# A New Puzzling Periodic Signal in GeV Energies of the $\gamma$-Ray Binary LS
I +61∘303

Pengfei Zhang  Department of Astronomy, School of Physics and Astronomy, Key
Laboratory of Astroparticle Physics of Yunnan Province, Yunnan University,
Kunming 650091, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract

LS I + 61∘303 is a high-mass X-ray binary system comprising a massive Be star
and a rapidly rotating neutron star. Its spectral energy distribution across
multi-wavelengths categorizes it as a $\gamma$-ray binary system. In our
analysis of LS I + 61∘303 using _Fermi_ -LAT observations, we not only
confirmed the three previously discussed periodicities of orbital,
superorbital, and orbital-superorbital beat periods observed in multi-
wavelength observations, but also identified an additional periodic signal.
This newly discovered signal exhibits a period of $\sim$26.3 day at a $\sim
7\sigma$ confidence level. Moreover, the power spectrum peak of the new signal
gradually decreases as the energy increases across the energy ranges of
0.1–0.3, 0.3–1.0, and 1.0–500.0 GeV. Interestingly, a potential signal with a
similar period was found in data obtained from the Owens Valley Radio
Observatory 40 m telescope. We suggest that the newly discovered periodic
signal may originate from a coupling between the orbital period and the
retrograde stellar precession period.

Gamma-rays(637); Gamma-ray sources(633); Periodic variable stars(1213)

##  1 Introduction

LS I + 61∘303 is a high-mass X-ray binary system, composed of a young massive
Be star (Grundstrom et al., 2007) and a rapidly rotating neutron star (Weng et
al., 2022), with a non-thermal electromagnetic emission extending from MHz
radio frequencies to TeV $\gamma$-ray energies (Dubus, 2013). Its properties
of the spectral energy distribution (SED) in multi-wavelength emissions
indicate that LS I + 61∘303 is dominated by the MeV-GeV $\gamma$-rays, this
makes it to be a $\gamma$-ray binary. Until now, very few $\gamma$-ray binary
systems have been found to produce detectable $\gamma$-ray emissions, with
only a handful in our Galaxy (Aharonian et al., 2005a, b; Albert et al., 2006;
Hinton et al., 2009; Fermi LAT Collaboration et al., 2012a; Corbet et al.,
2019) and one in the Large Magellanic Cloud (Corbet et al., 2016).

LS I + 61∘303 has a orbital period of $P_{1}\sim 26.496$ day (Gregory, 2002)
with an eccentricity $e\sim 0.54$ (Aragona et al., 2009) and a companion mass
of 12.5 $M_{\sun}$ (Casares et al., 2005), and locates at a distance of 2.0
kpc (Frail & Hjellming, 1991). The zero point of its orbital phase ($\phi$)
has historically been defined at JD = 2,443,366.775 (i.e. $\phi_{0}$) in
Gregory (2002), and the orbital phase of its periastron position is $\phi\sim
0.275$, which adopted from Aragona et al. (2009). In addition to the orbital
period of LS I + 61∘303, a long-term modulation period of 1667 day ($P_{2}$)
was discovered in GHz radio observations by Gregory (2002). With the
increasing accumulation of observational data for LS I + 61∘303, Massi & Jaron
(2013) reported a third modulation period of 26.92 day ($P_{3}$) in 6.7 years
Green Bank Interferometer radio database at 2.2 GHz and 8.3 GHz, and this
period was also revealed in 0.1–300.0 GeV with the _Fermi_ -LAT observations
by Jaron & Massi (2014); Chernyakova et al. (2023). Regarding the potential
origin of the final two periods, previous literature offered extensive
discussions (Gregory, 2002; Massi & Jaron, 2013; Jaron & Massi, 2013; Massi &
Torricelli-Ciamponi, 2014; Jaron & Massi, 2014; Massi et al., 2015; Massi &
Torricelli-Ciamponi, 2016). Based on _Fermi_ -LAT GeV observations for LS I +
61∘303, Xing et al. (2017) suggested that a non-axisymmetric circumstellar
disk may be present around the Be companion, which rotates with a period of
1667 days, leading to the long-term modulation, and the period of 26.92 days
is a result of the beat frequency ($f_{3}=\frac{1}{f_{1}-f_{2}}$) between the
orbital ($P_{1}$) and long-term ($P_{2}$) periods.

In GeV, LS I + 61∘303 has a $\gamma$-ray counterpart named J0240.5+6116 in the
first _Fermi_ -LAT source catalog (1FGL; Abdo et al., 2010a) and J0240.5+6113
in the fourth catalog Data Release 4 (4FGL-DR4; Fermi-LAT collaboration et
al., 2022; Ballet et al., 2023). Based on it, we analyzed the $\gamma$-ray
events around LS I + 61∘303 from the $\sim$15 year _Fermi_ -LAT observations.
In our timing analysis, besides three periodicities that have been reported in
previous literature, an additional new periodic signal reveals at $26.301\pm
0.037$ day at a $\sim 7\sigma$ confidence level. Interestingly, a potential
signal has a similiar period at $\sim 26.16\pm 0.11$ day shown in the Owens
Valley Radio Observatory (OVRO) data in the Figure 3 (d) of Jaron et al.
(2018), but it is not significant. The following presents our data analysis
and results.

