Analytic heating rate of neutron star merger ejecta derived from Fermi's theory of beta decay

0.22 MB·

by  Kenta Hotokezaka

Mon.Not.R.Astron.Soc.000,1–??(2015) Printed12January2017 (MNLaTEXstylefilev2.2)
Analytic heating rate of neutron star merger ejecta derived
from Fermi’s theory of beta decay
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1 Kenta Hotokezaka1,2∗, Re’em Sari1, and Tsvi Piran1
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1Racah Institute of Physics, The HebrewUniversity of Jerusalem, Jerusalem, 91904, Israel
2
2Centerfor Computational Astrophysics, 162 5th Ave, New York, NY, 10010, USA

n
a
J
10January2017
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ABSTRACT
]
E Macronovae (kilonovae) that arise in binary neutron star mergers are powered by
H radioactive beta decay of hundreds of r-process nuclides. We derive, using Fermi’s
theory of beta decay, an analytic estimate of the nuclear heating rate. We show that
h. the heating rate evolves as a power law ranging between t−6/5 to t−4/3. The overall
p magnitude ofthe heatingrate is determinedby the meanvaluesofnuclear quantities,
- e.g., the nuclear matrix elements of beta decay. These values are specified by using
o
nuclearexperimentaldata.Wediscusstheroleofhigherorderbetatransitionsandthe
r
t robustness of the power law. The robust and simple form of the heating rate suggests
s
a that observations of the late-time bolometric light curve ∝ t−34 would be a direct
[ evidence of a r-process driven macronova. Such observations could also enable us to

estimate the total amount of r-process nuclei produced in the merger.
1
v Key words: stars:neutron−gamma-rayburst:general
5
8
7
2
0 1 INTRODUCTION 2012; Wanajo et al. 2014; Lippuner& Roberts 2015;
.
1 Hotokezakaet al. 2016; Barnes et al. 2016).
Li & Paczyn´ski (1998) proposed that neutron star merg-
0
ers will be accompanied by macronovae (kilonovae), which Another approach to calculate the heating rate is
7
1 are optical–infrared transients powered by radioactive to consider the r-process material as a statistical assem-
: decay of the merger’s debris. These macronovae are bly of radioactive nuclei. Incorporating Fermi’s theory of
v among the most promising electromagnetic counterparts beta decay with such an approach provides a clear phys-
i
X togravitational-wave mergerevents(e.g. Metzger & Berger ical understanding of the nuclear heating rate (see, e.g.,
2012; Nissankeet al. 2013). Recently, macronova candi- Way & Wigner 1948 for a discussion on the energy gen-
r
a dates have been discovered in the afterglows of several eration by fission products). We follow this approach and
short gamma-ray bursts (Tanvir et al. 2013; Berger et al. use the Fermi theory to estimate the heating rate in neu-
2013;Yanget al.2015;Jin et al.2016).Jin et al.(2016)re- tron star mergers’ ejecta. Colgate & White (1966, see also
analyzedtheafterglowlightcurvesofhistoricalnearbyshort Metzger et al. 2010) considered this approach to estimate
gamma-rayburstsandsuggestedthatmacronovaeareubiq- theradioactiveluminosityofsupernovae.Buttheyusedonly
uitous in short GRBs’ afterglows. the relativistic regime of Fermi’s theory, which is not rele-
Theradioactiveheatgeneratedbyr-processnucleiplay vant on the macronova peak timescale as we discuss later.
an essential role in powering macronovae. Due to strong Furthermore, they assumed that each element decays to a
adiabaticcoolingtheejecta’sinitialinternalenergyispracti- stable one rather than following a decay chain, which we
cally negligible atthetimethattheejecta becomeoptically consider in thispaper.
thin, i.e. when τ ≈c/v, where τ is the optical depth and v
isthevelocityoftheejecta.Aswediscusslater,withtypical Inthispaper,weanalyticallyderivethenuclearheating
parameters the peak emission time is around a few days rateofmacronovaebasedonFermi’stheory.In§2,webegin
and hence this is the important timescale to focus on. De- withabriefsummaryofthebasicconceptsandtheoutcome
tailed computations using nuclear database and numerical of thiswork. In§3,we introducethekeyingredients of the
simulationshavebeenwidelyusedtoobtaintheradioactive theory needed to calculate the heating rate. We derive the
heatingrates(Freiburghauset al.1999;Metzger et al.2010; heating rate of the beta decay chains of allowed transitions
Goriely et al. 2011; Roberts et al. 2011; Korobkin et al. in§4.Wediscuss,in§5,theroleofforbiddentransitionsand
other effects that we ignore and we estimate their possible
effect.Wesummarizetheresultsanddiscusstheimplication
∗ E-mail:[email protected] to macronova studies in § 6.
