# Timing analysis of Swift J0243.6+6124 with _NICER_ and _Fermi_ /GBM during
the decay phase of the 2017–2018 outburst

M. M. Serim1, Ç. K. Dönmez2, D. Serim1, L. Ducci1 , A. Baykal2, and A.
Santangelo1
1Institut fur Astronomie und Astrophysik, Eberhard Karls Universität, Sand 1,
D-72076 Tübingen, Germany
2Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a timing and noise analysis of the Be/X-ray binary system Swift
J0243.6+6124 during its 2017–2018 super-Eddington outburst using _NICER_ /XTI
observations. We apply a synthetic pulse timing analysis to enrich the _Fermi_
/GBM spin frequency history of the source with the new measurements from
_NICER_ /XTI. We show that the pulse profiles switch from double-peaked to
single-peaked when the X-ray luminosity drops below $\sim$$7\times 10^{36}$
erg s-1. We suggest that this transitional luminosity is associated with the
transition from a pencil beam pattern to a hybrid beam pattern when the
Coulomb interactions become ineffective to decelerate the accretion flow,
which implies a dipolar magnetic field strength of $\sim$$5\times 10^{12}$ G.
We also obtained the power density spectra (PDS) of the spin frequency
derivative fluctuations. The red noise component of the PDS is found to be
steeper ($\omega^{-3.36}$) than the other transient accreting sources. We find
significantly high noise strength estimates above the super-Eddington
luminosity levels, which may arise from the torque fluctuations due to
interactions with the quadrupole fields at such levels.

###### keywords:

accretion, accretion discs – pulsars: individual: Swift J0243.6+6124 –
methods: data analysis

††pubyear: 2023††pagerange: Timing analysis of Swift J0243.6+6124 with _NICER_
and _Fermi_ /GBM during the decay phase of the 2017–2018 outburst–Timing
analysis of Swift J0243.6+6124 with _NICER_ and _Fermi_ /GBM during the decay
phase of the 2017–2018 outburst

##  1 Introduction

A new transient binary system in our galaxy, Swift J0243.6+6124, was
discovered at the onset of the outburst phase (Kennea et al., 2017). Initial
analyses showed that the system consists of a pulsar with a $\sim$9.8 s
spinning period (Kennea et al., 2017) and an O9.5Ve type companion with long-
term optical and infrared variabilities similar to the common Be/X-ray binary
systems (Kouroubatzakis et al., 2017; Reig et al., 2020). Using optical
observations, Reig et al. (2020) estimated the distance of the system to be
$\sim$5 kpc, whereas the _Gaia_ EDR2 estimated distance was
6.8${}_{-1.1}^{+1.5}$ kpc (Bailer-Jones et al., 2018). Adopting the _Gaia_
EDR2 distance, the maximum brightness of Swift J0243.6+6124 was estimated to
be $\sim$2 $\times$ 1039 erg s-1 at the peak of the outburst (Tsygankov et
al., 2018; Doroshenko et al., 2020). On the other hand, the source distance
(id: 465628193526364416) is revised as $5.2\pm 0.3$ kpc in the _Gaia_ EDR3
catalogue (Bailer-Jones et al., 2021). When the distance of $\sim$5 kpc is
taken into account, the peak luminosity would be $\sim$1 $\times$ 1039 erg
s-1, which is still higher than the Eddington limit for such a neutron star
(Reig et al., 2020); thus, Swift J0243.6+6124 is classified as an
ultraluminous X-ray Pulsar (ULXP), the first ever detected in our own galaxy.

Numerous studies regarding the temporal and spectral properties of Swift
J0243.6+6124 have been conducted in attempt to comprehend the physical
dynamics of this unique source (Wilson-Hodge et al., 2018; Tsygankov et al.,
2018; van den Eijnden et al., 2018; Jaisawal et al., 2018; Tao et al., 2019;
Doroshenko et al., 2020; Sugizaki et al., 2020; Kong et al., 2020; Wang et
al., 2020; Kong et al., 2022; Liu et al., 2022b; Bykov et al., 2022). Detailed
investigations revealed that both temporal and spectral features, including
shape of the power spectra, pulse profiles and energy spectra, change
systematically at two different transitional luminosity levels (for 6.8 kpc),
$L_{1}\sim 1.5\times 10^{37}$ erg s-1 and $L_{2}\sim 4.5\times 10^{38}$ erg
s-1 (Wilson-Hodge et al., 2018; Doroshenko et al., 2020; Kong et al., 2020).
Thus, these transitional luminosity levels $L_{1}$ and $L_{2}$ are interpreted
as transitions of subcritical to supercritical accretion regime and
supercritical to radiation pressure dominated disc (RPD) accretion regime,
respectively (Doroshenko et al., 2020). As a reminder for the discussions
throughout the paper, using the new Gaia distance of 5.2 kpc, the transitional
luminosity levels are calculated as $L_{1}$ = $8.8\times 10^{36}$ erg s-1 and
$L_{2}$ = $2.6\times 10^{38}$ erg s-1, respectively.

Despite the extensive studies, the magnetic field configuration of Swift
J0243.6+6124 is not yet clear. Initial studies have demonstrated that the
source pulsations are still detectable at luminosities as low as
10${}^{34}-10^{35}$ erg s-1, which indicates that the propeller regime has not
yet been attained at such low luminosities; consequently, the pulsar should
have a very compact magnetosphere to allow accretion to continue, which
confines the upper limit of the magnetic field strength to 3 $\times$ 1012 G
(Tsygankov et al., 2018; Doroshenko et al., 2020). Phase-resolved spectral
analysis of _NuSTAR_ observations at different luminosity levels hints for a
thick super-Eddington disc with an inner radius of 2–3 $\times$ 107 cm and a
weakly variable reflection component, signifying a magnetic field strength
$3\times 10^{12}$ G if the field is dipolar (Bykov et al., 2022). On the other
hand, the discovery of a cyclotron resonance scattering feature (CRSF) in the
spectrum of Swift J0243.6+6124 at $\sim$120–146 keV, which is only visible in
certain phases around the peak of the outburst (Kong et al., 2022) implies a
magnetic field strength of $\sim$1.6 $\times$ 1013 G near the surface of the
pulsar. Nevertheless, it is suggested that the observed CRSF is actually
associated with multipole fields (Kong et al., 2022) and the dipolar component
of the field strength should be in the range of 3–9 $\times$ 1012 G in order
to describe the observed properties of the source coherently (Doroshenko et
al., 2020). The accretion disc possibly penetrates into the magnetosphere more
than expected, and the disc interactions are dominated by multipole components
of the field at high luminosities (Doroshenko et al., 2020; Kong et al.,
2022).

With its ultraluminous episode and unique properties, the source has been the
target of many studies, especially in probing the nature of neutron star
accretion at very high luminosities (van den Eijnden et al., 2018; Doroshenko
et al., 2018; Wilson-Hodge et al., 2018; Jaisawal et al., 2019; Kong et al.,
2020, 2022; Bykov et al., 2022). In this study, we investigate the timing
properties of Swift J0243.6+6124, focusing mostly on its moderately luminous
stages ($\sim$10${}^{36}-10^{37}$ erg s-1) towards the end of the outburst in
2017–2018, during which the source remained in a subcritical accretion state.
We describe the data and the relevant screening processes used for timing
analysis in Section 2. In Section 3, we represent the pulse timing analyses
that are used for measuring spin frequencies and generating pulse profiles. In
addition, we also demonstrate our results on the torque fluctuations on
different timescales and luminosities. Lastly, in Section 4, we review and
discuss the results of our study in the light of the systematic luminosity-
dependent evolution of pulse profiles.

##  2 Data

Neutron Star Interior Composition Explorer (_NICER_) is stationed on
International Space Station (ISS) since 2017 June and operated by NASA. Its
primary instrument, X-Ray Timing Instrument (XTI), consists of an aligned
array of 56 X-ray concentrators and focal plane modules (FPM) collecting
photons from a $\sim$30 arcmin2 field onto silicon field detectors in each
FPM. These detectors are capable of soft X-ray spectroscopy with 0.2–12 keV
energy range and <300 ns timing precision with $\sim$1900 cm2 cumulative
effective area at 1.5 keV (Gendreau et al., 2016).