##  2 Data Analysis and Results

###  2.1 Data Reduction

Table 1: Best-fit results of likelihood analysis

Models | Parameter values
---|---
LP | $\alpha$ | $\beta$ | $E_{\rm b}$ | TS | $F_{\rm ph}$ | $F_{\rm en}$
| 2.461$\pm$0.006 | 0.112$\pm$0.002 | 1.517 | 227111.0 | $7.861\pm 0.030$ | 4.535$\pm$0.014
| 2.445$\pm 0.015^{\phi}$ | 0.117$\pm 0.006^{\phi}$ | 1.517ϕ | 31694.1ϕ | 7.283$\pm$0.083ϕ | 4.308$\pm$0.047ϕ
PLEC | $\Gamma$ | $b$ | $E_{\rm c}$ | TS | $F_{\rm ph}$ | $F_{\rm en}$
| 1.979$\pm$0.015 | 0.605$\pm$0.026 | 2.927$\pm$0.351 | 226614.0 | 7.793$\pm$0.032 | 4.498$\pm$0.017

Notes. Best-fit parameter values of the likelihood analysis in 0.1–500.0 GeV
for LP and PLEC models, ϕvalues derived from the dip data, $E_{\rm b}$ and
$E_{\rm c}$ in units of GeV. $F_{\rm ph}$ is the integrated photon flux in
units of $10^{-7}$ photons cm-2 s-1 and $F_{\rm en}$ is the integrated energy
flux in units of $10^{-10}$ erg cm-2 s-1.

We carried out a whole data analysis by selected the _Fermi_ -LAT Pass 8
_Front+Back_ events (evclass = 128 and evtype = 3) in the energy range of
0.1–500.0 GeV within a $20^{\circ}\times 20^{\circ}$ region of interest (RoI)
centered at 4FGL J0240.5+6113 (R. A. = $\rm 02^{h}40^{m}34^{s}.22$ and decl. =
$+61^{\circ}13^{\prime}43\farcs 30$). The data observations span from 2008
August 4 to 2023 September 21 (MJD: 54682.687–60208.046). We removed the
events with zenith angles $>90^{\circ}$ to exclude the $\gamma$-ray
contamination from the Earth Limb, those with quality flags of “bad”, by a
expression of “DATA_QUAL$>$0 && LAT$\\_$CONFIG==1” to save high-quality events
in good time intervals. The instrumental response function “P8R3_SOURCE_V3”
and the latest available Fermitools-2.2.0 were used in the following analysis.

Figure 1: $\gamma$-ray SEDs of LS I + 61∘303 in 0.1–500.0 GeV. The best-fit LP
models of the dip and whole LAT data are shown as red dashed and black solid
lines, respectively. The PLEC model is shown with a blue dotted line. The gray
dashed-dotted and red dotted histograms stand for TS values of data from the
dip and whole LAT.

Figure 2: TS maps in 0.1–500.0 GeV covering a $4^{\circ}\times 4^{\circ}$
region centered at 4FGL J0240.5+6113 with a pixel of $0^{\circ}.1$. The
$\gamma$-ray sources reported in the 4FGL-DR4 are shown with the green
crosses. Left panel: TS map standing for the $\gamma$-ray emissions from LS I
+ 61∘303 was created by fixing all model parameters in the new model and
removing 4FGL J0240.5+6113 from the model. Right panel: residual TS map was
created based on the same model with target having a LP spectral shape, with
the exception that 4FGL J0240.5+6113 is retained.

A model file was created by a python script make4FGLxml.py based on the newest
4FGL-DR4 catalog, which includes all the parameters of the sources within
$25^{\circ}$ centered at 4FGL J0240.5+6113. Then we modified the file to free
the flux normalizations and spectral parameters for the sources within
$5^{\circ}$, the normalizations for the sources within
$5^{\circ}$–$10^{\circ}$, together with the ones outside $10^{\circ}$ but
identified as variable sources. The two normalizations for the diffuse
emission components of Galactic and extragalactic were also set free. All
other parameters were fixed to be their values as them provided in 4FGL-DR4.
Then a binned maximum likelihood analysis was performed to update the free
parameters by employed the 15 yr _Fermi_ -LAT data. The best-fit parameters
were saved as a new model file, the following analysis based on this model.
For LS I + 61∘303, a spectral shape of a log-parabola model (LP),
$dN/dE=N_{0}(E/E_{b})^{-[\alpha+\beta\log(E/E_{b})]}$, is provided in 4FGL-
DR4. The best-fit parameters of LS I + 61∘303 are summarized in Table 1.
Considering that a pulsar hosted in LS I + 61∘303, we also used typical pulsar
$\gamma$-ray model of a power-law with an exponential cutoff (PLEC),
$dN/dE=N_{0}(E/E_{0})^{-\Gamma}\exp[-(E/E_{\rm c})^{b}]$, to describe the
target’s $\gamma$-ray emission. The best-fit parameters of the PLEC model are
also listed in Table 1.

Table 2: Values of flux data points of SEDs

| Whole data
---|---
Flux | 13.90(14) | 12.98(10) | 12.25(8) | 10.59(8) | 7.98(8) | 5.50(9) | 3.22(9) | 1.46(9) | 0.71(8) | 0.46(9) | 0.27(10) | 0.56(6) | 0.13(2)
TS | 31232.1 | 39091.2 | 48564.4 | 47672.0 | 33271.5 | 17115.8 | 6326.9 | 1574.0 | 420.3 | 141.5 | 36.4 | 58.7 | 6.8
| Dip data
Flux | 12.31(44) | 11.76(26) | 11.76(21) | 10.21(21) | 7.74(21) | 5.39(23) | 3.13(24) | 1.69(50) | 0.55(20) | 0.70(29) | 0.28(27) | 0.72(19) |  1.06a
TS | 3777.0 | 4975.9 | 6941.4 | 6806.6 | 4856.0 | 2455.7 | 959.7 | 289.0 | 35.5 | 39.7 | 6.7 | 9.3 | —

Notes. Values of flux data points of SEDs for the whole and dip data in
0.1–500.0 GeV based on LP model, the fluxes in units of $10^{-11}$ erg cm-2
s-1. Last data pointa of dip data is the 95% flux upper limits. Numbers in
parentheses represent uncertainties on the last digit.

