---
abstract: 'A theoretical light curve is constructed for the quiescent phase of the recurrent nova U Scorpii in order to resolve the existing distance discrepancy between the outbursts ($d \sim 6$ kpc) and the quiescences ($d \sim 14$ kpc). Our U Sco model consists of a very massive white dwarf (WD), an accretion disk (ACDK) with a flaring-up rim, and a lobe-filling, slightly evolved, main-sequence star (MS). The model properly includes an accretion luminosity of the WD, a viscous luminosity of the ACDK, a reflection effect of the MS and the ACDK irradiated by the WD photosphere. The $B$ light curve is well reproduced by a model of 1.37 $M_\odot$ WD $+$ 1.5 $M_\odot$ MS (0.8—2.0 $M_\odot$ MS is acceptable) with an ACDK having a flaring-up rim, and the inclination angle of the orbit $i \sim 80 \arcdeg$. The calculated color is rather blue ($B-V \sim 0.0$) for a suggested mass accretion rate of $2.5 \times 10^{-7} M_\odot$ yr$^{-1}$, thus indicating a large color excess of $E(B-V) \sim 0.56$ with the observational color of $B-V = 0.56$ in quiescence. Such a large color excess corresponds to an absorption of $A_V \sim 1.8$ and $A_B \sim 2.3$, which reduces the distance to 6—8 kpc. This is in good agreement with the distance estimation of 4—6 kpc for the latest outburst. Such a large intrinsic absorption is very consistent with the recently detected period change of U Sco, which is indicating a mass outflow of $\sim 3 \times 10^{-7} M_\odot$ yr$^{-1}$ through the outer Lagrangian points in quiescence.'
author:
- Izumi Hachisu
- Mariko Kato
- Taichi Kato and Katsura Matsumoto
- 'Ken’ichi Nomoto'
title: A MODEL FOR THE QUIESCENT PHASE OF THE RECURRENT NOVA U SCORPII
---

INTRODUCTION
============

U Scorpii is one of the best observed recurrent novae, the outbursts of which were recorded in 1863, 1906, 1936, 1979, 1987, and the latest in 1999. Especially, the 1999 outburst was well observed from the rising phase to the cooling phase by many observers (e.g., [@mun99]; [@kah99]; [@lep99]) including eclipses (Matsumoto, Kato, & Hachisu 2000). Based on Matsumoto et al.’s (2000) observation, Hachisu et al. (2000) have constructed a theoretical light-curve model for the 1999 outburst of U Sco and obtained various physical parameters of the recurrent nova. Their main results are summarized as follows: (1) A direct light-curve fitting of the 1999 outburst indicates a very massive white dwarf (WD) of $M_{\rm WD}= 1.37 \pm 0.01 M_\odot$. (2) The envelope mass at the optical maximum is estimated to be $\Delta M \sim 3 \times 10^{-6} M_\odot$. (3) Therefore, the mass accretion rate of the WD is $\dot M_{\rm acc} \sim 2.5 \times 10^{-7} M_\odot$ yr$^{-1}$ during the quiescent phase between 1987 and 1999. (4) An optically thick wind blows from the WD and plays a key role in determining the nova duration because it reduces the envelope mass ([@kat94]). About 60% of the envelope mass is carried away in the wind, which forms an expanding shell as observed in T Pyx (e.g., [@shr89]). The residual 40% ($1.2 \times 10^{-6} M_\odot$) is added to the helium layer of the WD. (5) As a result, the WD can grow in mass at an average rate of $\sim 1 \times 10^{-7} M_\odot$ yr$^{-1}$.

The above physical pictures are exactly the same as proposed by Hachisu et al. (1999b) as a progenitor system of Type Ia supernovae (SNe Ia). However, the distance to U Sco is still controversial because the direct light-curve fitting results in a relatively short distance of $\sim 6$ kpc ([@hac2000]), which is incompatible with the distance of $\sim 14$ kpc at the quiescent phase (e.g., [@web87]; [@war95]; [@kah99], for a summary). If the distance of $\sim 14$ kpc is the case, it could be hardly consistent with the results (1) to (5) mentioned above.

Our purpose in this Letter is to construct a light-curve model for the quiescent phase and to rectify the distance to U Sco. Our numerical method to obtain light curves has been described both in Hachisu & Kato (1999) to explain the second peak of T CrB outbursts and in Hachisu et al. (2000) to reproduce the light curve for the 1999 outburst of U Sco. Therefore, we mention only new parts of our numerical method in §2. In §3, by directly fitting our theoretical light curve to the observations, we derive the distance to U Sco. Discussions follow in §4, especially in relation to the recently detected orbital-period change of U Sco and a systemic mass loss through the outer Lagrangian points. We also discuss the relation to a progenitor system of SNe Ia.

