A LIGHT CURVE ANALYSIS OF CLASSICAL NOVAE: FREE–FREE EMISSION VERSUS PHOTOSPHERIC EMISSION

Andreae et al. (1991) obtained E(B − V) = 0.58 ± 0.06 from the He ii λ1640/λ4686 ratio and E(B − V) = 0.55 ± 0.1 from the interstellar absorption feature at 2200 Å for the reddening toward PW Vul. Saizar et al. (1991) reported E(B − V) = 0.60 ± 0.06 from the He ii λ1640/λ4686 ratio. Duerbeck et al. (1984) estimated the extinction to be E(B − V) = 0.45 ± 0.1 from galactic extinction in the direction toward the nova, whose galactic coordinates are (l, b) = (610983, +51967). For the galactic extinction, we examined the galactic dust absorption map in the NASA/IPAC Infrared Science Archive,³ which is calculated on the basis of recent data from Schlafly & Finkbeiner (2011). It gives E(B − V) = 0.43 ± 0.02 in the direction of PW Vul. The arithmetic mean of these four values is E(B − V) = 0.55 ± 0.05.

Recently, Hachisu & Kato (2014) proposed a new method for determining reddening of classical novae. They identified a general course of UBV color–color evolution and determined reddenings of novae by matching the track of a target nova with their general course. They obtained E(B − V) = 0.55 ± 0.05 for PW Vul, which agrees well with the above mean value.

As for the distance to PW Vul, a reliable estimate, d = 1.8 ± 0.05 kpc, was obtained by Downes & Duerbeck (2000) through the nebular expansion parallax method. Adopting this value, together with E(B − V) = 0.55 ± 0.05, the distance modulus of PW Vul is

$$\begin{eqnarray} (m-M)_V &=& 5 \log \left({d \over {\rm 10\, pc}}\right) + 3.1 E(B-V)\cr &=& 5 \log (180\pm 5) + 3.1 (0.55\pm 0.05)\cr &=& 13.0\pm 0.2. \end{eqnarray} \tag{ 1 }$$

We also obtained (m − M)_{V, PWVul} = 13.0 ± 0.1 (see Appendix B) from "the time-stretching method" (Hachisu & Kato 2010) of nova light curves. This value is consistent with Equation (1).

Figure 6 shows various distance-reddening relations for comparison. A horizontal magenta thick solid line with flanking thin lines represents the distance estimate of d = 1.8 ± 0.05 kpc (Downes & Duerbeck 2000) mentioned above. A vertical solid black line with flanking thin lines represents the color excess of E(B − V) = 0.55 ± 0.05. The distance-reddening relation of Equation (1) is plotted by a thick solid blue line with flanking thin solid lines.

Zoom In Zoom Out Reset image size

Marshall et al. (2006) published a three-dimensional dust extinction map of our galaxy in the direction of −1000 ⩽ l ⩽ 1000 and −100 ⩽ b ⩽ +100, with grids of Δl = 025 and Δb = 025, where (l, b) are the galactic coordinates. Four sets of data with error bars in Figure 6 show distance-reddening relations in four directions close to PW Vul: (l, b) = (6100, 500) (red open squares), (6125, 500) (green filled squares), (6100, 525) (blue asterisks), and (6125, 525) (magenta open circles). The closest one is the blue asterisk relation, which gives E(B − V) = 0.42 ± 0.08 at d ≈ 1.8 ± 0.05 kpc. This value is consistent with E(B − V) = 0.43 ± 0.02 calculated from the NASA/IPAC dust map in the direction of PW Vul. Our value of E(B − V) = 0.55 obtained above is larger than these values. However, the reddening trend of blue asterisks suggests a large deviation from the other three trends by ΔE(B − V) ≈ 0.1, i.e., reddening has patchy structure in this direction, and further variation of ΔE(B − V) ∼ 0.1 is possible. Thus, we adopt E(B − V) = 0.55 and d ≈ 1.8 kpc in this paper.

One of the most intriguing properties of classical novae is the metal enrichment of ejecta (e.g., Gehrz et al. 1998), which is ascribed to mixing with WD core material during outburst (e.g., Prialnik & Kovetz 1995). PW Vul is not an exception to this general trend, as summarized in Table 1. There is a noticeable scatter in the abundance estimates, from X = 0.47 to X = 0.69. The arithmetic mean is X = 0.57, Y = 0.22, and X_CNO = 0.19 for Z = 0.02. Here, X, Y, Z, and X_CNO are hydrogen, helium, heavy elements with solar abundance, and carbon–nitrogen–oxygen fractions in weight, respectively.

We took a simple parameterization for the degree of mixing between core material and accreted matter as η_mix = (0.7/X) − 1. Here we assume the solar composition for the accreted matter. Table 2 shows seven representative cases of degree of mixing, i.e., 100% (denoted by "CO nova 1," "CO nova 2," and "Ne nova 1"), 55% ("CO nova 3"), 25% ("CO nova 4" and "Ne nova 2"), and 8% ("Ne nova 3"). We first adopt "CO nova 4" because it is closest to the above averaged values of PW Vul. Then, we discuss the dependence of light curves on the chemical composition.

We now analyze the spectra of PW Vul, assuming that the continuum flux F_ν is simply the sum of a blackbody spectrum of the temperature T_ph and an optically thick free–free emission with the electron temperature of T_e, i.e.,

$$\begin{equation} F_{\nu } = f_1 B_{\nu }(T_{\rm ph}) + f_2 S_{\nu }(T_{\rm e}), \end{equation} \tag{ 2 }$$

where ν is the frequency, B_ν(T_ph) is the Planckian of the photospheric temperature T_ph = T_BB, and S_ν(T_e) is the free–free spectrum of the electron temperature T_e; f₁ and f₂ are numeric constants (e.g., Nishimaki et al. 2008). Following Wright & Barlow (1975), the free–free spectrum can be expressed as

$$\begin{equation} S_{\nu }(T_{\rm e}) = B_{\nu }(T_{\rm e}) K^{2/3}_{\nu }(T_{\rm e}), \end{equation} \tag{ 3 }$$

where the linear free–free absorption coefficient K_ν(T_e) is given by

$$\begin{equation} K_{\nu }(T_{\rm e}) = 3.7 \times 10^8 \left[ 1- \exp \left(-{{h \nu } \over {k T_{\rm e}}}\right) \right] Z^2 g_{\nu }(T_{\rm e}) T^{-1/2}_{\rm e} \nu ^{-3} \end{equation} \tag{ 4 }$$

in cgs units; g_ν(T_e) is the Gaunt factor. In general, the Gaunt factor depends weakly on the frequency and temperature, but we assume it to be unity, following Hachisu & Kato (2014). Therefore, there are four fitting parameters, i.e., f₁, f₂, T_ph = T_BB, and T_e. When hν ≪ kT_e, Equation (3) can be expressed as (Wright & Barlow 1975)

$$\begin{equation} S_\nu = 23.2 \left({{\dot{M}_{\rm wind}} \over {\mu v_{\infty }}} \right)^{4/3}{{\nu ^{2/3}} \over {D^2}} \gamma ^{2/3} g^{2/3} Z^{4/3}\, { \rm Jy}, \end{equation} \tag{ 5 }$$

where 1 Jy =10⁻²⁶ W m⁻² Hz⁻¹; the ion number density is assumed equal to the total gas number density, n, and the electron number density is equal to γ times the ion number density; $\dot{M}_{\rm wind}$ is the wind mass-loss rate in units of M_☉ yr⁻¹; D is the distance in units of kiloparsecs; v_∞ is the terminal wind velocity in units of km s⁻¹; μ is the mean molecular weight; Z is the charged degree of ion (only in this formula); ν is the frequency in units of Hz; and g is the Gaunt factor. Equation (5) can be further simplified as

$$\begin{equation} S_\nu \propto \nu ^{2/3}, {\rm or }\; S_\lambda \propto \lambda ^{-8/3}, \end{equation} \tag{ 6 }$$

where λ is the wavelength.

Figure 5 schematically shows the contributions of B_λ and S_λ. When the wind mass-loss rate is small, contribution of free–free emission is relatively small in Equation (2). Hachisu & Kato (2014) decomposited the spectrum of PW Vul about 64 days after the outburst using Equation (2) and concluded that the free–free flux is comparable to the pseudophotospheric flux in the V band. Here, we reanalyzed the same data and showed them in Figure 7, assuming four different extinctions, i.e., (1) E(B − V) = 0.45, (2) E(B − V) = 0.50, (3) E(B − V) = 0.55, and (4) E(B − V) = 0.60. In our decomposition process, we simply assume that T_BB = T_e and change the temperature in steps of 1000 K. We see that the blackbody emission gives a good approximation to the UV region, and the free–free emission is a good fit to the IR region. In the region between them, we have a comparable contribution from the blackbody and free–free components.

