Photoionization Heating of Nova Ejecta by the Post-Outburst Supersoft Source

0.86 MB·English

by  Timothy Cunningham

Submitted to the Astrophysical Journal
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
PHOTOIONIZATION HEATING OF NOVA EJECTA BY THE POST-OUTBURST SUPERSOFT SOURCE
Timothy Cunningham1,2, William M. Wolf2, and Lars Bildsten2,3
Submitted to the Astrophysical Journal
ABSTRACT
Theexpandingejectafromaclassicalnovaremainshotenough(∼104K)tobedetectedinthermal
radio emission for up to years after the cessation of mass loss triggered by a thermonuclear instabil-
ity on the underlying white dwarf (WD). Nebular spectroscopy of nova remnants confirms the hot
temperatures observed in radio observations. During this same period, the unstable thermonuclear
5
burning transitions to a prolonged period of stable burning of the remnant hydrogen-rich envelope,
1
causing the WD to become, temporarily, a super-soft X-ray source. We show that photoionization
0
heatingoftheexpandingejectabythehotWDmaintainstheobservednearlyconstanttemperatureof
2
(1−4)×104 Kforuptoayearbeforeaneventualdeclineintemperatureduetoeitherthecessationof

n the supersoft phase or the onset of a predominantly adiabatic expansion. We simulate the expanding
a ejecta using a one-zone model as well as the Cloudy spectral synthesis code, both incorporating the
J
time-dependentWDeffectivetemperaturesforarangeofmassesfrom0.60M to1.10M . Weshow
(cid:12) (cid:12)
3 that the duration of the nearly isothermal phase depends most strongly on the velocity and mass of
2 the ejecta and that the ejecta temperature depends on the WD’s effective temperature, and hence its
mass.
] Subject headings: stars: novae – stars: cataclysmic variables – stars: white dwarfs – X-rays: binaries
R
S
. 1. INTRODUCTION nosity as modeled by Starrfield et al. (1974), Fujimoto
h
(1982), Sala & Hernanz (2005), Hachisu & Kato (2010),
p Classical novae (CNe) are caused by the ejection of
and Wolf et al. (2013). This phase has been decteced in
- ∆M ∼ 10−4M of matter from an accreting white
o ej (cid:12) many galactic novae (Oegelman et al. 1984; Shore 2008;
dwarf (WD) triggered by a thermonuclear instability
r Krautter 2008; Schwarz et al. 2011), and observations in
t in recently-accreted hydrogen-rich matter (Gallagher &
s M31 are making it clear that a SSS phase is likely to
Starrfield 1978; Starrfield et al. 2008). The expanding
a follow every classical nova outburst (Henze et al. 2011,
[ ejecta are bright in the radio for hundreds of days after
the optical peak (Seaquist & Bode 2008) and are well 2014b). During the SSS phase, the WD has L≈104L(cid:12)

1 modeled by assuming the radiation comes from thermal andsinceitiscompact,isastrongsourceofradiationat
v bremsstrahlung emission from an expanding medium of kTeff ≈ 50−100 eV. This SSS phase can be as short as
0 nearlyconstantelectrontemperatureT ≈(1−4)×104K a few days for novae with the most massive WDs (Tang
9 e et al. 2014; Henze et al. 2014a) and as long as a decade
(Seaquist&Palimaka1977;Hjellmingetal.1979;Taylor
6 for low-mass WDs like V723 Cas (Ness et al. 2008). We
et al. 1988; Eyres et al. 1996). Such temperatures are
5 show here that the SSS phase is a sufficiently powerful
also observed via nebular spectroscopy long after opti-
0 sourceofphotoionizingheatingtosustainthehighejecta
cal peak in novae like V1974 Cygni 1992 (Austin et al.
1. 1996) and T Pyxidis V351 (Shore et al. 2013). How- temperature long after the optical peak.
0 ever, between the fading of the optical emission (which We start in Section 2 by reviewing the physics of pho-
5 may correspond to the end of mass ejection) and the toionization balance and establish our simplified model
1 endoftheradioemissionseveralhundreddays later, the of hydrogen and helium ionization in nova ejecta. We
: derive the temporal evolution of the ejecta temperature
v ejecta undergo an expansion in volume by several orders
inSection3anddiscusscomputationalmodelsinSection
i of magnitude. In the absence of a heating source, the
X 4. Wecompareoursemi-analyticresultstothecomputa-
ejecta would undergo adiabatic expansion, leading to a
tional results in Section 5, and end by showing how this
r large temperature decline to values orders of magnitude
a less than the observed 104 K. The cause of these much new understanding can inform us about the properties
of CNe ejecta, and, potentially, the WD mass.
higher temperatures for such prolonged periods is the
focus of this paper.