Based on the new model, we performed spectral analysis to derive the SED of LS
I + 61∘303 in 0.1–500.0 GeV. The whole LAT data was segmented into 13 equally
logarithmically spaced energy bins. The data points of SED were extracted by
performed the maximum likelihood analysis. LS I + 61∘303’s SED is shown in
Figure 1 and the accurate numbers are listed in Table 2. For ease of
comparison, we show the LP and PLEC models in Figure 1 with black solid and
blue dotted lines. From them, we know that the LP model performs better than
PLEC for target’s $\gamma$-ray emission.

Figure 3: LSP power spectrum (black histogram) constructed from 0.1–500.0 GeV
AP light curve of LS I + 61∘303\. The three periodicities of $P_{\rm new}$,
$P_{\rm orb}$, and $P_{\rm beat}$ are too close, resulting in overlap, for
clarity, please refer to Figure 4. The $P_{\rm sorb}$ is marked with a gray
dotted line. The orbital and survey repeat periods are marked with two red
arrows, and the first and second harmonics of $P_{\rm orb}$ are shown with two
purple arrows. The pink shaded region stands for the portion of power spectrum
shown in Figure 4.

In order to reveal the $\gamma$-ray emissions around LS I + 61∘303, a TS map
with a region of $4^{\circ}\times 4^{\circ}$ was created by employed _gttsmap_
based on the new model file. And a residual TS map was also created to exclude
the contamination from the new possible nearby $\gamma$-ray sources, that not
included in 4FGL-DR4. We show the two TS maps in Figure 2. From them, we
believe that the $\gamma$-ray events around LS I + 61∘303 are well described
by the new model, and no new $\gamma$-ray source has been identified beyond
4FGL-DR4.

###  2.2 Timing Analysis

An initial light curve was constructed by employed a modified version of
aperture photometry (AP) method centered at 4FGL J0240.5+6113. Taking into
account the instrument performance of LAT to maximize the signal-to-noise
ratio, an aperture radius of $3^{\circ}.16$ is adopted with a selection
criteron of an angle $\theta<$ max(6.68$-$1.76log${}_{10}(E_{\rm
MeV}),1.3)^{\circ}$, as that performed in Abdo et al. (2010b). The light curve
has an energy range of 0.1–500.0 GeV with time bins of 600 s. We excluded the
periods when 4FGL J0240.5+6113 was within $5^{\circ}$ of the Sun and Moon by
_gtmktime_. Exposures were calculated using _gtexposure_ to mitigate the
impact of significant exposure variations across different time bins. And
$\gamma$-rays arrival times are also barycenter corrected using _gtbary_. We
assigned weights to events with their probabilities of originating from 4FGL
J0240.5+6113 by employed _gtsrcprob_ based on the new model. The light curve
was then constructed by summing these probabilities, as opposed to simply
counting the number of photons within the aperture in each time bin (Kerr,
2011; Fermi LAT Collaboration et al., 2012b; Corbet et al., 2019).

Power spectrum was created for the AP light curve by employed a method of a
Lomb–Scargle periodogram (LSP; Lomb, 1976; Scargle, 1982), and we show it in
Figure 3. It covers a frequency range from $f_{\rm max}$ = 1/0.05 day-1 to the
entire _Fermi_ -LAT observations ($f_{\rm min}$ = 1/5525 day-1), and the
number of independent frequencies (i.e., the trial factor) was calculated by
$N=(f_{\rm max}-f_{\rm min})/\delta f$ = 110499, where $\delta f$ is frequency
resolution determined by the length of the _Fermi_ -LAT observations. In the
spectrum, the $\gamma$-ray periodicities of LS I + 61∘303 that have been
reported in previous works, i. e. the orbital ($P_{\rm orb}$), superorbital
($P_{\rm sorb}$), and orbital–superorbital beat periods ($P_{\rm beat}$),
reveal at $26.493\pm 0.058$, $1817.54\pm 252.84$, and $26.927\pm 0.055$ day,
respectively. We marked them with purple dashed, gray dotted, and orange
dashed-dotted lines, respectively. Interestingly, besides these $\gamma$-ray
periodicities, an additional periodic signal also reveals at $26.301\pm 0.037$
day, this new periodic signal ($P_{\rm new}$) is marked with a green solid
line in Figure 3. Their uncertainties of the $\gamma$-ray periodicities were
taken from their half-widths at half-maximum of each power peak. In our AP
analysis the background $\gamma$-rays are not modeled out for each time bin,
hence two artifact signals (marked with red arrows) originate from the 96
minutes orbital period and the survey repeat period at twice of them of
_Fermi_
satellite111https://fermi.gsfc.nasa.gov/ssc/data/analysis/LAT_caveats_temporal.html
can be seen at their corresponding periods.

Figure 4: LSP power spectrum zoomed in Figure 3 (i.e. the pink shaded region).
The new detected periodicity $P_{\rm new}$ is marked with a green solid line,
and others for $P_{\rm orb}$ and $P_{\rm beat}$ are shown with purple dashed
and orange dashed-dotted lines, respectively. The red dashed and blue dashed-
dotted lines stand for $5\sigma$ and $7\sigma$ confidence levels. The inset
plot shows that the schematic LSP power of LS I + 61∘303 drawn from the Figure
3 (d) of Jaron et al. (2018).