THEORETICAL LIGHT CURVES
========================

Our U Sco model is graphically shown in Figure \[uscofig\_q35\]. Schaefer (1990) and Schaefer & Ringwald (1995) observed eclipses of U Sco in the quiescent phase and determined the orbital period ($P= 1.23056$ days) and the ephemeris (HJD 2,451,235.777 $+$ 1.23056$E$) at the epoch of mid-eclipse. Thus, the companion is a main-sequence star (MS) which expands to fill its Roche lobe after a most part of the central hydrogen is consumed. We call such a star “a slightly evolved” MS. The inclination angle of the orbit ($i \sim 80\arcdeg$) is a parameter for fitting.

We have assumed that (1) $M_{\rm WD}= 1.37 M_\odot$, (2) the WD luminosity of $$L_{\rm WD} = {1 \over 2} {{G M_{\rm WD} \dot M_{\rm acc}} 
\over {R_{\rm WD}}} + L_{\rm WD,0},
\label{accretion-luminosity}$$ where the first term is the accretion luminosity (e.g., Starrfield, Sparks, & Shaviv 1988) and the second term $L_{\rm WD,0}$ is the intrinsic luminosity of the WD, and $R_{\rm WD}= 0.0032 R_\odot$ the radius of the $1.37 M_\odot$ WD, and (3) a black-body photosphere of the WD. The accretion luminosity is $\sim 1700 L_\odot$ for a suggested mass accretion rate of $\dot M_{\rm acc} \sim 2.5 \times 10^{-7} M_\odot$ yr$^{-1}$. Here, we assume $L_{\rm WD,0}=0$ because the nuclear luminosity is smaller than the accretion luminosity for this accretion rate, but we have examined other two cases of $L_{\rm WD,0}=2000$ and 4000 $L_\odot$ and found no significant differences in the distance as shown below. We do not consider the limb-darkening effect for simplicity.

It is assumed that the companion star is synchronously rotating on a circular orbit and its surface fills the inner critical Roche lobe as shown in Figure \[uscofig\_q35\]. We neglect both the limb-darkening effect and the gravity-darkening effect of the companion star for simplicity. Here, we assume a 50% irradiation efficiency of the companion star ($\eta_{\rm ir,MS}=0.5$). We have examined the dependence of the distance on the irradiation efficiency (i.e., $\eta_{\rm ir,MS}=0.25$ and 1.0) but found no significant differences in the distance as shown below. The non-irradiated photospheric temperature $T_{\rm ph, MS}$ of the companion star is a parameter for fitting. The mass of the secondary is assumed to be $M_{\rm MS}= 1.5 M_\odot$.

The size of the accretion disk is a parameter for fitting and defined as $$R_{\rm disk} = \alpha R_1^*,
\label{accretion-disk-size}$$ where $\alpha$ is a numerical factor indicating the size of the accretion disk, and $R_1^*$ the effective radius of the inner critical Roche lobe for the WD component (e.g., Eggleton 1983). We also assume that the accretion disk is axisymmetric and has a thickness given by $$h = \beta R_{\rm disk} \left({{\varpi} 
\over {R_{\rm disk}}} \right)^{\nu},
\label{flaring-up-disk}$$ where $h$ is the height of the surface from the equatorial plane, $\varpi$ the distance on the equatorial plane from the center of the WD, $\nu$ the power of the surface shape, and $\beta$ a numerical factor showing the degree of thickness and also a parameter for fitting. We adopt a $\varpi$-squared law ($\nu=2$) simply to mimic the flaring-up effect of the accretion disk rim (e.g., Schandl, Meyer-Hofmeister, & Meyer 1997), and have examined the dependence of the distance on the power ($\nu=1.25$ and 3.0) without finding any significant differences as shown below.