With the rise of frequent X-ray monitoring of classi- 2. PHOTOIONIZATIONBALANCE
cal novae, it is becoming increasingly clear that most, if For the late time evolution of interest to us, we chose
not all, CNe are followed by a super-soft X-ray source tomodelthenovaejectaatagetasonezoneofconstant
(SSS) phase during which the WD shrinks in radius to mass ∆M in a uniform density sphere (i.e. not a shell)
ej
nearly its pre-nova size, but continues to burn a rem- of radius r(t) = v t expanding at constant velocity v .
ej ej
nant hydrogen envelope at nearly the Eddington lumi- A hot WD of mass M and radius R resides at the
WD
center. TheejectahasacompositionX,hererepresented
1DepartmentofPhysics,King’sCollegeLondon asavectorofmassfractions. Thenumberdensityofany
2DepartmentofPhysics,UniversityofCalifornia,SantaBar- given isotope, i, in the ejecta is
baraCA93106
3Kavli Institute for Theoretical Physics, University of Cali- 3X ∆M
fornia,SantaBarbara,CA93106 n(X )= i ej , (1)
i A m 4πv3t3
i p ej
2 Cunningham et al.
where A is the mass number of isotope i and m is the
i p
proton mass. The electron-ion coupling time is always Table 1
Str¨omgrenBreakoutTimesa
muchshorterthantheage,t,duringthetimerelevantfor
our calculations, so there is only one temperature, T(t).
H He
Weexpressthefundamentalequationofthermodynamics
intermsofejectavolumeV,entropyS,andtotalnumber MWD tSb QH,47c tSd QHe,47e
of particles N as (M(cid:12)) (d) (1047 s−1) (d) (1047 s−1)
(cid:18) (cid:19)
3dT dV 0.60 31 8.0 95 1.2
TdS =NkT + . (2) 0.80 28 11.0 69 3.6
2T V 1.00 31 8.0 60 5.6
1.10 37 5.0 67 4.4
For a spherical ejecta of pure ionized hydrogen (N =
1.20 41 4.0 214 3.6
2N ) with r =vt, this becomes 1.30 132 2.9 387 2.7
e
(cid:18) (cid:19)
ds d 6
T dt =nekT 3dtlnT + t . (3) ab∆StMr¨oemjg=re1n0−br5eaMk(cid:12)ou;tvteijm=e1fo0r3pkumres−h1ydrogen
c EmissionrateofH-ionizingphotons
wherewenowconsiderthespecificentropys=S/V and d Str¨omgrenbreakouttimeforpurehelium
ne is the number density of electrons. e EmissionrateofHe-ionizingphotons
Theonlysourceofheatingwewillconsideristhatfrom
breaks out at an age of
the central, hot, WD in the SSS phase, which is a sys-
tem that has been modeled extensively (Starrfield et al. (cid:18) v (cid:19)−1(cid:18) ∆M (cid:19)2/3(cid:18) Q (cid:19)−1/3
1974;Fujimoto1982;MacDonald&Vennes1991;Sala& t ≈30 d ej ej H .
S 103 km/s 10−5 M 1048 s−1
Hernanz 2005; Hachisu & Kato 2010; Wolf et al. 2013). (cid:12)
(6)
We extended the work of Wolf et al. (2013) to generate
where we have scaled v , ∆M , and Q to values typi-
the time-dependence of T for a range of WD masses, ej ej H
eff cal of novae on intermediate-mass WDs. Table 1 shows
as shown in Figure 1. These simulations used the 1-D
that the H Str¨omgren sphere breaks out by 50 days for
stellar evolution code MESA (Paxton et al. 2011, 2013),
all but the most massive WDs for ∆M = 10−5 M
following the time-dependent evolution of WDs accret- ej (cid:12)
and v = 103 kms−1. This confirms a time after which
ingsolarcompositionmaterialat10−9 M yr−1 through ej
(cid:12) all of the ejecta is exposed to the photoionizing radia-
several nova flashes with super-Eddington winds as the
tion field. The pure He breakout times for singly ion-
primary mode of mass loss. In particular, we only dis-
ized He Str¨omgren spheres are systematically later due
play the results from the end of mass loss until the end
to less ionizing radiation available for a pure He ejecta.
of quasi-stable hydrogen burning for one representative
The more massive WDs have longer Str¨omgren break-
nova cycle. The original simulations did not have data
out times at the same ejecta parameters because while
fora0.8or1.1M WD,sowegeneratedadditionaldata
(cid:12) theyproduceharderspectrathanlower-massmodels,the
for this work. At any given age since mass loss ended,
total luminosity is not significantly higher, so the to-
we know the WD radius and T , and hence the radia-
eff tal number of ionizing photons emitted, Q, decreases,
tion field that powers the photoionization heating. This
requiring more time for a complete ionization of the
is the only information we use from the MESA models.
ejecta. Additionally, the SSS phases for the most mas-
The ejecta parameters ∆M and v used are indepen-
ej ej sive models are short enough that the ionization front
dent of the MESA models, and, in general, would vary
never makes it to the ejecta radius before the ionizing
significantly for novae on WDs of different masses.
source fades greatly, causing excessively long Str¨omgren
2.1. Str¨omgren Breakout breakout times. We explore the ionization state of the
ejecta at late times in Section 5.