Zoom In Zoom Out Reset image size

These four decompositions of different sets of T_e = T_BB and E(B − V) more or less show similarly good agreement. Among these four cases, T_BB = 27, 000 K in Figure 7(c) is in best agreement with the temperature deduced from the light curve analysis in Section 3.4. It should be noted that the dereddened spectrum with E(B − V) = 0.55 is closest to the straight line (thin solid red line) of F_λ∝λ^−2.67 of Equation (6). Hauschildt et al. (1997) calculated synthetic NLTE nova spectra (see their Figure 10), in which the continuum flux has a slope of F_λ∝λ^−2.7 in the range of λ = 0.2–2 μm for T_eff = 25, 000 K and T_eff = 30, 000 K. If we apply this slope directly to PW Vul, the spectrum is in best agreement with the reddening of E(B − V) = 0.55 in Figure 7(c).

The decomposition in Figure 7(c) indicates that the photospheric emission may substantially contribute in the optical light curve. In the next subsection, we calculate theoretical light curves from the sum of free–free plus photospheric emission for the optical bands and from photospheric emission alone for the UV 1455 Å band.

We now make light curve models for PW Vul, assuming the chemical composition of "CO Nova 4." We calculated nova light curves for various WD masses and fitted them to the observational data. The flux in the UV 1455 Å band (a narrow band of 1445–1465 Å, see Cassatella et al. 2002) was calculated as blackbody emission from the nova pseudophotosphere (R_ph and T_ph), using the optically thick wind solutions in Kato & Hachisu (1994). In our fitting process, we changed the WD mass from 0.80 M_☉ to 0.90 M_☉ in steps of 0.01 M_☉. In Figure 8, the 0.83 M_☉ WD model (thin solid black lines) shows reasonable agreement with the optical, NIR, and UV observations, in particular with the UV 1455 Å observations. An arrow labeled "spectral energy distribution (SED)" indicates the date (Day 64) at which the spectrum in Figure 7 was secured. Our 0.83 M_☉ WD model has T_ph = 28, 000 K on this day, which is consistent with the blackbody temperature of T_BB = 27, 000 K determined in Section 3.3.

Zoom In Zoom Out Reset image size

From the UV 1455 Å light curve fitting of the 0.83 M_☉ WD in Figure 8, we obtained the following distance-reddening relation, i.e.,

$$\begin{eqnarray} && 2.5 \log F_{\lambda 1455}^{\rm mod} - 2.5 \log F_{\lambda 1455}^{\rm obs}\nonumber\\ &&\quad = 2.5 \log (3.38 \times 10^{-12})\nonumber\\ && \quad\quad - 2.5 \log ((1.35\pm 0.27) \times 10^{-12}) \nonumber\\ && \quad = 5 \log \left({d \over {10{\rm kpc}}} \right) + 8.3 \times E(B-V), \end{eqnarray} \tag{ 7 }$$

where $F_{\lambda 1455}^{\rm mod}= 3.38 \times 10^{-12}$ erg cm⁻² s⁻¹ Å⁻¹ is the calculated UV 1455 Å band flux at maximum of the 0.83 M_☉ WD model for an assumed distance of 10 kpc, and $F_{\lambda 1455}^{\rm obs}= (1.35 \pm 0.27) \times 10^{-12}$ erg cm⁻² s⁻¹ Å⁻¹ is the maximum observed flux (Cassatella et al. 2002). Here we assume an absorption of A_λ = 8.3 × E(B − V) at λ = 1455 Å (Seaton 1979). The distance-reddening relation of Equation (7) is plotted by solid magenta lines in Figure 9 (labeled "UV 1455 Å"). Two relations of Equations (1) and (7) cross each other at the point of (E(B − V), d) = (0.57 ± 0.05 mag, 1.75 ± 0.3 kpc), consistent with the observations summarized in Section 3.1. Therefore, we safely conclude that the WD mass of PW Vul is as massive as ∼0.83 M_☉ if the chemical composition is close to X = 0.55, Y = 0.23, Z = 0.02, and X_CNO = 0.20.

Zoom In Zoom Out Reset image size

Contrary to the UV 1455 Å blackbody flux, our free–free model light curves are not yet calibrated. Optical light curves in Figure 8 are freely shifted in the vertical direction to fit the observation because the proportionality constant in Equation (9) of Hachisu & Kato (2006) is unknown for "CO nova 4." To fix the absolute magnitude of each light curve, we use the absolute magnitude of PW Vul as follows.

First, we calculate the light curve model of the 0.83 M_☉ WD and obtain the blackbody light curve in the V band (solid green line labeled "BB"), as shown in Figure 10. Here, we adopt the distance modulus of (m − M)_V = 13.0. Assuming a trial value for the proportionality constant C in Equation (A1) of Appendix A.1, which is the same as Equation (9) of Hachisu & Kato (2006), we obtain the absolute magnitude of the free–free model light curve (solid black line labeled "FF"). The total flux (solid red line labeled "TOTAL") is the sum of these two fluxes. However, in general, this total V-magnitude light curve does not fit well with the observed data. Then, we change the proportionality constant until the total V flux fits well with the observed V light curve. Figure 10 shows our final best-fit model. We directly read m_w = 16.0 from Figure 10, where m_w is the apparent magnitude at the end of the wind phase (open circle at the end of solid black line labeled "FF"). Then, we obtain M_w = m_w − (m − M)_V = 16.0–13.0 = 3.0, where M_w is the absolute magnitude of the free–free model light curve at the end of the wind phase. Thus, the proportionality constant can be specified by M_w = 3.0 of the 0.83 M_☉ WD for PW Vul.

Zoom In Zoom Out Reset image size

Based on the result of the 0.83 M_☉ WD and applying the time-scaling law of free–free light curves to other WD mass models, we obtain the absolute magnitudes of free–free light curves for other WD masses with the chemical composition of "CO nova 4" (see Appendix A). The absolute magnitudes are specified by the value of M_w and listed in Table 4 for 0.55–1.2 M_☉ WDs in steps of 0.05 M_☉.

It should be noted that our model light curve fits reasonably with the early V light curve but deviates from the visual observation (small blue dots in Figure 10) in the nebular phase. On the other hand, our free–free model light curve almost perfectly fits with the NIR light curves, even in the later phase. This deviation in visual magnitudes is owing to strong emission lines, such as [O iii], which are not included in our model (see Hachisu & Kato 2006, for details). In the NIR region, free–free emission dominates the spectrum, and our free–free light curve works well.

Thus, we may conclude that the effect of photospheric emission in the V band can be neglected in novae much faster than PW Vul because the mass-loss rate is large enough for free–free emission to dominate the spectrum in the optical and NIR regions. We will discuss such examples of fast novae in Sections 7.1.1 and 7.1.2. However, in less massive WDs or in novae much slower than PW Vul, we must take into account the contribution of photospheric emission. In this sense, PW Vul lies on a border of speed class between them.

The chemical composition of ejecta is usually not so accurately constrained as described in Section 3.2 and as tabulated in Table 1. If we adopt a chemical composition different from the true one, we could miss the WD mass and distance modulus of a nova. Therefore, we examine the dependence of our model light curve on the chemical composition, i.e., the degree of mixing. We adopted two other chemical compositions of "CO nova 2," a high degree of mixing, η_mix = 1.0 (100%), and "Ne nova 2," a low degree of mixing, η_mix = 0.25 (25%), mainly because their absolute magnitudes of free–free emission model light curves were already calibrated by Hachisu & Kato (2010) independently of the light curve of PW Vul.

In a similar way to that given in Section 3.4, we have obtained best-fit models for the two chemical compositions mentioned above. Figure 11 shows our model light curves for (1) "CO Nova 2" and (2) "Ne Nova 2." In our fitting process, we changed the WD mass from 0.70 M_☉ to 0.90 M_☉ in steps of 0.01 M_☉. In Figure 11(a), the 0.78 M_☉ WD model shows reasonable agreement with the V and UV observations. Additionally, a good agreement is found for the 0.85 M_☉ WD in Figure 11(b). It should be noted again that our model light curve fits well with the early V light curve but deviates from the visual observation in the later phase, i.e., in the nebular phase. This deviation is due to strong emission lines such as [O iii], which are not included in our model light curves.

Zoom In Zoom Out Reset image size

The flux in the UV 1455 Å band was calculated as blackbody emission from the nova pseudophotosphere, using the optically thick wind solutions in Kato & Hachisu (1994) and Hachisu & Kato (2006, 2010). From the UV 1455 Å flux fitting, we obtained the distance-reddening relation, which is plotted by solid magenta lines (labeled "UV 1455 Å") in Figure 12. Both the distance-reddening relations of UV 1455 Å in Figure 12 are similar to that for "CO nova 4" in Figure 9.

Zoom In Zoom Out Reset image size

The total V fluxes are calculated from the sum of the free–free and blackbody fluxes. From the fitting, we also obtained the distance-reddening relation of (1) (m − M)_V = 12.8 for the 0.78 M_☉ WD and (2) (m − M)_V = 13.2 for the 0.85 M_☉ WD. Both are barely consistent with our adopted value of (m − M)_V = 13.0 ± 0.2 for PW Vul. The two distance-reddening relations of total V-magnitude fitting are plotted by solid blue lines (labeled "TOTAL") in Figure 12. It is remarkable that our fitting misses the distance modulus only by 0.2 mag, even if we assume a different chemical composition of ΔX = 0.55–0.35 = 0.2.