Because the three periodicities of $P_{\rm new}$, $P_{\rm orb}$, and $P_{\rm
beat}$ have similar periods and result them in overlap each other in Figure 3,
for clarity we zoom in the power spectrum in the pink shaded region in Figure
4. The heights of the peaks of $P_{\rm new}$, $P_{\rm orb}$, $P_{\rm sorb}$
and $P_{\rm beat}$ are $\sim 39.1$, $\sim 465.9$, $\sim 121.5$ and $\sim
105.0$ compared to the mean power level, respectively. The normalization
method utilized here is detailed in Horne & Baliunas (1986). Using the method
provided by Lomb (1976) and Scargle (1982), the probability ($p_{{}_{\rm
lsp}}$) to obtain the power level of $P_{\rm new}$ equal or higher than $39.1$
from a chance fluctuation (a noise) is $\sim 1.0\times 10^{-17}$. This method
is often used for detecting periodic signals in the white noise, as
demonstrated in the search for $\gamma$-ray binary systems by Corbet et al.
(2016, 2019); Fermi LAT Collaboration et al. (2012a). Taking into account the
trial number $N$, the False Alarm Probability (FAP) is estimated at
FAP=$1-(1-p)^{N}\sim p\times N\sim 1.1\times 10^{-12}$, which corresponding to
a $\sim 7.1\sigma$ confidence level. In Figure 4, we show $5\sigma$ and
$7\sigma$ confidence levels with red dashed and blue dashed-dotted lines,
respectively. Interestingly, in radio OVRO data, a potential signal exhibits a
period ($26.16\pm 0.11$ day, the error derived from the half-widths at half-
maximum of the power peak) similar to the new signal. For easy reference, we
show the schematic LSP power of LS I + 61∘303 in the inset plot of Figure 4,
that drawn from Jaron et al. (2018).

In addition, a potential periodic signal of nearby $\gamma$-ray sources can
also cause a modulation for LS I + 61∘303 because of the broad PSF of the
_Fermi_ -LAT, particularly at lower energies. To prevent this situation, we
also constructed the power spectra for the two closest sources (4FGL
J0248.4+6021 and 4FGL J0243.3+6319) with the same process, and no similar
signal was identified for the new periodic signal claimed here.

Figure 5: LS I + 61∘303’s energy-dependent LSP power spectra in 0.1–0.3,
0.3–1.0, and 1.0-500.0 GeV. The black, red, and green histograms stand for the
LSP power spectra constructed with 0.1–0.3, 0.3–1.0, and 1.0–500.0 GeV AP
light curves, respectively.

To explore the energy-dependent periodicity of $P_{\rm new}$, we also created
three power spectra based on their AP light-curves in 0.1–0.3, 0.3–1.0, and
1.0–500.0 GeV. And they are shown in Figure 5 with black, red, and green
histograms, respectively. From it, we know that their power peak of $P_{\rm
new}$ signal gradually decreases as the energy increases across the three
energy ranges.

Figure 6: Phase-resolved light-curves. Flux varies with the corresponding
phase in the four energy intervals in 0.1–500.0, 0.1–0.3, 0.3–1.0, and
1.0–500.0 GeV. And four panels share same y-axis labels. For clarity two
period cycles are shown. The pink shade denotes the dip in $\phi=0.05-0.20$.

###  2.3 Phase-resolved Analysis

On the basis of period of $P_{\rm new}$, we divided the 0.1–500.0 GeV _Fermi_
-LAT events into 20 phase intervals and performed likelihood analysis for each
bin to obtain the phase-resolved light-curve based on the new model file,
except only freeing the flux normalizations for the sources within 5∘ and two
diffuse components. The folded light-curve is shown in Figure 6 (A) with a
phase zero corresponding to MJD 43,366.275, as it in Gregory (2002);
Chernyakova et al. (2023). From it, an obvious dip in $\phi=0.05-0.20$ can be
seen in the folded light-curve.

In order to investigate the phase-resolved light-curve depending on energy, we
constructed phase-resolved light-curves for the three previous energy
intervals. And we show them in B-D panels respectively. Their amplitudes of
the variabilities, i.e. the maximum flux minus the minimum, for the four
phase-resolved light-curve are 0.42, 0.60, 0.10, and 0.03$\times 10^{-8}$
photons cm-2 s-1. As shown in the Figure 6, the amplitudes of the phase-
resolved light-curves of $P_{\rm new}$ in the three energy ranges decrease as
the energy increases.

We also carried out a maximum likelihood analysis for the events in the phase
intervals in the dip (pink shade in Figure 6). Its best-fit LP model is shown
in Figure 1 with a red dashed line and the parameters are listed in Table 1. A
spectral analysis was performed for the data in the dip, the data points of
SED are displayed in Figure 1 and colored red. For the SED of dip data, we
retained data points with TS values $\geqslant$5, while others are displayed
their 95% flux upper limits. Based on Figure 1 and Table 2, it is evident that
there are no significant differences in the SEDs between the entire LAT data
and the dip data.

##  3 Summary and Discussion

LS I + 61∘303 is one of the unusual $\gamma$-ray binaries and composed of a
massive star and a rapidly rotating neutron star (Weng et al., 2022). The main
of the electromagnetic emissions from LS I + 61∘303 are at MeV-GeV energies
(Dubus, 2015; Dubus et al., 2017). The principal emission mechanisms of this
system are thought to be $\gamma$-ray emissions that could originate from
interactions between the relativistic wind coming from a rapidly rotating
neutron star (Dubus, 2006) and the stellar wind from its companion, or from
the relativistic jets generated by accretion onto a neutron star or black hole
(Mirabel & Rodríguez, 1998). In $\gamma$-rays, for LS I + 61∘303, three
periodicities corresponding to the orbital, superorbital, and
orbital–superorbital beat periods have been extensively discussed in previous
works. Using the events from _Fermi_ -LAT spanning from 2008 August to 2023
September, we carried out a timing analysis in 0.1–500.0 GeV for LS I +
61∘303, and a new periodic signal with a period of $26.301\pm 0.037$ day was
detected at a $\sim 7\sigma$ confidence level. Interestingly, in radio OVRO
data, a potential signal (not significant) with a similar period is revealed
at $26.16\pm 0.11$ day (Jaron et al., 2018). Furthermore, their error ranges
also partially overlap. Their results further strengthen our $\gamma$-ray
findings independently. The new signal is relatively weak compared to the
other known periodicities.