Ourfirstconcerniswhetherthephotoionizationsource
Beck et al. (1990) performed a much more detailed
is adequate to maintain a large region of mostly ion-
calculation of this breakout for a He Str¨omgren sphere
ized plasma in the ejecta. Hence, we start by finding
for a mixed ejecta with heavier elements (Z >2), and a
the Str¨omgren sphere radius, R , and compare it to the
ejecta dimension, vejt. For simSplicity, we assume pure lower ejected mass and velocity (∆Mej =1.3×10−6 M(cid:12)
ionizedhydrogen(ne =np)withintheStr¨omgrensphere, andvej =6×102 km s−1 thanweusedforTable1. Soas
andequatetherateofemissionofH-ionizingphotonsby to make a close comparison, we used their parameters,
the WD, QH, with the rate of recombinations in that includingtheirWDTeff (seeFigure6ofBecketal.1990)
volume, as inputs to our homogenous model. We found that the
4 H and He Str¨omgren breakouts occur at 31 days and
Q = πR3αn n , (4)
H 3 S p e 77 days, respectively, very close to the breakout they
reported at t=76 d for the He Str¨omgren sphere.
where α is the radiative recombination rate of hydrogen.
At early times R is far inside the outer ejecta radius,
S 2.2. Ionization Timescale
v t, and the ratio scales as
ej
AftertheStr¨omgrenbreakout,weallowthattheejecta
R (cid:18)4πQ (cid:19)1/3 may no longer absorb the majority of the photoionizing
v St = 3αNH2 (vejt), (5) radiation. Wecharacterizethisviatheopticaldepth,τν,
ej p through the ejecta at frequency ν,
whereNp isthetotalnumberofprotons. Hence,atearly (cid:88)
times,theStr¨omgrensphereisinsidetheejecta,andonly τν(r)= n(Xi) σpi(ν,Xi) r, (7)
i
Nova Ejecta Heating 3
where σ (ν,X ) is the photoionization cross-section and
pi i
thesumisoverallisotopesinX. Thetimeanisotopecan 0.60 M(cid:12)
spend in the neutral state is set by the photoionization 0.80 M
(cid:12)
timescale,whichwedefineattheejectaouterradius(r = ) 1.00 M
K (cid:12)
vejt), giving ( 1.10 M(cid:12)
1 (cid:18)R (cid:19)2 (cid:90)∞ B (T ) ure 60 11..2300 MM(cid:12)
=π WD dν ν eff σ (ν) , (8) t 1 (cid:12)
t (X ) r hν pi a
ion i r
IXi/h pe
m
where B (T ) is the Planck function evaluated for the
ν eff e
WDT ,andI istheionizationenergyforisotopeX . T
eff Xi i
Weconsideronlythetwomostsignificantelementsfor e
v
photoionizationheating;HandHe. Weassumethatafter
i
t
the breakout time n(X ) is low enough that the ejecta is c
i e
optically thin for Lyman limit photons (hν = 13.6 eV). ff
In this limit we neglect any secondary ionization caused E
byarecombinationemission(Baker&Menzel1938)and 5
0
consideronlyhydrogenicionizationprocesses(essentially 1
assumingtherearenoneutralheliumatoms). Thisgives
two dynamical expressions for the hydrogenic number 10 102 103
density of hydrogen and helium, respectively
Age (days)
n(H)
n˙(H)=− +n n(H+)α(T,H+) ; (9) Figure 1. ThecalculatedtimedependentT ’sofdifferentmass
tion(H) e WDs as simulated by Wolf et al. (2013), incleuffding two new cases
n(He+) (0.80&1.10M(cid:12))here. Theageisthetimesincetheendofmass
n˙(He+)=− +n n(He++)α(T,He++) . (10) loss driven by the thermonuclear instability. The rapid decline at
t (He+) e latertimesforthethreemostmassiveWDscorrespondstotheend
ion oftheirSSSphases.
