centrations, the relation Eq.(19) still holds. The only differ-ence is in the value of β . For x s = 1 /
3, the numerical sim-ulations lie on the β = 4 / x s = 1 / β . For the most uniform heating, x s = 0 . β in-creases by about a factor of 2 compared to a central source.For the central exponential source used in Fig.(7) and a con-stant opacity, we find that β depends on x s approximately as β ( x s ) ≈ (cid:0) + x s (cid:1) (24)Note that we assume local deposition of the heating source.In reality, for the case of e.g. Ni decay, there is the ad-ditional effect of gamma-ray deposition, which introduces anon-locality to the heating. In particular, gamma rays emit-ted closer to the center may deposit their energy farther out(or may escape entirely). Exploring this effect is outside thescope of this work (although see e.g. Dessart et al. (2016)). NON-CONSTANT OPACITY AND RECOMBINATIONWhile the previous results assumed a constant opacity,for certain compositions the opacity drops sharply when the K
HATAMI AND K ASEN ejecta cools and ions recombine. As the ejecta is typicallyhotter at the center, a cooling “recombination front” propa-gates from the surface inward (Grassberg et al. 1971; Grass-berg & Nadyozhin 1976; Popov 1993). The photosphereis nearly coincident with the recombination front, with atemperature set by the ionization/recombination temperature T ion .To account for recombination effects in our numerical cal-culations, we prescribe a temperature dependence that mim-ics the behavior of the opacity in hydrogen and helium-richcompositions, for which electron scattering dominates for T > T ion κ ( T ) = κ + κ − (cid:15)κ (cid:20) + tanh (cid:18) T − T ion ∆ T ion (cid:19)(cid:21) . (25)The opacity κ = κ for temperatures T > T ion but drops to κ = (cid:15)κ for T < T ion . The tanh function ensures a smoothtransition over a temperature range ∆ T ion . We use (cid:15) = 10 − and ∆ T ion = 0 . T ion , although our results are not sensitiveto the exact values. We take the temperature T to be equalto the radiation temperature T rad = ( E / a ) / , where E is theenergy density and a the radiation constant. This is ap-propriate for a radiation-dominated ejecta and wavelength-independent opacity.Fig.(6) shows the effect of changing the recombinationtemperature T ion , while keeping the heating source and ejectaproperties fixed. These runs use a central exponential sourcewith timescale t s = 10 days and energy E s = 10 ergs, anduniform ejecta with mass M ej = 5 M (cid:12) , velocity v ej = 10 cms − , and opacity κ = 0 . g − . For low T ion , most of theejecta remains ionized at and after peak and the light curveresembles the constant opacity case, with the exception of alate-time “bump" that occurs when recombination sets in andallows radiation to escape more easily. For higher T ion re-combination occurs earlier; for T ion (cid:38) E s , impacts the light curve morphology. This is incontrast to constant opacity models, where E s simply setsthe normalization of the light curve but leaves the shape thesame. Fig. (9) shows a set of numerical light curves whereonly E s is varied. At sufficiently large values of E s , theheating source keeps the ejecta ionized until very late times,and the light curve shape resembles a constant opacity lightcurve. As E s decreases, the ejecta recombines earlier, result-ing in a “bump” at late times. For sufficiently low E s , theheating source is unable to keep the ejecta temperature above T ion , and so recombination impacts the light curve peak.To determine whether recombination effects are importantor not, we can compare the heating rate to the luminositynecessary to keep the ejecta ionized. The ionizing luminosityis set by the ejecta radius and the recombination temperature
25 50 75 100 125 150 175 200
Time (Days) B o l o m e t r i c L u m i n o s i t y ( e r g s ) Figure 10.
Light curves of a central magnetar heating source with t s = 10 days and fixed ejecta properties and recombination temper-ature T ion = 6000K, but varying the heating energy E s . The inputheating rate for each light curve is shown (dashed grey). t peak (Days) L p e a k / L T ion = 6000K= 4/3= 0.94> 1< 1 Figure 11.
