# Independent Sign Language Recognition
with 3D Body, Hands, and Face Reconstruction

###### Abstract

Independent Sign Language Recognition is a complex visual recognition problem
that combines several challenging tasks of Computer Vision due to the
necessity to exploit and fuse information from hand gestures, body features
and facial expressions. While many state-of-the-art works have managed to
deeply elaborate on these features independently, to the best of our
knowledge, no work has adequately combined all three information channels to
efficiently recognize Sign Language. In this work, we employ SMPL-X, a
contemporary parametric model that enables joint extraction of 3D body shape,
face and hands information from a single image. We use this holistic 3D
reconstruction for SLR, demonstrating that it leads to higher accuracy than
recognition from raw RGB images and their optical flow fed into the state-of-
the-art I3D-type network for 3D action recognition and from 2D Openpose
skeletons fed into a Recurrent Neural Network. Finally, a set of experiments
on the body, face and hand features showed that neglecting any of these,
significantly reduces the classification accuracy, proving the importance of
jointly modeling body shape, facial expression and hand pose for Sign Language
Recognition.

Index Terms—  Independent Sign Language, 3D Body Reconstruction, Hand Shape,
Facial Characteristics

##  1 Introduction

With the term ”Sign Language” we refer to a language that employs signs made
with the hands and other movements, including facial expressions and postures
of the body, used primarily by people who are deaf. Humans are able, due to
their natural ability, to identify continuous and independent sign language
after they have been trained to identify and understand it. Unfortunately,
that is not the case with computers since Sign Language Recognition (SLR) is
considered a very hard task due to the need of combining information from
three different channels; face, body and hands.

Each independent task has already been successful in the past. But combining
all these features in the requisite detail for SLR is far from being
perfected. Using raw frames as input has been rigorously tested since it
includes all necessary information. However, it is extremely hard for the
network to focus on the proper features, especially when the environment
alters significantly, while it is also extremely time-consuming and resource-
expensive.

The importance of SLR is indisputable and managing to train a tool that can
efficiently and sufficiently identify Sign Language can be extremely
beneficial for the Sign Language community. To eliminate the constrains that
raw frames include, we investigate 3D body reconstruction, aiming to encompass
information coming from gestures, facial expressions and body structure. We
experiment with SMPL-X, a sophisticated parametric body model that can
reconstruct with notable precision the human body from a single RGB image. In
this paper, we compare i) raw images ii) 2D skeleton reconstruction with
Openpose and iii) 3D body reconstruction with SMPL-X, to evaluate the
efficiency of our proposed method. A secondary contribution of this work is
the study of the connection between the three channels of information (i.e.,
hand pose, facial expressions, body pose) and their importance in SLR. Many
works have already highlighted the significance of adding facial expressions
and body structure features to the gesture features, since many similar words
in Sign Language have exactly the same gesture sequence differing only in the
expression of the face or in the positioning of the hands with regard to the
body. We conduct experiments to show that the optimal results are achieved
only after all three channels of information are being included.

The rest of this paper is organized as follows: Section 2 provides a review of
related work in the recent literature. The proposed SMPL-X method and its
parameters that are being used for SLR, are analyzed in Section 3. Section 4
deals with the experimental setup, describing the sign language dataset used,
the feature extraction, their pre-processing and the model architectures we
exploit. In Section 5 we present the results of our experiments and evaluate
the three techniques and finally, in Section 6 we conclude our remarks and
propose plausible future directions.

Fig. 1: i) First image: Raw RGB frame, ii) Second Image: Optical flow of a
frame, iii) Third Image: Openpose 2D Skeleton, iv) Fourth image: 3D Body
Reconstruction produced by SMPL-X.

##  2 Related Work

Sign languages are natural languages communicable purely by vision via
sequences of time-varying 3D shapes. They serve for communication in the Deaf
communities, as well as among deaf and hearing people if the latter learn to
sign. They convey information and meaning via spatio-temporal visual patterns,
which are formed by manual (handshapes) and non-manual cues (facial
expressions and upper body motion). Computer-based processing and recognition
of sign videos is also broadly related to vision-based human-computer and
human-robot interaction using gesture recognition.

While significant progress exists in the field of automatic sign language
recognition from the computer vision and pattern recognition fields, (e.g.,
see [VoMe01, OnRa05, Agr+08, TPM14, KZNB18] and the references therein), it
still remains a quite challenging task especially for continuous sign
language. In addition to signs having a complex multi-cue 4D space-time
structure, the difficulty in their automatic recognition is also due to the
large variability with respect to inter-signer or intra-signer variations of
signing while expressing the same concept-word. Due to the above variability,
instead of recognizing each sign as a whole ‘visual word’, a more efficient
approach (inspired by speech recognition) is to decompose signs into
_subunits_ , resembling the phonemes of speech, and recognize them as a
specific sequence of subunits by using some statistical model, e.g., via
Hidden Markov Models (HMMs). Clearly, the subunits approach performs much
better on large vocabularies and continuous language; further, the subunits
are reusable and help with signer adaptation. In lack of a lexicon,
computational techniques that find such subunits are _data-driven_ , i.e.,
perform unsupervised clustering on a large database and use the cluster
centroids as subunits. This performs well in several instances, especially
when the subunits are pre-classified and statistically modeled based on visual
features into dynamic vs. static, as done in [TPM14]. A further improved
performance accompanied with phonetic interpretability may be obtained if the
chosen subunits are also based on the phonetic structure of a sign, as for
example by incorporating the Posture-Detention-Transition-Steady Shift (PDTS)
system [JoLi11] of phonetic labels. In [PTVM11] the phonetic information
provided by the PDTS transcriptions of sign videos was combined with the
automatically extracted visual features to create statistically trained
_phonetic subunits_ and a corresponding lexicon, which were then used for
optimally aligning (via Viterbi decoding) the data with the phonetic labels
and hence providing the missing temporal segmentation, as well as better sign
recognition.

While information and meaning in sign languages are mainly conveyed by moving
handshapes, they are also communicated in part by non-manual cues such as
facial expressions. These expressions can be visually modeled by deformable
models that encode both geometric shape and brightness texture information.
Deformable masks provided by active appearance models (AAMs) [CET01] can
successfully help with detecting and tracking several types of informative
events of a sign sequence, e.g., eye blinking, as done in [APM14]. AAMs
[RTPM13] have also significantly boosted the performance of handshape
recognition in sign language videos.

With the advancement of deep neural networks, much progress has been made in
independent and continuous SLR, with the use of CNN and LSTM networks. Koller
et al. [cvpr17] used a pretrained GoogleNet CNN architecture followed by two
Bidirectional LSTM layers to achieve, a minimum of 26.8% word error rate in
the RWTH-PHOENIX-Weather 2014 continuous sign language dataset [phoenix]. In
[ms-asl], Joze and Koller have experimented with different deep learning
methods in independent SLR, like the I3D [i3d] that consists of a plethora of
Inception Conv3D modules, or the hierarchical co-occurrence network (HCN)
[hcn] for body key-points. Moreover, Koller et al. [pami19] combined CNN-LSTM
architectures with multi-stream HMMs in the context of weakly supervised
learning for sing language videos. More recently, Camgoz et al. [cvpr2020]
utilized transformers for the task of SLR, while in [eccvw20] they proposed a
multi-channel transformer architecture that combined information for body,
face and hands.

Fig. 2: The architecture used for the Convolutional I3D-type model. On the
left is the proposed arcitecture with the 3D CNN Cells followed by one
Bidirectional LSTM layer. On the right are the interior layers of each 3D CNN
cell.

An integral component of our approach is the use of a recently introduced
parametric body model, SMPL-X [pavlakos2019expressive] that can jointly model
the body, the hands and the face of the person. With the exception of Adam
[joo2018total], this is the only available model that can jointly capture
these three channels of information. Previous statistical models, focus only
on the body (e.g., SCAPE [anguelov2005scape] and SMPL [loper2015smpl]), or add
hands, but still miss the facial expression (e.g., SMPL+H
[romero2017embodied]), which is crucial for the task of sign language
recognition. Conveniently, SMPL-X is also accompanied by a method that allows
us to reconstruct the model parameters for a person from a single image. The
method is called SMPLify-X and is based on the SMPLify approach by Bogo
[bogo2016keep].

##  3 Technical Approach

In this Section, we provide a short introduction to the SMPL-X
[pavlakos2019expressive] model and then give some details about the SMPLify-X
method that enables the estimation of SMPL-X model parameters given a single
RGB image as input. Next, we present the dataset we employ to evaluate our
approach and finally we analyze the feature extraction methods, the different
model architectures and the training hyper-parameters.

