A Theory for the Maximum Magnitude versus Rate of Decline (MMRD) Relation of Classical Novae
Izumi Hachisu, Hideyuki Saio, Mariko Kato, Martin Henze, Allen W. Shafter
aa r X i v : . [ a s t r o - ph . S R ] S e p Draft version September 8, 2020
Typeset using L A TEX twocolumn style in AASTeX63
A Theory for the Maximum Magnitude versus Rate of Decline (MMRD) Relation of Classical Novae
Izumi Hachisu, Hideyuki Saio, Mariko Kato, Martin Henze, and Allen W. Shafter Department of Earth Science and Astronomy, College of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo153-8902, Japan Astronomical Institute, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan Department of Astronomy, Keio University, Hiyoshi, Kouhoku-ku, Yokohama 223-8521, Japan Department of Astronomy, San Diego State University, San Diego, CA 92182, USA
ABSTRACTWe propose a theory for the MMRD relation of novae, using free-free emission model light curvesbuilt on the optically thick wind theory. We calculated ( t , M V, max ) for various sets of ( ˙ M acc , M WD ),where M V, max is the peak absolute V magnitude, t is the 3-mag decay time from the peak, and ˙ M acc is the mass accretion rate on to the white dwarf (WD) of mass M WD . The model light curves areuniquely characterized by x ≡ M env /M sc , where M env is the hydrogen-rich envelope mass and M sc isthe scaling mass at which the wind has a certain wind mass-loss rate. For a given ignition mass M ig ,we can specify the first point x = M ig /M sc on the model light curve, and calculate the correspondingpeak brightness and t time from this first point. Our ( t , M V, max ) points cover well the distributionof existing novae. The lower the mass accretion rate, the brighter the peak. The maximum brightnessis limited to M V, max & − . M acc & × − M ⊙ yr − . Asignificant part of the observational MMRD trend corresponds to the ˙ M acc ∼ × − M ⊙ yr − linewith different WD masses. A scatter from the trend line indicates a variation in their mass-accretionrates. Thus, the global trend of an MMRD relation does exist, but its scatter is too large for it to bea precision distance indicator of individual novae. Keywords: novae, cataclysmic variables — stars: individual (V1668 Cyg) — stars: winds INTRODUCTIONA typical classical nova shows a rapid rise of op-tical brightness until its peak followed by a slowdecline. There is a statistical trend that a fasterdecline nova shows a brighter optical peak. Thescatter around the main trend, however, is notsmall. It has long been debated whether or not ameaningful relation actually exists (e.g., Mclaughlin1945; Schmidt 1957; Cohen 1985; della Valle & Livio1995; Downes & Duerbeck 2000; Kasliwal et al. 2011;Shafter et al. 2011; Cao et al. 2012; Shafter 2013;Shara et al. 2017; Schaefer 2018; ¨Ozd¨ormez et al.2018; Selvelli & Gilmozzi 2019; della Valle & Izzo2020). Such a relation is called the maximum magnitudeversus rate of decline (MMRD) relation. If an MMRDrelation exists and is a simple monotonic relation, it canbe used to obtain the absolute peak brightness of a nova [email protected] from the rate of decline, and thus, it would be a usefultool to obtain the distance to a nova. Although thereare early attempts to theoretically explain the MMRDrelation (e.g., Livio 1992), we need a convincing the-oretical background to understand the main trend andlarge scatter of the existing MMRD distribution of no-vae.A nova is a thermonuclear runaway event on a mass-accreting white dwarf (WD). Hydrogen ignites and re-leases nuclear energy. The hydrogen-rich envelope ex-pands to a giant size. The subsequent nova evolutionwas theoretically followed by Kato & Hachisu (1994)based on the assumption of spherical symmetry. Strongoptically-thick winds are accelerated deep inside thephotosphere. The wind stops after a significant part ofthe hydrogen-rich envelope is ejected by the wind. Thetimescale of a nova in the early phase is determined bythe wind mass-loss rate and the amount of the hydrogen-rich envelope mass.Kato & Hachisu (1994) calculated optically thickwinds in the decay phase of novae and obtained the
Hachisu et al. photospheric radius R ph , temperature T ph , luminosity L ph , velocity v ph , and wind mass-loss rate ˙ M wind againstthe decreasing envelope mass M env . Observationally,early spectra of novae are dominated by free-free emis-sion (e.g., Ennis et al. 1977; Gallagher & Ney 1976).Therefore, Hachisu & Kato (2006) calculated free-freeemission model light curves with F ν ∝ ˙ M / ( v R ph ),where F ν is the flux at the frequency ν . These modellight curves well reproduce many nova light curves (e.g.,Hachisu & Kato 2006, 2010, 2016, 2018, 2019a,b).The theoretical free-free emission light curves showa homologous shape independent of the WD mass andchemical composition. Hachisu & Kato (2006) calledthis property of nova model light curves “the universaldecline law.” These properties, i.e., homologous andfrequency independent shapes of light curves, indicatethat the model light curves are expressed by a uniquefunction of a parameter. We find that this parameter isthe ratio of the envelope mass and the scaling envelopemass having a certain wind mass-loss rate, that is, x ≡ M env /M sc as will be explained in Section 2.Hachisu & Kato (2010) found that two differentmodel light curves, e.g., corresponding to the two differ-ent WD masses, can overlap each other if the timescaleof one of them is squeezed by a factor of f s , i.e., t/f s .The normalization factor is f s < f s > V brightnesses is normalized to be M V − . f s . Thus,the two different light curves overlap each other in the( t/f s )–( M V − . f s ) plane (see, e.g., Figures 48 and49 of Hachisu & Kato 2018). Hachisu & Kato (2019b)reformulated this property: if the V light curve of a tem-plate nova (time t ) overlaps with that of a target nova(time t ′ = t/f s ), we have the relation( M V [ t ]) template = ( M ′ V [ t ′ ]) target = ( M V [ t/f s ] − . f s ) target , (1)where M V [ t ] is the original absolute V brightness and M ′ V [ t ′ ] is the time-normalized brightness after time-normalization of t ′ = t/f s . This property was cal-ibrated on many novae (e.g., Hachisu & Kato 2016,2018, 2019a).Hachisu & Kato (2010) also presented a theoreticalexplanation of the MMRD relation based on the uni-versal decline law and Equation (1). Their interpreta-tion on the main trend of the MMRD distribution isthat V1668 Cyg is a typical classical nova, and thatthe novae having the same time-normalized light curves(i.e., the same normalized peak brightnesses) as that ofV1668 Cyg but the different WD masses form the maintrend line in the (log t )– M V, max diagram, i.e., M V, max = 2 . t − .
94 (Equation (25) in Hachisu & Kato2016). Here, t is the 3-mag decay time from the V peak and M V, max is the absolute V peak magnitude.This main trend line is located in the middle of the ob-servational MMRD distribution of novae in the (log t )– M V, max diagram and its peak becomes brighter alongEquation (1) with the decreasing f s . On this main trendline, a faster decline nova with a shorter t time (smaller f s ) corresponds to a more massive WD while a slowerdecline nova with a longer t time (larger f s ) does to aless massive WD.In the present work, we clarify the physics of MMRDpoints. Then, we explain the main trend of MMRD re-lation as a typical ˙ M acc with the different M WD ’s. Thescatter from the main trend line is explained by the dif-ference of ˙ M acc from a typical ˙ M acc .Our paper is organized as follows. First we pro-pose several timescaling laws and clarify the physics ofMMRD relation in Section 2. Then, we approximatethese timescaling laws with analytic expressions in Sec-tion 3, which simplify the calculations of ( t , M V, max ).In Section 4, we explain our theoretical ( t , M V, max ) re-lation on the base of ( ˙ M acc , M WD ), the main character-istic properties of cataclysmic binaries. Discussion andour conclusions are given in Sections 5 and 6, respec-tively. We tabulate our numerical results in AppendixA. TIMESCALING LAW OF FREE-FREEEMISSION MODEL LIGHT CURVESKato & Hachisu (1994) calculated envelope solutionsof wind mass-loss for various WD masses (rangingfrom 0 . M ⊙ to 1 . M ⊙ ) and chemical compositions.They provide the wind mass-loss rate ˙ M wind , photo-spheric temperature T ph , velocity v ph , and radius R ph for a specific envelope mass M env and WD mass M WD .Hachisu & Kato (2006, 2010) calculated the nova op-tical and infrared (IR) light curves based mainly onthe free-free emission model of winds. We plot suchexamples in Figures 1, 2, 3, and 4 for the chemi-cal composition of typical CO novae, i.e., CO nova 3(CO3; Hachisu & Kato 2016), each element of which is( X, Y, Z, X C , X O ) = (0 . , . , . , . , .
20) by massweight.2.1.
Brightnesses of Optical/IR Light Curves
Hachisu & Kato (2006, 2010) calculated the nova op-tical/IR light curves based mainly on the free-free emis-sion model of winds. Their free-free emission flux isapproximately calculated from F ν ( t : M WD ) = A ff ˙ M v R ph , (2) aximum Magnitude versus Rate of Decline Figure 1.
Comparison of our model V light curves with the V1668 Cyg light curve. A 0 . M ⊙ WD (CO3) model (solid redline) reasonably reproduces the V1668 Cyg optical y (unfilled magenta squares) and V (filled blue squares) light curves. Weadd the visual magnitudes (red dots) of V1668 Cyg. The distance modulus in V band of µ V ≡ ( m − M ) V = 14 . V lightcurves for 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.0, 1.05, 1.1, 1.15, and 1 . M ⊙ WDs (solid black lines) are plottedin the ( t/f s )-( M V − . f s ) plane. The time-normalization factor f s of each model, tabulated in Table 3 of Hachisu & Kato(2016), is measured against that of the V1668 Cyg light curves. We place three points, A, B, and C, on the model V light curve(0 . M ⊙ ), corresponding to three different initial envelope masses, M env , = 1 . × − M ⊙ , 1 . × − M ⊙ , and 0 . × − M ⊙ ,respectively. Point B is the optical peak of V1668 Cyg, m V, max = 6 . M V, max = 6 . − . − . where they assumed A ff to be a constant among vari-ous novae. The effects of the electron temperature, ion-ization degree, and chemical composition on the free-free flux are included through the envelope solutions,i.e., the wind mass-loss rate ˙ M wind , photospheric veloc-ity v ph , and photospheric radius R ph . Hachisu & Kato(2010, 2016) determined A ff by fitting the 0 . M ⊙ WD (CO3) model light curve with the V1668 Cyglight curves, where the distance modulus in V band µ V ≡ ( m − M ) V = 14 . . M ⊙ WDs (CO3). Allthe model light curves overlap well with each other in the( t/f s )-( M V − . f s ) plane. Hachisu & Kato (2006) called this property the universal decline law. Here, thetimescaling factor f s is measured against the timescaleof V1668 Cyg. The light curve of this nova is well repro-duced with a 0 . M ⊙ WD (CO3) model (solid red line).See Hachisu & Kato (2016, 2019a,b) for the model lightcurve fitting of V1668 Cyg.To deeply understand the physics of nova light curves,we break the scaling process shown in Figure 1 into twosteps. We plot the first step in Figure 2(a)(b) for threeWD mass models of 0 . M ⊙ (green), 1 . M ⊙ (red),and 1 . M ⊙ (blue), that is, the free-free flux parame-ter ˙ M / ( v R ph ) against ˙ M wind . Then, we plot thesecond step in Figure 2(c)(d), that is, the wind mass-loss rate ˙ M wind against the real time t and normalizedtime t/f s , respectively. The third step is the combina- Hachisu et al.
