# A study on the efficacy of model pre-training in developing neural text-to-
speech system

###### Abstract

In the development of neural text-to-speech systems, model pre-training with a
large amount of non-target speakers’ data is a common approach. However, in
terms of ultimately achieved system performance for target speaker(s), the
actual benefits of model pre-training are uncertain and unstable, depending
very much on the quantity and text content of training data. This study aims
to understand better why and how model pre-training can positively contribute
to TTS system performance. It is postulated that the pre-training process
plays a critical role in learning text-related variation in speech, while
further training with the target speaker’s data aims to capture the speaker-
related variation. Different test sets are created with varying degrees of
similarity to target speaker data in terms of text content. Experiments show
that leveraging a speaker-independent TTS trained on speech data with diverse
text content can improve the target speaker TTS on domain-mismatched text. We
also attempt to reduce the amount of pre-training data for a new text domain
and improve the data and computational efficiency. It is found that the TTS
system could achieve comparable performance when the pre-training data is
reduced to 1/8 of its original size.

Index Terms—  Text to Speech, Pre-training, Data Reduction

##  1 Introduction

In recent years, neural text-to-speech technology has demonstrated significant
successes in generating high-quality speech with good naturalness and
expressiveness for a target speaker [1, 2, 3, 4, 5]. In research and
development of neural TTS, the text content of training and test data are
often highly similar and in the same text domain. For many real-world
applications, TTS systems need to deal with text input with arbitrary content
across a wide range of domains. Their performance may deteriorate
substantially on domain-mismatched text [6] due to the limited content and
domain coverage of training data. It is generally costly or impractical to
increase the quantity and diversity of training data for a specific target
speaker, whilst speech data from other “non-target” speakers may be easily
accessible and available. Leveraging large amount of non-target speakers’ data
from different sources has become a common and appealing approach to
developing high-performance TTS systems when training data from the target
speaker(s) are limited [7, 8, 9, 10, 11]. However, it was noticed that the
actual benefits of using training speech from other speakers could be
uncertain and unstable [7]. Understandably the benefits depend very much on
the content and quality of non-target speakers’ data. There are three aspects
of consideration: (1) coverage and domain of text content; (2) speaker
similarity with respect to the target speaker; and (3) acoustic condition. The
present study mainly focuses on the aspect of text content.

Our preliminary explorations showed that TTS systems trained solely on target
speaker’s data did not perform well in predicting appropriate prosody for
domain-mismatched input text. Prosody is utmost important in determining the
naturalness and expressiveness of speech. We postulate that speech prosody can
be seen as the combination of two components of variation in speech, namely
text-based variation and speaker-based variation. The text-based component,
termed as text prosody henceforth, refers to the general prosodic
characteristics that are basic and essential to expressing the intended text
content, e.g., lexical tone, lexical stress, sentence intonation [12]. If text
prosody is not realized or controlled properly, the synthesized speech would
sound unnatural and inappropriate, even if the pronunciation is largely
correct. In [13], native listeners were found to be capable of detecting
abnormality in the non-native speech that has International Phonetic Alphabet
(IPA) transcription identical to that produced by native speakers. The
speaker-based component of prosodic variation[14], termed as speaker prosody,
is concerned primarily with an individual’s speaking style. In particular, we
differentiate it from timbre, which refers mainly to voice (phonation)
characteristics.

We conjecture that text prosody can be captured by pre-training the TTS model
on a large amount of speech data with diverse text content and involving
multiple non-target speakers. A speech generation system fine-tuned from such
pre-trained model with target speaker’s data is expected to have better
performance on domain-mismatched text input than a corresponding system
without using pre-trained model. In a typical multi-speaker TTS model
design[15, 16], text content and speaker identity are processed by separate
modules. It adopts a speaker-independent TTS model with text embeddings and
speaker embeddings. This speaker-independent TTS model deals with only text
information regardless of speaker variation. After being fine-tuned on target
speaker data, the speaker-independent TTS model would supposedly retain the
text prosody learned during pre-training. In this way, effective pre-training
with diverse text content can contribute to the performance of target speaker
TTS system in the aspect of handling domain-mismatched text.

In some cases, it may not be convenient to collect extensive speech data with
diverse content for model pre-training. Traditionally large amounts of text-
audio parallel data could be acquired by: (1) studio recording with the target
speaker(s) reading text scripts for many hours; or (2) downloading long speech
recordings from the Internet, dividing them into sentence-level utterances and
aligning with given text transcriptions [17, 18, 19]. Both approaches are
tedious, labour-intensive and costly. Using an excessive amount of pre-
training data might also cause other concerns, namely long training time and
high computational resources. For applications of speech generation in
specific text domains, we investigate different approaches to reducing the
required amount of pre-training data while maintaining desired TTS
performance.

The contributions of this paper are as follows. First, we show that using
diverse speech data to pre-train a speaker-independent TTS model can improve
the performance of the target speaker TTS on domain-mismatched text. Second,
test sets with different degrees of similarities to the text domain of target
speaker data are designed. The designed test sets are used to study how the
pre-trained speaker-independent TTS model improves target speaker TTS
performance; Third, We propose a method to improve the data and computational
efficiency by reducing the pre-training data for a specific new text domain.

Fig. 1: Diagram of TTS model

##  2 Data Description

###  2.1 Pre-training Data

Pre-training data are used to train a speaker-independent model, which is then
fine-tuned on target speaker data. As our research questions concern the
content coverage and domain of the pre-training data, a large-scale English
speech corpus created for automatic speech recognition (ASR), namely
LibriSpeech, is adopted [18]. LibriSpeech contains around $1000$ hours of
speech data from $2484$ speakers. The speech content are based on the
LibriVox’s audiobooks, which have a comprehensive coverage of topics. The
complete LibriSpeech corpus is used as pre-training data. For the
investigation on pre-train data reduction, 40,000 text-audio pairs (around 1/8
of the data in LibriSpeech) are used.

###  2.2 Target Speaker Data

The target speaker data for fine-tuning of the pre-trained speaker-independent
TTS model are obtained from the LJSpeech corpus, which contains $13,100$
(about $24$ hours) audio clips of high-quality English speech produced by a
female speaker reading passages from 7 non-fiction books. Experiments will be
carried out to evaluate the performance of TTS systems fine-tuned with the
whole LJSpeech corpus or its subsets of smaller size.

##  3 Effect of Pre-training on target speaker TTS performance

Two TTS models are used to show how the pre-trained speaker-independent TTS
model improves target speaker TTS performance. The first system is trained
only on target speaker data, i.e., without pre-training with other data. This
system is named as TTS w/o pre-training. The second system, named TTS w/ pre-
training, is built by applying target speaker data (LJSpeech) to fine-tune a
speaker-independent model pre-trained with LibriSpeech. The two systems are
evaluated on three specially designed test sets, which have different degrees
of similarity to the text content of target speaker data. The three test sets
are named as T-SIM, T-RAN, T-DIFF, in the order from highest to lowest degree
of similarity to the text content of target speaker data.