As reported by Chernyakova et al. (2023), two periods of LS I + 61∘303,
corresponding to the orbital and beat orbital/superorbital periods, exhibit
strong energy dependence. We also conducted an energy-dependent analysis for
the new signal. Our results in the three same energy intervals indicates that
the power spectrum peak of the new signal gradually decreases from the low
energy range to high. We speculate that the cause of our results may be due to
statistical effects, as there are significantly more photons at lower energies
compared to higher energies.

From the phase-resolved light curves of energy dependent intervals (B, C, and
D panles of Figure 6), we know that their amplitudes of the light-curve
decreases from low energy to high. From them, an obvious dip can be seen at
$\phi=0.05-0.20$, especially in the whole and low-energy data (A and B panels
of Figure 6). While the orbital periodic periodicity of LS I + 61∘303 is
characterized by a single peak in radio to X-ray and $\gamma$-ray bands
(Chernyakova et al., 2023; Xing et al., 2017). Hence, we speculate that the
origin of the new periodic signal may be different from the three
periodicities discussed in previous literature. Perhaps long-term, high-
cadence, and multi-wavelength observations can reveal it in the light curve.

Figure 7: Schematic illustration of the orbit of LS I + 61∘303, drawn from
Dubus (2013). The purple arrow indicates the direction of the neutron star’s
motion, the blue lines represent the spin axis of the Be star, and the green
arrow denotes the direction of the spin axis’s precession.

In our analysis, we checked the various combinations of frequencies involving
known periods (i.e. $P_{\rm average}=\frac{2}{f_{\rm x}+f_{\rm y}}$, $P_{\rm
beat}=\frac{1}{f_{\rm x}-f_{\rm y}}$) or possible harmonics. We found that the
new periodic signal is not a combination of frequencies from any known
periods. It appears to be a new, independent periodic signal. The spin
precession of Be stars in X-ray binaries has long been studied (Lubow &
Ogilvie, 2000; Bate et al., 2000; Ogilvie & Dubus, 2001; Martin et al., 2011).
We suggest that the Be star in the LS I + 61∘303 has a spin axis’s precession,
and the precession direction is opposite to the orbital motion of the neutron
star. To facilitate understanding of our hypothesis, we show this scenario and
the Schematic illustration of orbit of LS I + 61∘303 in Figure 7. Then we can
derive the $P_{\rm prec}$ with a formula of $1/P_{\rm new}=1/P_{\rm
orb}+1/P_{\rm prec}$. The period of $P_{\rm prec}$ is derived to be $\sim
3629$ day, which falls within the timescale of Be star’s precession estimated
by Martin (2023). Therefore we speculate that the newly discovered periodic
signal ($P_{\rm new}$) may originate from a coupling effect between the
orbital period ($P_{\rm orb}$) and the retrograde stellar precession period
($P_{\rm prec}$).

Regardless, the real originating for the new periodic signal still remains
unclear. Based on our results, LS I + 61∘303 may possess some properties that
are currently unknown. More observations in the multi-wavelengths are
encouraged to reveal the origin of the new periodic signal reported here.

We thank anonymous referee for very helpful suggestions and Z. Wang for
discussion about potential origins of the new periodic signal. This work is
supported in part by the National Natural Science Foundation of China Nos.
12163006 and 12233006, the Basic Research Program of Yunnan Province No.
202201AT070137, and the joint foundation of Department of Science and
Technology of Yunnan Province and Yunnan University No. 202201BF070001-020.
P.F.Z. acknowledges the support by the Xingdian Talent Support Plan - Youth
Project.

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instrument. In particular, $\Gamma$ has proven to be more reliable in
producing the desired outcome. This observation motivates the instalment of
thresholds, which filter out insufficient experience, ensuring a certain
quality standard of the instruments evaluated.

Thresholds for best, worst, and better. We consider two types of thresholds.
First, we can impose a threshold $n$ that states the minimum amount of past
witnesses of an instrument’s application. Second, we can impose a threshold on
the minimum success ratio of potential instruments. We begin by considering
the first threshold and adopt the second approach in Section 5.2.2.

Let $n$ refer to the threshold that needs to be met by the total amount of
witnesses
$|\textit{Achiev}^{w}_{\alpha}({\varphi},{\Delta})\cup\textit{Fail}^{w}_{\alpha}({\varphi},{\Delta})|$.
We write $\textit{Instr}^{w}_{\alpha}({\varphi,n})$ to denote the set of
$\varphi$-instruments satisfying threshold $n$, as defined in
($\mathsf{d24}$).