As long as the WD is emitting, we found that the
timescale over which these quantities relax is orders of with Eion(Xi) being the average photoelectron energy.
magnitude smaller than the age of the ejecta, so we can Under the optically thin assumption Eion is given by
consider an instantaneous steady state solution, where Draine (2011) as
n˙(X )=0, yielding
i ∞
(cid:18) 1 (cid:19) (cid:82) dν B(hνν,T)σpi(ν,Xi)(hν−Z2IH)
n(Xi) tion +Pci(T,Xi) =nen(Xi+1)α(T,Xi+1)(,11) Eion(Xi)=Z2IH/h (cid:82)∞ dν B(ν,T)σ (ν,X ) .
hν pi i
with Pci(T,Xi) the temperature-dependent collisional Z2IH/h
ionization rate which we adopt from Lang (1980). This (14)
equation yields the very small neutral fraction that we Using equations (3) and (12) we find the equation for
must know to calculate the heating rate from photoion- T(t) as
ization. dT 1 2T
= [Γ −Λ(T)]− . (15)
dt 3kn pi t
3. TIME-DEPENDENTENTROPYEVOLUTION e
Now that we have shown that photoionization balance Finding a temperature evolution, T(t), requires solving
drivestheneutralfractiontoalowvalueafterthebreak- equations (13) & (15) simultaneously.
outoftheStr¨omgrensphere,wecanconsidertheentropy Beforeshowingthedetailedsolutionsofequation(15),
evolution of the expanding ejecta, which will lead us to it is best to first exhibit a few limits. The first limit is
a temperature evolution. The photoionization heating to assume that the plasma is nearly fully ionized, and
timescale is much less than the age initially after break- that ionization balance is set by equation (11) but with
outsothatanyinformationregardingtheinitialentropy little impact from collisional ionization. This yields a
is lost. This will simplify our calculation as we con- simple relation of n /t ≈n n α and a heating rate of
H ion e p
siderthestateoftheplasmaunderphotoionizationheat- Γ ≈E n n α. Sincet (cid:28)twhentheWDishot,we
pi ion e p ion
ing and radiative cooling, where we use a cooling func- found that ejecta of different initial temperatures would
tionΛ(T)determinedbycomparativesimulationsrunin veryrapidly(onatimescalemuchlessthantheage)reach
Cloudy as detailed in Section 4. The resulting rate of a state of Γ ≈Λ(T) and a temperature, T nearly in-
pi iso
change of specific entropy s is then dependent of the electron density. The scale of these
heating and cooling terms (which nearly cancel in equa-
ds
T =Γ −Λ(T) , (12) tion (15)) are so much larger than the 2T/t adiabatic
dt pi expansion term that the ejecta evolves from one isother-
where the rate of photoionization heating, Γ , is given mal (heating balances cooling) state to the next under a
pi
by condition of thermal and photoionization balance. The
Γpi =(cid:88)tn(X(Xi))Eion(Xi) , (13) raersisuelstifnrgomnetahrelycoisnodtihteiormnaolfΓevpoil≈utΛio(nTo)ffotrhidsiffpehraesnettWheDn
ion i
i
4 Cunningham et al.
100000
T ’s. There is nearly no sensitivity to the actual elec-
eff
tron density when in this limit, so ejecta clumping will constant; solar
not matter, and all initial entropy information is lost.
inverse square; solar
However, the heating term, Γ /n , in equation (15) is 90000
pi e
∝ t−3, implying that there will come a time, which we constant; nova
call t , beyond which the heating term is comparable
iso inverse square; nova
to, orlessthan, theadiabaticexpansionterm, 2T/t, and ) 80000
K
we expect to see a temperature decline. If we neglect
(
radiative cooling, this critical timescale, tiso, is when re
Γ /3kn = E n α/3k = 2T/t. For the case of pure, u
pi e ion p t 70000
ionized Hydrogen this gives a
r
e
p
(cid:18) ∆M (cid:19)1/2(cid:18) v (cid:19)−3/2 m
tiso ≈250d 10−5 Mej 103 kejm/s Te 60000
(cid:12)
(cid:18) E (cid:19)1/2(cid:18) kT (cid:19)−1/2(cid:16) α (cid:17)1/2
× ion .
10 eV 1 eV 10−13 cm3s−1 50000
(16)
t = 204 days
Wewillshowinthefollowingsectionthatourdetailed
40000
thermalevolutioncalculationsfortheseremnantscanbe
0.0 0.2 0.4 0.6 0.8 1.0
simplydescribedbytheargumentsabove,withnearlyall
solutionsexhibitinganearlyisothermalphasefort<t , Fractional Mass Coordinate
iso
followed by a temperature decline. Figure 2. Temperature profiles of Cloudy simulations of nova
ejectawithvaryingdensityprofilesandcompositions. Eachrepre-
4. CLOUDYSIMULATIONS sents 10−5 M(cid:12) of ejected material with an outer velocity of 1000
We also simulated the nova ejecta using the radiative kms−1 beingheatedbyradiationfroma1.00M(cid:12) WDSSS.The
abscissa measures the total enclosed mass at a given shell radius,
transfercodeCloudy(version13.03,Ferlandetal.2013),
withtheoutermostpointbeingthepositionenclosingallthemass
whichhasbeensuccessfullyusedtomodelnovaejectain andaradiusofvejt.
