Long-term Period Changes and Brightness Variations for the Deeply Eclipsing Cataclysmic Variable SW Sex

The observed eclipse minima of SW Sex have been presented previously (e.g., Penning et al. 1984; Ashoka et al. 1994; Dhillon et al. 1997; Groot et al. 2001; Boyd 2012). Based on the published data (Penning et al. 1984), Ashoka et al. (1994) deduced that the orbital period increases in the rate of 2.2 × 10⁻¹¹ s s⁻¹. Given that there are only a few eclipse minima and most of them cluster, the investigation of longer-term observations is urgently required. Coupled with all archive mid-eclipse times, collected from the literature and the O − C Gateway database,⁸ there are 190 eclipse minima in total within the time span of about 40 yr. In order to eliminate the Earth's motion and light-travel time effects, times for all the observations have been converted to Barycentric Julian Day (BJD) in Barycentric Dynamical Time (Eastman et al. 2010). These minima can be used to derive a more accurate linear ephemeris

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{Min}.{\rm{I}}(\mathrm{BJD}\_\mathrm{TDB}) & = & 2444339.650996(183)\\ & & +0.134938472(2)\times E.\end{array}\end{eqnarray} \tag{ 1 }$$

We assigned different weights to these eclipse times during the fitting process because of their different uncertainties, where the weights were inversely proportional to the rms errors. The residues of the linear fit (denoted as O − C) are plotted in the upper panel of Figure 5, which shows a possible cyclic oscillation. Orbital period oscillation is often attributed to the mechanism of Applegate (1992) or/and the presence of a third body orbiting the binary. The model involving the third body urges at least a brown dwarf along in a highly eccentric orbit (∼0.8) and indicates the presence of a forth body, which complicates the system and is beyond the prediction of the data (overfitting), and thus is not reliable. Considering the secular light-curve variation, we find that the O − C values fall more sharply after 95,000 runs synchronize with reaching a higher state (see Figure 2), so the O − C data is divided into two parts by the boundary (95,000 runs). The first part is modeled by an cyclic oscillation

$$\begin{eqnarray}&&{(O-C)}_{1}={\rm{\Delta }}{T}_{0}+{\rm{K}}\,\sin (\omega \times E+\varphi ),\end{eqnarray} \tag{ 2 }$$

and the other one by a quadratic function

$$\begin{eqnarray}&&{(O-C)}_{2}={\rm{\Delta }}{T}_{1}+{\rm{\Delta }}{P}_{0}\times E+\beta {E}^{2},\end{eqnarray} \tag{ 3 }$$

where β is the coefficient of the quadratic term. As shown in the upper panel of Figure 5, the dark solid line denotes the best fitting of the oscillation, and the red solid line represents the quadratic fit. The residuals are shown in the bottom panel; there is no other variation that can be traced suggesting that these equations could model the O − C data enough. The fitting parameters are listed in Table 2.

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Table 2.  Parameters of the Orbital Period Changes in SW Sex

Parameters	Sine Fitting: (O − C)1
Revised epoch, ${\rm{\Delta }}{T}_{0}$ (days)	7.04(±2.67) $\times \,{10}^{-5}$
Semiamplitude, K(days)	0.000973 (±0.000031)
Orbital period, P3 = $\tfrac{2\pi }{\omega }\times P$ (yr)	36.57 (±1.16)
Orbital phase, $\varphi (\deg )$	−203.66 (±3.69)
Parameters	Quadratic fitting: (O − C)2
Revised epoch, ${\rm{\Delta }}{T}_{1}$ (days)	−0.27342(±0.02888)
Revised period, ${\rm{\Delta }}{P}_{0}$	5.60(±0.56) $\times \,{10}^{-6}$
Rate of period decrease, 2β (days/cycle)	−5.72(±0.53) $\times \,{10}^{-11}$

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SW Sex has been observed by DASCH (Digital Access to a Sky Century at Harvard). DASCH is a project to digitize and analyze the collection of  500,000 glass photographic plates held at the Harvard College Observatory. The plates were exposed from 1880 to 1990 (Laycock et al. 2010). It covers a much longer historical period, albeit at a lower sensitivity; however, we are only able to provide the long-term photometric behavior. The data of the object from DASCH is shown in Figure 6 ranging from JD2415000–JD2447000, and the distribution of the data is sparse and random. Considering that the occultation interval, approximately 40 minutes, only occupies 20% of the whole orbital period, it is highly possible that most data corresponds to the out-of-eclipse magnitude of the system. We supplement the latest data with wide band V to JD2459000. Since AAVSO data are in V band, while DASCH magnitudes are in B band, their mean brightness is not at the same level. Given that most of the radiation comes from the accretion disk and the effective temperature of the disk is about 14,000(±6000) K (Rutten et al. 1992), we should subtract about 0.15(±0.09) mag in AAVSO V mag to the DASCH B mag (Cox 2000).