Swift J0243.6+6124 has 214 _NICER_ /XTI observations in the _NICER_ master
catalogue between 2017 October 03 and 2019 June 07 (MJD 58029–58641),
corresponding to the outbursts in this study. Among those, we utilise the
observations prior to the rapid decline of the source luminosity at the end of
2019 February. Data reduction of the observations is done with heasoft v6.29
using the most recent calibration files at the time (CALDB release
xti20210707) for _NICER_. The clean events and filter files for screening data
are reproduced by employing the standard level 2 data processing steps
provided by the nicerl2 tool. Good time intervals (GTI) are selected adopting
the default screening parameters recommended by the NICER
team111https://heasarc.gsfc.nasa.gov/lheasoft/ftools/headas/nicerl2.html: ISS
is outside the predefined Southern Atlantic Anomaly (SAA) region, a minimum of
38 of the 56 detectors are enabled, the pointing offset is less than 0.015∘,
the source is at least 15∘ away from the dark Earth limb and 30∘ away from the
bright Earth limb. The event files between MJD 58029–58531 are merged using
the nimpumerge tool, event time-series are barycentered using barycorr with
the JPL ephemeris DE430, and the light curves with a time resolution of 0.1 s
are extracted with XSELECT. We also corrected the photon arrival times of the
generated _NICER_ /XTI light curve prior to the timing analyses described
below with the orbital solution provided by the _Fermi_ /GBM Accreting Pulsars
Program (APP) team (See Table 1).

Table 1: Orbital parameters provided by the _Fermi_ /GBM team, which are used to correct the photon arrival times prior to our timing analysis. $P_{\textrm{orb}}$ | 27.698899 | days
---|---|---
$T_{\pi/2}$ | 58116.097 | MJD
$a_{x}$ sin $i$ | 115.531 | lt-s
$\omega$ | -74.05 | degrees
$e$ | 0.1029 | –

Additionally, we make use of the pulse frequency history of Swift J0243.6+6124
which is publicly shared through _Fermi_ /GBM (Gamma-ray Burst Monitor)
monitoring program
website222https://gammaray.nsstc.nasa.gov/gbm/science/pulsars/lightcurves/swiftj0243.html
(Malacaria et al., 2020) and the regularly updated _Swift_ /BAT light curves
maintained by the _Swift_ /BAT
team333https://swift.gsfc.nasa.gov/results/transients/weak/SwiftJ0243.6p6124
(Krimm et al., 2013). The compiled _Fermi_ /GBM frequency history is orbit-
corrected and encapsulates the range between 2017 October 01 and 2019 January
14 (MJD 58027–58497). We used the _Swift_ /BAT daily average light curve,
which has an energy range of 15–50 keV, from the discovery of the source in
2017 October up to 2019 February. Doroshenko et al. (2020) argued that 2–150
keV count rates for _Insight-HXMT_ appear to be consistent with those measured
by _Swift_ /BAT, and the _Swift_ /BAT count rates can be roughly converted to
bolometric luminosity using a scaling factor $\sim$8.2 $\times$ 1038, assuming
a source distance of 6.8 kpc. In this article, we utilise the _Gaia_ EDR3
distance (5.2 kpc) and revise scaling factor for the _Swift_ /BAT count
rate–luminosity conversion to $\sim$4.8 $\times$ 1038 to estimate the
bolometric luminosity, unless otherwise stated.

##  3 Timing Analysis and Results

###  3.1 Synthetic pulse timing

During its outburst phase in 2017–2018, the X-ray luminosity of Swift
J0243.6+6124 varies by five orders of magnitude. At the same time, the
accretion geometry, and consequently the pulse profiles, drastically alter at
different accretion regimes (Wilson-Hodge et al., 2018; Doroshenko et al.,
2020). In particular, the pulse profiles are shown to be double-peaked at
subcritical regime ($L_{x}<L_{1}$) and evolve into a single-peaked shape at
supercritical regime ($L_{1}<L_{x}<L_{2}$), then again transform into a
double-peaked structure at the highest luminosities ($L_{2}<L_{x}$)
(Doroshenko et al., 2020). Moreover, the spin-up rate during the initial
stages of the outburst is very strong, reaching up to 2.2 $\times$ 10-10 Hz
s-1 (Doroshenko et al., 2018). The frequency derivative leads to a phase shift
of one cycle on a timescale of $\sqrt{2/|\dot{\nu}|}$ (Acuner et al., 2014);
and with the reported high spin-up rate during the outburst (Wilson-Hodge et
al., 2018; Doroshenko et al., 2018), this timescale becomes as short as
$\sim$1.1 days. Combined with the pulse profile variations, employing phase-
coherent timing technique becomes unfavourable at the luminous stages of the
outburst.

Figure 1: Upper panel: _Fermi_ /GBM spin frequency history (black dots) and
the synthetic timing solution (red curve) obtained from polynomial fitting in
the interval 1 (see Table 2 for timing parameters). Middle panel: The
frequency residuals of the synthetic timing solution. Lower panel: The
synthetic phase residual model obtained from _Fermi_ /GBM residuals (blue
curve), the residuals of time of arrivals (TOAs) of the _NICER_ observations
(gray dots) when folded with the corresponding timing solution and the shifted
_NICER_ TOA residuals (red dots) according to the expected phase residual
model.

Making use of the refined orbital solution provided by _Fermi_ /GBM team, we
used the following approach to measure the pulse frequencies from the _NICER_
data, which reside within the same time interval as the _Fermi_ /GBM
measurements: We first divide the _Fermi_ /GBM pulse frequency measurements
into three different segments, each of which is fitted with a different
polynomial model to represent the frequency evolution over time, and obtain a
synthetic timing solution444At this stage, it should be noted that the choice
of a polynomial model order is rather arbitrary; nonetheless, the synthetic
residual reconstruction compensates for the possible deviations from the
model. In principle, the procedure can be applied for any polynomial order,
provided that the reconstructed pulse profiles are compatible with the actual
profiles. (see Table 2). Using these timing solutions, we then calculate the
deviations of the _Fermi_ /GBM frequencies from the model to extract its
residuals. Utilising a linear spline interpolation of the _Fermi_ /GBM
frequency residual data set $\nu_{r}$, we convert them to a synthetic phase
residual model $\Phi_{r}$ using integration:

$\Phi_{r}(t)=\int_{t_{0}}^{t}\nu_{r}(t^{\prime})dt^{\prime}$ (1)

where $t_{0}$ indicates the start time of the segment. Next, we fold the
orbitally-corrected _NICER_ light curve with the same synthetic timing
solution to generate its phase residuals. Finally, we shift the _NICER_ phase
residuals to match with the synthetic phase residual model obtained from
_Fermi_ /GBM (see Figure 1).

In the first interval, the luminosity of the source changes substantially,
resulting in significant deviations from the polynomial description of the
rapid frequency evolution. However, the phase residuals of _NICER_
observations become compatible with the synthetic residual model when they are
shifted with expected integers in phase domain. The only exceptions are the
pulse profile variations at different episodes (Doroshenko et al., 2020) that
are needed to be taken into account. Thus, we further allow phase shifts for
the pulses in the supercritical regime by $\Delta\phi\sim 0.5$, corresponding
to the phase difference between the peaks of the double-peaked and one-peaked
profiles (see Figure 4 of Doroshenko et al. (2020)) to accord them with the
expected synthetic phase residuals (See Figure 1, bottom panel). On the other
hand, during the late stages of the outburst (at the interval 2, 3 and 4), the
source luminosity is rather low ($L_{x}\lesssim 8\times 10^{37}$ erg s-1), and
Swift J0243.6+6124 continues to accrete only in subcritical regime (i.e.
$L_{x}<L_{1}$). Therefore, the synthetic phase residuals reside within a
single cycle for the corresponding synthetic timing solutions given in Table
2.

Finally, in order to convert synthetic timing solutions to pulse frequency
measurements, we use each consecutive pair of pulse arrivals in the _NICER_
residual set. Each pair is fitted with a linear function
$\delta\nu=\delta\phi/(t_{2}-t_{1})$ where $t_{1}$ and $t_{2}$ are the arrival
times of the first and second pulses in the pair, and $\delta\phi$ is the
phase difference between the pair used for fitting. Each fit is transformed
into a spin frequency measurement at the corresponding interval’s midpoint by
using the frequency correction $\delta\nu$ over the synthetic timing solutions
(for applications, see Çerri-Serim et al. (2019); Serim et al. (2022)). The
1$\sigma$ error ranges of the slope are used as a gauge of the uncertainty in
the spin frequency measurements. Figure 2 demonstrates the spin frequency
history measured from _NICER_ observations whose results are consistent with
the spin frequency history shared by the _Fermi_ /GBM APP team.

As it can be seen from the frequency history presented in Figure 2, apart from
the initial stages of the Type II outburst, Swift J0243.6+6124 also spins up
between MJD $\sim$58470–58510. Afterwards, as the flux diminishes over time,
the frequency evolution trend returns back to the spin-down stage with an
average frequency derivative of $\sim-1.6\times 10^{-12}$ Hz s-1, which is
comparable to the average spin-down rate $\sim-1.8\times 10^{-12}$ Hz s-1
observed between MJD 58150–58460.