The surface of the accretion disk also absorbs photons from the WD photosphere and reemits with a black-body spectrum at a local temperature. We assume a 50% irradiation efficiency of the companion star, i.e., $\eta_{\rm ir,DK}=0.5$ (e.g., [@sch97]). We have examined other two cases of $\eta_{\rm ir,DK}=0.25$ and 1.0, and found no significant differences in the distance as shown below. The non-irradiated temperature of the disk surface is assumed to be determined by the viscous heating of the standard accretion disk model. Then, the disk surface temperature is given by $$\sigma T_{\rm ph, disk}^4 = {{3 G M_{\rm WD} \dot M_{\rm acc}} 
\over {8 \pi \varpi^3}} + \eta_{\rm ir,DK} 
{{L_{\rm WD}} \over {4 \pi r^2}} \cos\theta,$$ where $r$ the distance from the WD center, and $\cos\theta$ the incident angle of the surface (e.g., [@sch97]). The temperature of the disk rim is assumed to be 3000 K.

RESULTS
=======

Figure \[bmag\_bv\_color\_paper\] shows the observational points (open circles) by Schaefer (1990) together with our calculated $B$ light curve (thick solid line) for $\dot M_{\rm acc}= 2.5 \times 10^{-7} M_\odot$ yr$^{-1}$. To fit our theoretical light curves with the observational points, we calculate $B$ light curves by changing the parameters of $\alpha=0.5$—1.0 by 0.1 step, $\beta=0.05$—0.50 by 0.05 step, $T_{\rm ph, MS}= 3500$—8000 K by 100 K step, and $i=75$—$85\arcdeg$ by $1\arcdeg$ step and seek for the best fit model. The best fit parameters obtained are shown in Figure \[bmag\_bv\_color\_paper\] (see also Table \[tbl-1\]).

There are five different contributions to the $B$-light ($L_B$) in the system: the white dwarf ($L_{B1}$), the non-irradiated portions of the accretion disk ($L_{B2}$) and the donor star ($L_{B3}$), and the irradiated portions of the accretion disk ($L_{B4}$) and the donor star ($L_{B5}$). In order to show each contribution, we have added two light curves in Figure \[bmag\_bv\_color\_paper\], that is, a non-irradiation case of the ACDK ($\eta_{\rm ir, DK}=0$, dash-dotted), and a non-irradiation case of the MS ($\eta_{\rm ir, MS}=0$, dashed). The light from the WD is completely blocked by the accretion disk rim, thus having no contribution, $L_{B1}=0$. The depth of the primary eclipse, 1.5 mag, means $L_{B3}= 0.25 L_{B}$ because the ACDK is completely occulted by the MS. The difference of 1 mag between the thick solid and dash-dotted lines indicates $L_{B4}=0.60 L_B$. The difference of 0.1 mag between the thick solid and dashed lines indicates $L_{B5} = 0.10 L_B$. Thus, we obtain each contribution: $L_{B1} = 0$, $L_{B2} = 0.05 L_B$, $L_{B3} = 0.25 L_B$, $L_{B4} = 0.60 L_B$, and $L_{B5}= 0.10 L_B$.

Then we calculate the theoretical color index $(B-V)_c$ for these best fit models. Here, we explain only the case of $\dot M_{\rm acc}= 2.5 \times 10^{-7} M_\odot$ yr$^{-1}$. By fitting, we obtain the apparent distance modulus of $m_{B, 0}= 16.71$, which corresponds to the distance of $d= 22$ kpc without absorption ($A_B=0$). On the other hand, we obtained a rather blue color index of $(B-V)_c= 0.0$ outside eclipses. Together with the observed color of $(B-V)_o= 0.56$ outside eclipses ([@sch90]; [@sch95]), we derive a color excess of $E(B-V)= (B-V)_o - (B-V)_c= 0.56$ Here, suffixes $c$ and $o$ represent the theoretically calculated and the observational values, respectively. Then, we expect an absorption of $A_V= 3.1 ~E(B-V)= 1.8$ and $A_B= A_V + E(B-V) = 2.3$. Thus, we are forced to have a rather short distance to U Sco of 7.5 kpc.

In our case of $\alpha=0.7$ and $\beta=0.30$, the accretion disk is completely occulted at mid-eclipse. The color index of $(B-V)_c= 0.53$ at mid-eclipse indicates a spectral type of F8 for the cool component MS, which is in good agreement with the spectral type of F8$\pm$2 suggested by Johnston & Kulkarni (1992). Hanes (1985) also suggested that a spectral type nearer F7 is preferred.

