A Phenomenological Model for the Light Curve of Three Quiescent Low-inclination Dwarf Novae and One Pre-cataclysmic Variable

Dwarf novae (DN) are a subtype of primarily non-magnetic cataclysmic variables (CVs), in which a white dwarf primary accretes matters from a Roche-lobe filling late-type star via the inner (L1) Lagrange point (Warner 2003). An accretion disk can extend to the white dwarf surface via viscous processes between adjacent accretion annuli (e.g., friction and shear) due to the weak magnetic field of the white dwarf in DN (B < 10⁶ G). Systems in which the magnetic field of the white dwarf is large enough (10⁶ < B ≤ 10⁷ G) for the accretion disk to be disrupted inside the white dwarf magnetosphere and for material to begin to follow the magnetic field lines are called Intermediate Polars (IPs). In the following, we assume that the disk outside of the magnetosphere in an IP is equivalent in structure to the disk in a regular non-magnetic DN. The interaction between the ballistic stream leaving the L1 point and the accretion disk forms a region of energy release at the edge of the disk, which is called a hotspot. The combination of an accretion disk and a hotspot (hereafter just called the disk model) has been used as the typical accretion model for CVs (e.g., Smak 1970; Warner 2003) and has been quite successful in describing the asymmetrical eclipse profile of several high-inclination CVs during quiescence (e.g., Wood & Crawford 1986; Bruch 1996). This model can also be used to explain a symmetrical CV eclipse profiles during outburst (e.g., Kato et al. 2003; Ba̧kowska & Olech 2015), which is caused by the significant flux increase of the disk (i.e., the luminous accretion disk almost overwhelms the relatively faint white dwarf and hotspot).

Several high-inclination CVs in quiescence have been comprehensively studied using this disk model (e.g., Cook & Waner 1984; Bailey & Cropper 1991; McAllister et al. 2015). By detecting variations in the mid-eclipse times, substellar objects have been suggested to exist in several DN (e.g., V2051 Oph, Qian et al. 2015; and EM Cyg, Dai & Qian 2010a). In addition, many synthetic light curve analysis methods (e.g., BINSYN program, Linnell et al. 2012; Eclipsing Light Curve Code, Orosz & Hauschildt 2000; and the cool-disk model, Khruzina 2011) have been developed to analyze CV eclipse light curves. The irregular eclipse light curves of quiescent CVs are composed of the occultations of multiple components,including the white dwarf, the accretion disk and hotspots (e.g., Smak 1994; Feline et al. 2004; Littlefair et al. 2014; McAllister et al. 2015). High-inclination CVs only show a single eclipse in one orbit (i.e., the secondary eclipse of the red dwarf is invisible; e.g., Krzeminski 1965; Bailey et al. 1988) because the red dwarf in a CV system is usually regarded to be a very faint component compared with the accretion disk and the white dwarf. This single-eclipse feature implies that CV eclipse light curves cannot reveal full information about the red dwarf companion. Due to the complexity of the quiescent CV eclipse light curves, high time resolution is necessary to decompose all components. However, over 70% of CVs with short orbital periods (<3 hr) are fainter than 17 mag in quiescence based on the updated CV catalog (RKcat Edition 7.24, which was first published in Ritter & Kolb 2003). Therefore, the majority of CVs have only low time-resolution photometry comprised of blended flux from many components, which makes modeling these systems difficult. For example, recent discussions concerning two new-found CVs with deep eclipses carried out by Kjurkchieva et al. (2015) and Kennedy et al. (2016) have clearly indicated that the model light curves cannot perfectly fit the observed eclipsing light curves during ingress and outside of eclipse. Assuming that a deep eclipse is a common feature of CVs with orbital inclination higher than 80°, the fraction of low-inclination CVs can be simply estimated to be around 90%. To determine a general model independent of inclination, it is necessary to consider low-inclination CVs.

The unprecedented light curves from the Kepler K2 mission (Howell et al. 2014), with nearly continuous photometric coverage for 1–3 months at different pointings (Campaigns) along the ecliptic provide an excellent database to study quiescent CV light curves. K2 Campaign 0 (K2-C0) was an engineering test in the early stage of the K2 program and only covered ∼35 days because the spacecraft was not in fine point during the beginning of the campaign, while K2 Campaign 1 (K2-C1) covered a complete period of 80 days. In this paper, we focus on the phased light curves of four systems: 1RXS J0632+2536 (hereafter J0632+2536) and TW Vir, which are both DNs; RZ Leo, which is an unusual IP that also has displayed DN outbursts, making it one of the few systems to be both a magnetic system and a DN; and WD 1144+011, which is a pre-CV (meaning that the secondary is probably not filling its Roche lobe). J0632+2536 was observed in K2-C0, while TW Vir, RZ Leo and WD 1144+011 were observed in K2-C1. The phased light curves are investigated in detail using the synthesis methods XRBinary and NMfit described in Section 3. Due to the lack of any eclipse feature, all four objects are likely low-inclination systems. The preset model parameters are discussed in Sections 4 and 5. The details of the white dwarf accretion structure during quiescence and the physical parameters of the stars in each system are further discussed and visualized in Section 6.

Dai et al. (2016) used the PyKE suite of software tools developed by the Guest Observer Office (Still & Barclay 2012) to extract the K2 light curves of J0632+2536, RZ Leo and WD 1144+011, and derive their orbital periods using traditional period finding techniques (e.g., Lomb–Scargle periodogram, Lomb 1976; Scargle 1982; and phase dispersion minimization, Stellingwerf 1978) and their corresponding phased light curves. Due to the unstable orbital modulations of TW Vir, Dai et al. (2017) applied a phase-correcting method to obtain its orbital period and the phased data from the quiescent data.

Based on the continuous K2 data, we attempted to model the mean orbital light curves of each system by using an XRBinary light curve synthesis code that was developed by E. L. Robinson.¹² The default phase zero of a light curve generated by XRBinary is an inferior conjunction of the Roche-lobe filling secondary (i.e., the accretion disk around the white dwarf is occulted by the red component), which is in accord with the phases specified in eclipsing CVs. Compared with the typical CV eclipse light curve with a narrow and deep white dwarf eclipse (e.g., Littlefair et al. 2014; McAllister et al. 2015), all four light curves derived by Dai et al. (2016, 2017) only show wide and shallow modulations with an amplitude of several hundredths up to tenths of a magnitude. Note that the phases of the obtained light curves are arbitrary. The three DN (J0632+2536, RZ Leo and TW Vir) light curves clearly show double-hump modulations, with a nearly constant phase difference of ∼0.5 between the two minima dips. This means that the light minima at the lower (primary dip) and higher (secondary dip) flux levels are at phases 0.5 or zero, respectively. In principle, an irradiation effect is significant in CV systems and commonly results in a higher flux level at the phase 0.5 dip than at phase zero. However, phase zero of the double-hump modulation cannot be simply identified from this flux difference due to possible changes caused by hotspots on the disk. Therefore, for each DN with a double-hump modulation discussed in this paper, two phased light curves corresponding to phase zero at the primary and secondary dips, respectively, were analyzed to search for a final convergent solution. Since the secondary dip in TW Vir cannot be accurately measured due to large scatter in the data, the primary maximum was set to be at the reference phases 0.75 and 0.25, guaranteeing that minima would occur around phases 0.5 and zero. For the pre-CV WD 1144+011, phase zero was set to be the minimum of the light curve since the light curve shows a single-hump modulation.

