11institutetext: Quasar Science Resources SL for ESA, European Space Astronomy
Centre (ESAC), Science Operations Departement, 28692 Villanueva de la Cañada,
Madrid, Spain 22institutetext: Centre for Astrophysics Research, University of
Hertfordshire, College Lane, Hatfield AL10 9AB, UK 33institutetext: Institute
of Astronomy, Madingley Road, Cambridge CB3 0HA, UK 44institutetext: INAF –
Osservatorio Astronomico di Roma, via Frascati 33, 00078 Monteporzio Catone,
Italy 55institutetext: INAF - Osservatorio Astronomico di Cagliari, via della
Scienza 5, 09047 Selargius, Italy 66institutetext: CNRS, IRAP, 9 Av. colonel
Roche, BP 44346, 31028 Toulouse cedex 4, France 77institutetext: Cahill Center
for Astronomy and Astrophysics, California Institute of Technology, Pasadena,
CA 91125, USA 88institutetext: European Southern Observatory, Karl-
Schwarzschild-Strasse 2, 85748 Garching bei München, Germany 99institutetext:
Department of Physics, University of Rome “Tor Vergata,” Via della Ricerca
Scientifica 1, 00133 Rome, Italy 1010institutetext: Department of Physics and
Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, UK
1111institutetext: INAF - IASF Palermo, Via U. La Malfa 153, 90146 Palermo,
Italy 1212institutetext: INAF - IASF Milano, Via Corti 12, 20133 Milano, Italy
1313institutetext: Centre for Extragalactic Astronomy & Department of Physics,
Durham University, South Road, Durham DH1 3LE, UK

# Probing the nature of the low state in the extreme ultraluminous X-ray
pulsar NGC 5907 ULX1

F. Fürst 11 D. J. Walton 2233 G. L. Israel 44 M. Bachetti 55 D. Barret 66 M.
Brightman 77 H. P. Earnshaw 77 A. Fabian 33 M. Heida 88 M. Imbrogno M. J.
Middleton 99441010 C. Pinto 1111 R. Salvaterra 1212 T. P. Roberts 1313 G. A.
Rodríguez Castillo 1111 N. Webb 66

(Received XX.XX.XX / Accepted XX.XX.XX)

NGC 5907 ULX1 is the most luminous ultra-luminous X-ray pulsar (ULXP) known to
date, reaching luminosities in excess of $10^{41}$ $\text{erg\,s}^{-1}$. The
pulsar is known for its fast spin-up during the on-state. Here, we present a
long-term monitoring of the X-ray flux and the pulse period between 2003–2022.
We find that the source was in an off- or low-state between mid-2017 to
mid-2020. During this state, our pulse period monitoring shows that the source
had spun down considerably. We interpret this spin-down as likely being due to
the propeller effect, whereby accretion onto the neutron star surface is
inhibited. Using state-of-the-art accretion and torque models, we use the
spin-up and spin-down episodes to constrain the magnetic field. For the spin-
up episode, we find solutions for magnetic field strengths of either around
$10^{12}$ G or $10^{13}$ G, however, the strong spin-down during the off-state
seems only to be consistent with a very high magnetic field, namely,
$>10^{13}$ G. This is the first time a strong spin-down is seen during a low
flux state in a ULXP. Based on the assumption that the source entered the
propeller regime, this gives us the best estimate so far for the magnetic
field of NGC 5907 ULX1.

###### Key Words.:

X-rays: Binaries – X-rays: individual (NGC 5907 ULX1) – accretion, accretion
disk – magnetic fields – stars:neutron

##  1 Introduction

A growing number of ultraluminous X-ray sources (ULXs), namely, off-nuclear
X-ray binaries with apparent luminosities that exceed $10^{39}$
$\text{erg\,s}^{-1}$(see Kaaret et al. 2017 for a review), are now known to be
powered by highly magnetized neutron star accretors (with magnetic fields of
$B\gg 10^{9}$ G). Six such sources have been discovered to date through the
detection of coherent X-ray pulsations and thus classified as ultra-luminous
X-ray pulsars (ULXPs): M82 X-2 (Bachetti et al. 2014), NGC 7793 P13 (Fürst et
al. 2016; Israel et al. 2017b), NGC 5907 ULX1 (Israel et al. 2017a), NGC 300
ULX1 (Carpano et al. 2018), NGC 1313 X-2 (Sathyaprakash et al. 2019) and M51
ULX7 (Rodríguez Castillo et al. 2020). A few more candidates have been found
through the tentative identification of pulsations (e.g., NGC 7793 ULX-4,
Quintin et al., 2021) or the possible detection of cyclotron resonant
scattering lines (e.g., M51 ULX-8, Brightman et al., 2018). Additionally, the
galactic source Swift J0243.6+6124 reached luminosities significantly above
$10^{39}$ $\text{erg\,s}^{-1}$and can therefore also be classified as a ULXP
(Wilson-Hodge et al., 2018; Tsygankov et al., 2018). Other similar sources
include transient neutron stars with Be-star mass donors in the SMC, like SMC
X-3 (Townsend et al., 2017; Liu et al., 2022) and RX J0209.6$-$7427
(Vasilopoulos et al., 2020; Hou et al., 2022), which also reached ULX level
luminosities briefly during giant outbursts. While it is currently unclear if
the accretion geometry during these outbursts is similar to persistent ULXPs
or not (e.g., all ULXPs currently exhibit much higher luminosities), it shows
that there is a clear connection between Be-X-ray binaries and ULXPs.

There is speculation that neutron stars could make up a significant fraction
of the total ULX population (e.g., Pintore et al. 2017; Koliopanos et al.
2017; King et al. 2017; Middleton & King 2017; Walton et al. 2018c).
Furthermore, there is significant debate over how exactly these neutron stars
are able to reach such extreme luminosities and most of this debate is focused
on the strength of the magnetic fields in these systems. Some authors argue
for extremely strong, magnetar-level fields ($B>10^{13}$ G; e.g., Ekşi et al.
2015; Mushtukov et al. 2015), as such fields suppress the electron scattering
cross-section (Herold 1979) and permit higher luminosities for a given
accretion rate, while others argue for much lower fields and accretion rates
($B\approx 10^{10}$–$10^{11}$ G; e.g., Kluźniak & Lasota 2015; King et al.
2017) based on the large spin-up rates ($\dot{P}$) observed.

Both scenarios, however, present some issues. The very strong magnetic field
scenarios struggle to explain how accretion at luminosities below $\sim
10^{40}\,\text{erg\,s}^{-1}$ is possible since (in theory) the magnetosphere
would stop accretion and the source would be in a permanent propeller state
(see below). The low-magnetic field case on the other hand typically also
assumes a very high beaming factor to explain the apparent extreme
luminosities. This beaming would result in a very narrow funnel, which appears
to be in contradiction to the observed sinusoidal pulse profiles with high
pulsed fractions (Mushtukov et al., 2021). Additionally, the very high spin-up
rate requires an intrinsically high accretion rate, setting up an argument
against an extreme beaming factor.

It is also possible that a combination of these two explanations is present,
with a very strong quadrupolar field acting close to the neutron star, whereas
further away, the weaker dipolar field dominates (e.g., Israel et al., 2017a;
Middleton et al., 2019a; Kong et al., 2022). This scenario has been claimed
for Swift J0243.6+6124, driven by the discovery of a cyclotron resonant
scattering feature (CRSF) at up to 146 keV (Kong et al., 2022). This line
energy implies a filed strength in the line-forming region (which is likely to
be very close to the neutron star surface) of around $1.6\times 10^{13}$ G.
Kong et al. (2022) have argued that such a strong field has to be in the
multipolar component, as a dipole of this magnitude would lead to
contradictions with other measurements. Nonetheless, it is currently unclear
if the same scenario can explain the persistent ULXPs at even higher
luminosities.

Reliably determining the magnetic fields of ULXPs has proven challenging to
date. In general, the most robust method for determining field strengths in
accreting neutron stars is via the study of cyclotron resonant scattering
features (CRSFs; see Staubert et al. 2019 for a recent review). However, such
features are challenging to detect in ULXs given their relatively low fluxes
and the limited bandpass available for sensitive searches (even in the NuSTAR
era). Only two potential features have been seen from the ULX population to
date and these limited results paint something of a mixed picture (Brightman
et al. 2018; Walton et al. 2018a; Middleton et al. 2019b). Other potential
means of estimating the magnetic field strengths in ULXPs are thus clearly of
interest.

One thing a number of the known ULXPs have in common is that they show strong
long-term variability, sometimes including transitions to ”off-states” in
which their X-ray flux drops by orders of magnitude (e.g., Motch et al. 2014;
Earnshaw et al. 2016; Brightman et al. 2016). One possible explanation for
these events is that they represent transitions to the propeller regime, in
which the magnetic field of the neutron star (temporarily) acts as a barrier
to accretion, resulting in a precipitous drop in the observed X-ray flux
(Illarionov & Sunyaev 1975; Tsygankov et al. 2016; Earnshaw et al. 2018). If
this is correct, then they may offer an independent means to estimate the
dipolar magnetic fields of these sources. However, the nature of these events
is not yet clear, and may differ for different systems and events. For
example, in M82 X-2 X-ray monitoring suggests that low-states are related to
the $\sim$64 d super-orbital period seen in that system (Brightman et al.
2019), which would be somewhat challenging to reconcile with a propeller-based
interpretation. Furthermore, other authors have suggested that these low-flux
periods may be related to obscuration, instead of a cessation of accretion
(Motch et al. 2014; Vasilopoulos et al. 2019; Fürst et al. 2021).