For other mass accretion rates of $\dot M_{\rm acc}=$ (0.1—5.0)$\times 10^{-7} M_\odot$ yr$^{-1}$, we obtain similar short distances to U Sco, as summarized in Table \[tbl-1\]. It should be noted that, although the luminosity of the model depends on our various assumptions of the irradiation efficiencies, the $\varpi$-powered law of the disk, and the intrinsic luminosity of the WD, the derived distance to U Sco itself is almost independent of these assumptions, as seen from Table \[tbl-2\]. Therefore, the relatively short distance to U Sco ($\sim$ 6—8 kpc) is a rather robust conclusion, at least, from the theoretical point of view.

DISCUSSION
==========

Matsumoto et al. (2000) observed a few eclipses during the 1999 outburst and, for the first time, detected a significant period-change of $\dot P / P = (-1.7 \pm 0.7) \times 10^{-6}$ yr$^{-1}$. If we assume the conservative mass transfer, this period change requires a mass transfer rate of $\gtrsim 10^{-6} M_\odot$ yr$^{-1}$ in quiescence. Such a mass transfer for 12 years is too high to be compatible with the envelope mass on the white dwarf, thus implying a non-conservative mass transfer in U Sco.

We have estimated the mass transfer rate for a non-conservative case by assuming that matter is escaping from the outer Lagrangian points and thus the specific angular momentum of the escaping matter is $1.7 a^2 \Omega_{\rm orb}$ ([@saw84]; [@hac99a]), where $a$ is the separation and $\Omega_{\rm orb} \equiv 2 \pi /P$. Then the mass transfer rate from the companion is $\dot M_{\rm MS}= (-5.5 \pm 1.5) \times 10^{-7} M_\odot$ yr$^{-1}$ for $M_{\rm MS}= 0.8$—2.0 $M_\odot$ under the assumption that the WD receives matter at a rate of $\dot M_{\rm acc} = 2.5 \times 10^{-7} M_\odot$ yr$^{-1}$. The residual ($\sim 3 \times 10^{-7} M_\odot$ yr$^{-1}$), which is escaping from the system, forms an excretion disk outside the orbit of the binary. Such an extended excretion disk/torus may cause a large color excess of $E(B-V)= 0.56$.

Kahabka et al. (1999) reported the hydrogen column density of (3.1—4.8)$\times 10^{21}$ cm$^{-2}$, which is much larger than the Galactic absorption in the direction of U Sco (1.4$\times 10^{21}$ cm$^{-2}$, [@dic90]), indicating a substantial intrinsic absorption. It should also be noted here that Barlow et al. (1981) estimated the absorption toward U Sco by three ways: (1) the Galactic absorption in the direction of U Sco, $E(B-V) \sim 0.24$ and $A_V \sim 0.7$, (2) the line ratio of He II during the 1979 outburst ($t \sim$ 12 days after maximum), $E(B-V) \sim 0.2$ and $A_V \sim 0.6$, and (3) the Balmer line ratio during the 1979 outburst ($t \sim$ 33—34 days after maximum), $E(B-V) \sim 0.35$ and $A_V \sim 1.1$. The last one is significantly larger than the other two estimates. They suggested the breakdown of their case B approximation in high density regions. However, we may point out another possibility that the systemic mass outflow from the binary system has already begun at $t \sim$ 33 days and, as a result, an intrinsic absorption is gradually increasing.

The mass of the companion star can be constrained from the mass transfer rate. Such a high transfer rate as $\dot M_{\rm MS} \sim 5.5
\times 10^{-7} M_\odot$ yr$^{-1}$ strongly indicates a thermally unstable mass transfer (e.g., [@heu92]), which is realized when the mass ratio is larger than 1.0—1.1, i.e., $q= M_{\rm MS}/ M_{\rm WD} >$ 1.0—1.1 for zero-age main-sequence stars ([@web85]). This may pose a requirement $M_{\rm MS} \gtrsim 1.4 M_\odot$. We estimate the most likely companion mass of 1.4—1.6 $M_\odot$ from equation (11) in Hachisu et al. (1999b).

If the distance to U Sco is $\sim$ 6.0—8.0 kpc, it is located $\sim$ 2.3—3.0 kpc above the Galactic plane ($b=22\arcdeg$). The zero-age masses of the progenitor system to U Sco are rather massive (e.g., $8.0 ~M_\odot + 2.5 ~M_\odot$ from Hachisu et al. 1999b) and it is unlikely that such massive stars were born in the halo. Some normal B-type main-sequence stars have been found in the halo (e.g., PG0009+036 is located $\sim$ 5 kpc below the Galactic disk, [@smt96]), which were ejected from the Galactic disk because of their relatively high moving velocities $\sim$100—200 km s$^{-1}$. The radial velocity of U Sco is not known but it is suggested that the $\gamma$-velocity is $\sim$50—100 km s$^{-1}$ from the absorption line velocities ([@joh92]; [@sch95]). If so, it seems likely that U Sco was ejected from the Galactic disk with a vertical velocity faster than $\sim 20$ km s$^{-1}$ and has reached at the present place within the main-sequence lifetimes of a $\sim 3.0 M_\odot$ star ($\sim 3.5\times 10^8$ yr).