Peak time-luminosity relation with recombination ef-fects, for a central exponential heating source with t s = 10 days.Points correspond to numerical simulations with T ion = 6000K andvarying ejecta properties M ej , v ej , κ ej , as well different values of E s ,whose relative value is indicated by the point size. Blue and orangecircles correspond to Eq.(29) with η > η <
1, respectively.Also shown is Eq.(19) for different values of β (lines). as L ion ≈ π R σ sb T (26)From Section 3, the luminosity will roughly scale as L ∼ E s t s t d (27) UMINOUS T RANSIENT L IGHT C URVES AT P EAK Time Since Explosion (Days) L o g B o l o m e t r i c L u m i n o s i t y ( e r g s ) M N i = . MM N i = . M M N i = . M L dec ( t ) L pred ( t peak ) SN1987A
Figure 12.
The light curve of SN1987A (blue points) from Suntzeff& Bouchet (1990), compared to input Ni+Co decay of M Ni =0 . M (cid:12) (solid black) and 0 . M (cid:12) (dashed black). Also shown isEq.(19) for 0 . M (cid:12) of Ni and β = 0 .
82 (red dashed).
We define a ratio of the luminosity to the critical ionizingluminosity η ≡ LL ion ∝ c πσ E s t s κ M T (28)We calibrate the proportionality based on numerical simula-tions of an exponential heating source to find η ∼ . E s , t s , κ − . M − T − (29)where E s , = E s / erg, t s , = t s /
10 days, κ . = κ/ . g − , M = M ej / M (cid:12) , and T = T ion / K. For η (cid:46)
1, theheating luminosity is too low to keep the ejecta sufficientlyionized and so recombination effects become important.In Fig.(11), we show the results of a set of numerical sim-ulations with T ion = 6000K and various ejecta properties and E s . Interestingly, the numerical simulations with recombina-tion still fall on the relation Eq. (19). The only differenceis that the value of β changes. Specifically, recombinationtends to decrease the value of β . Also shown in Fig. (11)are the respective values of η for the numerical simulations.Points with η > β = 4 /
3, appropriate for a con-stant opacity. For η <
1, recombination is important and thepoints fall on a smaller β = 0 .
94 curve. DISCUSSION AND CONCLUSIONSWe have shown how the light curve peak time and luminos-ity are related for luminous transients, and derived analyticrelations that can be used to infer the physical properties of Peak Time (Days) P e a k L u m i n o s i t y ( e r g s ) . M . M . M M M Radioactive Ni Transients
Figure 13.
Peak time vs. peak luminosity for radioactive Ni-powered transients. Eq.(19) is shown for different values of β = 0 . β = 9 / β = 4 / the heating mechanism. In particular, Eq.(19) L peak = 2 β t (cid:90) β t peak t (cid:48) L heat ( t (cid:48) ) dt (cid:48) captures the relationship between t peak and L peak where thelight curve physics (i.e. recombination and concentration) iscontained in the β parameter. Furthermore, Eq.(24) β ( x s ) = 43 (cid:0) + x s (cid:1) gives the approximate dependence of β on the spatial distri-bution of heating. Another useful result is given in Eq.(23)for central sources, t peak t d = 0 .
11 ln (cid:18) + t s t d (cid:19) + . t peak depends not only on the diffusion time t d , but also the heating timescale t s . In addition, recombi-nation will change the value of β compared to a constantopacity. In Table 1, we give approximate values of β for avariety of transients. In Appendix A, we evaluate the peaktime-luminosity relation for specific heating sources.For example, one of the common sources of heating inluminous transients is the radioactive decay chain of Ni(Stritzinger et al. 2006; Valenti et al. 2007). In particu-lar, the decay chain of Ni → Co → Fe dominates theheating at timescales of interest. Fig.(13) shows the peaktime-luminosity parameter space of Ni-powered transients,where the relation is given in Appendix A. For a given peaktime and luminosity, one can thus infer an approximate valuefor the Ni mass for an appropriate choice in β .0 K HATAMI AND K ASEN
Time (Days) B o l o m e t r i c L u m i n o s i t y ( e r g s ) M n i = . MM n i = . MM n i = . MM n i = . MM n i = . MM n i = . M Ni+Co Source
Figure 14.