Figure 2. (a)(b) The free-free flux parameter of ˙ M / ( v R ph ) versus wind mass-loss rate ˙ M wind for 1 . M ⊙ (blue), 1 . M ⊙ (red), and 0 . M ⊙ (green) WD (CO3) models, which are the same models as in Figure 1. The three (blue, red, and green) linesalmost overlap each other. The dotted lines indicate the global trend of these three lines. (c)(d) The wind mass-loss rates areplotted against the real time t and the normalized time t/f s , respectively, for the same three WD mass models. (e)(f) Free-freeemission model V light curves for the 0 . M ⊙ (green), 0 . M ⊙ (black), 1 . M ⊙ (red), and 1 . M ⊙ (blue) WDs are plottedagainst the real time t and the normalized time t/f s , respectively. In panel (e), the right edge of 1 . M ⊙ light curve (blueline) corresponds to the end of wind phase. Note that the ordinate in panel (f) is M ′ V [ t ′ ] in Equation (1), corresponding to theabscissa of t ′ = t/f s while the ordinate in panel (e) is M V [ t ] − . f s because the abscissa is t . The symbols in panels (e) and(f) are the same as those in Figure 1. See the text for more details. aximum Magnitude versus Rate of Decline . M ⊙ WD (CO3)model (black line). Note that the ordinate in panel (e)is M V [ t ] − . f s because the abscissa is t while the or-dinate in panel (f) is M ′ V [ t/f s ](= M V [ t/f s ] − . f s ).The right column, Figure 2(b)(d)(f), are the same asthose in the left column, but in the normalized timescale, t/f s . Note that Figure 2(a) and 2(b) are essentially thesame because the time does not explicitly appear. Com-bining Figure 2(b) with 2(d), we obtain Figure 2(f). Theproportionality constant A ff is determined by fitting the0 . M ⊙ WD model light curve with the V1668 Cyg lightcurve. Figure 2(f) is essentially the same as Figure 1.The overlapping of envelope solutions in Figure2(a)(b) directly means that, when the wind mass-lossrates are the same, the fluxes of free-free emission arethe same irrespective of the WD mass and chemical com-position. We express this with a function of F ν ( ˙ M wind ).This expression does not explicitly include the WD massbut F ν ( t : M WD ) of Equation (2) depends on the WDmass through the wind mass-loss rate ˙ M wind as shownin Figure 2(c).After the optical maximum, the free-free flux de-creases as the wind mass-loss rate drops. This rela-tion, F ν ( ˙ M wind ), is common among the various WDmasses and chemical compositions. In the real timescaleof Figure 2(c) and 2(e), however, the wind mass-lossrate and flux decrease more rapidly in more massiveWDs. To clarify the difference in the timescale be-tween the 1 . M ⊙ and 1 . M ⊙ WDs, we designate theirtimes t and t ′ , respectively. For example, we plot a re-lation between t and t ′ at the vertical dashed line oflog( ˙ M wind /M ⊙ yr − ) = − . t and t ′ .This t and t ′ relation similarly holds for the horizon-tal dash-dotted lines of the free-free flux parameter˙ M / ( v R ph ) and M V − . f s in Figure 2(a) and2(e).We formulate the conversion from ( t, F ( t )) to( t ′ , F ′ ( t ′ )) by the time-normalization of t ′ = t/f s , thatis, F ′ ( t ′ ) = f s F ( t/f s ) . (3)This simply means that, if the timescale is squeezed byten times ( t ′ = t/ F ′ = 10 F ) because of energy conservation, i.e., ∆ E = F ′ ( t ′ )∆ t ′ = f s F ( t ′ )∆ t ′ = F ( t/f s )∆ t . In other words, weobserve the same outburst with two different temporalscalings, t and t ′ . Then, the flux is different betweenthese two systems, F ( t ) and F ′ ( t ′ ), while the energy isthe same, ∆ E , during the same intrinsic time-interval,∆ t ′ = ∆ t/f s , at the same intrinsic time, t ′ = t/f s . From Figure 2(a)(c), for example, we obtain F ( t : 1 . M ⊙ ) = A ff ˙ M v R ph = F ′ ( t ′ : 1 . M ⊙ )= f s F ( t/f s : 1 . M ⊙ ) . (4)Then, we convert the flux of Equation (4) to the absolute V magnitude as M V ( t : 1 . M ⊙ ) = M ′ V ( t ′ : 1 . M ⊙ )= M V ( t/f s : 1 . M ⊙ ) − . f s , (5)where M V is the absolute V magnitude of the free-freeemission light curve. The same equation holds for the0 . M ⊙ and 1 . M ⊙ WDs. Thus, we derive Equation(1). 2.2.
Timescales of Optical/IR Light Curves
The hydrogen-rich envelope mass decreases with time.We plot − ˙ M env against M env for the three WD massesof 1 . M ⊙ (blue), 1 . M ⊙ (red), and 0 . M ⊙ (green) inFigure 3(a). Here, the decreasing rate of the envelopemass is the summation of the wind mass-loss rate andmass-decreasing rate by nuclear burning, i.e., − ˙ M env = ˙ M wind + ˙ M nuc . (6)Time goes on from the upper-right to lower-left of eachline. The decrease in the brightness corresponds to thesame color line in Figure 2(e)(f). The starting pointis somewhere on the line at M env = M env , = M ig ,which corresponds to the maximum brightness of a novaoutburst. Here, we specify the envelope mass at themaximum brightness by M env , which we regard to beequal to the ignition mass, M ig . The ignition mass is de-fined by the hydrogen-rich envelope mass at the start ofhydrogen burning. The wind mass-loss rate ˙ M wind de-creases with decreasing envelope mass M env , and finallyvanishes when the envelope mass reaches the critical en-velope mass M cr required to drive a wind. After that,the envelope mass decreases slowly by nuclear burning.The nova ends when the nuclear burning extinguishes atthe bottom of each line. The sudden flattening on eachline corresponds to the end of the wind phase.If we normalize the envelope mass by each scalingmass, M env /M sc , these three lines almost perfectly over-lap each other for log( − ˙ M env /M ⊙ yr − ) ≥ − . − ˙ M env /M ⊙ yr − ) = − . . M ⊙ (magenta) and 1 . M ⊙ (black). Weshow only five WD masses in this figure, but obtained Hachisu et al.
Figure 3. (a) The envelope mass decreasing rate versus hydrogen-rich envelope mass of optically thick wind (or static) solutionsfor three WD masses with the chemical composition of CO nova 3 (CO3). The solid blue, red, and green lines denote 1 . M ⊙ ,1 . M ⊙ , and 0 . M ⊙ WD models, respectively. The break of each line corresponds to the M cr at which optically thick windsstop. The dotted lines indicate the global trends of ˙ M env - M env relations. (b) The ordinate is the same, but the horizontal axis isscaled by each scaling mass M sc , where the scaling mass is defined by each envelope mass having log( − ˙ M env /M ⊙ yr − ) = − . . M ⊙ (magenta) and 1 . M ⊙ (black). The horizontal dashed line denotes theenvelope mass decreasing rate of log( − ˙ M env /M ⊙ yr − ) = − . the similar tendency of envelope solutions for other WDmasses (ranging from 0 . M ⊙ to 1 . M ⊙ ) with the sameor different chemical compositions (see, e.g., Figure 6 ofKato & Hachisu 1994).Overlapping of the five lines means that we can ex-press − ˙ M env ( x ) ≈ ˙ M wind ( x ) as a function of single pa-rameter of x , i.e., x ≡ M env M sc , (7)independently of the WD mass or chemical composi-tion, while the M sc itself depends both on the M WD andchemical composition, but is almost independent of themass accretion rate for ˙ M acc . × − M ⊙ yr − . For alarger mass accretion rate of ˙ M acc & × − M ⊙ yr − ,the WD radius is slightly larger compared with thatof the cold core. As a result, the M sc for ˙ M acc & × − M ⊙ yr − is slightly larger than that for thecold core of ˙ M acc . × − M ⊙ yr − . In the presentpaper, however, we assume that M sc is independent ofthe mass accretion rate.The increase with M env /M sc in the wind mass-lossrate seems to saturate at the upper-right end of thethree lines. In this region, − ˙ M env ≈ ˙ M wind because˙ M nuc ≪ ˙ M wind . The numerical method adopted byKato & Hachisu (1994) requires convergence of numeri-cal iterations to precisely obtain the wind mass-loss rate.The convergence becomes very slow or fails near the re-gion where the lines seem to saturate. Therefore, we suppose that the true wind mass-loss rate does not sat-urate but increases along the dotted line.Then, the elapsed time is calculated from the decreas-ing rate of the envelope mass, that is, t = Z dM env ˙ M env = M sc Z d ( M env /M sc )˙ M env = M sc ( M WD ) M sc , τ = f s τ, (8)where M sc , is a given envelope mass (we adopt M sc , =0 . × − M ⊙ later in Equation (11)) and τ ≡ M sc , Z d ( M env /M sc )˙ M env = M sc , Z dx ˙ M env ( x ) . (9)Therefore, we obtain f s = M sc M sc , , (10)from the last equality in Equation (8). We explicitlywrite M sc ( M WD ) because the scaling mass depends onthe WD mass, i.e., a function of M WD for a given chem-ical composition, as shown later in Figure 7(a). Theoverlapping of lines for log( − ˙ M env /M ⊙ yr − ) ≥ − . f s ∝ M sc . The fact that the envelope massdecreasing rate ˙ M env ( x ) is approximately a unique func-tion of x ≡ M env /M sc and that the timescale is propor- aximum Magnitude versus Rate of Decline f s ∝ M sc are the first important conclusions ofthe present paper.2.3. Universal Decline Law and Peak Brightness
Figure 4 shows the time-normalized absolute V mag-nitude, M V − . f s , against the scaled envelope mass, M env /M sc , for 0 . M ⊙ (green), 0 . M ⊙ (black), 1 . M ⊙ (red), and 1 . M ⊙ (blue) WDs. These four lines almostoverlap with each other. Therefore, f s F ν ( x ) is approxi-mately a unique function of x = M env /M sc irrespectiveof the WD mass and chemical composition. This is thesecond important conclusion of the present paper.This can be understood as follows: we obtain theuniversal decline law in Figure 2(f) combining Figure2(b) and Figure 2(d). Similarly, we obtain the over-lap of each line in Figure 4 by combining Figure 2(b)and Figure 3(b). This is because − ˙ M env ≈ ˙ M wind forlog( M env /M sc ) = log x ≥ − . M env /M sc ) & .