Among the known ULXPs NGC 5907 ULX1 is the most luminous, with an astonishing
peak luminosity of $L_{\rm{X}}\sim 10^{41}$ $\text{erg\,s}^{-1}$ ($\sim$500
times the Eddington luminosity for a standard neutron star assuming a distance
of $d=17.1$ Mpc; Israel et al. 2017a; Tully et al. 2016), making it a case of
particular interest. Coordinated observations with XMM-Newton and NuSTAR show
that when the source is bright, it shows broadband spectra that are well
described with a combination of a super-Eddington accretion disk that
contributes thermal emission below $\sim$10 keV and an accretion column that
dominates at higher energies (Walton et al. 2015, 2018c; Fürst et al. 2017).
Studies of the short-timescale evolution of its pulse period revealed a
potential orbital period of $\sim$5 d (Israel et al. 2017a; however, this
period is not yet confirmed), and when in its ULX state the source is also
known to exhibit a $\sim$78 d X-ray period (Walton et al. 2016), which is
therefore likely to be super-orbital in nature. NGC 5907 ULX1 is also one of
the ULXPs that is known to exhibit intermittent off-states (Walton et al.
2015). However, unlike the case of M82 X-2, the low-flux periods in NGC 5907
ULX1 are not related to its super-orbital period and so, they could plausibly
be related to the propeller transition.

As reported by Israel et al. (2017a), the earliest measurement of pulsations
from NGC 5907 ULX1 are from February 2003, with a spin period $P$ around 1.43
s. Eleven years later, the pulsar had spun-up to $P\approx 1.14$ s (Israel et
al., 2017a). Assuming a constant spin-up over this period implies a rate of
change of the pulse period $\dot{P}=-8.1\times 10^{-10}$ s s-1. This long-term
spin-up indicates that the pulsar was not at spin equilibrium and that the
high mass accretion rate also resulted in a large increase of angular
momentum. Similar long-term spin-ups are observed for other ULXPs, like NGC
7793 P13 ($\dot{P}\approx-4\times 10^{-11}$ s s-1, Fürst et al., 2021) and NGC
300 ULX1 ($\dot{P}\approx-4\times 10^{-10}$ s s-1, Vasilopoulos et al., 2019).

As the spin-up is driven by transfer of angular momentum through accretion,
this can also be theoretically used to measure the magnetic field strength.
This fact has been discussed in detail by the seminal papers by Ghosh & Lamb
(1979a) and Ghosh & Lamb (1979b), which were subsequently updated by Wang
(1995). However, those calculations are based on various simplifications, in
particular, they assume a geometrically thin, optically thick accretion disk
as expected for sub-Eddington accretion rates (Shakura & Sunyaev 1973). In
ULXPs, however, we expect the accretion disk to be geometrically thick due to
their super-Eddington accretion rates (Shakura & Sunyaev, 1973; Abramowicz et
al., 1988). Nonetheless, using the Ghosh & Lamb (1979b) model for ULXPs NGC
7793 P13 and NGC 300 ULX1, magnetic fields around $10^{12}$ G were implied
(Fürst et al., 2016; Carpano et al., 2018), which is very well in line with
the magnetic fields observed in high-mass X-ray binaries in our own galaxy
(see, e.g., Staubert et al., 2019).

For NGC 5907 ULX1, much higher magnetic fields have been postulated based on
the extreme luminosity, with a multipolar magnetic field component as high as
a few $10^{14}$ G (Israel et al., 2017a). More recently, using an updated
description of the coupling between the magnetic field and the accretion flow
intended to be more suitable for a super-Eddington disk, Gao & Li (2021)
estimated a maximal field between 2–3 $\times 10^{13}$ G. These authors,
however, find that the observations presented by Israel et al. (2017a) are
also consistent with a field around $6\times 10^{11}$ G.

We are therefore in need of another approach to constrain the magnetic field.
One possibility is to study the expected spin-down during a low flux phase, in
which accretion might be inhibited due to the fast rotation of the neutron
star. In this regime, the spin-down torque onto the neutron star is dominated
by the magnetic field interacting with the residual accretion disk (Parfrey et
al., 2016). NGC 5907 ULX1 entered a suitable low-flux state in mid-2017,
which, broadly speaking, lasted until mid-2020, with brief episodes of higher
flux in between. In this paper, we discuss the pulse period evolution before,
during, and after this extended off-state, as observed with XMM-Newton, and
show that the pulsar spun-down significantly during the off-state.

The remainder of the paper is structured as follows. In Sect. 2, we describe
the data used and extraction methods. In Sect. 3 we discuss the flux and
period evolution and describe our pulsation search in detail. In Sect. 4, we
discuss our results and use the measurements to estimate the magnetic field.
Section 5 provides a conclusion, along with a summary of our main results and
an outlook for future measurements.

Figure 1:  Flux and period evolution of NGC 5907 ULX1 between 2014 to 2022.
Top: Swift XRT light curve (0.3–10.0 keV). In green the upper limit for the
point source luminosity as measured with Chandra (Belfiore et al., 2020) is
shown, using the right $y$-axis. The XMM-Newton luminosities are shown as red
diamonds and orange squares for observations with and without detect
pulsations, respectively, also using the right-hand $y$-axis. The horizontal
green line indicates the estimated propeller luminosity based on a magnetic
field strength of $B=2.6\times 10^{13}$ G (see Table 3). The blue curve shows
an extrapolation of the 78 d X-ray period seen from NGC 5907 ULX1 during its
ULX state (Walton et al. 2016). The shaded pink areas indicate where the
source was conceivably in the low state and spinning down. Bottom: Pulse
period measurements, as listed in Table 1 and Israel et al. (2017a). The gray
vertical dotted lines indicate times of observations with XMM-Newton. The blue
dashed line shows a possible fiducial model of epochs of spin-up during bright
states interspersed with spin-down during off-states. This line is only a
suggestion for the evolution of the period. For details see text.

##  2 Observations and data reduction

###  2.1 Swift

NGC 5907 ULX1 has been extensively monitored by the Neil Gehrels Swift
Observatory and, in particular, the X-ray telescope (XRT; Burrows et al.
2005). The Swift XRT light curve (Figure 1) is extracted using the standard
online pipeline (Evans et al. 2009, primarily using time bins of four days
while the source was bright (following Walton et al. 2016). The exception to
this is when NGC 5907 ULX1 drops its flux significantly. During these periods,
we revert to an approximately monthly binning. The average count rates for
these broader bins are determined by extracting XRT images and exposure maps
integrated across them, also using the online XRT pipeline, and calculating
the net count rate observed using a circular source region of radius
$10^{\prime\prime}$ and a much larger neighboring region to estimate the
background. Uncertainties are calculated following Gehrels (1986). Where the
source is not detected, an upper limit at the 3$\sigma$-level is calculated
using the method outlined by Kraft et al. (1991). These rates and upper limits
are then corrected for the fraction of the XRT PSF that falls outside of the
source extraction region. The Swift monitoring snapshots do not provide enough
photons to search for or measure the pulse period.

###  2.2 XMM-Newton

In order to understand the nature of the strong variability seen by Swift, we
executed a series of XMM-Newton observations over the last few years designed
to monitor the spin period of the source, both across and after the extended
low-flux period seen from $\sim$2017–2020. In addition, XMM-Newton also
performed a series of earlier observations extending as far back as 2003.
Details of these observations are given in Table 1.

For all of these XMM-Newton observations (see Table 1), the EPIC detectors
(EPIC-pn, EPIC-MOS; Strüder et al. 2001; Turner et al. 2001) were operating in
full-frame mode. We therefore focus on the data taken by EPIC-pn here
(temporal resolution of 73.4 ms), as the EPIC-MOS detectors do not have
sufficient timing capabilities to probe the $\sim$1 s spin period of the
neutron star (temporal resolution of $\sim$2.6 s). Data reduction was carried
out with the XMM-Newton Science Analysis System (SAS v19.1.0), largely
following standard procedures. The raw observation data files were processed
using epchainand the cleaned event files corrected to the solar barycenter
using the DE200 solar ephemeris with barycen. Source light curves were then
extracted on the time resolution of the EPIC-pn detector with xmmselect. We
typically used circular source regions of radius 25–30′′, depending on the
brightness of NGC 5907 ULX1, although for some of the extremely faint
observations, even smaller regions were occasionally used (ranging down to
$\sim$15′′). As recommended we only considered single and double patterned
events.

In all cases, the background was estimated from larger regions of blank sky on
the same detector as the source region. Given the large number of XMM-Newton
observations considered, it is unsurprising to notice that a broad range of
background activity is seen among them; some suffer from severe flaring, some
show brief, modest periods of enhanced background and some show a stable
background level throughout. In order to determine whether additional
background filtering is required – and, if so, then subsequently establishing
the level of background emission that is acceptable – we utilized the method
outlined in Piconcelli et al. (2004). This determines the background level at
which the signal-to-noise (S/N) of the source is maximized. We make this
assessment for each observation individually, and maximize the S/N over the
0.3–10.0 keV band. For cases where additional background filtering is
required, custom good-time-intervals are generated to exclude background
levels above that which gives the maximum S/N for the source.

Table 1: Details of the XMM-Newton observations of NGC 5907 ULX1 considered in
this work. The uncertainties given for the pulse period and its derivative are
statistical only and are dominated by the larger uncertainties of the orbital
ephemeris ($\approx 0.2$ ms). The luminosity is based on the absorption
corrected flux in the 0.3–10 keV band and the pulsed fraction is given for the
1–10 keV energy band.