The four K2 light curves are expressed in Simple Aperture Photometry flux (i.e., electrons per second), while the light curves calculated by XRBinary are given in ergs per second. Hence, the K2 light curves were normalized before the calculations of XRBinary. Since the huge number of data points in the observed light curves (i.e., the number of data points in the K2 light curves observed in long cadence (LC; 30 minutes sampling) and in short cadence (SC; 1 minutes sampling) are more than 10³ and 10⁵, respectively) can take a long calculation time with XRBinary, the K2 phased light curves are moderately binned with a uniform phase resolution 0.01. Dai et al. (2017) demonstrated that the orbital modulations of the three CVs are stable, and the large-amplitude dispersion of ∼0.43 mag in the phased light curve of RZ Leo (Figure 5 of Dai et al. 2016) is only caused by a uniform drift of the system light. Thus, while the phased light curve of TW Vir has the highest stability (Dai et al. 2017), all four binned and normalized K2 light curves represent their orbital modulations.

Although the XRBinary program was initially designed to calculate the light curves of low-mass X-ray binary stars (LMXBs) (Gomez et al. 2015), it can also be used to model low-inclination CVs based on the following two reasons. First, XRBinary sets the primary star as a much smaller sphere than all other dimensions in the binary system and is unresolved by any of the grids used in calculating the light curves. Like the neutron star in LMXBs, the white dwarf in both CVs and pre-CVs is very small compared with the companion red dwarf. Second, XRBinary simply assumes that the primary star only emits black body radiation. This is a reasonable assumption for low-inclination CVs and pre-CVs because the flux contribution from the visible white dwarf is approximately constant. XRBinary may not be appropriate to accurately model complicated eclipse light curves but it should be able to reproduce low-inclination CV light curves. In fact, XRBinary is a powerful tool to analyze the accretion disk around the central compact star by constructing a complex accretion disk model consisting of a disk, a disk rim, a disk torus and an inner disk. Ratti et al. (2013) have successfully applied XRBinary to reproduce an ellipsoidal light curve of the CV CXOGBS J174444.7-260330 in a low state.

In this paper, a standard CV model (i.e., a semi-detached close binary system with a Roche-lobe filling red dwarf and, in the case of the 3 DN, an accretion disk around the white dwarf) with a set of geometric and physical parameters is applied to model the phased light curves of four low-inclination systems. Based on a complete set of preset parameters, XRBinary calculates a theoretical light curve and a relative measurement of goodness of fit, χ² (i.e., the variance between the calculated and observed light curves). Taking advantage of the improved Nelder–Mead method (i.e., Multi-directional Search method (MDS), Nelder & Mead 1965; Torczon 1989; Geng 2008), which was successfully applied to fit the O-Cs of AM Her (Dai et al. 2013) and UZ For (Dai et al. 2010b), we developed a new program called NMfit to carry out a search for the best binary model within a given parameter space. After obtaining the best binary model, NMfit sets a series of small deviations around each of the parameters to test the variations in χ² caused by these deviations. Note that when an adjustable parameter is intentionally set to deviate from its optimal value, then the other parameters are fixed. Moreover, this optimization method ignores degeneracies between parameters, which is a big issue. Since the best binary model means the minimal χ², the deviations give rise to an increase of χ². When χ² increases to 5% larger than the minimal value, this tested parameter deviation is arbitrarily regarded as the uncertainty. All of the adjustable parameters are tested one-by-one to estimate their errors. The error estimates based on χ² are only lower limits to the true uncertainties in the derived parameters and the true uncertainties could be quite large because the physics of compact binaries is much more complicated and XRBinary arbitrarily makes many tacit assumptions (e.g., the disk is modeled as an object with sharp, well-defined edges and surfaces, and the disk surface elements emit and absorb like black bodies). The uniform design proposed by Fang (1980) is used to decide the initial parameter set before the search iterates. A good initial parameter set is crucial for the success of the iterations. In addition, since Phoebe¹³ is able to calculate and plot an accretion disk around a compact star (Prša et al. 2016), it was used to visualize the system configuration based on the best-fitting parameters derived by NMfit and XRBinary.

The 23 model parameters that were used to generate the light curves are listed in Table 1, which are divided up into seven fixed parameters and 16 adjustable parameters. Our assumptions for these parameters are as follows.

• 1.  
  Although the input and output light curves of XRBinary are normalized, the typical luminosity of the white dwarf can provide an important reference for the system luminosity of the output CV model. Since the mass range for isolated white dwarfs is 0.3 M_⊙ to 1.4 M_⊙ and over 30% of white dwarfs are centered on ∼0.56 M_⊙ (Berg et al. 1992; Provencal et al. 1998), the white dwarf masses of all four objects were searched in this mass range. By using the relation ${L}_{\mathrm{wd}}=4\pi {R}_{\mathrm{wd}}^{2}\,\sigma {T}_{\mathrm{wd}}^{4}$, the white dwarf radius R_wd is solely dependent on the white dwarf luminosity L_wd and temperature T_wd. According to the white dwarf mass–radius relationship, as shown in Figure 1 (e.g., Wood & Horne 1990; Provencal et al. 1998), L_wd can only be determined by T_wd for a given M_wd. During the iterations of NMfit, T_wd calculated from the adjustable parameter L_wd is preset to be a fixed parameter.
• 2.  
  To improve the reliability of the model, a simple disk with minimal free parameters consisting of up to two hotspots (one at the vertical side of the edge of the disk (es) and the other one on the surface of the disk (ss)) is used to describe the low-inclination CV light curves. This CV quiescent disk was assumed to normally extend to the surface of the white dwarf (i.e., R_in = R_wd). A boundary layer was not considered because it dominates the flux at ultraviolet wavelengths rather than at optical wavelengths. A hotspot on the vertical side of the edge of the disk can be completely described by three parameters: a uniform temperature T_es, the centering phase ζ_esmid and the full width ζ_eswidth. Considering that the other hotspot on the disk surface can be visible and modulate the light curves of low-inclination CVs, this hotspot is described using five parameters: ζ_ssmin, ζ_ssmax, R_ssmin, R_ssmax and T_ratio. The first four parameters are the boundaries of the hotspot, which are the angles and radii over which the hotspot extends, respectively. The last parameter T_ratio is a fractional change in T_disk (i.e., T_ratio = T_ss/T_disk).
• 3.  
  A different temperature distribution may exist in different quiescent accretion disks. The temperature distribution is assumed to be a power law in disk radius R_disk; i.e., ${T}_{\mathrm{disk}}\propto {R}_{\mathrm{disk}}^{\xi }$. For a steady-state disk, typically ξ = −0.75 (Wade & Hubeny 1998). However, subsequent papers suggested that accretion disks in CVs have much flatter temperature profiles with ξ > −0.75 (e.g., Marsh 1999; Orosz & Wade 2003). According to the disk instability model (Osaki & Kato 2013a, 2013b, 2014), the quiescent DN disk is an optically thin and cool disk with a flat radial temperature profile. Observations of various quiescent DN (e.g., Z Cha, Wood & Crawford 1986; OY Car, Wood et al. 1989; V2051 Oph, Rutkowski et al. 2016) confirm that their disk temperature distributions are much flatter than the prediction of ${T}_{\mathrm{disk}}\,\propto {R}_{\mathrm{disk}}^{-0.75}$. We initially assume ξ = −0.15 but allow this value to be adjusted by NMfit.
• 4.  
  The adjustable parameter L_d0 indicating the disk luminosity is only calculated from the temperature distribution of the disk. This parameter should satisfy a default condition that the temperature at the inner edge of disk (i.e., T_in) cannot be much larger than 10⁴ K because the average temperature of observed quiescent disks is below 10⁴ K, which is consistent with the typical temperature range of an accretion disk (from several 10³ to 10⁴ K) predicted by the limit-cycle oscillation between hot and cold states (Lasota 2001).
• 5.  
  Since both theories and observations suggest that the accretion disk around a white dwarf is geometrically thin (e.g., Pringle 1991; Frank et al. 1992; Wade & Hubeny 1998; Marsh 1999), the height of the disk, H_disk, is assumed to obey a power low with a power index H_pow = 1.1 for all four objects, but H_edge is set to be adjustable.
• 6.  
  Since the components of the model calculated in XRBinary (e.g., two component stars, the disk and hotspots on the disk) are independent, we can freely construct the necessary components to model the observed light curves. Three types of models, as listed in Table 2, were attempted to verify the necessity of a disk around the white dwarf and the hotspots on the disk. Model-0 is a detached binary model without a disk. Model-1 contains an accretion disk with a hotspot at the edge of the disk. Model-2 adds a second hotspot on the disk surface. The relative flux contributions in percentage from the different model components to the synthetic light curves calculated by XRBinary are shown in Figure 2. Their zero points are listed in Table 3.