2 K. Hotokezaka, R. Sari, and T. Piran
2 A SHORT SUMMARY OF BETA-DECAY sufficiently small compared to E and assume that there
0
HEATING RATE isnoangularcorrelationbetweentheelectronandneutrino.
Hereweimaginethatthewholesystemisenclosedinalarge
Radioactive nuclei that are far from the stability valley are
box with a volume V. Hereafter we change the notation as
producedinr-processnucleosynthesis.Thesenucleiundergo
p →p and E →E.
betadecaywithoutchangingtheirmassnumber.Aseriesof e e
In Fermi’s theory, the four particles interact at a sin-
beta decays in each mass number is considered as a decay
gle point with a coupling constant G so that the matrix
chain. Because the mean lives of radioactive nuclides typi- F
element is written as
cally become longer when approaching the stability valley,
thenucleiinadecaychainatgiventimetstayatsomespe-
cific nuclide with a mean life τ ∼ t. This means that the Hf′i=GFZ (ψe†OLψν)(ψp†ONψn)dV, (4)
number of decaying nuclei in a logarithmic time interval is
where ψ is the wave function of each particle involved in
constant, i.e, the decay rate is ∼N/t, where N is the total i
the beta disintegration, O and O are operators acting
numberof nucleiin thechain. Then thebetadecay heating L N
on the light particle’s spin, nucleon’s spin and isospin (see,
rate per nucleus is given by q˙ ∼ E(t)/t, where E(t) is the
e.g., Feynman& Gell-Mann 1958 for a discussions on beta
disintegration energy of thebeta decay as a function of the
interaction).Thewavefunctionofthelightparticlescanbe
mean life. As we will see later, two important concepts in
evaluatedatr∼0becausetheirdeBroglie wavelengthsare
beta decay theory enable usto determine E(t).First, there
much larger than the nuclear size. When the light particles
arefourphysicalconstantsintheproblem,theFermi’scon-
do not carry off orbital angular momentum with respect to
stant G , the electron mass m , the speed of light c, and
F e thecentral nucleus, the wave function of each light particle
the Planck constant ~. The fundamental timescale of beta
decayt ≈9·103scanbeobtainedfromthesephysicalcon- at r = 0 is just a normalization factor of V−1/2 with a
F Coulomb correction for the electron’s wave function. Thus
stants. Second, there is a well known relation between the
thesquare of thematrix element can bewritten as
disintegration energy and mean life as τ ∝E−5. Therefore,
theheating rate per nucleuscan be roughly estimated as G2
q˙(t)∼ mec2 t −1.2. (1) |Hf′i|2 = VF2F(Z,E)|MN|2, (5)
tF (cid:18)tF(cid:19) where F(Z,E) is the Coulomb correction factor, Z is the
protonnumberofthedaughternucleus,andM isthenu-
This gives a correct order of magnitude and a reasonable N
clear matrix element.Thetransitions described hereare al-
estimate of the time dependence of the beta decay heating
lowed transition. More specifically, allowed transitions are
rate of r-process material. In the following we refine these
transitionswhich satisfy bothconditionsthatthelight par-
ideas.
ticles don’t carry off orbital angular momentum and the
parity of thenucleusdoes not changevia its disintegration.
Otherwisethetransition is aforbiddentransition.1 Because
3 BASICS OF FERMI’S THEORY OF BETA
the population of allowed transitions is larger and because
DECAY
of their simplicity, we focus on allowed transitions in this
The nature of beta decay was successfully described by and the next sections. We will discuss the role of forbidden
Fermi (1934). Here we briefly describe key ingredients of transitions in §4.