thepastbySchwarz(2002),Vanlandinghametal.(2005),
and Schwarz et al. (2007). Cloudy allows a user to spec- incorporatedynamiceffectsotherthantheartificiallyin-
ifythedensitystructureofagas,itscomposition,andan serted adiabatic cooling.
incident radiation field. It then computes the ionization For these simulations, we found that the temperature
state of the material, yielding line strengths, tempera- was ratherconstantwith radius, leading toasingle tem-
tures, and many other details, including realistic cooling perature for each model that we defined as the mass-
curves. average over the outer 50% of the mass. These tempera-
To make the models in Cloudy as similar to our semi- ture histories are what we compare to the semi-analytic
analytic approach as possible, we set the radiation field models in Section 5. For later use in our semi-analytic
at a given time to be a blackbody spectrum with the work (e.g. equation (15)) we also extracted a cooling
T and radius set by the MESA models for that WD curve by finding the volumetric cooling rate at each grid
eff
mass. The constant density gas reaches an outer radius point, subtracting off the portion of that cooling due
r (t)=v t and an inner radius r (t)=0.01r (t) es- to expansion from equation (17), and then dividing by
out ej in out
sentiallyreproducingourconstantdensityspheremodel. n n . The resulting cooling curves depend on the WD
H e
To account for temperature decline from adiabatic ex- T (andhenceWDmass)duetodifferentdegreesofion-
eff
pansion, we insert a “cooling” term izationandlinecooling. Wethusgeneratedcharacteristic
Cloudy cooling curves for each WD mass by choosing a
Λ ≈2×10−16∆M v−3 T t−4 erg s−1 cm−3,
exp ej,−5 ej,3 4 8 time in the Cloudy calculations about halfway through
(17) the observed temperature plateau in log age. At times
that matches the second term in equation (3), where beyondthese,theheatingandcoolingtimescalesbecome
∆M =∆M /(10−5 M ),v =v /(1000km s−1), comparable to the age, and we can no longer trust the
ej,−5 ej (cid:12) ej,3 ej
T =T/(104 K), and t =t/(108 s). The ejecta material equilibrium Cloudy calculation, but rather should solve
4 8
was set to have a solar (Cloudy’s default abundance set) the fully time-dependent problem. We make progress on
compositionsothatcoolingbymetalswouldbeincluded. that aspect in the Section 5.
We then divide the WD evolution histories into twenty- Since Cloudy is inherently a multi-zone code, we can
one times, divided equally in log age, and use Cloudy use it to test our assumption that the density profile
to compute a temperature/ionization structure for the and composition has little influence on the temperature
ejecta. These calculations assume that the cloud has structure of the ejecta. We ran four models with our
reached both a photoionization and thermal equilibrium fiducial ejecta parameters of ∆Mej = 10−5 M(cid:12) and
with the applied radiation field. We already established v = 1000 kms−1 with the radiation field given by the
ej
in Section 3 that these are valid assumptions at early 1.00 M WD model. These four models used two dif-
(cid:12)
times while t (cid:28) t, so these Cloudy simulations pro- ferentdensitydistributions,constantandinversesquare,
ion
vide physically motivated snapshots at those early times as well as two sets of elemental abundance ratios, solar
of concern for us. Essentially, these simulations in their and nova. The nova abundances are a built-in option
current form solve (15) assuming dT/dt = 0 and do not in Cloudy and are intended to model the abundanes ob-
Nova Ejecta Heating 5
served in the ejecta of nova V1500 Cygni (Ferland &
Shields 1978). Since the adiabatic cooling is necessar- Table 2
IsothermalEvolutionofNovaEjectaa
ily radius-dependent for a non-constant density profile,
thesimpleextracoolingofequation(17)couldnotprop-
erly model expansive cooling. To be consistent, we in- MWD Tisob tisoc tthind
cluded no extra cooling from expansion in this compar- (M(cid:12)) (104 K) (d) (d)
ision study, resulting in hotter temperatures and longer 0.60 2.2 229 36
0.80 3.2 155 33
“plateau” periods. The temperature profiles of all four
1.00 4.7 152 47
casesatatimeapproximately200daysafterejectionare
1.10 4.5 217 74
shown in Figure 2. Importantly, the temperature range
changesnomorethanbyorderunityintheconstantden- a ∆Mej=10−5 M(cid:12); vej=103 kms−1
b Time-averagedtemperatureinisothermalphase
sitymodelsandisnearlyexactlyisothermalintheinverse c Timeaftertemperaturedrops20%fromearlytimes.
squarecases. Also,thetemperaturesforthesemodelsdo d Durationofopticallythick(tolyman-limitphotons)phase
not change greatly as the density profile and/or com-
theresultinginconsistentresultsfromhigher-massWDs.