OBSID | Start | Exposure | Luminosity | $P_{\rm{spin}}$ | $\dot{P}_{\rm{spin}}$ | PF
---|---|---|---|---|---|---
| Date | (ks) | ($10^{39}$ erg s-1) | (ms) | ($10^{-10}$ s s-1) | %
0145190201aa$a$Pulse period based on values presented by Israel et al. (2017a). | 2003-02-20 | 41.3 | $81\pm 4$ | $1427.76^{+0.11}_{-0.09}$ | $-90^{+40}_{-50}$ | $19.4\pm 2.5$
0729561301aa$a$Pulse period based on values presented by Israel et al. (2017a). | 2014-07-09 | 42.0 | $93.4^{+2.5}_{-2.8}$ | $1137.43^{+0.06}_{-0.07}$ | $-50\pm 28$ | $21.5\pm 2.3$
0804090301 | 2017-07-02 | 33.5 | $24.1^{+2.4}_{-2.6}$ | $945.79^{+0.08}_{-0.05}$ | $23^{+24}_{-29}$ | $19\pm 5$
0804090401 | 2017-07-05 | 36.0 | $14.1\pm 1.3$ | $946.17^{+0.04}_{-0.08}$ | $0^{+37}_{-20}$ | $30\pm 6$
0804090501 | 2017-07-08 | 40.0 | $5.7\pm 0.9$ | – | – | $<36.0$
0804090701 | 2017-07-12 | 40.0 | $1.2^{+0.5}_{-0.8}$ | – | – | $<44.0$
0804090601 | 2017-07-15 | 37.5 | $5.1^{+0.6}_{-1.0}$ | – | – | $<43.0$
0824320201 | 2019-06-12 | 59.7 | $62.4^{+1.9}_{-2.1}$ | $1019.142^{+0.026}_{-0.046}$ | $-29^{+16}_{-9}$ | $20.1\pm 2.3$
0824320301 | 2019-06-19 | 49.1 | $54.7^{+2.0}_{-2.1}$ | $1018.10^{+0.05}_{-0.07}$ | $-13^{+23}_{-18}$ | $13.5\pm 2.8$
0804090801 | 2019-06-22 | 38.7 | $48.8^{+2.7}_{-2.6}$ | – | – | $<13.0$
0804090901 | 2019-06-24 | 40.2 | $41.7^{+1.7}_{-1.9}$ | – | – | $<16.0$
0824320401 | 2019-06-26 | 56.4 | $42.9^{+1.6}_{-1.7}$ | – | – | $<12.0$
0804091001 | 2019-06-29 | 46.0 | $44.9^{+1.7}_{-1.8}$ | – | – | $<15.0$
0804091101 | 2019-07-05 | 34.8 | $37.5^{+1.6}_{-2.1}$ | – | – | $<19.0$
0804091201 | 2019-07-06 | 37.3 | $34.7\pm 1.9$ | – | – | $<18.0$
0851180701 | 2019-08-10 | 56.2 | $26.5^{+0.8}_{-1.3}$ | – | – | $<18.0$
0851180801 | 2019-08-12 | 60.8 | $25.7^{+0.8}_{-0.9}$ | – | – | $<17.0$
0824320501 | 2019-12-08 | 43.5 | $1.12^{+0.21}_{-0.39}$ | – | – | $<31.0$
0824320601 | 2020-07-23 | 44.5 | $59.5^{+2.1}_{-2.2}$ | $1032.94^{+0.08}_{-0.05}$ | $-23^{+21}_{-33}$ | $13.0\pm 2.7$
0824320701 | 2020-11-06 | 34.5 | $6.0^{+0.4}_{-1.0}$ | – | – | $<19.0$
0884220201 | 2021-02-20 | 54.7 | $51.3^{+2.5}_{-2.6}$ | $1012.620^{+0.025}_{-0.033}$ | $-36^{+10}_{-9}$ | $16.9\pm 2.2$
0884220301 | 2021-02-26 | 61.4 | $24.4^{+1.1}_{-1.2}$ | – | – | $<16.0$
0884220401 | 2021-03-04 | 59.0 | $35.5^{+2.0}_{-2.2}$ | $1009.563^{+0.027}_{-0.035}$ | $-31^{+11}_{-9}$ | $12.9\pm 2.3$
0891801501 | 2021-08-27 | 59.0 | $20.1^{+1.1}_{-1.2}$ | $987.558^{+0.030}_{-0.036}$ | $-11^{+10}_{-11}$ | $30\pm 4$
0893810301 | 2022-01-06 | 40.0 | $19.3^{+1.1}_{-1.5}$ | – | – | $<17.0$
111

##  3 Timing analysis

###  3.1 Light curve and flux evolution

The evolution of NGC 5907 ULX1 from early 2014 to early 2022 as seen by Swift
is shown in Figure 1. Before 2014, no dense monitoring of the source is
available, although there are a small number of observations with XMM-Newton,
Swift, and NuSTAR prior to this. The source faded significantly in 2017 – to
the extent that it became challenging to detect with Swift and only returned
to full activity briefly in $\sim$May 2019; we note that there was also a
brief re-brightening in $\sim$March 2018, but the source did not fully return
to its ”normal” ULX state. This low-flux period ended in mid 2020, and since
then NGC 5907 ULX1 has been detected in almost every Swift snapshot obtained.
Despite the broad recovery, comparing the behavior before 2017 and after 2020,
we can see that the source is still exhibiting a much larger variability in
the more recent data, with flux changes of up to a factor of $\sim$10 within a
few months.

Previously, NGC 5907 ULX1 was also in a low flux state between 2012–2014 (see
Walton et al. 2015), but its duration and exact luminosities are unclear due
to the lack of monitoring. Between 2003–2012 the source seems to have largely
been in a bright on-state, as seen by Swift, XMM-Newton, and Chandra (Israel
et al., 2017a).

Fortuitously, during the initial decline in flux in 2017 a series of five XMM-
Newton observations were taken, allowing us to obtain a precise measurement of
the neutron star spin just before the low state (Table 1). The activity period
in early 2019 was sufficiently bright and long-lived for us to trigger another
series of ten XMM-Newton observations in order to compare the properties of
NGC 5907 ULX1 in 2017 and 2019 and determine how the source has evolved across
the extended low-flux period in between. Furthermore, another series of seven
XMM-Newton observations of NGC 5907 ULX1 have been performed since its more
persistent recovery in mid-2020, allowing for further comparisons of the
recent behavior with that seen in 2019 as well as prior to 2017. In
particular, we focus on timing the pulsar and tracking the evolution of its
spin period in this work.

###  3.2 Pulsations

As a simple first step in the timing analysis we performed a Fast-Fourier
Transform (FFT) on all full-band EPIC-pn light curves, extracted with a time
resolution of 73.4 ms. This approach revealed significant pulsations at
$\sim$0.9812 Hz in ObsID 0824320201 (2019-06-12). In all other observations,
no significant signal was found using this basic analysis.

Based on the variation of the pulsed fraction as function of energy available
in the literature (Israel et al. 2017a), we find that below 1 keV the
pulsations are barely detectable. We therefore subsequently filtered all data
on energies $>1$ keV for the following analysis to increase the S/N. Following
this additional filtering, as a next step, we performed a more in-depth
pulsation search using an accelerated Fourier method, which searches a grid in
the frequency, $\nu,$ and frequency-derivative $\dot{\nu}$ space. We used the
implementation HENaccelsearch from the HENDRICS software package (Bachetti,
2015) and performed the search between $\nu$=0.9–1.2 Hz. This analysis
revealed good pulsation candidates in ObsIDs 0804090301, 0804090401,
0824320201, 0823430401, and 0891801501, with a detection significance of
$>3\sigma$.

For each of these observations, the best $\nu$-$\dot{\nu}$ combinations from
this search were then analyzed in more detail, with a grid search using epoch
folding (Leahy et al., 1983). The data were searched in temporal space across
a range of $\Delta P=\pm 0.3$ ms and $\Delta\dot{P}\pm=1\times 10^{-8}$ s s-1,
respectively, centered on the best values from HENaccelsearch. We oversampled
the search in $P$ by a factor of 5 compared to the number of independent
frequencies and used the same number of grid points also in $\dot{P}$ space.
This analysis is performed with ISIS (Houck & Denicola, 2000), following a
similar procedure to that described in Fürst et al. (2016). Uncertainties on
$P$ and $\dot{P}$ are given as the value where $\chi^{2}$ has dropped to half
its maximum value, namely, an FWHM width based on the $\chi^{2}$ landscape in
the $P$–$\dot{P}$ plane.

The results of this search are given in Table 1 and shown in the bottom panel
of Fig. 1. It is clear from these results that the neutron star was spinning
significantly slower in 2019 compared to 2017 and again slightly slower in
2020 compared to 2019. After mid 2020, NGC 5907 ULX1 can be seen to have
restarted its steady spin-up trend.

We then also performed the same epoch folding search for all ObsIDs where
HENaccelsearch did not find a significant signal. The searches were centered
on a period obtained from linear interpolation or extrapolation between the
neighboring pulsation detections. We increased the search range to $\Delta
P=3$ ms and $\Delta\dot{P}=5\times 10^{-8}$ s s-1 and again oversampled the
$P$ space by a factor of 5. Larger grids are computationally prohibitive for
this kind of brute force method. We did not find significant pulsations in any
of these observations.

The pulse periods we give here are not corrected for the orbital motion of the
neutron star. We chose not to perform this correction given the large
uncertainties on the current ephemeris (Israel et al., 2017a). However, taking
the values given by Israel et al. (2017a) and their respective uncertainties,
we find that the maximal influence of their orbital solution on the measured
spin-period is on the order of 0.2 ms. This is smaller than the difference in
spin period that we find between the two 2019 observations – and much smaller
than the differences between the 2017 and 2019 observations.

###  3.3 Pulsed Fraction

Where pulsations have been detected, we calculate the pulsed fraction based on
the pulse profile (PP) as

$PF=\frac{\max(PP)-\min(PP)}{\max(PP)+\min(PP).}$ (1)

For observations without detected pulsations, we instead calculate the upper
limits on the pulsed fraction. To do so, we simulate a series of event lists
with the same GTIs and average count rate as the real data but adding
sinusoidal pulsations with an increasing pulsed fraction. The period assumed
for these pulsations is based on the closest measured periods and is
determined by linearly interpolating the evolution of the pulse period between
these measurements to the time of the observation in question. For each pulsed
fraction, we simulated 50 individual event lists and performed the same search
for pulsations as applied to the real data. We consider pulsations to be
detected in our simulated datasets when we find a peak at the 99.9%
significance level or above. Starting from 5%, the pulsed fraction is
increased in steps of 0.1% up to the point where the fraction of simulations
that return significant pulsations increases above 90% (i.e., the pulsations
are recovered in at least 45 simulated event lists). We take this pulse
fraction to be the upper limit for the observation in question (Table 1).