Bolometric light curves of toy Ia models, with M ej =1 . M (cid:12) , v ej = 10 cm s − , κ = 0 . g − and assuming uniform den-sity. Shown are light curves for different amounts of nickel masses M Ni (heating rate shown as dashed grey lines) and, therefore, con-centration. We now consider the implications of our findings to ana-lyzing observed SNe. In particular, Arnett’s rule is often usedto infer the nickel mass of Type Ia SNe. In Fig. (14) we showthe effects of varying the nickel mass M Ni in a set of toy Iamodels with a total mass M ej = 1 . M (cid:12) and constant densityand opacity. The inner layers of ejecta in these models arecomposed of pure Ni, and so higher M Ni corresponds toa larger nickel core and less centrally concentrated heating.As expected, Arnett’s rule works better for larger M Ni andbecomes progressively worse for the more centrally concen-trated low M Ni models. This suggests that analyses of SNe Iausing Arnett’s rule may be systematically biased, with thenickel mass of sub-luminous Ia’s being overestimated.As another case study, we show in Fig. (12) the ob-served bolometric light curve of SN1987A, a Type II su-pernova whose primary peak is powered by radioactive Ni(Woosley 1988; Suntzeff & Bouchet 1990). The late-timelight curve behavior gives a constraint on the Ni massto be M Ni ≈ . M (cid:12) . Arnett’s rule predicts a M Ni a fac-tor of 2 too large, whereas using the new relation Eq.(19)with β = 0 .
82 (appropriate for hydrogen recombination T ion ≈ M Ni ≈ . M (cid:12) , in agreement with the late-time determina-tion.As another example, we show in Fig.(15) the peak time-luminosity relation for the Type Ib/c SNe models presentedin Dessart et al. (2016). As noted in their work, Arnett’s ruleseems to overestimate the Ni mass of their models. Us-ing Eq.(19), we find that the models lie on a β = 9 /
10 15 20 25 30 35 40 45 50
Peak Time (Days) L p e a k / N i M N i Arnetts Rule= 9/8Dessart+16 (Ibc)
Figure 15.
Peak time-luminosity relation of Eq.(19) compared tothe Ibc models of Dessart et al. (2016). tion. Given that the models do not have much mixing, wecan assume centrally located heating and attribute any devi-ation from β = 4 / β = 9 / T ion = 4000K, which is roughly that for a C- and O-richcomposition. On the other hand, helium has a much higherrecombination temperature and would imply a much smaller β ; this indicates that the Ni in the Dessart et al. (2016)models are primarily diffusing out from the much denser car-bon/oxygen inner ejecta rather than the outer helium ejecta.This is in agreement with the results in Piro & Morozova(2014), who similarly showed that light curve modeling is abetter constraint on the C/O core rather than the helium.The above examples demonstrate that, in principle, thepeak time-luminosity relation may allow one to infer thecomposition of the ejecta solely from photometric observa-tions . Suppose we know from observations of the radioac-tive tail of SN1987A that it is powered by 0 . M (cid:12) of Ni,and we assume the nickel to be largely centrally concen-trated. From the peak time and luminosity we can solve for β ≈ .