  1. Available $\varphi$-instruments for $\alpha\in\mathtt{Agent}$ at $w$ (with threshold $n$) ($\mathsf{d24}$)
$\textit{Instr}^{w}_{\alpha}({\varphi,n}):=\\{\Delta\ |\
\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi})$ and
$|\textit{Achiev}^{w}_{\alpha}({\varphi},{\Delta})\cup\textit{Fail}^{w}_{\alpha}({\varphi},{\Delta})|\geq
n\\}$

Based on ($\mathsf{d24}$), we can refine the notion of ‘betterness’ as
follows:

  1. Better $\varphi$-instruments, relative to threshold $n\in\mathbb{N}$ ($\mathsf{d25}$)
For each $\Delta,\Gamma\in\textit{Instr}^{w}_{\alpha}({\varphi,n})$,
$\Delta^{\alpha}\succeq^{w}_{\varphi,n}\Gamma^{\alpha}$ iff
$\textit{Succ}^{w}_{\alpha}({\varphi},{\Delta})\geq\textit{Succ}^{w}_{\alpha}({\varphi},{\Gamma})$

We interpret $\Delta^{\alpha}\succeq^{w}_{\varphi,n}\Gamma^{\alpha}$ in
($\mathsf{d25}$) as ‘$\Delta^{\alpha}$ is a weakly better $\varphi$-instrument
than $\Gamma^{\alpha}$ at $w$ given threshold $n$’. Imposing a threshold
installs a quality control in providing axiological judgments of
instrumentality. The undesirable consequences of the $0.999$ success rate
example in the previous section can now be excluded by stipulating a threshold
of any value $n>1$, identifying $\Gamma$ as the best $\varphi$-instrument for
$\alpha$. In ($\mathsf{d26}$) and ($\mathsf{d27}$) below, we define the sets
$\textit{Best}^{w}_{\alpha}({\varphi},{n})$ for ‘among the best’ and
$\textit{Worst}^{w}_{\alpha}({\varphi},{n})$ for ‘among the worst’, which
contain those instruments in the upper, respectively lower bound of the
ordering, satisfying the imposed threshold.

  1. Best $\varphi$-instruments, relative to threshold $n\in\mathbb{N}$ ($\mathsf{d26}$)
$\textit{Best}^{w}_{\alpha}({\varphi},{n}):=\\{\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi,n})\
|\ $ for each
$\Gamma\in\textit{Instr}^{w}_{\alpha}({\varphi,n}),\Delta^{\alpha}\succeq^{w,n}_{\varphi}\Gamma^{\alpha}\\}$

  2. Worst $\varphi$-instruments, relative to threshold $n\in\mathbb{N}$ ($\mathsf{d27}$)
$\textit{Worst}^{w}_{\alpha}({\varphi},{n})\ \\!{=}\
\\!\\{\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi,n})\ |$ for each
$\Gamma\in\textit{Instr}^{w}_{\alpha}({\varphi,n}),\Gamma^{\alpha}\succeq^{w,n}_{\varphi}\Delta^{\alpha}\\}$

Reconsider the example in Figure 4. Depending on the threshold applied, either
$\Delta$ or $\Gamma$ will qualify as among the best (worst)
$\varphi$-instruments at $w_{6}$. For instance, a threshold of $n=3$ excludes
$\Gamma$ as a potential ‘best’ $\varphi$-instrument, i.e.,
$\Gamma\in\textit{Instr}^{w_{6}}_{\alpha}({\varphi,2})$, but
$\Gamma\not\in\textit{Instr}^{w_{6}}_{\alpha}({\varphi,3})$. In fact, we find
that $\Gamma\in\textit{Best}^{w_{6}}_{\alpha}({\varphi},{2})$ and
$\Delta\in\textit{Worst}^{w_{6}}_{\alpha}({\varphi},{2})$, whereas
$\Delta\in\textit{Best}^{w_{6}}_{\alpha}({\varphi},{3})$. Furthermore, observe
that $\Delta\not\in\textit{Worst}^{w_{6}}_{\alpha}({\varphi},{3})$ since for
$\Theta=\Delta\cup\Gamma$ we have
$\Theta\in\textit{Instr}^{w_{6}}_{\alpha}({\varphi,3})$ and
$\textit{Succ}^{w_{6}}_{\alpha}({\varphi},{\Delta})>\textit{Succ}^{w_{6}}_{\alpha}({\varphi},{\Theta})$.

####  5.2.2 Thresholds and Good Instruments

In this section, we suggest possible ways to formally address the assessment
of good-instruments (item (4) of Definition 3). As discussed in Section 2, for
von Wright, comparative judgments are objective since they depend on empirical
data (e.g., collecting past experiences) and logical orderings (e.g., success
ratio orderings). However, such judgments become more problematic when we
address the label ‘good’.

Von Wright determines good instruments on the basis of their ‘good-making
properties’. Recall, how well a knife cuts depends on the good-making property
associated with ‘cutting’: the sharpness of the knife. Such value judgments
concerning knives, von Wright argues, can be objectively assessed, especially
when we order an available set of knives according to their sharpness. Without
such a comparative set, things become more difficult since we need to define a
minimal degree of ‘sharpness’ that enables ‘cutting well’. Von Wright
addresses this degree through causal properties [35, p. 26], which determine
the causal relation between ‘sharpness’ and ‘cutting’—a relationship that may
be objectively assessed through empirical investigations. Then, a knife cuts
‘well’ or qualifies as a good cutting-instrument, whenever it has the causal
property of sharpness needed for cutting (note that this property depends on
what needs to be cut).

We mention two problems related to the above approach. First, such judgments
of goodness are a special case of comparative judgments. That is, the goodness
reflected in a causal property is a goodness relative to (compared to) a
threshold, for instance, the minimum amount of sharpness for cutting
vegetables. In other words, judgments of instrumental goodness remain
comparative but objective with respect to an external criterion, i.e., a
threshold. Therefore, the term ‘good’ employed here does not differ from any
other use of ‘good’ which, for instance, is defined relative to some theory of
ethics. Still, the fact that such a threshold (the causal property) is rooted
in experience allows for the judgment to be objectively evaluated. This
observation is compatible with von Wright’s ideas due to his emphasis on the
logical nature of such judgments.