###  3.1 TTS Model Description

The TTS model as shown in Figure 1 consists of a trainable speaker table and a
speaker-independent TTS model. The speaker table models the voice
characteristics for individual speakers. The speaker-independent model follows
the non-autoregressive TTS system FastSpeech 2[5]. In this study, the TTS
model predicts pitch at phoneme level, which is found to give better TTS
performance than frame-level prediction in FastSpeech 2. The text encoder
converts the phoneme sequence into text embeddings, combined with speaker
embeddings to predict the pitch and duration using the respective predictor
modules. The length regulator (LR) module up-samples all phoneme-level
features to frame-level features, which are then transformed into mel-
spectrogram by the decoder. The Melgan vocoder[20] is used to generate speech
waveform from the mel-spectrogram.

###  3.2 Design of Domain-mismatched Test Sets

We collect around 50,000 sentences from various sources that act as general
text set to show the TTS performance on domain-mismatched text. Three test
sets, each has $60$ sentences, are chosen from the general text set for
subjective evaluation. A phoneme-based subword bigram language model is
trained on the text domain of the target speaker data (LJSpeech). The phoneme-
level subwords for each word are obtained by applying Byte Pair Encoding
(BPE)[21] to the phoneme sequence of the word. The vocabulary size of subwords
is 200. The language model is applied to calculate the perplexity scores for
all 50,000 sentences in the general text set. A sentence similar to the text
domain of target speaker data will achieve low perplexity with the language
model trained on that text domain. The text with the top 60 lowest perplexity
scores make up the test set T-SIM, which is considered to lie in the same or
similar domain of target speaker text data. The test set T-DIFF, which has a
low degree of similarity to the text domain of target speaker data, comprises
text with the top $60$ highest perplexity scores. The $60$ text in subset
T-RAN are randomly sampled from the general text set.

##  4 Pre-training Data Reduction for a new text domain

In the previous section, we aim to investigate how the pre-trained speaker-
independent TTS model improves the target TTS system on the general text set.
While in some cases, we want to improve the target speaker TTS performance on
a specific new text domain. We assume that the speaker-independent model pre-
trained on the data similar to the new text domain can effectively transfer
the text-based prosody to the target speaker TTS system. To show this idea,
around 9,000 text sentences from the novel books are prepared as the new text
domain, which differs from the text domain of target speaker data. We aim to
select a subset of LibriSpeech as pre-training data whilst achieving
comparable performance for the target speaker TTS system. Two methods are
developed to select data with high text similarity to the new text domain from
the LibriSpeech corpus.

###  4.1 Perplexity-Based Method

In this study, a phoneme-based subword bigram language model is trained on the
new text domain. The text in the LibriSpeech corpus with low perplexities on
the language model is assumed to have a high degree of similarity to the new
text domain. Then, the corresponding pairs data in LibriSpeech will be
selected as pre-training data.

###  4.2 BERT-Based Method

BERT[22] is a method that learns general-purpose text representation, which is
trained on a large amount of open-domain text data. In this study, a pre-
trained BERT model takes the text input to generate token representations for
each sentence. The sentence-level vector is obtained by performing average
pooling on token representations from the last encoder layer of BERT over a
sentence. The new text domain can then be represented by the centroid of the
sentence-level vectors belonging to the new text domain. The degree of
similarity between each text and the new text domain is measured by the L2
distance between the sentence-level vector of that text and the centroid
vector of the new text domain. The data with high text similarities to the new
text domain will be selected as pre-training data.

##  5 Results and Analysis

All the subjective tests are evaluated via our internal crowdsourced listening
test platform, with at least $15$ native judges for Comparison MOS (as CMOS)
[23] and $15$ judges for five-score MOS test for each test case. Readers are
recommended to listen to demo examples 111https://patrick-g-
zhang.github.io/pt-reduction/.

###  5.1 Subjective Evaluations on Domain-Mismatched Text

Table 1 shows the MOS and CMOS comparison between the TTS w/o pre-training and
TTS w/ pre-training on three test sets described in subsection 3.2. The target
speaker data is 24-hour LJSpeech, and the pre-training data is LibriSpeech.
The TTS w/o pre-training achieves the highest MOS score on T-SIM while the
lowest on T-DIFF. This result indicates that the TTS w/o pre-training
performance will drop as the text is different from the text domain of target
speaker data. On T-DIFF and T-RAN, the TTS performance of TTS w/ pre-training
is better than TTS w/o pre-training by a large margin, showing that the pre-
trained speaker-independent model improves the performance of target speaker
TTS on the domain-mismatched text. Moreover, the MOS gap and CMOS between TTS
w/o pre-training and TTS w/ pre-training on T-DIFF are more prominent than
T-RAN, which means the improvement is more significant as the test text is
more different from the text domain of the target speaker data. Concerning
T-SIM, the TTS w/ pre-training shows no performance gain (CMOS $-0.033$) in
contrast to TTS w/o pre-training, which might result from the fact that T-SIM
is similar to the text domain of target speaker data (LJSpeech), thus the
text-based prosody learned from the pre-trained speaker-independent model does
not benefit the TTS w/ pre-training on T-SIM. Since T-RAN is randomly sampled
and can represent the general text set, we can claim the TTS w/ pre-training
has superior performance to the TTS w/o pre-training on the domain-mismatched
text.

Table 1: The MOS and CMOS comparison on three test sets when the target
speaker data is 24-hour LJSpeech.

Set |  TTS w/o pre-training |  TTS w/ pre-training |  CMOS
---|---|---|---
T-SIM |  $\mathbf{3.95\pm 0.06}$ |  $3.93\pm 0.07$ |  $-0.033$
T-DIFF |  $3.67\pm 0.07$ |  $\mathbf{3.79\pm 0.06}$ |  $+0.318$
T-RAN |  $3.88\pm 0.08$ |  $\mathbf{3.98\pm 0.07}$ |  $+0.287$

As shown in Table 2, the CMOS results of two TTS systems, in terms of
pronunciation and prosody, are compared on T-RAN. When comparing the two
systems, the raters only focus on pronunciation or prosody (tone and rhythm)
rather than the overall impression of the speech. Compared with the baseline
TTS w/o pre-training system, the TTS w/ pre-training mainly improves prosodic
variation of speech instead of pronunciation. This result agrees with our
assumption that the main reason for TTS w/ pre-training achieves improvement
on the domain-mismatched text is that the text-based prosody learnt from pre-
training data can be transferred to the target speaker TTS system.