###  3.1 SMPL-X model and SMPLify-X reconstruction

SMPL-X is based on a template mesh which has $N=10475$ vertices, and a
kinematic tree with a set of 54 joints $J$ which are defined by the kinematic
tree and depend on the body shape parameters $\beta$. The model is controlled
by pose parameters $\theta$, shape parameters $\beta$ and expression
parameters $\psi$. The pose parameters $\theta$ correspond to the 3D rotation
of each model joint, with respect to its parent in the kinematic tree. With 3
DoF per joint, we require 162 pose parameters. Both shape parameters and
expression parameters correspond to 10 coefficients of a PCA shape and
expression shape respectively that are trained to express the majority of body
shapes and facial expressions. Given these parameters as input, the model
defines a function $M$ that returns a mesh with the selected articulation,
body shape and expression:

$M\left(\theta,\beta,\psi\right):\mathbb{R}^{|\beta|,|\theta|,|\psi|}\rightarrow\mathbb{R}^{3N}.$
(1)

This function $M$ is based on Linear Blend Skinning [Lewis_lbs] and includes
some learnable parameters for the pose, shape and expression spaces. These
parameters are trained from alignments of the template mesh to 3D scans, as
described in [pavlakos2019expressive].

SMPLify-X [pavlakos2019expressive] is a two stage method, capable of
efficiently and effectively reconstructing humans from image data. In the
first stage, a set of features is detected on the image, typically 2D
keypoints detected by Openpose [cao2019openpose, cao2017realtime,
simon2017hand, wei2016convolutional], which include body keypoints, hand
keypoints and facial landmarks. We refer to these detected 2D keypoints with
$J_{est}$, and we assume that each keypoint $J_{est,i}$ is detected with
confidence $\omega_{i}$. For the second stage, we fit SMPL-X to these 2D
landmarks, encouraging the projection of the 3D joints of the model to agree
with the detected 2D locations, which is expressed by the data term:

$\sum_{joint~{}i}\omega_{i}\rho(\Pi_{K}(R_{\theta}J(\beta)_{i}))-J_{est,i}),$
(2)

where $R_{\theta}$ is the rotation applied to the joints based on the pose
parameters $\theta$, $\Pi_{K}$ is the projection operation and $\rho$ is a
robust function used for the fitting. The complete objective also includes a
penalty for mesh intersections and a set of priors for regularization and is
optimized with an annealing scheme for the weights of each term to finally
produce the corresponding SMPL-X parameters.

Although SMPL-X can reconstruct with very high accuracy the person in a
specific sequence, our ultimate goal is to recognize the signs for each image
sequence. Our main insight is that the low dimensional parametric
representation of SMPL-X should capture the majority of the information that
is transmitted during a sign, i.e., the body pose, the hand pose and the
facial expressions. This should make it a very effective intermediate
representation for sign language recognition. In specific, SMPL-X takes each
frame of a video and reconstructs the 3D body, face and hands of the signer
conveying it in 88 parameters, converting each sign to a sequence of vectors
of length 88. This sequence of SMPL-X parameters across the frames of a sign
is being used as input to an RNN to classify the sign to one of the
corresponding categories.

GSLL Subset | Videos | Frames | TrainSet | DevSet | TestSet
---|---|---|---|---|---
50 classes | 538 | 22808 | 318 | 106 | 114
100 classes | 1038 | 45437 | 618 | 206 | 214
200 classes | 2038 | 92599 | 1218 | 406 | 414
300 classes | 3038 | 140771 | 1818 | 606 | 614
347 classes | 3464 | 161050 | 2066 | 695 | 703

Table 1: Statistics for the Greek Sign Language Lemmas Dataset and its respective subsets. Method \ GSLL Subset | Subset 50 | Subset 100 | Subset 200 | Subset 300 | Full Dataset | Parameters
---|---|---|---|---|---|---
3D RGB & Optical Flow Images | 90.41% | 86.85% | 80.79% | 71.36% | 65.95% | 43.41 million
2D Openpose Skeleton | 96.49% | 94.39% | 93.24% | 91.86% | 88.59% | 1.55 million
3D SMPL-X Reconstruction | 96.52% | 95.87% | 95.41% | 95.28% | 94.77% | 0.88 million
Table 2: Comparison of the three methods for training: i) Raw RGB images and
their Optical Flow ii) Openpose skeleton key-points and iii) 3D Body
Reconstruction key-points.

###  3.2 Greek Sign Language Lemmas Dataset

Since our goal is to test the ability of the proposed method to adequately
extract 3D hand, face and body features, we limit our approach to non-
continuous sign language recognition. Continuous SLR contains syntactic and
linguistic structure that is beyond the focus of this work. This means that we
exclude from our experimentation datasets like RWTH-PHOENIX-Weather 2014
[phoenix] and SIGNUM [signum] that consist of full sentences. Instead, we
focus on the Greek Sign Language Lemmas Dataset (GSLL) 111To be made public
upon publication of the paper. [TPM14] which proved to be ideal for our
experiments. The MS-ASL [ms-asl] dataset consists of 222 signers and extremely
varying backgrounds, which makes it challenging for Conv3D networks to
converge. In order to make a more fair comparison between 3D reconstruction
and 3D convolutional networks we choose the GSLL dataset which consists of
only two signers and 347 different signs (classes) in almost 3500 videos and a
steady blue background cloth. Table 1 provides more details for the dataset
and our selected subsets.

###  3.3 Training Methodology

Independent sign language recognition can be considered a task that is similar
to action recognition. Thus we expect similar techniques to work well on SLR
too. We decided to not intervene on the length of features’ sequence. Thus our
features vary in sequence length from a minimum of 10 frames to a maximum of
300. Next, we present the methods with which we confront this problem.

Openpose: We extract 411 parameters for each frame and feed the sequence in an
RNN consisting of one Bi-LSTM layer of 256 units and a Dense layer for
classifying, after applying standard scaling to our features. We believe that
by providing a recurrent network with these features will eliminate any
redundant information (e.g background, clothes, lighting) that a raw image
contains.

Raw Image and Optical flow: A 3D state-of-the-art method for action
recognition and signing is the I3D network [ms-asl, i3d]. Table 2 shows the
architecture of the 3D convolution module used both for raw RGB frames and for
the optical flow of these. We reshape each frame to a $175\times 175$ array
and normalize its pixels to $[0,1]$. We feed raw images to a VGG16-LSTM model
as well which is initialized with Imagenet weights, for further
experimentation.

SMPL-X: Due to its ability to interpret the structure of the body in detail,
we strongly believe that this method will provide key features for this task.
Moreover, SMPL-X provides 3D information, in comparison to Openpose that
results to 2D only keypoints, so the extracted features should be strictly
more informative. This method extracts 88 features per frame, creating a
(length of sequence) $\times$ 88 array for each sequence, which is being
standard scaled as in the Openpose experiments. Similarly to Openpose, we
employ the same neural network architecture so that we can directly compare
the two methods independently of the type of architecture.

We train all networks using categorical cross-entropy loss. SGD is used to
optimize the loss function, with an initial learning rate of 0.0001 and 10%
decay rate per epoch, while the batch size is set to 1, due to varying
sequence length. We perform Learning Rate Reduction and Early Stopping by
monitoring the validation loss with a patience of 3 and 5 epochs respectively.

##  4 Results and Discussion

According to Table 3, Openpose and SMPL-X models, which consist of 1.6 and 0.9
million parameters respectively, outperform the Conv3D-LSTM and VGG16-LSTM
model, which consist of 43 and 15 million parameters respectively. This can be
attributed to the fact that the former two eliminate the redundant information
from each frame, keeping only the essential body features. Specifically, VGG16
fails to converge and reduce its loss, achieving an accuracy below 10% for all
classes. This does not come as a surprise to us since Joze and Koller in [ms-
asl] have trained a VGG16-LSTM model for the MS-ASL dataset which achieved
13.33% for the ASL100 Subset and just 1.47% for the ASL500 Subset. As
mentioned earlier, GSLL Dataset is characterised by a uniform environment
between each sign and each signer (two signers in front of a blue cloth). The
MS-ASL dataset consists of 222 distinct signers where each signer performs in
a completely altered environment. We strongly believe that Openpose and mainly
SMPL-X will by far outperform convolutional models in these datasets, which
simulate more accurately the real world. Finally, SMPL-X seems to outperform
the features produced by Openpose especially with the increase of different
signs, dictating that a more detailed and qualitative representation of the
human body is needed for the Sign Language Recognition task. While varying and
more complex signs are being added to the train set, Openpose fails to convey
the small details that differentiate these signs, while SMPL-X holds its
accuracy almost fixed.

To further examine the features produced by SMPL-X, we experiment with a
combination of a subset of features produced by it. Specifically, as mentioned
in Section 3, the SMPL-X method produces a total of 88 features, 10 for shape
parameters, 3 for global orientation, 24 for left and right hand pose, 3 for
jaw pose, 6 for left and right eye pose, 10 for expression and 32 for the body
pose. Moreover, it is widely known that sign language does not depend solely
on gestures but fundamentally on body pose and facial expressions as well. To
demonstrate this fact, we proceed to a couple of more experiments. First, we
remove all information that comes from facial expressions (jaw pose, left and
right eye pose and expression) and train the model again with a total of 69
features. Secondly, we only remove the body pose information and train the
model with a total of 50 features. Finally, for the sake of completeness, we
remove left and right hand pose and train the model with a total of 64
parameters. We conduct the same experiments for Openpose by separating pose
keypoints (75 parameters), face keypoints (210 parameters), and left and right
hand kepoints (126 parameters). Table 4 sums up the results from all the
aforementioned experiments.