Table 2: The timing parameters obtained for four different intervals. Note that the synthetic timing solutions for all the intervals are established by fitting the existing _Fermi_ /GBM frequencies. | Interval 1 | Interval 2 | Interval 3 | Interval 4
---|---|---|---|---
Timing range (MJD) | 58027–58162 | 58162–58239 | 58299–58497 | 58450–58531
Epoch (MJD) | 58050 | 58200 | 58350 | 58500
$\nu$ (Hz) | 0.1015335(20) | 0.1021289(2) | 0.1021046(2) | 0.10210146(7)
d$\nu$/dt (10-10 Hz s-1) | 1.02(2) |  $-$0.0161(4) |  $-$0.0180(9) | 0.0730(10)
d${}^{2}\nu$/dt2 (10-17 Hz s-2) | 9.85(31) |  $-$0.0109(45) | 0.0265(36) |  $-$0.408(13)
d${}^{3}\nu$/dt3 (10-24 Hz s-3) |  $-$49.4(35) | - |  $-$0.0158(32) |  $-$4.52(23)
d${}^{4}\nu$/dt4 (10-31 Hz s-4) |  $-$9.86(90) | - |  $-$0.0139(22) | 35.9(29)
d${}^{5}\nu$/dt5 (10-36 Hz s-5) | 158.2(109) | - | 0.0556(51) | 3.69(41)
d${}^{6}\nu$/dt6 (10-41 Hz s-6) |  $-$9.77(66) | - | - |  $-$0.114(59)
d${}^{7}\nu$/dt7 (10-47 Hz s-7) | 2.41(17) | - | - | -
Figure 2: Top panel: The spin frequency history of Swift J0243.6+6124
generated for all intervals. The shaded regions represent corresponding time
intervals. The inset panels show the same frequencies in the marked regions
with a different scale for better viewing. Bottom panels: The resultant
_NICER_ TOA residuals of the synthetic timing solutions.

Interestingly, when the pulse profiles obtained from the timing analysis at
low flux states are examined, the pulses seem to exhibit single-peaked
profiles at very low flux levels. To illustrate this behaviour more clearly,
we present the luminosity-sorted pulse profiles (normalized to [0, 1] range)
of all observations after MJD 58300 in Figure 3. A systematic change in the
profiles emerges at a luminosity level of $\sim$$7\times 10^{36}$ erg s-1,
marking a potential new transitional level for the alteration of the accretion
geometry. As the luminosity decreases, the main peak gradually fades away and
the secondary peak grows stronger. It is also interesting to note that the
source tends to exhibit spin-down episodes below this luminosity level.

Figure 3: Luminosity-dependent pulse profile evolution of Swift J0243.6+6124
for the observations after MJD 58300. The pulse profiles are normalized to [0,
1] and plotted for two cycles for clarity.

###  3.2 Timing noise

Using the whole frequency history enriched with _NICER_ measurements, we
investigate the temporal noise behaviour of Swift J0243.6+6124. To estimate
the amplitude of the timing noise at different timescales, we proceed with the
rms-value technique developed by Boynton et al. (1972); Deeter (1984); Cordes
& Downs (1985). This technique utilises the rms values of the timing residuals
$\langle\sigma_{r}(m,T)\rangle$ that are acquired after eliminating the
polynomial trend of order $m$ from the data set of duration $T$. Then, the
associated noise strength $S_{r}$ can be calculated via:

$S_{r}=\frac{\langle\sigma_{r}(m,T)\rangle}{\langle\sigma_{r}(m,1)\rangle_{u}}\frac{1}{T^{2r-1}}$
(2)

where $r$ specifies the red noise order, and
$\langle\sigma_{r}(m,1)\rangle_{u}$ denotes the unit noise strength
normalization factor for $T=1$ d and $S_{r}=1$. Our calculations are performed
with the associated normalization factors gauged through direct evaluations
(Deeter, 1984, Table 1). We start by estimating the noise strength of the
maximal time span of the data $T_{max}$ and iterate the calculations for
halved timescales ($T_{max}/2^{n}$ for $n=1,2,3,..$). Then, we aggregate the
noise strength measurements in each timescale into logarithmically-binned
power density estimates. The uncertainties of power density estimates are
determined from 1$\sigma$ confidence intervals, depending on the number of
independent noise strength estimates enclosed during generation of the power
density estimate in each timescale, as described in Deeter et al. (1987). The
distribution of the power density estimates as a function of timescale (or
analysis frequency $\omega\equiv 1/T$) generates a power density spectrum
(PDS) of the spin frequency derivative fluctuations $P_{\Delta\dot{\nu}}$. We
check for the stability of the PDS profile for different polynomial orders and
proceed with the lowest stable order, $m=2$, quadratic polynomial trends, for
the input frequency series to characterize the regular spin evolution of Swift
J0243.6+6124, assuming the residues after the removal of the trend constitute
the timing noise. In order to check the validity of the power density
estimates in each time scale, we also present the corresponding measuremental
noise levels by taking the measuremental uncertainties of the frequency data
set ($\sigma_{i}$) into account (green crosses in Figure 5), which are
calculated via
$\frac{\sum_{i}^{N}\sigma_{i}^{2}}{N\,T\langle\sigma_{r}(m,1)\rangle_{u}}$.
The measuremental noise level provides a precursor to a noise level at which
the measuremental error range becomes dominant over the fluctuations in the
data set.

Figure 4: Upper panel: The spin frequency history of Swift J0243.6+6124.
Middle panel: Bolometric source luminosity converted from 15–50 keV
_Swift/BAT_ count rates, assuming $d=5.2$ kpc. Bottom panel: Frequency
residuals after eliminating $\nu_{\textrm{model}}(t)$.

In addition to the aforementioned standard method for PDS generation for
torque fluctuations, we also follow the approach described in Serim et al.
(2022) to see the effects of the accretion torques on the PDS. In principle,
this approach offers a different perspective on the same PDS, with the only
distinction being the minimization of torque fluctions arising from disk
accretion. It should be noted that in both cases, the input frequency data set
is already decoupled from orbital Doppler delays using the orbital parameters
given in Table 1. Therefore, we assume that the orbital modulations in the
frequencies are completely removed and they no longer contribute to the noise
strength measurements. In this case, we utilise a simple power law relation
between the spin-up rate and the luminosity, which is modified with a constant
spin-down rate ($\dot{\nu}_{model}=\beta L^{\alpha}+\dot{\nu}_{0}$) to account
for the stable spin-down episodes observed in the frequency history of the
source. Then, the luminosity-dependent frequency evolution model is built as
(Serim et al., 2022):

$\nu_{model}(t)=\nu_{0}+\int_{t_{0}}^{t}\dot{\nu}_{model}(t^{\prime})dt^{\prime}$
(3)

where $\nu_{0}$ is the spin frequency at the time of the burst onset $t_{0}$.
Instead of polynomial driven residuals built in the standard approach, the
residuals obtained from the elimination of $\nu_{model}$ is assumed to inherit
the noise component for this case (see Figure 4).

Figure 5: PDS of the spin frequency derivatives using quadratic polynomial
trends (red) and luminosity-dependent intrinsic spin frequency evolution model
(blue), along with the measuremental noise levels (green). The uncertainties
of power density estimates are expressed as 1$\sigma$ confidence intervals,
determined by the number of independent estimates present within.
Corresponding fits of the PDSs are shown as maroon and dark blue lines.

In both cases, generated PDSs of spin frequency derivative fluctuations are
modeled with a broken power law model:

$P_{\Delta\dot{\nu}}=\begin{cases}\,S_{r,1}\,\omega^{\Gamma}\hskip
28.45274pt\textrm{if}\,\omega<\omega_{\textrm{b}}\\\ \,S_{r,2}\hskip
42.67912pt\textrm{if}\,\omega>\omega_{\textrm{b}}\\\ \end{cases}$ (4)

where $\omega_{b}$ is the break analysis frequency and $\Gamma$ is the power
law index of the red noise component (Figure 5). The fitting procedure is
carried out by orthogonal distance regression (ODR) using the Python library
of SciPy. We report the uncertainties of the best fit parameters with
$1\sigma$ confidence level.

For the standard approach, in which polynomial trends are used, we find that
the PDS of the frequency derivatives is evolving as $\omega^{-3.36\pm 0.64}$
within the range $1/46\gtrsim\omega\gtrsim 1/500$ days-1, which points out a
steeper red noise component when compared with the other accreting sources
(Bildsten et al., 1997; Baykal et al., 2007; Serim et al., 2022, 2023). The
steepness of this red component is comparable to the case of 4U 1626–67
(Bildsten et al., 1997; Serim et al., 2023); however, the timescales within
which they are observed are dissimilar. The PDS continuum break occurs at
$\omega_{b}\gtrsim 1/46$ days-1 and evolves toward a flatter continuum at
higher analysis frequencies (i.e., becomes a white noise component,
S${}_{r,2}=(6.76\pm 0.16)\times 10^{-19}$ Hz2 s-2 Hz-1), implying uncorrelated
torque fluctuations at shorter timescales. When the regular frequency
evolution model is substituted for the luminosity-dependent model, the power
density estimate at the longest timescale is reduced by a factor of $>$100\.
The steepness of the red noise component is also reduced to $\omega^{-0.91\pm
0.38}$ but it does not completely vanish unlike the case of 2S 1417–624 (Serim
et al., 2022). It implies that either the luminosity-dependent model (at least
through a simple power law relation) does not remove all of the red noise
component of the PDS, or merely the standard disc component, which generally
contributes to PDS spectra as $\omega^{-2}$ (Bildsten et al., 1997; Serim et
al., 2023), is subtracted from the PDS continuum. At higher analysis
frequencies ($\omega_{b}\approx 1/27$ days-1), the PDS carries the same
structure as the former case, with the white noise normalization
S${}_{r,2}=(5.25\pm 0.17)\times 10^{-19}$ Hz2 s-2 Hz-1.