Table 2: The pronunciation and prosody CMOS comparison on T-RAN.

Focus setting |  CMOS
---|---
pronunciation |  $-0.041$
tone and rhythm |  $+0.157$

Two TTS systems are also compared when we use a 1.5-hour subset of the
LJSpeech corpus for fine-tuning, shown in Table 3. On all three test sets, the
TTS w/ pre-training performs much better than the TTS w/o pre-training.
Although 1.5 hours of speech data is sufficient for TTS system to produce
intelligible speech[7], the speech generated from system TTS w/o pre-training
still sounds unstable and jittery due to model overfitting. The improvement
tendency among the three subsets is the same as that in Table 1, where the
improvement is the most notable on T-DIFF and the least marked on T-SIM. The
TTS w/ pre-training performs better than TTS w/o pre-training on T-SIM in this
case, different from the corresponding result in Table 1 where target speaker
data is over 20 hours. After listening to the audios, we found the improvement
on T-SIM might result from that TTS w/ pre-training has more stable voice,
which means the TTS w/ pre-training mitigates the overfitting problem when the
target speaker data is limited.

Table 3: The MOS and CMOS comparison on three test sets when the target
speaker data is 1.5-hour LJSpeech.

Set |  TTS w/o pre-training |  TTS w/ pre-training |  CMOS
---|---|---|---
T-SIM |  $3.72\pm 0.07$ |  $\mathbf{3.81\pm 0.06}$ |  $+0.394$
T-DIFF |  $3.34\pm 0.08$ |  $\mathbf{3.65\pm 0.07}$ |  $+0.958$
T-RAN |  $3.66\pm 0.08$ |  $\mathbf{3.84\pm 0.07}$ |  $+0.591$

###  5.2 TTS Performance Evaluation on Pre-training Data Reduction for a New
Text Domain

For pre-training data reduction task, $60$ sentences are randomly sampled from
the new text domain for subjective evaluation. Four target speaker TTS systems
are evaluated, which is shown in Table 4. All four target speaker TTS systems
are fine-tuned from pre-trained speaker-independent TTS models. The only
difference between those systems is the pre-training data: i) Random: $40,000$
pairs data randomly sampled from the LibriSpeech, which is used as the
baseline system; ii) Full: LibriSpeech, which serves as the topline system;
iii) Perplexity-based: $40,000$ pairs data sampled from LibriSpeech with the
method described in subsection 4.1; iv) BERT-based: 40,000 pair data selected
from LibriSpeech using the method described in subsection 4.2.

Table 4: The TTS performance comparison among four target speaker TTS systems.

Pre-training data |  MOS |  CMOS1 |  CMOS2
---|---|---|---
Random |  $3.84\pm 0.08$ |  $0.0$ |  -
Full |  $\mathbf{3.94\pm 0.07}$ |  $+0.304$ |  $0.0$
Perplexity-based |  $3.90\pm 0.08$ |  $+0.246$ |  $-0.063$
BERT-based |  $\mathbf{3.91\pm 0.08}$ |  $+0.26$ |  $-0.035$

We conduct CMOS tests to compare the TTS performance between the two target
speaker TTS model with pre-training data selected by proposed methods and
target speaker TTS model with Random pre-training data, shown in column CMOS1
in Table 4. It shows that the target speaker TTS systems with both BERT-based
and Perplexity-based pre-training data can significantly improve TTS
performance on the new text domain compared with the baseline system (target
speaker TTS with Random pre-training data). This confirms the effectiveness of
the proposed pre-training data reduction approaches. The CMOS tests are also
performed to compare the systems with the topline system, as shown in column
CMOS2 in Table 4. It shows that target speaker TTS systems with both BERT-
based and Perplexity-based pre-training data are comparable to the topline
system.

##  6 Conclusion

In this study, we show that a pre-trained speaker-independent TTS model can
improve the performance of the target speaker TTS model on domain-mismatched
text compared with the TTS model trained only on the target speaker data. The
subjective evaluation shows that the improvement is mainly on the prosody side
instead of pronunciation with the help of the abundant text domains in pre-
training. In order to improve the data efficiency of pre-training, two methods
are applied to reduce the pre-training data of the speaker-independent TTS
model for a new text domain. The target speaker TTS system with a selected
subset of LibriSpeech as pre-training data can perform comparably to the
target speaker TTS system with complete LibriSpeech as pre-training data,
which significantly reduces the huge cost in pre-training data preparation.