For a lower value of the hydrogen content X, the evolution timescale becomes shorter, even if the WD mass is the same (see Kato & Hachisu 1994; Hachisu & Kato 2001). Therefore, a less massive WD of 0.78 M_☉ is fitted with the observation for a lower value of X = 0.35, as shown in Figure 11(a). The wind mass-loss rate is smaller for a less massive WD of 0.78 M_☉. The lower wind mass-loss rate leads to fainter free–free emission and, as a result, a fainter total V light curve. This is the reason for (m − M)_V = 12.8, which is a bit smaller than the original value of (m − M)_V = 13.0.

For the chemical composition of "Ne nova 2," however, the hydrogen content of X is the same as that of "CO nova 4." The difference is between X_CNO = 0.20 and X_CNO = 0.10. The CNO abundance is relevant to the nuclear burning rate, and a lower value of X_CNO = 0.10 makes the evolution timescale longer. This requires a more massive WD of 0.85 M_☉ than the 0.83 M_☉ WD of X_CNO = 0.20, as shown in Figure 11(b). The more massive WD blows stronger winds. This results in a brighter light curve of free–free emission and, as a result, a brighter total V light curve. This is the reason for (m − M)_V = 13.2, which is a bit larger than the original value of (m − M)_V = 13.0.

In Figure 12(a), the two distance-reddening relations, i.e., "UV 1455 Å" and "TOTAL fit," cross each other at the point of (E(B − V), d) = (0.63 mag, 1.5 kpc), not consistent with E(B − V) = 0.55 ± 0.05 and d = 1.8 ± 0.05 kpc. The degree of mixing may not be as high as 100% (X ≈ 0.35) in PW Vul.

Figure 12(b) shows that the two relations cross at the point of (E(B − V), d) = (0.54 mag, 2.0 kpc). This value is consistent with the reddening estimate of E(B − V) = 0.55 ± 0.05, although the distance estimate is a bit larger than the distance estimate of d = 1.8 ± 0.05 kpc. We may conclude that the lower degree of mixing (25% mixing) is more reasonable in PW Vul.

To summarize, we reached a reasonable distance-reddening result for a 25% mixing of "Ne Nova 2" but not for a 100% mixing of "CO Nova 2." Note that enrichment of neon with the hydrogen mass fraction being unchanged hardly influences the nova light curves because neon is not relevant to either nuclear burning (CNO cycle) or opacity (e.g., Kato & Hachisu 1994; Hachisu & Kato 2006, 2010). Therefore, the agreement in the lower 25% mixing model suggests that a lower degree of mixing (∼25%) is reasonable, rather than a higher degree of mixing (∼100%). Unfortunately, there is significant scatter in the abundance determinations (see Table 1), but their averaged values of chemical composition, which show a 23% mixing, are close enough to those of "CO nova 4." Therefore, we may conclude that through our method of model light curve fitting, one might discriminate between different degrees of mixing, at least in terms of the hydrogen mass fraction X. We summarize our fitting result for PW Pul in Table 3.

RR Pic was discovered by Watson at about 2.3 mag on UT 1925 May 25. The details of the visual light curve of RR Pic were found in Spencer Jones (1931), in which the photographic magnitude prior to 1925 was m_pg = 12.75, and the nova brightened up to third magnitude between February 18 (fainter than 11th mag) and April 13 (3rd mag), and the first maximum was reached on UT June 7 (m_v = 1.18). Spencer Jones (1931) discussed a possibility of another peak between April 13 (3rd mag) and May 25 (2.3 mag) before the first peak on June 7 (m_v = 1.18). Such a multiple peak has been observed also in V5558 Sgr, HR Del, and V723 Cas, as is clearly shown in Figure 1. We superpose these four novae in Figure 3 to confirm that these novae have very similar decline rates. We shift their times horizontally and their magnitudes vertically by +5.3, 0.0, +3.6, and +0.1 mag, respectively, as indicated in the figure. We also superpose these four novae in a logarithmic timescale in Figure 4. These four nova light curves almost overlap each other.

The distance to RR Pic was obtained using the trigonometric parallax, i.e., $d=521^{+54}_{-45}$ pc (Harrison et al. 2013). The distance modulus in the V band is calculated to be $(m-M)_V=5\log 521^{+54}_{-45}/10 + 0.13 = 8.7\pm 0.2$, where we used A_V = 0.13 after Harrison et al. (2013). Adopting this distance modulus, we plot, in Figure 19, our model light curves for four WD masses, i.e., (1) 0.51 M_☉, (2) 0.55 M_☉, (3) 0.60 M_☉, and (4) 0.65 M_☉, as well as the visual observation. Here, we assumed that the transition was completed at the first peak (UT 1925 June 7), and the outburst day was JD 2,424,170.0, about 140 days before the first peak. Figure 20 shows the same model light curves as those in Figure 19 but only the total V light curves of different WD masses.

It is remarkable that two models of (1) 0.51 M_☉ and (2) 0.55 M_☉ fit nicely with the visual magnitude until the nebular phase begins about 450 days after the outburst (∼300 days after the first peak, see Spencer Jones 1931; Iijima 2006). Note that the nova outburst begins quasi-statically and then undergoes a transition from static to wind evolution at the first peak. We think that during the transition, the nova accompanies oscillatory activity of relaxation. This corresponds to the second and third peaks of the light curve. Thus, we confirm that the absolute magnitudes of our model light curves (total flux of free–free plus blackbody) are consistent with that of very slow novae, even if they do not follow the universal decline law. Therefore, we confidently apply our absolute magnitude estimate to very slow novae.

It should be noted that the peak brightness of each model light curve depends on the initial envelope mass, which is closely related to the ignition mass of outburst: the larger the envelope mass, the brighter the peak. Therefore, we can estimate the envelope mass by adjusting the peak brightness to the observation if the WD mass is fixed.

Our model light curves of total V flux have a similar brightness in the early phase of outburst for the WD mass range of 0.5 M_☉ ≲ M_WD ≲ 0.7 M_☉ (see Figure 20). This property also can be seen in the light curve analysis of the slow nova DQ Her (see Figure 30 below). This is a problem in our light curve analysis because we are not able to identify the WD mass from only our model V light curve fitting in the early phase. On the other hand, we are able to estimate the absolute magnitudes of slow novae independently of the WD mass by directly comparing them with a nova with known distance, such as RR Pic.

Using the distance modulus of RR Pic, i.e., (m − M)_V = 8.7, we obtain the distance moduli for the other three novae. Because these four novae have very similar decline shapes and should have similar brightnesses in the early phase of outbursts, we simply assumed that their brightnesses are all the same in the overlapping region of the light curves in Figures 3 and 4. The difference in apparent V magnitude against V723 Cas is −5.3 for RR Pic, −3.6 for HR Del, and −0.1 for V5558 Sgr. Therefore, the difference ΔV from RR Pic is calculated as ΔV = −3.6 + 5.3 for HR Del, ΔV = −0.0 + 5.3 for V723 Cas, and ΔV = −0.1 + 5.3 for V5558 Sgr. Thus we have

$$\begin{eqnarray} (m-M)_{V, \rm RR Pic} &=& 8.7\cr &=&(m-M)_{V,\rm HR\ Del} - \Delta V \cr &=& 10.4 -(-3.6+5.3) = 8.7\cr &=&(m-M)_{V,\rm V723\ Cas} - \Delta V \cr &=& 14.0 -(-0.0+5.3) = 8.7\cr &=&(m-M)_{V,\rm V5558\ Sgr} - \Delta V \cr &=& 13.9 -(-0.1+5.3) = 8.7. \end{eqnarray} \tag{ 12 }$$

Then, the distance moduli of these three novae are (m − M)_{V, HRDel} = 10.4, (m − M)_{V, V723Cas} = 14.0, and (m − M)_{V, V5558Sgr} = 13.9.

V5558 Sgr was first detected by Sakurai (Nakano et al. 2007) at mag 10.3 on UT 2007 April 14.777. Sakurai also reported that nothing is visible on an image taken on UT April 9.8 (limiting mag 11.4). The star was also detected by Haseda (Yamaoka et al. 2007) at mag 11.2 on UT April 11.792. Because the outburst day is not known, we adopted UT 2007 April 8.5 as the outburst day, i.e., t_OB = JD 2454199.0, in this paper. The distance modulus in the V band was already obtained to be

$$\begin{eqnarray} (m-M)_V &=& 13.9 \cr &=& 5 \log \left({d \over {\rm 10\, pc}} \right) + 3.1 \times E(B-V), \end{eqnarray} \tag{ 13 }$$

in Equation (12) of Section 6.1. We plot this distance-reddening relation of Equation (13) in Figure 21. We found two different values of reddening in the literature; one is E(B − V) = 0.36, obtained by Munari et al. (2007) from the Na i D lines, and the other is E(B − V) = 0.8, obtained by Rudy et al. (2007b) from the O i lines. Because these two values are largely different, we examine other reddening estimates. Figure 21 also shows the distance-reddening relations taken from Marshall et al. (2006) in four directions close to V5558 Sgr, (l, b) = (116107, +02067). The closest one is that of the blue asterisks, which crosses our line of (m − M)_V = 13.9 at E(B − V) ≈ 0.7 and d ≈ 2.2 kpc. This reddening value is consistent with E(B − V) = 0.7 ± 0.05 estimated by Hachisu & Kato (2014), who obtained the reddening by assuming that the three novae, V5558 Sgr, HR Del, and V723 Cas, have the same intrinsic (B − V)₀ color in the premaximum phase. We adopt (m − M)_V = 13.9, E(B − V) = 0.7, and d = 2.2 kpc in this paper.