##  4 Results & discussion

###  4.1 Magnetic field based on spin-up strength

As seen in Fig. 1, we can identify two epochs of relatively stable spin-up,
the first one between 2014 and 2017 (starting around MJD 56847) and the second
one after mid 2020 (starting around MJD 59053). In addition, based on the
results presented in Israel et al. (2017a), the source was also spinning up on
average between 2003 to 2014 (starting around MJD 52690). However, we do not
know how stable the spin-up and X-ray flux was between 2003 and 2014, due to
the lack of regular monitoring. Using these three spin-up episodes, we can try
to estimate the magnetic field strength based on theoretical calculations
describing how the accreted matter couples to the neutron star via the
magnetic field, as well as how much angular momentum is transferred by this
coupling. Throughout the remainder of this paper, we assume a neutron star
radius $R=10^{6}$ cm and a neutron star mass of $M=1.4\,\rm M_{\odot}$, unless
otherwise noted.

To understand the accretion torque on the neutron star, it is important to
analyze the relationship between a number of important radii at which
significant changes in the geometry or the dominating physical processes
occur. These are detailed below.

The corotation radius, $R_{c}$, is the radius at which the Keplerian orbital
speed is equal to the rotational speed of the pulsar. High accretion rates can
only occur if the accreted material couples inside this radius to the magnetic
field. The corotation radius is given by:

$\begin{split}R_{c}&=\left(\frac{G}{4\pi^{2}}\right)^{1/3}\,P^{2/3}\,M^{1/3}\leavevmode\nobreak\
\mathrm{cm}\\\ &\approx 1.7\times 10^{8}\leavevmode\nobreak\
P^{2/3}\,M_{1.4}^{1/3}\leavevmode\nobreak\ \mathrm{cm}\end{split}.$ (2)

Next, there is the magnetospheric radius, $R_{M}$, which is the radius at
which the magnetic field dominates over the ram pressure of the accreted
material and within which the material has to follow the magnetic field lines.
This is equivalent to the inner radius of the accretion disk, and is related
to the Alfvén radius computed for spherical accretion with a factor $\xi$:

$R_{M}=\xi\mu^{4/7}\left(2GM\dot{M}_{\text{in}}^{2}\right)^{-1/7}\leavevmode\nobreak\
\mathrm{cm.}$ (3)

Then we have the spherization radius, $R_{\text{sph}}$, the radius at which
the accretion becomes locally super-Eddington and geometrically thick.
Following Shakura & Sunyaev (1973), this is given by:

$R_{\text{sph}}=3GM\dot{M}_{\text{tot}}\left(2L_{\text{Edd}}\right)^{-1}\leavevmode\nobreak\
\mathrm{cm.}$ (4)

In the above expressions, $P$ is the pulse period in seconds, $M_{1.4}$ is the
neutron star mass in units of 1.4 $\rm M_{\odot}$, $\mu$ is the magnetic
moment, $G$ is the gravitational constant, $\dot{M}_{\text{in}}$ is the
accretion rate at the inner edge of the accretion disk, $\dot{M}_{\text{tot}}$
is the total mass transfer rate, and $L_{\text{Edd}}$ is the Eddington
luminosity. For a dipolar magnetic field, $\mu=BR^{3}$, where $B$ is the
magnetic flux density of the dipolar field.

The problem in calculating these radii lies in the fact that they depend on
$\dot{M}_{\text{tot}}$ as well as $\dot{M}_{\text{in}}$, which, in principle,
can be inferred from the observed luminosity, $L$. However, the relation
between $\dot{M}_{\text{in}}$ and the luminosity itself depends on the
relative location of $R_{\text{sph}}$, $R_{M}$, and $R_{C}$.

Within $R_{\text{sph}}$, the outflows will ensure that the local Eddington
luminosity is not exceeded, leading, at the same time, to the formation of a
funnel that will go on to collimate the observed emission. As discussed by
King (2009), for high accretion rates in the super-Eddington regime, we have:

$L=\frac{L_{\text{Edd}}\left(1+\ln\dot{m}\right)}{b},$ (5)

where $\dot{m}=\dot{M}_{\text{tot}}/\dot{M}_{\text{Edd}}$,
$\dot{M}_{\text{Edd}}$ is accretion rate that corresponds to the Eddington
limit and $b$ is the beaming factor, which can be approximated as $b\approx
73/\dot{m}^{2}$. High accretion rates in this case mean $\dot{m}>8.5$ (King et
al., 2017), which is the case for all observations of NGC 5907 ULX1 considered
here.

The estimated mass accretion rate $\dot{M}_{\text{tot}}$ from Eq. 5 can be
used to calculate $\dot{M}_{\text{in}}$, which is the relevant accretion rate
for determining $R_{M}$. For the same reasons as discussed for the luminosity,
$\dot{M}_{\text{in}}$ is rising slower than $\dot{M}_{\text{tot}}$ inside of
$R_{\text{sph}}$ to guarantee that the disk does not locally exceed its
Eddington limit. In particular, following Shakura & Sunyaev (1973, cf ):

$\dot{M}_{\text{in}}=\begin{cases}\dot{M}_{\text{tot}}(R_{M}/R_{\text{sph}}),\quad&\text{for
}R_{M}<R_{\text{sph}},\\\ \dot{M}_{\text{tot}},\quad&\text{for
}R_{M}>R_{\text{sph}}.\end{cases}$ (6)

Finally, in order to calculate $R_{M}$, we have to determine $\xi$, the
conversion factor for the Alfvén radius calculated under the assumption of
spherical symmetry. The value of $\xi$ depends strongly on our assumptions of
how the accreted material couples and interacts with the magnetic field,
depending on the magnetic field geometry. Gao & Li (2021) distinguish two
cases of the ratio between the azimuthal and vertical B-field strength which
describe different physical regimes of the coupling between the accretion disk
and the magnetosphere (see also Wang, 1995). These two cases lead to different
expressions of the total dimensionless torque $n(\omega),$ where $\omega$ is
the so-called fastness parameter. The exact expressions of these equations can
be found in Gao & Li (2021). Based on the description of the transfer of
angular momentum onto a rigidly rotating neutron star, we can solve for
$\omega$. These equations are at least quadratic in $\omega$, leading to two
solutions for each case. We can then solve for the magnetic field, using the
following equations (Gao & Li, 2021).
For $R_{M}\geq R_{\text{sph}}$:

$B=1.371\times
10^{12}\xi^{-7/4}\omega^{7/6}M_{1.4}^{5/6}R_{6}^{-3}\dot{M}^{1/2}P^{7/6}\,\text{G.}$
(7)

And for $R_{M}\leq R_{\text{sph}}$:

$B=1.234\times
10^{13}\xi^{-7/4}\omega^{3/2}M_{1.4}^{1/2}R_{6}^{-3}P^{3/2}\,\text{G.}$ (8)

Here, $\xi$ can be approximated with:

$\xi\approx 2^{-1/7}\left(1-\omega\right)^{2/7},$ (9)

which is the same regardless of whether the magnetospheric radius is larger or
smaller than the spherization radius. Also, $R_{6}$ is the neutron star radius
in units of $10^{6}$ cm.

Given the high luminosity of NGC 5907 ULX1, we expect that the spherization
radius tends to be outside of $R_{M}$. However, for certain combinations of
magnetic field strengths and coupling assumptions, we can also find that this
may not be the case. For these potential solutions, the magnetic field is so
strong that it disrupts the accretion disk before it can become locally super-
Eddington. We have up to four $B$-field estimates in each case, however, not
all estimates fulfill the assumptions (i.e., for an overly strong magnetic
field, $R_{M}$ might be outside of $R_{\text{sph}}$, making the solutions for
$R_{M}\leq R_{\text{sph}}$ invalid in this case ). In Table 2, we list all
possible solutions for the three spin-up epochs, that is, for each set of
input values, $L$, $P$, and $\dot{P}$.

As can be seen in Table 2, we find roughly similar magnetic field strengths
for epochs 2 and 3, but a different value for epoch 1. The long baseline for
estimating $\dot{P}$ in this epoch and the lack of flux monitoring can explain
this difference: the average $\dot{P}$ and $L$ are likely to be inaccurate,
leading to unreliable estimates. Epochs 2 and 3 show a particularly good
agreement for the high $B$-field solution, with a variation of only $<10\%$
for both cases, while the difference for the low $B$-field solution is around
$\sim 50\%$. Nonetheless, without further information, we cannot determine
which solution describes the true nature of the system better.

If we assume that the neutron star has a $B$-field of $2.0\times 10^{13}$ G
(roughly the average of the high-B solutions based on epochs 2 and 3), we can
also estimate the average luminosity during epoch 1, which comes out to about
$7\times 10^{39}$ $\text{erg\,cm}^{-2}\,\text{s}^{-1}$. While this value may
be a bit on the low side, it could be realistic, given that NGC 5907 ULX1 did
seemingly exhibit low fluxes throughout much of 2013 (Walton et al. 2015). In
this case, $R_{M}>R_{\text{sph}}$, as the luminosity is very low compared to
the magnetic field.

Even with this lower estimated luminosity for epoch 1, the low $B$-field
solution does not agree with the other epochs. This discrepancy is mainly due
to the fact that the $B$-field estimate is independent of the luminosity as
long as $R_{M}<R_{\text{sph}}$, as the mass accretion rate at the inner
accretion disk edge is regulated by the Eddington limit.