position are changed, allowing us to use our simple so-
Most of these results can be motivated and explained by
larcompositionandconstantdensityassumptionsinour
ouranalyticworkthatderivedequation(16),yieldingthe
semi-analytic model. One important difference between
dependenceoft ontheejectaparametersandtheaver-
the different density profiles is that the opacity at the iso
agephotoelectronenergy. Suchanunderstandingwillbe
lyman-limitdivergesaswelettheinnerradiusgotozero
keytousingtheseobservedquantitiestoconstraineither
in the inverse square case. While a vanishing central ra-
the WD mass or the ejecta properties.
dius is certainly unphysical, we must be aware that the
To exhibit this physics, we explored the sensitivity of
time to become optically thick is sensitive to the choice
the plateau phase properties (T and t ) by varying
for an inner radius. In this comparison we continued to iso iso
v ,∆M andE insimulationsofapureHejectapho-
user =0.01v t,whichcausestheinitialopticallythick ej ej ion
in ej toionized by a 1.00 M WD. We found that variations
phase to last significantly longer than the similar con- (cid:12)
inejectaproperties(v and∆M )affectthedurationof
stant density models, sometimes approaching 100 days. ej ej
the plateau but do not alter its temperature. This is re-
5. RESULTS vealed in Figure 4 where the ejecta mass varies between
10−6 M (cid:46) ∆M (cid:46) 10−4 M with velocities in range
The results of the one zone semi-analytic model and (cid:12) ej (cid:12)
Cloudy simulations for WD mass 0.60, 0.80, 1.00, and 102 kms−1 ≤vej ≤5×103 kms−1. For calculating Fig-
1.10 M(cid:12) are shown in Figure 3 and Table 2 from the ure 4 the Eion was held constant in time for simplicity.
time-dependent WD models described earlier. All tem- Thishighlightsthatthedynamicbehavioroftheejecta
peraturesareinthe104 Kregion,consistentwiththatin- is described by these two quantities, vej and ∆Mej, that
ferredfromradioobservationsandnebularspectroscopy, we can parametrize into a factor
and show a long plateau of nearly constant temperature ∆M (cid:16) v (cid:17)−3
at the value expected for photoionization and thermal ζ ≡ ej ej . (18)
10−5 M 103 kms−1
equilibrium. Results for the two higher-mass WD mod- (cid:12)
els are not shown for reasons described below. UsingEquation(16),ζ canbeinferredfromtheobserved
The four WD evolutions shown exhibit a period of parameters t and T as
iso iso
isothermal expansion lasting around 100-200 days that
we will refer to as the plateau. Numerically, we de- (cid:18) t (cid:19)2(cid:18)kT (cid:19)
fined this as the duration over which the the tempera- ζ(tiso, Tiso)= 25i0sod 1 eiVso ×
ture stayed within 20% of the plateau temperature T .
iso
These durations are in accord with the timescale pre- (cid:18) E (cid:19)−1(cid:16) α (cid:17)
dicted by Equation (16), and show that, even with the ion , (19)
10 eV 10−13 cm3s−1
WD still providing a high level of photoionization, the
adiabatic expansion term eventually plays an important potentially yielding insights into the ejecta properties
role. from observations. The average photo-electron energy
The two methods - one-zone model and Cloudy simu- E depends on the WD’s T and is the main determi-
ion eff
lations - agree well for all four cases. The two higher nantofT ,asisevidentinFigure5. Hence,theplateau
iso
WD mass simulations (1.20M and 1.30M ) showed temperatureismostsensitivetotheWDT (andhence
(cid:12) (cid:12) eff
much shorter plateau phases (up to 80 days). However, WD mass).
this was due to the sharp decrease in T at early times Theplateauphasespacebehaviorofζ isshowninFig-
eff
relative to the 4 lower mass WDs, see Figure 1. The ure6withdashedlinesofconstantζ. Fornearlyallcases,
plateau phases were so short in these cases that the weseethatT ishighlyindependentoftheejectainfor-
iso
ejecta remained optically thick at the photoionization mation,v and∆M . Thisimpliesthatchangesmadeto
ej ej
edge throughout and in our model these solutions would the dynamic properties of the ejecta should be relatively
nothavebeenself-consistent. Forthisreasonweconsider insignificantforthetemperature,makingitapotentially
only the WDs in the mass range 0.60−1.10 M . Note clean diagnostic for WD mass. This also implies that
(cid:12)
that for lower ejecta masses and higher ejecta velocities changes to the ejecta profile (e.g. non-spherically sym-
expected of novae on higher-mass WDs, it is possible metric ejections) or homogeneity would unlikely impact
that the plateau phase would occur after the ejecta has T .