Second, for an agent in a time-limited decision-making situation, determining
the causal properties of an instrument (such as sharpness) may be an
overburdening task. An agent that aims at cutting, say, a piece of paper may
not have access to determining the causal properties of a different knife’s
sharpness relative to the robustness of the piece of paper. Although causal
properties form objective criteria for obtaining accurate judgments of
instrumentality from a theoretical point of view, from the agent’s point of
view, such criteria are unrealistic and impractical. We find this to be a
strong reason for using the agent’s personal experience with an instrument
(such as the knives available for cutting) as a criterion for judging
instrumentality.

The question that remains is: What qualifies as sufficient experience for
judging an instrument as good for a given purpose? We provide two notions of
threshold that serve to denote sufficient experience: first, we propose a
threshold on the success ratio of a given instrument and, second, we combine
the former with the notion of a minimum amount of past witnesses (see Section
5.2.1).

  1. Goodn $\varphi$-instruments, for $\alpha\in\mathtt{Agent}$ at $w$ (with $n\in\mathbb{R}$ and $0\leq n\leq 1$)
$\textit{Good}^{w}_{\alpha}({\varphi,n}):=\\{\Delta\ |\
\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi})$ and
$\textit{Succ}^{w}_{\alpha}({\varphi},{\Delta})\geq n\\}$ ($\mathsf{d28}$)

Definition ($\mathsf{d28}$) tells us that an action $\Delta$ is a goodn
$\varphi$-instrument for $\alpha$ at $w$, whenever ‘$\Delta$ qualifies as a
$\varphi$-instrument for $\alpha$ at $w$, and its success ratio satisfies
threshold $n$’. Observe that the initial definition of (excellent) instruments
($\mathsf{d16}$) and ($\mathsf{d17}$)—i.e., items (1) and (2) of Definition
3—are limiting cases of goodn instruments, namely, where the success ratios
are $n>0$ and $n=1$, respectively. The relation between ‘good’,
($\mathsf{d16}$) and ($\mathsf{d17}$) is expressed through the following:

  1. $\Delta\in\textit{Good}^{w}_{\alpha}({\varphi,1})$ iff $w\models[\Delta^{\alpha}]^{ex-instr}_{\ast}\varphi$ iff $\Delta\in\textit{Best}^{w}_{\alpha}({\varphi},{1})$

  2. $\Delta\in\textit{Good}^{w}_{\alpha}({\varphi,n})$ iff $w\models[\Delta^{\alpha}]^{instr}_{\ast}\varphi$ (with $0<n\leq 1$)

A poor instrument can be defined in two ways: it is either an instrument
failing to meet a ‘poorness’-threshold $n$ or an instrument that fails to
qualify as good. We opt for the first approach—represented by
($\mathsf{d29}$)—to leave room for instruments that are considered neither
good nor poor.

  1. Poorn $\varphi$-instruments, for $\alpha\in\mathtt{Agent}$ at $w$ ($n\in\mathbb{R}$ and $0\leq n\leq 1$)
$\textit{Poor}^{w}_{\alpha}({\varphi,n}):=\\{\Delta\ |\
\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi})$ and
$\textit{Succ}^{w}_{\alpha}({\varphi},{\Delta})\leq n\\}$ ($\mathsf{d29}$)

We face the same objection encountered while discussing naive betterness in
Section 5.2.1. Here too, an instrument that proved itself a certain number of
times could be considered more reliable and thus more eligible for the label
‘good’. Therefore, in some cases, we may need to expand definition
($\mathsf{d28}$) with a second threshold imposing a minimum amount of
experience with the instrument in question. The resulting definition is
presented below ($\mathsf{d30}$). We read
$\Delta\in\textit{Good}^{w}_{\alpha}({\varphi,n,m})$ as ‘at $w$ for $\alpha$,
action $\Delta$ is a good $\varphi$-instrument having a success ratio meeting
$n$ and a minimum amount of past witnesses $m$’.

  1. Good${}^{m}_{n}$ $\varphi$-instruments, for $\alpha\in\mathtt{Agent}$ at $w$ ($n\in\mathbb{R}$, $0\leq n\leq 1$, $m\in\mathbb{N}$)
$\textit{Good}^{w}_{\alpha}({\varphi,n,m}):=\\{\Delta\ |\
\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi,m})$ and
$\textit{Succ}^{w}_{\alpha}({\varphi},{\Delta})\geq n\\}$ ($\mathsf{d30}$)

The modified definition of ‘poor’ instruments is similarly obtained from
($\mathsf{d29}$).

Last, as an illustration of the above machinery, consider the
$\mathsf{TLAE}$-model provided in Figure 4. Suppose agent $\alpha$ desires to
secure $\varphi$ at $w_{6}$. Then, suppose the minimal required success ratio
is $n=0.75$. In that case, at $w_{6}$ we find that only
$\Gamma\in\textit{Good}^{w_{6}}_{\alpha}({\varphi,n})$ since
$\textit{Succ}^{w_{6}}_{\alpha}({\varphi},{\Gamma})=1\geq 0.75$. Action
$\Delta$ fails to qualify due to
$\textit{Succ}^{w_{6}}_{\alpha}({\varphi},{\Delta})=\frac{2}{3}<0.75$. Suppose
we impose the additional constraint that the minimum amount of witnesses is
$3=m$. In that case, both
$\Gamma,\Delta\not\in\textit{Good}^{w_{6}}_{\alpha}({\varphi,n,m})$ since
$\Gamma$ does not satisfy $m$ and $\Delta$ doesn’t satisfy $n$. Still, despite
$\Delta$ not being able to qualify as a ‘good’ $\varphi$-instrument, it is
nevertheless the ‘best’ $\varphi$-instrument for $\alpha$ at $w_{6}$, i.e.,
$\Delta\in\textit{Best}^{w_{6}}_{\alpha}({\varphi},{m})$.

