Because of the previously observed stability of the 171day period the superorbital modulation of the lowmass Xray binary 4U 182030 was considered a consequence of a third star orbiting around the binary. This study aims to further verify this triple model by testing the stability of superorbital period using the light curves collected by Xray sky monitoringscanning telescopes from 1987 to 2023. Both power spectral and phase analysis results indicate a significant change in the superorbital period from 171 days to 167 days over this 36year span. The evolution of the superorbital phase suggests that the superorbital period may have experienced an abrupt change between late 2000 and early 2023 or changed gradually with a period derivative of dot Psup3.58 pm 0.72 times 104 dayday. We conclude that the superorbital period of 4U 182030 was not as stable as anticipated by the triple model which strongly challenges this hypothesis. Instead we propose an irradiationinduced mass transfer instability scenario to explain the superorbital modulation of 4U 182030.
['Astrophysics - High Energy Astrophysical Phenomena']
The Puzzling Superorbital Period Variation of the Lowmass Xray Binary 4U 182030
{'The Puzzling Superorbital Period Variation of the Low-mass X-ray Binary 4U 1820-30': 'Yi Chou ( /uni5468 /uni7FCA ), 1 Jun-Lei Wu ( /uni5433/uni541B /uni78CA ), 1 Bo-Chun Chen ( /uni9673 /uni6CCA /uni931E ), 1 and Wei-Yun Chang ( /uni5F35 /uni744B /uni82B8 ) 1  \n1 Graduate Institute of Astronomy, National Central University 300 Jhongda Rd. Jhongli Dist. Tauyuan, 32001, Taiwan', 'ABSTRACT': 'Because of the previously observed stability of the 171-day period, the superorbital modulation of the low-mass X-ray binary 4U 1820-30 was considered a consequence of a third star orbiting around the binary. This study aims to further verify this triple model by testing the stability of superorbital period using the light curves collected by X-ray sky monitoring/scanning telescopes from 1987 to 2023. Both power spectral and phase analysis results indicate a significant change in the superorbital period from 171 days to 167 days over this 36-year span. The evolution of the superorbital phase suggests that the superorbital period may have experienced an abrupt change between late 2000 and early 2023 or changed gradually with a period derivative of ˙ P sup = ( -3 . 58 ± 0 . 72) × 10 -4 day/day. We conclude that the superorbital period of 4U 1820-30 was not as stable as anticipated by the triple model, which strongly challenges this hypothesis. Instead, we propose an irradiation-induced mass transfer instability scenario to explain the superorbital modulation of 4U 1820-30.', '1. INTRODUCTION': '4U 1820-30, discovered by Giacconi et al. (1974), is an ultra-compact low mass X-ray binary (LMXB) located near the center of globular cluster NGC 6624. It was the first X-ray source known to exhibit TypeI X-ray burst Grindlay et al. (1976), indicating that the accretor in this binary system is a neutron star. Its 685 s orbital period, discovered by Stella et al. (1987) from its sinusoidal-like orbital modulation in the X-ray light curve, makes 4U 1820-30 being the most compact LMXB. The mass-losing companion is a Roche-lobe filling helium white dwarf with a mass of 0.06-0.08 M /circledot (Rappaport et al. 1987). Mass transfer in the system is induced by the orbital angular momentum loss through gravitational radiation which should result in a positive orbital period derivative with a lower limit of ˙ P orb /P orb > 8 . 8 × 10 -8 yr -1 (Rappaport et al. 1987). However, observed orbital period derivatives reported by Tan et al. (1991); van der Klis et al. (1993a,b), Chou & Grindlay (2001) (hereafter CG01), and Peuten et al. (2014) were negative with the latest value of ˙ P orb /P orb = ( -5 . 21 ± 0 . 13) × 10 -8 yr -1 updated by Chou & Jhang (2023), evaluated from ∼ 46 years of orbital phase evolu-  \nCorresponding author: Yi Chou  \nyichou@astro.ncu.edu.tw  \ntion. This contradiction is believed due to the binary system accelerating by the gravitational potential in NGC 6624 (Tan et al. 1991; Chou & Grindlay 2001; Peuten et al. 2014; Chou & Jhang 2023). Additionally, superhump modulation with a period of 691 . 6 ± 0 . 7 s, ∼ 1% significantly longer than the orbital period, was observed in the both FUV (Wang & Chakrabarty 2010) and X-ray (Chou & Jhang 2023) bands. From the superhump period, the mass of companion of 4U 182030 is estimated as 0.07 M /circledot (Wang & Chakrabarty 2010; Chou & Jhang 2023).  \nIn addition to orbital and superhump variations, 4U 1820-30 exhibits superorbital modulation with a period much longer than the orbital period. Priedhorsky & Terrell (1984) discovered the Xray flux modulation by a factor of 2 with a period of 176 . 4 ± 1 . 3 days using the light curve detected by Vela 5B from 1969 to 1976. This periodicity was further confirmed by Smale & Lochner (1992). However, by analyzing the light curve collected between 1996 and 2000 by All Sky Monitor on-board Rossi X-ray Timing Explorer (RXTE ASM), CG01 revised the superorbital period to 171 . 39 ± 1 . 93 days. Combining the times of flux minima of the superorbital modulation (hereafter superorbital minima) detected by Vela 5B and All Sky Monitor onboard Ginga (Ginga ASM), CG01 further constrained the period to be 171 . 033 ± 0 . 326 days and claimed that the superorbital period was stable over ∼ 30 years with | ˙ P sup /P sup | < 2 . 2 × 10 -4 yr -1 . Based on the stabil-  \nthe superorbital period, GC01 proposed that this long-term variability is due to a hierarchical third star orbiting around the binary system (Grindlay 1986, 1988) (hereafter the triple model). The hierarchical third component can cause the eccentricity of inner binary system to vary with a period ( P ecc ) as  \nP ecc = K P 3 2 P orb (1)  \nwhere P 3 is the orbital period of third star, P orb is the binary orbital period and K is a constant of unity (Mazeh & Shaham 1979). Because the mass transfer rate is highly sensitive to the Roche lobe radius, which is proportional to the binary separation, the variation of binary eccentricity can cause the mass loss rate and the accretion rate to change with a period of P ecc and thus P sup = P ecc . For the 4U 1820-30 system, the orbital period of the third companion is estimated to be ∼ 1.1 days for K ∼ 1, and beat sidebands resulting from coupling binary modulation and ∼ 1.1 day periodicity may be observable in the power spectrum. Although these beat sidebands were not detected in RXTE observations (CG01), Chou & Jhang (2023) suggested that the 691 . 6 ± 0 . 7 s periodicity observed in the X-ray band might be caused by a hierarchical triple orbiting around the binary system with an orbital period of 0.8 days. Moreover, CG01 found that the active times of Type-I X-ray bursts were clustered within ± 23 days of expected superorbital minima, which aligns with the observation that the bursts can be seen only in low state (Clark et al. 1977; Stella et al. 1984). This fact implies that the superorbital modulation of 4U 1820-30 is due to changes in the accretion rate rather than external occultation or absorption effects, which is consistent with the triple model.  \nThe periodicity of 171 days was further confirmed by ˇ Simon (2003); Wen et al. (2006); Zdziarski et al. (2007a); Kotze & Charles (2012) using additional RXTE ASM data and by Farrell et al. (2009) using the data collected by Burst Alert Telescope onboard the Neil Gehrels Swift Observatory (Swift BAT). Applying the triple model, Zdziarski et al. (2007a) demonstrated that the factor of 2 superorbital modulation in X-ray light curve can be explained by the eccentricity of inner binary oscillating between 0 and 0.004. The discovery of the dependence of orbital modulation profile on the accretion rate Zdziarski et al. (2007b) also supports the triple model. The hard X-ray light curve collected from Swift BAT showed that the superorbital modulation can be observable only for the energy bands less than 24 keV (Farrell et al. 2009). Conversely, by comparing the peak widths of the power spectra made from light  \ncurves detected by RXTE ASM and Swift BAT with the corresponding simulated light curves, Farrell et al. (2009) found that the peak widths from real data are marginally wider than the ones from simulated data, concluding that this may be caused by the superorbital period change. Kotze & Charles (2012) adopted the dynamic power spectrum technique to analyze the superorbital variability of several X-ray binaries, and found no significant superorbital period change for 4U 1820-30 except for a weakening of power during MJD ∼ 5120052200.  \nOwing to the monitoring/scanning X-ray telescopes, 4U 1820-30 has been observed for decades and is still being monitored by the Swift BAT and the Monitor of All-sky X-ray Image (MAXI). In this work, we aim to further verify the stability of the superorbital period, which is the crucial evidence for the triple model of 4U 1820-30 system, and to establish an updated ephemeris for superorbital modulation. In this paper, we introduce the instruments used to obtain the light curves for this research, including Ginga ASM, RXTE ASM, Swift BAT and MAXI, as well as the light curve collected by RXTE Proportional Counter Array (RXTE PCA) while it processed the monitoring observations of the galactic center and plane (Markwardt 2006), in Section 2. A preliminary test of superorbital period stability test was performed using the power spectrum made by the entire light curve of each instrument (Section 3.1). A more detail measurement of superorbital period variation was obtained by analyzing the superorbital phase evolution and updating the ephemerides (Section 3.2). The new ephemerides allows us to verify whether the Type-I Xray bursts occur clustered around the expected superorbital minima. In Section 4, we discuss the instability of superorbital period, which poses a serious challenge of the triple model, and explore the possible interpretations for the superorbital period variation of 4U 1820-30.', '2. OBSERVATIONS': "The Ginga ASM consists of two identical gas proportional counters with six fan-beam collimators to restrict field of view (FOV) of 1 · × 45 · . It is sensitive to the X-ray photons with energies between 1 and 20 keV, and has a total effective area of 420 cm 2 . It provided realtime alerts of X-ray transient phenomena and long-term historical records of X-ray sources. The Ginga ASM monitored the sky from 1987 February to 1991 October. Further details of the Ginga ASM are described by Tsunemi et al. (1989). The Ginga ASM light curve of 4U 1820-30 collected from MJD 46861 to 48532 was analyzed in this study.  \nThe RXTE ASM (Levine et al. 1996) is an instrument mounted on RXTE to monitor the variable and the transient X-ray sources. It consists of three scanning shadow cameras, each containing a position-sensitive proportional counter, to observe the sky through a onedimensional coded mask with an FOV of 6 · × 90 · . It is designed to detect the cosmic X-rays in the photon energy range of 1.5-12 keV, which can be further divided into 1.5-3, 3-5 and 5-12 keV energy bands. In addition to these energy bands, the light curves with two different time resolutions, dwell (a 90 sec exposure) and one-day binned, were also archived. During its mission, from the beginning of 1996 to early 2012, the RXTE ASM scanned the entire sky every 90 minutes. In this research, the 1.5-12 keV RXTE ASM light curve collected between MJD 50088 and 55831 was selected to analyze the superorbital modulation of 4U 1820-30.  \nIn addition to the regular pointing observations, the RXTE PCA also conducted the monitoring observations of the galactic center and plane starting from 1999 (Markwardt 2006). The PCA is an instrument with an effective area of 6500 cm 2 designed to detected the X-ray photons in the energy range of 2-60 keV (Jahoda et al. 1996). Despite being a non-imaging instrument, its 1 · FOV, constrained by collimators, allowed for identification and detection of X-ray sources. It scanned over galactic bulge and plane approximately twice per week (Markwardt 2006), providing sufficient time resolution to resolve the superorbital modulation of 4U 1820-30. The light curve of 4U 1820-30 collected by PCA in this program from February 5, 1999 (MJD 51214) to October 30 2011 (MJD 55846) is available on the program website 1 .  \nThe BAT, an instrument on Swift, is a coded-mask telescope with a large FOV (1.4 steradian) to monitor the hard X-ray sky in the energy range 15-150 keV since 2004 (Barthelmy et al. 2005). Apart from triggering alerts for gamma-ray bursts, its angular resolution ( ∼ 20') and large photon collecting area (5200 cm 2 ) enable monitoring of the known cosmic X-ray sources as the Swift satellite orbits around the Earth every ∼ 96 minutes. This capability allows for the study of longterm variability these sources. In this work, we analyzed the daily binned light curve of 4U 1820-30 observed from February 14, 2005 (MJD 53415) through August 1, 2023 (MJD 60157).  \nThe MAXI, installed on the Japanese Experiment Module of International Space Station (ISS), is designed to alert the transient X-ray sources  \nFigure 1. Light curves collected by five instruments for analysis in this work. The bin size of these light curves is 10 days.  \n<!-- image -->  \nand monitor the long-term variations of the X-ray sources (Matsuoka et al. 2009). It contains two types of slit cameras with two different detectors: a gas proportional counter with an effective area of 5250 cm 2 for detecting the X-ray photons in the energy range of 230 keV, and a solid state camera of an effective area of 200 cm 2 sensitive to the X-ray photons in the energy range of 0.5-12 keV. MAXI can scan almost the entire sky twice during each ISS orbit ( ∼ 90 minutes). In this study, we analyzed the daily binned light curve of the energy range of 2-20 keV collected between 2009 August 12 (MJD 55055) and 2023 August 1 (MJD 60157), available on the MAXI website 2 , to study the superorbital modulation of 4U 1820-30.  \nThe light curves of 4U 1820-30 collected by these five instruments are shown in Figure 1.", '3.1. Power Spectral Analysis': 'In the power spectral analysis, all the light curves were rebinned into daily averages for consistency. All the data points with a signal-to-noise ratio less than 2 σ were filtered out for further analysis. To probe the superorbital periods of various observations, the Lomb-Scargle (LS) periodogram (Scargle 1982) was applied to generate the power spectra. The errors of signal frequencies were estimated by the method proposed by Horne & Baliunas (1986):  \nδf = 3 σ N 4 N 1 / 2 0 TA (2)  \nFigure 2. Power spectra derived from the light curves collected five instruments in this work. The vertical dashed line indicates the superorbital frequency of the CG01 ephemeris (f=2.136 cycle/year).  \n<!