5. We suppose that thesedeviations are due to the effect of numerical convergenceof iterations as mentioned above in Section 2.2. Theblue line (1 . M ⊙ WD) happens to be located below theother lines of 0 . M ⊙ (black) and 1 . M ⊙ (red) WDswhile the green line (0 . M ⊙ ) slightly diverges upward atlog( M env /M sc ) & .
5. We assume that the true M V − . f s for the 1 . M ⊙ and 0 . M ⊙ WDs follow thedotted line like the other two lines.Figure 1 shows our free-free emission model lightcurves for 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90,0.95, 1.0, 1.05, 1.1, 1.15, and 1 . M ⊙ WDs (CO3). Thetimescale of each WD is measured against that of the0 . M ⊙ WD (CO3), i.e., f s ( M WD ) = M sc ( M WD ) M sc (0 . M ⊙ , CO3)= M sc ( M WD )0 . × − M ⊙ . (11)We confirmed that the timescaling factor f s defined byEquation (10) or (11) is in good agreement with thetimescaling factor f s defined directly by the light curvesof the universal decline law in Figure 1.In short, the brightness of the universal decline law inFigure 1 can be specified by the two parameters, f s and τ ≡ t/f s , where f s is related to M sc through Equation(11) and τ is related to x with M V [ τ ] − . f s = M V [ x ] − . f s in Figure 2(f) and Figure 4. Thus,the two parameter set of ( f s , τ ) is equivalent to the setof ( M sc , x ).In Figure 1, the optical peak of V1668 Cyg corre-sponds to point B. Point A (C) indicates a much brighter(fainter) nova. These points A, B, and C are also plotted Table 1.
Scaling Masses for CO Novae a M WD M sc (CO2) M sc (CO3) M sc (CO4)( M ⊙ ) (10 − M ⊙ ) (10 − M ⊙ ) (10 − M ⊙ )0.55 2.80 2.74 3.340.60 2.22 2.17 2.630.65 1.84 1.80 2.160.70 1.53 1.50 1.800.75 1.21 1.18 1.410.80 0.961 0.935 0.1120.85 0.769 0.746 0.8870.90 0.617 0.598 0.7090.95 0.514 0.498 0.5880.98 0.462 0.448 ...1.00 0.430 0.416 0.4911.05 0.339 0.329 0.3851.10 0.270 0.260 0.3041.15 0.215 0.207 0.2411.20 0.171 0.165 0.192 a chemical composition of the hydrogen-richenvelope is assumed to be those of “COnova 2”, “CO nova 3”, and “CO nova 4”in Table 2 of Hachisu & Kato (2006), i.e.,( X, Y, Z, X C , X O ) = (0 . , . , . , . , . . , . , . , . , . . , . , . , . , . in Figure 2(e)(f) against the real time t and normalizedtime τ = t/f s , respectively. Points A, B, and C cor-respond to different initial envelope masses as shown inFigure 4. Point A, the brightest one, has M V, max = − . M env , = 1 . × − M ⊙ ( x ≡ M env , /M sc = 4 . M V, max = − . M env , = 1 . × − M ⊙ ( x = 3 . M V, max = − . M env , =0 . × − M ⊙ ( x = 2 . . M ⊙ ), the brighter peak corresponds to a largerignition mass. Here, we approximate the initial enve-lope mass M env , by the ignition mass M ig of a novaoutburst. In general, the ignition mass depends on themass accretion rate on to the WD (e.g., Nomoto 1982;Townsley & Bildsten 2004; Wolf et al. 2013a,b). Thelower the mass accretion rate, the larger the ignitionmass for a given WD mass. Thus, the nova is brighterfor a smaller mass-accretion rate even if the WD massis the same. APPROXIMATE ANALYTIC RELATIONSBETWEEN VARIOUS PHYSICAL QUANTITIESThe peak brightness of a nova is calculated from theinitial envelope mass, M env , , as shown in Figure 4. Wemake an approximate analytic relation for this (blackdotted line), i.e., M V − . f s = − . (cid:18) M env M sc (cid:19) − . . (12) Hachisu et al.
Figure 4.
Free-free emission model V light curves are plotted against M env /M sc for the 0 . M ⊙ (green), 0 . M ⊙ (black),1 . M ⊙ (red), and 1 . M ⊙ (blue) WDs. The absolute V magnitude is calibrated with the V1668 Cyg light curve (Hachisu & Kato2016). The three points of A, B, and C are the same as those in Figure 1 and specified by x ≡ M env , /M sc = 4 .
0, 3.1, and 2.1,respectively, on the 0 . M ⊙ WD model. The dotted line represents an approximate relation of Equation (12).
We tabulate M sc for each M WD in Tables 1 (CO novae),2 (Ne novae), and 3 (Solar abundance). Then, the 1-mag, 2-mag, and 3-mag decays from the peak are definedalong this approximate line, that is,1 = 9 . (cid:18) M env , M env , (cid:19) , (13)2 = 9 . (cid:18) M env , M env , (cid:19) , (14)and 3 = 9 . (cid:18) M env , M env , (cid:19) , (15)where M env , is the envelope mass at the peak, M env , atthe 1-mag decay, M env , at the 2-mag decay, and M env , at the 3-mag decay from the peak. We regard M env , tobe the same as the ignition mass. We derive M env , = 0 . M env , , (16) M env , = 0 . M env , , (17)and M env , = 0 . M env , . (18)This means that the brightness drops by 1, 2, and 3mag when the envelope mass is lost in the wind and decreases to 0.785, 0.616, and 0.483 times the initial en-velope mass, respectively. This property is independentof the WD mass or chemical composition. Thus, we usethe envelope mass that characterizes the evolution of anova outburst instead of the time since the optical/IRmaximum.Using the above property, we can rewrite the t , t ,and t times as a function of the envelope mass. Here, t , t , and t times are t = Z M env , M env , dM env ˙ M env , (19) t = Z M env , M env , dM env ˙ M env , (20)and t = Z M env , M env , dM env ˙ M env . (21)We analytically approximate the wind mass loss rate bythe dotted line in Figure 3(b), that is,log − ˙ M env M ⊙ yr − ! = 2 . (cid:18) M env M sc (cid:19) − . . (22)This equation can be rewritten as − ˙ M env = f s C wind (cid:18) M env M sc (cid:19) . , (23) aximum Magnitude versus Rate of Decline Table 2.
Scaling Masses forNeon Novae a M WD M sc (Ne2) M sc (Ne3)( M ⊙ ) (10 − M ⊙ ) (10 − M ⊙ )0.70 2.10 2.730.75 1.64 2.130.80 1.30 1.680.85 1.03 1.330.90 0.823 1.060.95 0.685 0.8761.00 0.568 0.7241.05 0.448 0.5681.10 0.353 0.4471.15 0.280 0.3531.20 0.223 0.2801.25 0.163 0.2041.30 0.113 0.1411.33 0.0810 0.1001.35 0.0603 0.0743 a chemical composition of thehydrogen-rich envelope is as-sumed to be those of “Nenova 2” and “Ne nova 3” inTable 2 of Hachisu & Kato(2006), i.e., ( X, Y, Z, X O , X Ne ) =(0 . , . , . , . , .
03) and(0 . , . , . , . , . together with the proportionality constant of C wind = 10 − . M ⊙ yr − . (24)Using Equation (23) together with the x -parameter, x ≡ M env /M sc , we derive the elapsed time from the opticalpeak, i.e., t = M sc C wind Z x x dxx . = M sc . C wind ( x − . − x − . ) , (25)and τ ≡ t/f s = 54 . x − . − x − . ) days . (26)Then, the t time is calculated from t = M sc C wind Z x x dxx . = M sc . C wind ( x − . − x − . )= M sc . f s C wind x . (0 . − . − . M sc . C wind x . , (27)and the t time is t = M sc . f s C wind x . (0 . − . − Table 3.
Scal-ing Masses for Solarabundance a M WD M sc ( M ⊙ ) (10 − M ⊙ )0.55 7.070.60 5.880.65 4.290.70 3.350.75 2.660.80 2.220.85 1.760.90 1.390.95 1.151.00 0.9481.05 0.7391.10 0.5811.15 0.4561.20 0.3601.25 0.2611.30 0.1801.33 0.1271.35 0.0940 a chemical compositionof the envelope isassumed to be that of“Solar” in Table 2 ofHachisu & Kato(2006), i.e.,( X, Y, Z ) =(0 . , . , . = 1 . M sc . C wind x . , (28)and the t time is t = M sc . C wind x . (0 . − . − . M sc . C wind x . . (29)From Equations (27) and (28), we have a simple relationbetween t and t as t = 0 . t , (30)and, from Equations (28) and (29), we have t = 0 . t . (31)It should be noted that Equations (22) and (23) areapproximately valid for log( − ˙ M env /M ⊙ yr − ) & − . Hachisu et al. shown in Figure 3(b). Therefore, our estimates by Equa-tions (27), (28), and (29) are approximately valid for x & . x & .
5, and x & .
9, respectively. Theselower bounds correspond to log( − ˙ M env /M ⊙ yr − ) = − . M env , , M env , and M env , , respectively.We compare our approximate relations of Equations(28) and (29) with the V1668 Cyg light curve. If weadopt a 0 . M ⊙ WD (CO3) model, we have x = 3 . M sc = 0 . × − M ⊙ from Table1. Then, we obtain t = 10 . t = 19 . t /t = 10 . / . .