Figure 6: The distribution of noise strength estimates using the standard
method on the timescale of 15 days as a function of luminosity. The dashed
grey lines indicate the transitional luminosity levels $L_{1}$ (left) and
$L_{2}$ (right), which are calculated for a distance of 5.2 kpc. The shaded
regions reflect the uncertainties in the interpretation of the corresponding
transitional luminosities in the literature. The measurements below $L_{2}$
are rebinned by a factor of 2 for visual purposes.

In order to understand the nature of the strong red noise component in the PDS
of torque fluctuations, we further check the luminosity dependence of the
timing noise strengths. Hence, we split the frequency history into $\sim$15
days long segments and calculate the noise strengths for each of them using
the standard method described above. Next, using the _Swift_ /BAT count rates,
we calculate the luminosity range for each interval555See Section 2 for the
conversion of _Swift_ /BAT count rates to luminosity.. The distribution of the
noise strength estimates as a function of luminosity is illustrated in Figure
6. The luminosity dependence of the noise strength estimates yields an
intriguing distribution. The noise strength amplitudes remain more or less
constant up to the transitional luminosity level $L_{2}$ with a slight de-
escalation between $L_{1}$ and $L_{2}$. The $S_{r}$ values significantly rise
(by a factor of $\sim$10) above $L_{2}$. It indicates a possible change in the
nature of torque fluctuations above $L_{2}$.

##  4 Discussion and Conclusion

We analyse the _NICER_ /XTI data set and enrich the spin frequency history of
Swift J0243.6+6124 with new measurements. The late-stage evolution of spin
frequency indicates another torque reversal around MJD $\sim$58510, after
which the source entered a new spin-down phase. When the frequency evolution
of Swift J0243.6+6124 is examined, the spin-down phases seem to occur
systematically at luminosities below $\sim$$7\times 10^{36}$ erg s-1. It has
already been shown that the source pulsations were observable at luminosities
down to 1034–1035 erg s-1, implying that propeller stage is not yet attained
at such low levels (Tsygankov et al., 2018; Doroshenko et al., 2020), and
therefore the spin-down phase is not associated with propeller regime.

The pulse profile evolution of Swift J0243.6+6124 is very intriguing. At
luminosities below $\sim$$7\times 10^{36}$ erg s-1, the pulse profiles are
single peaked. Between $\sim$ 7 $\times$ 1036 erg s${}^{-1}<L_{x}<L_{1}$, a
secondary peak component emerges and gains strength with increasing
luminosity; thus, the profiles become double peaked. Furthermore, when
$L_{x}>L_{1}$, the pulse profiles become single-peaked again. The
transformation of the pulse profiles around $L_{x}\sim 7\times 10^{36}$ erg
s-1 indicates a new transition in the accretion geometry. According to Becker
et al. (2012), the critical X-ray luminosity ($L_{1}$) specifies the onset of
the transition from fan beam to pencil beam; however, the transition does not
immediately take place. There is an intermediate accretion regime
$L_{coul}<L_{x}<L_{1}$ where the final phase of the deceleration of the
accreted material is experienced through Coulomb braking in the plasma. In
such a regime, a hybrid combination of both fan and pencil beam patterns is
expected (Becker et al., 2012; Blum & Kraus, 2000). They specify a limiting
luminosity below which Coulomb interactions are no longer effective enough to
stop the accretion flow. This transition luminosity is given by:

$L_{coul}\simeq 1.17\times
10^{37}B_{12}^{-1/3}\Lambda_{0.1}^{-7/12}\tau_{20}^{7/12}M_{1.4}^{11/8}R_{10}^{13/24}\,\textrm{erg
s}^{-1}$ (5)

where $B_{12}\equiv B/10^{12}$ G is the dipolar magnetic field strength of the
pulsar, $\Lambda_{0.1}\equiv\Lambda/0.1$ is a dimensionless parameter
accounting for various physical processes such as the possible role of plasma
shielding, $\tau_{20}\equiv\tau/20$ is the Thomson optical depth,
$M_{1.4}\equiv M/1.4$ $M_{\odot}$ is the pulsar mass, and $R_{10}\equiv R/10$
km is the pulsar radius. Below this luminosity, the accretion flow is
suggested to be decelerated via gas-mediated shock near the stellar surface
and the radiation from the polar caps fully transforms to a pencil beam
pattern. In addition, according to this model, the pencil beam pattern should
also persist at lower luminosity levels ($L_{x}<<L_{coul}$). Actually, such a
single-peaked pulse profile was observed by Doroshenko et al. (2020) with an
80 ks _NuSTAR_ observation around the luminosity level of $\sim$3 $\times$
1034 erg s-1. If we consider the transition at $7\times 10^{36}$ erg s-1 as
$L_{coul}$, neglecting the normalized dimensionless parameters of about unity
and using typical neutron star parameters, then the magnetic field of the
source can be estimated as $4.7\times 10^{12}$ G. Furthermore, when the
previously reported critical luminosity level (Wilson-Hodge et al., 2018;
Doroshenko et al., 2020), $L_{1}$, of the onset of the transition from hybrid
pattern to fan beam is updated for the same distance, it results in a magnetic
field strength of $5.3\times 10^{12}$ G. Thus, magnetic field strength
estimations obtained from both transitional levels become consistent at 5.2
kpc. Therefore, we suggest that the dipolar magnetic field strength of Swift
J0243.6+6124 can be confined to a range of $\sim$$(4.7-5.3)\times 10^{12}$ G.

On the other hand, we also investigated the PDS of the spin frequency
derivative fluctuations using the fairly sampled spin frequency data set which
is improved with new measurements obtained from _NICER_ /XTI observations. We
extract two different PDSs using different models to describe regular
rotational evolution. The first one utilizes the standard polynomial-driven
approach, and the second one makes use of a luminosity-dependent spin
frequency evolution model. Both PDSs exhibit bimodal behaviour in which the
high analysis frequency ($\omega_{b}\sim 1/46$ days-1 for the former,
$\omega_{b}\sim 1/27$ days-1 for the latter case) noise components are flat
while the low analysis frequency components carry red noise. It should be
noted that the observed break frequencies are rather close to the orbital
period of the source ($\sim$27.7 d). The white noise components in the PDS of
the spin frequency derivative fluctuations are generally attributed to the
uncorrelated torque fluctuations generated via wind accretion from the
companion (Boynton et al., 1972; Deeter & Boynton, 1985; Bildsten et al.,
1997; Serim et al., 2023). Hence, the high analysis frequency white noise
component of Swift J0243.6+6124 may hint at the accretion from the stellar
wind of its companion, which is effective on timescales less than the orbital
period of the source. Nevertheless, the long-term spin evolution and
fluctuations are governed by the disc interactions.