iso
become optically thin. In this work, though, we hold Thethermalevolutionoftheejectaatlatetimesissim-
the ejecta parameters constant and thus do not consider ple if the SSS phase has ended, as the ejecta undergoes
6 Cunningham et al.
M = 0.60 M M = 0.80 M
WD WD
)
K (cid:12) (cid:12)
(
e
r
u 104
t
a
r
e
p
m
e
T
103
M = 1.00 M M = 1.10 M
WD WD
)
K (cid:12) (cid:12)
(
e
r
u 104
t
a Cloudy
r
e
p Optically Thick
m
e Optically Thin
T
103
101 102 103 101 102 103
Age (days) Age (days)
Figure 3. TemperatureevolutionofthenovaejectaforfourWDmassesusingoursemi-analyticmodel(blue)withcoolingcurvesextracted
fromCloudyandsimulationsruninCloudy(dashedgold). ∆Mej=10−5M(cid:12)andvej=1000kms−1forallmodels,andthetimeevolution
oftheincidentradiationissetbytheunderlyingWDmassviathehistoriesshowninFigure1. Thedottedlineindicatesthattheejectais
opticallythickattheLymanlimitedgeatearlytimes.
an adiabatic temperature decline, and T ∝ t−2 there- T ∝ t−2 versus the more complicated relation of equa-
after. The recombination time is far longer than the age tion (20). Alternatively, a shut-off of the X-ray source
when t > t , so that this plasma would maintain itself could be confirmed by direct measurements like those of
iso
inahighlyionizedstateeventhoughphotoionizationhas Schwarz et al. (2011) and Henze et al. (2011, 2014b).
ended. Hence,itwouldstillbeathermalbremsstrahlung Equation (20) assumes a constant ionizing source, but
radio emitter during the adiabatic decline. If the SSS the WD models used in Figure 3 are still increasing
remainsonatt>t , wecanstillapproximatethether- in effective temperature when adiabatic expansion takes
iso
mal evolution. In this limit, photoionization heating is over,sotheirdeclineisevenlesssteepthenequation(20)
still operative, so the temperature decline would not be would indicate.
as steep as the adiabatic relation. Rather, for the cases
we considered here, the photoionization timescale is still 6. CONCLUSIONS
much less than the age, tion (cid:28)t, so that the heating re- We have shown that the photoionizing emission from
mains operative, and the required small neutral fraction the hot white dwarf following a classical nova is ade-
is easily maintained. If we assume negligible radiative quate to explain the inferred ejecta temperatures > 104
cooling, rather safe as T < 104 K, then equation (15) K that persist for nearly a year after the optical out-
simplifies to an integrable form. If the ejecta tempera- burst. For the typical WD mass, ≈ 0.8M , of a CNe
(cid:12)
ture was T0 at t=t0, then its value at a time t>tiso is thisnearlyisothermalphaselastsforaboutoneyear(see
simply equation(16))andhasatemperatureofT ≈(2−3)×104
T t2 (cid:18)t2 t2 (cid:19) K, in good agreement with many observed novae. Even
≈ 0 exp iso − iso , (20)
T t2 t2 t2 when the SSS phase lasts for decades, the balance be-
0 0 tweenheatingandcoolingeventuallyends,andtheejecta
exhibiting the less steep than adiabatic decline that is transitions into a phase of temperature decline roughly
evident in nearly all our results. Differentiating be- approximated analytically by equation (20). If the SSS
tween a decline from SSS shutoff versus onset of dom- phase ends early, then the ejecta temperature would
inance of adiabatic expansion with an active photoion- evolve adiabatically, as T ∝t−2.
izationsourcecouldbeattemptedbycomparingT(t)in- Though our preliminary calculations allow us to see
ferredfromradioobservationsornebularspectroscopyto the basics of how photoionization heating is balanced
Nova Ejecta Heating 7
T ∝ t−2
T ∝ t−2
4
K) 10 40
( K) 1
e
ur 200kms−1 (
at 500kms−1 ure 4 eV
er 1000kms−1 t 10 eV
p a
m 2000kms−1 er 16 eV
Te 5000kms−1 mp 22 eV
∆M = 10−5 M
ej (cid:12) e
T
E = 22 eV ∆M = 10−5 M
ion ej (cid:12)
3
10 Pure H vej = 1000 kms−1
3 Pure H
0
T ∝ t−2 1
10 102 103 104
4
K) 10 1×10−4 M(cid:12) Age (days)
( 5×10−5 M
ature 21××1100−−55 MM(cid:12)(cid:12)(cid:12) Fecraigrgireuiesrdeofo5up.thoaEtsojseeuclmteacitntrgeomnpspuefrreroamHtur4eejeeecVvtoa≤luwtEiitoihnoncfoo≤nrsv2ta2arnyetVinE.gioSanivmearunaldgaetnioeenngs--