###  4.2 The off-state

In the simplest picture, an accreting neutron star will enter the propeller
(or centrifugal inhibition) regime as soon as the corotation radius is inside
the magnetospheric radius ($R_{M}>R_{c}$). At the magnetospheric radius,
matter couples to the magnetic field and is forced to rotate at the angular
speed of the neutron star. In the case of $R_{M}>R_{c}$, this rotation is
faster than the Keplerian speed and therefore the matter experiences a net
outward force, all but halting direct accretion. The accretion luminosity will
then drop by orders of magnitude, however, residual accretion at the
magnetosphere might still be present, that is, the source does not have to
appear completely off (see, e.g., Corbet, 1996).

We can find the critical mass accretion rate for the transition to propeller
regime, $\dot{M}_{\text{in,crit}}{}$ by equating $R_{c}=R_{M}$ (using Eqs. 2
and 3), leading to:

$\displaystyle\dot{M}_{\text{in,crit}}$
$\displaystyle=\left(\frac{1}{4\pi^{2}}\right)^{-7/6}\,\frac{1}{\sqrt{2}}G^{-5/3}P^{-7/3}M^{-5/3}\xi^{7/2}\mu^{2},$
(10) $\displaystyle\approx 4.69\times 10^{13}\cdot
P^{-7/3}M^{-5/3}\xi^{7/2}\mu^{2}\,\text{g\,s}^{-1}.$ (11)

Based on this calculation, we give the critical luminosity for each epoch and
case in Table 2, which is $L_{\text{crit}}=\dot{M}_{\text{in,crit}}\epsilon
c^{2}$, where $\epsilon=0.15$, the accretion efficiency of a typical neutron
star on its surface. Given that the critical luminosity depends strongly on
the magnetic field, the estimates differ more strongly, for instance, by a
factor of 2 for the low $B$-field case (again, only comparing epochs 2 and 3).

Observationally, we can also estimate the luminosity of a possible onset of
the propeller effect from the Swift/XRT light curve, by finding the flux at
which the source drops below the detection limit of XRT. To estimate this
luminosity we transferred the Swift count-rate to flux and luminosity based on
the contemporaneous XMM-Newton data. We use a conversion factor of:

$C=\frac{\mathcal{F(\textsl{XMM-Newton})}}{\text{Rate({Swift})}}=6.516\times
10^{-11}\,\frac{\text{erg\,cm}^{-2}\,\text{s}^{-1}{}}{\text{cts\,s}^{-1}}\quad.$
(12)

As can be seen on the right-hand $y$-axis of Fig. 1, the source drops below
the detection limit of Swift around 1–2$\times 10^{39}\,\text{erg\,s}^{-1}$.
XMM-Newton detects the source around $1\times 10^{39}$
$\text{erg\,s}^{-1}$during the lowest flux states, but no pulsations were
found. This luminosity is therefore a good upper limit for the onset of the
propeller effect and we refer to it as the propeller luminosity. That value is
consistent with the fact that pulsations were only detected down to $14\times
10^{39}\,\text{erg\,s}^{-1}$ in July 2017 (ObsID 0804090401).

While XRT is not very sensitive, we know that accretion was heavily suppressed
in November 2017, due to a deep Chandra observation. This Chandra observation
revealed a diffuse nebula around the ULXP and provided a stringent upper limit
of $L<1.2\times 10^{38}\,\text{erg\,s}^{-1}$ for the point source (Belfiore et
al., 2020). We include this upper limit in Fig. 1.

The observational limit of $1\times 10^{39}\,\text{erg\,s}^{-1}$ for the
propeller luminosity is a little bit below the luminosity implied by the
highest estimated $B$-field from the spin-up (Tab. 2), namely,
$L_{\text{crit}}=2.2\times 10^{39}\,\text{erg\,s}^{-1}$ for a magnetic field
of $B=2.6\times 10^{13}$ G (orange dotted-dashed line in Fig. 1).

It is worth stressing that the initial decline in flux seen during the dense
XMM-Newton monitoring in July 2017 is not itself related to the propeller
transition in our interpretation. Rather, this must be related to some other
physical effect that resulted in a gradual reduction in the overall accretion
rate through the disk and the disk itself is also expected to make a
significant contribution in the XRT band (see, e.g., Walton et al., 2018b).
For example, density waves in the accretion disk might have reduced the
reduced the inner accretion rate or perhaps the mass transfer rate from the
stellar companion decreased slightly. We consider this initial stage of the
decline to be similar to the behavior seen in 2021 and 2022, where the source
also shows significant flux variability without fully entering an off-state.
However, in 2017 the flux continued to drop even further, to the point where
it reached our proposed propeller luminosity, resulting in a transition to the
propeller regime. This transition would naturally explain the very stringent
upper limit observed in November 2017 by Chandra. However, as this observation
occurred about 4 months later, we cannot make any firm statement with regards
to how rapidly the actual propeller transition would have occurred with these
data and, unfortunately, the XRT does not have the sensitivity to meaningfully
shed light on this issue either (the source is already not detected by the XRT
at our propeller luminosity). We note, however, given the rise by two orders
of magnitude (or more) in flux in only four days, as seen by Walton et al.
(2015) using XMM-Newton and NuSTAR, these observations (taken in 2013) may
provide the most stringent constraints on this issue.

We can also follow a more detailed description of the super-Eddington
accretion disks put forward by Chashkina et al. (2017, 2019). Their model in
particular takes advection of material in the accretion disk into account, as
well as illumination of the disk by the luminosity emitted close to the
compact object.

For a radiation-dominated disk, they find an updated description of the inner
disk radius, which does not depend on the mass accretion rate (Eq. 61 in
Chashkina et al., 2017):

$R_{M}=\left(\frac{73}{24}\alpha\lambda\mu_{30}^{2}\right)^{2/9}\cdot\frac{GM}{c^{2}}\,\text{cm}\quad{,}$
(13)

where $\alpha$ is the viscosity of the disk, based on the standard Shakura &
Sunyaev (1973) description and $\lambda\approx 4\times 10^{10}M_{1.4}^{-5}$.
We assume $\alpha=0.1$. Here $\mu_{30}$ is the magnetic moment of the neutron
star in units of $10^{30}$ G cm-3 and $c$ is the speed of light.

Assuming that we reach the equilibrium spin-period just before entering the
off-state, that is, a period of $P=0.946$ s as measured in July 2017, we can
equate $R_{M}=R_{c}$ and then estimate a magnetic field of around $6\times
10^{13}$ G. A similar estimate is found when following the description of King
et al. (2017), who use the standard formula for the magnetospheric radius (Eq.
3).

Based on our assumption that NGC 5907 ULX1 is spinning close to equilibrium
and assuming that it entered the propeller state at the lowest measured
luminosity where pulsations were still detected (in July 2017), we can rewrite
the limit on the mass-accretion rate given Eq. 33 of Chashkina et al. (2019)
to estimate the B-field:

$\mu^{2}_{30}=\frac{1}{1.8}\,\dot{m}\,\xi^{-7/2}M_{1.4}^{5/3}P^{7/3}\,\text{G}^{2}\,\text{cm}^{-6}\quad{.}$
(14)

Again, we use the measurements of July 2017 (ObsID 0804090401), that is,
$P=0.946$ s and based on Eq. 5, $\dot{m}=31.9$ and we approximate $\xi=0.75$
(based on Eq. 9), we find a B-field of around $B\approx 1.7\times 10^{13}$ G.
This estimate is slightly lower due to the effects of irradiation and
advection in this model.

If we assume that NGC 5907 ULX1 entered a propeller state when it became
undetectable by Swift, we would also expect that the source spins-down during
this period. This behavior is indeed what we observe, as the spin period in
2019 is significantly slower than in 2017, just before the off-state.

However, the source rebrightened briefly in 2018, which likely means that it
started accreting and spinning-up again. We do not have a pulsation
measurement for this period, thus estimating the spin-down strength is not
straight forward. In the bottom panel of Fig. 1, we indicate a possible spin-
history of the source. We define regions (shaded pink) in which the source is
off and spinning-down, thereby splitting up the data into seven time slices
that alternate between spin-up and spin-down.

In this model, we assume a spin-up strength of $\dot{P}=-2.03\times 10^{-9}$ s
s-1 between 2014–2017 and a spin-up strength of $\dot{P}=-1.50\times 10^{-9}$
s s-1 during all other spin-up episodes. For the spin-down, we assume a value
of $\dot{P}=2.25\times 10^{-9}$ s s-1, which is based on a spin-down estimate
during the off-state in 2020.

This model is of course highly simplified and averages over long periods of
time. It is possible that during the periods where the source was detected in
XRT at $\sim 10^{39}$ $\text{erg\,s}^{-1}$, active accretion occurred and the
source was spinning-up slightly (or at least, was not spinning down further)
if $R_{M}\approx R_{C}$. However, the current data do not provide the required
sensitivity and time resolution to model the spin history on shorter time-
scales.

Theoretical estimates of the spin-down strength during the propeller state are
difficult and depend on various assumptions of the interaction between the
magnetic field and the residual surrounding matter (e.g., Davies et al., 1979;
Urpin et al., 1998; Ikhsanov, 2001; D’Angelo & Spruit, 2010). Here we follow
calculations presented by Parfrey et al. (2016), which were originally
motivated by millisecond pulsars. In particular, these calculations assume
that the spin-down torque is completely dominated by the torque exerted by the
open field-lines and that there is no interaction between the field and the
disk inside the co-rotation radius. Based on equation 18 of Parfrey et al.
(2016) we can write an estimate for the magnetic field:

$B_{\text{down}}=\frac{2}{R^{3}}\frac{R_{M}}{\zeta}\sqrt{\frac{I\dot{P}c}{P}}\,\text{G},$
(15)

where $I$ is the moment of inertia of the neutron star
($I=\frac{2}{5}MR^{2}$). The parameter $\zeta$ describes how efficiently the
magnetic field lines are opened by the star-disk interaction. Here, we assume
maximum efficiency, namely, $\zeta=1$, which implies that all field lines
intersecting the disk are opened.