Fermi’s theory needed for obtaining the macronova heating IntegratingEq.(2) overtheaccessible phasespace, the
rate.Inabetadecayoneoftheneutronsinanucleusdisinte- mean-life of a beta-unstable nuclidewith thedisintegration
gratedtoaproton,anelectron,andananti-neutrino.Using energy of E is obtained as
0
Fermi’s Golden rule, the beta-disintegration probability of
a beta-unstable nucleus per unit time in a unit momentum 1 = |MN|2 p(E0)dpF(Z,E)p2(E−E )2, (6)
intervalof theelectron is written as τ tF Z0 0
ddpwedpe= 2~π (cid:12)Hf′i(cid:12)2ρ(Ee), (2) awnhderteFtihsetvhaerifaubnldeasminentthaelitnimteegsrcaallearoefibneutanidtsecoafym: e and c
(cid:12) (cid:12)
where p and E are momentum and kinetic energy of the
e e 2π3 ~7
etolencitarnonr,esHpf′oinissibthleefmorattrhiexbeleetmaednitsionftetghreaitniotenr,aactnidonρH(Eaem)iils- tF ≡ G2F m5ec4 ≈8610 s.
the number density of final states of the light particles be-
Notethat,althoughthisfundamentaltimescale isacharac-
tween E and E +dE . The number of final states is as-
e e e
teristic timescale of allowed betadecay,the lifetime of beta
sumed to be proportional to the volume of the accessible
unstable nuclides spreads over many orders of magnitude
phase space of the light particles:
because of the phasespace factor of Eq. (6).
ρ(E )dE = (4π)2V2p2dp p2dp TheCoulombcorrectionfactorinthematrixelementis
e e (2π~)6 e e ν ν obtained by evaluating the electron’s wave function at the
(4π)2V2
≈ p2dp (E −E )2dE , (3)
(2π~)6c3 e e 0 e e
whereE isthetotaldisintegrationenergy,p ismomentum
0 ν
of the neutrino, and energy conservation E0 ≈ Ee + Eν 1 We employ Konopinski’s classification of beta de-
has been used. We use the fact that the neutrino mass is cay(Konopinski1966).
Beta-decay heating rate of macronova 3
nuclear radius rn (Fermi1934): 102
|ψ (r )|2
F(Z,E) ∼= e n Z , (7) 101
|ψe(rn)|2Z=0 t≈ τ1
warnnh/de(r~αe/mη≈e=c1=)/,Z1s3q7e22=[/((i~21s(sv1+t!,)h−2sve])((ifiZs2npαteρh)-)s2e2t)sr1v−u/e2c2l,teouπcqrηiete|yi(scsoot−nfhset1tha+eenletie.cηlteF)rc!oo|t2nrro,Ecnh,a>ρrg=1e,, dN /dt [1/s]i111000--021 t≈ τ2 ∝ 1/tt≈ τ3
the Coulomb correction factor slowly increases with E as 10-3
F(Z,E)∝E2s−2. t≈ τ4
A simple form of the Coulomb correction factor is ob- 10-4
tained in the non-relativistic limit of Eq. (7), η ≫ 1 and 0.01 0.1 1 10 100 1000
(Zα)2 →0: t [s]
2πη Figure1.Adecaychainnormalizedbythetotalnumberofnuclei
F (Z,E)= . (8)
N 1−exp(−2πη) in the chain. The dashed line depicts e−1/t, where e is Euler
number.
The Coulomb correction factor is unity for η ≪ 1 and ap-
proaches to 2πη for η ≫ 1. This enhances the transition
probability at lower energies. At theseenergies theelectron ratherproportionaltoE−4orE−3 ontherelevanttimescale
is pulled by the nucleus due to the Coulomb force and the of macronovae, i.e., a few days.
amplitude of the electron’s wave function is larger near the In the context of macronovae, we are interested in the
nucleus.Asaresult,thelifetime of betaunstablenucleibe- relationbetweenthelifetimeandthemeanelectron’senergy
comesshorterthantheoneestimated withouttheCoulomb since the neutrinos don’t contribute to the heat deposition
correctionandthedependenceofthelifetimeonE isweak- inthemergerejecta. Thefraction ofenergyoftheelectrons
0
ened. Note that one can also obtain an identical form to to thetotal energy is:
Eq. (8) by solving the Schr¨odinger equation and evaluating hE i
theelectron’s wave function at r=0 . ǫe ≡ Ee , (11)
0
As the integral in Eq. (6) is easily calculated for given 1 p0
Eco0mapnadrisZon, cwoimthpathraetievxepehraimlf-elinvteasl fdat1t/a2: are often used for = E0 Z0 F(Z,p)p2E(E0−E)2dp.