-- image -->  \nwhere A is the amplitude of the signal, σ 2 N is the variance of the noise after the signal being removed, T is the time span of the light curve and N 0 is the number of data points. A was evaluated by fitting a single sinusoidal function to the light curve with the frequency fixed at the signal frequency obtained from the power spectrum and σ 2 N was estimated by the root-mean-square (rms) of the noise after the best fitted sinusoidal function was removed from the light curve.  \nThe power spectra are depicted in Figure 2, and the detected superorbital periods are outlined in Table 1. It is apparent that the superorbital period deviates from the expected stability suggested by triple model, showingt a tendency to decrease over time. By incorporating the superorbital period reported by (Priedhorsky & Terrell 1984) from the Vela5B light curve, we estimated the timescale of the superorbital change by fitting a linear function to the detected superorbital periods over time (see Figure 3). This result in a period derivative of ˙ P sup /P sup = ( -7 . 37 ± 0 . 33) × 10 -4 yr -1 , corresponding to an evolution timescale of 1,357 years. This period derivative exceeds the upper limit proposed by CG10 ( | ˙ P sup /P sup | < 2 . 2 × 10 -4 yr -1 ). However, the linear fitting yielded a reduced χ 2 of 10.2, suggesting that the superorbital evolution of 4U 1820-30 is likely more complex than the constant period derivative model suggests. Further variations in the superorbital period from phase analysis will be demonstrated in Section 3.2.  \nAdditionally, to compare the amplitudes of superorbital modulation, we folded these five light curves using two kinds of linear ephemerides. The first one is the  \nFigure 3. Superorbital periods measured from 6 different instruments, including the one from Vela5B observation reported by Priedhorsky & Terrell (1984). The horizontal lines indicate the durations of the corresponding light curves, and the dashed line represents the best fit of a linear model to estimate the period change rate, which yields a period derivative of ˙ P/P = ( -7 . 37 ± 0 . 33) × 10 -4 yr -1  \n<!-- image -->  \noptimal ephemeris proposed by CG01 (hereafter CG01 ephemeris),  \nT N = JD 2 , 450 , 909 . 9 + 171 . 033 × N = MJD 50909 . 4 + 171 . 033 × N (3)  \nThe other one is the local ephemeris, with a folding period corresponding to the best period obtained by the power spectrum (see Table 1),along with an arbitrary phase zero epoch for each light curve. The rms amplitudes folded by both types of ephemerides are listed in Table 1. The rms amplitudes of the profiles folded by the corresponding local ephemerides are larger than the those folded by CG01 ephemeris, indicating that the CG01 ephemeris is no longer suitable. This shows that the superorbital period of 4U 1820-30 has undergone significant changes during 1987 to 2023.', '3.2. Superorbital Phase Evolution': 'In this research, we aimed to trace the long-term evolution of the superorbital phase of 4U 1820-30, necessitating the analysis of superorbital phases measured from different instruments. However, time lags between different energy bands are often observed in astronomical time series. For instance, soft phase lags of pulsed emissions are commonly noted in accreting millisecond X-ray pulsars (Cui et al. 1998; Patruno & Watts 2021). Hence,a coherence test was conducted to verify if there was a significant time lag between any of two light curves from different instruments. However, this test could be  \nTable 1. The Superorbital Period and RMS Amplitude Measured from the Light Curves Collected by Different InstrumentsTable 2. Coherence Test for RXTE ASM, RXTE PCA, Swift BAT and MAXI Light Curves  \nonly performed on the light curves with overlapping observation times. Among the five light curves we analyzed in this study, except for Ginga ASM, other 4 light curves had overlapping observation times for each other, resulting in 6 pairs of light curves for the coherence test. For each pair of light curves, only overlapping parts were selected for coherence test. The power spectra were obtained the superorbital periods for the corresponding light curves. The superorbital modulation profiles of both light curves were conducted by folding the mean period measured from the power spectra with an arbitrary but fixed phase zero epoch. We discovered that all the profiles could be well fitted with a four-component sinusoidal function, that is, r ( φ ) = a 0 + ∑ 4 k =1 [ a k cos (2 πkφ ) + b k sin (2 πkφ )]. To measure the possible time delay between the two instruments, we applied the cross-correlation between the best-fitted modulation profiles. The test results are shown in Table 2. The phase difference is generally no more than 0.026 cycle, which is much smaller the phase jitters ( ∼ 0.1 cycle, see CG01). We conclude that there is no significant time delay is observable among these 4 instruments for the superorbital modulation of 4U 182030.  \nTo trace the evolution of superorbital phase, we segmented the light curves and folded them to derive the modulation profiles. For instruments highly sensitive to superorbital modulation, like RXTE ASM and MAXI light curves, two cycles (2 × 171 days) per seg-  \nent sufficed to yield clear profiles. In the case of Swift BAT observations, where no superorbital modulation was detected for the photon energy larger than 24 keV (Farrell et al. 2009), we selected four cycles as a data segment to ensure significant profile detection. As for the RXTE PCA light curve, due to the observation gaps, we adopted four cycles per segment to create the profiles. However, only three data segments provided sufficient phase coverage for further analysis. Given the very low sensitivity of Ginga ASM, a clear profile could only be obtained by folding the entire light curve.  \nFollowing the approach of CG01, we selected the superorbital minimum as the fiducial point of the superorbital phase. Ideally , we would fold a light curve segment using a fix ephemeris, such as CG01 ephemeris (Eq. 3) to determine the phase (i.e. φ CG 01 ). However, as indicated in Section 3.1, the CG01 ephemeris is unlikely to be an optimal ephemeris for the all observations, particularly for recent ones (e.g. Swift BAT and MAXI observations), which could lead to profile deformation. To precisely determine the φ CG 01 , we folded the light curve segments using the optimal linear ephemeris specific to each instrument (local ephemeris). This involved folding the data by the period obtained from power spectral analysis (see Table 1) and an arbitrary but fixed phase zero epoch. A typical modulation profile is depicted in Figure 4. The phase ( φ local ) of a data segment was determined by fitting a four-component sinusoidal function and identifying the phase corresponding to the intensity minimum (fiducial point). This phase value, along with the local ephemeris, facilitated the evaluation the superorbital minimum time ( t m ) closest to the mid of observation time of the data segment. Subsequently, t m was then folded by CG01 ephemeris (Eq. 3) to obtain the phase φ CG 01 .  \nThe superorbital orbital phases ( φ CG 01 ) are listed in Table 3, and their evolution is illustrated in Figure 5. It is noteworthy that while the superorbital modulation displays strong periodicity in the power spectra, the modulation profile varies from cycle to cycle, exhibiting the quasi-periodic nature as described in Zdziarski et al. (2007a). This variability induces phase jitters of ∼ 0.1 cycle, evident in Figure 5 and CG01.  \nFigure 4. A typical superorbital modulation profile of a data segment created by folding the light curve collected by RXTE ASM from MJD 54158.64 to 54497.86 ( ∼ 2 cycles) with a folding period of 169.09 days from power spectral analysis (Section 3.1) and an arbitrary phase zero epoch. The solid line represents the optimal fit of a 4-component sinusoidal function to locate the superorbital minimum phase φ local . A BLS feature found by ˇ Simon (2003) can be also observed.  \n<!-- image -->  \nThese phase jitters are considerably larger than the error estimated from photon statistics ( ∼ 0.005 cycle). Despite the presence of phase jitters, a discernible phase evolution trend can be discerned in Figure 5. However, independently evaluating errors from phase jitters is difficult, which depend on the evolution model. In the subsequent analysis, we utilized the unweighted fitting method outlined by Press et al. (2002) to update the ephemeris for the superorbital modulation of 4U 1820-30.', '3.2.1. Linear Model': 'According to the triple model, The period should be remain stable from long-term perspective because the superorbital modulation is induced by a hierarchical third component orbiting around the binary system. Therefore, our initial approach involved fitting a linear function to the phase evolution as depicted in Figure 5. The parameters of the optimal linear function are listed in Table 4 yielding a period of 168 . 21 ± 0 . 15 days with a phase zero epoch of MJD 50920 ± 4 . 56. We assessed the root-mean deviation (RMSD), defined as:  \nRMSD ≡ √ √ √ √ 1 ν N ∑ i =1 [ φ i -φ ( t i ) ] 2 (4)  \nwhere φ i is the detected phase, φ ( t i ) is the expected phase value at t i evaluated from best fit model, and ν is the degree of freedom. The RMSD is 0.1 for the linear model. However best-fitted period in this model  \nTable 3. Superorbital Phase ( φ CG 01 ) of 4U 1820-30  \nFigure 5. Evolution of superorbital phases folded by the CG01 ephemeris from 1987 to 2023. The dotted, solid and dashed lines represent the best fits for linear, glitch and quadratic models, respectively. The shaded area indicates the low power state between MJD 50773 and 52627, and the vertical dash-dot line represents the glitch time MJD 52264 evaluated by the glitch model, with the horizontal error bar indicating the 1 σ uncertainty of glitch time.  \n<!-- image -->  \nsignificantly differs from the reported superorbital periods that detected in early RXTE ASM observations, as listed in Table 5, as well as the superorbital period of 176 . 4 ± 1 . 3 days reported by Priedhorsky & Terrell (1984) from Vela5B observation. Furthermore, the expected phase at the midpoint of Ginga ASM observation time (MJD 47677.56) is 0 . 382 ± 0 . 033, about 7.3 σ deviated from the detected value of 0.14 (see Figure 5). Therefore, the linear model is unlikely to describe the superorbital phase evolution of 4U 1820-30.', '3.2.2. Glitch Model': "Table 5 presents the reported superorbital periods detected by early RXTE ASM observations, which are roughly consistent with the period in the CG01 ephemeris (171 days). However, for later observations, particularly those from Swift BAT and MAXI, the period is approximately 167.4 days. One the possibility is that the superorbital period underwent an abrupt change (glitch), likely between years 2000 and 2005 (see Figure 5).  \nOnthe other hand, Kotze & Charles (2012) conducted a dynamic power spectrum analysis and observed weaker superorbital modulation power during the period MJD 51200-52200, hereafter, referred to as the low power state. Additionally, signals with shorter periods, ∼ 85 days (first harmonic) and ∼ 65 days emerged in the dynamic power spectrum, indicating a change in the modulation profile during that time interval. However, considering the window size used to generate the dynamic  \nLinear model  \nφ = a 0 + a 1 ( t - T 0 )  \na 0 = ( T 0 - T 0 ,CG 01 ) /P CG 01 a  \na  \n1  \n= (  \nP  \n0  \n-  \nP  \nCG  \n01  \n)  \n/  \n(  \nPP  \nCG  \n01  \n)  \nParameter  \nValue  \na 0  \n0 . 065 ± 0 . 027  \na 1 (cycle/day)  \n( - 9 . 83 ± 0 . 54) × 10 - 5  \ncov ( a 0 , a 1 ) (cycle/day)  \n- 1 . 21 × 10 - 7  \nT 0 (MJD)  \n50920 . 48 ± 4 . 65  \nP (days)  \n168 . 21 ± 0 . 15  \nGlitch model  \nφ  \n=  \n{  \na  \na  \n0  \n'  \n0  \n+  \na  \n+  \na  \n1  \n'  \n1  \n(  \nt  \n-  \nT  \n(  \nt  \n-  \nT  \n0  \n0  \n)  \nif  \nt  \n≤  \nT  \n)  \nif  \nt > T  \na 0 = ( T 0 -T 0 ,CG 01 ) /P CG 01  \na 1 = ( P 1 -P CG 01 ) / ( P 1 P CG 01 )  \na ' 0 = ( T 0 -T 0 ,CG 01 ) /P CG 01 + n g ( P 1 -P 2 ) /P 2  \na ' 1 = ( P 2 -P CG 01 ) / ( P 2 P CG 01 )  \nT g = T 0 + n g P 1  \nTable 4. Parameters of Superorbital Modulation of 4U 1820-30  \nQuadratic model  \nφ = a 0 + a 1 ( t - T 0 ) + a 2 ( t - T 0 ) 2  \na 0 = ( T 0 - T 0 ,CG 01 ) /P CG 01  \na 1 = ( P 0 - P CG 01 ) / ( P 0 P CG 01 )  \na 2 = 1 / 2 ˙ P/ ( P 0 P CG 01 )  \na T 0 ,CG 01 =JD 2,450,909.9=MJD50909.4 and P CG 01 =171.033 days from Eq.9 of GC01.  \ng  \ng  \n;  \n.  \nTable 5. Superorbital periods of 4U 1820-30 from Early RXTE ASM Observations a  \npower spectrum of 4U 1820-30 in Kotze & Charles (2012) (5 cycles), this low power state time interval should be extended to approximately MJD 50773 to 52627.  \nCompared with the the phase evolution (i.e. Figure 5), it appears likely that the glitch occurred around MJD 52500, near end of low power state. Marginal evidence supports this assumption. As listed in Table 5, the reported superorbital period by Zdziarski et al. (2007a) was 170 . 6 ± 0 . 3 days, slightly smaller than those reported by CG01, ˇ Simon (2003), and Wen et al. (2006). This discrepancy may be due to that significant portion of data ( ∼ 37%) analyzed by Zdziarski et al. (2007a) were collected after MJD 52500. Similarly, the power spectral analysis of the entire RXTE ASM light curve yielded a superorbital period of 169.09 days (see Table 1), falling between 171 and 167.4 days, because about half ( ∼ 56%) of the data were collected after MJD 52500. Conversely, the periods detected form the power spectra of Swift BAT and MAXI were nearly identical at approximately 167.4 days, because both observations were made after MJD 52500 (see Table 1).  \nWe therefore fitted the phase evolution with the glitch model using the ephemeris described by Eq.5 in (Wolff et al. 2009)  \nT N = { T 0 + P 1 N if N ≤ n g ; T 0 + P 1 n g + P 2 ( N -n g ) if N > n g . (5)  \nwhere T 0 is the phase zero epoch, P 1 and P 2 are the periods before and after glitch, respectively, n g is the glitch cycle count, and the glitch time T g ≡ T 0 + P 1 n g . The fitting results are shown in Figure 5, and the parameters are listed in Table 4. We obtained significantly different superorbital periods of 170 . 67 ± 0 . 64 days and 167 . 66 ± 0 . 18 days before and after the glitch time MJD 52264, respectively, and ∆ P sup /P sup = -0 . 018 ± 0 . 007.  \nFigure 6. Superorbital modulation profiles folded by glitch ephemeris listed in Table 4 for the light curves collected by (a) Ginga ASM, (b) RXTE ASM, before glitch, (c) RXTE PCA, before glitch, (d) RXTE ASM, after glitch, (e) RXTE PCA, after glitch, (f) Swift BAT, and (g) MAXI.  \n<!