Figure 22 shows optical and NIR light curves of V5558 Sgr and our model light curves for (a) 0.51, (b) 0.55, (c) 0.6, and (d) 0.65 M_☉ WDs on a logarithmic timescale. Here, we assumed the distance modulus of (m − M)_V = 13.9. Solid blue lines show the total V fluxes (labeled "TOTAL") of our model light curves, whereas solid green lines correspond to the free–free V fluxes (labeled "FF"), and solid black lines represent the blackbody V fluxes (labeled "BB"). From the V light curve shape, we assumed that the transition from static to wind evolution occurred about 90 days after the outburst (at the first optical peak). For comparison, we add another case of the transition at the third peak. Because the peak brightness of our model light curves depends on the initial envelope mass at the outburst, we tune the initial envelope mass to the peak brightness for each model. We adopted a less massive envelope for the model light curve of the thin solid line that starts at the third peak of Figure 22. Figure 23 shows the same model light curves as those in Figure 22 but only the total V light curves of different WD masses for comparison.

Among the four WD mass models, the 0.51 and 0.55 M_☉ WD models are in good agreement with the observation, whereas the 0.60 and 0.65 M_☉ WDs may be too steep to be compatible with the observation. During the transition, the nova accompanies oscillatory activity of relaxation. This corresponds to the second, third, and fourth peaks of the light curve. This conclusion is unchanged even if we adopt the transition time at the third peak (thin solid lines in Figures 22 and 23). These fitting results are summarized in Table 3. This good agreement of the brightness supports that the absolute magnitudes of our model light curves (total flux of free–free plus blackbody) are reasonable even for very slow novae. It should be noted that the nebular phase started about 450 days after the outburst (a year after the first peak, see Poggiani 2012). However, [O iii] emission lines are too weak to contribute significantly to the V flux, so that the V light curve does not deviate so much from the model light curves, as shown in Figures 22(a) and 23.

As already mentioned above, Kato & Hachisu (2009) modeled the premaximum phase of these very slow novae with a static evolution followed by the transition from a static to a wind structure. They predicted that this transition occurs in a narrow range of WD masses, 0.5 M_☉ ≲ M_WD ≲ 0.7 M_☉. Thus, the brightness at the premaximum phase of these novae should be similar to the flat peak of the symbiotic nova PU Vul (M_WD ∼ 0.6 M_☉), whose brightness is M_V = −5.4 in stage 1 (in 1979) and M_V = −5.7 in stage 2 (in 1981–1983), respectively (see Figure 15 of Hachisu & Kato 2014). We plot these two absolute magnitudes of PU Vul in Figures 22 and 23 (horizontal thin dash-dotted and dash-three-dotted lines). These brightnesses are in perfect agreement with the brightness of V5558 Sgr at the premaximum halt (flat) phase just before the first peak, i.e., before the transition started.

The slow nova HR Del was discovered by Alcock (Candy et al. 1967) at m_v = 5.0 on UT 1967 July 8.94 (JD 2439680.44). Because the outburst day is not known, we adopt UT 1967 June 8.5 as the outburst day, i.e., t_OB = JD 2439653.0, from the light curve of Robinson & Ashbrook (1968). The distance modulus of HR Del was already obtained to be

$$\begin{eqnarray} (m-M)_V &=& 10.4 \cr &=& 5 \log \left({d \over {\rm 10\, pc}} \right) + 3.1 \times E(B-V), \end{eqnarray} \tag{ 14 }$$

in Equation (12) of Section 6.1. We plot this distance-reddening relation of Equation (14) in Figure 24 by a solid blue line. Verbunt (1987) obtained the extinction toward HR Del to be E(B − V) = 0.15 ± 0.03. The galactic dust absorption map of NASA/IPAC gives E(B − V) = 0.11 ± 0.006 in the direction toward HR Del, whose galactic coordinates are (l, b) = (634304, −139721), roughly consistent with Verbunt's value. Two lines of E(B − V) = 0.15 and Equation (14) cross at a distance of d = 0.97 kpc, as shown in Figure 24.

Downes & Duerbeck (2000), on the other hand, obtained the distance to HR Del to be d = 0.76 ± 0.13 kpc from the nebular expansion parallax. More recently, Harman & O'Brien (2003) obtained a new value of the distance d = 0.97 ± 0.07 kpc, also from the expansion parallax method of HST imaging. Other older estimates are all between the above two estimates, i.e., d = 0.940 ± 0.155 kpc for various expansion parallax methods (Malakpur 1975; Kohoutek 1981; Duerbeck 1981; Solf 1983; Cohen & Rosenthal 1983; Slavin et al. 1994, 1995) or d = 0.835 ± 0.092 kpc for the other techniques (Drechsel et al. 1977). Here we adopt the distance of d = 0.97 ± 0.07 kpc after Harman & O'Brien (2003) and the extinction of E(B − V) = 0.15 ± 0.03 after Verbunt (1987). These two values are consistent with Equation (14) in Figure 24.

We plot the light curve of HR Del in Figures 1 and 3 on a linear timescale and in Figures 4 and 25 on a logarithmic timescale. Figure 25 shows two model light curves for (1) 0.51 M_☉ and (2) 0.55 M_☉ WDs, in which we assumed that the transition completed 170 days after the outburst. Solid red lines show the total V flux of free–free (solid blue lines) plus blackbody (solid green lines). We calculated four model light curves of 0.51, 0.55, 0.60, and 0.65 M_☉ WDs but did not plot the 0.60 and 0.65 M_☉ WDs because these two are too steep to be compatible with the observation. The 0.51 M_☉ WD model shows good agreement with the observation, whereas the 0.55 M_☉ WD model is marginal, as shown in Figure 25(b). Note that the nebular phase started about 420 days after the outburst (∼250 days after the first peak, see Iijima 2006). During the transition, the nova accompanies oscillatory activity of relaxation. This corresponds to the second peak of the light curve. The brightness of M_V = −5.7 in PU Vul is in good agreement with the brightness of HR Del at the premaximum halt (flat) phase just before the optical maximum, i.e., before the transition started. This fact also confirms that our absolute magnitudes of optical light curves for slow novae are reasonable.

V723 Cas is also a very slow nova. It was discovered at mag 9.2 by Yamamoto on UT 1995 August 24.57 (JD 2449954.07). Munari et al. (1996) proposed UT July 20.5 as the outburst day, i.e., t_OB = JD 2449919.0; therefore we adopt this day in this paper. We plot the visual, V, R, I, J, H, and K light curves of V723 Cas in Figure 26 together with the UV 1455 Å and X-ray light curves. The distance modulus of HR Del was already obtained to be

$$\begin{eqnarray} (m-M)_V &=& 14.0 \cr &=& 5 \log \left({d \over {\rm 10\, pc}} \right) + 3.1 \times E(B-V), \end{eqnarray} \tag{ 15 }$$

in Equation (12) of Section 6.1. We plot this distance-reddening relation of Equation (15) in Figure 27 by a solid blue line. The distance to V723 Cas was estimated by Lyke & Campbell (2009) to be $d=3.85^{+0.23}_{-0.21}$ kpc using the expansion parallax method. Hachisu & Kato (2014) obtained the reddening toward V723 Cas to be E(B − V) = 0.35 ± 0.05 by fitting the general tracks with the observed track of V723 Cas in the UBV color–color diagram. These three trends, i.e., (m − M)_V = 14.0, d = 3.85 kpc, and E(B − V) = 0.35, cross consistently, as shown in Figure 27. Therefore, we adopt these values in this paper.

Figure 26 shows our model light curves of the 0.51 M_☉ WD for the chemical composition of "CO nova 4." Here, we assumed the distance modulus in the V band to be (m − M)_V = 14.0 and that the transition from static to wind evolution occurred 155 days after the outburst because the UV 1455 Å flux started to rise on this day. A solid black line shows the total V flux of free–free (solid blue line) plus blackbody (solid sky-blue line) emission calculated from the 0.51 M_☉ WD model, which is the one showing the best agreement with the observation compared with the other three WD mass models of 0.55, 0.60, and 0.65 M_☉. During the transition, the nova accompanies oscillatory activity of relaxation. This corresponds to the second, third, and fourth peaks of the light curve. Note that the model light curves were fitted to the lower envelope of the V light curve to avoid local photospheric fluctuations until the nebular phase started about 700 days after the outburst (∼550 days after the first peak, see Iijima 2006).

