Fireballs, Flares and Flickering: a Semi-Analytic, LTE, Explosive

arXiv:astro-ph/0410278v1 11 Oct 2004 ag fatohsclssesfo tr oactive to stars from systems astrophysical of range physical them. the in for conditions values deriving very systems; our two different of fit lightcurves We multiwavelength to to results simulations. results numerical our detailed compare analytic more LTE and improved an structure with an fireball ionization here expanding an out and for set calculation lightcurves We observed to spectra. con- and and spectra lightcurves tinuum optical the ap- for analytic proximations to fireball results resulting the the between comparing modelled numerically, collision We the clouds. of Aqr gas aftermath two AE the system of activity terms (CV) in flaring variable cataclysmic unusual the the of explained we 2003), Introduction 1. ieal,Fae n lceig eiaayi,LE Ex LTE, Semi-analytic, A Flickering: and Flares Fireballs, lceigadflrn cusars h whole the across occurs flaring and Flickering Skidmore & Horne (Pearson, paper recent a In colo hsc n srnm,Uiest fS.Andrews, St. of University Astronomy, and Physics of School oiin tt nvriy eateto hsc n Astr and Physics of Department University, State Louisiana ujc headings: Subject depen ne temperature a ionized. strong indicate partially the data SN is from the with arising to also Fits temperature and involved. spheric Cygni material SS variable the sy for cataclysmic astrophysical parameters the of from range taken wide data a to over occur that events flickering rnfr-sas niiul SCgi-sproa:indivi supernovae: - Cygni SS individual: stars: - transfer edrv ipeaayi xrsin o h otnu ihcre a lightcurves continuum the for expressions analytic simple derive We oe rmAceinDsst Supernovae to Disks Accretion from Model crto,aceindss-bnre:ls oa:catac novae: - binaries:close - disks accretion accretion, atc,Mi oe152,Psdn,C 91125-24 CA Pasadena, 105-24, code Mail Caltech, arnSkidmore Warren ABSTRACT .J Pearson J. K. A70803-4001 LA et Horne Keith and 1 onaylyrrgo fteds.SmlryPat- Similarly with disk. innermost the associated of the region is and layer point instead boundary impact but stream-disk distributed disk the uniformly the not is across CVs in flickering the supersonic. material become rapidly would With expansion lo- material. this adiabatically, a disk cooling the to of rise expansion give cal the deposition would and temperature localised overpressure gas sudden consequent the raise a would Such Bal- energy of & (1991). Hawley of that bus mechanisms as viscosity such the represent reconnections from magnetic and of systems effect these the in high anomalously present process the flickering viscosity with associated This be well present. may an is where disk those particular, ac- accretion of in feature and, recurrent systems a creting be to appears It galaxies. ul N1987A SN dual: bevtosb rc 20)sgetthat suggest (2000) Bruch by Observations nm,NcosnHl,BtnRouge, Baton Hall, Nicholson onomy, ot ag,S.AdesK1 9SS KY16 Andrews St. Haugh, North tm.W opr hs results these compare We stems. ec foaiywe hydrogen when opacity of dence rytm-needn photo- time-independent arly ymcvrals-radiative - variables lysmic 97,drvn physical deriving 1987A, dsetao aigand flaring of spectra nd plosive terson (1981) found that the flickering in HT Cas SS Cyg exhibits optical variability over a large was associated with the inner part of the accretion range of timescales e.g. outbursts (∆m 3.5 mag, disk. These findings, however, are contradicted by t 40 days), orbital modulations (∆m ≈ 0.5 mag, Welsh & Wood (1995) for HT Cas, by Baptista P≈= 0.275130 days), flickering (∆m =≈ 0.01 0.2 (2004) for V2051 Oph and Baptista (2002) for the mag, t 1 min) and Dwarf Nova Oscillations− Low-Mass X-ray Binary (LMXB) X1822-371. All (∆m ∼0.02 mag, t 2.8 10.9 s) (Mauche & these studies show flickering to arise from a range Robinson≈ 2001; van Teeseling≈ − 1997; Honey et al. of disk radii. 1989; Warner 1995, 2004). We shall concentrate The terms “flickering” and “flaring” are often our modelling on the lightcurve from a flickering used interchangably when describing the stochas- event. tic variability of accreting sources and, even when SN 1987A was the brightest supernova observed defined in a particular field, are not necessarily since Kepler’s in 1604. The lightcurve can be bro- used consistently. Warner (1995) and Baptista ken into three phases: a “flash” lasting a few (2004) describe flickering in CVs as the continu- hours, a subsequent “bump” lasting around 4 ous, random brightness fluctuations of 0.01–1 mag months and a final exponential “tail” powered by on timescales of seconds to dozens of minutes al- radioactive decay of unstable nuclei in the ejecta. though the exact numerical ranges differ between It was a type II supernova with an identified pro- the two. In contrast the unusal CV AE Aqr is genitor (Sk-69o202) that was a blue supergiant. 7 universally described as a “flaring” source since it It is believed that it lasted 10 y as a 20-30M⊙ has rarer by brighter events that often occur in main sequence blue supergiant∼ star, before becom- batches. These events can raise the total optical ing a red supergiant for 106 y during which ∼ luminosity of the system by factors of 2-3 contrast- phase it lost 3-6M⊙ before finally becoming a blue ing with the 5-20% typical of CV flickering (Bruch supergiant again for a few thousand years (Mc- 1992) but on a similar timescale. Consequently, we Cray 1993). The rebrightening “bump” phases adopt the convention of describing the small am- arises from a photosphere moving out with ex- plitude, continuous variations as “flickering” and pelled material and eventually reversing as the reserve the term “flaring” for larger scale events. material becomes optically thin again. It is in- The underlying model used to reproduce the teresting to note that the photosphere maintained a roughly constant temperature 6000 K close AE Aqr lightcurves was based on consideration ∼ of the flux emitted by a hot, spherically symmet- to the recombination temperature for hydrogen ric ball of gas as it expanded and cooled. While at the expected densities in the expanding shell. we saw that the “expanding fireball” situation in The models of AE Aqr flares in Pearson, Horne AE Aqr could arise from collisional heating of gas & Skidmore (2003) showed a similar isothermal blobs, the same situation might arise from the behaviour. very different causes outlined above. As a result, these “fireball” models may have a much wider 2. Model Assumptions range of applicablility than simply the unusual be- We briefly recap here the dynamical model de- haviour of AE Aqr. Accordingly, in this paper, we veloped in Pearson, Horne & Skidmore (2003) but will compare analytic expressions derived for the the interested reader is referred there for a more lightcurves of such expansions to observations of detailed derivation. Let us assume a spherically SS Cygni and SN 1987A. symmetric expansion of a Gaussian density profile SS Cyg is a member of the dwarf nova subclass with radial velocity proportional to the distance of cataclysmic variables. It consists of a 1.2M⊙ from the centre of the expansion. We define white dwarf accreting material through an accre- r tion disk from a 0.71M⊙ K5V secondary that loses η (1) material via Roche lobe overflow (Ritter & Kolb ≡ a 2003). SS Cyg was the second member of this as a dimensionless measure of the radius r in terms subclass identified (Wells 1896) and has had a of the current lengthscale of the Gaussian a. The near continuously monitored lightcurve for over a expansion factor β acts as a dimensionless time, century (Warner 1995). Like all dwarf novae, being the constant of proportionality between the