In thethreeregimes discussed earlier ǫ satisfies:
e
ln2
ft ≡ t . (9)
1/2 |M |2 F 1/2 (relativistic:E0 >1),
N
ǫe = 1/3 (nonrelativistic: (Ec <E0<1), (12)
Although MN of each beta transition cannot becalculated  1/4 (nrCoulomb: E0 <Ec),
withinFermi’stheory,M canbedeterminedfromthemea-
N wherewe assumed E <1.
surements of the lifetime and the electron’s spectrum. It is c
sufficientforourpurposetoknowthestatisticaldistribution
of this quantity.For allowed transitions, the distribution of
ft isknowntohaveapeakaround105scorrespondingto 4 THE HEATING RATE: THE IDEAL-CHAINS
1/2
|M |2 ∼0.05(e.g.Blatt & Weisskopf1958),whichwetake APPROXIMATION
N
as a reference valuein this paper.2
Neutron-richnucleiproducedviather-processundergobeta
Onecanshowthatf attainssimpleformsinthefollow-
decaytowardsthebeta-stablevalleywithoutchangingtheir
ing threeregimes:
massnumber.Aseriesofbetadecaysofnucleiineachmass
1 E5 (relativistic: E >1), number can be considered as a decay chain. Here we con-
 3106√20E7/2 (nonrelativistic0: siderideal-chainsofradioactivenucleiwithaseriesofmean
f(Z,E0)= 21π0Z5αE03 (nonrelatEivcis<ticE0C<ou1lo)m, b: (10) tliovteasl(nτu1m<bτe2r<ofτ3nu<cl.e.i.),thinrowughhicohuteatchhecdheacianycopnrsoecrevsessatnhde
3 0 sufficiently many chains exist. Within this approximation,
 E0<min(Ec,1)), thenumberofdecayingnucleiinalogarithmictimeinterval
where E = (2πZα)2/2. The non-relativistic regime ex- is constant and thebeta decays at a given time t are domi-
c
ists only for Z . 30, and thus, there is no such a regime natedbynuclideswithmean-livesofτ ∼t(seeFig.1).This
in r-process material. In previous a work Colgate & White is, of course, valid for t > τ1, where τ1 is the mean life of
(1966) applied only the relativistic regime τ ∝ E−5. How- thefirstnuclideinadecaychain.Theheatingrateperunit
ever, as we will see later, the mean-lives of the nuclei are mass is then
Q˙(t)=− Ee,i dNi ≈ e−1 hEe(t)i, (13)
hAim dt hAim t
Xi u u
2 For neutron and mirror nuclides such as 3H, the comparative
half-livesare ∼103 s corresponding to |MN|2 ∼1. Such transi- where hAi is the mean mass number of the r-process ma-
tionsarecalledassuperallowedtransitions.Thesetransitionsare, terial, and mu is the atomic mass unit. Note that e is the
however,absentinr-processmaterial. Eulernumber,whicharisesfromthefactthatthedecayrate
4 K. Hotokezaka, R. Sari, and T. Piran
1016 1
Formula Eq.(13)
1015 HW+16
e
g]1014 at
g/s/1013 ng r
e [er1012 heati
-e heating rat11110000118901 -ormalized e 0.1 Formula Eq.(13)
107 N NR-Coulomb
HW+16
106
100 101 102 103 104 105 106 107 100 101 102 103 104 105 106 107
time [s] time [s]
Figure 2. The heating rate of the ideal chains of allowed beta decay by electrons. Left panel: the specific heating rate derived by
Eq. (13). Right panel: the heating rate normalized by the relativistic regime of Eq. (14). Also shown in both panels is the electron
heatingratetakenfromHotokezaka etal.2016,wheretheheatingrateisobtainedbyusingEvaluatedNuclearDataFile.Hereweadopt
|MN|2=0.05,hAi=200,andhZi=50fortheanalyticmodel.
of each nuclide is proportional to e−t/τ. One can obtain Q˙ non-relativistic Coulomb regime (magenta dotted line) at
by using Eq.(6) and (11). late times.
Intherelativisticandnon-relativisticCoulombregimes, It is worthy noting that the formula with the non-
we can derive simple explicit forms of Eq. (13). As the relativistic Coulomb limit reproduces the full heating rate
lifetime of beta-unstable nuclides monotonically increases after103s,eventhoughitshouldbevalidonlyafter∼106s.
with decreasing E , the relativistic regime is valid at early Thiscanbeunderstoodasfollows. Themeanlifeisapprox-
0
times and the non-relativistic Coulomb regime is valid imately proportional to E0−4 between the relativistic and
at late times. More specifically, the relativistic regime is the non-relativistic regimes, and thus, the energy genera-
valid until tR ≈103s(0.05/|MN|2) and the non-relativistic tion rate evolves as E0/t ∝ t−45. In addition, in this stage,
CoulombregimeisvalidaftertNC≈106s(0.05/|MN|2).Us- ǫe changes from 1/2 to 1/4, which approximately corre-
ing Eqs. (10) and (12), we obtain the heating rate in these sponds to ǫe ∝ t−19. As a result, the electron heating rate
regimes: is ∝ t−1.35, which is quite similar to the one in the non-
6 relativistic Coulomb regime.