94. Since each recombination temperature has its ownunique value of β , we can then infer T ion ∼ β may indi-cate the spatial distribution of the heating source. For exam-ple, one can assume to good approximation a constant opac-ity for Type Ia SNe. Assuming central heating, this wouldpoint to a β = 4 /
3, yet SNe-Ia seem to obey Arnett’s rulefairly well, which corresponds to a larger value of β if us-ing Eq.(19). This is in agreement with the results presented UMINOUS T RANSIENT L IGHT C URVES AT P EAK β .The main result, Eq.(19), is general enough to be appliedto an arbitrary heating source, e.g. central-engine accretion,a magnetar, kilonovae, etc. Thus, for an observed transientpeak time/luminosity that might be powered by other meansthan Ni, one must simply choose a different L in ( t ) (whichneed not be analytic). Next, by choosing an appropriate β (e.g. 4 / t s andenergy E s . One is not able to break this degeneracy from thepeak time and luminosity alone. Such constraints require ad-ditional information/observations or by putting physical lim-its on allowed values.Several physical effects were neglected in our analysis hereso as to isolate the basic behavior of supernova light curves.The models presented assume spherical symmetry and adopta grey opacity. Asymmetries in the ejecta/heating as well asnon-grey effects likely play a role in the overall shape of thelight curve, and on the inferred β in the new relation. Dessartet al. (2018) show that clumping affects the recombinationrate, which would impact the inferred β . We also used asimple parameterization for the spatial distribution of heat-ing, which was taken to be uniform out to some radius; morecomplicated distributions (e.g. from accounting for gamma-ray deposition throughout the ejecta) warrant further investi-gation. We further made the assumption of homologous ex-pansion, which may be violated for supernovae interactingwith a circumstellar material. Because our relations apply tothe bolometric peak, errors can also arise from observationaleffects, such as uncertainties in the distance, reddening, orbolometric correction.Conclusions drawn from the relation presented here are ofcourse conditional on the specific form of the heating sourceassumed (e.g. radioactive decay vs. magnetar spindown). Itis thus important to perform consistency checks on the natureof the heating source using additional information aside fromthe properties at peak, such as examining the slope of thelate-time light curve tail. While our comparisons here have demonstrated the limi-tations of the Arnett-like models, there exist other analyticmodels of transient light curves that attempt to account forthe time-dependent diffusion effects that are important forsetting the luminosity before and around peak (e.g. Piro& Nakar (2013, 2014); Waxman et al. (2018)). In futurework, we will investigate a broader range of analytic models,and look for improved methods for calculating analytic lightcurves. Additionally, there exist other analytic techniques inestimating properties of the light curve, in particular the in-tegral relation of Katz et al. (2013). This requires knowingthe full shape of the light curve out to times well after peak,rather than the properties at peak. The Katz integral approachand the new relation presented here are thus complementaryin inferring the physical properties of the light curve.There is still much work to be done to understand how/whythe peak time-luminosity relation as well as it does, and inparticular to better calibrate its value for specific heatingmechanisms and ejecta properties. Nonetheless, the frame-work presented here will be useful for more detailed model-ing, as well as providing a fast way to characterize the largenumber of transients to be discovered in current and upcom-ing surveys.D.K.K was supported in part by the National ScienceFoundation Graduate Research Fellowship Program. D.K.Kand D.N.K are supported by the U.S. Department of Energy,Office of Science, Office of Nuclear Physics, under contractnumber DE-AC02-05CH11231 and DE-SC0017616; Sci-DAC award DE-SC0018297; the Gordon and Betty MooreFoundation through Grant GBMF5076; and the ExascaleComputing project (17-SC-20-SC), a collaborative effort ofthe U.S. Department of Energy and the National Nuclear Se-curity Administration.This research used resources of the National Energy Re-search Scientific Computing Center, a DOE Office of Sci-ence User Facility supported by the Office of Science ofthe U.S. Department of Energy under Contract No. DE-AC0205CH11231. Software:
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UMINOUS T RANSIENT L IGHT C URVES AT P EAK Table 1.