54. These t and t values are slightly shorter than, but approximatelyconsistent with, the observation, e.g., t = 12 . t = 24 . V band) in Mallama & Skillman(1979), or t = 12 days and t = 23 days (for V band)in di Paolantonio et al. (1981).Hachisu & Kato (2006) discussed the relation be-tween t and t based on the universal decline law be-cause it has a slope of F ν ∝ t − . . They obtainedthe relation t = 1 . t + 0 . t , where ∆ t is thetime from the outburst to optical maximum. Usually∆ t is short compared with t and t . Then, we have t = (1 / . t = 0 . t . This is approximately equiva-lent to the result of Equation (31). Hachisu & Kato(2006) compared their results with the observation.For example, Capaccioli et al. (1990) obtained t =(1 . ± . t + (1 . ± .
5) days for t <
80 days, or t = (1 . ± . t + (2 . ± .
6) days for t >
80 days.If t ≫ t = (1 / . t = 0 . t . Thisis also approximately equivalent to Equation (31).Recent work done by ¨Ozd¨ormez et al. (2018) con-cluded, however, that the relation between t and t is not unique but different among various types ofnova light curve shapes defined by Strope et al. (2010).¨Ozd¨ormez et al. (2018) obtained log t = 0 .
96 log t +0 .
32 for S (smooth) -type, log t = 0 .
92 log t + 0 .
43 forP (plateau) -type, log t = 0 .
72 log t + 0 . t = 0 .
46 log t + 1 .
29 for J (jitter) -type.The S-type relation corresponds to t ≈ . t ) . . Thisis consistent with Equation (31). We should note thatthe physical meaning of t or t is a local decline trendnear the optical peak. If a light curve has multiple peaks,secondary maximum, oscillations, early dust blackout,jitters, or flares, we should not apply t or t because t or t is greatly affected by such local variations.Not all but rather many novae broadly follow the uni-versal decline law (e.g., Hachisu & Kato 2006, 2007,2010, 2016, 2018, 2019a,b). Strictly speaking, the uni-versal decline law is well applied to S-type light curveshape novae defined by Strope et al. (2010). Such anexample is V1668 Cyg in Figure 1. The other types ofnova light curve shapes deviate from our model light curves in some part. However, their global trends ofdecline can be sometimes fitted with our model lightcurves. Our approximate formulae mentioned above arevalid for such novae. THEORETICAL MMRD RELATION4.1.
The MMRD Relation of the Universal DeclineLaw
In the previous section, we formulated the nova modellight curves for various WD masses and chemical compo-sitions by the two parameters of ( M sc , x ). In this sec-tion, we convert these two parameters to ( t , M V, max ).From Equation (29), we have t = 19 . (cid:18) M sc . × − M ⊙ (cid:19) (cid:16) x . (cid:17) − . days . (32)This is approximately valid for x & . M V, max = 2 . (cid:18) M sc . × − M ⊙ (cid:19) − . (cid:16) x . (cid:17) − . . (33)This is approximately valid for x & . x & . M sc lines and equi- x lines in Figure 5.The solid magenta lines represent each equi- M sc line cor-responding to the M sc values in Table 3. The thin solidblue lines denote each equi- x line, from bottom to top, x = 1, 1.5, 2, 2.5, 3, 3.5, 4, 5, 6, and 7. The thickyellow line corresponds to the x = 2 line, above whichEquations (32) and (33) are approximately valid.We add observational points taken fromDownes & Duerbeck (2000) with filled red circles andSelvelli & Gilmozzi (2019) with filled red stars. We alsoadd an unfilled red star at the position of V1500 Cygtaken from della Valle & Izzo (2020). For the dis-tance to a nova, Downes & Duerbeck (2000) usedthe expansion parallax method of nova shells whileSelvelli & Gilmozzi (2019) and della Valle & Izzo(2020) used the trigonometric parallaxes of Gaia DataRelease 2 (Gaia DR2). These two data show a similartrend in the (log t )- M V, max diagram. We also add twolinear trend lines of M V, max = − . .
12 log t (thicksolid black line) and M V, max = − . .
54 log t (thicksolid cyan line) that represent linear trends derived bySelvelli & Gilmozzi (2019) and Downes & Duerbeck(2000), respectively. The upper-left outlined red star isfor GK Per while the lower-right outlined red star is forV533 Her both from Selvelli & Gilmozzi (2019). Wediscuss these two novae in Section 5. aximum Magnitude versus Rate of Decline Figure 5.
The maximum V magnitude M V, max against the rate of decline t for various equi- M sc (solid magenta lines) andequi- x (solid blue lines) models. The M V, max and t are calculated from Equations (33) and (32), respectively. The filled redcircles and stars are for galactic novae obtained by Downes & Duerbeck (2000) and Selvelli & Gilmozzi (2019), respectively.We also add an unfilled red star at the position of V1500 Cyg taken from della Valle & Izzo (2020). The upper-left outlinedred star is for GK Per while the lower-right outlined red star is for V533 Her. The thick solid cyan line represents the “classical”MMRD relation defined by Downes & Duerbeck (2000), i.e., M V, max = − .
99 + 2 .
54 log( t ) while the thick solid black linerepresents the linear MMRD relation defined by Selvelli & Gilmozzi (2019), i.e., M V, max = − .
08 + 2 .
12 log( t ). The solidmagenta lines connect the same scaling mass M sc but for different x . The scaling masses are, from right to left, the same asthose tabulated in Table 3. The thin solid blue lines connect the same x , i.e., x = 1, 1.5, 2, 2.5, 3, 3.5, 4, 5, 6, and 7, fromlower to upper. The thick yellow line corresponds to the x = 2 line, above which Equation (32) is approximately valid. Themagenta lines of equi- M sc have a slope of 6.3 while the blue lines of equi- x have a slope of 2.5 in the (log t )- M V, max diagram. The equi- M sc lines have a slope of 6.3 (= 9 . / . x lines have a slope of 2.5 in the (log t )- M V, max diagram. The latter slope is close to 2.54of the cyan line, the trend MMRD line defined byDownes & Duerbeck (2000), but slightly steeper thanthe slope of 2.12 (thick solid black line) obtained bySelvelli & Gilmozzi (2019). This thick black line tra-verses the two blue lines of x = 3 . x ∼ x ∼
6, the center of which is x ∼ . x = 3 . f s ∝ M sc as shown in Equations (10) and(11). On the other hand, the t time is not a global2 Hachisu et al. timescale but a local timescale only near the peak of anova light curve. The t time is proportional to f s ( ∝ M sc ) but depends also on the x as in Equation (29) or(32). This is the reason why the MMRD points showa large scatter around the main trend in the (log t )- M V, max diagram. This can be easily understood if weeliminate M sc from Equations (32) and (33) and obtain M V, max = 2 . t − . − .
75 log (cid:16) x . (cid:17) . (34)Then, we have M V, max = 2 . t − . , for x = 3 . . (35)This MMRD relation is essentially the same as that(cyan line) obtained by Downes & Duerbeck (2000),but slightly steeper than the trend MMRD line (blackline) obtained by Selvelli & Gilmozzi (2019) in Figure5. The scatter from the main trend ( x ∼ . − . x , that is, thedifference in the ignition mass.4.2. The MMRD Relation for Novae
In this subsection, we obtain ( t , M V, max ) against novamodels, which are specified by the mass accretion rate˙ M acc and the WD mass M WD (or sometimes by the re-currence time t rec ). These parameters are determinedby the binary nature. We adopt the ignition mass M ig ,mass accretion rate ˙ M acc , and recurrence time t rec frompublished data available to the authors.4.2.1. Ignition mass model
The ignition masses have been calculated by many au-thors (e.g., Prialnik & Kovetz 1995; Kato et al. 2014;Hachisu et al. 2016; Chen et al. 2019). However, theignition masses can sometimes differ significantly eachother, depending not only on the model assumption butalso on the calculation method (numerical code, see, e.g.,Kato et al. 2017a).We adopt the result of Kato et al. (2014). They cal-culated the accumulation mass M acc and ignition mass M ig assuming solar abundance. They obtained mod-els for ˙ M acc ≥ × − M ⊙ yr − . In the presentpaper, we extend the mass accretion rate down to˙ M acc ≥ × − M ⊙ yr − taking into account thelower mass-accretion rate limit for cataclysmic variables(e.g., Knigge et al. 2011). We tabulate M acc (third col-umn) in Table 4 for various WD masses M WD (first col-umn) and mass accretion rates ˙ M acc (second column).We assume that the envelope mass at optical maximum M env , is almost the same as the ignition mass M ig , i.e., M env , = M ig . Using the scaling mass M sc in Table 3,we calculate the ratio of x ≡ M env , /M sc = M ig /M sc . These x are also tabulated on the fifth column in Table4. It should be noted that the accumulation mass M acc = t rec × ˙ M acc is slightly smaller than the envelope mass atignition M ig = M env , = x × M sc . When the mass ac-cretion starts, there is residual hydrogen-rich materialon the WD. This residual is leftover from the previousnova explosion. The ignition mass is defined by the sum-mation of the accreted mass and the envelope mass atthe epoch when hydrogen burning extinguishes, i.e., M ig = M env , = M acc + M env , min . (36)(See Figure 1 of Kato et al. (2014) for the relationamong these envelope masses.)For a given ignition mass, we obtain the absolute V magnitude at optical maximum M V, max from Equation(33). The values of M V, max are tabulated at 6th col-umn in Table 4. We obtain the t time from Equation(32) and t time from Equation (31). These are alsotabulated in Table 4.4.2.2. Global trend of MMRD relation
We plot these peak V brightness versus rate of declinerelation in Figure 6. Each thin solid blue line connectsthe same WD mass models with different mass accretionrates. The thick solid gray lines connect the models withthe same mass accretion rate. These gray lines have apeak of M V, max at M WD = 1 . M ⊙ . The thin red linesconnect the same recurrence period models.In this figure, we add the observational MMRDpoints, filled red circles, filled red stars, and an un-filled red star, obtained by Downes & Duerbeck (2000),Selvelli & Gilmozzi (2019), and della Valle & Izzo(2020), respectively. We can see that the distributionof MMRD points is covered by mass accretion ratesbetween ˙ M acc ∼ × − M ⊙ yr − and ˙ M acc ∼ × − M ⊙ yr − and centered on ˙ M acc ∼ × − M ⊙ yr − in the longer t region of t &
30 days,based on the result of Kato et al. (2014).The thick solid black line is a trend of MMRD distri-bution obtained by Selvelli & Gilmozzi (2019), whichis close to an equi-mass accretion rate line of ˙ M acc ∼ × − M ⊙ yr − in the longer t region of t &
30 days(or in the fainter region of M V, max ≥ − . M acc = 3 × − M ⊙ yr − ,1 × − M ⊙ yr − , 1 × − M ⊙ yr − , and 1 × − M ⊙ yr − in the brighter region of M V, max ≤− .