## References

  * [1] Xu Tan, Tao Qin, Frank Soong, and Tie-Yan Liu,  “A survey on neural speech synthesis,”  arXiv preprint arXiv:2106.15561, 2021.
  * [2] Yuxuan Wang, RJ Skerry-Ryan, Daisy Stanton, Yonghui Wu, Ron J Weiss, Navdeep Jaitly, Zongheng Yang, Ying Xiao, Zhifeng Chen, Samy Bengio, et al.,  “Tacotron: Towards end-to-end speech synthesis,”  in Proc. Interspeech, Aug. 2017, pp. 4006–4010.
  * [3] J. Shen, R. Pang, R. J. Weiss, M. Schuster, N. Jaitly, Z. Yang, Z. Chen, Y. Zhang, Y. Wang, R. Skerrv-Ryan, R. A. Saurous, Y. Agiomvrgiannakis, and Y. Wu,  “Natural tts synthesis by conditioning wavenet on mel spectrogram predictions,”  in Proc. ICASSP, 2018, pp. 4779–4783.
  * [4] Isaac Elias, Heiga Zen, Jonathan Shen, Yu Zhang, Ye Jia, Ron J Weiss, and Yonghui Wu,  “Parallel tacotron: Non-autoregressive and controllable tts,”  in Proc. ICASSP, 2021, pp. 5709–5713.
  * [5] Yi Ren, Chenxu Hu, Xu Tan, Tao Qin, Sheng Zhao, Zhou Zhao, and Tie-Yan Liu,  “Fastspeech 2: Fast and high-quality end-to-end text to speech,”  in International Conference on Learning Representations, 2020.
  * [6] Mutian He, Yan Deng, and Lei He,  “Robust Sequence-to-Sequence Acoustic Modeling with Stepwise Monotonic Attention for Neural TTS,”  in Proc. Interspeech, 2019, pp. 1293–1297.
  * [7] Yu-An Chung, Yuxuan Wang, Wei-Ning Hsu, Yu Zhang, and RJ Skerry-Ryan,  “Semi-supervised training for improving data efficiency in end-to-end speech synthesis,”  in Proc. ICASSP, 2019, pp. 6940–6944.
  * [8] Sercan Ö Arık, Jitong Chen, Kainan Peng, Wei Ping, and Yanqi Zhou,  “Neural voice cloning with a few samples,”  in Proc. NIPS, 2018, pp. 10040–10050.
  * [9] Yutian Chen, Yannis Assael, Brendan Shillingford, David Budden, Scott Reed, Heiga Zen, Quan Wang, Luis C Cobo, Andrew Trask, Ben Laurie, et al.,  “Sample efficient adaptive text-to-speech,”  in Proc. ICLR, 2018.
  * [10] Erica Cooper, Cheng-I Lai, Yusuke Yasuda, Fuming Fang, Xin Wang, Nanxin Chen, and Junichi Yamagishi,  “Zero-shot multi-speaker text-to-speech with state-of-the-art neural speaker embeddings,”  in Proc. ICASSP. IEEE, 2020, pp. 6184–6188.
  * [11] Daxin Tan, Hingpang Huang, Guangyan Zhang, and Tan Lee,  “Cuhk-ee voice cloning system for icassp 2021 m2voc challenge,”  arXiv preprint arXiv:2103.04699, 2021.
  * [12] Paul Alexander. Taylor,  Text-to-speech synthesis,  Cambridge University Press, 2009.
  * [13] Katsura Aoyama and Susan G Guion,  “Prosody in second language acquisition,”  Language experience in second language speech learning: In honor of James Emil Flege, vol. 17, pp. 281, 2007.
  * [14] Ye Jia, Yu Zhang, Ron Weiss, Quan Wang, Jonathan Shen, Fei Ren, Patrick Nguyen, Ruoming Pang, Ignacio Lopez Moreno, Yonghui Wu, et al.,  “Transfer learning from speaker verification to multispeaker text-to-speech synthesis,”  Proc. NIPS, vol. 31, 2018.
  * [15] Junichi Yamagishi, Takashi Nose, Heiga Zen, Zhen-Hua Ling, Tomoki Toda, Keiichi Tokuda, Simon King, and Steve Renals,  “Robust speaker-adaptive hmm-based text-to-speech synthesis,”  IEEE Transactions on Audio, Speech, and Language Processing, vol. 17, no. 6, pp. 1208–1230, 2009.
  * [16] Shan Yang, Zhizheng Wu, and Lei Xie,  “On the training of dnn-based average voice model for speech synthesis,”  in Proc. APSIPA. IEEE, 2016, pp. 1–6.
  * [17] Min Chu, Chun Li, Hu Peng, and Eric Chang,  “Domain adaptation for tts systems,”  in Proc. ICASSP, 2002, vol. 1, pp. I–453.
  * [18] Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur,  “Librispeech: an asr corpus based on public domain audio books,”  in Proc. ICASSP, 2015, pp. 5206–5210.
  * [19] Heiga Zen, Viet Dang, Rob Clark, Yu Zhang, Ron J Weiss, Ye Jia, Zhifeng Chen, and Yonghui Wu,  “Libritts: A corpus derived from librispeech for text-to-speech,”  in Proc. Interspeech, 2019, pp. 1526–1530.
  * [20] Kundan Kumar, Rithesh Kumar, Thibault de Boissiere, Lucas Gestin, Wei Zhen Teoh, Jose Sotelo, Alexandre de Brébisson, Yoshua Bengio, and Aaron C Courville,  “Melgan: Generative adversarial networks for conditional waveform synthesis,”  Proc. NIPS, vol. 32, 2019.
  * [21] Rico Sennrich, Barry Haddow, and Alexandra Birch,  “Neural machine translation of rare words with subword units,”  in Proc. ACL, 2016, pp. 1715–1725.
  * [22] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova,  “Bert: Pre-training of deep bidirectional transformers for language understanding,”  in Proc. NAACL, 2019, pp. 4171–4186.
  * [23] Philipos C Loizou,  “Speech quality assessment,”  in Multimedia analysis, processing and communications, pp. 623–654. Springer, 2011.


11institutetext:  University of Crete, Department of Physics & Institute of
Theoretical & Computational Physics, 70013 Heraklion, Greece
22institutetext: Institute of Astrophysics, Foundation for Research and
Technology-Hellas, 71110 Heraklion, Crete, Greece
33institutetext: Max-Planck-Institut für extraterrestrische Physik, Postfach
1312, 85741 Garching, Germany

#  Cyclotron line formation by reflection on the surface of a magnetic neutron
star

N. D. Kylafis J. E. Trümper 112233 N. A. Loudas 1122

(Received ; accepted )

###### Abstract

Context.  Accretion onto magnetic neutron stars results in X-ray spectra that
often exhibit a cyclotron resonance scattering feature (CRSF) and, sometimes,
higher harmonics of it. Two places are suspect for the formation of a CRSF:
the surface of the neutron star and the radiative shock in the accretion
column.

Aims.  Here we explore the first possibility: reflection at the neutron-star
surface of the continuum produced at the radiative shock. It has been proposed
that for high-luminosity sources, as the luminosity increases, the height of
the radiative shock increases, thus a larger polar area is illuminated, and as
a consequence the energy of the CRSF decreases because the dipole magnetic
field decreases by a factor of two from the pole to the equator. This model
has been specifically proposed to explain the observed anticorrelation of the
cyclotron line energy and luminosity of the high-luminosity source V 0332+53.

Methods.  We used a Monte Carlo code to compute the reflected spectrum from
the atmosphere of a magnetic neutron star, when the incident spectrum is a
power-law one. We restricted ourselves to cyclotron energies $\ll m_{e}c^{2}$
and used polarization-dependent scattering cross sections, allowing for
polarization mode change.

Results.  As expected, a prominent CRSF is produced in the reflected spectra
if the incident photons are in a pencil beam, which hits the neutron-star
surface at a point with a well-defined magnetic field strength. However, the
incident beam from the radiative shock has a finite width and thus various
magnetic field strengths are sampled. As a result of overlap, the reflected
spectra have a CRSF, which is close to that produced at the magnetic pole,
independent of the height of the radiative shock.

Conclusions.  Reflection at the surface of a magnetic neutron star cannot
explain the observed decrease in the CRSF energy with luminosity in the high-
luminosity X-ray pulsar V 0332+53. In addition, it produces absorption lines
much shallower than the observed ones.

###### Key Words.:

accretion – pulsars: general – stars: magnetars – stars: magnetic field –
stars: neutron – X-rays: stars

##  1 Introduction

Cyclotron lines observed in the X-ray spectra of accreting neutron stars
provide direct information for the magnetic field strength in these compact
objects. The first cyclotron line was discovered in Hercules X-1 (Trümper et
al. 1977, 1978). Today we know about 36 objects showing electron cyclotron
lines, sometimes with harmonics (Staubert et al. 2019). They are sometimes
called cyclotron resonance scattering features (CRSFs).