In general, for the sources that are presumed to have an accretion disc, the
red noise continuum with $\omega^{-2}$ dependence sets in at low timescales,
which are possibly saturated at viscous timescales (Bildsten et al., 1997;
Serim et al., 2023). Even though the number of studies is limited, the PDSs of
the torque fluctuations of transient accreting sources demonstrate that
steepness of red noise components also seem to occur as $\sim$$\omega^{-2}$
(e.g., SAX J2103.5+4545, Baykal et al. (2007); 2S 1417–624, Serim et al.
(2022)). Utilizing the standard PDS generation method, we find that the
steepness of the red noise component of Swift J0243.6+6124 is significantly
higher ($\sim$$\omega^{-3.36}$) when compared with other accreting sources
(Serim et al., 2023). Such a steep red noise component is only observed in
ultra-compact binary system 4U 1626–67 (Bildsten et al., 1997; Serim et al.,
2023) and in several magnetars (Woods et al., 2002; Çerri-Serim et al., 2019);
however, the timescales in which the component arise are different than the
case of Swift J0243.6+6124. To be more specific, the red noise component of
the PDS of Swift J0243.6+6124 develops approximately on the orbital
timescales, whereas the red noise in 4U 1626–67 is present on timescales
longer than $\sim$1000 days (Bildsten et al., 1997; Serim et al., 2023). In
the case of SGR 1806–20 and SGR 1900+14, the red noise components are observed
on timescales longer than $\sim$100 days and the onset timescale of the red
noise components are attributed to a threshold for which these magnetars
become burst active (Woods et al., 2002). Therefore, we believe that the
strong red noise component observed in Swift J0243.6+6124 originates from
different physical processes than the aforementioned cases. To understand the
nature of this component, we used the procedure described in (Serim et al.,
2022) where the rotational evolution is prescribed by a simple torque model.
In the case of 2S 1417–624 (Serim et al., 2022), this model almost completely
eliminates the red noise component associated with disc accretion; however,
for Swift J0243.6+6124, the results are slightly peculiar. The steepness is
reduced from $\sim\omega^{-3}$ to $\omega^{-1}$ but the red noise structure
does not entirely vanish. This situation may originate from different factors.
First, it is possible that the model used in Figure 4 provides an
oversimplistic view for $\dot{\nu}$–$L_{x}$ correlation, thus more complex
models (e.g., Karaferias et al. (2022)) are required to eliminate this
component. Secondly, if the $\omega^{-1}$ dependence has a physical origin,
then it may indicate that $\omega^{-2}$ dependence observed for the disc
component is subtracted. Noting that the steepness and strength of the torque
fluctuations are generally attributed to the nature of the magnetic field
(Woods et al., 2002; Çerri-Serim et al., 2019), it is possible that the
remaining red noise component might be of magnetic origin. Therefore, we
further investigate the luminosity dependence of the noise strength
estimations to inspect the nature of torque fluctuations at different levels
(see Figure 6). We find that the noise strengths remain roughly constant up to
the critical luminosity level $L_{2}$, above which the RPD accretion disc
regime sets in (Doroshenko et al., 2020). When the source luminosity exceeds
$L_{2}$, the noise strength estimates suddenly increase by a factor of 10,
which suggest a possible change in the nature of torque fluctuations above
this level. Moreover, it is recently shown that that the torque–luminosity
relation of Swift J0243.6+6124 flattens at the RPD regime (Karaferias et al.,
2022; Liu et al., 2022a). Hence, as the luminosity increases, the torque
exertions become less efficient and more noisy, which may originate from the
interactions with the quadruple components of the field (Long et al., 2007).
In addition, the observed CRSF was evident only in certain pulse phases at the
peak of the outburst, and it is attributed to the multipole component of the
field (Kong et al., 2022). Thus, the excess noise strength above the
transitional level $L_{2}$ bolsters the idea that multipole components should
play an important role in torque interactions at super-Eddington luminosity
levels (Doroshenko et al., 2020; Kong et al., 2022).

## Acknowledgements

The authors thank the referee for their valuable remarks that assisted in the
improvement of this manuscript. Authors acknowledge the support from TÜBİTAK
(The Scientific and Technological Research Council of Turkey) through the
research project MFAG 118F037. The authors also thank Prof. Dr. Sıtkı Çağdaş
İnam for his insightful comments.

## Data Availability

Whole X-ray data used in this study are publicly available. _NICER_ /XTI data
can obtained through the High Energy Astrophysics Science Archive Research
Center (https://heasarc.gsfc.nasa.gov). _Swift_ /BAT count rates can be
acquired via Swift transient monitoring project
(https://swift.gsfc.nasa.gov/results/transients/weak/) and _Fermi_ /GBM
frequency measurements are available at GBM Accreting Pulsars Project website
(https://gammaray.msfc.nasa.gov/gbm/science/pulsars.html).

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# How Hard Is Squash? - Towards Information Theoretic Analysis of Motor
Behavior in Squash

###### Abstract

Fitts’ law has been widely employed as a research method for analyzing tasks
within the domain of Human-Computer Interaction (HCI). However, its
application to non-computer tasks has remained limited. This study aims to
extend the application of Fitts’ law to the realm of sports, specifically
focusing on squash. Squash is a high-intensity sport that requires quick
movements and precise shots. Our research investigates the effectiveness of
utilizing Fitts’ law to evaluate the task difficulty and effort level
associated with executing and responding to various squash shots. By
understanding the effort/information rate required for each shot, we can
determine which shots are more effective in making the opponent work harder.
Additionally, this knowledge can be valuable for coaches in designing training
programs. However, since Fitts’ law was primarily developed for human-computer
interaction, we adapted it to fit the squash scenario. This paper provides an
overview of Fitts’ law and its relevance to sports, elucidates the motivation
driving this investigation, outlines the methodology employed to explore this
novel avenue, and presents the obtained results, concluding with key insights.
We conducted experiments with different shots and players, collecting data on
shot speed, player movement time, and distance traveled. Using this data, we
formulated a modified version of Fitts’ law specifically for squash. The
results provide insights into the difficulty and effectiveness of various
shots, offering valuable information for both players and coaches in the sport
of squash.

##  1 Introduction

Squash [jones2018review] is a fast-paced and physically demanding sport that
requires exceptional agility, quick reflexes, and precise shot-making
abilities. Played on a small, enclosed court with four walls and a ceiling,
squash provides a unique challenge for players as they navigate the confined
space and strategically outmaneuver their opponents. The combination of
intense movements, such as running, lunging, and jumping, makes squash a
thrilling and captivating sport to play and watch.

In a game of squash, players employ various shots to gain control of the rally
and outwit their opponents. Each shot carries its own purpose and technique,
adding an element of strategy and unpredictability to the game. The drive, one
of the most common shots, involves hitting the ball straight at a medium to
low height along the wall, aiming to maintain ball control and restrict the
opponent’s reach. The boast, on the other hand, entails hitting the ball off
the side wall at an angle, causing it to bounce towards the front wall and
away from the opponent, often surprising them and altering the rally’s
direction. The drop shot requires delicate touch and precision, landing the
ball softly near the front wall, just out of the opponent’s reach. The lob, a
defensive shot, involves hitting the ball high and deep into the back of the
court, buying the player time to recover and prepare for the next shot.

Understanding the dynamics and complexities of squash shots is essential for
players to excel in the sport. It not only requires technical proficiency but
also demands a deep comprehension of movement efficiency and decision-making.
In this regard, the application of Fitts’ law in squash can provide valuable
insights into the biomechanics and cognitive processes involved in shot
selection and execution.

Fitts’ law, originally formulated for human-computer interaction and small
tasks on digital devices, offers a framework to quantify the relationship
between movement time, target difficulty, and motor control. By adapting
Fitts’ law to the context of squash, we can measure the movement time required
to reach and return various shots and calculate an index of difficulty for
each shot. This allows us to determine the effort/information rate associated
with different shot movements. Exploring the application of Fitts’ law in
squash has significant implications for the field of sports, particularly in
understanding the efficiency and effectiveness of different shots. It provides
valuable insights into optimizing player training and coaching methodologies
by identifying shots that demand additional focus and practice.

However, it is important to note that the original formulation of Fitts’ law
may require adjustments to accommodate the unique characteristics of squash
shots and movements. Therefore, we conducted a series of experiments,
analyzing players of varying skill levels retrieving different shots. Through
data analysis and statistical modeling, we formulated and validated a modified
Fitts’ law equation specific to squash, capturing the relationship between
movement time, index of difficulty, and information processing rate.

In this paper, we present the findings of our study on the application of
Fitts’ law in squash. We delve into the background of Fitts’ law, discussing
its original formulation and close variants that address certain limitations.
We then outline the problem formulation, explaining how we adapted Fitts’ law
to measure the difficulty and effort of picking up squash shots. The
methodology section details our experimental setup, data collection process,
and the calculations used to determine the index of difficulty and movement
time for each shot. Finally, we present the results and discuss their
implications for players, coaches, and the overall understanding of squash as
a sport.

By leveraging the principles of Fitts’ law in the context of squash, this
research aims to enhance our understanding of shot dynamics, performance
optimization, and strategic decision-making. Ultimately, it contributes to the
ongoing exploration of sports science and human performance in the realm of
squash, providing a foundation for further research and practical applications
in training and competition.

To the best of our knowledge, this is the first work to systematically
investigate the application of Fitts’ law in the context of squash shots. By
adapting this well-established framework to the unique demands of squash, we
have made significant strides in understanding the biomechanics, cognitive
processes, and efficiency of the movement required to retrieve different
shots. This study represents a considerable effort in bridging the gap between
sports science and squash, offering valuable insights that can inform training
methodologies, enhance player performance, and contribute to the overall
advancement of the sport.

Figure 1: Dynamic gameplay in a Squash Court: Capturing the intensity of
competitive exchange between two skilled players.

##  2 Background

###  2.1 Fitts’ experiment

Fitts’ original experiment [fitts1954information], conducted by Paul M. Fitts,
played a pivotal role in formulating the fundamental equation that expresses
the relationship between throughput, the index of difficulty (ID), and
movement time (MT). The experiment involved a simple tapping task, wherein two
targets of width W were placed at a distance A from each other. By measuring
the time taken to tap each target alternately, Fitts calculated the index of
difficulty using the equation: $ID=\log_{2}(2A/W)$.