-- image -->  \nThe RMSD for this glitch model is 0.078. Comparing this with the RMSD of 0.1 from the linear model, the Ftest yielded a p-value of 0.04, indicating that the glitch model is better than the linear model. Figure 6 shows the modulation profiles folded by the glitch ephemeris. All the superorbital minima (fiducial points) are close phase zero, implying that this ephemeris effectively describes the superorbital phase evolution of 4U 1820-30 from 1987 to 2023.", '3.2.3. Quadratic Model': 'While the glitch model effectively describes the superorbital phase evolution, we cannot rule out the possibility that period difference between early RXTE ASM observations and recent ones stems from a smooth change in the superorbital period. Farrell et al. (2009) observed that the peak width of the superorbital signal in the power spectrum made from Swift BAT light curve was marginally wider than that from simulation, suggesting that a change in superorbital period. Hence, we apply a simple model assuming a constant period derivative ( ˙ P sup ) to fit a quadratic function to the superorbital phase evolution.The fitting results are depicted in Figure 5, and the parameters are listed in Table 4. A period derivative of ˙ P sup = ( -3 . 58 ± 0 . 72) × 10 -4 day/day, or ˙ P sup /P sup = ( -7 . 71 ± 1 . 54) × 10 -4 yr -1 was obtained from the fitting, and a quadratic ephemeris  \nT N = ( MJD 50914 . 41 ± 3 . 89) + (169 . 53 ± 0 . 29) × N +( -3 . 03 ± 0 . 61) × 10 -2 × N 2  \n(6)  \nFigure 7. Superorbital modulation profiles folded by quadratic ephemeris listed in Table 4 for the light curves collected by (a) Ginga ASM, (b) RXTE ASM, (c) RXTE PCA, (d), Swift BAT, and (e) MAXI.  \n<!-- image -->  \nwas established. This period derivative value is consistent with the one evaluated from power spectra in Section 3.1. The RMSD for the quadratic model is 0.083. Compared to this with the RMSD of 0.078 from glitch model, the F-test yielded a p-value of 0.34, which indicating that these two models are about equally adept at describing the superorbital phase evolution. Figure 7 illustrates the modulation profiles folded by the quadratic model. Similar to the glitch model, all the superorbital minima (fiducial points) are located around phase zero. It provides an evidence that the quadratic model is suitable for describing the superorbital phase evolution of 4U 1820-30.', '3.3. X-ray Burst Active Times': "As previously mentioned in Section 1, Type-I X-ray bursts of 4U 1820-30 are exclusively observable during the low state (Clark et al. 1977; Stella et al. 1984). Further confirmed by CG01, indicated the Type-I X-ray bursts were detected only within ± 23 days around superorbital minima for bursts reported before 1985. This supports the notation that the superorbital modulation stems from changes in the accretion rate changes rather than occultation effects. However, CG01's statistics only included four burst active dates. Subsequently, more X-ray bursts of 4U 1820-30 were detected. With the updated superorbital ephemerides, this evidence can be further substantiated. Although the possibility exists that the X-ray bursts occur in another low state, which may deviate significantly from the superorbital minima (e.g.the brief low state found by ˇ Simon (2003)), it is  \nlikely that most of X-ray burst active times would cluster around the superorbital minima.  \nIn this study, we collected the reported burst active dates of 4U 1820-30 from Grindlay et al. (1976); Vacca et al. (1986); Haberl et al. (1987); Zdziarski et al. (2007b); Garc'ıa et al. (2013), and (Yu et al. 2024), as well as four superbursts discovered by Strohmayer & Brown (2002); in't Zand et al. (2011); Serino et al. (2021a), and (Serino et al. 2021b). The deviations in burst active dates relative to the nearest superorbital minima predicted by CG01 and three ephemerides derived in this study are shown in Figure 8. It is evident that large deviations can be observed after the low power state (MJD 52630) when using the superorbital minima predicted by the CG01 ephemeris. The root-mean-square deviations are 41.2, 32.7 27.4 and 27.0 days for CG01, linear, glitch and quadratic ephemerides, respectively. This provides supportive evidence that glitch and quadratic ephemerides are better than CG01 and linear ephemerides in describing the superorbital phase evolution of 4U 1820-30. Notably, X-ray bursts occurring between MJD 50399 and MJD 50343 exhibited large deviations ( ∼ 60 days) for all ephemerides. Upon examining the RXTE ASM light curve, we observed that these X-ray bursts occurred around a BLS with a count rate of only 12.0 cts/s compared to the mean count rate of 21.0 cts/s. This observation reaffirms that superorbital modulation is primarily caused by accretion variations rather than occultation effects.", '3.4. X-ray Burst Active Times': "As previously mentioned in Section 1, Type-I X-ray bursts of 4U 1820-30 are exclusively observable during the low state (Clark et al. 1977; Stella et al. 1984). Further confirmed by CG01, indicated the Type-I X-ray bursts were detected only within ± 23 days around superorbital minima for bursts reported before 1985. This supports the notation that the superorbital modulation stems from changes in the accretion rate changes rather than occultation effects. However, CG01's statistics only included four burst active dates. Subsequently, more X-ray bursts of 4U 1820-30 were detected. With the updated superorbital ephemerides, this evidence can be further substantiated. Although the possibility exists that the X-ray bursts occur in another low state, which may deviate significantly from the superorbital minima (e.g.the brief low state found by ˇ Simon (2003)), it is likely that most of X-ray burst active times would cluster around the superorbital minima.  \nIn this study, we collected the reported burst active dates of 4U 1820-30 from Grindlay et al. (1976);  \nFigure 8. The deviations of Type-I X-ray burst active dates relative to the nearest expected superorbital minima evaluated by CG01, linear, glitch and quadratic ephemerides. The circles are the superbursts reported by Strohmayer & Brown (2002); in't Zand et al. (2011); Serino et al. (2021a,b). The region between two horizontal dashed lines is the ± 23 days burst active interval suggested by CG01, and the vertical dash-dot line in the plot of glitch ephemeris is the glitch time evaluated by the glitch model (Section 3.2.2).  \n<!-- image -->  \nVacca et al. (1986); Haberl et al. (1987); Zdziarski et al. (2007b); Garc'ıa et al. (2013), and (Yu et al. 2024), as well as four superbursts discovered by Strohmayer & Brown (2002); in't Zand et al. (2011); Serino et al. (2021a), and (Serino et al. 2021b). The deviations in burst active dates relative to the nearest superorbital minima predicted by CG01 and three ephemerides derived in this study are shown in Figure 8. It is evident that large deviations can be observed after the low power state (MJD 52630) when using the superorbital minima predicted by the CG01 ephemeris. The root-mean-square deviations are 41.2, 32.7 27.4 and 27.0 days for CG01, linear, glitch and quadratic ephemerides, respectively. This provides supportive evidence that glitch and quadratic ephemerides are better than CG01 and linear ephemerides in describing the superorbital phase evolution of 4U 1820-30. Notably, X-ray bursts occurring between MJD 50399 and MJD 50343 exhibited large deviations ( ∼ 60 days) for all ephemerides. Upon examining the RXTE ASM light curve, we observed that these X-ray bursts occurred around a BLS with a count rate of only 12.0 cts/s compared to the mean count rate of 21.0 cts/s. This observation reaffirms that superorbital modulation is primarily caused by accretion variations rather than occultation effects.  \nOn the contrary, out of the four superbursts detected, only the one on MJD 51430 (Strohmayer & Brown  \n2002) fell within ± 23 days region, whereas the other three occurred outside of this timeframe for both glitch and quadratic ephemerdes (see Figure 8). Upon examining the RXTE ASM and MAXI light curves, we observed that the count rates on the dates of superbursts were 57%, 150%, 121% and 138% of the corresponding mean count rates of the light curves for the superbursts detected on MJD 51430 (Strohmayer & Brown 2002), MJD 55272 (in't Zand et al. 2011), MJD 59449 (Serino et al. 2021a) and MJD 59543 (Serino et al. 2021b), respectively. It is likely that the low state constraint for the regular X-ray bursts of 4U 1820-30 (Clark et al. 1977; Stella et al. 1984) does not apply to the superbursts. More observations are required for further confirmation.", '4.1. Challenge of the Triple Model': "The initial aim of this study was to further validate the stability of superorbital period of 4U 1820-30, a crucial piece of evidence for the triple model as described in Section 1, which explains its superorbital modulation with a period of ∼ 170 days. Given that 4U 1820-30 resides in NGC 6624, a star-crowded region, previous studies suggested a high likelihood of the binary capturing a third star and forming a stable hierarchical triple system (Grindlay 1988). Black (1982) proposed a stability criteria for such a triple system as  \nµ ≤ µ crit = 0 . 175 ∆ 3 (2 -∆) 3 / 2 (7)  \nwhere µ = ( m 2 + m 3 ) / 2 m 1 , ∆ = 2( R -1) / ( R + 1), R = R 3 /R 1 , m 1 , m 2 and m 3 are the masses of binary primary, secondary and tertiary companion, respectively, R 1 is the binary separation and R 3 is the maximum separation of the binary primary and the tertiary companion. Applying Eq. 7 to 4U 1820-30 system with the assumption that m 1 = 1 . 4 M /circledot , m 2 = 0 . 07 M /circledot (Chou & Jhang 2023), m 3 = 0 . 5 M /circledot (CG01), binary period of 685 s and third star orbital period of 1.1 days 1, according to the Kepler's third law, we found that µ = 0 . 204 and µ crit = 24 . 18, which satisfies the stability criteria proposed by Black (1982). Additionally, CG01, based on their analysis of the RXTE ASM light curve collected between 1996 and early 2000 and in conjunction with fidicial points detected by Vela5B and Ginga, found no significant superorbital period derivative, setting an upper limit of | ˙ P/P | < 2 . 2 × 10 -4 yr -1 , thereby confirming its stability and lending support to the triple model. Therefore, for the 4U 1820-30 system, with additional subsequent observations after 2000, one would expect that the observed superorbital pe-  \nuld closely match the value found by CG01 (171 days) and that the period derivative could be further constrained.  \nHowever, upon analyzing X-ray light curves collected by the sky monitoring/scanning instruments from 1987 to 2023, we discovered a significant change in the superorbital period from 171 days to 167 days, identified through both power spectral analysis (Section 3.1) and phase analysis (Section 3.2) over a time span of ∼ 36 years. This suggests that the ephemeris proposed by CG01 is no longer suitable for describing the superorbital modulation of 4U 1820-30, and the period is not as stable as anticipated by triple model. By analyzing the superorbital phase evolution, we suggested that the superorbital period may have experienced an abrupt change during late 2000 to early 2003 ( T g =MJD 52264 ± 466) or may be constantly changing with a period derivative of ˙ P = ( -3 . 58 ± 0 . 72) × 10 -4 day/day.  \nThe significant difference between the period detected from Vela5B observation, 176 . 4 ± 1 . 3 days (Priedhorsky & Terrell 1984), and Ginga observation, 171 . 12 ± 1 . 99 days (see Table 1) suggests that the superorbital period may have experienced another glitch between 1976 and 1987. If 4U 1820-30 is a hierarchical triple system, from Eq. 1, a glitch in superorbital period may be induced by changes in either the binary orbital period or the third star orbital period as  \n∆ P sup P sup = 2 ∆ P 3 P 3 -∆ P orb P orb (8)  \nGlitches in orbital periods have been observed in some of total eclipsing LMXBs, like EXO 0748-676 (Wolff et al. 2009), XTE J1710-281 (Jain & Paul 2011; Jain et al. 2022) and AX J1745.6-2901 (Ponti et al. 2017). These glitches likely result from by magnetic, solar-type cycles of the companion star, affecting the mass distribution of companion and leading to variations in its quadruple moment (Wolff et al. 2009). However, the magnitudes of these glitches are typically in the order of milliseconds, with ∆ P orb /P orb ∼ 10 -7 -10 -6 . For 4U 182030 system, the superorbital period glitch was measured as ∆ P sup /P sup = 1 . 8 × 10 -2 . No glitch has ever been observed in the binary orbital phase evolution (see Figure 4 in Chou & Jhang 2023), implying that the orbital period glitch of the third companion was as high as ∆ P 3 /P 3 = 9 × 10 -3 , about 4 orders of magnitude larger than those from eclipsing LMXBs. However, the superorbital period change may not occur abruptly but within a finite short time interval. Suppose the timescale to be 900 days, estimated from the uncertainty of T g , the mean orbital period derivative of the third star would be as high as ˙ P 3 ≈ 2 × 10 -3 day/day. Thus, the triple  \nmodel is unlikely to explain this large superorbital period change in such a short time.  \nThe low power state is a particular phase during the superorbital modulation evolution of 4U 1820-30. The dynamic power spectrum demonstrated in Figure 24 of Kotze & Charles (2012) indicates that in addition to the weaker power detected in the superorbital period, the powers of its first harmonics and a signal of period ∼ 65 days became significant. This suggests that the modulation were more complicate than usual. Our phase analysis results also show that the superorbital phases had larger fluctuation during the low power state (see Figure 5) with an RMSD of 0.11, evaluated by the best glitch model fitting, compared to 0.075 for the phases outside low power state. It is probable that the superorbital period was 171 days before the low power state, became unstable during the low power state, and stabilized at 167 days later. However, it is unclear what the cause of this phenomenon is.  \nConversely, the superorbital phase evolution is also well-fitted by a quadratic model, although there is a significant difference (2.2 σ ) between the period evaluated as ephemeris extrapolated to the midpoint of Vela5B observation time (172 . 84 ± 0 . 94 days) and the observed period (176 . 4 ± 1 . 3 days) as reported by Priedhorsky & Terrell (1984). If 4U 1820-30 is a hierarchical triple system, from Eq. 1, the relation of period derivatives can be written as  \n˙ P sup P sup = 2 ˙ P 3 P 3 -˙ P orb P orb (9)  \nAlthough the exact binary orbital period derivative being unknown, it is believed it is ˙ P orb /P orb ∼ +10 -7 yr -1 (Chou & Jhang 2023). Therefore, the observed superorbital period derivative ˙ P sup /P sup = ( -7 . 71 ± 1 . 54) × 10 -4 yr -1 is contributed by the tertiary companion, with a value of ˙ P 3 /P 3 = -3 . 9 × 10 -4 yr -1 , corresponding to a variation timescale of only ∼ 2,600 years. From Eq. 7, we infer that the triple system should be stable; therefore, such a fast period change is unlikely to occur. Furthermore, the acceleration from the gravitational potential in NGC 6624, estimated as a c /c ∼ -10 -7 yr -1 (Peuten et al. 2014), is insufficient for the orbital period derivative of the third companion derived from Eq. 9. Thus, the triple model can hardly explain the observed superorbital period derivative.  \nIf 4U 1820-30 is not a triple system, the constraints regarding the observed value of the binary orbital period derivative may be relaxed. To explain the discrepancy between the positive theoretical value ( ˙ P orb /P orb > 8 . 8 × 10 -8 yr -1 , Rappaport et al. 1987) and the negative observed value ( ˙ P orb /P orb = ( -5 . 21 ± 0 . 13) × 10 -8 yr -1 ,  \nChou & Jhang 2023) of the binary orbital period derivative, it has been proposed that 4U 1820-30 is being accelerated by the gravitational potential within the globular cluster NGC 6624 (Tan et al. 1991; Peuten et al. 2014). However, Peuten et al. (2014) suggested that the maximum radial acceleration from the gravitational potential from NGC 6624 itself ( | a c,max /c | = 1 . 3 × 10 -9 yr -1 is an order of magnitude smaller than the value required to explain the observed binary period period derivative. Therefore, Peuten et al. (2014) proposed three possible scenarios to provide additional acceleration for 4U 182030, a flyby stellar mass dark remnant, a intermediatemass black hole at the center of NGC 6624, and a central concentration of dark remnants. Only the last scenario was preferred because the first two scenarios tend to destroy the triple system (Peuten et al. 2014). However, if 4U 1820-30 is a pure binary system, the first two scenarios become viable explanations for the observed binary orbital period derivative.", '4.2. Thermal Disk Instability': 'The fact that the type-I X-ray bursts of 4U 182030 can only be observed in the low state (Clark et al. 1977; Stella et al. 1984) has been reconfirmed in this work (see Section 3.4). This implies that the superorbital modulation of 4U 1820-30 is caused by variations in accretion rate rather than by external absorption or precession of accretion disk. Kotze & Charles (2012) listed eight possible mechanisms to account for the superorbital modulations observed in X-ray binaries, but only the third body (i.e., triple model) and the X-ray state changes can possibly be responsible for the superorbital modulation of 4U 1820-30. If the triple model is ruled out due to the instability of superorbital period, the only remaining mechanism is the X-ray state changes. X-ray state changes refer to variations in mass accretion rate between high and low states due to thermal disk instability, as observed in dwarf novae and soft X-ray transients. Priedhorsky & Terrell (1984) proposed that this mechanism could explain for the superorbital modulation of 4U 1820-30. However, Menou et al. (2002) pointed out that if thermal disk instability could occur in 4U 1820-30 system, the mass transfer rate ˙ m ≤ ˙ m crit = 4 . 4 × 10 16 g s -1 . From the mean flux of < F bol > = 8 . 7 × 10 -9 erg cm -2 s -1 for 4U 182030 (Zdziarski et al. 2007b) and the distance of 8.019 kpc for NGC 6624 (Baumgardt & Vasiliev 2021), we obtained a mean luminosity of < L > = 6 . 7 × 10 37 erg s -1 and a mass accretion rate of ˙ m 1 = 3 . 6 × 10 17 g s -1 for a neutron star with a mass of 1.4 M /circledot and a radius of 10 6 cm. It is approximately an order of magnitude larger  \nthan the ˙ m crit . Therefore, this mechanism is unlikely to explain the superorbital modulation of 4U 1820-30.', '4.3. Irradiation-induced Mass Transfer Instability': "Zdziarski et al. (2007b) discovered that the binary orbital modulation amplitude and the offset phase in 4U 1820-30 depend significantly on the accretion rate, which is highly related to the superorbital modulation phase. The orbital modulation in the X-ray band is believed to be caused by absorption from structures in the disk rim where the accretion flow from the companion impacts the outer edge of the disk (Stella et al. 1987). As the mass loss rate changes, variations of the accretion stream induce changes in the absorption of outer edge structures and the position of impact point. This makes the amplitude and phase of orbital modulation dependent on the mass loss rate, and subsequently, on the accretion rate after a viscous time of ∼ 10 5 s (Zdziarski et al. 2007b). The variation in mass loss rate could be explained by the triple model as described in Section 1, where the eccentricity variation of the binary system induced by the third companion result in changes to the mass loss rate. Although the superorbital modulation is probably not a consequence of a third companion, the the discovery by Zdziarski et al. (2007b). implies that the superorbital modulation of 4U 1820-30 is due to changes in mass loss rate.  \nOne possible cause of variations in the accretion flow, aside from the presence of a third companion, is irradiation-induced mass transfer instability. This model has ever proposed to explain the flux variation of soft X-ray transients (Hameury et al. 1986) and was included into the hybrid model proposed by ˇ Simon (2003). Due to small binary separation of 4U 1820-30, the irradiation on the companion by the X-ray emission from the neutron star and the inner part of accretion disk is strong. Because the companion of 4U 1820-30 is only partially degenerate (Rappaport et al. 1987), irradiation on the non-degenerate envelope enhances the mass loss of the companion. Chou & Jhang (2023) estimated that at least 40% of the mass lost from the companion is ejected from the binary system. Such a strong outflow is probably caused by the irradiation on the companion, as proposed by Tavani (1991). However, a part of X-ray irradiation on the companion is blocked by the accretion disk, with the area that depending on the scale height of disk rim. When the scale height of disk rim is small, a larger irradiation area enhances the mass loss rate and the accretion flow, which increases the scale height of the accretion disk rim. Conversely, when the scale height of accretion disk rim is large, a larger portion of the companion's surface is shielded by the disk. This results in  \na reduction in the mass loss rate, as well as in the accretion flow and the scale height of the accretion disk rim. This a cyclical process may explain the quasi-periodic superorbital modulation of 4U 1820-30.  \nSuppose the accretion disk in 4U 1820-30 is geometrically thin and optically thick (Pringle 1981). The shielded region on the companion can be estimated. For 4U 1820-30 with a neutron star mass of m 1 = 1 . 4 M /circledot , a companion mass of m 2 = 0 . 07 M /circledot (Chou & Jhang 2023), and an orbital period of P orb = 685 s, we derived the binary separation of a = 1 . 33 × 10 10 cm from Kepler's third law. The discovery of the superhump in 4U 1820-30 system (Wang & Chakrabarty 2010) indicates that the rim of the accretion disk reaches a 3:1 resonance radius, giving a disk radius of r d = 6 . 4 × 10 9 cm. The scale height of the accretion disk can be evaluated as √ kT/µm H / Ω k (Spruit 2010) where T and Ω k are the temperature and the Keplerian angular velocity at the disk radius r , m H is the mass of a hydrogen atom, µ = 4 for a helium-dominated disk, and k is the Boltzmann constant. The temperature at the disk rim is estimated as  \nT = ( 3 G ˙ m 1 m 1 8 πσr 3 d ) 1 4 (10)  \n(Pringle 1981) where σ is Stefan-Boltzmann constant and G is gravitational constant. The accretion rate of 4U 1820-30 ˙ m 1 = 3 . 6 × 10 17 g s -1 . Thus, the temperature at the disk rim is T = 2 . 7 × 10 4 K and the scale height is H = 2 . 8 × 10 7 cm. For a Roche lobe filled companion, the radius of companion is R 2 = R L = 2 / 3 4 / 3 [ q/ (1 + q )] 1 / 3 a (Paczy'nski 1971) where q = m 2 /m 1 . For 4U 1820-30 system, q=0.05, so R 2 = 1 . 68 × 10 9 cm. The scale height of the irradiation shielded by the accretion disk on the companion around L 1 point is h = H ( a -R 2 ) /r d = 5 . 