# Updated Orbital Ephemeris and Detection of Superhump Modulation in X-ray
Band for the Ultra-Compact Low Mass X-ray Binary 4U 1820-30

Yi Chou Graduate Institute of Astronomy, National Central University
300 Jhongda Rd. Jhongli Dist. Tauyuan, 32001, Taiwan  Yao-Wun Jhang  Graduate
Institute of Astronomy, National Central University
300 Jhongda Rd. Jhongli Dist. Tauyuan, 32001, Taiwan

###### Abstract

The 4U 1820-30 is a ultra-compact low mass X-ray binary (LMXB) near the center
of the globular cluster NGC 6624. Its negative orbital period derivative,
observed from the phase evolution of its sinusoidal-like orbital variation,
contradicts the positive value obtained from the theoretical prediction. In
this paper, we present the analysis of the 4U 1820-30 orbital modulation from
light curves obtained from the Neutron star Interior Composition ExploreR
(NICER) observations from 2017 to mid 2022. Combined with historical records,
the orbital derivative is measured from the orbital phase evolution between
1976 and 2002 is $\dot{P}/P=(-5.21\pm 0.13)\times 10^{-8}$ yr-1. No
significant second order orbital period derivative is detected with a
2$\sigma$ upper limit of $|\ddot{P}|<5.48\times 10^{-22}$ s s-2. We discuss
the possible intrinsic orbital period derivative of 4U 1820-30 and suggest
that this binary system may have a significant mass outflow similar to some
other LMXBs. In addition, a periodic modulation with a period of $691.6\pm
0.7$ s, which is consistent with the superhump period discovered in the far
ultraviolet band of the Hubble Space Telescope, was also detected in in the
X-ray light curves collected by NICER. We conclude that such modulation is
probably caused by a period of $0.8\pm 0.1$ day apsidal precession of
accretion disk similar to the SU UMa type dwarf novae and some LMXBs. However
we cannot exclude the possibility that it is induced by a hierarchical third
star orbiting around the binary system.

††facilities: ADS, HEASRAC, NICER††software: heasoft(v6.30), nicerdas(v009)

##  1 Introduction

The ultra-compact low mass X-ray binary (LMXB) 4U 1820-30, located near the
center of the globular cluster NGC 6624, was discovered by Giacconi et al.
(1974). The detection of its Type I X-ray burst (Grindlay et al., 1976)
indicated that the accretor of this accreting system is a neutron star. Stella
et al. (1987) first reported its sinusoidal-like orbital modulation in X-ray
band with a period of 685.0118 s. This orbital period is the shortest among
all known LMXBs making 4U 1820-30 the most compact X-ray binary whose mass-
losing companion is a Roche-lobe filling helium white dwarf with a mass of
0.06-0.08 $M_{\sun}$ (Rappaport et al., 1987). Priedhorsky & Terrell (1984)
discovered a long-term variation in 4U 1820-30 luminosity by a factor of
$\sim$3 with a period of of $\sim$176 days detected from the observations of
the Vela 5B spacecraft. In addition to revising the period to be
171.003$\pm$0.326 days, Chou & Grindlay (2001) found that this period is
stable for over $\sim$30 years with a period change rate
$|\dot{P}_{long}/P_{long}|<2.2\times 10^{-4}$ $yr^{-1}$ and proposed that 4U
1820-30 could be a hierarchical triple with a third star orbiting the binary
system with a period of $\sim$1.1 days, inducing binary orbital eccentricity
modulations with a period of $\sim$171 days (Mazeh & Shaham, 1979; Grindlay,
1988). This long-term periodicity was further confirmed by Šimon (2003).

In addition to the orbital modulation observed in the X-ray band, Anderson et
al. (1997) discovered a modulation with a period of 687$\pm$2.4 s in the
ultraviolet (UV) band from the Hubble Space Telescope (HST). Although this
period is slightly longer than the orbital period detected in the X-ray band,
the difference is not sufficiently significant to distinguish between them. On
the other hand, from the far-ultraviolet (FUV) light curve collected by HST,
Wang & Chakrabarty (2010) reported the discovery of a period of 693.5$\pm$1.3
s modulation, whose period is only $\sim$1% longer but significantly different
from the orbital period detected in the X-ray band. They suggested that this
period is superhump period although it might be consistent with the
hierarchical triple system model.

Since the discovery of orbital modulation by Stella et al. (1987), several
studies were conduct to refine the orbital period and constrain the orbital
period derivative from historical and newly observed data, such as Smale et
al. (1987), Morgan et al. (1988) and Sansom et al. (1989). The first
significant orbital derivative of 4U 1820-30 was detected by Tan et al. (1991)
with a value of $\dot{P}/P=(-1.08\pm 0.19)\times 10^{-7}$ $yr^{-1}$. This
value was further refined by van der Klis et al. (1993a, b) and Chou &
Grindlay (2001). After combining all the observations in the X-ray band,
Peuten et al. (2014) reported the latest value of orbital period derivative as
$\dot{P}/P=(-5.3\pm 0.3)\times 10^{-8}$ $yr^{-1}$ using the orbital modulation
phase observed from 1976 to 2010. All the aforementioned detected values
indicate that the orbital period derivative is negative, which contradicts to
the theoretical predicted lower limit ($\dot{P}/P>8.8\times 10^{-8}$
$yr^{-1}$) proposed by Rappaport et al. (1987), who considered 4U 1820-30 to
be an accreting system that consists of a neutron star and a mass-losing
helium white dwarf. Even when considering other effects, such as tidal
dissipation in the white dwarf driven by the eccentricity of the binary
system, the intrinsic orbital derivative may be lower but should still be
positive (Prodan & Murray, 2012).

To explain the contradiction between the observed (negative) and theoretically
predicted (positive) values of the orbital derivative of 4U 1820-30, several
scenarios have been proposed. Tan et al. (1991) first suggested that this
deviation might be caused by the binary system accelerating in the
gravitational potential of globular cluster, which was supported by Chou &
Grindlay (2001). However, with consideration of the projected location of 4U
1820-30 in the globular cluster, distance of NGC 6624 and mass distribution
evaluated by the mass-to-light ratio across the cluster, the maximum
acceleration from the gravitational potential of the globular cluster
($a_{max}/c\approx 10^{-9}yr^{-1}$) is approximately an order of magnitude
smaller than the one that is required to explain the disparity between
theoretical and observed orbital period derivative values ($a/c\approx
10^{-8}yr^{-1}$) (Prodan & Murray, 2012; Peuten et al., 2014). Peuten et al.
(2014) discussed some other possibilities got further accelerating the binary
system, including stellar mass dark remnant, intermediate-mass black hole and
central concentration of dark remnants. To preserve the stability of the
triple system, which is considered the cause of 171 days of long-term
modulation (Chou & Grindlay, 2001), as well as the abnormal period derivative
of three pulsars (J1823-3021A, B, C) in NGC 6624, the central concentration of
dark remnants is the most likely scenario to explain the discrepancy. On the
other hand, Jiang et al. (2017) suggested that the negative orbital period
derivative observed in 4U 1820-30 is possibly produced by the temporary
circumbinary disk which can effectively extracted the orbital angular momentum
from the binary system through resonant interaction.