Despite the fact that a large body of observational data has been collected
from these sources, it is still not clear where the CRSFs are produced. It has
been generally assumed that the cyclotron lines are produced at the radiative
shock (Basko & Sunyaev 1976) in the accretion column above the surface of a
magnetic neutron star, but no calculation has been performed so far for the
formation of a cyclotron line in a radiative shock. Typical calculations
involve CRSF formation in a slab illuminated from one side (Ventura et al.
1979; Nagel 1981; Nishimura 2008; Araya & Harding 1999, 2000; Schoenherr et
al. 2007). A slab, however, is significantly different from a radiative shock
in an accretion column, on one side of which there is supersonic, free-falling
matter and on the other subsonic, thermal plasma. In such a situation, both
the power-law continuum and the cyclotron line are produced by a first order
Fermi process at the shock. In other words, bremsstrahlung photons from below
the shock criss-cross the shock several times and, as a result, the in-falling
matter is slowed down and a power-law hard X-ray spectrum is produced. This
has been demonstrated by Kylafis et al. (2014), but no resonant scattering
cross section was taken into account there. A calculation with resonant
scattering is currently underway.

For five of the 36 cyclotron line sources, significant correlations of the
line energy with X-ray luminosity have been found by long-term observations
with different X-ray satellites. The line energy $E_{c}$ is positively
correlated with X-ray luminosity $L_{x}$, when the luminosity is relatively
low ($L_{x}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}10^{37}$ erg s-1). The following four
sources of this type are known: Hercules X-1 (Staubert et al. 2007; Klochkov
et al. 2011), GX 304-1 (Klochkov et al. 2012; Malacaria et al. 2015;
Rothschild et al. 2017), A 0535+26 (Klochkov et al. 2011; Sartore et al.
2015), and Vela X-1 (Fürst et al. 2014; La Parola et al. 2016). The only
object which has been found to show a secure negative correlation is V 0332+53
(Mowlavi et al. 2006; Tsygankov et al. 2010; Vybornov et al. 2017), which has
a rather large luminosity (up to $1.5\times 10^{38}$ erg s-1). This source
shows a transition to a positive correlation at a luminosity of $\sim 10^{37}$
erg s-1 (Doroshenko et al. 2017; Vybornov et al. 2018).

For an explanation of these correlations the work of Basko & Sunyaev (1976)
has been instrumental. It predicts that below a critical luminosity $L_{x}\sim
10^{37}$ erg s-1, the infalling matter in the accretion column is stopped
close to the neutron-star surface by a radiative shock; whereas for larger
luminosities ($L_{x}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
5.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}10^{37}$ erg s-1),
the radiative shock rises to a height $H$, which is proportional to
luminosity.

Several explanations have been put forward for the positive correlation (c.f.
Mushtukov et al. 2015 or Mukherjee & Bhattacharya 2012). The negative
correlation can be explained qualitatively by the decrease in the magnetic
field, if, in the high luminosity case, the radiative shock rises with
luminosity. However, it has been pointed out by Poutanen et al. (2013) that in
a dipole field, the predicted rate of change of $B$ (and $E_{c}$ ) with $H$
(and $L_{x}$) is much larger than the observed one. As a remedy, the authors
propose that the observed cyclotron line is not produced in the accretion
shock, but from reflection of its X-ray beam in the atmosphere of the polar
cap. Since with increasing $H$ the illuminated polar cap area is increased and
the strength of the dipole field decreases (up to a factor 2) with distance
from the magnetic pole, a modest decrease in the cyclotron line energy with
luminosity is expected. In the following, we check on this by means of Monte
Carlo simulations. In § 2 we present our model, in § 3 we describe briefly our
Monte Carlo code, in § 4 we show the results of our calculations, in § 5 we
compare our results with the observations of V 0332+53 by Doroshenko et al.
(2017), which cover a wide range of luminosities, and in § 6 we comment on our
results and draw our conclusions.

##  2 The model

With the aid of a Monte Carlo code, we study the reflection of continuum X-ray
photons at the atmosphere of a magnetic neutron star. The photons are emitted
at the radiative shock, they have a power-law distribution of energies, and
one can think of them as being emitted at height $H$ above the magnetic pole
in the form of a wide beam. The larger $H$ is, the larger the polar area on
the neutron star that gets illuminated, and the larger the variation of the
magnetic field strength in this area.

It is generally assumed that the beam that emerges from the radiative shock is
directed mainly toward the stellar surface, due to relativistic beaming
(Kaminker et al. 1976; Mitrofanov & Tsygan 1978; Poutanen et al. 2013). This,
however, is not correct, because it assumes that the escaping photons from the
shock have had their last scattering with a freely falling electron, while in
a radiative shock, it is equally likely that the last scattering is due either
to an in-falling electron above the shock or to a thermal electron below the
shock. It is well accepted observationally that a fan beam is emitted at the
shock, but its shape is unknown. For this reason, in our calculations below,
and for purposes of demonstration, we assume beaming functions (Trümper et al.
2013) of the form

$dN/d\theta^{\prime}\propto\sin^{m}\theta^{\prime},$ $None$

where $\theta^{\prime}$ is the polar angle of the direction of the photon (see
Fig. 1), $m=1$ for an isotropic distribution, and $m=2$ or $3$ for strong
beaming. The polar angle $\theta^{\prime}$ and the angle $\alpha$ between the
photon momentum and the infalling electron velocity are related by
$\theta^{\prime}+\alpha=\pi$. We have also examined the downward beam
distribution of Lyubarskii & Sunyaev (1988) and have found that our
conclusions are unaffected. It is easy to verify that, for angles
$\theta^{\prime}$ such that the photons encounter the neutron-star surface,
the downward beam distribution of Lyubarskii & Sunyaev (1988) falls between
the $m=1$ and $m=3$ beaming functions.

Figure 1:  Schematic of a photon emitted at the radiative shock, at height $H$
above the surface of a neutron star of radius $R$, and hitting the surface.
The polar angle of the photon is $\theta^{\prime}$ and the corresponding one
at the point of incidence is $\theta$. The relation between $\theta$ and
$\theta^{\prime}$ is derived in the Appendix.