2 current and fiducial scalelength a0. Hence, where I is the intensity of the emerging radiation, a B is the Planck function and τ is the optical depth β (2) measured along the line of sight from the observer. ≡ a0 For cases where the source function is everywhere consistent with our above assumptions regarding the same this becomes velocity and implying I = B (1 e−τ ). (11) β =1+ H (t t0) (3) − − where t0 is the time at which all the fiducial values We define x as the distance from the fireball are determined and H is an “expansion constant” centre toward the observer, and y as the distance setting the speed of the expansion. For simplic- perpendicular to the line of sight. The above line ity, we consider only the case of uniform 3-D ex- integral (equation (11)) gives the intensity I(y) for pansion, avoiding factors such as the angle of the lines of sight with different impact parameters y. observer to lines of symmetry. We also restrict The fireball flux, obtained by summing intensities ourselves to a spatially uniform temperature dis- weighted by the solid angles of annuli on the sky, tribution since, for example, a power law distribu- is then ∞ 2π y tion gives a result for the later integration in (22) f(λ)= I(y) dy (12) d2 in terms of the hypergeometric function 2F1. We Z0 assume a power law with index Γ for the temporal where d is the source distance. dependance of temperature. Γ = 0 corresponds to We can calculate the evolution of the continuum an isothermal expansion and Γ = 2 to the adia- lightcurve using expressions for the linear absorp- batic case. tion coefficient. The free-free absorption coeffi- In summary then, we have cient (per unit distance) can be written as −Γ T = T0 β , (4) −( hν ) − 1 −3 κff = κ0 1 e kT T 2 ν ne ni (13) −3 −η2 − ρ = ρ0 β e , (5) h i where where M −2 2 ρ0 = 3 (6) κ0 =3.692 10 Q gff (SI) (14) 2 2 × (π a0) and M is the total mass involved in the expansion. (Keady & Kilcrease 2000), gff is the free-free Gaunt factor (Gaunt 1930) and Q the ionic Using v(r) = Hr = Hrβ−1 we can integrate 0 charge. We wish to retain the explicit frequency 1 ρv2 over all space to derive the total kinetic en- 2 dependence but otherwise follow a parallel deriva- ergy of the expansion tion as for Kramers’ opacity. For a mixed ele- ∞ 2 2 mental composition, then, we sum over all species E = M a2 H2 η4 e−η dη (7) kin √π 0 (assumed fully ionized) Z0 3 = M a2 H2 (8) 2 ρ 0 Qi ni gi,ff (1 Z)g ¯ff (15) 4 ≈ mH − all ions 3 2 X M v0 (9) ≡ 4 (Bowers & Deeming 1984). Hereg ¯ff is a mean defining v0 = Ha0, the speed of expansion at r = Gaunt factor (close to unity) and Z the metal mass a. fraction. We also have ρ ρ (1 + X) 3. Theoretical Lightcurves and Spectral n = = (16) e µ m 2 m Distributions e H H (Bowers & Deeming 1984) where X is the hy- The radiative transfer equation has a formal so- drogen mass fraction. Combining and writing the lution under conditions of LTE correction for stimulated emission as ∞ − I = B e τ dτ, (10) − hν ǫ =1 e ( kT ) (17) Z0 −