In this paper, we present our analysis results for the orbital modulation of
4U 1820-30 using the Neutron star Interior Composition ExploreR (NICER)
observations from 2017 to 2022. Combined with historical records, the orbital
phase evolution of $\sim$46 years enable us not only to further refine the
orbital period but also measure or constrain the second order orbital
derivative ($\ddot{P}$)in order to verify the possible scenarios for the
negative orbital derivative observed in 4U 1820-30. In section 2, we describe
the NICER observations of 4U 1820-30, as well as data screening and grouping.
The data analysis results are demonstrated in section 3, including the update
of the orbital period derivative ($\dot{P}/P$), constraint on the second order
orbital derivative ($\ddot{P}$), and detection of a period of $691.6\pm 0.7$ s
superhump modulation in X-ray light curve. In section 4.1, we discuss the
possible intrinsic orbital period derivative and suggest that the observed
orbital period derivative of 4U 1820-30 is improper for inferring the
gravitational acceleration in NGC 6624. Finally, the superhump and
hierarchical triple interpretations for the period of $691.6\pm 0.7$ s
modulation are also demonstrated in Section 4.2.

##  2 Observations and Data Reduction

NICER (Gendreau & Arzoumanian, 2017), aboard the International Space Station,
is a soft X-ray telescope designed to investigate neutron stars through X-ray
timing. Its X-ray Timing Instrument (XIT) is composed 56 X-ray concentrators
(XRCs) and associated Focal Plane Modules (FPMs). The 56 FPMs are further
divided into seven groups, each group consisting of eight FPMs, controlled by
a Management and Power Unit (MPU). There are four FPMs, including detector ID
11, 20, 22 and 60, which have been inactive since launched as indicated in the
NICER Mission
Guide111$https://heasarc.gsfc.nasa.gov/docs/nicer/mission\\_guide/$.

NICER made 173 observations of 4U 1820-30 between June 26 2017 and July 9 2022
(Figure 1). All data files analyzed in this study were reduced using NICERDAS
version 9, as a part of HEASOFT v6.30. The events were filtered using the
standard screening criteria described in the NICER Mission Guide. Their photon
arrival times were corrected to the barycenter of the solar system using the
tool barycorr by applying the JPL solar planetary ephemeris DE430. However,
the barycenter correction could not be processed for observations with ObsID
0050300118, 1050300103, 2050300119, 5604010304 and 5604010305 because the
orbital files did not match the corresponding data222See
$https://heasarc.gsfc.nasa.gov/docs/nicer/data_{a}nalysis/workshops/NICER-
Workshop-QA-2021.pdf$, therefore these observations were removed from the
analysis. To reduce the noise and background, only the events with both slow
and fast chains triggered and an energy range between 1 and 10 keV (i.e. PI
values between 100 and 1000) were selected for further analysis.

However, we found that the number of active FPMs was less than 52 after data
screening in certain observations, for example, only 47 active FPMs for the
observation with ObsID 4680010126. Therefore, for the light curves, we
normalized the count rate to the number of active FPMs, i.e., cts/s/FPM, in
this work. Furthermore, some observations showed significantly lower exposure
times in an MPU (mostly in MPU3) than in others. Hence, we performed the
following process to identify low-exposure MPU. First, all events in an
observation were binned into 1 s bin where $N_{i}$ is the number of the events
of the ith MPU in this bin. $n_{i}$ is defined as the $N_{i}$ being normalized
by the number of active FPMs in this MPU, i.e., $n_{i}\equiv N_{i}/fpm_{i}$
where $fmp_{i}$ is the number of active FPMs in the ith MPU and the
uncertainty of $n_{i}$ can be written as $\sigma_{i}=\sqrt{N_{i}}/fpm_{i}$.
Taking MPU0 as a reference because the low exposure has never been observed in
this MPU, we defined deviation as $\Delta n_{i}\equiv n_{i}-n_{0}$ thus, the
corresponding uncertainty is $\sigma_{\Delta
n_{i}}=\sqrt{\sigma_{i}^{2}+\sigma_{0}^{2}}$. We expected that $<\Delta
n_{i}>=0$ if the exposure time of the ith MPU was the same as that for MPU0.
Figure 2 shows the distributions of $\Delta n_{i}$ for an observation in which
the centroid of $\Delta n_{3}$ is significantly lower than the others.
Therefore, we considered the mean deviation larger than 1$\sigma$ as the low
exposure time and the events in this MPU were removed from the analysis.
Finally, several Type-I X-ray bursts were detected in all observations. Burst
events, detected 100 s after the beginning of the bursts, were also excluded
from analysis.

Because the orbital variation of 4U 1820-30 in the X-ray band is rather weak (
3% peak-to-peak), to significantly detect the modulation profile, a light
curve with sufficient long exposure time is essential. However, the exposure
times of many observations were too short to detect the orbital modulation.
Therefore we attempted to combine consecutive observations with short exposure
times to create a group to make the light curve. The total exposure time of
each group was at least 2000 s with the total time span of less than six days
to minimize the trend due to long-term (171 days) variation. All other
sporadic observations with short exposure times were excluded from the
analysis. Table 1 shows the grouping of observations for the analysis
described in the following section.

##  3 Data Analysis

###  3.1 Orbital Period Derivative

The events selected for each group were binned to a light curve with a bin
size of 1 s. Because there were 171 days of long-term modulation, to correctly
obtain the orbital modulation profile, we removed the trend by fitting a
polynomial up to 4th order for each the light curve. The de-trended light
curves were folded using the revised linear ephemeris proposed by Peuten et
al. (2014).

$T_{N}={BJD_{TDB}2442803.63600}+{\big{(}{{685.0118}\over{86400}}\big{)}}\times
N$ (1)

where $T_{N}$ is the time of the maximum intensity of the Nth cycle and $N$ is
the cycle count. However, a small number of the folded light curves showed
insignificant orbital modulation. We applied analysis of variance (AoV)
(Schwarzenberg-Czerny, 1989) to filter them out. In AoV, the variable
$\Theta_{AoV}$ is defined as

$\displaystyle(r-1)s_{1}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{r}n_{i}({\bar{x}}_{i}-{\bar{x}})^{2}$
$\displaystyle(n-r)s_{2}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{r}\sum_{j=1}^{n_{i}}(x_{ij}-{\bar{x}}_{i})^{2}$
$\displaystyle\Theta_{AoV}$ $\displaystyle\equiv$ $\displaystyle
s_{1}^{2}/s_{2}^{2}$ (2)

where $n$ is the total number of the data points in the light curve, $\bar{x}$
is the average count rate of the light curve, $r$ is the number of bins in the
folded light curve, $n_{i}$ is the number of data points in the ith bin of the
folded light curve, ${\bar{x}}_{i}$ is the average count rate in the ith bin
of the folded light curve and $x_{ij}$ is the jth count rate of the light
curve in the ith bin of the folded light curve. If the light curve is pure
Gaussian white noise, $\Theta_{AoV}$ follows the F-distribution with $r-1$ and
$n-r$ degrees of freedom. We selected $r=32$ for the testing process. Only the
folded light curves with false alert probability less than 0.0053
(approximately more than 3$\sigma$ significance) were accepted for further
analysis. Each folded light curves with significant orbital modulation
detection was then fitted by a single sinusoidal function and the maximum
value of the fitted curve was taken as fiducial point for the orbital phase. A
typical orbital modulation profile is shown in Figure 3. In a manner similar
to that done by Chou & Grindlay (2001) and Peuten et al. (2014), we combined
the orbital phases of each year using weighted averaging. The phase jitter, as
seen in Rossi X-ray Timing Explorer (RXTE) data Chou & Grindlay (2001), can be
observed in the phases of each year. Therefore, we estimated the phase jitter
value for each year based on the standard deviation of the phases of the year.
The final error for the phase of each year was evaluated by quadratically
adding the phase jitter and average phase error from the fitting of the folded
light curve. The mean arrival time of the average phase for each year was
calculated using the Equation 2 in Chou & Grindlay (2001). The average orbital
phases measured from 2017 to 2022 NICER observations are listed in Table 2.