Now, we can understand the current evolutionary status and a further evolution of U Sco system. The white dwarf has a mass $1.37 \pm 0.01 M_\odot$. It is very likely that the WD has reached such a large mass by mass accretion. In fact the WD is currently increasing the mass of the helium layer at a rate of $\dot M_{\rm He} \sim 1.0 \times 10^{-7} M_\odot$ yr$^{-1}$ ([@hac2000]). We then predict that the WD will evolve as follows. When the mass of the helium layer reaches a critical mass after many cycles of recurrent nova outbursts, a helium shell flash will occur. Its strength is as weak as those of AGB stars because of the high mass accretion rate ([@nom82]). A part of the helium layer will be blown off in the wind, but virtually all of the helium layer will be burnt into carbon-oxygen and accumulates in the white dwarf ([@kat99h]). Therefore, the WD mass can grow until an SN Ia explosion is triggered ([@nom84]).

We thank the anonymous referee for many critical comments to improve the manuscript. This research has been supported in part by the Grant-in-Aid for Scientific Research (07CE200, 08640321, 09640325, 11640226, 20283591) of the Japanese Ministry of Education, Science, Culture, and Sports. KM has been financially supported as a Research Fellow for Young Scientists by the Japan Society for the Promotion of Science.

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[ccccccccccc]{} 5.0$\times 10^{-7}$ & 0.7 & 0.30 & 5900 & 17.18 & $-0.08$ & 0.45 (F5) & 0.64 & 2.01 & 2.65 & 8.0 2.5$\times 10^{-7}$ & 0.7 & 0.30 & 5500 & 16.71 & $+0.00$ & 0.53 (F8) & 0.56 & 1.76 & 2.32 & 7.5 1.0$\times 10^{-7}$ & 0.7 & 0.25 & 5000 & 16.02 & $+0.12$ & 0.66 (G4) & 0.44 & 1.38 & 1.82 & 6.9 5.0$\times 10^{-8}$ & 0.7 & 0.25 & 4600 & 15.45 & $+0.24$ & 0.78 (G9) & 0.32 & 1.01 & 1.33 & 6.7 2.5$\times 10^{-8}$ & 0.7 & 0.25 & 4200 & 14.72 & $+0.37$ & 0.91 (K2) & 0.19 & 0.60 & 0.79 & 6.1 1.0$\times 10^{-8}$ & 0.7 & 0.25 & 3700 & 13.58 & $+0.58$ & 1.13 (K5) & & & & 5.2

[lcccccccc]{} $L_{\rm WD,0}=2000 L_\odot$ & 6100 & 17.31 & $-0.09$ & 0.41 (F3) & 0.65 & 2.04 & 2.60 & 8.4 $L_{\rm WD,0}=4000 L_\odot$ & 6400 & 17.59 & $-0.13$ & 0.36 (F1) & 0.69 & 2.17 & 2.87 & 8.8 $\eta_{\rm ir,MS}=1.0$ & 5500 & 16.76 & $+0.00$ & 0.53 (F8) & 0.56 & 1.76 & 2.32 & 7.7 $\eta_{\rm ir,MS}=0.25$ & 5500 & 16.66 & $+0.00$ & 0.53 (F8) & 0.56 & 1.76 & 2.32 & 7.4 $\eta_{\rm ir,DK}=1.0$ & 6000 & 17.16 & $-0.06$ & 0.43 (F4) & 0.62 & 1.95 & 2.57 & 8.3 $\eta_{\rm ir,DK}=0.25$ & 5300 & 16.39 & $+0.08$ & 0.58 (F9) & 0.48 & 1.51 & 1.99 & 7.6 $\nu=3.0$ & 5600 & 16.90 & $-0.04$ & 0.51 (F7) & 0.60 & 1.88 & 2.48 & 7.6 $\nu=1.25$ & 5300 & 16.39 & $+0.09$ & 0.58 (F9) & 0.47 & 1.47 & 1.95 & 7.7