Parameters | Openpose | SMPL-X
---|---|---
All |  88.59% |  94.77%
Without Face | 88.34% | 93.19%
Without Hands | 70.20% | 89.58%
Without Body | 84.21% | 85.02%
Table 3: Experiments with subset of features produced by Openpose and SMPL-X.

First of all, we can see that omitting any of these three channel indeed
reduce the accuracy of our model. In fact, we expect the omission of facial
characteristics to affect even more the accuracy in the continuous sign
language where the face plays a crucial role into expressing the intensity of
a word. For example, ”rain” and ”snow” have the exact same hand
configurations, whereas only the mouth shape changes. Furthermore, we observe
that removing hand information in SMPL-X is less harmful than removing body
pose. That can be attributed to the fact that when few and simple signs are
available, the sign can be mainly conveyed through the movement of the arms
while the hands often remain straight. Nonetheless, both hands and body
structure (chiefly due to arms) are of vital importance for SLR while at the
same time, omitting facial expression affect the model’s performance. On the
other hand, Openpose due to the fact that has very few parameters for body,
i.e. only 75 out of 411, it is much more destructive to remove hands features
than body.

##  5 Conclusion and Future Work

In this paper we investigated the extraction of 3D body pose, face and hand
features for the task of Sign Language Recognition. We compared these
features, to Openpose key-points, the most famous method for extracting 2D
skeleton parameters and features from raw RGB frames and their optical flow
that are fed in a state-of-the-art Deep Learning architecture used in action
and sign language recognition. The experiments revealed the superiority of
SMPL-X features due to the detailed and qualitative features extraction in the
three aforementioned regions of interest. Moreover, we exploited SMPL-X to
point out the significance of combining all these three regions for optimal
results in SLR. Future work on 3D body, face and hands extraction for SLR
includes further experiments in different independent datasets with more
signers and varying environment. Furthermore, we strongly believe that
applying SMPL-X in continuous SLR will give further prominence to this method,
where facial expressions and body structure are even more crucial. Finally,
applying SMPL-X in different action recognition tasks is an interesting
experiment to examine the universality of SMPL-X success.


# Luminosity dependence of the cyclotron line energy in 1A 0535+262 observed
by Insight-HXMT during 2020 giant outburst

L. D. Kong*<EMAIL_ADDRESS>Key Laboratory for Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan
Road, Beijing 100049, China  University of Chinese Academy of Sciences,
Chinese Academy of Sciences, Beijing 100049, China  S. Zhang*
<EMAIL_ADDRESS>Key Laboratory for Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  L. Ji*<EMAIL_ADDRESS>School of Physics and Astronomy, Sun
Yat-Sen University, Zhuhai, 519082, China  P. Reig  Institute of Astrophysics,
Foundation for Research and Technology-Hellas, 71110 Heraklion, Crete, Greece
University of Crete, Physics Department, 70013 Heraklion, Crete, Greece  V.
Doroshenko  Institut für Astronomie und Astrophysik, Kepler Center for Astro
and Particle Physics, Eberhard Karls, Universität, Sand 1, D-72076 Tübingen,
Germany  Space Research Institute of the Russian Academy of Sciences,
Profsoyuznaya Str. 84/32, Moscow 117997, Russia  A. Santangelo  Institut für
Astronomie und Astrophysik, Kepler Center for Astro and Particle Physics,
Eberhard Karls, Universität, Sand 1, D-72076 Tübingen, Germany  R. Staubert
Institut für Astronomie und Astrophysik, Kepler Center for Astro and Particle
Physics, Eberhard Karls, Universität, Sand 1, D-72076 Tübingen, Germany  S. N.
Zhang  Key Laboratory for Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China
University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  R. Soria  5College of Astronomy and Space Sciences,
University of the Chinese Academy of Sciences, Beijing 100049, China  Sydney
Institute for Astronomy, School of Physics A28, The University of Sydney,
Sydney, NSW 2006, Australia  Z. Chang  Key Laboratory for Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  Y. P. Chen  Key Laboratory for
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China  P. J. Wang  Key Laboratory
for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy
of Sciences, 19B Yuquan Road, Beijing 100049, China  University of Chinese
Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China  L.
Tao  Key Laboratory for Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China
J. L. Qu  Key Laboratory for Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China
University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China

###### Abstract

We report on a detailed spectral analysis of the transient X-ray pulsar 1A
0535+262, which underwent the brightest giant outburst ever recorded for this
source from November to December 2020 with a peak luminosity of $1.2$ $\times
10^{38}\ \rm erg\ s^{-1}$. Thanks to the unprecedented energy coverage and
high cadence observations provided by Insight-HXMT, we were able to find for
the first time evidence for a transition of the accretion regime. At high
luminosity, above the critical luminosity $6.7\times 10^{37}$ erg s-1, the
cyclotron absorption line energy anti-correlates with luminosity. Below the
critical luminosity, a positive correlation is observed. The 1A 0535+262
becomes, therefore, the second source after V 0332+53, which clearly shows an
anti-correlation above and transition between correlation and anti-correlation
around the critical luminosity. The evolution of both the observed CRSF line
energy and broadband X-ray continuum spectrum throughout the outburst exhibits
significant differences during the rising and fading phases: that is, for a
similar luminosity the spectral parameters take different values which results
in hysteresis patterns for several spectral parameters including the cyclotron
line energy. We argue that, similarly to V 0332+53, these changes might be
related to different geometry of the emission region in rising and declining
parts of the outburst, probably due to a changes in the accretion disk
structure and its interaction with the magnetosphere of the neutron star.

pulsars: individual (A0535+26), X-rays: binaries, accretion: accretion pulsar

††journal: ApJL

##  1 Introduction

Accreting pulsars are neutron stars (NS) with a magnetic field of $B\sim$
$10^{12}$ G, orbiting and powered by accretion from a massive (spectral type
late O or early B) companion star (Reig & Nespoli, 2013). The strong magnetic
field implies a relatively large margnetospheric radius of $\sim 10^{7-10}$
cm. At the magnetosphere the accreting plasma is forced to follow the magnetic
field lines, and matter is channeled mostly onto the magnetic poles (mounds or
columns depending on the luminosity). In the hot, highly magnetized plasma the
kinetic energy of the in-falling material is converted to heat and radiation.
Typical cutoff power law continuum spectra and absorption features are
generated in these structures.

Magnetic field in many of these objects can be measured directly through
measurement of energies of so-called cyclotron resonant scattering features
(CRSFs) modifying the continuum X-ray spectrum of X-ray pulsars. These are
characteristic absorption-like features found in the spectra of highly
magnetized accreting pulsars usually from $\sim$ 10 keV to 100 keV
corresponding to so-called Landau levels (Staubert et al., 2019) and directly
related to field strength in the line forming region. The centroid line
energies $E_{\rm cyc}$ are expected, i.e. at:

$E_{cyc}\ =\ \frac{n}{(1+z)}\frac{\hbar eB}{m_{\rm e}c}\ \approx\ 11.6\
\frac{n}{(1+z)}\times B_{12}\ (\rm keV),$ (1)

where $B_{12}$ is the magnetic field strength in units of $10^{12}$ G, $z$ is
the gravitational redshift due to the NS mass and $n$ is the number of the
Landau level involved: e.g., $n\ =\ 1$ for the fundamental line and $n\ \geq\
2$ for the harmonics.

With the increase in precision of CRSF measurements, the dependence of CRSFs
on luminosity and spin phase has become obvious and is one of the important
areas of current X-ray pulsar research as it potentially allows probing of
emission region geometry and properties (Staubert et al., 2019). Of particular
interest is luminosity dependence of line properties as it might be used to
probe mechanism of plasma deceleration and transition between so-called sub-
and super- critical accreition regimes associated with onset of an accretion
column first suggested by Basko & Sunyaev (1976). As a result, two main
accretion regimes, super-critical and sub-critical accretion, are expected
depending on whether the source luminosity is higher or lower than a “critical
luminosity” $L_{\rm crit}$, which strongly depends on the magnetic field of
the neutron star (Basko & Sunyaev, 1976; Becker et al., 2012; Mushtukov et
al., 2015a). In the super-critical regime, for $L>L_{\rm crit}$, an accretion
column forms, and the falling plasma is decelerated by a radiation shock that
forms at a certain distance from the neutron star surface. In this case, the
emission height increases with increasing luminosity. In the sub-critical
regime, for $L<L_{\rm crit}$, the in-falling matter is presumably decelerated
by Coulomb interaction forming a region whose height decreases with increasing
luminosity. At even lower luminosity, the description of the deceleration
process of falling material is not very conclusive. A reasonable scenario is
that there exists a transition for the regime from Coulomb-stopping to gas-
shock dominance beyond which only a small accretion mound forms on the NS
surface. Based on this picture, and considering that local magnetic field
strength decreases with height, the correlation between the CRSF line energy
and luminosity can be used to trace the accretion regimes. For instance, a
transition between the sub and super-critical regimes has been recently
reported by Doroshenko et al. (2017) and Vybornov et al. (2018) for V 0332+53
during a giant outburst.