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Table 1.  All Parameters Used to Calculate Light Curves by Using XRBinary

Parameters	Values	Statements
Fixed
STAR2TILES	5000	tiles covering star surface
DISKTILES	20,000	tiles covering disk surface
BANDPASS (Å)	4200 ∼ 9000	bandpass of K2 light curve
Hpow	1.1	power index of Hdisk
Lwd	⋯	white dwarf luminosity
Twd	⋯	white dwarf temperature
Rina	⋯	inner radius of disk
Adjustable
Components
qorb	⋯	mass ratio
iorb	⋯	orbital inclination
Mwd	⋯	white dwarf mass
Trd	⋯	red dwarf temperature
Rout	⋯	outer radius of disk
Hedge	⋯	Hdisk at the outer edge
ξ	⋯	power index of Tdisk
Ld0	⋯	disk luminosity
Hotspot at the edge of disk
Tes	⋯	temperature
ζesmid	⋯	centering phase
ζeswidth	⋯	full width
Hotspot on the surface of disk
ζssmin	⋯	lower limit of boundary in ζ direction
ζssmax	⋯	upper limit of boundary in ζ direction
Rssmin	⋯	lower limit of boundary in radius direction
Rssmax	⋯	upper limit of boundary in radius direction
${T}_{\mathrm{ratio}}^{\mathrm{ss}}$	⋯	fractional change in Tdisk

Note.

^aR_in can be adjustable for IPs due to the disrupted inner disk.

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The irradiation effect is only calculated by XRBinary when setting the parameter IRRADIATION to ON. Since a luminous secondary may weaken the illumination from the vicinity of the white dwarf and the disk, the parameter IRRADIATION is preset to OFF if the summation of the luminosity of the white dwarf and the disk (i.e., L_wd+L_d) is smaller than the luminosity of the secondary L_rd. Although XRBinary can calculate the heating effect due to the irradiation, the output parameter T_rd of XRBinary is simply derived from L_rd by using the formula ${L}_{\mathrm{rd}}\sim 4\pi {R}_{\mathrm{rd}}^{2}\,\sigma \,{T}_{\mathrm{rd}}^{4}$. This means that T_rd represents the temperature of the secondary averaged over its entire surface, which can be significantly higher than the true night side temperature of the secondary if irradiation in the systems is significant. Here, the night side is the surface of the secondary facing away from the WD and disk, and the day side is the surface facing the WD and disk.

Davey & Smith (1992) found that the size of the irradiation region is usually as large as the entire day side surface of the secondary star. The numerical work by Kirbiyik (1982) suggested that the difference in the effective temperature between the day and night sides of the secondary is very large. Considering that the typical difference may reach close to 10⁴ K (Warner 2003), irradiation would be easily observed if this tremendous disparity of the effective temperature exists. Therefore, an estimation of the size of the irradiation region on the surface of the secondary can help to understand the flux contribution from the irradiation region in both CV and pre-CV systems. The effective temperature of the secondary on the night side can be specified by the semi-empirical CV donor sequence (Knigge 2006; Knigge et al. 2011). The derived T_rd can be regarded as a lower limit to the effective temperature of the irradiation region.

Due to orbital rotation, our viewing angle of the irradiated region can change with the orbital phase. Assuming that the irradiation region on the leading side of the star¹⁴ is axisymmetric with the line between the center points of two component stars, the flux ratio should reach a maximum between phases 0.5 and 0.0; i.e., F_irr = f_0.5/f_0.0, where f_0.5 and f_0.0 are the irradiation flux at phases 0.5 and 0.0, respectively. The details of the calculations of F_irr are described in Appendix A. By investigating the variations in F_irr along with a normalized area of the irradiation region (i.e., S_irr = A_irr/A_star, where A_irr and A_star are the areas of the irradiation region and the whole star, respectively), a relation between F_irr and S_irr based on the physical model derived by XRBinary and NMfit can be estimated and used to compare with the observed flux ratio F_obs between phases 0.5 and 0.0.

Since the K2 light curves of the four binary systems are from the broad bandpass listed in Table 1, their K2 magnitudes (hereafter Kp₂) translated by Dai et al. (2016) are commonly different from their apparent visual magnitudes. Assuming this magnitude difference to be a systematic error, the calculated magnitudes of the four systems (hereafter CKp₂) based on the luminosities of the models can be set to be a reference parameter for the comparisons with the results from the Gaia mission (Gaia Collaboration et al. 2016). Table 4 lists the distances of all four systems based on the absolute stellar parallax in the Gaia database (Gaia Collaboration et al. 2018).¹⁵ CKp₂ can be estimated from the Gaia distance D_g derived from the Gaia parallax (Luri et al. 2018) and the system luminosity L_all, which is summed for all model components and corresponds to the zero point of the normalized K2 light curves, by using the following formula,

$$\begin{eqnarray}&&{\mathrm{CKp}}_{2}=2.5\,\mathrm{log}\left[\displaystyle \frac{{L}_{\mathrm{bol}0}}{{L}_{\mathrm{all}}}\right]\,-\,{{BC}}_{v}\,+\,5\,\mathrm{log}\left[\displaystyle \frac{{D}_{g}}{10\,\mathrm{pc}}\right],\end{eqnarray} \tag{ 1 }$$

where L_bol0 = 3.0128 × 10³⁵ erg s⁻¹,¹⁶ and BC_v is a bolometric correction of the main sequence star in the typical Johnson-Cousins V band with the solar log g, [Fe/H] and [α/Fe]. Note that BC_v corresponding to the derived T_rd can cause an uncertainty of CKp₂ for CV and pre-CV systems because BC_v is improperly used for the white dwarf and disk. This calculated uncertainty may explain the discrepancy between CKp₂ and Kp₂.