In combination with the historical records of the orbital phases listed in
Table 1 of Peuten et al. (2014), the evolution of the orbital phases of 4U
1820-30 measured between 1976 and 2022 are shown in Figure 4. Given that a
significant orbital period derivative has been detected ($>17\sigma$, Peuten
et al., 2014), we directly fitted the orbital phase evolution with the
quadratic function shown in Eq 3 assuming that the orbital period derivative
($\dot{P}$) was a constant over the entire time span of $\sim 46$ years

$\Phi(t)={{T_{0}-T_{fold}}\over
P_{fold}}+{\Big{(}{{P_{0}-P_{fold}}\over{P_{0}P_{fold}}}\Big{)}}t+{1\over
2}{\dot{P}\over{{P_{0}}}^{2}}t^{2}$ (3)

where $P_{0}$ is the orbital period at the expected phase zero epoch $T_{0}$,
$T_{fold}=BJD_{TDB}2442803.63600$, $P_{fold}=685.0118$ s and
$t=T_{mean}-T_{0}$ where $T_{mean}$ is mean arrival time of the average phase
of each year. We obtained a reasonable fitting with $\chi^{2}/\nu=33.72/37$.
This fitting results give an orbital period derivative of $\dot{P}=(-1.131\pm
0.029)\times 10^{-12}$ s s-1, or $\dot{P}/P=(-5.21\pm 0.13)\times 10^{-8}$
yr-1, which is almost identical to those found by van der Klis et al. (1993b)
and Peuten et al. (2014) but with smaller uncertainty. The quadratic ephemeris
can be updated as follows:

$T_{N}={(BJD_{TDB}2442803.636101\pm 0.00094)}+{\big{(}{{685.011968\pm
0.000021}\over{86400}}\big{)}}\times N+(4.48\pm 0.12)\times 10^{-15}\times
N^{2}$ (4)

We also attempted to probe the second order orbital period derivative
($\ddot{P}$) suggested by Peuten et al. (2014). We fitted the phase evolution
with a cubic function and obtained a second order orbital period derivative of
$\ddot{P}=(9.5\pm 27.4)\times 10^{-23}$ s s-2 with $\chi^{2}/\nu=33.60/36$. In
comparison with quadratic model, the F-test gave an F-value of 0.13 and a
p-value of 0.72, which implied that the cubic model improved the fitting with
confidence level only 28%. We therefore conclude that there is no significant
second order orbital period that can be detected with a $2\sigma$ upper limit
of $|\ddot{P}|<5.48\times 10^{-22}$ s s-2

###  3.2 Search for Superhump Modulation in X-ray Light Curves

Wang & Chakrabarty (2010) discovered a periodic modulation in the FUV light
curves of 4U 1820-30 with a period of 693.5 s, using data collected by the
HST. They suggested that this modulation, which has a period about 1.2% longer
than the orbital period, is the superhump period, based on similarities
between 4U 1820-30 and the dipping LMXB X 1916-053. In X 1916-053, the
superhump period is not only seen in the optical band but also in X-ray light
curves (Chou et al., 2001). For 4U 1820-30, the sinusoidal-like orbital
modulation in the X-ray band is believed to be caused by absorption of
structures in the disk rim, similar to X-ray dipping sources (Stella et al.,
1987). Therefore, it is likely that the superhump modulation is detectable in
the X-ray light curves of 4U 1820-30.

To resolve the superhump modulation signal ($\sim$693 s) from the orbital
variation ($\sim$685 s) in the power spectrum, as well as to avoid too much
alias induced by the observation gaps, a continuous observation with a time
span much more than $\sim$0.8 day was preferred. We found that the NICER
observations made between September 1 to 5 2017 with ObsId from 1050300111 to
1050300115 were suitable for searching for superhump modulation. The trend-
removed light curve, with a total exposure time of $\sim$104 ks and time span
of $\sim$4 days is shown in Figure 5. To search for the periodicity of this 1
s binned light curve, we adopted the Lomb-Scargle (LS) periodogram (Scargle,
1982) to obtain the power spectrum as shown in Figure 6. The most significant
peak is located at $\sim$1.46 mHz, which is caused by the orbital modulation.
Similar to the analysis performed by Wang & Chakrabarty (2010), we fitted a
Gaussian function to the main peak and obtained the peak frequency
$f_{0}=1.460\pm 0.001$ mHz, corresponding to a period of $684.9\pm 0.5$ s. In
addition to the main peak, several smaller peaks were observable. To identify
whether these peaks were true signal or alias induced by observation gaps and
orbital modulation, we fitted the light curve by a constant plus a sinusoidal
function with a frequency fixed at 1.46 mHz to model the sinusoidal-like
orbital modulation, and then subtracted the model from it. The residual light
curve was analyzed again using the LS periodogram. The main peak from the
orbital modulation was completely eliminated, but two significant peaks, which
were evidently not the alias caused by the main peak and observation gaps,
could be clearly observed in the power spectrum of residual (shown in Figure
6). The Gaussian fitting to these two peaks resulted in a frequency of
$f_{1}=1.446\pm 0.001$ mHz, corresponding to a period of $691.6\pm 0.7$ s,
which was close to the $693.5\pm 1.3$ s superhump period proposed by Wang &
Chakrabarty (2010)333Actually, in Wang & Chakrabarty (2010), the superhump
frequency was found to be $f_{1}=1.4442\pm 0.0027$ mHz, whose corresponding
period should be $692.4\pm 1.3$ s instead of $693.5\pm 1.3$ s. This revised
period is consistent to what we found in this work. for the most significant
peak and $f_{2}=1.432\pm 0.002$ mHz, whose corresponding period is $698.3\pm
0.8$ s, for the other peak. We noticed that the relation between the frequency
of the main peak and the frequencies of the two peaks from the residual light
curve could be written as $f_{n}=f_{0}-0.014\times n$ mHz where n=1 or 2. This
indicates that the two peaks detected in the residual light curve power
spectrum are the beat sidebands of a period of $0.8\pm 0.1$ days modulation.
We concluded that 4U 1820-30 is a superhump LMXB with accretion processing in
a period of $\sim$0.8 days and the period of the most significant peak (i.e.
$691.6\pm 0.7$ s) is the superhump period, as the one detected in the FUV band
(Wang & Chakrabarty, 2010). Figure 7 shows the residual light curve folded by
the superhump period where the peak-to-peak amplitude is approximately 1%.
Further discussions about the superhump in 4U 1820-30 are presented in section
4.2.