We measured an average spin-down $\dot{P}=1.19\times 10^{-9}$ s s-1 between
mid 2017 to mid 2019, that is, over the two first off-states. The spin-down is
likely a bit faster, given the short re-brightening of NGC 5907 ULX1 in 2018
(see Fig. 1). Nevertheless, with this value for $\dot{P}$, we find
$B_{\text{down}}=6.83\times 10^{13}$ G. This magnetic field strength is
slightly higher than the largest value based on the spin-up calculations, but
only by a factor of about 2–3. Given the significant number of simplifications
and approximations going into this estimate, it is probably not surprising
that we don’t find a perfect agreement. An overview of all our $B$-field
estimates is given in Table 3.

It is interesting to note that the source already showed a significant spin-
down between two XMM-Newton observations in 2017 (OBSIDs 0804090301 and
0804090401). These were taken while the source was on its decline into the
off-state – and were separated by only $\sim$3 d. The evolution between these
two observations implies $\dot{P}\approx 1.4\times 10^{-9}$ s s-1, which is
similar to what we measured between 2017–2019. We note, however that both
observations are still significantly above the propeller luminosity that would
correspond to a $10^{13}$ G magnetic field, so there would appear to be an
inconsistency here. Assuming an orbital period of 4.4 d, the lower limit of
the period proposed by Israel et al. (2017a), we find that the observed spin-
down is barely consistent with being due to the orbital motion within the
uncertainties. However, the orbital period is not confirmed, so we cannot draw
firm conclusions on the nature of this spin-down.

Another possibility for the observed spin-down is that the source is entering
the subsonic propeller (or magnetic inhibition) regime. In this regime,
$R_{M}<R_{C}$, but the matter is still too hot to enter the magnetosphere and
accrete. This regime was discussed in detail by Davies et al. (1979); Davies &
Pringle (1981) and Ikhsanov (2007). Following Davies & Pringle (1981) and
Henrichs (1983), we can write a simple expression for the expected magnetic
field given an observed spin-down rate:

$B_{\text{SSP}}=\sqrt{\dot{P}\frac{GMI}{4\pi}}\,\text{G}.$ (16)

With the same spin-down rate assumptions as above, this would imply
$B_{\text{SSP}}=8.85\times 10^{12}$ G. While still high, this estimate is
significantly lower than the ones obtain before, as expected from the
different assumptions regarding the state of the source. It is difficult to
estimate the luminosity at which the subsonic propeller would start, given
that it strongly depends on the wind density, temperature, and turbulence just
outside the magnetosphere – which are all unknown.

Table 2: Implied magnetic field strength for three different epochs following
the method given by Gao & Li (2021). The values for $L$, $\dot{m}$, and $P$
are based on the observation at the given date (MJD). We note that for epoch
1, we do not know if the given luminosity is representative of the average
luminosity during this epoch, due to the lack of X-ray flux monitoring during
that time.

|  | MJD |  $L$aa$a$in $10^{39}$ $\text{erg\,cm}^{-2}\,\text{s}^{-1}$ in the 0.3–10 keV energy band |  $\dot{m}$bb$b$in $\dot{M}_{\text{Edd}}$ |  $P$cc$c$in seconds |  $\dot{P}$dd$d$in $10^{-9}$ s s-1 | low B | high B
---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  | $R_{M}/R_{\text{sph}}$ | Bee$e$in $10^{12}$ G |  $L_{\text{crit}}$ff$f$in $10^{37}$ $\text{erg\,s}^{-1}$ | $R_{M}/R_{\text{sph}}$ | Bee$e$in $10^{12}$ G |  $L_{\text{crit}}$ff$f$in $10^{37}$ $\text{erg\,s}^{-1}$
epoch 1 | case 1 | 52690.67 | 6.73 | 64.68 | 1.43 | -0.81 | 0.16 | 0.31 | 0.08 | 0.92 | 26.63 | 188.82
case 2 | 0.16 | 0.31 | 0.08 | 1.05 | 45.96 | 249.53
epoch 2 | case 1 | 56847.95 | 9.34 | 75.14 | 1.14 | -2.04 | 0.40 | 3.68 | 14.08 | 0.61 | 12.43 | 95.96
case 2 | 0.38 | 3.36 | 11.96 | 0.73 | 25.89 | 223.80
epoch 3 | case 1 | 59053.74 | 5.95 | 61.14 | 1.03 | -1.32 | 0.40 | 2.27 | 7.26 | 0.73 | 12.08 | 105.02
case 2 | 0.39 | 2.16 | 6.62 | 0.86 | 23.97 | 223.06

222

###  4.3 The super-orbital period

Walton et al. (2016) discovered flux variability with a period of $78.1\pm
0.5$ d in the Swift/XRT data of NGC 5907 ULX1, using data from roughly weekly
monitoring observations between 2014–2016. The period showed a peak-to-peak
amplitude of roughly a factor of 3 and was interpreted as a super-orbital
period. As can be seen in Fig. 1, the flux just before the off-state in 2017
follows the exact same period, with a very similar amplitude and phase. This
is also true for the recovery after the off-state in 2020: the XRT flux
measurements align very well within the uncertainties of the period with the
extrapolated profile of the 78 d period. Since 2020, NGC 5907 ULX1 shows a
somewhat larger variability in its flux, with variations of at least a factor
of 10. This increase in variability makes identifying the 78 d period more
difficult, nonetheless, the bright states in the data still largely line up
well with the peaks of the predicted profile. The Swift/XRT monitoring is
still ongoing to investigate the stability of this period. The current data do
not allow for a independent measurement of the period, so we currently cannot
say if there is a change in the period after the off-state or not.

##  5 Conclusions

Table 3: Summary of magnetic field estimations with the different methods presented in this work. Method | Description | Value [$10^{13}$ G]
---|---|---
Spin-Up |  |
Gao & Li | High B |  $\sim$1.2 or $\sim$2.4
Gao & Li | Low B | 0.2–0.3
Propeller transition |  |
Ghosh & Lamb |  $L$ of propeller at $2.2\times 10^{39}\,\text{erg\,s}^{-1}$ | 2.6
Chaskina | advection in disk | 6
Spin equilibrium |  |
Chaskina | based on July 2017 data | 1.7
Spin-Down |  |
Parfrey | using average spin-down between 2017-2019 | 6.8
Davies & Pringle | sub-sonic propeller | 0.89

We have studied the pulse period evolution of NGC 5907 ULX1 between 2003–2022.
In 2017, the source entered an extended off-state, during which it dropped
below the detection limit of Swift/XRT. During this off-state the secular
spin-up trend reversed, and the neutron star rotational period slowed down
significantly. After the source left the off-state in mid-2020, spin-up has
resumed albeit at a lower rate than before.

We have used different methods to estimate the magnetic field of the neutron
star, either based on the spin-up or spin-down strength. The main results are
summarized in Table 3. In particular, we used the torque transfer model
presented by Gao & Li (2021) during the spin-up. We find that the calculated
field strengths agree well for the two most recent spin-up episodes in
2014–2017 and 2020–2022, in particular, for the high $B$-field solution. The
first epoch between 2003–2014 gives very different estimates, but due to the
lack of continuous flux monitoring, the estimated X-ray flux is highly
uncertain. The highest estimate for the magnetic field strength in our data is
$\approx 2.5\times 10^{13}$ G, while for the low $B$-field solution, we find
values as low as $2\times 10^{12}$ G. While based on these numbers we cannot
distinguish which magnetic field is present in reality, we note that for a
$\sim$$10^{12}$ G field, we would expect to see a cyclotron resonant
scattering feature (CRSF) around 12 keV, which so far has not been observed in
the spectrum (Staubert et al., 2019; Fürst et al., 2017; Israel et al.,
2017a).

We also estimate the magnetic field based on the spin-down during the off-
state between 2017–2019. If we assume that the source entered the propeller
regime when it dropped below the detection limit of Swift/XRT at about
$L=$1–2$\times 10^{39}$ $\text{erg\,s}^{-1}$, we estimate a magnetic field of
$B\leq 2.5\times 10^{13}$ G. Using the update disk description of Chashkina et
al. (2019), we find that this luminosity for the propeller transition implies
a magnetic field $B\approx 1.7\times 10^{13}$ G. The spin-down torque itself
is difficult to estimate as it depends on the unknown interaction between the
magnetic field and the residual matter surrounding it, be it the stellar wind
or a residual accretion disk. We performed our calculations based on a
description by Parfrey et al. (2016) and again find that the strong spin-down
can only be explained with a very high magnetic field ($\approx 6.8\times
10^{13}$ G). If we assume that the source was spun down while in the subsonic
propeller regime, we find a magnetic field of around $8.9\times 10^{12}$ G.

While we cannot rule out a low magnetic field directly, circumstantial
evidence points clearly toward the direction of a magnetic field of a few
$10^{13}$ G in NGC 5907 ULX1. This value is in line with previous estimates of
the magnetic field (e.g., Israel et al., 2017a; Gao & Li, 2021) and implies
that the source is accreting at very high rates.

## ACKNOWLEDGEMENTS

The authors would like to thank the referee for their useful feedback, which
helped to improve the final version of the manuscript. DJW acknowledges
support from an STFC Ernest Rutherford Fellowship. TPR acknowledges support
from the Science and Technology Facilities Council (STFC) as part of the
consolidated grant award ST/000244/1 This research has also made use of data
obtained with XMM-Newton, an ESA science mission with instruments and
contributions directly funded by ESA Member States, and has also made use of
public data from the Swift data archive. We thank Kyle Parfrey for the useful
discussion. This work made use of data supplied by the UK Swift Science Data
Centre at the University of Leicester.