Considering that BC_v is a model-dependent quantity with many observational constraints, several numerous tabulations are provided in the literature. All three popular BC_v tables, respectively, proposed by Flower (1996), Bessell et al. (1998) and Casagrande & VandenBerg (2014) were used to calculate CKp₂ to minimize our errors. The relations of BC_v and T_eff plotted in Figure 3 clearly show that the three BC_v tabulations significantly differ for cool stars with T_eff ≤ 4000 K. Inspection of Figure 3 indicates that BC_v of RZ Leo, TW Vir and WD 1144+011 are obviously different for each table. Consequently, the difference between the three BC_v tabulations for the same T_eff is regarded as the uncertainty of BC_v. Compared with the two early BC_v tables of Flower (1996) and Bessell et al. (1998), the updated BC_v given by Casagrande & VandenBerg (2014) is moderate for cool stars. Therefore, they were used to calculate CKp₂ for all four systems to compare with the K2 magnitudes at the zero point of the normalized K2 light curves (Dai et al. 2016). The interpolated BC_v and the errors used to calculate CKp₂ for all four systems are listed in Table 4.

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7.1. J0632+2536

This is a poorly studied DN that experienced several DN outbursts in 2009 and 2012 (Korotkiy & Sokolovsky 2012; Masi 2012; Ohshima 2012). Dai et al. (2016) showed its double-hump light curve extracted from the K2 data archive and several quiescent spectra with strong double peaked Balmer emission lines obtained from the Large Binocular Telescope (LBT). Based on these spectra and the phased K2 light curve, the secondary of J0632+2536 is thought to be a K5V star and the orbital inclination cannot be lower than 50°. According to the MK spectral classes (Cox 2000), the initial temperature and mass of the secondary are set to be 4410 K and 0.67 M_⊙, respectively. Several high-precision measurements of white dwarf masses in CVs (Knigge 2006; Knigge et al. 2011; Zorotovic et al. 2011) have indicated that the mean CV white dwarf mass is usually larger than that of isolated white dwarfs (Berg et al. 1992; Kepler et al. 2007). Thus, for J0632+2536, which lacks an accurate measurement of the white dwarf mass, a mean CV white dwarf mass of 0.83 M_⊙ was preset to be the initial parameter (i.e., M_wd = 0.83 M_⊙). Finally, Urban & Sion (2006) used IUE spectra of 53 quiescent DN to show that the temperature of a white dwarf, T_wd, in a CV with an orbital period above the period gap is approximately 25,793 K (Sion 1999; Urban & Sion 2006). We assume this value for the temperature of the white dwarf in J0632+2536 because its orbital period of 0.314478 day (Dai et al. 2016) is above the gap.

A large number of trials calculated using XRBinary and NMfit suggest that a convergent solution using a normal CV configuration cannot be achieved when setting phase zero to the primary dip of the light curve. Hence, phase zero was set for the secondary dip. Model-1 was used to model the phased light curve of J0632+2536. At first, four parameters (M_wd, T_wd, mass ratio q_orb and T_rd) were fixed and the other nine parameters consisting of the orbital inclination i_orb and eight parameters of the accretion disk were set to be adjustable. By using NMfit, a preliminary accretion disk model indicated that the inner radius of the disk is basically close to the white dwarf radius, so we then fixed the parameters ξ and R_in to derive the uncertainties of the other 11 parameters. Since ξ is an insensitive parameter, its uncertainty cannot be obtained. By using Phoebe 2.0, a 2D CV configuration at phase 0.75 shown in Figure 2 visually indicates that J0632+2536 has a large and thick accretion disk with a small hotspot at the edge of the disk. Since the orbital inclination of J0632+2536 is not high enough for the white dwarf to be occulted by the secondary, the flux contribution from the white dwarf is constant and does not vary with the orbital period, so this constant flux contribution is simply added to the ellipsoidal modulations caused by the secondary. Compared with the small white dwarf, the top panel of Figure 2 clearly shows that the large and thick accretion disk can be partially eclipsed around phase zero. Moreover, the hotspot at the edge of the disk, which has a temperature of 6200 (±100) K and has a phase width of 0.033(2) at phase 0.844(2), only contributes a maximum of 5% of the flux from the whole system. Since CKp₂ of J0632+2536 listed in Table 4 is almost equal to the corresponding Kp₂, the obtained physical model of J0632+2536 is compatible with the Gaia distance. However, the T_eff of J0632+2536 shown in the Gaia database is obviously higher than that derived from the LBT spectra (Dai et al. 2016) and the derived T_rd listed in Table 5.

Table 5.  Photometric Solutions for Three DN

Parametera	J0632+2536	RZ Leo	TW Vir
Model typeb	Model-1	Model-2	Model-2irr
Orbit
qorb(Mrd/Mwd)	0.8(1)	0.080(6)	0.41(5)
iorb (degree)	60.3(5)	61.0(9)	44.3(5)
White dwarf
Mwd(M⊙)	0.81(8)	1.26(3)	1.10(3)
Rwd(R⊙)c	0.01	0.005	0.007
Twdd	25.4	15.0	19.1
Lwdc	1.5	0.045	0.23
Red dwarf
Mrd(M⊙)	0.7(1)	0.101(8)	0.45(6)
Rrd(R⊙)c	0.81	0.16	0.48
Trd	4.54(8)	3.71(3)	4.00(4)
Lrdc	9.6	0.17	2.0
Accretion disk
Rin(R⊙)	0.01d	0.211(2)	0.007d
Rout(R⊙)	1.22(1)	0.401(2)	0.536(3)
Hedge(R⊙)	0.091(4)	0.0801(6)	0.1174(3)
ξ	−0.11e	−0.12(6)	−0.11(4)
Ld0	3.9(3)	0.00061(6)	3.36(6)
Ldc	4.1	0.023	4.0
Hotspot at the edge of the disk
Tes	6.2(1)	3.71(2)	4.80(2)
ζesmid (phase)	0.844(2)	0.723(1)	0.727(6)
ζeswidth (phase)	0.033(2)	0.057(4)	0.296(3)
Hotspot on the surface of disk
ζssmin (phase)	⋯	0.38(2)	0.32(5)
ζssmax (phase)	⋯	0.56(2)	0.74(3)
Rssmin(R⊙)	⋯	0.211e	0.489(8)
Rssmax(R⊙)	⋯	0.217(5)	0.536e
${T}_{\mathrm{ratio}}^{\mathrm{ss}}$	⋯	7.2(2)	1.20(2)
χ2	11.9	2.3	0.66

Notes.

^aThe unit of temperature and luminosity is 10³ K and 10³² erg s⁻¹, respectively. ^bThe model with a superscript "irr" denotes that the irradiation effect is included in XRBinary. ^cCalculated by XRBinary. ^dFixed in NMfit program. ^eInsensitive to the observed light curves.