So far a clear negative correlation between the $E_{\rm cyc}$ and the X-ray
luminosity was only confirmed in V 0332+53 (Makishima et al., 1990, Tsygankov
et al., 2010, Cusumano et al., 2016, Doroshenko et al., 2017; Vybornov et al.,
2018), while for 4U 0115+63, the reported anti-correlation is debated
(Tsygankov et al., 2007; Nakajima et al., 2006; Müller et al., 2013; Iyer et
al., 2015). A positive correlation of the line centroid with luminosity is
more common and it has been found in Her X-1 (Staubert et al., 2007), Vela X-1
(Fürst et al., 2014), GX 304-1 (Klochkov et al., 2012), and Cep X-4 (Fürst et
al., 2015). For 1A 0535+262, the relationship between cyclotron energy and
luminosity is not settled, with some results supporting a positive correlation
(Reig & Nespoli, 2013, Sartore et al., 2015) and others reporting the absence
of any correlation (Terada et al., 2006, Caballero et al., 2007; Klochkov et
al., 2011). GX 304-1 and Vela X-1 also appear to show a flattening of the
correlation with increasing luminosity (Rothschild et al., 2017, La Parola et
al., 2016). Here we investigate luminosity dependence of line properties in
another Be XRB transient 1A 0535+262 at luminosities exceeding those reached
in previous investigations.

The transient X-ray binary 1A 0535+262 was discovered in outburst during
observations of the Crab by the Rotation Modulation Collimator on $Ariel\ V$,
through detection of the pulsations with period of $\sim$ 104 s (Rosenberg et
al., 1975). The system is composed of an O9.7IIIe donor star and a strongly
magnetized neutron star (Steele et al., 1998). The orbit has an eccentricity
of $e=0.47$ and an orbital period of $P_{\rm orb}\sim 110.3$ d (Finger et al.,
1996). The distance to 1A 0535+262 is 2 kpc, which has been recently confirmed
by Gaia (Bailer-Jones et al., 2018). The fundamental line and the first
harmonic have been detected at $\sim$ 46 keV and $\sim$ 100 keV by different
missions during different outbursts (Kendziorra et al., 1994; Grove et al.,
1995; Staubert et al., 2019). A somewhat higher CRSF line energy of $\sim$ 50
keV was found during the pre-outburst flare which took place about 5 days
before the peak of a normal (type-I) outburst (Caballero et al., 2008; Postnov
et al., 2008).

Since its discovery, 1A 0535+262 experienced a series of outbursts with peak
flux varying from a few 100 mCrab to $\sim$ 5.5 Crab (Camero-Arranz et al.,
2012, Caballero et al., 2013). After a long period of quiescence following the
2011 outburst, the source went through a giant outburst in November 2020, with
a peak flux recorded as $\sim$ 12.5 Crab in the energy range $15-50$ keV by
Swift/BAT. Owing to the unprecedentedly high intrinsic source flux,
observation cadence and wide energy range covered by Insight-HXMT, we can
investigate for the first time:

  * •

The relationship between the CRSF energy and luminosity at a higher luminosity
than ever before

  * •

The evolution of the spectral parameters, both during the rise and decline of
the outburst, allowing us to study whether the evolutionary paths are similar
or differ (hysteresis) at different phases of the outburst.

Sec 2 briefly describes the Insight-HXMT detectors and the data reduction
procedure. The details of the spectral analysis are presented in Sec 3.1,
while Sec 3.2 & Sec 3.3 focus on the evolution of the CRSF with luminosity and
time, respectively. Sec 3.4 focus on the variation of other spectral
components. Discussion and conclusions are included in Sec 4 and Sec 5.

##  2 Observation and Data reduction

The Hard X-ray Modulation Telescope, also dubbed as Insight-HXMT (Zhang et
al., 2014; Zhang et al., 2020), was launched on June 15, 2017, with a science
payload that allows observations in a broad energy band (1-250 keV) and with
large effective area at high energies. Insight-HXMT consists of three
collimated telescopes: the High Energy X-ray Telescope (HE, 18 cylindrical
NaI(Tl)/CsI(Na) phoswich detectors) (Liu et al., 2020), the Medium Energy
X-ray Telescope (ME, 1728 Si-PIN detectors) (Cao et al., 2020), and the Low
Energy X-ray Telescope (LE, Swept Charge Device (SCD)) (Chen et al., 2020),
with collecting-area/energy-range of 5000 $\rm cm^{2}$/20-250 keV, 952 $\rm
cm^{2}$/5-30 keV and 384 $\rm cm^{2}$/1-10 keV, and typical Field of View
(FoV) of $1.6^{\circ}\times 6^{\circ}$, $1^{\circ}\times 4^{\circ}$ and
$1.1^{\circ}\times 5.7^{\circ}$; $5.7^{\circ}\times 5.7^{\circ}$ for LE, ME
and HE, respectively (see Figure 1).

Figure 1: The FoVs for each detector are shown in black color for HE (three
boxes for small FoVs and two boxes for large FoVs), red color for ME and blue
color for LE, respectively. The blue cross shows the Crab location during the
ObsID P031431600101.

Insight-HXMT observed 1A 0535+262 from Nov 6, 2020 (MJD 59159) to Dec 24, 2020
(MJD 59207), for a total exposure of $\sim$ 1.910 Ms. For the analysis
reported in this article, we used the Insight-HXMT Data Analysis Software
(HXMTDAS) v2.04, together with the current calibration model v2.05
(http://hxmtweb.ihep.ac.cn/software.jhtml). The data are selected under a
series of criteria as recommended by the Insight-HXMT team. In particular,
data with elevation angle (ELV) larger than $10^{\circ}$, geometric cutoff
rigidity (COR) larger than 8 GeV and offset for the point position smaller
than $0.04^{\circ}$ were considered. In addition data taken within 300 s of
the South Atlantic Anomaly (SAA) passage outside of good time intervals
identified by onboard software was rejected. Because the Crab (ra=83.63308,
dec=22.0145) is located close to 1A 0535+262 (ra=84.727396, dec=26.315789),
and occasionally falls into the FOV of some of the Insight-HXMT detectors, we
exclude also the detector boxes which suffered from contamination by the Crab
(see Figure 1).

The energy bands adopted for the spectral analysis are: $2-7$ keV for the LE,
$8-35$ keV for the ME, and $30-120$ keV for the HE. Because of the uncertain
calibration at 22 keV of the ME, we ignore the $20-23$ keV range during the
spectral analysis for this instrument. The instrumental backgrounds are
estimated with the tools provided by Insight-HXMT team: LEBKGMAP, MEBKGMAP and
HEBKGMAP, version 2.0.9 based on the standard Insight-HXMT background models
(Liao et al., 2020a; Liao et al., 2020b; Guo et al., 2020). The XSPEC
v12.10.1f software package (Arnaud, 1996) was used to perform spectral
fitting. In order to improve the counting statistic of the energy spectra, we
combined the exposures within one day by _addspec_ and _addrmf_ tasks.
Considering current accuracy of the instrument calibration, we include
0.5$\%$, 0.5$\%$ and 1% systematic error for spectral analysis for LE, ME and
HE, respectively. The uncertainties of the spectral parameters are computed
using Markov Chain Monte Carlo (MCMC) with a length 10,000 and are reported at
90% confidence level.

##  3 Results

###  3.1 The spectral analysis

Considering the large number of observations carried out, we only show
detailed spectral fits for two representative observations carried out on Nov
18, 2020 and Dec 7, 2020, respectively. The observation of Nov 18, 2020 (MJD
59171) corresponds to the peak of the outburst at a luminosity of $11.67\times
10^{37}$ erg s-1, while the X-ray luminosity during the observation on Dec 7,
2020 (MJD 59191) was $4.74\times 10^{37}$ erg s-1 (here and throughout the
manuscript the luminosity is calculated in $2-150$ keV assuming a distance of
2 kpc (Bailer-Jones et al., 2018)). In Figure 2, the data from LE, ME, and HE
are plotted in black, red and green colors, respectively. We first fitted the
spectra with the following model: _tbabs_ $\times$(_cutoffpl_ +_gaussian_).
The _cutoffpl_ is a simple continuum with just three free parameters:

$F(E)=K\times E^{-\Gamma}exp(-E/E_{\rm fold})$ (2)

where $K$, $\Gamma$ and $E_{\rm fold}$ determine the normalization
coefficient, the photon index, and the exponential folding energy,
respectively. Absorption is taken into account through the _tbabs_ model, the
Tuebingen-Boulder interstellar medium (ISM) absorption model (Wilms et al.,
2000). The Galactic hydrogen column density toward the pulsar, $n_{\rm H}$, is
fixed at $0.59\times 10^{22}$ atom cm-2. The _gaussian_ line is needed to
model the iron emission line, usually at $E\sim 6.6$ keV.