3 gives It should be noted that in pulling κ1(µi,µe) out of 2 − 1 −3 κff = κ1 ǫ ρ T 2 ν (18) the integral (23) we have implicitly assumed that where we have defined the spatial variation of the ionization fraction has negligible impact on the behaviour. This is an 51 κ1 =6.695 10 (1 Z) (1+ X)¯gff (SI). (19) assumption that we shall return to later. × − Inserting our density profile (equation (5)) we ar- In Pearson, Horne & Skidmore (2003) we ap- rive at the result proximated the flux received from the fireball by splitting it into two components as viewed on the 2 κ1 ǫ M −2η2 κff = e . (20) plane of the sky: an optically thick central region 1 3 3 6 T 2 ν π a bounded by y = ym (see equation 44) and an op- On dimensional grounds if on no other, a sim- tically thin surrounding. Here, however, we cal- ilar expression to equation (18) and hence (20) culate the integral exactly. Combining equations must also exist for bound-free opacity when it (11), (12) and (24) with a change of integration dominates (when most species are fully recom- variable to u = τ(y), we have bined). Textbook derivations of Kramer’s opacity τ π a2 B 0 1 e−u (eg. Bowers & Deeming 1984) show us f = − du (27) 2 d2 u Z0 κ1,bf Z (1 + X)g ¯bf (21) ∝ Expressing the exponential in its series form we where gbf is the bound-free Gaunt factor (Gaunt integrate to find 1930). Useful tables for g have been calculated bf ∞ by Glasco & Zirin (1964). The constant of propor- π a2 B ( 1)(n+1) τ n f = − 0 (28) tionality, however, is more difficult to determine 2 d2 n n! n=1 than for free-free, since it must account for the X ionization edges in the absorption and, in partic- which can also be rewritten in the form of standard ular, the change of energy level populations with functions temperature for each ion. We shall assume a rela- 2 tion analagous to equation (18) also holds for the π a B f = 2 (E1(τ0)+ γ + ln(τ0)) (29) situation of mixed ionized and recombined species 2 d where both forms of opacity contribute. where E1 is the first order exponential integral The optical depth parallel to the observers line and γ 0.577216 is Euler’s gamma constant of sight (Abramowitz≈ & Stegun 1972; Jeffreys & Jeffreys −∞ 1956; Press et al. 1992). This has the form of τ(y) = κ dx (22) − ∞ f =Ω B S(τ0), (30) Z 2 ∞ κ1 ǫ M −2( y )2 −2( x )2 x = e a e a d πa2 1 3 5 where Ω = is the solid angle subtended by the T 2 ν3 π a −∞ a 2d2 Z (23) current standard deviation (a/√2) of the Gaussian y density profile and the “Saturation Function” −2( )2 = τ0 e a (24) S(τ ) = (E (τ )+ γ + ln(τ )) (31) where the optical depth on the line of sight 0 1 0 0 through the centre of the fireball is is plotted in Figure 1. We note the asymptotic 10−Γ limits β ( 2 ) τ = c (25) 0 β τ for τ 1 S(τ) ≪ (32) ≈ γ + ln τ for τ 1. and the time at which the fireball becomes opti- ≫ cally thin along this line of sight is The intensity of emitted radiation as a function of 2 impact parameter is plotted for different times in 2 10−Γ κ1 ǫ M Figure 2. We can see how the flux saturates at βc . (26) 1 5 ≡ 2 3 √2 π5 a the black body function for highly optically thick "T0 ν 0 #

4 impact parameters. In comoving coordinates this region gradually shrinks and at later times the en- tire emission becomes optically thin. The total flux given by equation (29) is plotted against time in Figure 3 for different values of Γ, using βc = 1. Figure 4 shows lightcurves at different wavelengths assuming Γ = 0 (isothermal) and 3 βc = (λ/5000 A)˚ 5 . The effect of the expanding photosphere can be seen from the optically thick contribution to the total flux, the latter of which rises to a peak at βpk. Initially the optically thick flux is the dominant source and the emission rises as the emitting area grows while the photosphere is advected outwards. Eventually, the decreasing density causes the photosphere to reverse and be- gin to shrink, reaching zero size at β = βc. The re- maining emission comes from the optically thin region surrounding the optically thick circle as seen Fig. 2.— Comparison of the intensity profiles at on the plane of the sky. This optically thin emis- different times calculated using (24) and (11) for sion dominates at late times. a Γ = 0 (isothermal) evolution. Recalling that a = βa0, for the special case of Γ = 0 (isothermal expansion) we can neglect the time variation of the Planck function and differentiate (29) with respect to β to find a condition for the maximum flux

4(E (τ )+γ+ln(τ )) (10 Γ)(1 e−τ0 )=0. (33) 1 0 0 − − −

This can be solved numerically to find τ0,pk = 6.8204. From (25) we then find βpk = 0.6811βc. We can also differentiate (29) with respect to β for Γ = 0 although the form is much more untidy and 6 we must solve for β directly. We plot βpk against Γ in Figure 5 continuing to neglect any time de-

Fig. 3.— The temporal behaviour of the total ﬂux at 5000 A˚ for diﬀerent cooling indices Γ = 2, 1, 0, 1, 2, assuming βc = 1 and T0 = 15 000− K.− The y-axis is parameterised in multiples πa0 ˚ of f0 = 2d2 Bν (5000 A, 15 000 K).

Fig. 1.— The Saturation Function S.

5 pendence of ǫ. We plot βpk against wavelength in Figure 6 for several values of Γ. Bruch (1992) found a correlation between the amplitude of flickering in different wavebands for several CVs (see inter alia his Figure 2). Such a correlation could arise from an isothermal evolution at a consistent temperature. This can be understood from (29). Since the peak flux occurs at the same central optical depth independent of wavelength, then for two wavelengths λ1 and λ2 the peak fluxes are related by

f β 2 B pk,1 = pk,1 λ,1 (34) f β B pk,2 pk,2 λ,2 − c2 2 5 λ T βc,1 λ1 e 2 0 1 = c2 − (35) βc,2 λ2 e λ1T0 1! − − 2 19 c2 5 c2 Fig. 4.— The temporal behaviour of the total flux 3 5 λ1T0 λ2T0 λ2 1 e e 1 λ 5 − c = − c2 2 − at several wavelengths assuming βc = , λ λ T λ T 5000 A˚ 1 1 e 2 0 ! e 1 0 1! − − T0 = 15 000 K and Γ = 0. The y-axis is again (36) parameterised in multiples of f0. The optically thick contribution at 5000 A˚ is plotted as a dashed where c = hc/k and assuming no complications 2 line and the optically thin contribution as a dot- such as a Balmer edge between them. We plot dashed line. the predicted values for different temperatures as stars; alongside the data taken from Table 4 of Bruch (1992), in a color-color diagram in Fig- ure 7 (nb. we used Bλ rather than Bν in the above derivation purely for comparison with this dataset). This figure shows that a simple application of our model enables us to reproduce the observed range of V-B reasonably well using T as a free paramemter. The U-B colour however, is generally underpredicted. This results from not mak- ing allowance for the increased opacity above the Balmer jump when evaluating βc in deriving 36. The size of the Balmer jump depends upon a non-trivial combination of mass, lengthscale and temperature to produce an effective value of κ1. Any individual system may produce flickers with a range of physical parameters causing it to produce points scattered across this diagram. Using numerical methods outlined later, we derived values for κ1 for several temperatures over a range of densities. These are plotted in Figure 8. Fig. 5.— The time of peak flux at 5000 A˚ plotted We can see how, for a large range of temperatures against Γ, assuming βc = 1 and T0 = 15 000 K. of interest and for densities ranging over several or- ders of magnitude, κ1 is well represented by a constant value. At lower temperatures the free-free

6 opacity transitions to bound-free opacity at lower densities than it does at higher temperatures. At these lower temperatures we could introduce significant errors if we estimate κ1 from a position that has one form of opacity when the other is in fact dominant. It is very unlikely that this would be the case, however, if we estimate the ionization state from conditions at a suitable position and consider how κ1 smoothly changes with the ionization. Qualitatively then, at the very least, we can be confident of the ability of the analysis to allow us to examine the behaviour of the proposed mechanism and to predict fluxes to within factors of order unity.