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The region where the mass transfer stream intersects the accretion disk (i.e., the hotspot at the edge of the disk) is simply assumed to be a rectangle with a uniform temperature T_es by XRBinary. A mass transfer rate ${\dot{M}}_{\mathrm{rd}}$ (hereafter, the parameters with the subscript rd related to the secondary star) can be roughly estimated by using the following formula,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{rd}}\,\simeq \,\displaystyle \frac{{L}_{\mathrm{acc}}\,{R}_{\mathrm{out}}}{G\,{M}_{\mathrm{wd}}},\end{eqnarray} \tag{ 2 }$$

where L_acc is the luminosity of the hotspot at the edge of the disk. The mass transfer rate can be estimated to be ${\dot{M}}_{\mathrm{rd}}\,\sim 2.5(\pm 0.2)\times {10}^{-10}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, corresponding to a mass loss timescale (i.e., ${\tau }_{\dot{M}}\sim {M}_{\mathrm{rd}}/{\dot{M}}_{\mathrm{rd}}$) of 2.8 × 10⁹ year. The top right-hand panel of Figure 4 indicates that M_rd = 0.7(1) M_⊙ is closer to the semi-empirical mass-period relation of Warner (2003) than that of Smith & Dhillon (1998). Furthermore, the secondary radius R_rd = 0.81 is consistent with the radius-period relations shown in the bottom right-hand panel of Figure 4. The thermal (or Kelvin–Helmholtz) timescale of the secondary (i.e., ${\tau }_{\mathrm{kh}}\sim {{GM}}_{\mathrm{rd}}^{2}/({L}_{\mathrm{rd}}{R}_{\mathrm{rd}})$) can be calculated to be 7.6 × 10⁷ year, as listed in Table 6, which is smaller than the derived ${\tau }_{\dot{M}}$. This means that the mass transfer via the L1 point is slow and the secondary is always able to maintain thermal equilibrium. Therefore, the secondary of J0632+2536 is almost indistinguishable from an isolated main sequence star of the same mass. The derived mass and radius shown in the bottom left-hand panel of Figure 4 confirm that the secondary of J0632+2536 is a typical K5V star (Dai et al. 2016). Inspection of the top panel of Figure 2 indicates that the ellipsoidal modulations of this K5 star dominate the observed double-hump modulations. Although the top left-hand panel of Figure 4 indicates that T_rd = 4540(80) K is nearly consistent with a normal K5V star based on the MK spectral classes (Cox 2000), this derived T_rd is around 350 K higher than the prediction of the semi-empirical CV donor sequence (Knigge 2006; Knigge et al. 2011). The most likely explanation for this discrepancy may be due to irradiation of the K5V star. Since the irradiation effect is not calculated by XRBinary as L_rd > L_wd + L_d, we investigated its strength in more detail. Based on our best-fitting model, the maximal F_irr is around 1.08, as shown in Figure 5. As long as 20% of the irradiation region can be occulted by the disk at phase 0.5, the observed flux ratio of J0632+2536 (i.e., F_obs ∼ 0.87) can be explained. Due to our derived large disk and moderate orbital inclination, a partial eclipse of the irradiation region at phase 0.5 can be expected.

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Table 6.  The Estimated Parameters of the Secondaries of Four Systems

Namea	${\dot{M}}_{\mathrm{rd}}$	${\tau }_{\dot{M}}$	τkh	${\tau }_{\dot{M}}/{\tau }_{\mathrm{kh}}$
	${10}^{-10}\,{M}_{\odot }$ yr−1	109 year	109 year
J0632+2536	2.5(±0.2)	2.8(±0.5)	0.076	37
RZ Leo	0.0311(±0.0008)	32.4(±2.7)	0.45	72
TW Vir	1.92(±0.05)	2.3(±0.3)	0.25	9
WD 1144+011b	67.8(±16.0)	0.1(±0.03)	0.006	18
WD 1144+011c	7.9(±1.8)	1.0(±0.3)	0.027	37

Notes.

^aCalculations by using the parameters listed in Tables 5 and 7. ^bFor the model-1. ^cFor the model-2.

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7.2. RZ Leo

Dai et al. (2016) confirmed that RZ Leo is a short-period DN with an orbital period of 0.07603 day, similar to that derived by Patterson et al. (2003) and Kato et al. (2009). Ishioka et al. (2001) and Mennickent et al. (1999) indicated that the secondary of RZ Leo should be a normal red dwarf, possibly a main sequence M0 star. Although this spectral type cannot be supported by the mass ratio of 0.14 derived from its superhump excess of 0.033 (Ishioka et al. 2001), we speculated that RZ Leo should consist of a massive white dwarf and a normal red dwarf with a small mass of <0.2 M_⊙. Thus, we assumed that the initial M_rd used in NMfit is 0.15 M_⊙, which means M_wd = 1.1 M_⊙ and T_rd ∼ 3500 K. Szkody et al. (2017) and Dai et al. (2016) detected a short white dwarf spin period of ∼220 s classifying RZ Leo as a member of the IP subtype of CVs. From an Ultraviolet (UV) study, Pala et al. (2017) determined that the white dwarf had a temperature of 15,014(±638) K and contributed 83% of the UV flux. Since the humps of the model-1 light curve are located exactly at phases 0.25/0.75, model-2 with an extra hotspot on the surface of disk can be used to describe the offset secondary hump of RZ Leo. For the phased light curve with phase zero at the primary minimum, a convergent solution was found. However, inspection of the bottom left-hand panel of Figure 4 shows that the derived secondary mass and radius obviously deviate from the predicted mass–radius relation of the semi-empirical CV donor sequence (Knigge 2006; Knigge et al. 2011). Based on the MK spectral classes (Cox 2000), the derived radius is significantly smaller than that of a normal isolated main sequence star. Furthermore, the mass and radius also conflict with the semi-empirical mass-period and radius-period relations of CV secondary (Smith & Dhillon 1998; Warner 2003) shown in the two right-hand panels of Figure 4.

Instead, we created a model with phase zero at the secondary dip of the light curve. Initially, we fixed the inner accretion disk radius to the radius of the white dwarf. However, a convergent solution could not be found. Instead, an accretion disk model with two hotspots and R_in much larger than R_wd was found to fit the data reasonable well. R_in > R_wd can be interpreted as a truncation of the accretion disk far from the WD, consistent with the IP classification of RZ Leo by Szkody et al. (2017) and Dai et al. (2016). This means that the physical structure of RZ Leo may be more complex than the simple disk model calculated by XRBinary because accretion curtains near the magnetic poles of the white dwarf may be involved. This may be the reason that the best-fitting light curve derived by XRBinary cannot perfectly describe the secondary hump of RZ Leo. Based on this disk model, CKp₂ calculated by using Equation (1) is around 1.5 mag larger than Kp₂. This discrepancy is likely to be caused by the lack of an accretion curtain in the disk model. In the future, a complete IP model including the light from an accretion curtain should help correct this difference. In spite of the deviation in the secondary hump and CKp₂, limits to two key parameters of the disk (R_in and ξ) can be obtained from the double-humped K2 phased light curve.