The following additional distance-reddening relation (labeled "UV 1455 Å " in Figure 27) can be deduced from our UV 1455 Å flux fitting, i.e.,

$$\begin{eqnarray} && 2.5 \log F_{1455}^{\rm mod} - 2.5 \log F_{1455}^{\rm obs}\cr &&\quad = 2.5 \log (3.2 \times 10^{-12}) \cr && \quad\quad - 2.5 \log (1.5 \times 10^{-12}) \cr && \quad = 5 \log \left({d \over {10{\rm kpc}}} \right) + 8.3 \times E(B-V), \end{eqnarray} \tag{ 16 }$$

where $F_{1455}^{\rm mod}= 3.2 \times 10^{-12}$ erg cm⁻² s⁻¹ Å⁻¹ is the calculated flux at the upper limit of the figure box at a distance of 10 kpc, and $F_{1455}^{\rm obs}= 1.5 \times 10^{-12}$ erg cm⁻² s⁻¹ Å⁻¹ is the observed flux corresponding to the upper limit of the figure box. The two distance-reddening relations, i.e., Equations (15) and (16), cross each other at the point of E(B − V) ≈ 0.34 and d ≈ 3.9 kpc, consistent with the distance of $d=3.85^{+0.23}_{-0.21}$ kpc (Lyke & Campbell 2009) and E(B − V) = 0.35 ± 0.05 (Hachisu & Kato 2014) mentioned above.

Ness et al. (2008) obtained (m − M)_V = 13.7, E(B − V) = 0.5 ± 0.1, and $d=2.7^{+0.4}_{-0.3}$ kpc by assuming that the absolute magnitude of V723 Cas is similar to that of HR Del. The main difference from ours comes from the reddening estimate. The interstellar extinction toward V723 Cas was estimated by many authors, but the values are quite scattered; that is, in increasing order, E(B − V) = 0.20 ± 0.12 in 1999 August and 0.25 ± 0.1 in 2000 July (Rudy et al. 2002) from the Paschen and Brackett lines, E(B − V) = 0.29 calculated from A_V = 0.89 (Iijima et al. 1998) from reddenings of field stars near the location of V723 Cas, E(B − V) = 0.45 (Munari et al. 1996) from the interstellar Na i D double lines, E(B − V) = 0.5 ± 0.1 (Ness et al. 2008) estimated from various values in the literature and their N_H value from X-ray spectrum model fits, E(B − V) = 0.57 (Chochol & Pribulla 1997) from intrinsic colors at maximum and at two magnitude below maximum, E(B − V) = 0.60 (González-Riestra et al. 1996) from the 2200 Å dust absorption feature, and E(B − V) = 0.78 ± 0.15 (Evans et al. 2003) from the IR H i recombination lines. Recently, González-Riestra revised the value to be E(B − V) = 0.30–0.35 (private communication, 2012, see also Hachisu & Kato 2014). Hachisu & Kato (2014) obtained E(B − V) = 0.35 ± 0.05 by fitting the general tracks with the observed track of V723 Cas in the UBV color–color diagram. The recent NASA/IPAC dust map gives E(B − V) = 0.34 ± 0.01 toward V723 Cas, whose galactic coordinates are (l, b) = (1249606, −88068). Therefore, we adopt E(B − V) = 0.35 and d = 3.85 kpc in this paper. These fitting results are summarized in Table 3.

We examine whether or not the absolute brightness of our model light curve is correct for classical novae with known distances. Harrison et al. (2013) determined the distances of four novae, V603 Aql, GK Per, DQ Her, and RR Pic, with Hubble Space Telescope (HST) annual parallaxes. We have already examined the case of RR Pic in Section 6.1 and showed that the total brightness of our model light curve reasonably reproduces the absolute brightness of RR Pic for the distance modulus of (m − M)_{V, RR Pic} = 8.7 (Harrison et al. 2013). In this subsection, we study the other three, i.e., GK Per, V603 Aql, and DQ Her.

7.1.1. GK Per 1901

The distance of GK Per is obtained to be $d=477^{+28}_{-25}$ pc by Harrison et al. (2013). The distance modulus in the V band is obtained to be $(m-M)_{V,\rm GK\ Per}=5\log 477^{+28}_{-25}/10 + 3.1\times 0.3 = 9.3\pm 0.1$, where we adopt E(B − V) = 0.3 (Wu et al. 1989) after Harrison et al. (2013). The WD mass of GK Per was estimated by Morales-Rueda et al. (2002) to be $M_{\rm WD}=0.77^{+0.52}_{-0.24} \, M_\odot$, being not accurately constrained.

For GK Per, we assumed the chemical composition of "Ne nova 2" because no estimates are available in the literature. For this chemical composition, the absolute magnitudes of free–free emission model light curves were already determined in Table 3 of Hachisu & Kato (2010). Using the absolute magnitudes of free–free model light curves, we calculated the total (free–free plus photospheric) V flux light curves for the WD masses of 1.05, 1.1, 1.15, and 1.2 M_☉. We plotted these four V light curves in Figure 28(a), where we adopted (m − M)_{V, GK Per} = 9.3. Among the four WD masses, we obtained a best fit for M_WD = 1.15 M_☉ (a thick solid black line). The other fluxes (blackbody and free–free fluxes) are also plotted only for M_WD = 1.15 M_☉ in Figure 28(b).

Zoom In Zoom Out Reset image size

In Figure 28, we plot optical and NIR light curves of the very fast nova V1500 Cyg as well as the fast nova GK Per. GK Per shows a transition phase in the middle part of the outburst. We do not know how to fit our model light curves with the observation in such an oscillatory light curve. For this purpose, we overlap the light curve of V1500 Cyg to that of GK Per and select which part of oscillatory brightness to be fitted. In the figure, we shift the light curve of GK Per horizontally by Δlog t = −0.20 and vertically ΔV = +2.6 mag to overlap it to the light curves of V1500 Cyg. The two V light curves reasonably overlap in the early phase and in the much later phase. In the middle part of the light curves, the lower bound of oscillatory brightness of GK Per reasonably overlaps that of V1500 Cyg. Therefore, we fit our model light curves to the lower bound of oscillatory brightness during the transition phase of GK Per. Note that our model light curve fits reasonably with the early V light curve but deviates from the visual observation in the nebular phase. This deviation in visual magnitudes is owing to strong emission lines such as [O iii], which are not included in our model (see Hachisu & Kato 2006 for details).

To summarize, we are able to reproduce the absolute brightness for the distance modulus of (m − M)_{V, GK Per} = 9.3 and M_WD = 1.15 M_☉. Photospheric emission (solid red line labeled "BB") does not contribute to the total V flux (solid black line labeled "TOTAL"), as shown in the Figure 28(b). Hachisu & Kato (2006) showed that nova light curves follow a universal decline law if free–free emission dominates the spectrum. Figure 28(b) confirms that the nova light curves follow the universal decline law.

7.1.2. V603 Aql 1918

The distance of V603 Aql is obtained to be $d=249^{+9}_{-8}$ pc by Harrison et al. (2013). The distance modulus in the V band is calculated to be $(m-M)_{V,\rm V603\ Aql}=5\log 249^{+9}_{-8}/10 + 3.1\times 0.07 = 7.2\pm 0.07$, where we adopt E(B − V) = 0.07 (Gallagher & Holm 1974) after Harrison et al. (2013). The WD mass of V603 Aql was obtained by Arenas et al. (2000) to be M_WD = 1.2 ± 0.2 M_☉.

The chemical composition of V603 Aql is not available; therefore we assume "Ne nova 2" in this paper, partly because we already estimated the absolute magnitudes of free–free emission model light curves for "Ne nova 2" (Hachisu & Kato 2010). We calculated the total (free–free plus photospheric) V flux light curves for the WD masses of 1.1, 1.15, 1.2, and 1.25 M_☉. We plotted these four V light curves in Figure 29(a), where we adopted (m − M)_{V, V603 Aql} = 7.2. Among the four WD masses, we obtained a best fit for M_WD = 1.2 M_☉ (a thick solid black line). The other fluxes (blackbody and free–free fluxes) are also plotted only for M_WD = 1.2 M_☉ in Figure 29(b). The brightness of our model light curves is consistent with both the distance modulus of (m − M)_V = 7.2 and the WD mass of M_WD = 1.2 M_☉. This confirms that the absolute magnitudes of our model light curves are reasonable.

Zoom In Zoom Out Reset image size

In Figure 29, we add optical and NIR light curves of the very fast nova V1500 Cyg. V603 Aql shows a transition phase in the middle part of the outburst, like GK Per. In order to know how to fit our model light curves with the oscillatory light curve, we again overlap the light curve of V1500 Cyg to that of V603 Aql and select which part of oscillatory brightness is to be fitted. In the figure, we shift the light curve of V603 Aql horizontally by Δlog t = −0.05 and vertically by ΔV = +5.2 mag to overlap it to the light curves of V1500 Cyg. Moreover, we set the start of the light curves about seven days later (indicated by an arrow) than the start of V1500 Cyg. The two V light curves reasonably overlap in the early phase and in the much later phase. In the middle part of the light curves, the upper bound of oscillatory brightness of V603 Aql reasonably overlaps that of V1500 Cyg. Therefore, we fit our model light curves to the upper bound of oscillatory brightness during the transition phase of V603 Aql. Note again that our model light curve fits reasonably with the early V light curve but deviates from the visual observation in the later nebular phase.