0. If we divide the data set (filled red stars)of Selvelli & Gilmozzi (2019) into two groups by thebrightness of M V, max = − .
0, the brighter (upper-left) group is located in between ˙ M acc = 1 × − aximum Magnitude versus Rate of Decline Figure 6.
Same as Figure 5, but for equi- M WD , equi- ˙ M acc , and equi- t rec models of solar abundance. The thin solid bluelines connect the same WD mass M WD but for different mass accretion rates ˙ M acc . The WD masses are, from right to left, M WD = 0 .
6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.25, 1.3, and 1 . M ⊙ . The thick yellow line corresponds to the x = 2 line. The thinsolid red lines connect the same recurrence time, i.e., t rec = 30, 100, 300, 1000, 10000, 10 , 10 , and 10 yr, from lower to upper.The thick solid gray lines represent the same mass accretion rate, from lower to upper, ˙ M acc = 3 × − , 1 × − , 5 × − ,3 × − , 1 × − , 1 × − , and 1 × − M ⊙ yr − . See Table 4 for each data. The four unfilled red circles denote theMMRD positions of the four recurrent novae, CI Aql, T CrB, U Sco, and V745 Sco. Other symbols and lines are the same asthose in Figure 5. See text for more details. and 1 × − M ⊙ yr − and the fainter (lower-right)group is located in between ˙ M acc = 3 × − , and3 × − M ⊙ yr − . This difference in the mass accre-tion rate broadly correspond to cataclysmic variablesbelow/above the period gap (e.g., Knigge et al. 2011).For an MMRD point of the observed nova, we are ableto broadly specify the WD mass and mass accretion rate.The main trend in the MMRD diagram corresponds tothe mass accretion rate of ˙ M acc ∼ × − M ⊙ yr − withthe different WD masses in the lower region of M V, max ≥− .
0. Selvelli & Gilmozzi (2019) estimated the massaccretion rates from the quiescent luminosities of their data set novae in Figure 6 (filled red stars). Their me-dian value is ˙ M WD = 3 . × − M ⊙ yr − , which is closeto our main trend value of ˙ M acc ∼ × − M ⊙ yr − .The scatter up to ± σ = ± . Thus, the global trend of an MMRD rela-tion does indeed exist, at least, in the fainter region of M V, max ≥ − . , but its scatter is too large for it to be aprecision distance indicator of individual novae. Selvelli & Gilmozzi (2019) reported a correlation be-tween ˙ M acc and the speed class t ( ˙ M acc increasing with t as in their Figure 5) for their 17 nova data set and4 Hachisu et al. wrote “we cannot find a simple explanation that couldaccount for this.” This correlation ( ˙ M acc increases with t ) can be easily seen up to t .
80 days for the filled redstars in our Figure 6. For t &
80 days, this correlationseems to be flat except for RR Pic and HR Del. Thisglobal trend of our ˙ M acc versus t relation is consistentwith their Figure 5.4.2.3. MMRD positions of individual novae
Finally, we point out that there is a − . V brightness if we limit the mass accretionrate above ˙ M acc ≥ × − M ⊙ yr − .Several novae are located outside our region of( t , M V, max ). They are, from left to right, V1500 Cyg(red filled circle: t = 3 . M V, max = − . − . − . − . M V, max = − . M V, max = − .
72 and t = 8 days (CP Pup, filled redstar), − .
51 and 250 days (RR Pic, filled red star), and − .
58 and 230 days (HR Del, filled red star). These fourcorrections are indicated by the red arrows in Figure 6.This explains that the brightest peaks of novaeare fainter than M V ∼ − . Thebrightness cap ( M V ∼ − . ) of classical novae is con-strained by the lowest mass-accretion rate of ˙ M acc ∼ × − M ⊙ yr − . In the longer t time, only two novae (HR Del andRR Pic) are outside our theoretical region between M WD = 0 . . M ⊙ and between ˙ M acc = 3 × − M ⊙ yr − and 1 × − M ⊙ yr − . This simplymeans (1) that the WD mass is smaller than 0 . M ⊙ or (2) that the concept of t and t time should notbe applied to these novae for some reasons. Thesetwo novae have multiple peaks (see, e.g., Figure 1 ofHachisu & Kato 2015). The t or t time is a localtimescale near the peak and is greatly affected by suchvariations. We should not apply t and t to such no-vae. Hachisu & Kato (2015) analyzed the light curvesof HR Del and RR Pic and globally fitted their modellight curves of M WD = 0 . M ⊙ and 0 . M ⊙ with theobserved light curves of these two novae except the mul-tiple peaks.GK Per and V446 Her belong to the shortest t andbrightest M V, max group of Selvelli & Gilmozzi’s novadata set, i.e., ( t , M V, max ) = (13 , − .
05) and (15 , − . − M ⊙ yr − (e.g., Knigge et al. 2011). Figure 6 indicates rela-tively low mass accretion rates of ∼ × − and ∼ × − M ⊙ yr − . Selvelli & Gilmozzi (2019) es-timated the mass accretion rates, ˙ M acc = 2 × − and0 . × − M ⊙ yr − , respectively. Their estimated rateof V446 Her seems to be consistent with our theoreticalvalue estimated from the position in the (log t )- M V, max diagram while that of GK Per is about ten times largerthan our value. This suggests that, for a certain kind ofnovae, the mass accretion rate during a post-nova phaseis rather high compared with that of a substantial pre-nova phase, or that the mass accretion rate graduallydecreases toward much lower rates in a timescale of re-currence period.CP Pup has a lowest value of ˙ M acc in Selvelli& Gilmozzi’s nova data set, i.e., ˙ M acc = 0 . × − M ⊙ yr − . On the other hand, Figure 6 indi-cates ˙ M acc = (1 . − × − M ⊙ yr − . This isabout 30 times smaller than that of Selvelli & Gilmozzi.Schaefer & Collazzi (2010) reported that the pre-novabrightness of CP Pup is ∆ m ∼ ∼
100 times smaller than thatduring a post-nova phase. If the ˙ M acc decreases toward˙ M acc = 0 . × − M ⊙ yr − at the pre-nova phase, theaverage mass-accretion rate during the quiescent phaseis close to ˙ M acc ∼ × − M ⊙ yr − . This averagevalue is consistent with our theoretical estimate.4.2.4. MMRD positions of recurrent novae
Recurrent novae (unfilled red circles) are located be-low the yellow line in Figure 6. Here, we plot fourMMRD positions of the recurrent novae, CI Aql ( t =32 days , M V, max = − . , − . . , − . , − . m V, max = 9 . A V = 3 . E ( B − V ) = 3 . × . . d = 3189 +949 − pc (Gaia distance, Schaefer 2018). Weexclude RS Oph and T Pyx. This is because the lightcurve of RS Oph is contaminated by the shock-heatingbetween the ejecta and circumstellar matter and, as aresult, the t time does not represent the timescale ofintrinsic decline (e.g., Hachisu & Kato 2018). T Pyxhas multiple peaks and we should not apply the conceptof t .The recurrence periods of these recurrent novae are ∼ −
80 yr (e.g., Schaefer 2010). It is obvious that theMMRD positions of these recurrent novae are not con- aximum Magnitude versus Rate of Decline Figure 7. (a) The scaling mass M sc against the WD mass M WD for various chemical compositions. The filled black starsconnected by a black line show the models in Table 3 (solar composition). The filled blue triangles connected by a blue linedenote the models in Table 2 (Ne nova 2). The unfilled red squares connected by a red line correspond to the models in Table1 (CO nova 3). The CO3 has X = 0 . Y = 0 . Z = 0 . X C = 0 .
15, and X O = 0 .
20 by mass weight. The Ne2 is composedof X = 0 . Y = 0 . Z = 0 . X O = 0 .
10, and X Ne = 0 .
03. (b) Same as in panel (a), but we shift the WD masses of thechemical composition Ne2 by +0 . M ⊙ , and of CO3 by +0 . M ⊙ as plotted in the figure. These three lines overlap well until1 . M ⊙ . sistent with the equi-recurrence period lines of t rec = 30and 100 yr in Figure 6. We give three reasons for thisexceptional case. The first point is that the presenttheory cannot be applicable to recurrent novae becauseEquation (32) is not valid for x . M acc & × − M ⊙ yr − in recurrent novae. Thismeans that the WD core is hot and its radius is largerthan that of a cold core as mentioned in Section 2.2.We must calculate wind solutions assuming the largeWD radius depending on the mass accretion rate. Thethird point is that a hot helium layer develops under-neath a hydrogen-rich envelope in recurrent novae (e.g.,Kato et al. 2017b). This possibly affects the thermalstate of hydrogen-rich envelope. We must take into ac-count at least these three effects for recurrent novae.The calculation is so complicated that we leave it tonear future. DISCUSSIONWe have obtained the relation between the peak V brightness M V, max and the rate of decline t (or t ) inSection 4.2 for solar composition ( X = 0 . Y = 0 . Z = 0 .
02 by mass weight). However, it has beenreported that nova ejecta are enriched by heavy elementssuch as carbon, oxygen, and neon (e.g., Gehrz et al. 1998). Here, we discuss the effect of enrichment in heavyelements.Kato & Hachisu (1994) calculated nova models forvarious chemical compositions. We have already tab-ulated the scaling masses M sc for three cases, CO nova2 (CO2), CO nova 3 (CO3), and CO nova 4 (CO4) inTable 1 and two cases, Ne nova 2 (Ne2) and Ne nova3 (Ne3) in Table 2. The CO3 chemical compositionhas X = 0 . Y = 0 . Z = 0 . X C = 0 .
15, and X O = 0 .
20 by mass weight. The Ne2 chemical compo-sition is composed of X = 0 . Y = 0 . Z = 0 . X O = 0 .
10, and X Ne = 0 .