1.2·1010 erg t−5
sg day Note that Colgate & White (1966) and Metzger et al.
Q˙(t)≈ 0.3·101×0 hesAr·ggi−2t0d−10a34y(cid:16)|M0.N05|2(cid:17)−15 (t.tR), (14) (dc2ae0sce1a0yt)hraeeastscouhtmeasletnthuhemavtbaaellrenyoufcolfreasudtsaiobtahilcaitttiyvueinnndauercsglieoniegsdleeacsrtreeaapds.ieoIsancwttiihvtihes
 ×hZi7−0·31hAi−2010(cid:16)|M0.N05|2(cid:17)−13 (t&tNC), tdiimster.ibTuhteisdajusssutmnpextitotnoitshveaslitdabifletnhuecrlaeid,ioi.aec.,tiavtelantueclteiimaerse.
wheret istimeinunitsofaday,hAi isthemeanmass Under such an assumption, the resulting heating rate de-
day 200
number normalized by 200, and hZi70 is the mean proton clines more steeply as ∝t−1.4 in the relativistic regime. As
numbernormalized by 70. Note that the overall magnitude wewilldiscussinthenextsection,theactualsituation isin
of theheating rate isdetermined by themean values of the between these two assumptions.
nuclear quantities, A, Z, and M . These values should be
N
constant within an order of magnitude, and thus, the mag-
nitude of the heating rate does not depend significantly on 5 DEVIATION FROM OUR ASSUMPTIONS
thedetailsoftheabundancepatternofther-processnuclei.
The analytic formula derived in the previous section re-
Furthermore, we emphasize that the formula of Eq. (14) is
produces remarkably well the result based on the nuclear
independentofthedistributionofthenucleardecayenergy.
database.However,therearetwoimportanteffectsthathave
Figure 2 depicts the heating rate obtained
notbeentakenintoaccount.Herewediscusstheroleofthese
from Eq. (13) and the one derived using a nuclear
effects.
database (Hotokezakaet al. 2016; see also similar heat-
ing rates in Metzger et al. 2010; Goriely et al. 2011;
Roberts et al. 2011; Korobkin et al. 2012; Wanajo et al.
5.1 The role of forbidden transitions
2014; Lippuner& Roberts 2015). We find that the heating
rate based on the simple analytic formula reproduces the Higher orbital-angular momentum transitions (unique for-
one based on the database remarkably well. In order to see bidden): The light particles’ wave function in the matrix
more details, the right panel of Fig. 2 shows the heating element Eq. (4) can be expanded in a series of spherical
rates normalized to the values obtained for the relativistic harmonics,ofwhichthelthtermisproportionalto(Pr/~)l,
regime (Eq. 14). The normalized analytic heating rate whereP~ isthetotalmomentumofthelightparticles.Thelth
(blue solid line) is flat at early times and it approaches the transition corresponds to the transition in which the light
Beta-decay heating rate of macronova 5
particles carry off orbital angular momentum of l~. This
expansion converges rapidly on theenergy scale of beta de- 2nd p Forbidden
1st u Forbidden
cay on the length scale of nucleus r ∼ fm. As a result,
n 10 1st p Forbidden
the lth transition probability is suppressed by a factor of Allowed
(Pr /~)2l . (0.1)2l. For first unique forbidden transitions,
anandditionalshapefactor2(p2+p2)shouldbemultipliedin 2c]
ν e e
m
tinheτe∝lecEtr0−on5 isnpetchterunmono-freElaqt.iv(i6s)t.icTChoisulsohmapberfeagcitmoer,rweshuilcths E [0 1
can be seen in Fig. 3 (a blue dotted line). Even though the
number of beta unstable nuclides that disintegrate mainly
via unique forbidden transitions is small, they may play a 0.1
role by increasing theheating rate after a few hours.