According to the typical disk-field interaction models (e.g., Pringle & Rees 1972; Ghosh & Lamb 1978; White & Stella 1988), the corotation radius of a magnetic white dwarf, R_co, can be calculated using,

$$\begin{eqnarray}&&{R}_{\mathrm{co}}={f}_{\mathrm{co}}{M}_{\mathrm{wd}}^{1/3}{P}_{\mathrm{rot}}^{2/3},\end{eqnarray} \tag{ 3 }$$

where ${f}_{\mathrm{co}}={({{GM}}_{\odot }/4{\pi }^{2}{R}_{\odot }^{3})}^{1/3}$, R_co and M_wd are in solar units, P_rot is the spin period of the magnetic white dwarf in unit of seconds. By using the parameters listed in Table 5 and a spin period of 220 s (Dai et al. 2016; Szkody et al. 2017), R_co of RZ Leo can be estimated to be 0.085 R_⊙, which is 17 times larger than R_wd = 0.005 R_⊙. Considering that the accretion process in RZ Leo is steady (Dai et al. 2017), R_in is required to be almost equal to R_co. However, R_in = 0.211(2) R_⊙ is much larger than R_co. In a typical IP (Ghosh & Lamb 1978), R_co is regarded as the inner radius of an unperturbed disk and the transition region of a truncated disk (i.e., located between R_co and R_in) may be actually invisible in the optical band due to a possible disruption of the accretion flow caused by the magnetosphere of the magnetic white dwarf. Since a hotspot on the disk surface always shows R_ssmin < R_in during the iterations of NMfit, the parameter of R_ssmin is fixed to be equal to R_in. The high effective temperature of this hotspot indicated by T_ratio = 7.2 indicates that more than 97% of the luminosity of the disk is from this slim and extended hotspot. Moreover, this hotspot is regarded to be a plausible second impact region of an inward and overflowing stream striking onto the magnetosphere of the magnetic white dwarf. This region was previously reported in several IP systems (e.g., EX Hya in outburst, Hellier et al. 1989; and QZ Vir in quiescence, Shafter & Szkody 1984). Note that all three systems are for an unusually short period (under the gap) IPs. Compared to this hotspot, the other hotspot at the edge of the disk is much smaller and cooler. Both edge and surface hotspots are located at the phases 0.723(1) and 0.47(2), respectively. This disk configuration derived in quiescence is consistent with the disk of EX Hya during outburst, with two hotspots at similar phases (Hellier et al. 1989). This disk configuration may be common for IP systems. A low-luminosity truncated disk with ${L}_{d}\,=2.3\times {10}^{30}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (i.e., the luminosity contribution of the disk is less than 10% of the whole CV system) may be a straightforward conclusion for an IP system like RZ Leo due to the lack of a hot inner part of the disk. Compared with the other four sources of light in RZ Leo (i.e., the two component stars and two hotspots on the disk), the relative flux contributions from this truncated disk shown in Figure 2 are almost negligible (close to zero).

The best-fitting CV model indicates that RZ Leo has a moderate orbital inclination of 610(±09) and contains a massive white dwarf and a low mass red dwarf. The mass ratio, q_orb = 0.080(6) is within the error bar of that estimated from its superhump excess (Ishioka et al. 2001). The secondary mass and radius (i.e., M_rd = 0.101(8) M_⊙ and R_rd = 0.16 R_⊙) are not only in accord with an isolated red dwarf later than M5V but also support the semi-empirical mass-period and radius-period relations of CV secondaries (Smith & Dhillon 1998; Warner 2003). The small mass transfer rate, ${\dot{M}}_{\mathrm{rd}}\simeq 3.01(\pm 0.07)\times {10}^{-12}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ calculated by using Equation (2), seems to explain the small and cool hotspot at the edge of the disk that is shown in the disk model. Recently, Dubus et al. (2018) showed the average mass transfer rates of a sample of ∼130 CVs with a parallax distance in the Gaia DR2 catalog. The estimated mass transfer rate of RZ Leo lies well within the region where a stable, cool disk exists within the system. This explains a lack of DN outbursts for RZ Leo. ${\tau }_{\dot{M}}\sim 3.4\times {10}^{10}$ year calculated from the mass transfer rate is larger than τ_kh of the secondary. The derived secondary mass and radius further confirm that the secondary of RZ Leo is in thermal equilibrium, since they are consistent with the mass–radius relation of the isolated main sequence stars shown in the bottom left-hand panel of Figure 4. Since T_rd = 3710(±30) K suggests a slightly earlier spectral type than M5V, this higher T_rd may be attributed to irradiation of the companion. Due to L_rd > L_wd + L_d, the irradiation effect is not calculated by XRBinary. However, the maximal F_irr ∼ 1.54 that is shown in Figure 5 indicates a significant irradiation effect in RZ Leo. A prominent irradiation effect can be expected in an IP system because the white dwarf primary is not embedded in the truncated thin disk. Furthermore, the small disk of RZ Leo implies that the irradiation region is visible at phase 0.5 despite the similar orbital inclination to that of J0632+2536. Although F_obs ∼ 0.93 does not seem to support a large irradiation effect, the presence of an accretion curtain and a brighter second impact region complicate an accurate calculation of the irradiation effect in RZ Leo.

7.3. TW Vir

7.4. WD 1144+011

This poorly understood variable star is classified as a DA+dMe binary from a single optical spectrum obtained by Berg et al. (1992). For comparison with this spectrum obtained over 20 years ago, a new spectrum was taken on 2017 January 22 by using the BFOSC spectrograph attached to the Xinglong Observatory 2.16 m telescope (XL 216, Fan et al. 2016). This is a better optical spectrum that shows some changes from the original. The details concerning our spectrum are presented in Appendix B. A synthetic analysis based on the single-hump light curve extracted by Dai et al. (2016) can test the possibility of the existence of a disk around the primary white dwarf of WD 1144+011. Considering that WD 1144+011 is a long orbital period system with an orbital period of 9.81 hr (Dai et al. 2016), the initial parameters of the white dwarf are set to be the same as those of J0632+2536. Combined with the parameters of a M dwarf, all three models were used to attempt to reproduce the stable single-hump modulation that is apparent in the K2 light curve.

7.4.1. Model Without a Disk

In principle, model-0 can only produce ellipsoidal modulation caused by the orbital motion of the Roche-lobe filling secondary. The asymmetry shown in the single-hump modulation of WD 1144+011 cannot be explained by a pure ellipsoidal modulation derived from model-0. Moreover, the irradiation effect in WD 1144+011 that was calculated by XRBinary cannot resolve this problem because its long orbital period may imply a bright secondary like J0632+2536 (i.e., L_rd > L_wd). In spite of this, an irradiation region on the surface of the secondary was added to allow for an asymmetrical single-hump modulation (i.e., a bump on the rise to the maximum). We assumed that this irradiation region is a circular bright starspot on the secondary, which can be easily calculated by XRBinary. Since the single-hump modulation of WD 1144+011 observed in K2-C1 is very stable (lasting at least a complete campaign periods of ∼3 months, Dai et al. 2016), a stable region due to irradiation of the secondary may be more plausible to explain this long-term steady modulation than a variable starspot on an active M type dwarf.