03. We regard the CO3 tobe a typical chemical composition for CO novae and theNe2 to be a typical one for neon novae. The M sc for theCO3 and Ne2 are plotted in Figure 7(a) together withthe solar composition case in Table 3.If the scaling mass is the same for two different chem-ical compositions, their timescales are also the same asinferred from Equation (11). We shift the lines of CO3and Ne2 in Figure 7(a) toward the right by +0 . M ⊙ and +0 . M ⊙ , respectively, in Figure 7(b). Then, thethree lines (CO3, Ne2, and Solar) almost overlap witheach other at least until M WD = 1 . M ⊙ . This meansthat, for the CO3 case, the timescale ( t ) and the peak V brightness ( M V, max ) of M WD , CO3 + 0 . M ⊙ are thesame as those of M WD , solar . In other words, we havethe same MMRD relation between the CO3 and solar6 Hachisu et al.
Figure 8.
Same as Figure 5, but for the chemical composition of CO nova 3 (CO3). Each scaling mass (magenta line)corresponds to the WD mass, from right to left, 0 . M ⊙ to 1 . M ⊙ by 0 . M ⊙ step except 0 . M ⊙ , the same as thosetabulated in Table 1(CO3). The thick solid magenta lines of equi- M sc (or equi- M WD ) correspond to the WD mass region forCO novae ( M WD ≤ . M ⊙ ). compositions at M WD , CO3 = M WD , solar − . M ⊙ , (37)and between the Ne2 and solar at M WD , Ne2 = M WD , solar − . M ⊙ . (38)This difference in the WD mass for different chemicalcompositions appears in the WD mass estimate fromthe light curve fitting (e.g., Hachisu & Kato 2016). Toconfirm these WD mass relations, we plot the (log t )– M V, max diagrams for the CO3 and Ne2 cases in Figures8 and 9, respectively, which are the same as Figure 5but for different set of M sc .To better understand these WD mass relations, weexamine the case of V533 Her (Nova Her 1963).Selvelli & Gilmozzi (2019) obtained t = 44 ± M V, max = − . ± .
22. The MMRD point ofV533 Her is the lower-right outlined red star in Fig-ures 5 and 6. The position is on the M WD , solar =0 . ± . M ⊙ and t rec = 4000 ± M WD , CO3 = M WD , solar − . M ⊙ = 0 . ± . M ⊙ . For the Ne2case, we obtain M WD , Ne2 = M WD , solar − . M ⊙ =0 . ± . M ⊙ . We can confirm that these two WDmasses are reasonable in Figures 8 and 9, respectively.For the case of M WD , solar & . M ⊙ , the WD massdifference between M WD , solar and M WD , CO3 or between M WD , solar and M WD , Ne2 for the same M sc becomessmaller as shown in Figure 7(a). Then, the WD massdifference should be measured directly from Figure 7(a).We examine the case of GK Per (Nova Per 1901).Selvelli & Gilmozzi (2019) obtained t = 13 ± aximum Magnitude versus Rate of Decline Figure 9.
Same as Figure 5, but for the chemical composition of Ne nova 2 (Ne2). Each scaling mass (magenta line)corresponds to the WD mass, from right to left, 0 . M ⊙ to 1 . M ⊙ by 0 . M ⊙ step except 1 . M ⊙ , the same as thosetabulated in Table 2(Ne2). The thick solid magenta lines of equi- M sc (or equi- M WD ) correspond to the WD mass region forneon novae ( M WD ≥ . M ⊙ ). and M V, max = − . ± .
16. The MMRD point is lo-cated at/near M WD , solar = 1 . M ⊙ and t rec = 8 × yrin Figure 6. If we assume the CO3 chemical composi-tion for GK Per, the WD mass difference is estimated tobe 0 . M ⊙ at M WD , solar = 1 . M ⊙ from Figure 7(a).Then, we have M WD , CO3 = M WD , solar − . M ⊙ =1 . M ⊙ . For the Ne2 case, we obtain the differ-ence of 0 . M ⊙ and the WD mass of M WD , Ne2 = M WD , solar − . M ⊙ = 1 . M ⊙ . We can also confirmthat these two WD masses are reasonable in Figures 8and 9, respectively.Hachisu & Kato (2007) estimated the WD mass tobe M WD , Ne2 = 1 . ± . M ⊙ for GK Per based on thenova model light curve fitting. This is approximatelyconsistent with the above estimate from the MMRDpoint. On the other hand, Hachisu & Kato (2019a) es-timated the WD mass of V533 Her to be M WD , Ne2 = 1 . ± . M ⊙ from the nova model light curve fitting.This value is larger than the above estimate from theMMRD relation, M WD , Ne2 = 0 . ± . M ⊙ . This ispartly because observational V data are very poor nearthe peak and there are jitters with a 0.5 mag amplitudeat 2-mag decay from the peak (see, e.g., Figure 33 ofHachisu & Kato 2019a). CONCLUSIONSHachisu & Kato (2006, 2010) constructed free-freeemission model light curves of novae based on theoptically-thick wind theory (Kato & Hachisu 1994).These light curves provided good fit to those of manyobserved novae. Hachisu & Kato (2006) also showedthat the theoretical nova light curves are homologousagainst a normalized time τ = t/f s and can be expressedwith one-parameter family of the timescaling factor f s .8 Hachisu et al.
We find the new parameter set of ( M sc , x ) equivalentto ( f s , τ ). Using this equivalent conversion, we proposea theory for the maximum magnitude versus rate ofdecline (MMRD) relation for novae based on the uni-versal decline law. It should be noted that our MMRDrelations from the universal decline law are applicableonly to specific light curves of novae such as S-typesdefined by Strope et al. (2010). Our main conclusionsare as follows: We adopt the dimensionless envelope mass x ≡ M env /M sc , as the parameter that describes the novalight curves. The peak V brightness is expressed by ananalytic form of x and M sc , i.e., Equation (33), where x is the ratio of the initial envelope mass and M sc , x = M env , /M sc . The decline rate of t (or t ) is alsocalculated from an analytic formula of x and M sc , i.e.,Equation (32) (or Equation (31)). The scaling masses M sc are defined by theenvelope mass having a wind mass-loss rate oflog( ˙ M wind /M ⊙ yr − ) = − .
7, which are summarizedin Tables 1, 2, and 3 for various WD masses and chem-ical compositions (taken from Kato & Hachisu 1994),and x , M V, max , t , and t are tabulated in Table 4 forvarious WD masses and mass accretion rates of solarabundance material (taken from Kato et al. 2014, withadditional calculation for the present work). We plot our model MMRD relations for the sameWD mass, same mass accretion rate, and same re-currence time (solar composition). The theoreticalrange of MMRD for expected nova parameters well con-strains the MMRD points of observed novae obtained byDownes & Duerbeck (2000) and Selvelli & Gilmozzi(2019) except the very long t &
200 days re-gion. The WD mass ranges mainly from 1 . M ⊙ to 0 . M ⊙ . The mass accretion rate is typically between3 × − M ⊙ yr − and 1 × − M ⊙ yr − centered at5 × − M ⊙ yr − in the longer t region of t > yr. From the MMRD point of an observed nova, weare able to broadly specify the WD mass and mass ac-cretion rate. The main trend in the observed MMRDdistribution corresponds to the mass accretion rate of˙ M acc ∼ × − M ⊙ yr − with the different WD massesin the longer t region of t >
30 days. The scatter of nova MMRD points from the main trend line can beattributed to a large scatter of the observed mass ac-cretion rates from that of the main trend defined by˙ M acc ∼ × − M ⊙ yr − . Thus, the global trend of anMMRD relation does exist, but its scatter is too largefor it to be a precision distance indicator of individualnovae. In the shorter t region of t <
30 days, themain trend MMRD relation traverses the four linesof mass-accretion rates, ˙ M acc = 3 × − M ⊙ yr − ,10 − M ⊙ yr − , 10 − M ⊙ yr − , and 10 − M ⊙ yr − .In general, the smaller the t time, the smaller the ˙ M acc .The recurrence time is typically between t rec = 10000yr and 10 yr. If we divide the data of Selvelli & Gilmozzi (2019)into two groups by the brightness of M V, max = − .