0 × 10 7 cm, which equivalent to a latitudes of sin -1 ( h/R 2 ) = 1 . 7 · on the companion surface. Although the shielded latitude is small, it covers the L 1 point if the orbital plane and the accretion disk are coplanar. The accretion stream flows from a small region around the L 1 point on the surface of companion. If the accretion disk rim partially obscures this region, even a marginal change in irradiation on this region due to variations in the scale height of accretion disk rim could induce a significant change in mass loss because of weak effective gravitational field around L 1 point. Such a large variation in mass loss rate could result in quasi-periodic superorbital modulation, causing a 2-3 fold change in X-ray flux of 4U 1820-30. However, more observations and theoretical studies are required to verify this irradiation-induced mass transfer instability  \nscenario, including the evolution of superorbital period discovered in this study.", '5. SUMMARY': 'The triple model was once considered a plausible explanation for the superorbital modulation observed in 4U 1820-30. The stability of the superorbital period is the crucial evidence for verifying this model. CG01 suggested that the superorbital period was stable at 171 days and early RXTE ASM data support this 171-day periodicity, indicating stability of superorbital period.  \nIn this study, we analyzed the data collected by Ginga ASM, RXTE ASM, RXTE PCA, Swift BAT, and MAXI over a time span of 36 years to verify the triple model for the 4U 1820-30 system. The superorbital periods derived from the power spectra of these five instruments show a significant change from 171 days to 167 days between 1987 and 2023, suggesting the instability of the superorbital period. Phase analysis revealed that the superorbital period may have experienced a period glitch between late 2000 and early 2003, or may have changed smoothly with a period derivative of ˙ P sup = ( -3 . 58 ± 0 . 72) × 10 -4 day/day. Two ephemerides, glitch and quadratic, were established to describe the expected superorbital minimum times of 4U 1820-30. These updated ephemerides accurately describe the superorbital minimum times with a mean phase jitters of ∼ 0 . 08 cycles. The fact that the Type-I X-ray bursts can be observed only in the low state implies that a high probability of detecting the bursts around the superorbital minimum. By examining previously reported burst detection dates with different ephemerides, we found that the burst dates are more clustered around the superorbital phase zero when folded with the glitch and quadratic ephemerides, rather than with linear and CG01 ephemerides. This is not only reconfirms the low state constraint for regular X-ray bursts as suggested by Clark et al. (1977) and Stella et al. (1984), but also provides supportive evidence that the glitch and quadratic ephemerides better at describe the superorbital minimum times.  \nThe instability of the superorbital periodicity in 4U 1820-30 discovered in this work seriously challenges the triple model. According to Eq. 1, the superorbital period change could be due to either the binary period variation or the orbital period change of the third companion. However, the binary orbital modulation has been monitored for over 46 years, and neither period glitch nor period derivative of an order of ˙ P orb /P orb ∼ 10 -4 yr -1 has ever been observed (see Chou & Jhang 2023). Therefore, the superorbital period changes likely reflect the orbital period variation of the third compan-  \n. While orbital period glitches have been observed in some eclipsing LMXBs, the magnitude of these change is much smaller than that the superorbital period glitch observed 4U 1820-30. The period derivative derived from the quadratic model indicates that the timescale of the orbital period evolution of the third companion is ∼ 2600 years, which is inconsistent with the stability expected in a hierarchical triple system. If the triple model does not apply to the 4U 1820-30 system, two previously unfavorable scenarios proposed by Peuten et al. (2014) - a stellar mass dark remnant and an intermediate mass black hole - can be reconsidered to explain the discrepancy between the theoretical and observed binary orbital period derivatives.  \nThe absence of regular Type-I bursts in the high state suggests that the superorbital modulation of 4U 1820-30 results from variations in the accretion rate rather than from the occultation effect caused by the precession of a tilting or warping accretion disk. Thermal disk instability is unlikely to be the cause of the superorbital modulation due to the high accretion rate of 4U 1820-30. On the other hand, because the amplitude of orbital modulation highly depends on accretion rate (Zdziarski et al. 2007b), the superorbital modulation could result from variation in mass transfer from companion. Given the instability of superorbital period of 4U 1820-30, such variation is unlikely to be induced by a third companion. We proposed that irradiation-induced mass transfer instability may be responsible for the superorbital modulation of 4U 1820-30. The accretion stream is expected to flow from a small region around the L 1 point on companion, where the effective gravitational field is weak. Therefore, the accretion stream is highly sensitive to the X-ray irradiation onto this region. The irradiation onto this region may be partly blocked by the accretion disk rim, whose scale height also depends on the accretion stream. Small variations in the scale height can lead to significant changes in accretion stream. A cyclical process could result in quasi-periodic superorbital modulation in 4U 1820-30.  \nUsing the data collected by X-ray monitoring/scanning X-ray telescopes, we discovered the instability of the superorbital period of 4U 1820-30. Our study, we found that both the glitch model and the quadratic model describe the superorbital phase evolution well. However, additional observations are necessary to validate these models or to provide a better ephemeris for the superorbital modulation of 4U 182030. This period instability suggests that the triple model is unlikely suitable to explain the superorbital modulation of 4U 1820-30. Although we proposed that the irradiation-induced mass transfer instability may be responsible for the superorbital modulation, further observations and theoretical works are required to verify this model, including the periodicity, modulation amplitude and profile, and the puzzling phase evolution, which identified in this study. Fortunately, Swift BAT and MAXI are continuously monitoring the X-ray sky. Additionally, the newly operational Wide-field X-ray Telescope on-board the Einstein Probe (Yuan & Osborne 2015), which is sensitive to 0.5 to 4.0 keV X-ray photons and scans the entire night sky in three satellite orbits, can provide further data to better understand the nature of the superorbital modulation of 4U 1820-30.  \nThis research has made use of data and software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC. We also express our gratitude to the RXTE team for archiving the RXTE PCA monitoring observations of the galactic center and plane data, and to MAXI team for archiving the MAXI data.  \nFacilities: ADS, HEASRAC, Ginga (ASM), RXTE (ASM), RXTE (PCA),Swift (BAT), MAXI  \nSoftware:  \nheasoft(v6.30)', 'REFERENCES': "Barthelmy, S. D., Barbier, L. M., Cummings, J. 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