In XRBinary, a starspot on the secondary can be described by the four parameters (i.e., θ^sp, ϕ^sp, R^sp and ${T}_{\mathrm{ratio}}^{\mathrm{sp}}$)¹⁷ . For WD 1144+011, this starspot is assumed to be a hotspot (i.e., ${T}_{\mathrm{ratio}}^{\mathrm{sp}}\gt$ 1). All of the parameters including four component parameters (M_wd, T_wd, q_orb and T_rd) and four parameters of the starspot were set to be adjustable in NMfit. The derived M_wd = 1.11(8)M_⊙ is almost the same as that of TW Vir shown in Figure 1, while T_wd = 27,100 K suggests a hotter and brighter white dwarf. Compared with the three DN, WD 1144+011 is the only object with a slightly cooler M dwarf (T_rd = 3500(±100) K) than the predicted mass–temperature relation of the semi-empirical CV donor sequence (Knigge 2006; Knigge et al. 2011). This is consistent with T_eff = 3657.5 K shown in the Gaia database. Compared with the higher T_eff of the two DN J0632+2536 and TW Vir, this compatible T_eff can be explained by the lack of a disk around the white dwarf. Since the companion star in WD 1144+011 is cooler than the secondaries in the other systems discussed here, it has the largest uncertainty in BC_v, as shown in Figure 3. This is reflected in the large range of derived CKp₂ (14.8 ∼ 16.2 mag), which is consistent with the corresponding Kp₂ value listed in Table 4. Like TW Vir, the ellipsoidal modulations caused by the M star never dominates the orbital modulation. The mass and radius of the secondary deviate from all the mass–radius, mass-period and radius-period relations shown in Figure 4. Note that both semi-empirical mass-period and radius-period relations are only available in the period range of 1.3 ∼ 9 hr (Smith & Dhillon 1998; Warner 2003). With an orbital period of 9.81 hr, WD 1144+011 is beyond this period range. The bottom left-hand panel of Figure 4 shows that the inflation of the secondary is about 38%. Besides the known inefficient mechanisms of donor bloating (e.g., tidal deformation, rotational deformation and irradiation of the secondary) discussed by Knigge et al. (2011), the deviation from thermal equilibrium of the secondary caused by the donor mass loss can result in a large donor inflation of up to 20% ∼ 30% (Patterson 2005; Knigge 2006). However, this mass loss mechanism is still not enough to explain the derived inflated secondary star.

Although both light curves calculated with and without irradiation by XRBinary are almost identical due to L_rd > L_wd, the final χ² of model-0^irr is slightly smaller than that of model-0. Thus, the best-fitting parameters and their uncertainties listed in Table 7 are calculated using model-0^irr. The reproduced light curves and the corresponding 2D pre-CV configuration are shown in the top left-hand and right-hand panels of Figure 6, respectively. This detached binary configuration with a low inclination of 143(±04) supports the previous classification of WD 1144+011 as a pre-CV system (i.e., a DA+dMe detached system). Our model also shows that the orbital variation seen in the light curve is dominated by the hotspot on the secondary's surface. Compared with the three DN, the pre-CV WD 1144+011 is the only object with F_obs = 1.03 larger than 1. Thus, ${T}_{\mathrm{ratio}}^{\mathrm{sp}}=1.097(\pm 0.003)$ combined with F_obs suggests that the calculated irradiation region may be a good representation of the physical picture of WD 1144+011. The effective temperature of the starspot can be estimated to be around 3900 K based on ${T}_{\mathrm{ratio}}^{\mathrm{sp}}$. A relation between F_irr and S_irr shown in Figure 5 indicates weak irradiation in WD 1144+011, similar to J0632+2536, and S_irr ∼ 0.5 corresponding to F_obs = 1.03 is consistent with the size of the starspot calculated from R^sp = 277(±07). Due to the lack of an accretion disk, this irradiation region is much smaller than the typical size of an irradiation region in a CV (Davey & Smith 1992).

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Table 7.  Three Photometric Solutions for the Pre-CV WD 1144+011

Parametersa	WD 1144+011
Model typeb	Model-0irr	Model-1	Model-2
Orbit
qorb(Mrd/Mwd)	0.49(2)	0.99(2)	0.92(9)
iorb (degree)	14.3(4)	17.7(4)	26.3(6)
White dwarf
Mwd(M⊙)	1.11(8)	0.68(16)	0.9(2)
Rwd(R⊙)c	0.007	0.012	0.009
Twdd	27.1	27.0	25.7
Lwdc	0.91	2.5	1.3
Red dwarf
Mrd(M⊙)	0.54(4)	0.67(16)	0.8(2)
Rrd(R⊙)c	0.87	0.97	1.03
Trd	3.5(1)	7.44(7)	5.23(6)
Lrdc	4.1	100.7	27.7
Bright starspot on the red dwarf
θsp (degree)	38(3)	⋯	⋯
ϕsp (degree)	0.2e	⋯	⋯
Rsp (degree)	27.7(7)	⋯	⋯
${T}_{\mathrm{ratio}}^{\mathrm{sp}}$	1.097(3)	⋯	⋯
Accretion disk
Rin(R⊙)	⋯	0.01d	0.009d
Rout(R⊙)	⋯	0.597(5)	0.808(8)
Hedge(R⊙)	⋯	0.363(8)	0.176(3)
ξ	⋯	−0.11e	−0.18(4)
Ld0	⋯	13.7(±2.7)	13.5(±2.3)
Ldc	⋯	23.0	19.4
Hotspot at the edge of the disk
Tes	⋯	6.50(5)	5.38(7)
ζesmid (phase)	⋯	0.623(4)	0.548(8)
ζeswidth (phase)	⋯	0.41(2)	0.17(1)
Hotspot on the surface of disk
ζssmin (phase)	⋯	⋯	0.129(6)
ζssmax (phase)	⋯	⋯	0.221(2)
Rssmin(R⊙)	⋯	⋯	0.605(1)
Rssmax(R⊙)	⋯	⋯	0.808e
${T}_{\mathrm{ratio}}^{\mathrm{ss}}$	⋯	⋯	1.90(4)
χ2	8.6	7.9	8.8

Note. All footnotes are the same as those of Table 5.

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7.4.2. Model With a Disk

Since an extra light source is imperative to explain the asymmetrical single-hump modulation, model-1 and model-2 were investigated. All of the adjustable parameters and their uncertainties are listed in Table 7. ${\dot{M}}_{\mathrm{rd}}$ is estimated for both models and is given in Table 6. The white dwarf mass of model-1 is smaller than the prediction of the average white dwarf mass in a CV with the same orbital period (Ritter & Burkert 1986; Zorotovic et al. 2011). T_wd from both of the disk models is consistent with T_wd from the non-disk model, suggesting that WD 1144+011 does contain a hot white dwarf. The final χ² of the two disk models are almost the same as that of the non-disk model-0. All three models imply an oversized secondary with generally consistent masses and radii. The top left-hand panel of Figure 4 shows that the secondary of model-1 has an extremely high effective temperature, which is over twice the average effective temperature of an M dwarf. Although the secondary of model-2 agrees with a G9V star based on the MK spectral classes (Cox 2000), a spectral type of G9V contradicts with our spectrum. Compared with the Gaia distance of WD 1144+011 listed in Table 4, both L_rd for the model-1 and model-2 are too large. Hence, the consistency of the effective temperature may be only coincidental. To recheck the deviations in T_rd in model-1 and model-2, T_rd was fixed to be 3500 K in line with the secondary temperature found using model-0. However, no convergent solution could be obtained. In spite of L_rd > L_wd + L_d, irradiation was still included in these models. Like model-0, the light curves calculated with and without irradiation for the two disk models only show small discrepancies around phases 0.5 and 0.0. Both calculated F_irr are far larger than F_obs. This means that the assumed large irradiation of the secondary is not supported by the K2 data. Compared with the pre-CV configuration obtained from model-0, the two models with an accretion disk are not convincing. As such, our modeling based on the K2 phased light curve provides further evidence for the classification of WD 1144+011 as a detached pre-CV system.