In Fitts’ experiment, the movement time ranged from 180ms to 731ms. The index
of performance for various experimental setups was calculated by dividing the
calculated ID by the movement time. Fitts observed that the mean index of
performance was 10.10 bits/s, with a standard deviation of 1.33 bits/s, which
he assumed to be the average information processing rate of the human motor
control system.

In the original Fitts’ law formulation, the focus was primarily on predicting
movement time based on the difficulty of the task represented by the index of
difficulty (ID), which is calculated using the target width (W) and distance
(A) between targets. The term throughput was defined to incorporate the
concept of information processing rate or task performance by taking into
account the trade-off between speed and accuracy in motor tasks. It provides a
more comprehensive understanding of the relationship between movement time,
index of difficulty, and task performance. The throughput (TP) in the context
of Fitts’ law refers to the rate at which information is processed and tasks
are completed. It is calculated by dividing the index of difficulty (ID) by
the movement time (MT), resulting in the formula: $TP=ID/MT$.

The throughput model recognizes that maximizing speed alone may compromise
accuracy, while maximizing accuracy alone may result in slower task
completion. By considering the throughput, researchers can evaluate the
efficiency of a task performance, taking into account both speed and accuracy.
It provides insights into the information processing capabilities of
individuals and can be used to compare performance across different tasks or
conditions. A higher throughput value indicates a more efficient performance,
as it reflects a greater amount of information processed per unit of time. The
throughput model has been widely used in the field of human-computer
interaction, ergonomics, and user interface design. It helps in understanding
the impact of task characteristics, such as target size and distance, on the
overall performance of motor tasks. By considering throughput, researchers and
designers can optimize task conditions to achieve the desired balance between
speed and accuracy.

Figure 2: Fitts’ Original Experiment

###  2.2 Close Variants of Fitts’ law

While Fitts’ law demonstrated a high correlation between movement time and
index of difficulty, there were some issues encountered. As the values for the
index of difficulty decreased, the movement time deviated from the regression
line, indicating non-compliance with Fitts’ formulation. To address this,
variants of Fitts’ law were proposed.

Mackenzie’s Variant: [mackenzie1992fitts] One variant, introduced by Scott
Mackenzie, adjusted the movement time formula to $MT=a+\log_{2}(2A/W+1)$,
ensuring that as the value of $2A/W$ approached 0, the index of difficulty
remained non-negative. This adjustment resulted in a more practical
formulation and improved the alignment of the graph for lower index of
difficulty values with the regression line.

Welford formulation: [mackenzie1993fitts] Another variant, known as the
Welford formulation, introduced the equation $MT=a+b_{1}\log
2(A)+b_{2}\log_{2}(1/W)$, which can be generalized to $ID=b\log_{2}(a/W_{k})$.

Zhai-Accot’s Steering law: [accot1997beyond] .[accot1999performance]
Additionally, Zhai and Accot developed the steering law as an extension of
Fitts’ law to account for trajectory tasks, as Fitts’ law is primarily
applicable to pointing tasks. The steering law considers both the targets and
the path between them, with the path width influencing the task difficulty.
This law modified the index of difficulty calculation by using the ratio of
$A/W$, representing the path width, in the equation $MT=a+b(A/W)$.

Multiple-Target Fitts’ Law: [balakrishnan2004beating] The multiple-target
Fitts’ law extends the original formulation to tasks involving multiple
targets. It accounts for the complexity and characteristics of tasks that
require multiple sequential movements. This variant considers the total
distance traveled and the time taken to complete all the movements in the
task.

Extended Fitts’ Law: [mackenzie1992extending] This variant incorporates
additional factors such as target distance, target amplitude, and target width
to provide a more comprehensive model for predicting movement time. It takes
into account the movement trajectory and the complexity of the task, resulting
in improved accuracy in predicting performance.

Multi-directional Fitts’ Law: [mackenzie2012fittstilt] The original Fitts’ law
assumes unidirectional movements, but in many real-world tasks, movements
occur in multiple directions. This variant extends Fitts’ law to include
movements in different directions, considering the angular aspect of the task.
It provides a more accurate prediction of movement time for tasks involving
movements in various directions.

Discrete Fitts’ Law: [jagacinski1980fitts] .[guiard1997fitts] Discrete Fitts’
law is a variant that applies Fitts’ law to discrete pointing tasks, where the
targets are presented one at a time. It is particularly relevant in situations
where sequential targeting is required, such as menu selection or scrolling.
This variant considers the time required to transition between targets,
improving the applicability of Fitts’ law to discrete tasks.

Compound Fitts’ Law: [milner1992model] Compound Fitts’ law extends the
original formulation to accommodate tasks with multiple subtasks or sequential
movements. It considers the time required to complete each subtask and
combines them to estimate the overall movement time. This variant is useful in
analyzing complex tasks that involve a series of discrete movements.

Model-based Fitts’ Law: [crossman1983feedback] Model-based variants of Fitts’
law incorporate additional parameters based on specific models of human
movement. These models may take into account factors such as acceleration,
deceleration, and target acquisition dynamics to improve the accuracy of
predicting movement time. Model-based approaches enhance the applicability of
Fitts’ law to a wide range of tasks and movement scenarios.

Expanding Targets Fitts’ Law: [mcguffin2005fitts] This variant extends Fitts’
law to accommodate targets that dynamically change in size during the task.
Instead of a fixed target width, the target width continuously expands or
contracts based on certain criteria. This variant is particularly relevant in
interactive user interfaces where targets may dynamically resize based on user
input or system feedback.

Time-Constrained Fitts’ Law: [zelaznik1988role] This variant incorporates time
constraints into the Fitts’ law formulation. It aims to predict the achievable
throughput or performance under specific time limitations. By considering the
movement time and index of difficulty along with a predefined time constraint,
this variant provides insights into the optimal performance that can be
achieved within a given time frame.

Bivariate Fitts’ Law: [accot2003refining] Bivariate Fitts’ law extends the
original formulation to include additional parameters that account for the
task’s spatial and temporal aspects. It considers factors such as movement
direction, movement speed, and the correlation between spatial and temporal
characteristics of the task. This variant provides a more comprehensive model
that captures the complex relationship between movement parameters and
performance.

Cognitive Fitts’ Law: [sleimen2013age] Cognitive Fitts’ law expands the
original formulation by incorporating cognitive factors into the model. It
takes into account cognitive load, attentional demands, and decision-making
processes during the task. By considering the cognitive aspects of motor
control, this variant provides insights into the influence of cognitive
factors on movement performance and efficiency.

Fitts’ Law in 3D Space: [cha2013extended, zeng2012fitts] While the original
Fitts’ law primarily applies to 2D pointing tasks, variants have been proposed
to extend it to three-dimensional (3D) space. These variants consider
additional dimensions, such as depth or height, along with horizontal and
vertical movements. They account for the spatial characteristics of tasks that
involve reaching or pointing in 3D environments, such as virtual reality or
robotic applications.

These variants and extensions of Fitts’ law provide researchers with a range
of tools to account for different task characteristics and improve the
applicability of the model in various domains.

##  3 Problem formulation, solution overview, and assumptions

In this study, we aimed to apply Fitts’ law to the specific scenario of
picking up various shots in our game. However, since the conventional
formulation of Fitts’ law is primarily designed for pointing tasks, we
recognized the need for a modified version tailored to our unique situation.
To achieve this, we conducted a comprehensive analysis of the factors
influencing the difficulty of picking up shots and performed a pilot study in
the squash court as a sanity test to verify its applicability.

Two key factors emerged as crucial determinants: the speed of the ball being
played and the distance that players need to cover to reach the desired shot
location. These factors directly impact the complexity and challenge involved
in successfully picking up shots. The speed of the ball influences the
player’s reaction time and the level of precision required to intercept the
shot effectively. Faster-moving balls demand quicker reflexes and more
accurate positioning, increasing the overall difficulty of shot selection.
Additionally, the distance that players need to travel to reach the desired
shot location affects the time available for movement and the positioning
required to retrieve the ball effectively. Longer distances can lead to more
challenging shot pickups, as they require swift and efficient movement across
the court.

By incorporating these factors into our index of difficulty formulation, we
can quantitatively assess the level of challenge associated with different
shot selections. Our modified version of Fitts’ law takes into account the
speed of the ball (v) and the distance to the shot location (D), resulting in
the equation:

$ID=\log_{2}(vD)$ (1)

Here, $v$ represents the speed of the ball, and $D$ corresponds to the
distance the player must cover to successfully pick up the shot. This equation
allows us to calculate an index of difficulty (ID) specific to our gameplay
scenario for each shot, providing a measure of the relative difficulty
involved in successfully picking up different shots in our squash game.