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As shown in Figure 6, the black dots represent the data from DASCH. The open circles are from the AAVSO database, and the light curves have been overlaid by the mean mag of the out-of-eclipse, which are donated by blue circles. We have subtract 0.15 mag in the plot without considering interstellar extinction E(B − V) = 0. The brightness variation of the system in the figure indicates that it wanders irregularly on a characteristic timescale of several years. DASCH data show two gradual increases and the AAVSO data show one. The DASCH data exhibit that the brightness quickly decayed, in less than 300 days, and then was eager to recover. The AAVSO data also shows that the brightness of the system increased, but after that, the luminosity had been stagnating for several years. These three enhancements show the same amplitude variations, about 0.6 mag, in general. We take 3535 days (about 9.7 yr) as a mag-changing cycle, which are split by the black dashed vertical lines shown in Figure 6, and the luminosity increase happens to occur within this time interval. If we boldly assume that the magnitude increases linearly and the rising time interval is about 9.7 yr, the brightness will decay and reach the peak magnitude again in the next decade, repeating the pattern that occurred before. However, the system exhibits the highest luminosity and currently shows stagnation. The newest rich data show that these oscillations saturate at a low amplitude. The luminosity is expected to be dropped rapidly in the near future. So it is necessary to be monitored continuously in the future to sum up the characteristics of the possible 10 yr period state changes and find out the mechanism behind it.

Based on the investigation of the (O − C) diagram, we found the orbital period of SW Sex undergoes a period oscillation with a timescale of about 36 yr. The period wiggle in binary systems is generally caused by Applegate's mechanism (Applegate 1992) or by the light-travel time effect due to the presence of a third companion (e.g., Li et al. 2017; Fang et al. 2019; Zhu et al. 2019). According to the Applegate's mechanism, the period variations are caused by the magnetic activity of the secondary star, which drives the redistribution of angular momentum in the interior of the star. From the semiamplitude (K) and the modulation period (P_mod) of the orbital period oscillation, using the equation

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}P}{P}=2\pi \displaystyle \frac{K}{{P}_{\mathrm{osc}}},\end{eqnarray} \tag{ 4 }$$

the fractional period change Δ P/P was determined to be 4.57 × 10⁻⁷. Penning et al. (1984) derived the masses of the primary and the secondary stars as M₁ = 0.58 ± 0.20 M_⊙ and M₂ = 0.33 ± 0.06 M_⊙, respectively, and the orbital inclination i = 79° ± 1° from the dynamical analysis of the system, while Dhillon et al. (1997) doubted their method and constrained the component masses to M₁ ∼ 0.3–0.7 M_⊙, M₂ < 0.3 M_⊙, and i > 75°. Assuming the typical mass of the primary as ${M}_{1}\,=0.6{M}_{\odot }$ and that of the secondary as ${M}_{1}=0.3{M}_{\odot }$, the radius of the secondary star is ${R}_{2}=0.39{R}_{\odot }$ (McAllister et al. 2019) and the binary separation is $a=1.06{R}_{\odot }$ from Kepler's law. According to

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}P}{P}=-9\displaystyle \frac{{\rm{\Delta }}Q}{{M}_{2}{a}^{2}},\end{eqnarray} \tag{ 5 }$$

we can calculate the variation of the quadrupole moment of the secondary star to be ${\rm{\Delta }}Q=1.67\times {10}^{47}\,{\rm{g}}\,{\mathrm{cm}}^{2}$. The secondary star may approach the full convective state so it can be modeled using the Lane–Emden equation for an n = 1.5 polytrope. We followed the same method of Brinkworth et al. (2006) to split the whole star into an inner core and an outer shell, and the required energy ${\rm{\Delta }}E$ to diver the variation of the quadrupole moment ${\rm{\Delta }}Q$ can be determined for a different range of the shell mass, as displayed in the right panel of Figure 7. The luminosity of the secondary star is ${L}_{2}=4\pi {R}_{2}^{2}\sigma {T}_{\mathrm{eff}}^{4}$, which, for ${T}_{\mathrm{eff}}\sim 3066\,{\rm{K}}$, gives the energy available over the 36.57 yr as $E\sim 5.28\times {10}^{40}\,\mathrm{erg}$. This value, indicated in the right panel of Figure 7, succeeds in supplying enough energy to modulate the orbital period. A plot of the ratio of the value of minimum energy required to drive the Applegate's mechanism over the energy supplied in 36.57 yr versus the mass of two components is shown in the left panel of Figure 7. Applegate's mechanism can be responsible for the period oscillation if the value of ${\mathrm{log}}_{10}$(ΔE_min/E) is greater than 0. The results indicate that Applegate's mechanism cannot be excluded to interpret the period variation.