100 102 104 106 108 1010 1012 1014 1016
Relativistic transitions (parity forbidden): Some inter-
t [s]
actionsmixthelargeandsmallcomponentsofDiracspinor 1/2
of the nucleon in the matrix element Eq. (4). A transition
duetosuchaninteractionisaparity forbiddentransitionas Figure 3. Energy and half-life of beta unstable r-process nu-
it changes the nucleus’ parity without removing the orbital clides. Open circles, closes, filled squares, and filled circles are
angularmomentum(seemagentacrossesinFig.3).Thecor- allowed,firstparityforbidden,firstuniqueforbidden,andsecond
responding amplitude is suppressed by a factor of O(v /c) parity forbidden transitions respectively. Here the data points
n
orO(Zα)comparedtoallowedtransitions.Herethevelocity are taken from Evaluated Nuclear Data File ENDF/B-VII.1 li-
brary (Chadwicketal. 2011). Each curve depicts the theoreti-
ofnucleonsv istypicallyv ∼0.1c.Asaresult,theproba-
n n
cal expectation with a constant nuclear matrix element of each
bility of these transitions is lower than the allowed ones by
typeoftransitions:theallowed(redsolid),thefirstparityforbid-
a factor of O(v2/c2) or O(Z2α2). The theoretical curves of
n den (magenta dashed and dot-dashed), the first unique forbid-
thefirstorderparityforbiddentransitionsareshown asthe
den(bluedotted),andthesecondparityforbidden(green).Here
dashed and the dot-dashed lines in Fig. 3. Here we use a weadopt|MN|2=0.05andhZi=50forallanalyticmodelsbut
suppression factor of (vn/c)2 ≈ 0.01 and (Zα)2 ≈ 0.25, re- |MN|2=0.01forthefirstuniquetransition.
spectively. The first order parity forbidden transitions have
anelectronspectralshapethatissimilartotheallowedtran-
sitions. As one can see in this figure, the curves of these 1
transitions have the same shapes to the allowed one with a
constant shift in the half-life. The existence of these transi-
tions in addition to the allowed ones increases the heating ∝T-0.1
rate. The lifetimes of second order parity forbidden transi- )2
tions,inwhichangularmomentumof~iscarried off bythe T1/
>
lightparticles,aretoolongtoberelevantforthemacronova (n
ai
h
heating rates (see green points in Fig. 3). c
N ∝T-0.2
101 102 103 104 105 106 107 108
5.2 Deviation from the ideal-chains T [s]
1/2
approximation
Beta decay chains terminate when they reach stable nu- Figure 4. The cumulative distribution of the lifetime of decay
clides.Oncethishappenstheseterminatedchainsdon’tcon- chains.T isthesumofthehalf-livesofbetaunstablenuclides
1/2
tributetotheheatingrateanymore.Theoveralllifetime of ofadecaychain.
a chain, T , can be estimated from the sum of the half-
1/2
livesofnuclidesinthechain.Thecumulativedistributionof
6 CONCLUSION AND DISCUSSION
the chains for A = 90–210 as a function of the chains’ life-
timeisshowninFig.4.Thenumberofthechainsbeginsto We derive an analytic form of the macronova heating rate
decrease slowly as ∝T1−/02.1 at ∼100s. After about 10 days by considering statistical assembly of radioactive r-process
it decreases slightly faster as ∝ T1−/02.2. This steep decline nuclidesandFermi’stheoryofbetadecay.Theresultingan-
at late times due to the termination of the decay chains is alyticformula reproducestheheatingratederivedfrom the
consistent with the assumption made by Colgate & White nuclear database remarkably well. Within the assumption
(1966) and Metzger et al. (2010). thattheidealdecaychainsofallowedbetatransitionsgener-
Insummary,thecontributionofforbiddentransitionsto ateradioactiveheats,weshowthattheheatingrateevolves
theheatingrateslightlyincreasestheheatgenerationatlate as ∝ t−6/5 at early times and ∝ t−4/3 at late times. The
times.Onthecontrary,atthesametime,theterminationof overallmagnitudeofheatingrateisdeterminedbythemean
the beta decay chains slightly decreases it. As a result, the valueofthenuclearmatrixelements,massandatomicnum-
combined effects on the heating rate somehow cancel out. berof beta unstablenuclides involved in the decay chains.
Note that these corrections to the heating rate depend on We discuss the role of forbidden transitions and the
theactual abundancedistribution of thechains. deviation from the ideal-chains approximation. The former
6 K. Hotokezaka, R. Sari, and T. Piran
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ACKNOWLEDGMENTS
We thank Michael Paul, Kohsaku Tobioka, and Shinya
Wanajo for useful discussions. This research was supported
byanERCadvancedgrant(TReX)andbytheI-COREPro-
gram of the Planning and Budgeting Committee and The
Israel Science Foundation (grant No 1829/12), and an ISF
grant.