0, theupper brighter group broadly correspond to cataclysmicvariables below the period gap ( ˙ M acc ∼ − M ⊙ yr − )while the lower fainter group correspond to binariesabove the period gap ( ˙ M acc ∼ × − M ⊙ yr − ). The lower the mass accretion rate, the larger theignition mass for a given WD mass and chemical com-position. Thus, the lower the mass accretion rate, thebrighter the V peak of a nova. There is a M V ∼ − . M acc ≥ × − M ⊙ yr − (e.g.,Knigge et al. 2011). This explains that the brightestpeaks of novae are at/around M V ∼ − . Thus, weclarified the reason for the brightness cap ( M V ∼ − . )of classical novae, which is constrained by the lowestmass-accretion rate of ˙ M acc ∼ × − M ⊙ yr − . Finally, we discussed the effect of enrichment ofheavy elements in nova ejecta. The enrichment has thesame effect as the WD mass decrease. For example,the difference between the abundances of solar and Ne2novae is 0.12 M ⊙ , i.e., M WD , solar = M WD , Ne2 +0 . M ⊙ ,for the same M V, max and t .ACKNOWLEDGMENTSI.H. and M.K. thank Department of Astronomy, SanDiego State University, for the warm hospitality, duringwhich we initiated the present work. We are gratefulto the anonymous referee for useful comments, whichimproved the manuscript. aximum Magnitude versus Rate of Decline A. NOVA IGNITION MODELSWe have calculated the MMRD points, M V, max and t (or t ), based on the result of Kato et al. (2014) for vari-ous WD masses and mass accretion rates (solar abundance) together with additional calculation that expanded theparameter range of mass-accretion rate for the present work. Corresponding parameters are the WD mass, M WD ,mass-accretion rate, ˙ M acc , accumulation mass, M acc (Section 4.2.1), scaling mass, M sc (Section 2.2), parameter x (Sections 2.2 and 2.3), peak absolute V magnitude, M V, max (Equation (33)), times t , t (Section 3, Equation (32)),and the recurrence time, t rec (Section 4.2, Figure 6). Table 4 . MMRD Relation for Kato et al.’s Model a M WD ˙ M acc M acc M sc x M V, max t t t rec ( M ⊙ ) ( M ⊙ yr − ) ( M ⊙ ) ( M ⊙ ) (days) (days) (yr)0.6 1.0E-11 2.52E-4 5.88E-5 4.56 -7.20 78.6 146. 2.52E+70.6 3.0E-11 2.42E-4 5.88E-5 4.39 -7.04 83.2 154. 8.08E+60.6 5.0E-11 2.39E-4 5.88E-5 4.33 -6.99 84.9 157. 4.78E+60.6 1.0E-10 2.37E-4 5.88E-5 4.31 -6.97 85.6 159. 2.37E+60.6 3.0E-10 2.31E-4 5.88E-5 4.21 -6.87 88.8 164. 7.71E+50.6 1.0E-9 2.17E-4 5.88E-5 3.97 -6.62 97.0 180. 2.17E+50.6 1.6E-9 2.07E-4 5.88E-5 3.79 -6.44 104. 192. 1.29E+50.6 3.0E-9 1.91E-4 5.88E-5 3.52 -6.12 116. 215. 6.35E+40.6 5.0E-9 1.75E-4 5.88E-5 3.24 -5.79 131. 243. 3.49E+40.6 1.0E-8 1.48E-4 5.88E-5 2.78 -5.16 165. 306. 1.48E+40.6 1.6E-8 1.32E-4 5.88E-5 2.52 -4.74 192. 356. 8235.0.6 2.0E-8 1.25E-4 5.88E-5 2.41 -4.56 205. 380. 6269.0.6 3.0E-8 1.15E-4 5.88E-5 2.24 -4.26 229. 424. 3846.0.7 1.0E-11 1.72E-4 3.35E-5 5.44 -8.54 34.4 63.6 1.72E+70.7 3.0E-11 1.63E-4 3.35E-5 5.17 -8.32 37.1 68.8 5.44E+60.7 5.0E-11 1.60E-4 3.35E-5 5.06 -8.24 38.3 70.9 3.19E+60.7 1.0E-10 1.55E-4 3.35E-5 4.93 -8.13 39.8 73.7 1.55E+60.7 3.0E-10 1.48E-4 3.35E-5 4.70 -7.93 42.8 79.2 4.92E+50.7 1.0E-9 1.35E-4 3.35E-5 4.33 -7.59 48.4 89.6 1.35E+50.7 3.0E-9 1.10E-4 3.35E-5 3.59 -6.82 64.1 119. 3.68E+40.7 1.0E-8 8.81E-5 3.35E-5 2.92 -5.98 87.2 161. 8808.0.7 3.0E-8 6.47E-5 3.35E-5 2.22 -4.85 131. 243. 2156.0.7 5.0E-8 5.68E-5 3.35E-5 1.99 -4.39 155. 288. 1136.0.7 6.0E-8 5.47E-5 3.35E-5 1.93 -4.25 163. 302. 911.0.8 1.0E-11 1.18E-4 2.22E-5 5.56 -9.07 22.1 40.9 1.18E+70.8 3.0E-11 1.10E-4 2.22E-5 5.20 -8.80 24.4 45.2 3.68E+60.8 5.0E-11 1.07E-4 2.22E-5 5.06 -8.68 25.4 47.1 2.15E+60.8 1.0E-10 1.06E-4 2.22E-5 4.98 -8.62 26.1 48.3 1.06E+60.8 3.0E-10 1.00E-4 2.22E-5 4.74 -8.41 28.1 52.0 3.34E+50.8 1.0E-9 9.18E-5 2.22E-5 4.37 -8.07 31.8 58.8 9.18E+40.8 1.6E-9 8.70E-5 2.22E-5 4.15 -7.86 34.3 63.5 5.44E+40.8 3.0E-9 7.90E-5 2.22E-5 3.79 -7.49 39.3 72.8 2.63E+40.8 5.0E-9 7.21E-5 2.22E-5 3.48 -7.14 44.7 82.7 1.44E+40.8 1.0E-8 6.01E-5 2.22E-5 2.94 -6.44 57.6 107. 6006.0.8 1.6E-8 5.19E-5 2.22E-5 2.57 -5.89 70.3 130. 3244.0.8 3.0E-8 4.29E-5 2.22E-5 2.17 -5.18 90.9 168. 1430. Table 4 continued Hachisu et al.
Table 4 (continued) M WD ˙ M acc M acc M sc x M V, max t t t rec ( M ⊙ ) ( M ⊙ yr − ) ( M ⊙ ) ( M ⊙ ) (days) (days) (yr)0.8 5.0E-8 3.67E-5 2.22E-5 1.89 -4.61 112. 207. 734.0.8 7.0E-8 3.34E-5 2.22E-5 1.74 -4.27 127. 234. 477.0.8 7.5E-8 3.28E-5 2.22E-5 1.71 -4.21 130. 240. 437.0.9 1.0E-11 7.93E-5 1.39E-5 5.90 -9.83 12.7 23.4 7.93E+60.9 3.0E-11 7.29E-5 1.39E-5 5.44 -9.49 14.3 26.5 2.43E+60.9 5.0E-11 7.04E-5 1.39E-5 5.26 -9.35 15.0 27.8 1.41E+60.9 1.0E-10 6.74E-5 1.39E-5 5.05 -9.18 16.0 29.6 6.74E+50.9 3.0E-10 6.31E-5 1.39E-5 4.74 -8.92 17.6 32.6 2.10E+50.9 1.0E-9 5.66E-5 1.39E-5 4.27 -8.49 20.5 38.0 5.66E+40.9 3.0E-9 4.84E-5 1.39E-5 3.69 -7.88 25.6 47.5 1.61E+40.9 1.0E-8 3.87E-5 1.39E-5 2.99 -7.01 35.2 65.1 3867.0.9 3.0E-8 2.72E-5 1.39E-5 2.17 -5.69 56.9 105. 908.0.9 5.0E-8 2.30E-5 1.39E-5 1.86 -5.06 71.6 133. 459.0.9 7.0E-8 2.06E-5 1.39E-5 1.69 -4.66 82.7 153. 294.0.9 9.0E-8 1.90E-5 1.39E-5 1.58 -4.38 91.6 170. 212.0.9 1.0E-7 1.84E-5 1.39E-5 1.53 -4.27 95.5 177. 184.0.9 1.1E-7 1.79E-5 1.39E-5 1.50 -4.17 99.0 183. 163.1.0 1.0E-11 5.07E-5 9.25E-6 5.68 -10.11 8.89 16.5 5.07E+61.0 3.0E-11 4.59E-5 9.25E-6 5.16 -9.71 10.3 19.1 1.53E+61.0 5.0E-11 4.41E-5 9.25E-6 4.96 -9.55 10.9 20.2 8.81E+51.0 1.0E-10 4.30E-5 9.25E-6 4.85 -9.46 11.3 20.9 4.30E+51.0 3.0E-10 4.00E-5 9.25E-6 4.53 -9.18 12.5 23.2 1.33E+51.0 1.0E-9 3.63E-5 9.25E-6 4.12 -8.79 14.4 26.6 3.63E+41.0 1.6E-9 3.44E-5 9.25E-6 3.92 -8.58 15.5 28.7 2.15E+41.0 3.0E-9 3.12E-5 9.25E-6 3.57 -8.20 17.9 33.1 1.04E+41.0 5.0E-9 2.85E-5 9.25E-6 3.27 -7.84 20.3 37.7 5689.1.0 1.0E-8 2.39E-5 9.25E-6 2.78 -7.16 26.0 48.2 2386.1.0 1.6E-8 2.05E-5 9.25E-6 2.41 -6.58 32.2 59.6 1280.1.0 3.0E-8 1.67E-5 9.25E-6 2.00 -5.80 42.6 78.9 555.1.0 5.0E-8 1.39E-5 9.25E-6 1.69 -5.12 54.6 101. 277.1.0 1.0E-7 1.09E-5 9.25E-6 1.37 -4.25 75.0 139. 109.1.0 1.2E-7 1.02E-5 9.25E-6 1.30 -4.03 81.1 150. 85.11.1 1.0E-11 3.01E-5 5.81E-6 5.36 -10.37 6.10 11.3 3.01E+61.1 3.0E-11 2.68E-5 5.81E-6 4.78 -9.91 7.23 13.4 8.93E+51.1 5.0E-11 2.56E-5 5.81E-6 4.58 -9.73 7.72 14.3 5.12E+51.1 1.0E-10 2.49E-5 5.81E-6 4.46 -9.62 8.03 14.9 2.49E+51.1 3.0E-10 2.31E-5 5.81E-6 4.14 -9.31 8.98 16.6 7.69E+41.1 1.0E-9 2.09E-5 5.81E-6 3.76 -8.92 10.4 19.2 2.09E+41.1 1.6E-9 1.98E-5 5.81E-6 3.57 -8.70 11.2 20.8 1.23E+41.1 3.0E-9 1.79E-5 5.81E-6 3.26 -8.32 12.9 23.8 5976.1.1 5.0E-9 1.64E-5 5.81E-6 2.99 -7.97 14.6 27.1 3277.1.1 1.0E-8 1.38E-5 5.81E-6 2.55 -7.31 18.6 34.5 1381.1.1 1.6E-8 1.19E-5 5.81E-6 2.22 -6.73 23.0 42.5 742.1.1 3.0E-8 9.62E-6 5.81E-6 1.83 -5.94 30.6 56.7 321.1.1 5.0E-8 7.95E-6 5.81E-6 1.54 -5.23 39.6 73.4 159.1.1 1.0E-7 6.13E-6 5.81E-6 1.23 -4.29 55.8 103. 