We assume a dipole magnetic field $\@vec{B}$ with its dipole moment
$\@vec{m}=m\hat{k}$ along the $z$ axis. The vector $\@vec{B}$ on the surface
of the neutron star of radius $R$ is given by

$\@vec{B}=B_{0}\left[3(\hat{m}\cdot\hat{r})\hat{r}-\hat{m}\right],$ $None$

where $B_{0}$ is the magnetic field at the equator of the neutron star and
twice this at its pole, $\hat{r}$ is the radial unit vector, and
$\hat{m}={\@vec{m}}/m=\hat{k}$ is the unit vector along $\@vec{m}$. For a
point in the $xz$ plane, on the surface of the neutron star, we write
$\hat{r}=\cos\theta\hat{k}+\sin\theta\hat{i}$, where $\theta$ is the polar
angle, that is the angle between the radial vector and the $z$ axis, and eq.
(2) becomes

$\@vec{B}(\theta)=B_{0}\left[(3\cos^{2}\theta-1)\hat{k}+3\cos\theta\sin\theta\leavevmode\nobreak\
\hat{i}\right],$ $None$

The strength of the magnetic field on the surface of the neutron star at polar
angle $\theta$ is

$B=B_{0}(1+3\cos^{2}\theta)^{1/2}.$ $None$

The possibility of an off-center dipole exists, but it is beyond the present
paper to investigate it. We have restricted ourselves to a centered dipole, as
Poutanen et al. (2013) did.

Consider a photon emitted at the shock, at height $H$ above the neutron-star
surface, with direction
$\hat{n}=\cos\theta^{\prime}\hat{k}+\sin\theta^{\prime}\hat{i}$, in the $xz$
plane. For angles $\theta^{\prime}>\pi/2$, such that
$\sin\theta^{\prime}<R/(R+H)$, the photon will hit the neutron-star surface at
a point $(x,z)$, which is the intersection of the straight line path of the
photon and the circle $x^{2}+z^{2}=R^{2}$. The polar angle $\theta$ of this
point $(x,z)$, which will give the local magnetic field strength $B(\theta)$
(see eq. 4), is determined from either $\cos\theta=z/R$, $z>0$ or
$\sin\theta=x/R$, $x>0$ (see Appendix). We do not take into account
gravitational bending, for two reasons. First, it is negligible. Using eqs.
(B8) and (B6) of Poutanen et al. (2013) and our Appendix, we have found that
for $H=10$ km and $m=1$ the magnetic field strength at the circumference of
the illuminated polar cap is $B=0.69B_{0}$ for the Newtonian case and
$B=0.62B_{0}$ when gravitational bending is taken into account. A $\sim 10\%$
lower magnetic field has no effect in our conclusions. Second, gravitational
bending brings the emitted photons at the radiative shock closer to the
magnetic pole, and thus strengthens our main conclusion.

If $\hat{n}=u\hat{i}+v\hat{j}+w\hat{k}$ is the direction of a photon, then the
angle of incidence with respect to the local magnetic field
$\@vec{B}(\theta)$, at the first or any subsequent scattering, is determined
from

$\cos\chi=\hat{n}\cdot{{\@vec{B}(\theta)}\over{B(\theta)}}.$ $None$

Here it is tacitly assumed that the region on the neutron-star surface sampled
by the photon before it escapes is very small, i.e., the mean free path is
much less than $R$, and $\@vec{B}$ is constant in this region.

For a proper treatment of the radiative transfer, one needs to consider the
complete, Quantum Electrodynamic, Compton cross sections (Herold 1979;
Daugherty & Harding 1986; Harding & Daugherty 1991; Sina 1996; Gonthier et al.
(2014); Mushtukov et al. 2016; Schwarm 2017). However, these cross sections
are quite cumbersome, because they contain infinite sums, and are not very
practical. A significant simplification of the above cross sections (though
still intimidating) was derived by Nobili, Turolla, & Zane (2008), in order to
treat resonant scattering of photons with energy approaching $m_{e}c^{2}$, in
a magnetic field comparable to the critical one
$B_{cr}=m_{e}^{2}c^{3}/e\hbar$. These cross sections were computed exactly at
resonance, i.e., the Lorentz profile was approximated by a delta function.

Since most of the cyclotron lines that have been observed so far (Staubert et
al. 2019) are at cyclotron energies $<<m_{e}c^{2}$, we restrict ourselves to
resonant scattering of photons with energy $E<<m_{e}c^{2}$ in a magnetic field
of strength $B<<B_{cr}$. For such a case, we showed analytically and
numerically (Loudas, Kylafis, & Truemper 2021) that the modified classical
cross sections given below in eq. (6) provide accurate representations of the
fully relativistic ones.

For a cyclotron energy $E_{c}(\theta)=\hbar\omega_{c}(\theta)=\hbar
eB(\theta)/m_{e}c\ll m_{e}c^{2}$, the modified classical polarization-
dependent resonant cross sections are given by (Loudas et al. 2021)

${{d\sigma_{11}}\over{d\Omega^{\prime}}}\approx{{3\pi r_{0}c}\over
8}\left(\cos^{2}\chi\cos^{2}\chi^{\prime}\leavevmode\nobreak\ L_{-}+{1\over
2}{B\over{B_{cr}}}\leavevmode\nobreak\ L_{+}\right),$ $None$
${{d\sigma_{12}}\over{d\Omega^{\prime}}}\approx{{3\pi r_{0}c}\over
8}\left(\cos^{2}\chi\leavevmode\nobreak\ L_{-}+{1\over
2}{B\over{B_{cr}}}\cos^{2}\chi^{\prime}\leavevmode\nobreak\ L_{+}\right),$
$None$ ${{d\sigma_{21}}\over{d\Omega^{\prime}}}\approx{{3\pi r_{0}c}\over
8}\left(\cos^{2}\chi^{\prime}\leavevmode\nobreak\ L_{-}+{1\over
2}{B\over{B_{cr}}}\cos^{2}\chi\leavevmode\nobreak\ L_{+}\right),$ $None$
${{d\sigma_{22}}\over{d\Omega^{\prime}}}\approx{{3\pi r_{0}c}\over
8}\left(L_{-}+{1\over
2}{B\over{B_{cr}}}\cos^{2}\chi\cos^{2}\chi^{\prime}\leavevmode\nobreak\
L_{+}\right),$ $None$

where the index 1 (2) stands for the ordinary (extraordinary) mode, $\chi$ and
$\chi^{\prime}$ are the incident and scattered angles, respectively, with
respect to the local magnetic field, $r_{0}$ is the classical electron radius,
and $c$ is the speed of light. The quantities $L_{-}$ and $L_{+}$ are the
normalized Lorentz profiles given by

$L_{\pm}={{\Gamma_{\pm}/2\pi}\over{(\omega-\omega_{r})^{2}+(\Gamma_{\pm}/2)^{2}}},$
$None$

where $\omega$ and $\omega_{r}$ are the photon frequency and the resonant
frequency, respectively, in units of $m_{e}c^{2}/\hbar$. The decay widths
$\Gamma_{\pm}$ are given by (Herold, Ruder, & Wunner 1982)

$\Gamma_{-}={4\over
3}{{m_{e}c^{2}}\over\hbar}\alpha{{B^{2}}\over{B_{cr}^{2}}},$ $None$
$\Gamma_{+}={2\over
3}{{m_{e}c^{2}}\over\hbar}\alpha{{B^{3}}\over{B_{cr}^{3}}},$ $None$

where $\alpha$ is the fine structure constant. We note that $\Gamma_{-}$ is
equal to the classical $\Gamma=(4e^{2}\omega_{c}^{2})/(3m_{e}c^{3})$, which
accounts for the finite transition life-time of the excited state (e.g.,
Daugherty & Ventura 1978; Ventura 1979).