## References

  * Abramowicz et al. (1988) Abramowicz M. A., Czerny B., Lasota J. P., Szuszkiewicz E., 1988, ApJ, 332, 646
  * Bachetti (2015) Bachetti M., 2015, MaLTPyNT: Quick look timing analysis for NuSTAR data
  * Bachetti et al. (2014) Bachetti M., Harrison F. A., Walton D. J., et al., 2014, Nat, 514, 202
  * Belfiore et al. (2020) Belfiore A., Esposito P., Pintore F., et al., 2020, Nature Astronomy, 4, 147
  * Brightman et al. (2016) Brightman M., Harrison F., Walton D. J., et al., 2016, ApJ, 816, 60
  * Brightman et al. (2019) Brightman M., Harrison F. A., Bachetti M., et al., 2019, ApJ, 873, 2, 115
  * Brightman et al. (2018) Brightman M., Harrison F. A., Fürst F., et al., 2018, Nature Astronomy, 2, 312
  * Burrows et al. (2005) Burrows D. N., Hill J. E., Nousek J. A., et al., 2005, Space Science Reviews, 120, 165
  * Carpano et al. (2018) Carpano S., Haberl F., Maitra C., Vasilopoulos G., 2018, MNRAS, 476, L45
  * Chashkina et al. (2017) Chashkina A., Abolmasov P., Poutanen J., 2017, MNRAS, 470, 2799
  * Chashkina et al. (2019) Chashkina A., Lipunova G., Abolmasov P., Poutanen J., 2019, A&A, 626, A18
  * Corbet (1996) Corbet R. H. D., 1996, ApJ, 457, L31
  * D’Angelo & Spruit (2010) D’Angelo C. R., Spruit H. C., 2010, MNRAS, 406, 2, 1208
  * Davies et al. (1979) Davies R. E., Fabian A. C., Pringle J. E., 1979, MNRAS, 186, 779
  * Davies & Pringle (1981) Davies R. E., Pringle J. E., 1981, MNRAS, 196, 209
  * Earnshaw et al. (2016) Earnshaw H. M., Roberts T. P., Heil L. M., et al., 2016, MNRAS, 456, 3840
  * Earnshaw et al. (2018) Earnshaw H. P., Roberts T. P., Sathyaprakash R., 2018, MNRAS, 476, 4272
  * Ekşi et al. (2015) Ekşi K. Y., Andaç d. C., Çıkıntoğlu S., Gençali A. A., Güngör C., Öztekin F., 2015, MNRAS, 448, L40
  * Erkut et al. (2019) Erkut M. H., Ekşi K. Y., Alpar M. A., 2019, ApJ, 873, 2, 105
  * Evans et al. (2009) Evans P. A., Beardmore A. P., Page K. L., et al., 2009, MNRAS, 397, 1177
  * Fürst et al. (2016) Fürst F., Walton D. J., Harrison F. A., et al., 2016, ApJ, 831, L14
  * Fürst et al. (2021) Fürst F., Walton D. J., Heida M., et al., 2021, A&A, 651, A75
  * Fürst et al. (2017) Fürst F., Walton D. J., Stern D., et al., 2017, ApJ, 834, 77
  * Gao & Li (2021) Gao S.-J., Li X.-D., 2021, Research in Astronomy and Astrophysics, 21, 8, 196
  * Gehrels (1986) Gehrels N., 1986, ApJ, 303, 336
  * Ghosh & Lamb (1979a) Ghosh P., Lamb F. K., 1979a, ApJ, 232, 259
  * Ghosh & Lamb (1979b) Ghosh P., Lamb F. K., 1979b, ApJ, 234, 296
  * Henrichs (1983) Henrichs H. F., 1983, in Accretion-Driven Stellar X-ray Sources, edited by W. H. G. Lewin, E. P. J. van den Heuvel, 393–429, Cambridge University Press, Cambridge, UK
  * Herold (1979) Herold H., 1979, Phys. Rev. D, 19, 2868
  * Hou et al. (2022) Hou X., Ge M. Y., Ji L., et al., 2022, ApJ, 938, 2, 149
  * Houck & Denicola (2000) Houck J. C., Denicola L. A., 2000, in ASP Conf. Ser. 216: Astronomical Data Analysis Software and Systems IX, vol. 9, 591
  * Ikhsanov (2001) Ikhsanov N. R., 2001, A&A, 375, 944
  * Ikhsanov (2007) Ikhsanov N. R., 2007, MNRAS, 375, 2, 698
  * Illarionov & Sunyaev (1975) Illarionov A. F., Sunyaev R. A., 1975, A&A, 39, 185
  * Israel et al. (2017a) Israel G. L., Belfiore A., Stella L., et al., 2017a, Science, 355, 817
  * Israel et al. (2017b) Israel G. L., Papitto A., Esposito P., et al., 2017b, MNRAS, 466, L48
  * Kaaret et al. (2017) Kaaret P., Feng H., Roberts T. P., 2017, ARA&A, 55, 303
  * King et al. (2017) King A., Lasota J.-P., Kluźniak W., 2017, MNRAS, 468, L59
  * King (2009) King A. R., 2009, MNRAS, 393, L41
  * Kluźniak & Lasota (2015) Kluźniak W., Lasota J.-P., 2015, MNRAS, 448, L43
  * Koliopanos et al. (2017) Koliopanos F., Vasilopoulos G., Godet O., Bachetti M., Webb N. A., Barret D., 2017, A&A, 608, A47
  * Kong et al. (2022) Kong L.-D., Zhang S., Zhang S.-N., et al., 2022, ApJ, 933, 1, L3
  * Kraft et al. (1991) Kraft R. P., Burrows D. N., Nousek J. A., 1991, ApJ, 374, 344
  * Leahy et al. (1983) Leahy D. A., Elsner R. F., Weisskopf M. C., 1983, ApJ, 272, 256
  * Liu et al. (2022) Liu J., Vasilopoulos G., Ge M., et al., 2022, MNRAS, 517, 3, 3354
  * Middleton et al. (2019a) Middleton M. J., Brightman M., Pintore F., et al., 2019a, MNRAS, 486, 1, 2
  * Middleton et al. (2019b) Middleton M. J., Brightman M., Pintore F., et al., 2019b, MNRAS, 486, 1, 2
  * Middleton & King (2017) Middleton M. J., King A., 2017, MNRAS, 470, L69
  * Motch et al. (2014) Motch C., Pakull M. W., Soria R., Grisé F., Pietrzyński G., 2014, Nat, 514, 198
  * Mushtukov et al. (2021) Mushtukov A. A., Portegies Zwart S., Tsygankov S. S., Nagirner D. I., Poutanen J., 2021, MNRAS, 501, 2, 2424
  * Mushtukov et al. (2015) Mushtukov A. A., Suleimanov V. F., Tsygankov S. S., Poutanen J., 2015, MNRAS, 454, 2539
  * Parfrey et al. (2016) Parfrey K., Spitkovsky A., Beloborodov A. M., 2016, ApJ, 822, 1, 33
  * Piconcelli et al. (2004) Piconcelli E., Jimenez-Bailón E., Guainazzi M., Schartel N., Rodríguez-Pascual P. M., Santos-Lleó M., 2004, MNRAS, 351, 161
  * Pintore et al. (2017) Pintore F., Zampieri L., Stella L., Wolter A., Mereghetti S., Israel G. L., 2017, ApJ, 836, 113
  * Quintin et al. (2021) Quintin E., Webb N. A., Gúrpide A., Bachetti M., Fürst F., 2021, MNRAS, 503, 4, 5485
  * Rodríguez Castillo et al. (2020) Rodríguez Castillo G. A., Israel G. L., Belfiore A., et al., 2020, ApJ, 895, 1, 60
  * Sathyaprakash et al. (2019) Sathyaprakash R., Roberts T. P., Walton D. J., et al., 2019, MNRAS, 488, 1, L35
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
  * Staubert et al. (2019) Staubert R., Trümper J., Kendziorra E., et al., 2019, A&A, 622, A61
  * Strüder et al. (2001) Strüder L., Briel U., Dennerl K., et al., 2001, A&A, 365, L18
  * Townsend et al. (2017) Townsend L. J., Kennea J. A., Coe M. J., et al., 2017, MNRAS, 471, 4, 3878
  * Tsygankov et al. (2018) Tsygankov S. S., Doroshenko V., Mushtukov A. A., Lutovinov A. A., Poutanen J., 2018, MNRAS, 479, 1, L134
  * Tsygankov et al. (2016) Tsygankov S. S., Mushtukov A. A., Suleimanov V. F., Poutanen J., 2016, MNRAS, 457, 1101
  * Tully et al. (2016) Tully R. B., Courtois H. M., Sorce J. G., 2016, AJ, 152, 50
  * Turner et al. (2001) Turner M. J. L., Abbey A., Arnaud M., et al., 2001, A&A, 365, L27
  * Urpin et al. (1998) Urpin V., Konenkov D., Geppert U., 1998, MNRAS, 299, 1, 73
  * Vasilopoulos et al. (2019) Vasilopoulos G., Petropoulou M., Koliopanos F., et al., 2019, MNRAS, 488, 4, 5225
  * Vasilopoulos et al. (2020) Vasilopoulos G., Ray P. S., Gendreau K. C., et al., 2020, MNRAS, 494, 4, 5350
  * Walton et al. (2018a) Walton D. J., Bachetti M., Fürst F., et al., 2018a, ApJ, 857, L3
  * Walton et al. (2018b) Walton D. J., Brightman M., Risaliti G., et al., 2018b, MNRAS, 473, 4377
  * Walton et al. (2016) Walton D. J., Fürst F., Bachetti M., et al., 2016, ApJ, 827, L13
  * Walton et al. (2018c) Walton D. J., Fürst F., Heida M., et al., 2018c, ApJ, 856, 128
  * Walton et al. (2015) Walton D. J., Harrison F. A., Bachetti M., et al., 2015, ApJ, 799, 122
  * Wang (1995) Wang Y. M., 1995, ApJ, 449, L153
  * Wilson-Hodge et al. (2018) Wilson-Hodge C. A., Malacaria C., Jenke P. A., et al., 2018, ApJ, 863, 1, 9


# Handwritten and Machine printed OCR for Geez Numbers Using Artificial Neural
Network

Eyob Gebretinsae Beyene
Addis Ababa University, AAiT, ITSC
Addis Ababa, Ethiopia

<EMAIL_ADDRESS>

###### Abstract

Researches have been done on Ethiopic scripts. However studies excluded the
Geez numbers from the studies because of different reasons. This paper
presents offline handwritten and machine printed Geez number recognition using
feed-forward back propagation artificial neural network. On this study,
different Geez image characters were collected from google image search and
three persons are instructed to write the numbers using pencil. In total we
have collected 560 numbers of characters. We have used 460 of the characters
for training and 100 are used for testing. Accordingly we have achieved
overall all classification $\sim$$89.88\%$

##  1 Introduction

In Ethiopia different Semitic languages are spoken. Semitic languages like
Tiginrya, Guraginya and the official working language Amharic are widely used
in Ethiopia. Researchers refer to such languages as Ethiopic languages [2]
.Language studies [1] upkeep that, the root of all the Ethiopic languages is
Ge’ez and currently it survives in church liturgy and literature. Starting
form the Aksumite kingdom Geez was the national language of Ethiopia for over
centuries [1].