##  4 Discussion

###  4.1 Orbital Period Change Rate

Using the phases obtained by the orbital modulations detected by the NICER
2017-2022 observations, we updated the orbital ephemeris of 4U 1820-30 by
combining all historical records listed in Table 1 of Peuten et al. (2014).
The orbital period derivative evaluated from $\sim$46 years’ baseline
observations is $\dot{P}/P=(-5.21\pm 0.13)\times 10^{-8}$ $yr^{-1}$, which is
almost identical but more precise to the values reported by van der Klis et
al. (1993b) and Peuten et al. (2014), and similar to the one proposed by Chou
& Grindlay (2001) whose deviation may be caused by phase jitter effect. No
significant second order orbital derivative was detected with
$|\ddot{P}|<5.48\times 10^{-22}$ s $s^{-2}$ (2$\sigma$ upper limit). This
indicates that the observed orbital period derivative of 4U 1820-30 was very
stable, at least from 1976 to 2022. The observed negative $\dot{P}$ is
incompatible to positive orbital period derivative ($\dot{P}/P>8.8\times
10^{-8}$ $yr^{-1}$) predicted by the standard scenario proposed by Rappaport
et al. (1987). Although the orbital period derivative may be lower if other
effects (e.g. tidal dissipation in the white dwarf driven by the eccentricity
of binary system (Prodan & Murray, 2012)) are considered, it should be still
positive (Peuten et al., 2014).

The most popular scenario for explaining the discrepancy between the
theoretical and observed values of the orbital period derivative is the
acceleration of the binary system by the gravitation potential in the globular
cluster NGC 6624 (Tan et al., 1991; Chou & Grindlay, 2001; Peuten et al.,
2014). In this scenario, the observed orbital period derivative can be express
as

${\dot{P}_{obs}\over P}\approx{\dot{P}_{int}\over P}+{a_{c}\over c}$ (5)

(Phinney, 1992; Perera et al., 2017) where ${\dot{P}_{obs}}$ is the observed
orbital period derivative, ${\dot{P}_{int}}$ is the intrinsic orbital period
derivative in the rest frame from the binary orbital evolution and $a_{c}$ is
the line-of-sight acceleration from the cluster potential where $a_{c}<0$
means that the acceleration is towards the observer. Here, we ignore some
other effects, such as the differential Galactic acceleration, acceleration
towards the Galactic plane, and Shklovskii effect (Shklovskii, 1970), whose
contributions to the observed orbital period derivative are only approximately
$10^{-11}$ to $10^{-10}$ $yr^{-1}$ (Peuten et al., 2014; Perera et al., 2017),
i.e. much smaller than the observed orbital derivative ($\sim 10^{-7}$
$yr^{-1}$). Our observation results show that the observed orbital period
derivative is stable over $\sim$46 years, which implies that it is highly
possible that both the intrinsic orbital period derivative and the external
effect (e.g. acceleration in the globular cluster) are stable because it is
very unlikely that the higher order terms of these two effects (i.e.
$\ddot{P}_{int}/P$ and ${\dot{a}}_{c}/c$) have almost identical values but
different signs to be canceled out to each other.

However, based on the cluster profile evaluated from the surface brightness
and project radius of 4U 1820-30 to the center of NGC 6624, Peuten et al.
(2014) found that the maximum acceleration of 4U 1820-30 was only
$a_{c,max}/c\approx-4.6\times 10^{-9}$ $yr^{-1}$, i.e. insufficient for
explaining the observed orbital period derivative. Peuten et al. (2014)
discussed several possibilities for further accelerate of this binary system
in the globular cluster NGC 6624. The first is that the binary system is
accelerated by a flyby stellar size dark remnant, which can be a white dwarf,
neutron star, or stellar size black hole. To preserve 4U 1820-30 as a triple
system, which is currently the only way to explain the 171 days’ periodic
modulation, the mass of the dark remnant cannot exceed 6 $M_{\sun}$, otherwise
the triple system will be destroyed due to the tidal force. Thus, a low mass
dark remnant is preferred. On the other hand, the time span to maintain
sufficient large acceleration to explain the binary’s orbital period
derivative would be shorter for lower mass dark remnant (20-160 years for
1-5.5 $M_{\sun}$, respectively), which makes it a short-lived phenomenon. From
the 2$\sigma$ upper limit of the second order orbital derivative, the
variation time scale of $\dot{P}$, $|{\dot{P}}/{\ddot{P}}|$, would be longer
than 65 years. However, if we consider the intrinsic orbital derivative, which
is at least $8.8\times 10^{-8}$ yr-1, from Eq 5, the variation time scale of
$a_{c}$ would be larger than 177 years under the assumption that
${\dot{P}}_{int}$ is a constant. Although this possibility cannot be
completely excluded, the flyby dark remnant scenario appears unlikely to
explain these observational results. In addition, it is scenario is very
difficult to explain the spin period derivatives detected in the other three
pulsars (PSR J1823-3021A,B,C) (see Peuten et al., 2014, Table 2) in NGC 6624
because the probability of the these four sources experiencing similar flyby
remnants simultaneously is very low (Peuten et al., 2014).

Another scenario discussed by Peuten et al. (2014) is that the binary system
may be accelerated by an intermediate-mass black hole (IMBH) located at the
center of the globular cluster. A mass of 7,500 $M_{\sun}$ IMBH is sufficient
to explain the observed negative orbital period derivative with
$a_{c}/c=a_{max}/c=-5.3\times 10^{-8}$ yr-1 (see Peuten et al., 2014, Figure
7). Together with the other three pulsars, it needs a mass of 19,000
$M_{\sun}$ IMBH to explain their observed spin period derivatives. However,
similar to the stellar dark remnant, such a large IMBH mass would destroy the
triple system very quickly to make what we observed is a short-lived event. In
addition, Peuten et al. (2014) proposed that a central concentration of dark
remnants, such as white dwarfs, neutron stars and stellar size black holes,
that sank into the center of NGC 6624 may be responsible for the negative
orbital period derivative of 4U 1820-30, as well as the other three pulsars.
They concluded that this scenario was preferable for explaining the observed
period derivatives for the four sources. However, from the precise timing
analysis of PSR J1823-3021A, Perera et al. (2017) found that this millisecond
pulsar was very likely orbiting around an IMBH of at least 7500 $M_{\sun}$
mass located at the center of NGC 6624 with high eccentricity and they argued
that it is not necessary to favor central dark remnant over an IMBH at center,
but Gieles et al. (2018) constructed mass models without IMBH at the center
that were able to explain the spin period derivatives of three millisecond
pulsars.

However, for any of the scenario used to interpret the negative orbital period
derivative or negative orbital phase drift, the internal orbital period
derivative must be considered. Because the orbital modulation of 4U 1820-30 is
likely a result of periodic occultation by the accretion stream of the
vertical structure at the edge of the accretion disk (Stella et al., 1987;
Morgan et al., 1988; Sansom et al., 1989; van der Klis et al., 1993b), van der
Klis et al. (1993b) suggested that the negative orbital phase drift of a -0.2
cycle detected from data collected between 1976 and 1991 may be due to
accretion disk size change that causes the azimuth point of impact shift.
Nevertheless, if the minimum intrinsic orbital period derivative, i.e.
$8.8\times 10^{-8}$ yr-1 (Rappaport et al., 1987), taken into account, the
azimuth point of impact has to shift 0.95 cycle, which makes this scenario
unlikely (Chou & Grindlay, 2001). A similar argument can also be applied to
scenarios in which the observed negative orbital period derivative is caused
by acceleration of the binary in the globular cluster. Considering the minimum
intrinsic orbital period derivative, the acceleration along the line-of-sight
of 4U 1820-30 should be at least $a_{c}/c=-1.41\times 10^{-7}$ yr-1.