The first terms in eqs. (6a) - (6d) are the classical (Thomson) differential
cross sections. The second terms, which are proportional to $B/B_{cr}$, are
purely quantum mechanical and are due to spin flip. It is amazing that just
these terms are enough to convert the classical cross sections into accurate
approximations of the fully relativistic quantum-mechanical ones for $B\ll
B_{cr}$.

For the resonant frequency $\omega_{r}$ and the new photon energy in the rest
frame of the electron after scattering, we use eqs. (8) and (6), respectively,
of Nobili et al. (2008).

The polarization averaged differential cross section is given by

${{d\sigma}\over{d\Omega^{\prime}}}\approx{{3\pi
r_{0}c}\over{16}}(1+\cos^{2}\chi)(1+\cos^{2}\chi^{\prime})\left(L_{-}+{1\over
2}{B\over{B_{cr}}}L_{+}\right).$ $None$

For comparison, we have performed a calculation with this polarization
averaged differential cross section.

The black-body temperature at the surface of the neutron star is typically
$\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}1$ keV, which is much smaller that the
cyclotron energy $E_{c}=25$ keV at the pole, that we have considered, and one
could consider the electrons at the surface of the neutron star as cold.
However, the temperature of the electrons in the atmosphere of the neutron
star can be up to an order of magnitude larger than the blackbody temperature
(Zampieri et al. 1995). Thus, we have taken into account the temperature of
the electrons. A one-dimensional (along the local $\@vec{B}$) nonrelativistic
Maxwellian is fine for our purposes.

Since the heavy elements on the surface of the neutron star will not be
completely ionized, for the absorption cross section, we use the hydrogenic
approximation (Bethe & Salpeter 1957)

$\sigma_{a}={{6.3\times 10^{-18}}\over Z^{2}}(I_{Z}/E)^{3}\leavevmode\nobreak\
{\rm cm}^{2},$ $None$

where $I_{Z}=13.6Z^{2}$ eV is the ionization potential and we assume that the
predominant element is Carbon ($Z=6$). We remark that our conclusions are not
affected at all if there are no heavy elements on the surface of the neutron
star.

##  3 The Monte Carlo code

Our Monte Carlo code is similar to previously used codes in our work and it is
based on Cashwell & Everett (1959) and Pozdnyakov et al. (1983). We emit
photons toward the neutron star from height $H$, with a polar angle
$\theta^{\prime}$, either in the form of a delta-function distribution or with
continuous distributions given by eq. (1) or that of Lyubarskii & Sunyaev
(1988). The directions of emission are restricted between
$\theta^{\prime}=\arcsin[R/(R+H)]$ (tangent) and $\theta^{\prime}=\pi$ (pole).

We consider a uniform atmosphere at the surface of the neutron star, with the
number density of the electrons ten times larger than the corresponding one
for absorbers. This ratio of densities was selected so that absorption and
scattering have comparable mean free paths at 15 keV. Below this energy,
absorption dominates, while above it scattering dominates.

In our calculations, we consider a value for $B_{0}$, such that $E_{c}=12.5$
keV $=11.6(B_{0}/1\times 10^{12})$ keV at the equator and 25 keV at the pole.
The variation of the strength of the magnetic field with polar angle $\theta$
is given by eq. (4). Our input spectra have the form of a power law

${{dN}\over{dE}}=\left({{E}\over{E_{0}}}\right)^{-\alpha},$ $None$

where $E_{0}=1$ keV is a reference energy and $\alpha$ is the power-law index.
For the Monte Carlo runs, we have used a minimum of $10^{7}$ photons. For high
resolution (e.g., Fig. 7), we have used $10^{9}$ photons.

##  4 Results

As a test of our Monte Carlo code, and in order to see clearly the reflected
spectrum, we first use a flat power-law spectrum as input, i.e., $\alpha=0$.
For the photons to see a specific magnetic field strength, we send all of them
to the pole, i.e., $\theta^{\prime}=\pi$. Equally well, we could have sent
them to the tangent ($\theta^{\prime}=\arcsin[R/(R+H)]$) or any intermediate
direction. The temperature of the electrons is taken equal to zero, as a
benchmark. In Fig. 2, we show our results. The black dashed line represents
the input spectrum and the black crosses give the reflected spectrum. However,
the observed spectra consist of two components: the photons that are emitted
at the radiative shock and come directly to the observer and those that come
to the observer after being reflected at the neutron-star surface. The
relative strength of these two intensities is difficult to compute, because it
depends on two angles, both unknown: the inclination of the spin axis of the
neutron star and the angle between the spin and the magnetic axes. For most
combinations of these two angles, the direct spectrum is more than 50% of the
spectrum of the photons emitted at the radiative shock. Also, the larger the
height $H$ of the radiative shock, the larger the direct component. Therefore,
in Fig. 2, we show mixtures of 50% direct and 50% reflected (blue stars) and
70% direct and 30% reflected (red pluses) as representative examples.

Figure 2:  Reflected spectrum $dN/dE$ in all directions (black crosses) in
arbitrary units, as a function of energy $E$ in keV. The input spectrum is
shown as a black dashed line. The other two lines show mixed spectra: 50%
direct and 50% reflected (line with blue stars) and 70% direct and 30%
reflected (line with red pluses).

The reflected spectrum in Fig. 2 (black crosses) has the expected
characteristics. First, the input photons near resonance have a small mean
free path in the neutron-star atmosphere and, as a result, a good fraction of
them get reflected. However, with every scattering these photons lose energy
due to electron recoil. Thus, the reflected photons appear at energies below
the resonance. Second, the resonant input photons that do not get reflected,
but instead go deeper into the neutron-star atmosphere, face a large optical
depth for escape and thus they escape only in the line wings. Similarly,
photons with energy somewhat larger than $E_{c}$ that go deep in the
atmosphere, loose energy with every scattering and can escape, if they are not
absorbed, only in the line wings. Thus, an absorption feature is generated at
the resonance. However, since the atmosphere of the neutron star near
resonance is an essentially pure scattering one, the number of photons is
conserved. The photons that are missing in an absorption feature, must appear
as an emission feature at lower energies, due to electron recoil.

At low energies, the reflected spectrum is significantly reduced due to
absorption (see eq. 9). At high energies, the reduction of the reflected
spectrum is due to down-scattering, followed by absorption.