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Another plausible explanation is the light-travel time effect of a third body. Assuming that the third body is coplanar with the central host systems (i.e., ${i}^{{\prime} }$ = i = 79°), the minimum mass of the third companion is estimated to be M₃sini' = 0.014 M_⊙ with a separation of 10.52 au, which indicates that it may be a giant planet. Parameters of the third body are shown in Table 3, the calculation is based on M₁ = 0.6 ± 0.2 M_⊙, M₂ = 0.3 ± 0.1 M_⊙.

Table 3.  Parameters for the Third Body of Light-travel Time Effects in SW Sex

Parameters	Values and Error
Eccentricity (${e}^{{\prime} }$)	0.0
Period (P3)	36.57 (±1.16) yr
Amplitude (K3)	0.000973 (±0.000031) day
${a}_{12}^{{\prime} }$ sin${i}^{{\prime} }$	0.17 (±0.01) au
f(m)	3.58(±0.32) × 10−6 M⊙
${M}_{3}\sin {i}^{{\prime} }$	0.014 (±0.002) M⊙
a3	10.52 (±3.97) au

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On the basis of the current data, both mechanisms can explain the variation of the orbital period and the time span of the observations is too short compared to the timescale of the period oscillation, so further monitoring of SW Sex is necessary.

As shown in Figure 5, the O − C curve falls sharply following 95,000 runs, which is beyond the prediction of the theoretical sine-wave (O − C)₁. Thus, a quadratic curve (O − C)₂ is applied to describe it. The binary shrinks its orbits due to an AML from the system. For a close star with the orbital period of 3.24 hr (≥3 hr), the orbital angular momentum of the system is taken away by the magnetic braking (MB) mainly due to the magnetized stellar wind (Verbunt & Zwaan 1981). Although gravitational radiation (GR) can also work at the same time in the system, it is relatively much smaller (Paczynski & Sienkiewicz 1981). Rappaport et al. (1983) proposed a formula to calculate the orbital period decrease due to MB,

$$\begin{eqnarray}\begin{array}{rcl}{\dot{P}}_{\mathrm{MB}} & = & -1.4\times {10}^{-12}(\displaystyle \frac{{M}_{\odot }}{{M}_{1}}){\left(\displaystyle \frac{{M}_{1}+{M}_{2}}{{M}_{\odot }}\right)}^{1/3}\\ & & \times {\left(\displaystyle \frac{{R}_{2}}{{R}_{\odot }}\right)}^{\gamma }{\left(\displaystyle \frac{d}{{P}_{\mathrm{orb}}}\right)}^{7/3}\,{\rm{s}}\,{{\rm{s}}}^{-1}.\end{array}\end{eqnarray} \tag{ 6 }$$

where γ is the MB index in a range from 0 to 4. The standard MB, γ = 4, predicts the orbital period decrease of $\dot{P}$_MB = −5.44 × 10⁻¹² s s⁻¹, it is about two orders of magnitude smaller than the period decay listed in Table 2. Even the highest decreasing $\dot{P}$_MB = −2.41 × 10⁻¹⁰ s s⁻¹ in the framework of the MB, γ = 0, fails to explain such an observed period decrease.

Given that the CV evolution is derived partially by AMLs via mass loss from the disk (King 1995) and SW Sex has a very high mass-accretion rate, we think the only factor leading to the fast period decrease is the wind emanating from the accretion disk. Comparison between Figure 2 and Figure 5 shows that the fast period decrease is accompanied by the luminosity maximum. When the system reaches the highest luminosity, the strong radiation induces a thick disk wind that can prevent the accretions onto the disk and takes away a lot of angular momentum from the system. The latest data of the brightness of the system shows a small amplitude oscillation (see Figure 2), and this phenomenon may also be evidence for the wind-driven instability in the accretion disk. Such a situation was originally put forward by Begelman et al. (1983) and Shields et al. (1986). To assess the role of the disk winds, they defined $\dot{{M}_{w}}$ and $\dot{{M}_{a}}$ as the wind mass-loss rate and mass-accretion rate onto an accretor, respectively. The ratio C = $\dot{{M}_{w}}$/$\dot{{M}_{a}}$ measures the efficiency of wind driving due to the accretion power and indicates how strongly the wind is coupled to the latter. For a large value of C, a small increase in $\dot{{M}_{a}}$ and hence luminosity can induce an increase in $\dot{{M}_{w}}$, which is large compared to $\dot{{M}_{a}}$. If the resultant $\dot{{M}_{a}}$ + $\dot{{M}_{w}}$ is larger than the mass input rate from the secondary star, the disk will be depleted. After a delay, both $\dot{{M}_{a}}$ and $\dot{{M}_{w}}$ will eventually be reduced. A reduction in $\dot{{M}_{w}}$ in turn will cause the disk to be replenished so that $\dot{{M}_{a}}$ and the luminosity will once again be increased. We now prove observational evidence of the wind, which takes away angular momentum in the form of mass loss from the disk's surface.