Furthermore, large deviations of orbital period derivatives between the
measured value and theoretical prediction, which considers orbital angular
loss mechanisms only by gravitational radiation and magnetic braking, are
often seen in LMXBs. Burderi et al. (2010) found that the observed orbital
period derivative of the LMXB pulsar X 1822-371 was three order of magnitudes
larger than the predicted value. The orbital period derivative of AX
J1745.6-2901, which was first detected by Ponti et al. (2017), was at least
one order of magnitude larger than the theoretical value. For the transient
neutron star LMXB MXB 1659-298, the orbital period derivative was likely
almost two order of magnitudes larger than the predicted value (Iaria et al.,
2018). Similar large deviations have also been observed in accreting
millisecond X-ray pulsars (AMXPs), such as SAX J1808.4-3658 (di Salvo et al.,
2008; Burderi et al., 2009; Sanna et al., 2017), IGR J17062-6143 (Bult et al.,
2021), and SWIFT J1749.4-2807 (Sanna et al., 2022). The most interesting
source is X 1916-053, an ultra-compact LMXB composed of a neutron star and a
white dwarf, similar to 4U 1820-30, with an orbital period of 3000 s. An
orbital period derivative of $\dot{P}/P=(1.62\pm 0.34)\times 10^{-7}$ yr-1,
i.e. $\sim$200 times larger than the one induced by gravitational radiation,
was reported by Hu et al. (2008). The most popular explanation for this large
discrepancy is non-conservative mass transfer, i.e. a significant amount of
mass transferred from the donor is ejected from the binary system instead of
accreting onto the accretor. The outflow can carry away more orbital angular
momentum to make the orbital period change faster than expected. For X
1916-053, Hu et al. (2008) found that approximately 60% - 90% of the mass was
transferred from the white dwarf companion outflowing from the binary system.
Chou et al. (2016) estimated that more than 60% of the mass lost from the
companion was ejected from X 1822-371. Moreover, the mass outflow ratios can
be as high as more than 90% for some LMXBs, such as IGR J17062-6143 (90%, Bult
et al., 2021), MXB 1659-298 (98%, Iaria et al., 2018), and SAX J1808.4-3658
(99%, Burderi et al., 2009). Tavani (1991) proposed a radiation-driven model
in which the non-conservative mass transfer may be caused by the irradiation
of the secondary and accretion disk by large a flux from the primary. It can
induce a strong evaporative wind from the secondary and accretion disk that
makes the orbital evolution time scale $\sim$100 times shorter than expected,
such as X1916-053 (Hu et al., 2008) and X 1822-371 (Chou et al., 2016).
Alternatively, for AMXPs, di Salvo et al. (2008) suggested that, similar to
the mechanism of black widow pulsars, the mass outflow can be driven by the
pulsar wind, even in a quiescence state.

We suggest that the radiation-driven mass transfer mechanism, which induces
non-conservative mass transfer, can also be applied to 4U 1820-30 that makes
its intrinsic orbital period derivative much larger than the one that
considers the orbital angular momentum lost only by gravitational radiation.
Therefore, considering the orbital angular momentum lost by the outflow, the
orbital period derivative, from Equation 3 in di Salvo et al. (2008), can be
expressed as

${\dot{P}_{orb}\over P_{orb}}=3\biggl{\\{}{\biggl{(}{\dot{J}_{orb}\over
J_{orb}}\biggr{)}_{GR}}-{\dot{M}_{2}\over M_{2}}\biggl{[}1-\beta
q-\biggl{(}1-\beta\biggr{)}\biggl{(}{{\alpha+q/3}\over{1+q}}\biggr{)}\biggr{]}\biggr{\\}},$
(6)

where $(\dot{J}_{orb}/J_{orb})_{GR}$ is the orbital angular momentum loss
driven by gravitational radiation, $M_{1}$ and $M_{2}$ are the masses of the
neutron star and the companion, respectively, $q$ is the mass ratio
($q=M_{2}/M_{1}$), $\beta$ is the ratio of the mass accreting onto the neutron
star, ${\dot{M}}_{1}=-\beta{\dot{M}}_{2}$, and $\alpha$ is the specific
angular momentum lost by the outflow expressed in units of the specific
orbital angular momentum of the companion. For the Roche lobe filling
companion, its radius $R_{2}$ is approximately equal to the Roche lobe radius
$R_{L}=2/(3^{4/3})[1/(1+q)]^{1/3}$(Paczyński, 1971). Combined with the
Kepler’s 3rd law and the mass-radius relation of the companion,
$R_{2}\propto{M_{2}}^{n}$, from Eq 6, we can have

${\dot{M}_{2}\over M_{2}}={{\biggl{(}{\dot{J}_{orb}\over
J_{orb}}\biggr{)}_{GR}}\over{{5\over 6}+{n\over 2}-\beta
q-(1-\beta){{\alpha+q/3}\over{1+q}}}}$ (7)

and

${\dot{P}_{orb}\over P_{orb}}={({{3n}\over 2}-{1\over
2})}{{\biggl{(}{\dot{J}_{orb}\over J_{orb}}\biggr{)}_{GR}}\over{{5\over
6}+{n\over 2}-\beta q-(1-\beta){{\alpha+q/3}\over{1+q}}}}$ (8)

For 4U 1820-30, taking $M_{1}=1.4M_{\sun}$, $M_{2}=0.07M_{\sun}$, and $n=-1/3$
for the fully degenerate secondary, we obtained $\dot{M}_{2}=-1.04\times
10^{-8}$ $M_{\sun}$ yr-1 for the mass conserved case ($\beta=1$). Thus we may
evaluate the luminosity as $L={{GM_{1}\dot{M}_{2}}/R_{1}}=1.22\times 10^{38}$
erg s-1 where $R_{1}=10^{6}$ cm is the neutron star radius that is larger than
the mean observed value of $<L>=(6.0\pm 0.2)\times 10^{37}$ erg s-1 (Zdziarski
et al., 2007b). If we assume that the deviation is caused by the non-
conservative mass transfer, the luminosity can be evaluated as

$L=-{{GM_{1}\beta\dot{M}_{2}}\over R_{1}}=-{{GM_{1}M_{2}}\over
R_{1}}{\beta{\biggl{(}{\dot{J}_{orb}\over J_{orb}}\biggr{)}_{GR}}\over{{5\over
6}+{n\over 2}-\beta q-(1-\beta){{\alpha+q/3}\over{1+q}}}}$ (9)

which enables us to constrain $\alpha$ and $\beta$ values. Figure 8 shows the
relation between the luminosity evaluated from Eq 9 and mass outflow ratio
(1-$\beta$). This indicates that the mass outflow ratio has a lower limit
whereas $\alpha$ has an upper limit. From Eq 9, $\alpha$ can be written as a
function of $1-\beta$, where $\alpha=f(1-\beta)$ is shown in Figure 9 for the
companion’s mass between 0.06$M_{\sun}$ and 0.08$M_{\sun}$. Evidently, at
least $\sim 40\%$ (for $M_{2}=0.06M_{\sun}$) of the mass lost from the
companion has to be ejected from the binary system. By applying the function
$\alpha=f(1-\beta)$ to Eq 8, orbital period derivative can be written as a
function of mass loss ratio shown in Figure 10. This implies that the
intrinsic orbital period derivative is no less than $1.44\times 10^{-7}$ yr-1
and can be very large for intense mass outflow.