4. Analysis

Fig. 6.— The time of peak flux plotted against Equation (29) gives us the complete temporal and frequency behaviour for the emergent flux in wavelength for various values of Γ assuming T0 = 15 000 K. The values are normalised to the peak terms of τ0. We must use the Planck function time at 6000 A˚ in each case. appropriate to the time-dependent behaviour of the temperature from (4). Equation (25) gives us τ0 as a function of β and βc. The behaviour of the flux thus rests on the character of the parameter βc. We know functionally that

−3 βc = βc(κ1,ǫ,ν ), (37)

κ1 = κ1(η,ν,β), (38) ǫ = ǫ(ν,β) (39)

and we shall examine each of these dependencies and appropriate levels of approximation in each case.

4.1. Global Behaviour βc = const In examining the behaviour of the flux as a function of time it is instructive to restrict ourselves initially to a single wavelength and ignore spatial or time dependency in βc. We see from equations (25) and (28) that the late-time behaviour (kT hν) of the lightcurve is ≪ Fig. 7.— The amplitude ratio (a U-B v V-B color- determined by the temperature index Γ such that color diagram) for flickering in several CVs ob- f β−5 or β−4 for isothermal or adiabatic cooling ∼ served by Bruch (1992). The predicted values are respectively. plotted as stars for different temperatures ranging from 8000 K (extreme right) to 24 000 K in 2000 K 4.2. Spatial Dependence βc = βc(η) steps. 4.2.1. One Zone Model

We know βc depends on κ1 which in turn depends on the ionization structure across the ex-

7 pansion. At the outset we acknowledge that a de- state and, hence, to find a suitable approximate tailed solution accounting for the changing ioniza- value for κ1. The effect of the density profile on tion fraction across the density profile would re- the optical depth is accounted for explicitly in the quire numerical integration such as that reported derivations in section 3. in Pearson, Horne & Skidmore (2003) for AE Aqr. However, an alternative ‘typical’ place to es- We are looking for simpler, more easily calculable timate κ1 is at the point of maximum density approximations that avoid such detailed methods. (x = 0) along the limiting impact parameter ym To this end we do not include here any explicit al- that remains optically thick. We can derive ym by lowance for the variation of ionization states across setting (24) equal to unity, giving the profile; instead we approximate the behaviour 1 by calculating the conditions at a suitable point ln τ 2 y = a β 0 (44) within the expansion. With a Gaussian density m 0 2 profile we might expect the flux to be dominated by the region close to the photosphere. As a result where τ0 > 1 ie. for times β<βc. Hence,

we could approximate the ionization structure by − (−2−Γ) (10 Γ) 4 the solution to the Saha equation under the con- ρ = ρ0 β 4 βc for β<βc eval −3 ditions prevailing there. ( ρ0 β otherwise. To find the position of the observed photo- (45) sphere we need to integrate along a given line of sight until optical depth unity. From our earlier 4.2.2. Pure hydrogen case - semi-analytic solu- derivation we can see the optical depth down to tion ′ any height x is For simplicity, let us consider a fireball with ∞ almost pure hydrogen composition (nb. if Z = √2 −2( y )2 −2( x )2 x τ(y)= τ0 e a e a d (40) 0 then equation (21) implies κ1,bf = 0), with an √π x′ a Z a ionization fraction ′ ni ni which, setting τ(y) = 1 for the special case x =xph ι = . (46) gives ≡ ni + nn n y 2( )2 1 xph − e a = τ0 erfc √2 . (41) Ignoring any contribution to opacity from H and 2 a 2 noting κ ρ , the total opacity coefficient be- This defines the locus of points on the photo- comes ∝ spheric surface. Since the largest contribution to 2 2 κ ι κ , + (1 ι) κ , . (47) the opacity integral comes from the region of high- 1 ≈ 1 ff − 1 bf est density, we will introduce least error by eval- For hydrogen, the Saha equation can be approx- uating κ1 there. This will clearly occur along the imated by central line of sight y = 0 and hence we arrive at ni ne 21 3 5 −1 −3 2.4 10 T 2 exp( 1.58 10 T ) m 2 nn ≈ × − × erfc(√2ηph)= (42) τ0 (48) n2 which we must solve numerically. However, we i = A(T ) (49) n n must remember that it is possible for the pho- − i tospheric surface to lie behind the density peak. (Lang 1980) where we have tidied the RHS into This will just occur when τ(0) = 1 at η = 0, im- the function A(T ). Hence, we derive the quadratic 2 ( − ) plying β =2 10 Γ βc. Ultimately then, we have 2 ni + A(T ) ni A(T ) n = 0 (50) 2 2 − −3 −η ( − ) ρ0 β e ph for β < 2 10 Γ βc ρeval = and by restricting ourselves to the positive root ρ β−3 otherwise. ( 0 arrive at (43) A 4n It must be emphasized that we are using this ι = 1+ 1 . (51) position solely to evaluate a typical ionization 2n r A − !