61.31.1 1.6E-7 5.14E-6 5.81E-6 1.06 -3.68 69.7 129. 32.21.2 1.0E-11 1.54E-5 3.60E-6 4.42 -10.10 5.04 9.34 1.54E+61.2 3.0E-11 1.35E-5 3.60E-6 3.90 -9.59 6.08 11.3 4.50E+5 Table 4 continued aximum Magnitude versus Rate of Decline Table 4 (continued) M WD ˙ M acc M acc M sc x M V, max t t t rec ( M ⊙ ) ( M ⊙ yr − ) ( M ⊙ ) ( M ⊙ ) (days) (days) (yr)1.2 5.0E-11 1.28E-5 3.60E-6 3.70 -9.37 6.59 12.2 2.55E+51.2 1.0E-10 1.24E-5 3.60E-6 3.61 -9.26 6.85 12.7 1.24E+51.2 3.0E-10 1.14E-5 3.60E-6 3.33 -8.93 7.73 14.3 3.81E+41.2 1.0E-9 1.03E-5 3.60E-6 3.02 -8.53 8.94 16.6 1.03E+41.2 1.6E-9 9.77E-6 3.60E-6 2.87 -8.31 9.67 17.9 6107.1.2 3.0E-9 8.89E-6 3.60E-6 2.62 -7.94 11.1 20.5 2962.1.2 5.0E-9 8.12E-6 3.60E-6 2.41 -7.59 12.6 23.4 1624.1.2 1.0E-8 6.85E-6 3.60E-6 2.05 -6.94 15.9 29.5 685.1.2 1.6E-8 5.89E-6 3.60E-6 1.79 -6.37 19.6 36.3 368.1.2 3.0E-8 4.76E-6 3.60E-6 1.47 -5.57 26.2 48.6 159.1.2 5.0E-8 3.90E-6 3.60E-6 1.24 -4.85 34.1 63.2 78.01.2 1.0E-7 2.97E-6 3.60E-6 0.978 -3.88 48.5 89.9 29.71.2 1.6E-7 2.45E-6 3.60E-6 0.832 -3.21 61.8 114. 15.31.2 1.8E-7 2.33E-6 3.60E-6 0.800 -3.05 65.6 122. 12.91.25 1.0E-11 9.93E-6 2.61E-6 3.95 -9.99 4.33 8.01 9.93E+51.25 3.0E-11 8.61E-6 2.61E-6 3.44 -9.42 5.32 9.84 2.87E+51.25 5.0E-11 8.15E-6 2.61E-6 3.27 -9.20 5.75 10.7 1.63E+51.25 1.0E-10 7.92E-6 2.61E-6 3.18 -9.09 5.99 11.1 7.92E+41.25 3.0E-10 7.27E-6 2.61E-6 2.93 -8.75 6.78 12.6 2.42E+41.25 1.0E-9 6.57E-6 2.61E-6 2.66 -8.35 7.84 14.5 6565.1.25 1.6E-9 6.21E-6 2.61E-6 2.52 -8.13 8.48 15.7 3881.1.25 3.0E-9 5.65E-6 2.61E-6 2.31 -7.77 9.69 17.9 1884.1.25 5.0E-9 5.16E-6 2.61E-6 2.12 -7.42 11.0 20.4 1031.1.25 1.0E-8 4.34E-6 2.61E-6 1.81 -6.76 14.0 25.9 434.1.25 1.6E-8 3.75E-6 2.61E-6 1.58 -6.20 17.1 31.7 234.1.25 3.0E-8 3.02E-6 2.61E-6 1.30 -5.40 22.9 42.4 101.1.25 5.0E-8 2.49E-6 2.61E-6 1.10 -4.70 29.6 54.8 49.81.25 1.0E-7 1.87E-6 2.61E-6 0.858 -3.69 42.8 79.2 18.71.25 1.6E-7 1.52E-6 2.61E-6 0.722 -2.98 55.3 103. 9.471.25 2.0E-7 1.37E-6 2.61E-6 0.668 -2.65 62.3 115. 6.861.3 1.0E-11 5.59E-6 1.74E-6 3.35 -9.75 3.69 6.84 5.59E+51.3 3.0E-11 4.83E-6 1.74E-6 2.91 -9.17 4.56 8.45 1.61E+51.3 5.0E-11 4.55E-6 1.74E-6 2.75 -8.94 4.96 9.18 9.11E+41.3 1.0E-10 4.40E-6 1.74E-6 2.66 -8.80 5.21 9.66 4.40E+41.3 3.0E-10 4.03E-6 1.74E-6 2.45 -8.46 5.90 10.9 1.34E+41.3 1.0E-9 3.65E-6 1.74E-6 2.23 -8.07 6.80 12.6 3646.1.3 1.5E-9 3.48E-6 1.74E-6 2.13 -7.89 7.26 13.5 2319.1.3 3.0E-9 3.14E-6 1.74E-6 1.94 -7.49 8.39 15.5 1046.1.3 1.0E-8 2.41E-6 1.74E-6 1.52 -6.49 12.1 22.3 241.1.3 3.0E-8 1.69E-6 1.74E-6 1.10 -5.16 19.6 36.2 56.21.3 1.0E-7 1.02E-6 1.74E-6 0.720 -3.40 37.1 68.7 10.21.3 1.6E-7 8.27E-7 1.74E-6 0.609 -2.71 47.7 88.3 5.171.3 2.0E-7 7.44E-7 1.74E-6 0.561 -2.38 53.9 99.8 3.721.3 2.2E-7 7.12E-7 1.74E-6 0.543 -2.24 56.6 105. 3.241.35 1.0E-11 2.38E-6 9.40E-7 2.64 -9.43 2.86 5.30 2.38E+51.35 3.0E-11 2.06E-6 9.40E-7 2.30 -8.86 3.51 6.50 6.86E+41.35 5.0E-11 1.95E-6 9.40E-7 2.19 -8.66 3.79 7.01 3.91E+41.35 1.0E-10 1.85E-6 9.40E-7 2.08 -8.45 4.09 7.57 1.85E+41.35 3.0E-10 1.71E-6 9.40E-7 1.93 -8.14 4.57 8.47 5700.1.35 1.0E-9 1.54E-6 9.40E-7 1.75 -7.74 5.28 9.78 1544. Table 4 continued Hachisu et al.
Table 4 (continued) M WD ˙ M acc M acc M sc x M V, max t t t rec ( M ⊙ ) ( M ⊙ yr − ) ( M ⊙ ) ( M ⊙ ) (days) (days) (yr)1.35 1.6E-9 1.46E-6 9.40E-7 1.66 -7.53 5.70 10.6 914.1.35 3.0E-9 1.34E-6 9.40E-7 1.53 -7.18 6.47 12.0 445.1.35 5.0E-9 1.22E-6 9.40E-7 1.41 -6.84 7.33 13.6 244.1.35 1.0E-8 1.03E-6 9.40E-7 1.21 -6.21 9.21 17.1 103.1.35 1.6E-8 9.06E-7 9.40E-7 1.07 -5.72 11.0 20.4 56.61.35 3.0E-8 7.33E-7 9.40E-7 0.889 -4.94 14.6 27.1 24.41.35 5.0E-8 6.03E-7 9.40E-7 0.751 -4.24 18.8 34.9 12.11.35 1.0E-7 4.50E-7 9.40E-7 0.587 -3.23 27.2 50.4 4.501.35 1.6E-7 3.62E-7 9.40E-7 0.494 -2.52 35.3 65.3 2.261.35 2.0E-7 3.25E-7 9.40E-7 0.455 -2.18 39.9 73.9 1.621.35 2.5E-7 2.91E-7 9.40E-7 0.419 -1.84 45.1 83.6 1.17 a chemical composition of the envelope is assumed to be that of “Solar” in Table 2 of Hachisu & Kato(2006). REFERENCES
Cao, Y., Kasliwal, M. M., Neill, J. D., et al. 2012, ApJ, 752,133Capaccioli, M., della Valle, M., D’Onofrio, M., Rosino, L.1990, ApJ, 360, 63Chen, H.-L., Woods, T. E., Yungelson, L. R., et al. 2019,MNRAS, 490, 1678Cohen, J. G. 1985, ApJ, 292, 90della Valle, M., & Izzo, L. 2020, The Astronomy andAstrophysics Review, 28, 3della Valle, M., & Livio, M. 1995, ApJ, 452, 704di Paolantonio, A., Patriarca, R., & Tempesti, P. 1981, Inf.Bull. Variable Stars, 1913, 1Downes, R. A., & Duerbeck, H. W. 2000, AJ, 120, 2007Ennis, D., Becklin, E. E., Beckwith, S., et al. 1977, ApJ,214, 478Gallagher, J. S., & Ney, E. P. 1976, ApJ, 204, L35Gehrz, R. D., Truran, J. W., Williams, R. E., & Starrfield,S. 1998, PASP, 110, 3Hachisu, I., & Kato, M. 2006, ApJS, 167, 59Hachisu, I., & Kato, M. 2007, ApJ, 662, 552Hachisu, I., & Kato, M. 2010, ApJ, 709, 680Hachisu, I., & Kato, M. 2015, ApJ, 798, 76Hachisu, I., & Kato, M. 2016, ApJ, 816, 26Hachisu, I., & Kato, M. 2017, in Proceedings of thePalermo Workshop 2017 on “The Golden Age ofCataclysmic Variables and Related Objects - IV”, ed. F.Giovannelli et al. (Trieste: SISSA PoS), 315, 47Hachisu, I., & Kato, M. 2018, ApJS, 237, 4Hachisu, I., & Kato, M. 2019a, ApJS, 241, 4Hachisu, I., & Kato, M. 2019b, ApJS, 242, 18Hachisu, I., Saio, H., & Kato, M. 2016, ApJ, 824, 22 Kasliwal, M. M., Cenko, S. B., Kulkarni, S. R., et al. 2011,ApJ, 735, 94Kato, M., & Hachisu, I., 1994, ApJ, 437, 802Kato, M., Hachisu, I., & Saio, H. 2017a, in Proceedings ofthe Palermo Workshop 2017 on “The Golden Age ofCataclysmic Variables and Related Objects - IV”, ed. F.Giovannelli et al. (Trieste: SISSA PoS), 315, 56 (arXiv:1711.01529)Kato, M., Saio, H., Hachisu, I., & Nomoto, K. 2014, ApJ,793, 136Kato, M., Saio, H., & Hachisu, I. 2017b, ApJ, 844, 143Knigge, C., Baraffe, I., & Patterson, J. 2011, ApJS, 194, 28Livio, M. 1992, ApJ, 393, 516Mallama, A. D., & Skillman, D. R. 1979, PASP, 91, 99McLaughlin, D. B. 1945, PASP, 57, 69Nomoto, K. 1982, ApJ, 253, 798¨Ozd¨ormez, A., Ege, E., G¨uver, T., & Ak, T. 2018, MNRAS,476, 4162Prialnik, D., & Kovetz, A. 1995, ApJ, 445, 789Schaefer, B. E. 2010, ApJS, 187, 275Schaefer, B. E. 2018, MNRAS, 481, 3033Schaefer, B. E., & Collazzi, A. C. 2010, ApJ, 139, 1831Schmidt, Th. 1957, Z. Astrophys., 41, 181Selvelli, P., & Gilmozzi, R. 2019, A&A, 622, A186Shafter, A. W. 2013, AJ, 145, 117Shafter, A. W., Darnley, M. J., Hornoch, K., et al. 2011,ApJ, 734, 12Shara, M. M., Doyle, T., Lauer, T. R., et al. 2017, ApJ,839, 109Strope, R., Schaefer, B. E., & Henden, A. A. 2010, AJ, 140,34 aximum Magnitude versus Rate of Decline23