According to the radiation-driven model proposed by Tavani (1991), the outflow
can be ejected from the companion and accretion disk. If we assume the mass
outflow is only ejected from the Roche-lobe filling companion, the minimum
specific orbital angular momentum of the secondary star is at the inner
Lagrangian point, which can be calculated using the formula
$\alpha=\\{1-[2/{(3^{4/3})q^{1/3}(1+q)^{2/3}}]\\}^{2}$ (di Salvo et al.,
2008). For 4U 1820-30, $\alpha=0.68$ when $M_{2}=0.07M_{\sun}$ and
$M_{1}=1.4M_{\sun}$. This means that if the mass outflow is ejected from the
companion only, we expect that $\alpha>0.68$. However, as shown in Figure 9,
we found that $\alpha<0.68$, which implies that a part of the mass outflow is
ejected from the accretion disk.

Although the observed negative orbital period derivative, $\dot{P}/P=(-5.21\pm
0.13)\times 10^{-8}$ yr-1, for 4U 1820-30 is very likely due to the
acceleration of the binary system by the gravitational potential in the
globular cluster NGC 6624 (Tan et al., 1991; Chou & Grindlay, 2001; Peuten et
al., 2014), when considering the acceleration value, the intrinsic orbital
period derivative (i.e. $\dot{P}_{int}/P$ in Eq 5) cannot be neglected. From
the above discussion, it is highly possible that 4U 1820-30 has significant
mass outflow ratio ($>40\%$) and the intrinsic orbital period derivative is at
least $1.44\times 10^{-7}$ yr-1 for $M_{1}=1.4M_{\sun}$ and
$M_{2}=0.6M_{\sun}$, which makes the minimum line-of-sight acceleration equal
to $|a_{c}/c|=1.96\times 10^{-7}$ yr-1. This value can be larger for a more
intense mass outflow, as shown in Figure 10. Therefore, we suggest that it is
inappropriate to infer the acceleration of 4U 1820-30 in NGC 6624 using its
observed orbital period derivative because there is too much uncertainty in
it. In contrast to 4U 1820-30, the millisecond pulsars in NGC 6624 are likely
better suited for inferring the accelleration in NGC 6624, as their intrinsic
spin period derivatives are relatively small ($\dot{P}/P\sim 10^{-10}$ yr-1).

###  4.2 Superhump Modulation in 4U 1820-30

Superhumps are periodic modulations that are commonly observed in SU UMa type
dwarf novae during their superoutburst states with periods 1%-7% longer than
the corresponding orbital periods (see Warner, 1995, for extensive review).
The superhump period is believed to be the beat period of the orbital period
and accretion disk prograde precession period. If the mass ratio
$q=M_{2}/M_{1}$ of a cataclysmic variable (CV) is less than 0.33, the
accretion disk radius may extends to more than the 3:1 resonance radius. Under
the influence of a tidal force from the companion, the accretion disk becomes
asymmetric and starts precessing in the inertial frame (Whitehurst, 1988;
Whitehurst & King, 1991). In addition to the SU UMa systems, superhumps are
also observed permanently (called permanent superhumps) in some CVs with no
extreme brightness changes (Patterson, 1999). Negative superhumps (or
infrahumps) have also been detected in permanent superhump systems, with
periods a few percent shorter than the corresponding orbital periods due to
disk nodal precession (e.g. AM CVn, Harvey et al., 1998).

In addition to the CVs, superhump modulations were observed in some LMXBs.
Haswell et al. (2001) indicated that the the 3:1 resonance condition
($q\lesssim 0.33$) could be more easily achieved in LMXBs than in CVs because
of their larger accretors’ masses. They proposed that superhumps might be
detectable for LMXBs with orbital period less than 4.2 hours. For the black
hole LMXBs, superhumps have been found in Nova Mus 1991, GRO JU0422+32
(O’Donoghue & Charles, 1996), XTE J1118+480 (Zurita et al., 2002), and
probably in GS 2000+25 (O’Donoghue & Charles, 1996) and GRS 1915+105 (Neil et
al., 2007). On the other hand, to explain the discrepancy between the periods
observed in the X-ray band (3000 s) and optical band (3027 s) of dipping-
bursting LMXB X1916-053, White (1989) suggested that the optical modulation is
similar to the superhumps seen in SU UMa type dwarf novae due to disk
precession in a period of $\sim$3.9 day whereas the X-ray dip period is the
orbital period. Chou et al. (2001) detected a series of $\sim$3.9 day beat
sidebands, including the optical period (3027 s), beside the X-ray dip period
in the X-ray light curves collected by RXTE 1996 observations. This $\sim$3.9
day periodicity is likely due to the dip shape change caused by the slow disk
precession in the inertial frame. Furthermore, Retter et al. (2002) discovered
a negative superhump with a period of 2979 s in an X-ray light curve and Hu et
al. (2008) found a 4.87 day periodic variation in the dip width from RXTE 1998
observations caused by the nodal precession of the accretion disk. These
findings imply that X 1916-053 is similar to a permanent superhump system in
the CVs (Retter et al., 2002; Hu et al., 2008).

The 693.5 s periodic modulation discovered in the FUV band is interpreted as a
superhump in the 4U 1820-30 system (Wang & Chakrabarty, 2010), that indicates
that the accretion disk radius exceeds the 3:1 resonance radius and precesses
with a period of $\sim$0.8 day. Although the superhumps in LMXBs are usually
observed in longer wavelength bands, such as optical band for black hole LMXBs
and X 1916-053, and FUV band for 4U 1820-30, the disk precesion effects are
also observable in X-ray light curves. For instance, the X-ray dip shape
changes with a $\sim$3.9 days disk apsidal precession (Chou et al., 2001) or
$\sim$4.8 days disk nodal precession (Hu et al., 2008) in X 1916-053. Although
4U 1820-30 is not a dipping LMXBs, orbital modulation in the X-ray band is
believed to be the result of absorption by the structure in the disk rim
(Stella et al., 1987). Therefore, as the disk undergoes processing, the beat
sidebands beside the orbital peak, as detected by X 1916-053 (see Chou et al.,
2001, Figure 5) with the period $P_{side}$ satisfied

${1\over{P_{side}}}={1\over{P_{orb}}}-{n\over{P_{prec}}}$ (10)

where $P_{orb}$ is the orbital period, $P_{prec}$ is the disk precession
period, and $n=\pm 1,\pm 2,...$ in the power spectrum. In this work, we have
detected two significant sidebands with periods of $P_{1}$=691.6$\pm$ 0.7 s
(n=1 in Eq 10), which was consistent with the proposed superhump period of
693.5$\pm$1.3 s (Wang & Chakrabarty, 2010), and $P_{2}$=698.3$\pm$ 0.8 s
(n=2). Combined with the orbital period detected in the power spectrum
($P_{orb}=684.9\pm 0.5$ s), we estimated that the accretion disk is precessing
in a period of 0.8$\pm$0.1 day.