8 We see then that the ionization fraction ι and desired, the integral in (23) thus becomes parameter βc are interdependant and should be ∞ 2 −2 x x solved for iteratively to ensure consistency. Thus κ e ( a ) d 1 a we can determine the lightcurve behaviour by ini- Z−∞ x0.1 2 x0.9 2 tially calculating a value for βc from (26) with a 2 −2( x ) 2 −2( x ) = κ1,bf e a dx + κ1,m e a dx judiciously chosen value for κ1. Thereafter we use a 0 a x0.1 Z ∞ Z the number density from (43) to calculate the ion- 2 2 −2( x ) ization fraction from (51). This then allows us to + κ1,ff e a dx (52) a x find κ1 from (47) and hence βc again from (26) to Z 0.9 x0.9 better accuracy. We can repeat this iteratively to =2κ1,ff erfc √2 achieve the desired accuracy. This value can then a x0.1 x0.9 be used to calculate the lightcurve behaviour at a +2κ , erfc √2 erfc √2 1 m a − a given wavelelength. h x0.1 i +2κ , 1 erfc √2 (53) 1 bf − a 4.2.3. Mixed composition h i For the more realistic case of mixed composi- which we use to replace a factor κ1 π/2 in equation (26). tion, we cannot calculate the ionization state at a p given position analytically. Instead we must iter- The question remains of how to rapidly find atively solve the network of Saha equations under values for η0.1 and η0.9. Again, in the spirit of the prevailing physical conditions to enable us to finding an easily calculable approximation, and determine κ1. This solution can then iterated with noting that even inaccurately determining these βc around the equations (26), (43) and κ1 loop in boundaries will still improve our integral calcula- a similar way to the above. tion, let us consider a situation where all species other than hydrogen remain fully ionized. Using 4.2.4. Three Zone Model standard methods, we derive The above method works well in situations 2 µ = . (54) where either free-free or bound-free opacity is e 1+ X (2ι 1) dominating. Unfortunately, as we noted at the − end of section 3, in the situation of mixed opacity Incorporating this into equation (48) along with sources we can introduce significant errors if we our density profile, we can show use a value for κ1 assuming the incorrect source. 2 2 m 1 ι We can rescue much of the above formalism, how- ρ e−η = H − A(T ) (55) 0 [1+(2 ι 1) X] ι ever, if we split the spatial profile into 3 zones. − We are almost bound to improve our calculation which we can rearrange and solve directly for ηι. regardless of where we place the boundaries of We note the particularly simple form this reduces the zones but clearly it makes sense to try to en- to for ι =0.5. sure that we have free-free dominated (outer re- For similar reasons to the single zone model, gions), bound-free dominated (inner regions) and we evaluate κ and κ at the point of highest mixed (intermediate regions) zones. Assuming bf ff density in their region (eg. either at η or η that hydrogen species provide the dominant opac- 0.9 max for free-free). Given the rapid density variation in ity source, we select the boundary between these the mixed zone, we evaluate κ at η or η as regions by the hydrogen ionization fractions ι = m 0.5 max appropriate. The possible integration schemes are 0.1 and ι =0.9 which occur at η and η . 0.1 0.9 illustrated schematically in Figure 9. Specifically then, we need to replace the integral in equation (23) with one over the three types 4.3. Frequency Dependence βc = βc(η,ν) of zone and rework equation (26) to redefine βc. The parameter β has a direct frequency de- Calculating x0.1 and x0.9 using y = 0 or ym as c pendence from the ν−3 in opacity and also dependence through both κ1 and ǫ. For regions where the Gaunt factors are slowly changing functions

9 of frequency eg. along the Paschen continuum in using the methods of Gray (1976) with the excep- the 4000–8000 A˚ range or when free-free opacity is tion of H− bound-free (Geltman 1962) and free- dominating, we ignore the small error introduced free (Stilley & Callaway 1970), He− (McDowell, by neglecting the contribution of gff and gbf to κ1. Williamson & Myerscough 1966) and HeI bound- Instead we need only correct for the direct and ǫ free (Huang 1948). Calculating βc from this opac- frequency dependence of βc. Thus, ity we can iterate to a consistent solution at each time. Altering the number of zones used yielded 2 6 10−Γ ν0 10−Γ ǫ virtually no discernable effect on the predicted βc(ν) βc(ν0) . (56) ≈ ν ǫ0 lightcurves. For temperatures around 16 000 K, the two lines of sight considered produced results With this correction we have only to calculate βc differing by at most around 0.5%. However, at accurately at a single wavelength with the iterative lower temperatures the predicted fluxes could dif- method outlined above. The lightcurve behaviour fer more, reaching around 5% at 10 000 K (see as a function of both wavelength and time then Figures 10 and 11). follows immediately from the combination of (56) and (29). 5. Comparison to Observation - Deducing Fireball Parameters 4.4. Time Dependence βc = βc(η,ν,β) The model was fitted to lightcurves for the flick- The dependence of β on time again comes c ering in the dwarf nova SS Cyg and the ‘bump through both κ and ǫ (if Γ = 0). In principle, we 1 phase’ of SN 1987A. We employed a χ2 minimiza- can use a similar expression6 to equation (56) also tion amoeba code (Press et al. 1992) to derive the to correct for the ǫ time-dependence. However, best-fit values for M, a , T , t , H, and Γ. We use the exponential nature of the expression renders 0 0 0 the approximation that the ionization fractions for the form of the lightcurves more complex and less all the species are well represented by the condi- instructive than that derived above. As a result, tions at x = 0,y = y and use the single zone this correction is probably best included only in m integration approach to arrive at a self-consistent numerical solutions. More seriously, we cannot, in solution for β and κ there at each time. general, predict the future ionization structure for c 1 a mixed composition gas from its current state. 5.1. SS Cygni Thus, the time-dependence of κ1 can, in general, only be included with numerical solution at a se- Rapid spectroscopy of SS Cyg was carried out ries of different times β. The exception to this rule using the Low Resolution Imaging Spectrograph is when we can be sure that the dominant opacity (LRIS: Oke et al. 1995) on the 10-m Keck II tele- is and will remain free-free throughout the time scope on Mauna Kea, Hawaii, between UT 09:38 considered. In this case κ1 is a constant in time and 09:56 on 6 July 1998, covering orbital phases and we can use the same rapid approximation as 0.8276 to 0.8715. The instrumental set and data the previous section. reduction were the same as described by Steeghs et al. (2001), O’Brien et al. (2001) and Skidmore 4.5. Comparison of Integral Methods et al. (2003). 14,309 spectra were obtained using the rapid data acquisition system. The spectra We can lift the restriction to the purely free-free − covered 3259-7985 A˚ with 2.4 A˚ pixel 1 dispersion opacity case by iterative numerical solution of the and a mean integration time of 72.075 ms and no Saha equations for a gas of mixed composition at dead-time between individual spectra. Further de- each time in the lightcurve. From the ionization tails of these observations are given in Skidmore profile of each species we can in principle deter- et al. (in prep). A particular flickering event was mine the opacity at any point. isolated between UT 9:39:10 and UT 9:43:44 and We compared the results achieved by the two flux from other sources removed by fitting, for each integration lines of sight y = ym or 0 and using wavelength, a low order polynomial to the fluxes 1- or 3-zone integration schemes. The total con- both before and after the event. Lightcurves were tinuum opacity was calculated numerically using formed from the mean flux in the regions 3590-