Peak Time-Luminosity Relation for Generic Heating SourcesHeating Type Source Function H ( t , t s ) f ( τ , β )Exponential exp (cid:2) − t / t s (cid:3) − (1 + βτ ) exp [ − βτ ]Magnetar n > n − + t / t s ) n n − (cid:2) − (1 + ( n − βτ ) (1 + βτ ) − n (cid:3) Magnetar n = 2 (cid:0) + t / t s (cid:1) − n ln (1 + βτ ) − (cid:0) + / βτ (cid:1) − Constant w/ Shutoff Θ ( t − t s ) 1 / E δ ( t − t s ) 1Power-Law α (cid:54) = 2 (cid:16) tt s (cid:17) − α − α (cid:2) ( βτ ) − α − (cid:3) Power-Law α = 2 (cid:16) tt s (cid:17) − ln βτ APPENDIX A. EXPRESSIONS FOR THE PEAKTIME-LUMINOSITY RELATIONFrom Section 3, we found an expression for the peak time-luminosity relation as L peak = 2 β t (cid:90) β t peak t (cid:48) L heat ( t (cid:48) ) dt (cid:48) (A.1)where L peak is the observed peak luminosity at the peaktime t peak , L heat ( t ) is the time-dependent heating rate, and β is a constant that depends on opacity/concentration effects.Eq.(A.1) can be evaluated analytically for several functionalforms of L heat ( t ).Suppose the heating source can be written most generallyas L heat ( t ) = E s t s H ( t , t s ) (A.2)where H ( t , t s ) is the time-dependent component of L heat ( t ), t s is the heating source timescale, and E s the characteristic heat-ing energy. Equivalently, the heating source can be expressedin terms of a characteristic luminosity by setting L = E s / t s .Let τ = t peak / t s be the ratio between the peak time andsource timescale. Then the peak luminosity can be evaluatedto get L peak = 2 E s t s β t × f ( τ , β ) (A.3)where f ( τ , β ) depends on the exact functional form of H ( t , t s ). For an exponential source, H ( t , t s ) = exp (cid:2) − t / t s (cid:3) (A.4)the integral can be evaluated to get f ( τ , β ) = 1 − (1 + βτ ) e − βτ (A.5) In Table 1, we give the analytic expressions of f ( τ , β ) for avariety of heating functions H ( t , t s ). The choice of β againdepends on opacity/recombination effects, as well as the spa-tial distribution of heating. In Table 2, we give approximatevalues of β based on numerical results.A.1. Radioactive Two-Decay Chain
Numerous transients are powered by the radioactive decayof synthesized elements, e.g. Ni in Type I and IIb/pec su-pernovae.Consider a decay chain consisting of 0 → → t and t , respectively (ignoring the contribution toheating of species 2). The total number of the species at time t is expressed as N ( t ) = Ne − t / t (A.6) N ( t ) = N t t − t (cid:16) e − t / t − e − t / t (cid:17) (A.7)where N = N (0), and it is assumed that N (0) = 0. Let Q , m , Q , and m be the decay energies and species mass. Defineheating rates per unit mass as ε = Q m t (A.8) ε = Q m ( t − t ) (A.9)Then the heating luminosity can be expressed as L heat ( t ) = M (cid:104) ( ε − ε ) e − t / t + ε e − t / t (cid:105) (A.10)where M = Nm is the total initial mass of species 0.4 K HATAMI AND K ASEN
Table 2.
Values of β for specific transientsTransient β Generic - Central source, constant opacity 4 / . . . / . With this expression we can derive the peak time-luminosity relation as L peak = 2 ε Mt β t (cid:20)(cid:18) − ε ε (cid:19) (cid:16) − (1 + β t peak / t ) e − β t peak / t (cid:17) + ε t ε t (cid:16) − (1 + β t peak / t ) e − β t peak / t (cid:17)(cid:21) (A.11)A.1.1. Radioactive Nickel Decay