Significant residuals at $2-20$ keV, $\sim 45$ keV and $\sim 100$ keV are
observed. The residuals at lower energies were accounted for by adding two
black-body components: a hotter one with $kT>1$ keV and a cooler one with
$kT<1$ keV. The flux in $2-10$ keV of two black body and cutoffpl components
are shown in Table 1. Within $2-10$ keV, the flux of two black body component
and that of cutoffpl component were comparable. The absorption features are
clearly visible in the residuals of best-fit spectra with the continuum model
described above at $\sim 45$ and $\sim 100$ keV which is consistent with the
previously determined energies of the fundamental CRSF and and its harmonic in
1A 0535+262 (Kendziorra et al., 1994; Grove et al., 1995; Staubert et al.,
2019), and thus were interpreted as such. To model these lines, we used a
multiplicative absorption model _mgabs_ with Gaussian profile:

$F^{{}^{\prime}}(E)=F(E)\times\emph{mgabs}=F(E)\times[(1-\tau_{1}e^{\frac{-(E-Ecyc)^{2}}{2\sigma_{1}^{2}}})\times(1-\tau_{2}e^{\frac{-(E-R\times
Ecyc)^{2}}{2\sigma_{2}^{2}}})],$ (3)

where $F^{{}^{\prime}}(E)$ is the spectrum modified by _mgabs_ , $E_{\rm cyc}$
is the cyclotron line central energy, $\tau_{1}$ and $\sigma_{1}$ characterize
the central optical depth and the width of the line. The parameters $R$
describe the ratio of the 1-st harmonic line and fundamental line, $\tau_{2}$
and $\sigma_{2}$ describe the central optical depth and the width of the 1-st
harmonic line.

The model _tbabs_ $\times$ _magbs_ $\times$ (_bbodyrad1_ +_bbodyrad2_
+_gaussian_ +_cutoffpl_) improved the fit considerably giving a reduced
$\chi^{2}$ of 0.97 and 0.93 for the two observations on Nov 18 and Dec 7,
2020, respectively (Table 1). Although some residuals are still present at
around 60 keV (Figure 2), these are not significant (F-value = 4.5 & 4.2, see
Table 1) and other parameters are not affected by the addition of an extra
absorption component. This structure may be caused by the non-Gaussian
character of the cyclotron line or the uncertainty of the calibration. Because
the significance of this structure is not high, it is difficult to draw any
firm conclusion.

The fundamental $E_{\rm cyc}$ was 43 keV during the peak of the outburst and
increased to 47 keV during the fading phase of the outburst. The ratio between
the fundamental and harmonic line is $R\approx$ 2.3, and this ratio is also
consistent with the results from INTEGRAL observations (Sartore et al., 2015).
The width of the fundamental is about half of that of the harmonic, again
consistent with previous observations by other satellites, where the width for
fundamental line was measured to be $\approx$ 10 keV, and $\approx$ 5 keV for
the harmonic (Sartore et al., 2015).

The parameters of other spectral components also changed significantly between
these two observations. For the observation at higher luminosity, the
temperatures of the black body components are $0.6_{-0.01}^{+0.01}$ keV and
$1.9_{-0.02}^{+0.03}$ keV, higher than those ($0.47_{-0.01}^{+0.01}$ keV and
$1.55_{-0.01}^{+0.03}$ keV) of the observation at lower luminosity. The
normalization factors of the black body components between the two
observations vary from $3999_{-224}^{+89}$ and $166_{-10}^{+6}$ to
$3151_{-300}^{+363}$ and $115_{-6}^{+4}$.

To study the evolution of the fundamental CRSF line, we initially let all
spectral parameters free, except the hydrogen column density that was fixed to
$5.9\times 10^{21}$ cm-2. Then we also tried fixing $R$, $\sigma_{1}$,
$\sigma_{2}$, $\sigma_{\rm Fe}$ to the average values 2.3, 10 keV, 5 keV and
0.3 keV, respectively. Whether these parameters were fixed or not made no
significant differences to the other parameters. We will show the details of
the spectral evolution in the following sections.

Figure 2: The panels show the spectra and residuals of Nov 18, 2020 and Dec 7,
2020. The spectra and residuals from LE, ME and HE detectors are shown in
black, red and green points. Two black-body, a cutoffPL and a iron emission
line are shown in the panels. The residuals without CRSF lines
($\tau_{1},\tau_{2}=0$) are shown at the bottom of the figure. The parameters
are shown in Table 1. Table 1: Parameters of the spectral fitting

Date |  | 2020-11-18 | 2020-12-07 |
---|---|---|---|---
MJD |  | 59171 | 59190 |
Model | Parameters | 2bb+cutoffPL | 2bb+cutoffPL |
tbabs | $n_{\rm H}\ (10^{22}\ \rm cm^{-2})$ |  $0.59$ (fixed) |  $0.59$ (fixed) |  $0.59$ (fixed) |  $0.59$ (fixed) |  $0.59$ (fixed) |  $0.59$ (fixed) |
gabs |  $E_{\rm abs}$ (keV) | … | … | $59.9_{-3.9}^{+7.2}$ | … | … | $61.0_{-2.5}^{+4.8}$ |
|  $\sigma_{\rm abs}$ (keV) | … | … | $3.8_{-3.3}^{+2.1}$ | … | … | $3.3_{-2.1}^{+5.6}$ |
| $\tau_{\rm abs}$ | … | … | $0.06_{-0.03}^{+0.03}$ | … | … | $0.07_{-0.05}^{+0.04}$ |
F-test | F-value | … | … | 4.5 | … | … | 4.2 |
| P-value | … | … | 0.0035 | … | … | 0.0063 |
mgabs |  $E_{\rm cyc}$ (keV) | $43.0_{-0.4}^{+0.8}$ | $43.0_{-0.2}^{+0.4}$ | $42.4_{-0.3}^{+0.8}$ | $47.0_{-0.2}^{+1.0}$ | $46.7_{-0.3}^{+0.2}$ | $46.2_{-0.4}^{+0.6}$ |
|  $\sigma_{1}$ (keV) | $13_{-1}^{+1}$ | 10 (fixed) | 10 (fixed) | $11_{-1}^{+1}$ | 10 (fixed) | 10 (fixed) |
| $\tau_{1}$ | $0.24_{-0.01}^{+0.01}$ | $0.18_{-0.01}^{+0.01}$ | $0.18_{-0.01}^{+0.01}$ | $0.24_{-0.01}^{+0.01}$ | $0.24_{-0.01}^{+0.01}$ | $0.24_{-0.01}^{+0.01}$ |
| R | $2.3_{-0.1}^{+0.1}$ | 2.3 (fixed) | 2.3 (fixed) | $2.3_{-0.1}^{+0.1}$ | 2.3 (fixed) | 2.3 (fixed) |
|  $\sigma_{2}$ (keV) | $5_{-2}^{+3}$ | 5 (fixed) | 5 (fixed) | $8_{-2}^{+7}$ | 5 (fixed) | 5 (fixed) |
| $\tau_{2}$ | $0.3_{-0.1}^{+0.1}$ | $0.3_{-0.1}^{+0.1}$ | $0.32_{-0.02}^{+0.04}$ | $0.6_{-0.1}^{+0.1}$ | $0.8_{-0.1}^{+0.1}$ | $0.8_{-0.1}^{+0.2}$ |
gaussian |  $E_{\rm Fe}$ (keV) | $6.58_{-0.02}^{+0.01}$ | $6.59_{-0.02}^{+0.02}$ | $6.59_{-0.01}^{+0.02}$ | $6.57_{-0.02}^{+0.01}$ | $6.58_{-0.02}^{+0.02}$ | $6.59_{-0.01}^{+0.01}$ |
|  $\sigma_{\rm Fe}$ (keV) | $0.19_{-0.03}^{+0.02}$ | 0.3 (fixed) | 0.3 (fixed) | $0.25_{-0.03}^{+0.03}$ | 0.3 (fixed) | 0.3 (fixed) |
| norm ($10^{-2}$) | $3.3_{-0.3}^{+0.1}$ | $4.8_{-0.3}^{+0.2}$ | $4.7_{-0.4}^{+0.3}$ | $2.6_{-0.3}^{+0.3}$ | $3.0_{-0.2}^{+0.1}$ | $3.0_{-0.1}^{+0.3}$ |
bbodyrad2 |  $kT$ (keV) | $0.60_{-0.01}^{+0.01}$ | $0.56_{-0.01}^{+0.01}$ | $0.57_{-0.01}^{+0.01}$ | $0.47_{-0.01}^{+0.01}$ | $0.44_{-0.02}^{+0.02}$ | $0.45_{-0.02}^{+0.01}$ |
| norm | $3999_{-224}^{+89}$ | $4439_{-284}^{+276}$ | $4348_{-171}^{+294}$ | $3151_{-300}^{+363}$ | $3799_{-681}^{+666}$ | $3625_{-173}^{+772}$ |
|  $log_{10}$(Flux) | … | $-8.63_{-0.02}^{+0.03}$ | … | … | $-9.31_{-0.04}^{+0.09}$ | … |
| ($2-10$ keV, erg cm-2 s-1) |  |  |  |  |  |  |
bbodyrad1 |  $kT$ (keV) | $1.90_{-0.02}^{+0.03}$ | $1.70_{-0.03}^{+0.03}$ | $1.73_{-0.03}^{+0.03}$ | $1.55_{-0.01}^{+0.03}$ | $1.50_{-0.02}^{+0.02}$ | $1.52_{-0.03}^{+0.01}$ |
| norm | $166_{-10}^{+6}$ | $175_{-8}^{+9}$ | $173_{-8}^{+8}$ | $115_{-6}^{+4}$ | $123_{-8}^{+5}$ | $122_{-3}^{+8}$ |
|  $log_{10}$(Flux) | … | $-7.90_{-0.02}^{+0.02}$ | … | … | $-8.24_{-0.02}^{+0.02}$ | … |
| ($2-10$ keV, erg cm-2 s-1) |  |  |  |  |  |  |
cutoffPL | $\Gamma$ | $-0.80_{-0.02}^{+0.02}$ | $-0.55_{-0.02}^{+0.01}$ | $-0.57_{-0.02}^{+0.02}$ | $-0.10_{-0.02}^{+0.02}$ | $-0.08_{-0.02}^{+0.01}$ | $-0.09_{-0.01}^{+0.02}$ |
| $E_{\rm fold}$ | $9.45_{-0.04}^{+0.07}$ | $10.04_{-0.06}^{+0.05}$ | $10.01_{-0.06}^{+0.05}$ | $12.3_{-0.1}^{+0.2}$ | $12.4_{-0.1}^{+0.1}$ | $12.4_{-0.1}^{+0.1}$ |
| $K$ | $0.16_{-0.01}^{+0.01}$ | $0.28_{-0.01}^{+0.01}$ | $0.26_{-0.01}^{+0.01}$ | $0.27_{-0.01}^{+0.02}$ | $0.28_{-0.01}^{+0.01}$ | $0.28_{-0.01}^{+0.02}$ |
|  $log_{10}$(Flux) | … | $-7.53_{-0.01}^{+0.01}$ | … | … | $-7.84_{-0.01}^{+0.01}$ | … |
| ($2-10$ keV, erg cm-2 s-1) |  |  |  |  |  |  |
Luminosity |  $L_{2-150}/10^{37}$ (erg s-1) | … | $11.67_{-0.05}^{+0.03}$ | … | … | $4.74_{-0.02}^{+0.01}$ | … |
Fitting |  $\chi_{\rm red}^{2}$/d.o.f | 0.97/510 | 1.03/514 | 1.01/511 | 0.93/510 | 0.95/514 | 0.93/511 |