10 3650 A,˚ 4165-4270 A,˚ 4520-4620 A,˚ 5100-5800 A,˚ 4 1014 kg s−1 and 2 10−4 of the total 5970-6500 A˚ and 7120-7550 A˚ at 2 s resolution. disk× mass M 7∼ 10×20 kg (Schreiber & disk ≈ × Model lightcurves were computed using a dis- Gänsicke 2002). Similarly, the lengthscale of 6 tance of 166 pc (Harrison et al. 1999) and assum- the expanding region, a 9.1 10 m is 1.5% of the estimated disk radius∼ R× 6 ∼108 m ing solar composition. It appears that the dis- disk ≈ × agreement is dominated by the systematic error of (Schreiber & Gänsicke 2002). The kinetic en- the oversimplicity of our model rather than obser- ergy is much greater than the thermal energy 25 M T vational errors. Accordingly, we carried out fits Eth 4 10 1017 kg 20 000 K J but com- both weighting the data points according to their ≈ × parable to that of a putative, moderately strong, formal observational errors and with equal weight- magnetic field in a sphere of radius a0, Emag = ing. We plot the derived analytic lightcurves 3 2 6 1028 a0 B J. alongside the observations in Figures 12 and 13. × 107 m 6 T From each set of derived parameters, we carried We generated full optical spectra from the ‘6- out a detailed numerical integration of the opacity parameter weighted’ set of values and compare calculated with an LTE ionization varying across them to the observed spectra at three differ- the density profile as described in Pearson, Horne ent times in Figure 16. The model data have & Skidmore (2003). These models allowed us been convolved with the instrumental blurring of ˚ to generate a timeseries of continuum spectra and 9.8 A FWHM. The results show how parameters “numerical” lightcurves for comparison to the data derived from the analytic forms for the continuum in each case. These numerical lightcurves are also lightcurves can subsequently be used to repro- plotted in the Figures. The fitted value of Γ is duce spectra. The agreement with observation at close to zero in both cases and so we refitted the early times is remarkable. At the peak flux and data fixing Γ = 0 exactly. These results are shown later, however, the models predict more metal in Figures 14 and 15. The best-fit parameters for and stronger lines than are observed, particularly ˚ each case are summarised in Table 5.1. in the 3000–5000 A range. Many of the relative line strengths, however, are still reproduced. Although we would expect the numerical lightcurves to more accurately represent reality, 5.2. SN 1987A the analytic lightcurves generally fit the data better. This is unsuprising since the parameters used Given the similarities in the mechanisms un- in both cases have been optimized for the ana- derlying the flickering lightcurves in our mod- lytic forms. The numerical lightcurves do seem els and those of supernovae, we attempted to fit closer to the data in the 3615 Awindow˚ in Fig- the UBVRI photometric data for SN 1987A ob- ures 12 and 13. However, Figures 14 and 15 and tained from Catchpole et al. (1987). The data our other unpublished lightcurves suggest that were converted to fluxes using the standard values this is serendipitous. The difficulty in reproduc- and effective wavelengths for the bands in Mur- ing the Balmer jump may well reflect that we have din (2001). The fits assumed an interstellar red- 5 fitting points on the Paschen Continuum (wave- dening E(B-V)=0.15 (Hamuy et al. 1988; Fitz- lengths longer than 3646 A)˚ and only one above patrick 1988) and assumed the same distance to the Balmer jump. Since the continuum level will the LMC of 50.1 kpc as Catchpole et al. (1987). have a close to ν−3 relationship between discon- Given the results of sophisticated NLTE spectral tinuities, fixing the level at several points may and lightcurve modelling (eg. Mitchell et al. 2002) well bias the parameters towards an accurate fit we should not expect an accurate match between here rather than the more complex interplay of our simple models and the SN 1987A observations. parameters required to accurately reproduce the However, it will allow us to highlight the under- Balmer jump. In ideal circumstances we would lying similarities of these problems and to gauge like several more points at shorter wavelengths to the robustness of our derived parameters. The fits address this issue. used only the data after JD 2446875.0 to elim- The derived masses, M 1.6 1017 kg are inate the flash phase and the data were all as- equivalent to 400 s of∼ the estimated× mean signed equal weights. For the sake of continuity mass transfer rate∼ from the secondary of M˙ with the SS Cyg fit, we continued to allow 6 fit ≈

11 Model M a0 T0 t0 H Γ v0 Ekin − − SS Cyg (1017 kg) (106 m) (104 K) (UT) (s 1) (kms 1) (1028 J) 6 param. weight. 1.6 9.8 1.91 9.6680 0.090 -0.092 890 9.7 6 param. unweight. 1.6 8.1 2.07 9.6677 0.106 -0.064 860 8.8 Isothermal weight. 1.5 7.4 2.38 9.6679 0.115 — 860 8.3 Isothermal unweight. 1.5 11.0 2.38 9.6691 0.078 — 860 8.4 − − − SN 1987A (1031 kg) (1013 m) (103 K) (MJD) (10 8 s 1) (kms 1) (1043 J) 5-band UBVRI 2.6 2.1 4.72 47018.7 7.2 0.066 1500 4.5 4-band BVRI 2.5 2.2 4.69 47025.3 7.0 0.071 1500 4.3

Table 1: Derived and auxiliary parameters from the analytic ﬁts to ﬂickering of SS Cyg and the lightcurve of SN 1987A.