The mass ratio can be inferred from the orbital and superhump periods. From
the periods we obtained in this work, the superhump excess is defined as
$\varepsilon=(P_{sh}-P_{orb})/P_{orb}$, where $P_{sh}$ is the superhump
period, i.e. $\varepsilon$=0.010$\pm$0.001. According to the empirical
equation $\varepsilon=0.18q+0.29q^{2}$ proposed by Patterson et al. (2005),
the mass ratio is estimated as $q$=0.05. On the other hand, for the low mass
ratio, Goodchild & Ogilvie (2006) derived a linear relation of the superhump
excess and mass ratio, i.e. $\varepsilon=(0.2076\pm 0.0003)q-4.1\times
10^{-4}$. From this relation, the mass ratio is $0.049\pm 0.005$ for 4U
1820-30. Both results indicate that the mass ratio is $q\approx 0.05$. If the
mass of the neutron star in 4U 1820-30 equals to its canonical value, i.e.
1.4$M_{\sun}$, the mass of the companion is 0.07$M_{\sun}$, which is
consistent with the companion’s mass range, i.e.
0.06$M_{\sun}$-0.08$M_{\sun}$, suggested by Rappaport et al. (1987).

Although no negative superhump signal is detected in the 4U 1820-30 NICER
observations, based on the similarity between 4U 1820-30 and X 1916-053, we
expect that 4U 1820-30 is also probably a permanent superhump system. If so,
from the equation derived by Iaria et al. (2021), the nodal precession
frequency can be expressed as

${\omega_{n}\over\omega_{orb}}=-{5\over 32}{q\over{1+q}}cos\delta$ (11)

where $\omega_{n}$ is the nodal precession frequency, $\omega_{orb}$ is the
orbital frequency, and $\delta$ is the tilt angle of the accretion disk. For a
small tilt angle $cos\delta\approx 1$ and $q$=0.05 from the superhump period,
the nodal precession period is evaluated as 1.07 days and the negative
superhump period should be 679.9 s. The detection of negative superhump could
further identify that 4U 1820-30 is a permanent superhump system as X
1916-053. However, the detection may be hampered by the $\sim$1 day
observation gaps because it is close to the nodal precession period. Probably
an uninterrupted 2-3 days observation is necessary to identify the existence
of a negative superhump.

Conversely, even though our detection of orbital period sidebands with periods
of $P_{1}$=691.6$\pm$0.7 s and $P_{2}$=698.5$\pm$ 0.8 s is possibly due to
disk precession as the superhump detected in FUV band (Wang & Chakrabarty,
2010), we cannot exclude the possibility that the sidebands are induced by a
distant third star orbiting around the binary system. To explain the $\sim$171
days long-term modulation of the 4U 1820-30 system, Grindlay (1986, 1988)
proposed that in the high-density star cluster core as NGC 6624, a star may be
captured by the binary system and form a hierarchical triple system. As the
third star orbiting the binary system, the eccentricity of the binary
modulates with a period $P_{long}=KP^{2}_{outer}/P_{orb}$ where $P_{outer}$ is
the third star orbital period, $P_{orb}$ is the binary orbital period, and $K$
is a constant of order unity, depending on the mass ratios and relative
inclination of the third star orbit and binary (Mazeh & Shaham, 1979). The
eccentricity variation changes the binary separation and thus the Roche lobe
radius leads to mass loss rate modulation with a period of $\sim$171 days.
This model was further confirmed by Zdziarski et al. (2007a). For 4U 1820-30,
the triple model implies that the orbital period of third star is $\sim$1.1
days for $K\simeq 1$. Chou & Grindlay (2001) attempted to search for possible
beat sidebands from the light curves collected by RXTE but failed, which was
likely due to either the $\sim$1.1 day observation gaps or the variation being
too small. Conversely, by applying the triple model, our detection of the beat
sidebands indicates that the orbital period of third companion is 0.8$\pm$0.1
day, a little smaller than the expected $\sim$1.1 days. However, this value
could be a factor of two larger or smaller owing to the different relative
inclination angles (Chou & Grindlay, 2001). The third companion may have
various effects on the binary modulation with the third companion orbital
period. As the binary system rotates, the beat sidebands from the power
spectrum near the orbital signal may be observed. For instance, as the third
companion star moving around the binary system in a prograde orbit, the small
tidal force from it may slightly alter the binary separation. Because the
Roche lobe radius is proportional to the binary separation and the mass loss
rate is sensitive to the Roche lobe radius, it may cause a small variation of
accretion stream from the $L_{1}$ point and change the vertical structure of
the disk rim with a period of $\sim 692$ s, i.e. the beat period of binary and
third star orbital period, thus the beat sidebands are observable in the X-ray
light curves. However, to identify if the $\sim 692$ s periodicity is induced
by the third companion, further evidence is required. For example, Chou &
Grindlay (2001) considered the third companion with a maximum mass of
0.5$M_{\sun}$ and the motion of the binary radius relative to the mass center
of the triple system was only $\sim 3.4sini_{3}$ lt-s where $i_{3}$ is the
inclination of the third companion. Although this small motion only induces an
orbital phase variation of less than 0.005, it is probably detectable by long-
term continuous observations with a highly sensitive X-ray telescope.

##  5 Summary

4U 1820-30 is the most compact LMXB located near the center of globular
cluster NGC 6624. In this work, we analyzed its orbital modulation from the
X-ray light curves collected by NICER between 2017 to mid of 2022. In
combination with the records since 1976, the orbital phase evolution in this
46 years can be well described by a quadratic curve with a detected orbital
period derivative of $\dot{P}/P=(-5.21\pm 0.13)\times 10^{-8}$ yr-1. In
addition to updating the orbital ephemeris, we found that no significant
second order period derivative can be observed with a 2$\sigma$ upper limit of
$|\ddot{P}|<5.48\times 10^{-22}$ s s-2.

The discrepancy between the detected (negative) and theoretically predicted
(positive) orbital period derivative may be owing to the binary system
accelerating in the gravitational potential in the globular cluster. However,
as discussing the intrinsic orbital period derivative, we concluded that the
4U 1820-30 may have significant amount of mass outflow, which makes the
intrinsic orbital derivative much larger than the one evaluated from the
orbital angular momentum loss driven by gravitational radiation only.
Therefore, it is difficult to infer the acceleration of the binary system by
using the observed orbital period derivative because there is too much
uncertainty in the intrinsic orbital period derivative.

On the other hand, from the X-ray light curves collected by NICER between
September 1 and 5 2017, a significant periodic modulation with a period of
$691.6\pm 0.7$ s, consistent with the superhump period discovered in the FUV
band of the Hubble Space Telescope, was detected. This modulation is likely
caused by the change of the absorption structure of the disk rim with a disk
precession period of $0.8\pm 0.1$ day coupling with the binary orbital motion.
From the superhump period, the mass ratio is evaluated as $q\approx 0.05$. If
the mass of neutron star is $1.4M_{\sun}$, the companion’s mass is
0.07$M_{\sun}$, which is consistent with the mass range 0.06 to
0.08$M_{\sun}$, suggested by Rappaport et al. (1987). While the $691.6\pm 0.7$
s periodicity in X-ray band is likely the superhump period, it is still
possible that the modulation is induced by a hierarchical third star orbiting
around the binary system with an orbital period of $0.8\pm 0.1$ day, such as
the tidal force from the third companion changing the accretion stream, as
well as the vertical structure of disk rim. Long-term observations with higher
sensitivity X-ray telescope may be helpful for the identification.