  * Note

: Uncertainties are reported at the 90% confidence interval and were computed
using MCMC (Markov Chain Monte Carlo) of length 10,000. The 0.5%, 0.5% and 1%
system error for LE, ME and HE respectively has been added during spectral
fittings.

###  3.2 The CRSF centroid energy evolution

To investigate the evolution of the spectral parameters with time and
luminosity, the multi-component model described in the previous section was
fitted to all Insight-HXMT spectra. The left panel in Figure 3 shows the
variation of the fundamental line $E_{\rm cyc}$ with the $L_{x}$ (hereafter
the luminosity is considered for the 2-150 keV energy range), where the
$E_{\rm cyc}$ during rising phase and fading phase of the outburst are
distinguished by blue and red points. The right panel in Figure 3 shows the
outline of the outburst with a resolution of one point per day. The finer
sampling of the fading phase allowed us to use a broken linear model to
describe the relationship between $E_{\rm cyc}$ and $L_{x}$, which is shown
with green line in the left panel of Figure 3. The form of the broken linear
model is:

$E_{\rm cyc}=\left\\{\begin{array}[]{l}k_{1}\times(\frac{L_{\rm x}}{10^{37}\rm
erg\ s^{-1}})+(\frac{E_{0}}{\rm 1\ keV}),\ \ \ \rm if\ \frac{\emph{L}_{\rm
x}}{10^{37}\rm erg\ s^{-1}}\leq\frac{\emph{L}_{\rm break}}{10^{37}\rm erg\
s^{-1}}\\\ k_{2}\times(\frac{L_{\rm x}-L_{\rm break}}{10^{37}\rm erg\
s^{-1}})+k_{1}\times(\frac{L_{\rm break}}{10^{37}\rm erg\
s^{-1}})+(\frac{E_{0}}{\rm 1\ keV}),\ \ \ \rm if\ \frac{\emph{L}_{\rm
x}}{10^{37}\rm erg\ s^{-1}}>\frac{\emph{L}_{\rm break}}{10^{37}\rm erg\
s^{-1}}\end{array}\right.$ (4)

where, $k_{1}$ and $k_{2}$ are the increasing/decreasing rate, and $L_{\rm
break}$ is the break point. From the fitting, we get $E_{0}=46.8\pm 0.14$ keV,
$k_{1}=-0.04\pm 0.04$, $k_{2}=0.49\pm 0.05$ and $L_{\rm break}=6.7\pm 0.4$.
The $E_{\rm cyc}$ and $L_{\rm x}$ follow a clear anti-correlation above
$L_{\rm break}$, whereas $E_{\rm cyc}$ does not significantly change with
$L_{x}$ below $L_{\rm break}$. Considering that $E_{\rm cyc}$ strongly depends
on the magnetic field and the conditions in the region near the neutron star
surface where the accretion flow is decelerated, the change in the
relationship between $E_{\rm cyc}$ and luminosity points to a change in the
accretion regime and allows to tentatively associate the break luminosity with
the critical luminosity corresponding to onset of an accretion column: $L_{\rm
break}=L_{\rm crit}/10^{37}\ \rm erg\ s^{-1}=6.7\pm 0.4$.

Note that there is an important difference in behavior of the line in the
rising and the declining parts of the outburst, even for comparable
luminosities. In particular, there is a positive correlation of $E_{\rm cyc}$
and $L_{x}$ below $L_{\rm crit}$ during the rising phase whereas $E_{\rm cyc}$
does not change significantly with luminosity during the fading phase,
remaining at $\approx$ 47 keV. We note that both types of behaviour were not
previously reported for 1A 0535+262, however, this is likely simply due to the
lack of suitable observations. Indeed, Insight-HXMT observations sampled the
rising phase of the outburst in detail for the first time while past
investigations of the line properties mostly focused on the declining part of
the outburst. This might explain the non-detection of the correlation of the
line energy with luminosity at low to moderate luminosities up to now. On the
other hand, luminosity range where the anti-correlation is observed was simply
not observed previously with instruments capable of accurate measurement of
the line energy since the current outburst of the source is the brightest in
recent history.

It is interesting to note also that $E_{\rm cyc}$ is systematically lower
during the rising phase than during the fading phase. Within the super-
critical zone, the luminosity dependence of the line energy can be fitted by
$E_{\rm cyc}=-0.54\pm 0.03\times(L_{x}/10^{37}\ \rm erg\ s^{-1})+49.5\pm 0.3$
keV for the rising phase and $E_{\rm cyc}=-0.53\pm 0.04\times(L_{x}/10^{37}\
\rm erg\ s^{-1})+50.2\pm 0.4$ keV for the fading phase. Although there is
uncertainty in the fit, a shift of 0.7 keV is shown in the left panel of
Figure 3, while the slope of the correlation appears to be constant throughout
the outburst. We note that this behavior appears to be similar to the line
evolution reported by Cusumano et al. (2016); Doroshenko et al. (2017);
Vybornov et al. (2018) for V 0332+53, however, with some important
differences. In particular, in V 0332+53 the line energy was: a) lower in the
declining phase than in the rising phase of the outburst (i.e. opposite to 1A
0535+262), and b) consistent with a linear decay in time which resulted in
slightly different anti-correlation slopes (Doroshenko et al., 2017).