On one hand, hundred years back Ethiopian history had been written in Geez for
centuries. Though, the language is now working on Ethiopian Orthodox Church
only, but enormous amount of Geez manuscripts are existing across the country.
Such manuscripts was written using ink and quill. In order to keep the
Ethiopian history, it is essential to digitize such manuscripts in to text
format. On the other hand, due to Amharic is an official working language of
Ethiopia, researches on Ethiopic languages focuses [2] on recognition of
Amharic text. Though, Amharic use almost all the alphabets of Geez but
normally it excludes Geez numbers and use Arabic numerals. In our view it is
essential to digitize and to research Geez numbers, since we can find the
symbols on different types of historical manuscripts. In addition Geez numbers
has been commonly used in Ethiopian calendars, Tigiringa and Amharic Bibles
and local publications.

Geez numbers represents by 1-9 symbols but there is no zero concept and as
such no symbol for zero. However, there are independent symbols for 10, 20,
30, 40, 50, 60, 70, 80, 90, and 100. Figure 1 shows the list of Geez numbers
with the corresponding Arabic numeral or English number equivalent.The Geez
character numbers have symbol similarities with the Geez letters. The main
difference in most of the symbols are the two dashes/horizontal lines above
and below the character of Geez. Figure 2 shows sample selected characters how
the Geez numbers have similarity with the Geez letters.

Figure 1: Geez number representations Figure 2: Similarity Geez letters and
Alphabets

From Figure 2 we comprehend that, researches done on the recognition of
Geez/Amharic letter characters will affect the recognition performance of the
letters when Geez number are incorporated. This study focused on the
recognition of scanned manual and machine printed Geez numbers using
artificial neural network.

Humans are gifted and can easily sense and recognize written textual
information, however it is still a challenge to detect and recognize textual
information by artificial machines [2].Text detection and recognition from
scanned images, photos, and different posters is an interesting and
challenging research area in computer vision and machine vision [3]. According
to [4][5], sufficient of researches have been done on optical character
recognition systems on Latin characters. According [1], OCR has been an active
research area for more than three decades, however the lack of precision in
historical documents recognition is still an issue. Moreover this problem is
much greater in script languages of economic developing countries. However, it
is fact that researches done on recognition of computer printed documents have
greater accuracy in detection and recognition [3][4].

However, there are few researches done on OCR for Ethiopic scripts [4][5][6].
Due to insufficient researches on Ethiopic scripts, there is no system or tool
for Ethiopic Script recognition. Hence, this research will have its own
contribution on building knowledge and method on Ethiopian script recognition.
The remaining topic of this paper is organized as follows. We present overview
of Neural network and its architecture on topic two, design and implementation
of feed-forward back propagation neural network on topic three, result and
discussion on part four and conclusion of the study on part five.

##  2 OCR and Neural Network

Artificial neural network use number of processing nodes called neurons linked
in connected layers [7]. Each node perform some computation and each link have
assigned weights and used train the network by adjusting such weights [8].
Researches have been done on both handwritten and scanned machine printed of
different language texts. Articles on optical character recognition use
different techniques and methods to convert characters to texts such us Neuro-
Fuzzy Model and Artificial Neural Network [3][7][8] .Studies on Ethiopic
character recognition recommend researchers to apply artificial neural network
as a future study direction in order to increase recognition performance [6].
From the different neural network architectures feed-forward network
architecture is commonly used for character recognition [7] [9]. Such network
are used a number of neurons as input layer, hidden layer and output layer. On
this study we apply multi layer back-propagation feed-forward neural network
architecture

##  3 Design and Implementation

###  3.1 Data Collection

As this study focus on both handwritten and machine printed Geez number
recognition, scanned images of Geez number characters are collected from
google image repository and from bible. Since limited images are available on
google image repository, typed Geez number characters are incorporated to the
data set using regular, italic and bold styles. In addition handwritten Geez
character numbers are collected by scanning written numbers which was
instructed to three individuals’ to write the numbers using pencil. Sample
collected images are shown in Figure 3. Accordingly 560 amount of data was
collected from different sources. Since we select the 20 distinct symbols of
Geez numbers, we have prepared 28 sample data set for each character
characters. Independent scanned Geez number of characters are stored on a
file. We use $82.2\%$ of the data for training and $17.8\%$ for testing. Each
file is acquired by the system for further processing. Figure 4 and Figure 5
presents each process before the image is ready for the neural network.

###  3.2 Image PreProcessing

The collected Geez number characters are loaded as independent images. Each
image is then ready for the next step which is image pre processing. Character
images are converted to gray scale image. The gray images then converted to
binary image. Before applying segmentation we inverse the image in order to
have one (1) value for the character.

###  3.3 Character Segmentation and Resize

Since the characters are placed on different area of the image, character
segmentation is required. Accordingly we apply a method to crop the character
based on the first and last row and column one (1) value of the image.
Moreover, for having the same number of pixel on each character we apply a
function to resize the image to 45x40.

Figure 3: Sample Collected images

Figure 4: Segmentation Process

Figure 5: Segmented Characters

Figure 6: Layers of the neural network

##  4 Result and Discussion

###  4.1 Implementation

The feed-forward back-propagation network is implemented using Matlab. The
input layer have inputs of the image matrix 45x40 which is 1800. Each image is
converted to a single column matrix array and feed to the network. The network
have 3 hidden layers with 20, 15 and 10 neurons respectively as shown on
Figure 6. The output layer have 20 neurons that represent the 20 Geez number
symbols. Output numbers are represented using 5 sequential binary numbers. On
this study we apply training function that updates weight and bias values
according to conjugate gradient back propagation with Polak-Ribiére updates.
The overall classification and recognition performance of the system is
achieved $89.88$%. This classification result is restricted by the amount of
the collected training and test data sets. However better recognition
performance can be achieved when the data set is more than the collected data
set on this study. Figure 7 show the overall classification performance of the
neural network.

###  4.2 Training and Testing

The network is trained using 460 training data set, which represent 23
independent characters for single Geez number symbol. Accordingly we achieved
$98.03$% training classification performance. This indicates that, the feed-
forward back propagation training classification result is outstanding. Figure
8 show the training classification result of the network.

The network is tested using 100 training data set, which represent 5
independent characters for single Geez number symbol. Based on the test data
we have, the network was tested and achieved a good result which is $65.3$ %
accuracy. Figure 9 show the test result of the network.

##  5 Conclusion

This paper implement a feed-forward back propagation network with four hidden
layer. The paper presented how Geez number character images prepared for the
training and test data sets, pre processing on the collected images, training
and testing of the neural network. Accordingly we achieved $89.88$ % of
overall classification and recognition performance.

Figure 7: Recognition result

Figure 8: Training Result

Figure 9: Test result

## References

[1] W. K. Chen, Linear Networks and Systems. Belmont, CA: Wadsworth Press,
2003.

[2] Yaregal Assabie, J. B. , “Ethiopic Character Recognition Using Direction
Field Tensor”, The 18th International Conference on Pattern Recognition
(ICPR’06), IEEE.,2006.

[3] Sachin Kumar, H. S. B., Annapurna Singh, "A Review on Character
Recognition and text Detection: Neural-Fuzzy Approach." Journal of Basic and
Applied Engineering Research, 2014, Vol 1, pp. 61-64 5.

[4] JOHN COWELL, F. H., “Amharic Character Recognition using a Fast Signature
Based Algorithm”, Proceedings of the Seventh International Conference on
Information Visualization, IEEE,2003.

[5] Million Meshesha, C. V. J., "Optical Character Recognition of Amharic
Documents ", African Journal of Information and Communication Technology, vol.
3, pp. 53-66, 2007.

[6] Yaregal Assabie, J. B., "Lexicon-based Offline Recognition of Amharic
Words in Unconstrained Handwritten Text." IEEE, 2008.

[7] Chirag I Patel, R. P., Palak Patel., "Handwritten Character Recognition
using Neural Network." International Journal of Scientific & Engineering
Research vol 2, pp.1- 6, 2011

[8] Gunjan Singh, A. P., Sushma Lehri, "Neuro-Fuzzy Model Based Classification
of Handwritten Hindi Modifiers " International Journal of Application or
Innovation in Engineering & Management (IJAIEM) vol.3, pp.311-325,2014.

[9] J.Pradeep, E. S., S.Himavathi. "Diagonal Based Feature Extraction For
Handwritten Character Recognition System Using Neural Network." IEEE, pp
.364-368, 2011