To further assess player performance, we integrated the calculated ID value
into the Fitts’ law equation, giving us the index of performance (IP):

$IP=ID/MT=\log_{2}(vD)/MT$ (2)

In this equation, ’MT’ denotes the time it takes for the player to reach the
shot and successfully pick it up. By applying this modified formulation, we
aim to gain insights into the relationship between the difficulty of shot
selection, player movement time, and overall performance.

Apart from the speed of the ball and the distance that players need to travel,
there are several other factors that can influence the difficulty of picking
up squash shots. In this experiment, we ensure the controlled manipulation of
these factors to maintain consistency in the difficulty level, minimizing
their impact on the results. However, if the factors vary significantly, one
should consider these factors alongside the speed of the ball and distance to
the shot location to get a more comprehensive and accurate assessment of the
difficulty involved in picking up squash shots. These factors include:

  1. 1.

Shot Angle: The angle at which the ball is played can affect the difficulty of
shot selection. Shots played at acute angles or extreme angles away from the
player’s position may require more precise movement and positioning,
increasing the overall difficulty.

  2. 2.

Shot Placement: The location on the court where the ball is played can impact
the complexity of shot retrieval. Shots placed near the corners or close to
the wall may require players to cover a larger distance or adopt challenging
body positions, adding to the difficulty level.

  3. 3.

Shot Velocity: In addition to the speed of the ball, the velocity at which the
ball is struck by the opponent can influence the difficulty. Higher ball
velocities demand quicker reactions and increased agility to intercept the
shot effectively.

  4. 4.

Shot Spin: The spin or rotation of the ball can introduce unpredictability and
variation in shot trajectories. Dealing with spin requires players to adjust
their positioning, footwork, and racket angle, thereby affecting the
difficulty of shot pickup.

  5. 5.

Player Positioning: The initial position of the player on the court relative
to the shot location plays a crucial role in shot retrieval difficulty. If the
player is out of position or too far from the desired shot location, it will
require more effort and time to reach and retrieve the ball.

  6. 6.

Player Fitness and Stamina: The physical fitness and stamina of the player can
influence their ability to quickly cover distances and maintain performance
over extended periods. Fatigue can increase the difficulty of shot pickups,
especially in demanding gameplay situations.

  7. 7.

Environmental Factors: External factors such as lighting conditions, court
surface, temperature, and humidity can also impact shot retrieval difficulty.
Poor visibility, slippery surfaces, or adverse weather conditions can pose
additional challenges for players.

##  4 Methodology

###  4.1 Participants

The experiment involved three subjects who were proficient in the sport of
squash, each representing a different skill level. The participants were
selected based on their experience and expertise in playing squash to ensure a
range of skill levels and to capture a comprehensive understanding of the
factors influencing shot difficulty.

###  4.2 Experimental Design

The experiment focused on four different types of squash shots, aiming to
evaluate the difficulty level of picking up these shots. Each participant
performed three trials for each shot, resulting in a total of 12 trials per
participant. The shots were performed by a designated player positioned at the
center of the court, while the subjects, also positioned at the center, were
responsible for picking up the shots. The experiment was carried out
sequentially for each participant, beginning with the young participant with
expert skill level, followed by the old participant with expert skill level,
and concluding with the young participant with a moderate skill level.

###  4.3 Court Configuration

The experiment took place in a standard squash court with well-defined
dimensions conforming to the official standards of the sport. The center court
area, where the shots were played and picked up, is located 4.2 meters from
the back wall and 5.55 meters from the front wall. It is situated between the
two side walls, with a distance of 3.2 meters from each side. This central
area forms a T-shaped region, as depicted in Figure 3.

###  4.4 Experimental Instruments

The squash racket used to play all the shots was the Tecnifibre Carboflex 125
X-Speed. The video recording device used was an iPhone 13 and the video
editing software (whose use is highlighted later) used was EaseUS. To measure
the distances, the in-built measurement software on the iPhone 13 was used.

###  4.5 Data Collection and Analysis

To capture relevant data for analysis, a video recording of the experiment was
taken by a designated observer positioned either at the back or to the side of
the court. This recording allowed for a detailed post-experiment analysis.

After picking up each shot, the player maintained a stationary position with
the racket in the same position where it made contact with the ball. The
distance between the player’s starting position and the pickup location was
measured and recorded as the distance traveled (D) by the player to reach the
shot.

The video analysis involved determining the point of contact between the ball
and the front wall. The distance between the player who hit the shot and the
point of contact on the front wall was measured to evaluate the distance
travelled by the ball to eventually calculate its velocity.

Furthermore, the video recording was slowed down by a factor of 10 to enable
precise measurement of the time taken by the ball to reach the front wall.
This, along with the calculation of distance travelled to the front wall
mentioned above, allowed for the calculation of the ball’s speed (v). It is
important to note that, due to the short duration of the time period, the
ball’s average speed was assumed to be equal to its instantaneous speed.

###  4.6 Data Analysis and Calculations

The collected data, including the distance traveled by the player (D) and the
speed of the ball (v), were used to calculate the index of difficulty (ID) for
each shot as expressed in Equation 1.

Subsequently, the obtained value of ID was utilized in the original Fitts’ law
equation to determine the index of performance (IP) for each shot. The index
of performance was calculated using the Equation 2.

The resulting data and performance metrics provided valuable insights into the
relationship between shot difficulty, player movement, and ball
characteristics, contributing to a comprehensive analysis of the factors
influencing the difficulty of picking up squash shots.

Note: Ethical guidelines and informed consent protocols were followed
throughout the experiment to ensure the well-being and rights of the
participants.

Figure 3: Task Space: Dimensions of a Squash Court

##  5 Results

Table 1: Data from the Squash Experiment on Fitts’ law - DB refers to the distance travelled by ball, T refers to the time taken by the ball to cover that distance, v refers to the speed of the ball, DP refers to the distance travelled by the player, ID refers to the index of difficulty, MT refers to the movement time of the player and IR refers to the information rate Person | Shot | Trial | DB(cm) | T (s) | V(m/s) | DP (cm) | ID (bits) | MT(s) | IR (bits/s)
---|---|---|---|---|---|---|---|---|---
1 | Drive | 1 | 586 | 0.197 | 29.75 | 374 | 6.8 | 1.22 | 5.57
1 | Drive | 2 | 587 | 0.204 | 28.77 | 386 | 6.8 | 1.21 | 5.62
1 | Drive | 3 | 580 | 0.198 | 29.29 | 454 | 7.01 | 1.23 | 5.7
1 | Drop | 1 | 615 | 0.395 | 15.57 | 355 | 5.79 | 1.06 | 5.46
1 | Drop | 2 | 614 | 0.3775 | 16.26 | 352 | 5.84 | 1.09 | 5.36
1 | Drop | 3 | 605 | 0.395 | 15.32 | 364 | 5.8 | 1.04 | 5.58
1 | Lob | 1 | 686 | 0.3125 | 21.95 | 380 | 6.38 | 1.89 | 3.38
1 | Lob | 2 | 665 | 0.3625 | 18.34 | 402 | 6.2 | 1.63 | 3.8
1 | Lob | 3 | 670 | 0.3275 | 20.46 | 361 | 6.21 | 1.78 | 3.49
1 | Boast | 1 | 971 | 0.792 | 12.26 | 491 | 5.91 | 1.013 | 5.83
1 | Boast | 2 | 980 | 0.732 | 13.39 | 360 | 5.59 | 1.208 | 4.63
1 | Boast | 3 | 961 | 0.715 | 13.44 | 350 | 5.56 | 1.27 | 4.38
2 | Drive | 1 | 598 | 0.258 | 23.18 | 386 | 6.48 | 1.361 | 4.76
2 | Drive | 2 | 567 | 0.21 | 27 | 388 | 6.71 | 1.256 | 5.34
2 | Drive | 3 | 593 | 0.204 | 29.07 | 316 | 6.52 | 1.321 | 4.94
2 | Drop | 1 | 636 | 0.446 | 14.26 | 260 | 5.21 | 1.12 | 4.65
2 | Drop | 2 | 679 | 0.47 | 14.45 | 330 | 5.58 | 1.08 | 5.17
2 | Drop | 3 | 587 | 0.413 | 14.21 | 285 | 5.34 | 1.11 | 4.81
2 | Lob | 1 | 686 | 0.283 | 24.24 | 308 | 6.22 | 1.45 | 4.29
2 | Lob | 2 | 720 | 0.276 | 26.09 | 362 | 6.56 | 1.89 | 3.47
2 | Lob | 3 | 710 | 0.335 | 21.19 | 329 | 6.12 | 1.46 | 4.19
2 | Boast | 1 | 982 | 1.02 | 9.63 | 469 | 5.5 | 1.35 | 4.07
2 | Boast | 2 | 999 | 0.964 | 10.36 | 425 | 5.46 | 1.63 | 3.35
2 | Boast | 3 | 991 | 0.94 | 10.54 | 440 | 5.54 | 1.42 | 3.9
3 | Drive | 1 | 625 | 0.201 | 31.09 | 332 | 6.69 | 1.322 | 5.06
3 | Drive | 2 | 596 | 0.196 | 30.41 | 351 | 6.74 | 1.34 | 5.03
3 | Drive | 3 | 610 | 0.24 | 25.42 | 367 | 6.54 | 1.27 | 5.15
3 | Drop | 1 | 711 | 0.452 | 15.73 | 373 | 5.87 | 1.09 | 5.39
3 | Drop | 2 | 690 | 0.411 | 16.79 | 350 | 5.88 | 1.081 | 5.44
3 | Drop | 3 | 662 | 0.43 | 15.4 | 360 | 5.79 | 1.113 | 5.2
3 | Lob | 1 | 600 | 0.291 | 20.62 | 360 | 6.21 | 1.78 | 3.49
3 | Lob | 2 | 640 | 0.334 | 19.16 | 371 | 6.15 | 1.64 | 3.75
3 | Lob | 3 | 701 | 0.29 | 24.17 | 388 | 6.55 | 1.52 | 4.31
3 | Boast | 1 | 937 | 0.999 | 9.38 | 483 | 5.5 | 1.13 | 4.87
3 | Boast | 2 | 962 | 0.91 | 10.57 | 460 | 5.6 | 1.56 | 3.59
3 | Boast | 3 | 969 | 1.08 | 8.97 | 490 | 5.46 | 1.44 | 3.79