We emphasize that for both sources the observed luminosity dependence of the
cyclotron line energy is inconsistent with simple predictions of the available
theoretical models. Indeed, regardless of the assumed model the line energy is
expected to be related to height of the column, which in turn is expected to
be proportional to accretion rate, so one expects same line energy for a fixed
luminosity, i.e. no hysteresis as observed in V 0332+53 or 1A 0535+262.
Doroshenko et al. (2017) argued that this discrepancy can be resolved if
column height is also influenced by additional factors, for instance,
magnetospheric radius defining the footprint area of the column and thus
indirectly its height. Doroshenko et al. (2017); Vybornov et al. (2018)
demonstrated through analysis of spin evolution and power spectra of the
aperiodc variability that gradual recession of the accretion disk throughout
the outburst (for a fixed luminosity) is indeed a plausible explanation for V
0332+53. To understand whether similar scenario can be realized also in 1A
0535+262, we considered, therefore, possible complex time and luminosity
dependence of the line centroid energy also for this source.

Figure 3:  Left panel: The evolution of the $E_{\rm cyc}$ with luminosity is
shown. The $E_{\rm cyc}$ in the rising and fading phase are shown in the red
and blue colors, respectively. The black dashed line shows the $L_{\rm break}$
with error range within grey shadow from a broken linear fitting, which
consists to the critical luminosity $L_{\rm crit}$. The red and blue dotted
lines show the fit of the rising and fading trends of $E_{\rm cyc}$ during the
super-critical state. Right panel: The evolution of the fundamental cyclotron
line $E_{\rm cyc}$ with time is shown with blue points. The luminosity
evolution with time is also shown with black points.

In particular, to decouple time and luminosity dependence of the CRSF energy
we adopted the same approach as (Doroshenko et al., 2017; Vybornov et al.,
2018), i.e. introduced a linear time drift of the centroid energy in addition
to the broken linear fit described above. The right panel of the Figure 3
shows the time dependence of the X-ray luminosity in $2-150$ keV (grey points)
and the fundamental CRSF energy (blue points). As a result we found that
additional linear increase of the line energy at a rate of $\sim 0.071$ keV
d-1 indeed allows to describe the overall evolution of the line energy
throughout the outburst. We emphasize again that none of the theoretical
models predict any hysteresis effects, so our modeling is just an attempt to
decouple the theoretically expected luminosity dependence of the line energy
from other, yet unidentified effects. This allows to discuss them separately
and better understand overall dependence of the line energy.

First of all, we note that after subtracting this linear trend from the
observed $E_{\rm cyc}-L_{\rm x}$ correlation the transition becomes more
prominent as illustrated in Figure 4. This allows better quantification of the
observed properties of this correlation and more accurate measurement of the
transition luminosity. We estimate, i.e. slopes for the positive and negative
correlation parts at $k_{1}=0.23\pm 0.05$ and $k_{2}=-0.41\pm 0.04$ with the
break at the $L_{\rm break}=6.37\pm 0.45$. We note that observed luminosity of
the break is consistent with theoretical expectations for onset of the
accretion column (Basko & Sunyaev, 1976; Becker et al., 2012; Mushtukov et
al., 2015a) and thus interpret the observed transition as the accretion regime
transition in 1A 0535+262. Furthermore, the fact that, similarly to V 0332+53
additional time linear drift of the line energy on top of the expected
luminosity depence is required, also deserves a discussion.

Figure 4:  Correlation of the CRSF centroid energy on luminosity with the
linear drift of $\Delta E_{\rm cyc}=0.071$ keV d-1 taken into account. The
$E_{\rm cyc}$ in the rising and fading phase are shown in the red and blue
colors, respectively. The green line shows the broken linear fit. From the
fit, the black dashed line shows the transition at $6.37\times 10^{38}$ erg
s-1. The grey area shows the error range of the transition luminosity.

###  3.3 Luminosity dependence of the broadband continuum spectrum

In Figure 5, we plot the evolution of all spectral parameters as a function of
luminosity during the outburst. The red and black points represent the rising
and fading phase, respectively. The grey dashed line denotes the critical
luminosity at $6.7\times 10^{38}$ erg s-1. As it is evident from the figure,
the behavior in the rising and fading phases of the outburst is different for
all parameters with the exception of $\tau_{1}$ of the fundamental line.
Indeed, although the evolution with X-ray luminosity is smooth, the values of
the parameters is significantly different at similar luminosities in the
rising and fading phases.

The most prominent changes are seen in the black body components at higher
luminosity. For both black body components, the temperatures are
systematically lower in the fading phase, and the normalisation parameters
evolve in the opposite way. From Figure 5, there is an unexpected hump of $\rm
norm_{\rm b,2}$ $\sim\ 4000-8000$ between $3-4\times 10^{37}$ erg s-1, and
this hump does not match the change of the general trend. At this time,
$kT_{2}$ evolves into a dip shape. We note that this phenomenon may be caused
by the degeneracy between temperature and normalization of the black body
model rather than of physical origin. By fixing $\rm norm_{\rm b,2}$ at 4000,
we find that the evolution of $kT_{2}$ becomes smooth and continuous, and
other parameters in the model did not change significantly (see blue points in
Figure 5). The spectrum at low energy, where the blackbody components dominate
also display different trends with luminoisty (Figure 5). The normalization of
the blackbody is related to the size (radius) of the emitting area as
norm=$R_{\rm BB}^{2}/d^{2}$, where $R_{\rm BB}$ is the radius if the emitting
region in km and $d$ is the distance to the source in units of 10 kpc. For
lower temperature black body, the radius size can be limited to $R_{\rm
BB}\sim 5-13$ km, while for the higher temperature black body, it dominated at
smaller radius size $R_{\rm BB}\sim 1-3$ km. The thermal emission radius
varies $R_{\rm BB}\sim 1-3$ km, which appears as a reasonable size for the
polar cap. The lower temperature black body could be associated with the top
of the column cap (Tao et al., 2019). The asymmetry in the evolution of these
two black body components throughout the outburst may indicate that the
accretion geometry is different in the rising and fading phase.

The spectrum tends to be harder (smaller $\Gamma$) and display smaller fold
energies in the rising than in the fading phases. We notice that the evolution
trends of the spectral parameters in the black body and cutoff components do
not remain at low luminosity, probably due to relatively poor statistics of
the observational data. The evolution of the harmonic CRSF at $\sim$ 100 keV
(fixed at 2.3$\times E_{\rm cyc}$ for all observations) was not addressed in
detail here because of the relatively high HE background at above 100 keV and
hence larger systematic uncertainty in precisely measuring the 1-st harmonic
CRSF parameter $\tau_{2}$.

Figure 5: The evolution of the all parameters with the luminosity $L_{\rm x}$.
The grey dashed line shows the critical luminosity $L_{\rm crit}$. The rising
and fading phase during the outburst is distinguish with red and black points.
The blue points show the corrected points by fixing $\rm norm_{\rm b,2}$ at a
reasonable value 4000

.

##  4 Discussion

We have investigated the brightest outburst of 1A 0535+262 observed so far
using the data from Insight-HXMT. As a result, we found for the first time a
clear anti-correlation between the cyclotron line energy and luminosity above
$\sim 6.7\times 10^{37}$ erg s-1 (see Figure 3 and Figure 4). For 1A 0535+262,
this trend has not been found in the previous outbursts (Staubert et al.,
2019), likely because the luminosity range where it occurs has not been
sampled adequately by past observations. This phenomenon is rare, and it has
only been seen once before, i.e. in V 0332+53 (Makishima et al., 1990;
Tsygankov et al., 2010; Cusumano et al., 2016; Doroshenko et al., 2017
Staubert et al., 2019). Thus, 1A 0535+262 is the second source that displays
this behavior. We emphasize that our investigation is the first to cover this
luminosity range, which explains non-detection of the anti-correlation by
previous studies.

The luminosity dependence of the observed line energy in both regimes has been
already extensively discussed in the literature (Tsygankov et al., 2007;
Becker et al., 2012; Poutanen et al., 2013; Mushtukov et al., 2015a, b;
Staubert et al., 2019). Above the critical luminosity a higher accretion rate
would result in larger radiation pressure and hence a higher accretion column
above the magnetic pole and a smaller $E_{\rm cyc}$ no matter whether the line
is observed directly from the column (Tsygankov et al., 2007) or in reflection
(Poutanen et al., 2013). In sub-critical regime the positive correlation
between $E_{\rm cyc}$ and $L_{\rm x}$ can be explained if accreting matter is
stopped via the Coulomb braking dominated process. In this case a
collisionless shock first proposed by Langer & Rappaport (1982) and Bykov &
Krasilshchikov (2004) is expected to form at a height inversely proportional
to the electron density in the emission region (Shapiro & Salpeter, 1975).
Therefore, height of the shock surface decreases with an accretion rate
leading to a higher local magnetic field and correlation of the line energy
with flux. This model was successfully applied to interpret the $E_{\rm
cyc}-L_{\rm x}$ correlations in Cep X-4 (Vybornov et al., 2017) and GX 304-1
(Rothschild et al., 2017). An alternative explanation for the observed CRSF
energy evolution in the sub-critical regime is the Doppler effect caused by
the radiation pressure which changes the velocity profile of the falling
matter within the line-forming region (Mushtukov et al., 2015b). When the
radiation pressure increases with luminosity, the velocity near the NS surface
decreases. Consequently, the lower electron velocity results in smaller
redshift and higher CRSF energy. As a result, a positive correlation between
the line centroid energy and the luminosity is expected in a regime with sub-
critical luminosities.