8.1. Synthetic Codes

8.2. Physical Models

8.3. The Secondaries

This work was partly supported by CAS Light of West China Program, the Chinese Natural Science Foundation (Nos. 11133007 and 11325315), and the Science Foundation of Yunnan Province (No. 2016FB007). P.S. acknowledges support from NSF grant AST-1514737. M.R.K is funded through a Newton International Fellowship provided by the Royal Society. We acknowledge the support of the staff of the Xinglong 2.16 m telescope. This work was partially supported by the Open Project Program of the Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences. We would like to thank the anonymous referee for their comments and suggestions in improving this paper.

Software: IRAF (Tody 1986, 1993), XRBinary (v2.4), NMfit (v1.0), Phoebe (v2.0, Prša et al. 2016).

Assuming that the secondary is a sphere and the irradiation region with a higher effective temperature of T_irr is axisymmetric with the line between the center points of the two component stars, the area of the complete irradiation region S_irr can be calculated by the following formula,

$$\begin{eqnarray}&&{A}_{\mathrm{irr}}=2\pi {R}_{\mathrm{rd}}^{2}(1-\cos \theta ),\end{eqnarray} \tag{ 4 }$$

where R_rd and θ are the radius of the secondary and the half opening angle of the irradiation region shown in Figure 7, respectively. Therefore, θ should be in a range of 0 ∼ π/2. Due to the projection effect (i.e., the orbital inclination less than 90°), the orbital modulation of the secondary will cause the fraction of the irradiation region seen by an observer to vary over the orbital period. The total flux at an orbital phase ϕ can be described as

$$\begin{eqnarray}&&{f}_{\phi }={A}_{\mathrm{irr}}^{{\prime} }(\phi )\sigma \,{T}_{\mathrm{irr}}^{4}+[2\pi {R}_{\mathrm{rd}}^{2}\,-\,{A}_{\mathrm{irr}}^{{\prime} }(\phi )]\,\sigma \,{T}_{\mathrm{star}}^{4},\end{eqnarray} \tag{ 5 }$$

where ${A}_{\mathrm{irr}}^{{\prime} }(\phi )$ and T_star are the visible area of the irradiation region at the phase ϕ and the effective temperature of the unirradiated part of the secondary, respectively. Note that Equation (5) never considers the occultation of irradiation region by the white dwarf or the disk. In principle, ${A}_{\mathrm{irr}}^{{\prime} }(0.0)$ is the minimum, while ${A}_{\mathrm{irr}}^{{\prime} }(0.5)$ is the maximum. An inspection of Figure 7 indicates that the former can be estimated by the formula,

$$\begin{eqnarray}&&{A}_{\mathrm{irr}}^{{\prime} }(0.0)=\left\{\begin{array}{l}2{R}_{\mathrm{rd}}^{2}{\displaystyle \int }_{i}^{\theta }\,{\theta }^{{\prime} }\cdot \sin \psi \,d\psi \quad \quad \theta \geqslant i\\ 0\qquad \qquad \qquad \qquad \qquad \theta \lt i\end{array}\right.\end{eqnarray} \tag{ 6 }$$

where i and ψ are the orbital inclination and the second coordinate of the spherical coordinate system, respectively. Moreover, θ' satisfies the following relation,

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{\mathrm{rd}}\sin \psi \cos {\theta }^{{\prime} }}{{R}_{\mathrm{rd}}\sin i}=\displaystyle \frac{{R}_{\mathrm{rd}}\cos \psi }{{R}_{\mathrm{rd}}\cos i},\end{eqnarray} \tag{ 7 }$$

By combining Equations (6) and (7), ${A}_{\mathrm{irr}}^{{\prime} }(0.0)$ can be expressed as,

$$\begin{eqnarray}&&{A}_{\mathrm{irr}}^{{\prime} }(0.0)=\left\{\begin{array}{l}2{R}_{\mathrm{rd}}^{2}{C}_{\mathrm{irr}}\quad \quad \theta \geqslant i\\ 0\qquad \qquad \theta \lt i\end{array}\right.\end{eqnarray} \tag{ 8 }$$

where ${C}_{\mathrm{irr}}={\int }_{i}^{\theta }\arccos (\tfrac{\tan i}{\tan \psi })\cdot \sin \psi \,d\psi$. Based on ${A}_{\mathrm{irr}}^{{\prime} }(0.0)$, ${A}_{\mathrm{irr}}^{{\prime} }$(0.5) can be expressed as,

$$\begin{eqnarray}&&{A}_{\mathrm{irr}}^{{\prime} }(0.5)=\left\{\begin{array}{l}{A}_{\mathrm{irr}}-{A}_{\mathrm{irr}}^{{\prime} }(0.0)\qquad \,\theta \geqslant i\\ {A}_{\mathrm{irr}}\qquad \qquad \qquad \quad \theta \lt i\end{array}\right.\end{eqnarray} \tag{ 9 }$$

By using Equation (5), the flux ratio between the phases 0.5 and 0.0 (i.e., F_irr = f_0.5/f_0.0) can be calculated for a given θ. Based on the normalized area of the irradiation region, S_irr = A_irr/A_star (i.e., cos θ = 1 − 2S_irr), F_irr can be described as,

$$\begin{eqnarray}&&{F}_{\mathrm{irr}}\\ &&\,=\,\left\{\begin{array}{l}\displaystyle \frac{(\pi -2\pi {S}_{\mathrm{irr}}-{C}_{\mathrm{irr}}){T}_{\mathrm{star}}^{4}+(2\pi {S}_{\mathrm{irr}}-{C}_{\mathrm{irr}}){T}_{\mathrm{irr}}^{4}}{(\pi -{C}_{\mathrm{irr}}){T}_{\mathrm{star}}^{4}+{C}_{\mathrm{irr}}\,{T}_{\mathrm{irr}}^{4}}\quad \theta \geqslant i\\ (1-2{S}_{\mathrm{irr}}){\left[\displaystyle \frac{{T}_{\mathrm{irr}}}{{T}_{\mathrm{star}}}\right]}^{4}-2(1+{S}_{\mathrm{irr}})\quad \theta \lt i\end{array}\right.\end{eqnarray} \tag{ 10 }$$

The relations of F_irr versus S_irr for the three DN and one pre-CV system discussed here are plotted in Figure 5.

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A 1.8 arcsec slit and the G4 grating with a low resolution of 1953 (i.e., 2.97 Å pixel⁻¹) was used with an exposure time of 1800 s. Flux standards and FeAr lamps were used along with IRAF¹⁹ reductions to produce the final calibrated spectrum shown in Figure 8.

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This spectrum is dominated by a strong and fairly broad H_α emission line and several molecular absorption bands (e.g., MgH5211 Å and CaH6358 Å), and is similar to the previous spectrum obtained by Berg et al. (1992). Thus, our spectrum supports the previous spectral classification of WD 1144+011 derived by (Berg et al. 1992). An active M star can show variable H_α in emission. However, our spectrum shows a flatter blue continuum with a higher red flux level. The measured average equivalent width of H_α is close to 100 Å, although this line is difficult to measure accurately due to the uncertainty of the continuum. It is clear from both our spectrum and that of Berg et al. (1992) that there is no prominent blue continuum from a hot white dwarf or disk and that the Balmer lines are in emission and of variable strength. The changes in flux levels between our spectrum and the spectrum presented in Berg et al. (1992) could be due to phase differences when the two spectra were taken.