Fig. 16.— Comparison of 3 spectra at UT 9.6722 (rise), 9.6830 (peak) and 9.6916 (fall) calculated from numerical integration across our ﬁreballs with the observed spectra of SS Cyg. The ‘6 parameter weighted’ values were used to generate the numerical spectra.

12 parameters rather than fixing the combination of rive which is consistent with our including both t0 and H to give a launch time of JD 2446849.82 optically thick and thin emission regions whereas when the associated neutrino event was detected they treat all the emission as optically thick. Our by Kamiokande-II (Koshiba 1987). Since we have lightcurves also sometimes show an intriguing fea- not attempted to remove a “background” level the ture (unfortunately most apparent in the poorly fit may be contaminated by the early stages of the fitted U band) of a point of inflexion at around radioactive tail. However, the flux in this region JD 2446960. This is reminiscent, in timing and is dominated by the expanding/collapsing photo- in character, of the break in the lightcurves at- sphere effect and any heating should show up in tributed to heating by radioactive decay. In our the fitted value of Γ differing from zero. lightcurves, however, it appears to arise from the We see in Figure 17 that we can achieve a qual- point at which the material becomes completely itative match between the analytic expression and optically thin with no contribution from an opti- the data. However, the U band is particularly cally thick core region. All these factors, showing poorly reproduced, representing the effect of blan- reasonable results for SN 1987A despite the inher- keting noted by Menzies et al. (1987). We refitted ent simplicity of our model, reinforce our confi- to the data using just the BVRI bands and show dence in the robustness of the parameters derived the results in Figure 18. The parameters for the for SS Cyg which had much better agreement be- fits are included in Table 5.1. We see that elimi- tween data and theory. nating the U band data has relatively little effect and derive a value for the mass of the material in 6. Summary 31 the expansion of M = 2.6 10 kg ( 13M⊙). × ≈ We have derived a formalism and analytic ex- This compares to an estimate of 7M⊙ < M < env pressions applicable to a variety of systems that re- 11M⊙ by Saio, Nomoto & Kato (1988). The produces the evolution of their optical lightcurves. typical expansion velocity of 1500 km s−1 com- The method has been tested for two widely dif- pares to a value derived from measurements of the ferent cases allows us to derive reasonable values blueshifts in P Cygni profiles of 2500 km s−1 for the physical parameters involved in the ex- at maximum light (McCray 1993).∼ Typical esti- pansion. There is encouraging agreement with mates of E 2 1044 J (Bethe & Pizzochero kin a method using a full integration of the opac- 1990; Woosley∼ 1988)× are somewhat larger than ity across the expanding region to generate both our E 4 1043 J. kin ≈ × lightcurves and spectra. We have seen how the It is unsuprising that the lightcurves do not flickering of close binary systems can be modelled match in detail, since the Gaussian density pro- as smaller, hotter analogues of the well known file we have assumed does not accurately repre- rebrightening “bump” in supernovae lightcurves. sent supernova ejecta. In particular, the reversal We hope to test this approach further by compar- of the fast rise and slow decline derived analyt- ison to data from LMXBs and other systems in ically can be understood from the way in which the future. a photosphere marching inwards through a shell (in comoving coordinates) evolves. In the shell case, the material will first become optically thin We thank the referee for the thoughtful and at low impact parameters once the photosphere helpful suggestions in response to an earlier ver- reaches the inner surface of the shell and later at sion of this paper. KJP has been supported, in higher impact parameters with their greater path- part, by the U.S. National Science Foundation lengths through the shell material. While these through grant AST-9987344 and, in part, through can not be considered in any sense a good fits, we LSU’s Center for Applied Information Technology do retrieve values for the mass, size and temper- and Learning. He also thanks Juhan Frank for ature (see Figure 19) for this phase comparable stimulating discussions that improved this paper. with those given by Catchpole et al. (1987). We REFERENCES also see the same isothermal behaviour from the lightcurve analysis as those authors. The size of Abramowitz M., Stegun I. A., 1972, Handbook our photosphere is less than the radius they de- of Mathematical Functions, Dover, New York,