In conclusion, whether an instrument is considered ‘good’ or ‘poor’ depends on
its evaluative criteria. Nevertheless, via adopting different, yet
combinatory, thresholds (i)-(iii), we can draw meaningful conclusions
concerning the (comparative) value of the instruments at an agent’s disposal.

  * (i)

the depth of past experience, e.g., ($\mathsf{d11}$);

  * (ii)

the minimum amount of witnesses, e.g., ($\mathsf{d23}$);

  * (iii)

the minimum rate of success, e.g., ($\mathsf{d28}$).

Furthermore, we also saw that a ‘best’ $\varphi$-instrument $\Delta$ is not
necessarily a goodn $\varphi$-instrument if the success ratio expressed by
threshold $n$ is not met by $\Delta$. Alternatively, one could consider
defining ‘best’ in terms of ordering only those instruments that meet the
threshold for qualifying as a good instrument. We leave such considerations
for future work.

###  5.3 Defeasibility of Instrumentality Judgments

Instrumentality, as defined in this work, is a defeasible notion in three
ways. First, depending on the length of the interval considered to evaluate
the past, an instrument $\Delta$ may fail to qualify as an (excellent)
$\varphi$-instrument once the interval is shortened. Furthermore, it can be
easily checked that the excellence of instruments may also fail to be
preserved when the interval is extended.

Second, concerning the future, an instrument may fail to remain classified as
an (excellent) instrument, either because an agent changes her expectations
or, in the case of excellent candidate instruments, because the instrument has
failed to produce the desired end in the meantime. (We note that this holds
independent of whether time continues, i.e., whether $\lozenge\top$ holds.)
Nevertheless, plain candidate instrumentality is preserved over time when
experience is accumulated. That is, once an instrument proves to be a
candidate instrument, it remains a candidate instrument.

The third and foremost defeasible aspect of our formalised instrumentality
relations arises through the use of expectations. Namely, the conjectural
element of instrumentality judgments imposes a defeasible characteristic on
such judgments: although universal statements can be objectively true for the
past, such statements remain uncertain with respect to the future. The
defeasibility of expectations consists in the possibility that, although an
agent $\alpha$ expects that the (excellent) instrument will serve its intended
end once again at the moment of evaluation $w$, the actual successor of $w$ is
such that that instrument fails to deliver the purpose. These cases reveal a
discrepancy between $\alpha$’s expectations and the actual future.
Defeasibility with respect to the actual future is demonstrated by the fact
that the formula below is not a theorem of $\mathsf{TLAE}$.

  1. $[\Delta^{\alpha}]_{n}^{ex-instr}\varphi\rightarrow\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip 0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}(t(\Delta^{\alpha})\rightarrow\varphi)$

We close this section with a clarification concerning expectations: the
conjectural element refers to the agent’s expectations at the moment of
evaluation, which is what an agent expects with respect to the immediate
future of that moment. By contrast, in evaluating the past, we must ignore the
agent’s past expectations in selecting the agent’s relevant experience. In
fact, a series of unexpected events in the past may have led the agent to the
conviction that a particular instrument is suitable for a specific purpose.
That is, the agent may learn about instruments through unexpected events. For
instance, suppose that the moment of evaluation $w$ is an immediate successor
of a moment $w^{\prime}$: then, it might be the case that at $w^{\prime}$, the
agent did not expect $w$ to obtain. Regardless, once $w$ obtains, the agent
gains new experience concerning instrumentality judgments, as she always does
whenever time passes.

##  6 Closing Remarks

We point out that we did not define von Wright’s notion of bad instruments
[35, p. 23, p. 35]. An available instrument (possibly a good or excellent
instrument) is qualified as bad whenever a (side-)effect of that instrument
would be undesirable. For example, negotiations may be a good instrument for
ending a war. Using an atomic bomb may even be excellent as an instrument.
Still, the latter is a bad instrument because it will additionally lead to the
destruction of the planet and the death of many (see [35] for a discussion).
Here, the label ‘bad’ is assigned to the instrument based on its (additional)
consequences. These may, for instance, be certain moral or social values that
are violated.

Moreover, there is the potential to provide deontic extensions of the logic
$\mathsf{TLAE}$ (the Greek déon refers to ‘what is binding’, i.e., duty). For
instance, deontic concepts such as ‘obligation’ and ‘prohibition’ can be
incorporated in $\mathsf{TLAE}$ through violation constants, e.g., an action
(outcome) is obligatory if and only if that action’s (outcome’s) complement
entails a violation. This approach is known as Anderson’s reduction of deontic
logic [1]. Such constants denote that the agent is in a violation state. A
deontic extension of the basic $\mathsf{LAE}$ logic from [6] was developed in
[7]. In that system, various instrumentality notions concerning obligations
and prohibitions were introduced. For instance, in addition to the traditional
distinction between ‘ought to be’ and ‘ought to do’—respectively, obligations
about states of affairs and obligations about actions—the involvement of
instrumentality statements allows for a third category called ‘norms of
instrumentality’. These are norms that oblige or prohibit a particular action
as a means to achieving a particular end. To give an example, the norm ‘It is
prohibited to use nonpublic information as an instrument to acquire financial
profit on the stock market’ (known as the law on ‘insider trading’) forbids
the use of such information only as a means to attain financial profit. The
only temporal operator employed in [7] is the immediate successor modality.

Combining the above two observations, one can extend the present work to a
deontic setting which allows for reasoning with prohibitions that forbid those
instruments that have deontically ‘bad’ consequences (e.g. violations or
sanctions) despite being ‘good’ instruments. Another direction would be to
define obligatory actions in terms of those ‘good’ instruments (or best) that
will secure an obligatory outcome. Here one can think of acquiring different
notions depending on whether the agent’s expectations will be involved.

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