To summarize, we are able to reproduce the absolute brightness for the distance modulus of (m − M)_{V, V603 Aql} = 7.2 and M_WD = 1.20 M_☉. Photospheric emission does not contribute to the total V flux, as shown in the Figure 29(b).

7.1.3. DQ Her 1934

The trigonometric parallax distance of DQ Her is $d=386^{+33}_{-29}$ pc (Harrison et al. 2013). Adopting A_V = 3.1 × E(B − V) = 3.1 × 0.1 = 0.31 (Verbunt 1987), we obtain the distance modulus in V band as $(m-M)_V=A_V+5\log (d/{\rm 10\, pc}) =0.31 + 5 \log (386^{+33}_{-29}/ 10)= 8.24\pm 0.18$. Thus, we adopt (m − M)_{V, DQ Her} = 8.2. The WD mass of DQ Her was obtained by Horne et al. (1993) to be M_WD = 0.60 ± 0.07 M_☉. The chemical composition of ejecta was estimated by Petitjean et al. (1990) and Williams et al. (1978), as listed in Table 1. Here, we adopt the chemical composition of "CO nova 2" because the averaged value is X = 0.31 and close to that of "CO nova 2."

For the chemical composition of "CO nova 2," the absolute magnitudes of the free–free emission model light curves were already determined in Table 2 of Hachisu & Kato (2010). Therefore we calculated the total (free–free plus photospheric) V flux light curves for the WD masses of 0.55, 0.60, 0.65, and 0.70 M_☉ and plotted them in Figure 30(a). It is remarkable that all the four V light curves fit reasonably to the observed visual magnitudes, at least in the early phase before the dust blackout started. Therefore, we cannot select a best one among these four light curves. However, this again confirms that the absolute brightness of our model light curves is reasonable, at least in the early decline phase before the dust blackout.

Zoom In Zoom Out Reset image size

In Figure 30(b), we adopt M_WD = 0.60 M_☉ from the central value estimated by Horne et al. (1993) and plot our total V flux, free–free, and photospheric blackbody light curves for the 0.6 M_☉ WD. The photospheric emission (solid red line labeled "BB") significantly contributes to the total V flux (solid black line labeled "TOTAL"). If we do not include the photospheric emission, our free–free model V flux (solid blue line labeled "FF") does not fit to the observed one. This good agreement with the observed brightness suggests that photospheric emission is necessary to reproduce the light curves of slow novae, such as DQ Her as well as free–free emission.

To summarize, our model light curves of total V flux reasonably reproduce the absolute brightnesses of optical light curves of novae with known distances, at least in the early phase before the nebular phase or dust blackout starts. For slower novae, photospheric emission dominates the spectrum in the V band. For faster novae, on the other hand, free–free emission dominates the spectrum in the V band, and therefore fast novae follow the universal decline law.

In this subsection we discuss whether or not slow novae follow the MMRD relation. Theoretical free–free emission light curves of novae clearly shows a trend that a more massive WD has a brighter maximum magnitude (smaller M_{V, max}) and a faster decline rate (smaller t₃ time). The relation between t₃ and M_{V, max} for novae is called the "Maximum Magnitude versus Rate of Decline" (MMRD) relation.

Figure 31 shows observed data points of (t₃, M_{V, max}) for many classical/recurrent novae. A solid blue line flanked with ±1.5 mag lines indicates the relation of "Kaler–Schmidt's law" (labeled "MMRD1," see Schmidt 1957), which is Equation (A13). A solid magenta line flanked with ±1.5 mag lines indicates the relation of "della Valle–Livio's law" (labeled "MMRD2," see della Valle & Livio 1995), which is Equation (A14). Red filled circles are novae taken from Downes & Duerbeck (2000), the distances of which were mainly derived from the nebular expansion parallax method. We show the six novae studied in the present work by large black open circles, i.e., PW Vul, V705 Cas, GQ Mus, V5558 Sgr, HR Del, and V723 Cas. Green filled squares are novae taken from Hachisu & Kato (2010), the distance moduli of which are based on the time-stretching method. Blue filled squares indicate the four novae, V603 Aql, DQ Her, GK Per, and RR Pic, the distances of which were determined by Harrison et al. (2013) with HST annual parallaxes.

Zoom In Zoom Out Reset image size

Hachisu & Kato (2006) found that nova light curves follow a universal decline law when free–free emission dominates the continuum spectrum in optical and NIR regions. Using this property, Hachisu & Kato (2010) found that if two nova light curves overlap each other after one of the two is squeezed/stretched by a factor of f_s (t' = t/f_s) in the time direction, the brightnesses of the two novae obey the relation of $m^{\prime }_V = m_V - 2.5 \log f_s$, which is the same as Equation (A6). Based on this property, they derived a MMRD relation of M_{V, max} = 2.5log t₃ − 11.6 for the chemical composition of "CO nova 2," i.e., Equation (35) in Hachisu & Kato (2010). We can again derive a similar trend, M_{V, max} = 2.5log t₃ − 11.65, i.e., Equation (A12), for another chemical composition, "CO nova 4," in Appendix A. Note that these two relations are in good agreement with Kaler–Schmidt's law. The main trend of the MMRD relation is governed by the WD mass: the more massive a WD, the steeper the decline of a nova light curve. Hachisu & Kato (2010) further showed that the maximum brightness of a nova also depends on the initial envelope mass (ignition mass). This initial envelope mass depends on the mass-accretion rate to the WD (see, e.g., Kato et al. 2014 for a recent result). Hachisu & Kato (2010) concluded that the scatter of individual MMRD points is due to various mass-accretion rates to the WD, even for the same WD mass: the brighter the nova, the smaller the mass-accretion rate to the WD (see Figure 15 of Hachisu & Kato 2010).

Harrison et al. (2013) concluded that DQ Her and GK Per almost follow the MMRD relation (MMRD1), but V603 Aql and RR Pic do not (see Figure 31). We examine the reason for about 1 mag faintness of V603 Aql compared with the solid blue line (MMRD1 relation). Hachisu & Kato (2014) analyzed the light curves of V1500 Cyg and V603 Aql and concluded that both of these novae harbor a ∼1.2 M_☉ WD for the envelope chemical composition of "Ne nova 2." We estimated the initial envelope masses for these two novae, i.e., M_env = 0.92 × 10⁻⁵ M_☉ for the 1.2 M_☉ WD model of V1500 Cyg and M_env = 0.47 × 10⁻⁵ M_☉ for V603 Aql. The initial envelope mass corresponds to the envelope mass at optical maximum (see Figure 29). Thus, the difference in the ignition masses makes the apparent difference in the start time of the outbursts and in the peak brightness, as shown in Figure 29. This is the main source for scatter of individual MMRD points around the proposed MMRD relation. Therefore, a smaller initial envelope mass is the main reason that V603 Aql is fainter by ∼1.2 mag than the solid blue line of MMRD1.

Now we first examine the case of PW Vul among the seven novae studied in the present work. The MMRD point of PW Vul is located slightly (0.2 mag) above Kaler–Schmidt's law (MMRD1), a thick solid blue line in Figure 31. Remember that our free–free emission model light curves usually follow the averaged MMRD relation of MMRD1 (Kaler–Schmidt's law). Thus, the agreement of PW Vul with the MMRD1 relation indicates that the initial envelope mass of PW Vul was typical for ∼0.83 M_☉ WDs and also that free–free emission dominates the spectrum in the V band (see Figure 10).

Next is V705 Cas. This MMRD point is much (∼0.8 mag) brighter than both of the MMRD1 and MMRD2 relations. The photospheric emission is not enough to make the t₃ time longer by a factor of 2.0 (because 2.5Δlog t₃ = 2.5log 2.0 ≈ 2.5 × 0.3 ≈ 0.8), as is clearly shown in Figure 16. Therefore, we attribute the difference in the maximum brightness to the difference in the initial envelope mass. To confirm this, we compared the rise time of the UV 1455 Å flux between PW Vul and V705 Cas in Figures 10 and 16. The UV 1455 Å flux had already risen at the optical maximum in PW Vul, whereas it had not yet risen in V705 Cas. This can more easily be seen in Figure 13. The longer time before the UV 1455 Å peak means that the envelope mass is more massive in V705 Cas than in PW Vul. The more massive initial envelope mass makes the brighter optical maximum of V705 Cas. Thus, it is located above the solid blue line of the MMRD1 relation.

The third object is GQ Mus. The MMRD point of this object is also much (∼0.8 mag) above both the MMRD1 (solid blue line) and MMRD2 (solid magenta line). GQ Mus shows a bump of ∼1 mag brighter than the model light curve in the very early phase. This feature is very similar to that of V1500 Cyg. One could suppose that this bump is the origin of the deviation from the solid blue line of the MMRD1 relation, but V1500 Cyg does not show such a large deviation (see Figure 31). Therefore, we suppose that photospheric emission is the main source of the deviation. We found, from Figure 17(a), that the t₃ time of the total flux (solid black line) is 2.3 times longer compared with the case of free–free alone (solid blue line). This effect makes the t₃ time longer by about Δlog t₃ = log 2.3 ≈ 0.35. This corresponds to the increase in brightness by 2.5Δlog t₃ = 2.5 × 0.35 = 0.9 mag in the MMRD diagram, roughly consistent with the present position of GQ Mus in Figure 31. Thus, we conclude that GQ Mus is above the MMRD relation because of the photospheric emission effect.