Supernovae of Type I and IIb/pec are powered primarily bythe radioactive decay of Ni followed by Co (Stritzingeret al. 2006; Valenti et al. 2007). The heating function can bewritten in terms of the nickel mass M Ni as L heat ( t ) = M Ni (cid:104) ( ε Ni − ε Co ) e − t / t Ni + ε Co e − t / t Co (cid:105) (A.12)where ε Ni = 3 . · erg g − s − and ε Co = 6 . · erg g − s − are the specific heating rates of Ni- and Co-decay, and t Ni = 8 . t Co = 111 . β as L peak = 2 ε Ni M Ni t β t (cid:20)(cid:18) − ε Co ε Ni (cid:19) (cid:16) − (1 + β t peak / t Ni ) e − β t peak / t Ni (cid:17) + ε Co t ε Ni t (cid:16) − (1 + β t p / t Co ) e − β t p / t Co (cid:17)(cid:21) (A.13)Let t p = t peak / day be the peak time in days. Using the numer-ical values of ε Ni , ε Co , t Ni , and t Co we get a more compactnumerical expression (accurate to within ∼ L peak = 10 (cid:18) M Ni M (cid:12) (cid:19) β t p × (cid:2) . − (cid:0) + . β t p (cid:1) exp (cid:0) − . β t p (cid:1) − . (cid:0) + . β t p ) exp (cid:0) − . β t p (cid:1)(cid:1)(cid:3) erg s − (A.14) A.1.2. Magnetar-powered Supernovae
The spindown luminosity of a magnetar is generally de-scribed by (Bodenheimer & Ostriker 1974; Gaffet 1977;Woosley 2010; Kasen & Bildsten 2010) L mag ( t ) = E mag t mag l − (cid:0) + t / t mag (cid:1) l (A.15)where l = 2 for magnetic dipole spin-down, E mag = I NS Ω × P − erg (A.16)is the magnetar energy with P = P / t mag = 6 I NS c B R Ω = 1 . B − P yr (A.17)is the spindown timescale with B = B / G the magneticfield strength. For l = 2, we get L peak = 2 E mag t mag β t (cid:104) ln(1 + βτ ) − (cid:0) + / ( βτ ) (cid:1) − (cid:105) (A.18)were τ = t peak / t mag .A.2. Accretion-Powered Transients
Another interesting heating source is that of an accretingcompact object (Dexter & Kasen 2013). Let ˙ M ∼ M acc / t acc be the accretion rate of mass M acc and timescale t acc . We hereconsider two functional forms of accretion luminosity. Thefirst is of constant heating that “shuts off” after a time t acc , L acc ( t ) = (cid:15) M acc c t acc Θ ( t − t acc ) (A.19)where Θ ( t − t acc ) is the Heaviside step function, and (cid:15) is theradiative efficiency. Substituting this into Eq.(19) we get L peak = (cid:15) M acc t acc β t (A.20) UMINOUS T RANSIENT L IGHT C URVES AT P EAK n = − / L acc ( t ) = (cid:15) M acc c t acc (cid:18) tt acc (cid:19) − / (A.21)Again evaluating this source in Eq.(19) we get L peak = 6 (cid:15) M acc c t acc β t (cid:34)(cid:18) β t peak t acc (cid:19) / − (cid:35) (A.22)A.3. Kilonovae
The kilonova heating rate from the radioactive decay of r-process elements can be parameterized as (Li & Paczy´nski1998; Metzger et al. 2010; Roberts et al. 2011; Lippuner &Roberts 2015). L heat ( t ) = ε M ej (cid:18) tt (cid:19) − η × f ( t ) (A.23)where ε ≈ erg g − s − is the specific heating rate, M ej is the ejecta mass, t ≈ η ≈ .
3. The function f ( t )gives the thermalization efficiency with which the radioactivedecay energy is able to deposit as heat in the ejecta.(Kasen & Barnes 2018) suggest an analytic approximationof the thermalization efficiency of electrons f ( t ) = (cid:18) + tt e (cid:19) − (A.24)where t e ≈ . M / . v − . days (A.25)is the electron thermalization timescale, M . = M ej / . M (cid:12) ,and v . = v ej / . c the ejecta velocity.Evaluating the integral for η = 1 . L peak = 2 . ε M ej (cid:18) t β t peak (cid:19) . F (cid:0) . , , . , − β t peak / t e (cid:1) (A.26)where F ( a , b , c , x ) is the hypergeometric function which weapproximate by a functional fit of F (cid:0) . , , . , − β t peak / t e (cid:1) ≈ (cid:0) + β t peak / t e (cid:1) − / (A.27)Thus, the peak time-luminosity relation for kilonova is ap-proximately L peak = 2 . ε M ej (cid:18) t β t peak (cid:19) . · (cid:0) + β t peak / t e (cid:1) − / (A.28)Assuming the r-process heating is uniformly mixed through-out the ejecta, we choose an approximate β ≈ B. DERIVATION OF THE PEAK TIME-LUMINOSITYRELATIONIn Section 3, we showed that a simple relation holds be-tween the peak time and luminosity of a light curve L peak = 2 β t (cid:90) β t peak t (cid:48) L heat ( t (cid:48) ) dt (cid:48) (B.1)assuming there exists some time t = β t peak such that (cid:15) ( t ) = 0,where we defined (cid:15) ( t ) = (cid:20) t L peak − (cid:90) t t (cid:48) L ( t (cid:48) ) dt (cid:48) (cid:21) − tE ( t ) (B.2)Our goal, then, is to motivate that such a time when (cid:15) ( t ) = 0exists. We define two quantities F ( t ) = t L peak − (cid:90) t t (cid:48) L ( t (cid:48) ) dt (cid:48) (B.3)and E ( t ) = tE ( t ) (B.4)Thus, the time when (cid:15) ( t ) = 0 also implies F ( t ) = E ( t ) (B.5)Initially, E ( t ) will rise as heat is deposited and trapped inthe optically thick ejecta. Eventually, the expanding ejectabecomes optically thin, allowing radiation to freely escape.As a result, any heating goes directly into the light curve, L ( t ) = L heat ( t ). This implies that E ( t ) = 0 at late times t (cid:29) t peak .Since E ( t ) is continuous, if F ( t ) is a monotonically in-creasing function of time and F ( t ) < E ( t ) for some t , thenit follows that F ( t ) and E ( t ) must intersect.Taking the derivative in time of F ( t ) we get F (cid:48) ( t ) = t (cid:2) L peak − L ( t ) (cid:3) (B.6)Since L ( t ) ≤ L peak by definition, it follows that F (cid:48) ( t ) ≥ F ( t ) is indeed a monotonically increasing function oftime.Next, we need to show that there exists a time such that F ( t ) < E ( t ). At t = 0 we have F ( t ) = E ( t ) = 0. For small t , wecan expand the derivative of E ( t ) to get E (cid:48) ( t ) ≈ t (cid:2) L heat (0) + tL (cid:48) heat (0) − t L (cid:48) (0) (cid:3) (B.8) ≈ tL heat (0) + O ( t ) (B.9)where we make use of the fact that L (0) = 0. Similarly, for F ( t ), we get F (cid:48) ( t ) ≈ t (cid:2) L peak − t L (cid:48) (0) (cid:3) (B.10) ≈ tL peak + O ( t ) (B.11)6 K HATAMI AND K ASEN
If the condition L heat (0) > L peak is satisfied, then E (cid:48) ( t ) > F (cid:48) ( t ) (B.12)for small t . Since F (0) = E (0) = 0, we have that, at earlytimes, E ( t ) > F ( t ). Combined with the monotonicity of F ( t )and the fact that E ( t ) → E ( t )and F ( t ) must intersect. In other words, there exists a timesuch that (cid:15) ( t ) = 0 and the peak time-luminosity relation holds.There is a-priori mathematical justification for L heat (0) > L peak , however in physical cases of interest (e.g. radioac-tive decay, magnetar spindown, etc.), this seems to be a validassumption, whereby L heat ( t ) is monotonically decreasing intime. Furthermore, both diffusion and adiabatic degradationact to spread out and decrease the heating luminosity in time.This is confirmed in our numerical simulations for a wide va-riety of heating functions, wherein all the light curves seemto indicate L heat (0) > L peak (see e.g. Figs. (1) and 2).Another mathematical possibility is that F ( t ) and E ( t ) in-tersect more than once. Assuming L heat (0) > L peak , since F ( t ) is monotonically increasing, such behavior requires E ( t )to “oscillate”, i.e. there exists more than one time that d E / dt = 0. It is unclear whether such behavior is physical.In summary, it is difficult to prove definitively that the peaktime-luminosity relation holds for an arbitrary heating func-tion. For a variety of monotonically decreasing heating func-tions, we have confirmed numerically that L heat (0) > L peak .Furthermore, this intersection appears to occur only once. InFig.(16), we show the behavior of E ( t ) and F ( t ) for a subsetof numerical simulations, assuming a central source and con-stant opacity. We pick different functional forms (exponen-tial and magnetar-like) as well as different source timescales.In all cases, the time when (cid:15) ( t ) = 0 is nearly identical, with β ≈ . t / t peak [( t ) , ( t )] / L p e a k t p e a k Exp. t s = 10 dExp. t s = 30 dExp. t s = 100 dMagnetar t s = 10 d Figure 16.
Comparison of the quantities E ( t ) (solid lines) and F ( t )(dashed lines) as a function of time, for a central heating source withdifferent functional forms and source timescales. The time when E ( t ) = F ( t ) gives the value of ββ