The NICER data for this research were obtained from the High Energy
Astrophysics Science Archive Research Center (HEASARC) online service,
provided by the NASA/Goddard Space Flight Center. This work was supported by
the National Science and Technology Council of Taiwan through the grant MOST
110-2112-M-008-029-.

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Figure 1:  NICER observation distribution of 4U 1820-30. The light curve is
detected by MAXI and the black bars represent the NICER observation times.

Figure 2: Distribution of $\Delta n_{i}/\sigma_{\Delta n_{i}}$ for the
observation with ObsId 1050300126. All centroids (dashed lines) are close to
the expected value ($<\Delta n_{i}>=0$) except $\Delta n_{3}$, which indicates
that MPU3 has significant lower exposure time than the others.

Figure 3: Typical orbital modulation profile observed on September 3 2017
(ObsID 1050200113) folded by the linear ephemeris as shown in Eq. 1. The solid
line is the best sinusoidal fitting for the modulation

Figure 4: Orbital phase evolution of 4U 1820-30 from 1976 to 2022 folded by
the linear ephemeris shown in Eq 1. The filled diamond are the orbital phases
from NICER observations and the dashed line is the optimal quadratic fit
result.

Figure 5: Trend removed light curve collected by NICER between September 1 and
5, 2017 for the superhump modulation search.

Figure 6: Top panel: power spectrum (dashed line) of trend removed light curve
collected by NICER between September 1 and 5, 2017. The main peak is from the
orbital modulation. A Gaussian function fitting (solid line) resulted in a
frequency $f_{0}=1.460\pm 0.001$ mHz (or $P_{0}=684.9\pm 0.5$ s). Bottom
panel: power spectrum (dashed line) of residual light curve (after the orbital
modulation being removed). The frequencies of the two peaks are
$f_{1}=1.446\pm 0.001$ mHz (or $P_{1}=691.6\pm 0.7$ s) and $f_{2}=1.432\pm
0.002$ mHz (or $P_{2}=698.3\pm 0.8$ s) evaluated by Gaussian fittings (solid
lines). The $f_{1}$ is consistent with the superhump frequency proposed by
Wang & Chakrabarty (2010).

Figure 7: Residual light curve folded by the 691.6 s superhump period detected
in this work with an arbitrary phase zero epoch.

Figure 8: Expected luminosity from Eq 9 vs. mass outflow ratio (1-$\beta$)
with different $\alpha$ values for $M_{2}=0.07M_{\sun}$ . The dashed line is
the observed luminosity $<L>=6.0\times 10^{37}$ erg s-1

Figure 9: Constraints of $\alpha$ and mass outflow ratio ($1-\beta$) for
$L=6.0\times 10^{37}$ erg s -1

Figure 10: Possible intrinsic orbital period derivative vs. mass outflow
($1-\beta$) for $L=6.0\times 10^{37}$ erg s-1

Table 1:  NICER Observation Grouping ObsId | Start Time | End Time | Exposure Time
---|---|---|---
| (yy-mm-dd) | (yy-mm-dd) | (s)
0050300108-109 | 17-06-26 | 17-06-27 | 6143
0050300113-117 | 17-07-09 | 17-07-13 | 5289
0050300119-120, 1050300101-102 | 17-07-14 | 17-07-18 | 3664
1050300105 | 17-08-22 | 17-08-22 | 4146
1050300107-108 | 17-08-28 | 17-08-29 | 33234
1050300109-110 | 17-08-30 | 17-08-31 | 28501
1050300111-112 | 17-09-01 | 17-09-02 | 26775
1050300113 | 17-09-03 | 17-09-03 | 28161
1050300114 | 17-09-04 | 17-09-04 | 26849
1050300115 | 17-09-04 | 17-09-05 | 22491
1050300116-118 | 17-09-06 | 17-09-08 | 22161
1050300121-126 | 17-11-07 | 17-11-12 | 10635
1050300128-130 | 18-02-13 | 18-02-16 | 3231
1050300134-138 | 18-03-12 | 18-03-16 | 3416
2050300102-103 | 19-06-05 | 19-06-06 | 6677
2050300106-107 | 19-06-09 | 19-06-10 | 13820
2050300108-109 | 19-06-12 | 19-06-13 | 18960
2050300110-111 | 19-06-13 | 19-06-15 | 9868
2050300112 | 19-06-15 | 19-06-16 | 12539
2050300114-115 | 19-06-19 | 19-06-21 | 16008
2050300116-118 | 19-06-23 | 19-06-25 | 13903
2050300120-124 | 19-06-27 | 19-07-01 | 11050
2050300125-127 | 19-07-03 | 19-07-05 | 13085
2050300128-129 | 19-07-06 | 19-07-07 | 11671
3050300101-102 | 20-05-20 | 20-05-22 | 2630
3050300105-108 | 20-10-07 | 20-10-11 | 9975
3586010101-105 | 20-11-10 | 20-11-14 | 10185
4050300101 | 21-04-27 | 21-04-27 | 2928
4680010101-103 | 21-05-02 | 21-05-04 | 5583
4050300104 | 21-05-19 | 21-05-19 | 4620
4663010101-102 | 21-06-19 | 21-06-20 | 19185
4050300105-106 | 21-08-23 | 21-08-24 | 11590
4663020101-102 | 21-10-08 | 21-10-09 | 8270
4680010105-106 | 21-10-13 | 21-10-14 | 2788
4680010107-109 | 21-10-17 | 21-10-20 | 3824
4680010112-115 | 21-10-24 | 21-10-27 | 12627
4680010116 | 21-10-28 | 21-10-28 | 11937
4680010117-119 | 21-10-29 | 21-10-31 | 11012
4680010120-123 | 21-11-01 | 21-11-05 | 11783
4680010124-128 | 21-11-06 | 21-11-11 | 5615
5050300101-102 | 22-03-30 | 22-03-31 | 4465
5050300107-111 | 22-04-04 | 22-04-09 | 8663
5604010103,5050300112-116 | 22-04-16 | 22-04-20 | 8595
5604010301-303 | 22-04-30 | 22-05-02 | 9453
5604010401-402 | 22-06-12 | 22-06-13 | 10535
5604010501-503 | 22-06-26 | 22-06-28 | 5993
5604010601-602 | 22-07-08 | 22-07-09 | 610349

Table 2: Average Orbital Phase of 4U 1820-30 from 2017-2022 NICER Observations Mean Arrival Time | Phase | Error
---|---|---
($BJD_{TDB}$) |  |
2457999.65 | -1.58 | 0.03
2458178.08 | -1.76 | 0.05
2458656.07 | -1.74 | 0.05
2459078.61 | -1.86 | 0.07
2459429.82 | -1.96 | 0.11
2459719.61 | -2.03 | 0.03