As already mentioned in Section 3.2, however, the observed evolution of line
energy appears to be more complex than simple predictions outlined above. In
particular, it is obvious from Figure 3 that below $6.7\times 10^{37}$ erg
s-1, the CRSF energy follows two different paths in the rising and fading
phases. In the rising phase, $E_{\rm cyc}$ is positively correlated with
$L_{\rm x}$, while it is approximately constant in the fading phase. We note
that past studies of the source in this luminosity range were inconclusive,
i.e. some had claimed a positive correlation (Sartore et al., 2015), while
others reported the absence of any correlation (Terada et al., 2006; Caballero
et al., 2007; Caballero et al., 2013). The observed behavior in the current
outburst is consistent with these findings as the source was mostly observed
during its outburst decline. The HXMT data confirms the absence of a
correlation during the final stages of the outburst.

The overall observed line evolution throughout the outburst can be described
as a combination of a time linear increase with a rate of $\Delta E_{\rm
cyc}=0.071$ keV d-1, and a broken linear dependence on luminosity shown in
Figure 4. This allows us to estimate the transitional luminosity at
$6.37\times 10^{38}$ erg s-1. We emphasize that this value is only valid under
assumption above, i.e. that luminosity dependence of the line enrgy is
modified by a time linear drift throughout the outburst. Nevertheless, the
obtained value appears to be consistent with expected critical luminosity
value associated with onset of an accretion column (Basko & Sunyaev, 1976;
Becker et al., 2012; Mushtukov et al., 2015a), which is estimated to be a
broad range $2.0-6.7\times 10^{37}$ erg s-1 (Becker et al., 2012; Mushtukov et
al., 2015a). We conclude, therefore, that the observed break in luminosity
dependence of the line energy is associated with the transition from sub-to
super-critical accretion regime.

On the other hand, the origin of the time linear increase of the line energy
remains unclear. The asymmetric CRSF behavior was also reported for the 2005
outburst of V 0332+53 (Cusumano et al., 2016; Doroshenko et al., 2017),
however, the CRSF energy in the rising part of the outburst was larger than in
the fading phase in this case. Cusumano et al. (2016) interpreted the observed
decrease result as a rapid dissipation of external magnetic field of the
neutron star due to diamagnetic screening effects caused by accreted matter.
Alternatively, Doroshenko et al. (2017) argued that the energy drift of CRSF
was related to the different emission geometries because of different
magnetosphere sizes in rising and fading phase presumably associated with
changes of internal structure of the accretion disk. We emphasize that unlike
V 0332+53 the time linear drift in 1A 0535+262 implies increase rather than
decrease of the line energy with time. This implies that the scenario invoked
by Cusumano et al. (2016) can definitively not be applied in this case and the
observed changes in line energy must be associated with some change of the
geometry or intrinsic conditions in the emission region rather any intrinsic
field changes.

To qualitatively understand what kind of changes might be responsible to
account for the observed behavior of the line one could start by adopting
scenario proposed by Doroshenko et al. (2017) for V 0332+53 reversing the
rising and parts of the outburst. That is, to assume that for some reason the
inner disk radius is larger and the polar cap footprint is smaller during the
rising part of the outburst compared to those in the declining part. We
emphasize that in this scenario it is not clear what could lead to different
evolution of the inner disk radius in the two sources, and this discussion is
out of scope of the current work. Therefore, below we only try to understand
whether such change could in principle explain the observed line evolution or
other explanation needs to be found. With that said, one could anticipate (for
a given luminosity) a taller accretion column in the rising phase, which is
expected to result in a lower line energy no matter whether the line forms
within the accretion column or in reflection off the neutron star surface as
proposed by Poutanen et al. (2013). We note that up to now no other scenarios
have been proposed to explain anti-correlation of line energy with luminosity
in super-critical state, so assumption that the polar cap indeed has a smaller
footprint is, in fact, required in both scenarios to explain observed increase
of the line energy in super-critical regime in the declining part of the
outburst.

Considering that in the case of 1A 0535+262 the hysteresis in line energy is
observed for both sub- and super-critical regimes, it is equally important,
however, to consistently explain also the observed increase of the line energy
in sub-critical part of the declining part of the outburst. As already
mentioned, two scenarios to explain correlation of line energy with luminosity
in the sub-critical regime have been proposed in the literature, i.e. change
of the height of the collision-less shock (Langer & Rappaport, 1982; Vybornov
et al., 2017), and Doppler shifts due to scattering in the accretion flow
decelerated by radiative pressure (Mushtukov et al., 2015b). We note that in
the latter model the lower footprint column in the rising phase, which can be
characterized in terms of a larger $h/d$ (h is the height above the NS
surface; d is the radius of the hotspot), is actually expected to lead to a
lower observed line energy (see i.e. lower panel of Figure 2 in Mushtukov et
al. 2015b). On the opposite, larger polar cap size required to explain higher
line energy in super-critical state in the declining phase of the outburst,
implies also lower $h/d$ and higher observed line energy also in sub-critical
state. We conclude, therefore, that higher observed line energy (for a given
luminosity) in the declining part of the outburst can be qualitatively
explained by an increased footprint of the accretion flow. The larger
normalization of the black body components during the declining phase would
agree with this interpretation.

On the other hand, for collisionless shock model with higher observed line
energy in the declining part corresponds to a stronger field in the line
forming region and thus requires lower shock height resulting higher electron
density in the scenario considered by Vybornov et al. (2018). Recently,
Kulsrud & Sunyaev (2020) proposed that the accretion matter can slip across
the magnetic lines under a strong ideal Schwarzschild instability and hence
increase the size of the polar cap. This would also imply lower column height
of the collisionless shock, i.e. would also be qualitatively in agreement with
observations. We note that increase of the polar cap size would also be
consistent with the scenario by Mushtukov et al. (2015b), i.e. it would be
hard to disentangle between the two interpretations. On the other hand,
leakage of matter through sides of the column proposed by Kulsrud & Sunyaev
(2020) could provide physical explanation for the increase of the polar cap
size in both cases.

We conclude, therefore, that the observed line energy is qualitatively
consistent with the assumption of a larger inner disk radius in the rising
phase of the outburst. We emphasize, however, that this conclusion is only one
possibility and a detailed quantitative and self-consistent modeling of the
observed line energy is required to really understand the behavior of the line
in 1A 0535+262 and differences between this source and V 0332+53. Discussion
of the latter is beyond the scope for the current outburst, however, we’d like
to mention that the only obvious difference between the two pulsars is shorter
spin period of V 0332+53. This implies that the latter source is likely much
closer to co-rotation during an outburst, which could affect the interaction
of the accretion flow with the magnetosphere and potentially alter
magnetosphere size and footprint of the accretion column.

##  5 Conclusions

The high cadence and counting statistics provided by observations of a giant
outburst of 1A 0535+262 with Insight-HXMT in a broad energy range revealed for
the first time a series of new findings in context of variability of the
properties of the cyclotron line observed in this source with luminosity and
time. First, we were able to detect for the first time a clear anti-
correlation of the observed line energy with luminosity above $\sim 6.7\times
10^{37}$ erg s-1. Furthermore, also for the first time, a clear correlation of
the line energy with flux was observed below this luminosity in the rising
part of the outburst.

Overall, the observed line energy was found to exhibit complex behaviour
characterized by apparent hysteresis of line parameters between rising and
declining parts of the outburst. This hysteresis can be accounted for by a
time-linear increase of the line energy at a rate of $0.071$ keV d-1 on-top of
a combination of a correlation with flux below critical luminosity of $\sim
6.7\times 10^{37}$ erg s-1 and anti-correlation above this level. This
behaviour is similar to that previously reported for another Be-transient V
0332+53 (Doroshenko et al., 2017; Vybornov et al., 2018), although in the
latter case a time linear decay rather than increase was observed. The origin
of this discrepancy is unclear and subject for further investigations,
however, already now our results clearly show that the observed variability of
cyclotron lines in X-ray pulsars is clearly more complex than was foreseen by
theoretical models.

This work made use of data from the Insight-HXMT mission, a project funded by
China National Space Administration (CNSA) and the Chinese Academy of Sciences
(CAS). This work is supported by the National Key R&D Program of China
(2016YFA0400800) and the National Natural Science Foundation of China under
grants U1838201, 11473027, U1838202, 11733009, U1838104, U1938101, U2038101,
U1938103 and Guangdong Major Project of Basic and Applied Basic Research
(Grant No. 2019B030302001).

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