per 52××1100−−66 MM(cid:12)(cid:12) olitghibelrewicsoenitnridbiucatitoendf∆roMmejc=oll1is0i−on5aMl i(cid:12)on,ivzeajti=on1s0(3Pkcmi =s−01).. Unless
m 1×10−6 M
(cid:12)
Te v = 1000 kms−1 0
ej 0 0.60 M
E = 22 eV 0 (cid:12)
ion 0 0.80 M
6 (cid:12)
310 Pure H K) 1.00 M(cid:12)
0 1.10 M
( 0 (cid:12)
10 102 103 104 re 00
u 5
Age (days) t
a
Figure 4. Ejecta temperature evolution for variations in ejecta er 0
velocity (top panel) and ejecta mass (bottom panel). Simulations p 00
m
cliagribrileedcoounttriabsusutimoninfgropmurecoHlliseiojencatlaiwonitizhatcioonnssta(nPtciE=ion0)a.nUdnnleegss- Te 40
oEtihoner=wis2e2ienVd.icated ∆Mej = 10−5 M(cid:12), vej = 103 kms−1 and au 00
e 0
by cooling at early times, and how the heating is weak t 0
a 3
compared to adiabatic expansive effects at late times, l
P
work remains to realize the quantitative accuracy ade- 0 ζ =0.1
quate to infer CNe ejecta properties or WD masses. We 0 ζ =10 ζ =1
0
certainly identified a degeneracy in how the ejecta infor- 0
2
mation(∆M andv )affecttheplateauphaseduration,
ej ej
which should allow observers to make predictions about 102 103
the properties of a given nova, complementing existing Plateau Duration (days)
radio techniques (Seaquist & Palimaka 1977; Hjellming
etal.1979;Heywoodetal.2005). Forexample,equation Figure 6. Characteristic plot of the plateau phase with dotted
(19)makespossibletheinferenceofanejectamassrange linesofconstantejectaparameterζ=1014× ∆Mej spacedevenly
v3
based on the length of the isothermal temperature evo- ej
inlogspace. Alineofaparticularcolorconnectsmodelsofidentical
lution inferred in the radio combined with the measured WDmass.
velocity of the ejecta and temperature.
However, specific events would be best modeled by a wouldinturnimpactthemappingoftheobservedejecta
uniqueCloudysimulationthatreflectstheknownproper- temperaturetotheWDmass. Theidealcalcuationcould
tiesoftheeventasinferredfromtheoptical,aswellasthe use a more realistic ejecta density profile, like a Hubble
X-ray supersoft phase. That would also allow for inclu- flow(Seaquist&Palimaka1977)withanappropriatefill-
sion of unusual abundances constrained from modeling ing factor, though our efforts imply that the ejecta tem-
thenebularspectroscopyasinSchwarz(2002),Vanland- perature is relatively independent of the specific density
ingham et al. (2005), and Schwarz et al. (2007), which structure for the first year or so. We hope that the ex-
8 Cunningham et al.
pandedeffortswiththeKarlG.JanskyVeryLargeArray Hjellming,R.M.,Wade,C.M.,Vandenberg,N.R.,&Newell,
tomonitornearbygalacticnovaewillenableaneweraof R.T.1979,AJ,84,1619
quantitative analysis of the late time radio observations Krautter.2008,ClassicalNovae,2ndedn.,ed.M.F.Bode&
A.Evans(CambridgeUniversityPress),232–251
(Roy et al. 2012).
Lang,K.R.1980,AstrophysicalFormulae:acompendiumforthe
We thank Laura Chomiuk, Tommy Nelson, Jeno
physicistandastrophysicist,2ndedn.(Berlin;NewYork:
Sokoloski and Dean Townsley for discussions on the ra- Springer-Verlag)
dioobservationsthatinspiredourefforts,andOrlyGnat MacDonald,J.,&Vennes,S.1991,ApJ,373,L51
for advice on using Cloudy. We also thank our referee, Ness,J.U.,Schwarz,G.,Starrfield,S.,etal.2008,AJ,135,1328
Oegelman,H.,Beuermann,K.,&Krautter,J.1984,ApJ,287,
Steve Shore, for his many helpful insights that improved
L31
the paper. This work was supported by the National Paxton,B.,Bildsten,L.,Dotter,A.,etal.2011,ApJS,192,3
Science Foundation under grants PHY 11-25915, AST Paxton,B.,Cantiello,M.,Arras,P.,etal.2013,ApJS,208,4
11-09174, and AST 12-05574. Most of the MESA sim- Roy,N.,Chomiuk,L.,Sokoloski,J.L.,etal.2012,Bulletinofthe
ulations for this work were made possible by the Triton AstronomicalSocietyofIndia,40,293
Sala,G.,&Hernanz,M.2005,A&A,439,1061
Resource, a high-performance research computing sys-
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atUCSanDiego. WearealsogratefultoBillPaxtonfor K.M.2007,ApJ,657,453
his continual development of MESA. Schwarz,G.J.,Ness,J.-U.,Osborne,J.P.,etal.2011,ApJS,
197,31
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