The remaining objects are static to wind transition novae, RR Pic, V5558 Sgr, HR Del, and V723 Cas. These MMRD points are also much (0.8–1.2 mag) above the solid blue line of MMRD1, but HR Del and V723 Cas are consistent with the MMRD2. From Figure 26 of V723 Cas, we see that the t₃ time estimated along our total V flux model (solid black line) is ∼2.2 times longer than the case of free–free alone (solid blue line). Note again that our free–free emission model light curves usually follow the averaged MMRD relation of MMRD1 (Kaler–Schmidt's law). The photospheric emission effect raises the brightness by 2.5Δlog t₃ = 2.5log 2.2 ≈ 0.8 mag compared with the MMRD1 (solid blue line), roughly consistent with the position of V723 Cas in Figure 31. Similarly, for HR Del we obtain, from Figure 25(a), a factor of ∼2.2 and a rise of 2.5Δlog t₃ = 2.5log 2.2 ≈ 0.8 mag, owing to the photospheric emission effect. This is also consistent with the position of HR Del in Figure 31. V5558 Sgr and RR Pic showed much brighter (∼1.2 mag) optical maxima than the MMRD1 (solid blue line). These two novae showed prominent amplitudes of oscillations during the multiple peak. We think that these large peaks are related to more massive envelopes compared with those of HR Del and V723 Cas. In fact, the amplitude of multiple peak is decreasing in V5558 Sgr, suggesting reduction of the envelope mass due to mass loss. The same explanation is possible in RR Pic, whose MMRD point is also 1.2 mag brighter than the MMRD1 relation (solid blue line).

It is interesting to see the position of the recurrent nova RS Oph (red filled star) and the 1 yr recurrence period M31 nova, M31N2008-12a (red open diamond) in Figure 31. RS Oph is located 1.2 mag below the MMRD1. This faintness corresponds to a much smaller envelope mass at optical maximum, suggesting a massive WD and very high mass-accretion rate. This situation is very consistent with the total picture of recurrent novae; a very massive WD close to the Chandrasekhar mass and a high mass-accretion rate to the WD (e.g., Hachisu & Kato 2001). In this figure, we adopt M_{V, max} = −7.8 and t₃ = 10.5 days with the distance of d = 1.4 kpc (Hachisu et al. 2006; Barry et al. 2008; Hachisu & Kato 2014), absorption of A_V = 3.1E(B − V) = 3.1 × 0.65 = 2.0 (Hachisu & Kato 2014), m_{V, max} = 5.0 (Rosino & Iijima 1987), and t₃ = 10.5 days from optical light curve fitting with our free–free model light curves (Hachisu & Kato 2001; Hachisu et al. 2006, 2007). The 1 yr recurrence period M31 nova, M31N2008-12a, is depicted by a red open diamond. It is very faint, i.e., M_{g, max} = −6.6 (maximum in the g-band) and t_{3, g} ≈ 4.2 days (measured in the g-band light curve), taken from Tang et al. (2014). The 1 yr recurrence period is close to the shortest recurrence period of novae, suggesting a very massive WD close to the Chandrasekhar mass and a very high accretion rate (see, e.g., Tang et al. 2014; Kato et al. 2014), thus a very small envelope mass. These support our conclusion that the peak brightness of a nova depends on the initial envelope mass as well as the WD mass itself.

To summarize, the primary parameter of the MMRD relation is the WD mass, and the secondary parameter is the initial envelope mass. Variations in the initial envelope mass are the origin of scatter from the averaged MMRD relation, MMRD1. More massive envelopes correspond to the region above the MMRD1 line, and less massive envelopes correspond to the region below the MMRD1 line. Photospheric emission is the third factor of the MMRD relation but becomes more important in slow novae because it makes the t₃ time longer in low-mass WDs.

In terms of free–free emission, Hachisu & Kato (2006) obtained model light curves of novae in 0.05 M_☉ steps for masses in the range M_WD = 0.55–1.2 M_☉ with the chemical composition "CO nova 4." The flux is calculated from

$$\begin{equation} F_\nu ^{\lbrace M_{\rm WD}\rbrace } (t) = C \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace M_{\rm WD}\rbrace }_{(t)}, \end{equation} \tag{ A1 }$$

where $\dot{M}_{\rm wind}$ is the wind mass-loss rate, v_ph is the wind velocity at the photosphere, R_ph is the photospheric radius of each wind solution, and C is the proportionality constant (see Equation (9) of Hachisu & Kato 2006). Note that the flux F_ν is independent of the frequency ν in the case of optically thin free–free emission. The details of the calculations are presented in Hachisu & Kato (2006, 2010). Then the magnitude of model light curves are calculated as

$$\begin{equation} m_{\rm ff} = -2.5 \log \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace M_{\rm WD}\rbrace }_{(t)} + G^{\lbrace M_{\rm WD}\rbrace }. \end{equation} \tag{ A2 }$$

The numerical data entering into Equation (A2) are tabulated in Table 4. Subscript (t) denotes the time dependence, whereas superscript {M_WD} indicates a model parameter. The last row (15th mag) of each column in Table 4 corresponds to the end of the wind phase in each light curve sequence. In other words, we define the constant $G^{\lbrace M_{\rm WD}\rbrace }$ in Equation (A2) such that the last (lowest) point of each light curve (the end of an optically thick wind phase) is 15th mag. This helps to shorten the table. The magnitudes of free–free emission m_ff are plotted in Figure 32.

The free–free emission model light curves in Figure 32 have a very similar shape. For the chemical compositions of "CO nova 2" and "Ne nova 2," Hachisu & Kato (2010) showed that model light curves corresponding to different masses have a homologous behavior, in the sense that they overlap each other if properly squeezed or stretched along time. Here we show that the same property applies to the case of chemical composition, "CO nova 4." Figure 33 demonstrates that the free–free emission model light curves in Figure 32 overlap each other if they are properly squeezed/stretched along time.

Zoom In Zoom Out Reset image size

The time-scaling factor, f_s, of each model was determined by increasing or decreasing f_s until the model UV 1455 Å light curve shape matches the observational points. We also normalize the peak flux of our model UV 1455 Å light curve to match the observational peak. For example, the evolution of the 0.8 M_☉ model evolves 1.15 times slower than the PW Vul observation, so f_s ≈ 1.15. The 0.85 M_☉ model evolves 1.1 times faster than that, so f_s ≈ 0.9. The scaling factors log f_s thus obtained are tabulated in Table 4.

If we squeeze the timescale of our model light curves with t' = t/f_s, as shown in Figure 33, these light curves are written as

$$\begin{equation} {m^{\prime }}_V^{\lbrace M_{\rm WD}\rbrace } (t^{\prime })= -2.5 \log \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace M_{\rm WD}\rbrace }_{(t^{\prime }f_s)} +K_V, \end{equation} \tag{ A3 }$$

where K_V is a constant common for all WD masses. Because they all overlap each other (i.e., the universal decline law), we regard all these as the same phenomena. Then, this indicates that

$$\begin{equation} \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace M_{\rm WD}\rbrace }_{(t^{\prime }f_s)} = \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace 0.83 \, M_\odot \rbrace }_{(t^{\prime }f_s)} = \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace 0.83 \, M_\odot \rbrace }_{(t^{\prime })}, \end{equation} \tag{ A4 }$$

for all M_WD. Note that f_s = 1 for the 0.83 M_☉ WD.

In general, if we squeeze the timescale of a physical phenomenon by a factor of f_s (i.e., t' = t/f_s), we convert the frequency to ν' = f_sν and the flux of free–free emission to $F^{\prime }_{\nu ^{\prime }} = f_{\rm s} F_\nu$ because

$$\begin{equation} {{d} \over {d t^{\prime }}} = f_{\rm s} {{d} \over {d {t}} }. \end{equation} \tag{ A5 }$$

Substituting $F^{\prime }_{\nu ^{\prime }} = F^{\prime }_\nu$ (independent of the frequency in optically thin free–free emission) into $F^{\prime }_{\nu ^{\prime }} = f_{\rm s} F_\nu$ and integrating $F^{\prime }_\nu = f_{\rm s} F_\nu$ with the V-filter response function, we have the following relation, i.e.,

$$\begin{eqnarray} &&m^{\prime }_V (t/f_s) = m_V (t) - 2.5 \log f_{\rm s}. \end{eqnarray} \tag{ A6 }$$

Substituting Equation (A4) into (A3) and then Equation (A3) into (A6), we obtain the apparent V magnitudes of

$$\begin{equation} m_V^{\lbrace M_{\rm WD}\rbrace } (t) = 2.5 \log f_{\rm s} -2.5 \log \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace 0.83 \, M_\odot \rbrace }_{(t/f_s)} + K_V, \end{equation} \tag{ A7 }$$

where note that f_s is the time-scaling factor for the WD with mass of M_WD, and not for the 0.83 M_☉ WD.