13 10th Edition Koshiba M., 1987, IAU Circ. No. 4338 Baptista R., Bortoletto A., Harlaftis E. T., MN- Lang K. R., 1980, Astrophysical Formulae, RAS, 335, 665 Springer Verlag, New York, 2nd Edition Baptista R., Bortoletto A., ApJ, 128, 411 McCray R., 1993, ARA&A, 31, 175 Bethe H. A., Pizzochero P., 1990, ApJL, 350, 33 McDowell M. R. C., Williamson J. H., Myerscough V. P., 1966, ApJ, 144, 827 Bowers R. L., Deeming T., 1984, Astrophysics I: Stars, Jones and Bartlett, Boston Mauche C. W., Robinson E. L., 2001, ApJ, 562, 508 Bruch A., 1992, A&A, 266, 237 Bruch A., 2000, A&A, 359, 998 Menzies J. W., Catchpole R. M., van Vuuren G., Winkler H., Laney C. D., Whitelock P. A., Catchpole R. M., Menzies J. W., Monk A. S., War- Cousins A. W. J., Carter B. S., Marang F., gau W. F., Pollacco D., Carter B. S., Whitelock Lloyd Evans T. H. H., Roberts G., Kilkenny P. A., Marang F., Laney C. D., Balona L. A., D., Spencer Jones J., Sekiguchi K., Fairall A. Feast M. W., Lloyd Evans T. H. H., Sekiguchi P., Wolstencroft R. D., 1987, MNRAS, 227, 39p K., Laing J. D., Kilkenny D. M., Spencer Jones J., Roberts G., Cousins A. W. J., van Vuuren Mitchell R. C., Baron E., Branch D., Hauschildt G., Winkler H., 1987, MNRAS, 229, 15p P. H., Nugent P. E., Lunqvist P., Blinnikov S., Pun C. S. J., 2002, ApJ, 574, 293 Fitzpatrick E. L., 1988, ApJ, 335, 703 Murdin P., 2001, Encyclopedia of Astronomy and Gaunt J. A., 1930, Phil. Trans., 229, 163 Astrophysics, IOP Publishing Ltd. and Nature Publishing Group, London Geltman S., 1962, ApJ, 136, 935 O’Brien K., Horne K., Boronson B., Still M., Gray D. F., 1976, Observations and Analysis of Gomer R., Oke J. B., Boyd P., Vrtilek S. D., Stellar Photospheres, New York, Wiley 2001, MNRAS, 323, 484 Glasco H. P., Zirin H., 1964, ApJS, 9, 193 Oke J. B., Cohen J. G., Carr M., Cromer J., Dini- Hamuy M., Suntzeff N. B., Gonzalez R., Gabriel gizian A., Harris F. H., Labrecque S., Lucinio M., 1999, AJ, 95, 63 R., Schall W., Epps H., Miller J., 1995, PASP, 107, 375 Harrison T. E., McNamara B. J., Szokdy P., McArthur B. E., Benedict, G. F., Klemola A. Patterson J., 1981, ApJS, 45, 517 R., Gilliland R. L., 1999, ApJ, 515, L93 Pearson K. J., Horne K. D., Skidmore W., 2003, Hawley J. E., Bablus S. A., 1991, ApJ, 376, 223 MNRAS, 388, 1067 Honey W. B., Bath G. T., Charles P. A., White- Press W. H., Teukolsky S. A., Vetterling W. T., hurst R., Jones D. H. P., Echevarria J., Arevalo Flannery B. P., 1992, Numerical Recipes in For- M. J., Solheim J.-E., Tovmassian G., Takagishi tran, Cambridge Univ. Press, Cambridge, 2nd K., 1989, MNRAS, 236, 727 Edition Huang S., 1948, ApJ, 108, 354 Ritter H., Kolb U., 2003, A&A, 404 301 Jeffreys H., Jeffreys B. S., 1956, Methods of Math- Saio H., Nomoto K., Kato M., 1988, Nature, 334, ematical Physics, Cambridge Univ. Press, Cam- 508 bridge, 3rd Edition Schreiber M. R., Gänsicke B. T., 2002, A&A, 382, Keady J. J., Kilcrease D. P., 2000, in, Cox A. 124 N., Allen’s Astrophysical Quantities, Chapter 5, Springer Verlag, New York, 4th Edition

14 Skidmore W., O’Brien K., Horne K., Gomer R., Oke J. B., Pearson K. J., 2003, MNRAS, 338, 1057 Skidmore W., O’Brien K., Pearson K. J., Horne K., Gomer R., Oke J. B., in prep.

Steeghs D., O’Brien K., Horne K., Gomer R., Oke J. B., 2001, MNRAS, 323, 484 Stilley J. L., Callaway J., 1970, ApJ, 160, 245 van Teeseling A., 1997, A&A, 324, L73 Warner B., 1995, Cataclysmic Variable Stars, Cambridge Univ. Press, Cambridge Warner B., 2004, PASP, 116, 115 Wells L. D., 1896, Harv. Coll. Obs. Circ., No. 12 Welsh W. F., Wood J. H., 1995, in IAU Colloq. 151, Flares and Flashes, ed. J. Greiner, H. W. Duerbeck, R. E. Gershberg (Lecture Notes in Physics, 454), Springer, Berlin, 300 Woosley S. E., 1988, ApJ, 330, 218 Wynn G. A., King A. R., Horne K. D., 1997, MN- RAS, 286, 436

Fig. 8.— The value of κ1 at 5000 A˚ for temperatures ranging in 2000 K steps from 8000 K (upper left curve) to 24 000 K.

This 2-column preprint was prepared with the AAS LATEX macros v5.2.

15 Fig. 10.— Comparison of predicted lightcurves using the 4 combinations of 1- or 3-zone and y = 0 or ym line of sight integration methods. The 4 methods produce virtually indistinguishable results. The parameters of the model are 16 6 M =3.0 10 kg, a0 =7.5 10 m, T0 = 16000K and Γ =× 0. A lightcurve× calculated from a full Fig. 9.— Schematic representation of the alter- numerical integration of the opacity through the native integration schemes. The dot-dashed lines ﬁreball is plotted as a dashed line. A distance of mark the two possible lines of sight considered for 166 pc has been assumed. an observer (at x = ). Along the y = 0 line of ∞ sight the one-zone model would calculate βc using κ(= κbf ) evaluated at the photosphere. For a three zone scheme βc would be calculated using a combination of the relevant opacities evaluated at the indicated positions. The boundaries for each zone occur at the intersection of the dot- dashed line with the solid lines. Similarly the position where the opacities would be calculated for an optically thick/thin transition line of sight are indicated. While the circles of ηι have been placed purely for illustrative convenience, the position of the photosphere has been calculated assuming τ0 = 10. Other possible combination exist depending on the relative positions of the photosphere, peak density and ionization boundaries.

Fig. 11.— The same comparison as Figure 10 but with T0 = 10 000 K.The 1- and 3- zone models are indistinguishable in either case. The y = 0 models produce slightly lower peak ﬂuxes.

16 Fig. 12.— Analytic fits (solid) to the SS Cyg data points and lightcurves generated from numerically Fig. 14.— Same comparison as Figure 12, Γ = calculated spectra using the best-fit parameters 0 (isothermal) was fixed and the points were (dashed). Γ was allowed to be a fit parameter weighted according to the observational errors. and the points were weighted according to the observational errors.

Fig. 15.— Same comparison as Figure 12, Γ = Fig. 13.— Same comparison as Figure 12, Γ was 0 (isothermal) was ﬁxed and the points were allowed to be a ﬁt parameter and the points were weighted equally. weighted equally.

17 Fig. 17.— Analytic ﬁts (solid) to the UBVRI SN 1987A data of Catchpole et al. (1987). Γ was allowed to be a ﬁt parameter and all the dat- apoints with t > JD 2446875.0 were given equal weight.

Fig. 19.— The temperature and radius derived by Catchpole et al. (1987) (asterisks) and temperature, optically thick radius (in the V band) and current lengthscale from our UBVRI ﬁt parameters.

Fig. 18.— Same comparison as Figure 17 ﬁtting only BVRI band data.