In this section, we report the experimental results depicting the relationship
between index of difficulty (ID) and movement time (MT) along with the
corresponding index of performance/information rate for each person and each
shot.

For Person 1, the average ID for drives was 6.87 bits (SD = 0.1), while the
mean ID for drops was 5.81 bits (SD = 0.02). This indicates that, on average,
the drop shot is easier to retrieve. Similarly, the mean ID for lobs was 6.26
bits (SD = 0.08), and for boasts, it was 5.69 bits (SD = 0.16), suggesting
that boasts were the easiest shot for Person 1.

For Person 2, the average ID for drives was 6.57 bits (SD = 0.1), while the
average ID for drops was 5.38 bits (SD = 0.15). The average ID for lobs was
6.3 bits (SD = 0.2), and for boasts, it was 5.5 bits (SD = 0.03).

For Person 3, the average ID for drives was 6.66 bits (SD = 0.08), while the
mean ID for drops was 5.85 bits (SD = 0.04). The mean ID for lobs was 6.3 bits
(SD = 0.2), and for boasts, it was 5.52 bits (SD = 0.06).

When considering all the players, the mean ID for drives was 6.7 bits (SD =
0.16), for drops it was 5.68 bits (SD = 0.23), for lobs it was 6.29 bits (SD =
0.16), and for boasts it was 5.57 bits (SD = 0.13).

The correlation between MT and ID is illustrated in the graph below. The
positive correlation is evident from the upward sloping trend line, indicating
that as ID increases, so does MT. The data points are scattered around the
trend line but remain relatively close to it, confirming the relationship
between MT and ID. The equation of the trend line is given as MT = 0.456 ID -
1.3.

Figure 4: Overall Relationship Movement Time (s) vs Index of Difficulty (bits)
Figure 5: Movement Time (s) vs Index of Difficulty (bits) for drives Figure 6:
Movement Time (s) vs Index of Difficulty (bits) for boasts Figure 7: Movement
Time (s) vs Index of Difficulty (bits) for lobs Figure 8: Movement Time (s) vs
Index of Difficulty (bits) for drops

##  6 Discussions

###  6.1 Implications of Index of Difficulty

The findings indicate that the Index of Difficulty (ID) is a reasonable
measure that effectively captures the difficulty of squash shots. The low
standard deviation in the average ID suggests that the formulation of ID
adequately considers the essential factors influencing shot difficulty,
specifically the ball speed and its landing position. The ID appears to be
primarily influenced by these factors, which aligns with the intuitive
understanding that better shots are designed to challenge opponents by
increasing the time required to reach them.

###  6.2 Overall relationship between movement time and index of difficulty

The increasing index of difficulty (ID) indicates an improvement in the
quality of shots played which makes it harder to pick up, thus resulting in
increased movement time. This observation aligns with the intuitive
understanding among squash players that better shots are designed to
disadvantage opponents by prolonging the time required to reach them.

The positive relationship between movement time (MT) and ID is illustrated by
the upward trend observed in the MT vs. ID graph presented in Figure 4. This
graph represents the relationship for all shots except drops, lobs, and
boasts. Drives are excluded from this analysis, likely due to their distinct
nature as shots characterized by high velocity, which places them in a
separate shot family.

The correlation coefficient between the trend line and the actual plotted
values indicates a medium correlation, allowing us to infer a positive
relationship between MT and ID. While the correlation is only moderate, a
clear upward trend can be observed visually.

The relatively modest correlation between the trend line and the actual MT vs.
ID graph may be attributed to the presence of various minor factors, as
discussed earlier in the paper, that can slightly influence shot difficulty.
However, these factors are too insignificant and random to be incorporated
into the formulation of the ID, as evident from the clear visual trend
observed in the data.

###  6.3 Shot-specific relationship between movement time and index of
difficulty

The individual relationship between movement time (MT) and index of difficulty
(ID) for each shot is illustrated in Figures 5, 6, 7, and 8. These figures
reveal that as the ID of a particular shot increases, the movement time
decreases. This phenomenon can be attributed to the fact that harder shots
require the receiving player to move faster and exert more effort, enabling
them to reach the ball more swiftly. However, this negative relationship does
not hold true for lobs, as they are slow and defensive shots that do not exert
significant pressure on the opponent. Consequently, lobs follow the expected
positive relationship between MT and ID.

Moreover, the lines in the figures appear relatively straight for all shots
except boasts, indicating that the index of difficulty within a particular
shot type has minimal influence on movement time. This suggests that movement
time remains largely consistent for each shot type, and it is only when
transitioning between different types of shots that the index of difficulty
starts to impact movement time. However, boasts exhibit a distinct downward
slope, signifying that the index of difficulty does affect movement time for
this particular shot type. This deviation from the relatively constant
movement time observed in other shots can be attributed to the unique
characteristics of boasts, which are two-wall shots that significantly differ
from the other shot types.

###  6.4 Analyzing the effort required for different shots and its practical
implications

When considering movement time within a specific shot type, there is generally
minimal variation. However, the throughput, which refers to the effort
required to retrieve a shot, may vary slightly based on the index of
difficulty within each shot type. For instance, a drop shot with a higher
index of difficulty would likely take the same amount of time to retrieve as a
drop with a lower ID, but the effort (throughput) needed to retrieve the more
difficult drop would be higher. Nonetheless, these variations in throughput
within shot types are relatively small. The significant changes occur when
transitioning between different shot types, such as drives and lobs, or drives
and boasts.

Table 1 demonstrates that the throughput for picking up drives and drops was
similar and considerably higher compared to the other two shots. This can be
attributed to the characteristics of these shots. For drives, the player must
cover a greater distance as the ball reaches the back corner, while the ball
simultaneously travels at a fast pace, necessitating quick movement. Drops, on
the other hand, are often targeted towards the front corner, requiring less
overall movement than drives. However, drops are delicately placed shots, and
their slower nature results in a shorter reaction time for the player to reach
the ball. Consequently, players exert significant effort and quickly spring
from their position to retrieve drops. It should be noted that the drops used
in this experiment were moderately good in order to ensure the players could
reach them. Thus, high-quality drops would likely require even greater
throughput and may have the highest index of difficulty. This aligns with the
practical understanding that drops are frequently used as attacking shots to
exert pressure on opponents. Based on the finding that drives and drops
require the most effort to retrieve, it can be inferred that these shots are
the most effective to play. This information enables players to focus their
training on effectively executing these shots and improving their movement to
retrieve them.

In terms of results, lobs and boasts exhibit similar, significantly lower
throughput compared to drives and drops. This is because lobs and boasts are
generally defensive shots designed to provide players with an opportunity to
recover from a pressured situation. As observed, lobs and boasts have
relatively longer movement times compared to the other two shots, with lobs
having the longest. This is due to the time it takes for these shots to reach
the front wall, allowing the player ample time to retrieve them. Additionally,
this provides the player who played the lob or boast sufficient time to regain
a better position on the court.

##  7 Conclusion

In this work, we performed a quantitative assessment of picking up different
squash shots. We aimed to investigate the most challenging shots to pick up
and the shot with the longest retrieval time. The findings revealed that the
drives posed the greatest difficulty, requiring significant effort to
retrieve. Conversely, the lob shot had the longest retrieval time due to its
slow speed and deep placement. Furthermore, during the investigation, a
potentially suitable equation for measuring the index of difficulty (ID) of
any shot was identified.