The corresponding absolute magnitudes of the light curves can be readily obtained from Equation (A7) and from the distance modulus of PW Vul, i.e.,

$$\begin{eqnarray} M_V^{\lbrace M_{\rm WD}\rbrace } (t) &=& m_V^{\lbrace M_{\rm WD}\rbrace } (t) - (m-M)_{V, \rm PW \,Vul} \cr \cr &=& 2.5 \log f_{\rm s} -2.5 \log \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace 0.83 \, M_\odot \rbrace }_{(t/f_s)} \cr \cr & & + K_V - (m-M)_{V, \rm PW \,Vul} \cr \cr &=& 2.5 \log f_{\rm s} + {m^{\prime }}_V^{\lbrace M_{\rm WD}\rbrace } (t/f_s) - (m-M)_{V, \rm PW \,Vul}, \nonumber\\ \end{eqnarray} \tag{ A8 }$$

where (m − M)_{V, PW Vul} = 13.0 is the distance modulus of PW Vul harboring a 0.83 M_☉ WD, and we use Equation (A4), i.e.,

$$\begin{equation} \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace M_{\rm WD}\rbrace }_{(t)} = \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace 0.83 \, M_\odot \rbrace }_{(t/f_s)}, \end{equation} \tag{ A9 }$$

and Equation (A3), i.e.,

$$\begin{equation} {m^{\prime }}_V^{\lbrace M_{\rm WD}\rbrace } (t/f_s)= -2.5 \log \left[ {{\dot{M}_{\rm wind}^2} \over {v_{\rm ph}^2 R_{\rm ph}}} \right]^{\lbrace M_{\rm WD}\rbrace }_{(t)} +K_V, \end{equation} \tag{ A10 }$$

to derive the last line of the above equation. The last line in Equation (A8) simply means that the model light curve ${m^{\prime }}_V^{\lbrace M_{\rm WD}\rbrace }(t)$ in Figure 33 is shifted horizontally by log f_s and vertically by 2.5log f_s − (m − M)_{V, PW Vul} to retrieve the absolute magnitude and real timescale. These retrieved absolute magnitudes are plotted in Figure 34 on the real timescale. We also tabulate the absolute magnitude, M_w, at the end point of winds in Table 4 and plot them in Figure 35. These values of "CO nova 4" are in between those for "CO nova 2" and "Ne nova 2." This confirms that our calibration of absolute magnitude is reasonable. Then, we retrieve the absolute magnitudes of all model light curves in Table 4 as

$$\begin{equation} M_V = m_{\rm ff} - (m_{\rm w} - M_{\rm w}) = m_{\rm ff} - (15.0 - M_{\rm w}). \end{equation} \tag{ A11 }$$

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

It should be noted that K_V is a constant common to all WD masses (see Hachisu & Kato 2010). Using Equations (A2) and (A3), we obtain $m_{\rm ff} - m^{\prime }_V=15.0 - 16.0 = -1.0=G^{\lbrace 0.83 \, M_\odot \rbrace }-K_V$ at the bottom of the light curve (end of winds), where we directly read $m^{\prime }_V=m_{\rm w}=16.0$ for the 0.83 M_☉ model from Figure 8. We recall that $G^{\lbrace M_{\rm WD}\rbrace }$ was defined to satisfy m_ff = 15 at the end point of optically thick winds in Equation (A2). We determine the value of K_V from $K_V = G^{\lbrace 0.83 \, M_\odot \rbrace } + 1.0$.

It is interesting to verify the consistency of our theoretical light curves with the empirical finding that the absolute magnitude 15 days after optical maximum, M_V(15), is almost constant for novae. The value was first proposed by Buscombe & de Vaucouleurs (1955) with M_V(15) = −5.2 ± 0.1, followed by Cohen (1985) with M_V(15) = −5.60 ± 0.43, van den Bergh & Younger (1987) with M_V(15) = −5.23 ± 0.39, Capaccioli et al. (1989) with M_V(15) = −5.69 ± 0.42, and Downes & Duerbeck (2000) with M_V(15) = −6.05 ± 0.44. The decline rates of our model light curves depend slightly on the chemical composition. We have already obtained M_V(15) = −5.95 ± 0.25 for 0.55–1.2 M_☉ WDs with "CO nova 2" and M_V(15) = −5.6 ± 0.3 for 0.70–1.3 M_☉ WDs with "Ne nova 2" (Hachisu & Kato 2010). This value is M_V(15) = −5.4 ± 0.4 for 0.7–1.05 M_☉ WDs with "CO nova 4," as shown in Figure 34. In this figure, we plot, by magenta filled circles, the V maxima of each model light curve corresponding to the V maximum of PW Vul in Figure 33 and, by blue crosses, the absolute magnitudes of each model light curve 15 days after V maximum, i.e., 15 days from each magenta filled circle. The obtained values are roughly consistent with the above empirical relations.

The clear trend appearing from Figure 34 is that a more massive WD is systematically brighter at maximum (smaller M_{V, max}; see magenta filled circles near the mark B) and has a faster decline rate (smaller t₂ or t₃ time). The relation between t₃ (or t₂) and M_{V, max} for a nova is usually called the "Maximum Magnitude versus Rate of Decline" (MMRD) relation. Here t₃ (t₂) time is defined by 3 mag (2 mag) decay time from its maximum in units of days. Now we derive a theoretical MMRD relation for the "CO nova 4" chemical composition. The apparent maximum brightness, m_{V, max}, of each WD mass model light curve is expressed as $m_{V,{\rm max}} = m^{\prime }_{V,{\rm max}} + 2.5 \log f_{\rm s}$ when the t₃ time is squeezed as $t_3 = f_{\rm s} t^{\prime }_3$. Eliminating f_s from these two relations, we have $m_{V,{\rm max}} = 2.5 \log t_3 + m^{\prime }_{V,{\rm max}} - 2.5 \log t^{\prime }_3$. We obtained $t^{\prime }_3 = 80$ days and $m^{\prime }_{V,{\rm max}} = 6.1$, where we measured $t^{\prime }_3$ and $m^{\prime }_{V,{\rm max}}$ along our model light curves in Figure 33. Then, we obtained our MMRD relation as

$$\begin{eqnarray} M_{V,{\rm max}} & = & m_{V,{\rm max}} - (m-M)_V \cr & = & 2.5 \log t_3 + m^{\prime }_{V,{\rm max}} - 2.5 \log t^{\prime }_3 - (m-M)_V \cr & = & 2.5 \log t_3 -11.65, \end{eqnarray} \tag{ A12 }$$

where we used (m − M)_V = 13.0 for PW Vul. Figure 36 shows this theoretical MMRD relation. This figure also shows the MMRD relation calibrated with V1668 Cyg (black dashed line taken from Hachisu & Kato 2010). We also indicated the time-scaling factor f_s against the PW Vul light curves in the upper axis of the same figure.

Zoom In Zoom Out Reset image size

For comparison, two empirical MMRD relations are plotted in the same figure, i.e., Kaler–Schmidt's law (solid blue line labeled "MMRD1"; Schmidt 1957), i.e.,

$$\begin{equation} M_{V, {\rm max}} = -11.75 + 2.5 \log t_3, \end{equation} \tag{ A13 }$$

and della Valle–Livio's law (solid magenta line labeled "MMRD2": della Valle & Livio 1995), i.e.,

$$\begin{equation} M_{V, {\rm max}} = -7.92 -0.81 \arctan \left({{1.32-\log t_2} \over {0.23}} \right), \end{equation} \tag{ A14 }$$

where we used a relation of t₂ ≈ 0.6 × t₃ for the optical light curves that follow the universal decline law (Hachisu & Kato 2006).

Figure 36 also shows observational points for individual novae (red filled circles), taken from Table 5 of Downes & Duerbeck (2000). Note that we remeasured the t₃ time of PW Vul along our model light curve, which resulted in t₃ = 80 days and M_{V, max} = −6.9, as shown by a large black open circle. This data point is to the left of Downes & Duerbeck's (2000) estimate (t₃ = 126 day and M_{V, max} = −6.7), denoted by a large red filled circle.

The scatter of the observed data with respect to the MMRD formulae in Figure 36 is generally large and larger than the observational errors. This strongly suggests that the scatter is due to the presence of a second parameter. Hachisu & Kato (2010) pointed out that the main parameter governing the MMRD relation is the WD mass (represented by the time-scaling factor f_s in Figure 36), the second parameter being the initial envelope mass (or the mass-accretion rate of the WD). In turn, the initial envelope mass (ignition mass) depends on the mass-accretion rate to the WD (see, e.g., Figure 3 of Kato et al. 2014) such that the lower the mass-accretion rate, the larger the envelope mass. This simply means that for the same WD mass, novae are brighter/fainter for lower/higher mass-accretion rates. Hachisu & Kato (2010) clearly showed the dependence of maximum brightness on the initial envelope mass (see their Figures 8 and 15). We further show fainter examples of maximum brightness, RS Oph and M31N2008-12a, both of which are recurrent novae with ∼20 and ∼1 yr recurrence periods, respectively. We conclude that this second parameter, the initial envelope mass, can reasonably explain the scatter of individual novae from the empirical MMRD relations so far proposed (see Figure 15 of Hachisu & Kato 2010).