Furthermore, we see in Fig. 2 that, despite the fact that the reflected
spectrum has a sharp and prominent absorption feature, this feature is not so
prominent when the reflected spectra are mixed with nonreflected ones, i.e.,
with input spectra that reach the observer directly. Naturally, the larger the
direct component in the mixture, the shallower the absorption features and
vice versa.

We repeated the calculation of Fig. 2, but with the polarization averaged
differential cross section of eq. (8), instead of the mode-dependent ones of
eqs. (6a - 6d). The results are indistinguishable from those of Fig. 2.

In Fig. 3, we show the results of the same calculation, but with the thermal
motion of the electrons taken into account. We consider values of $kT_{e}$
equal to 1 keV (line with black stars), 3 keV (line with blue circles), and 10
keV (line with red diamonds). The input spectrum is shown as a black dashed
line.

Figure 3:  Reflected spectrum $dN/dE$ in all directions in arbitrary units, as
a function of energy $E$ in keV for $kT_{e}=1$ keV (line with black stars),
$kT_{e}=3$ keV (line with blue circles), and $kT_{e}=10$ keV (line with red
diamonds). The input spectrum is shown as a black dashed line.

Contrary to our intuition regarding the thermal broadening of an absorption
line, where the Lorentz profile is blue-shifted and red-shifted by the thermal
motion of the absorbers, here the thermal motion of the electrons has a small
and rather undetectable effect on the CRSF. The explanation is as follows.

Independent of the temperature of the electrons (assuming
$kT_{e}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}10$ keV, as it is thought appropriate
for a neutron star atmosphere), the discussion that we have given above for
the formation of the absorption feature still holds. Photons at resonance
cannot escape from deep in the atmosphere unless they change their energy and
escape in the line wings. If the electrons are cold, resonance photons lose
energy with every scattering and escape in the red wing. However, if the
electrons are hot, they can give energy to the photons, which then escape in
the blue wing. The hotter the electrons, the larger the blue wing. This is
exactly what is seen in Fig. 3.

In the subsequent calculations and for concreteness, we take $kT_{e}=1$ keV,
though the spectra are indistinguishable from those with $kT_{e}=0$. We stress
that our conclusions are independent of the temperature of the electrons.

In Fig. 4, we show the dependence of the reflected spectrum on the direction
of escape. The incident photons are again in the form of a pencil beam toward
the pole. The line with the black circles gives the spectrum of the reflected
photons that escape with angle $\chi^{\prime}$ with respect to the local
$\@vec{B}=2B_{0}\hat{k}$, such that $0\leq\cos\chi^{\prime}<0.33$. The line
with the blue diamonds corresponds to $0.33\leq\cos\chi^{\prime}<0.66$ and the
line with the red triangles corresponds to $0.66\leq\cos\chi^{\prime}\leq
1.0$. The three reflected spectra are very similar.

Figure 4:  Reflected spectrum $dN/dE$ in three direction bins, as follows:
$0\leq\cos\chi^{\prime}<0.33$ (line with black circles),
$0.33\leq\cos\chi^{\prime}<0.66$ (line with blue diamonds), and
$0.66\leq\cos\chi^{\prime}\leq 1.0$ (line with red triangles). The angle
$\chi^{\prime}$ is measured with respect to $\@vec{B}$ at the point of
incidence.

In Fig. 5, we use as input spectrum a steep power law with index $\alpha=5$,
as suggested by the spectra in Tsygankov et al. (2006), Cusumano et al.
(2016), and Doroshenko et al. (2017). Also, instead of $dN/dE$ we show
$E^{2}\leavevmode\nobreak\ dN/dE$ versus energy. The angle of incidence is
$\theta^{\prime}=\pi$ and the symbols are the same as in Fig. 2. The reflected
spectrum has a prominent absorption feature, but in the mixed spectra (lines
with stars and pluses) it is not as prominent.

Figure 5:  Reflected spectrum in all directions in $E^{2}dN/dE$ form
(crosses), with arbitrary units, as a function of energy $E$ in keV. The input
spectrum is shown as a dashed line. The other two lines show mixed spectra:
50% direct and 50% reflected (stars) and 70% direct and 30% reflected
(pluses). The angle $\theta^{\prime}=\pi$.

In all of the above cases, we have considered a delta-function beam from the
shock to the magnetic pole. However, as we have discussed in sections 2 and 3,
the beam that is emitted at the shock is wide and it samples various magnetic
field strengths on the surface of the neutron star. For a rather large shock
height $H=10$ km and $R=12.5$ km, the beam extends from
$\theta^{\prime}=\arcsin[R/(R+H)]=146.3$ degrees (tangent) to
$\theta^{\prime}=180$ degrees (pole). At $\theta^{\prime}=146.3$ degrees, the
photon hits the neutron-star surface at a polar angle (see Fig. 1)
$\theta=53.4$ degrees (see Appendix). The local magnetic field strength is
$B=1.44B_{0}$, with a corresponding cyclotron energy $E_{c}=17.97$ keV.

In Fig. 6, we show the reflected spectra from a number of delta-function beams
from $\theta^{\prime}=147$ degrees (grazing incidence) to
$\theta^{\prime}=180$ degrees (vertical incidence) for a shock height $H=10$
km. All delta-function beams have the same number of photons and their
spectrum is that of the black dashed line. The red dotted line corresponds to
$\theta^{\prime}=147$ degrees, with the local cyclotron energy $E_{c}=19.63$
keV at the point of incidence. The green dot - dashed line corresponds to
$\theta^{\prime}=149$ degrees (local $E_{c}=21.34$ keV), the blue double-dot -
dashed line corresponds to $\theta^{\prime}=152$ degrees (local $E_{c}=22.59$
keV), the magenta double-dash - dotted line corresponds to
$\theta^{\prime}=155$ degrees (local $E_{c}=23.33$ keV), the black solid line
corresponds to $\theta^{\prime}=160$ degrees (local $E_{c}=24.09$ keV), and
the brown long-dashed line corresponds to $\theta^{\prime}=172$ degrees (local
$E_{c}=24.88$ keV), As expected, the energy of the absorption feature
increases with increasing $\theta^{\prime}$ from 17.97 keV to 25 keV.

Figure 6:  Reflected spectra in all directions in $E^{2}dN/dE$ form, with
arbitrary units, as a function of energy $E$ in keV. For all cases, the input
spectrum is given by the black dashed line. The red dotted line corresponds to
$\theta^{\prime}=147$ degrees, the green dot - dashed line corresponds to
$\theta^{\prime}=149$ degrees, the blue double-dot - dashed line corresponds
to $\theta^{\prime}=152$ degrees, the magenta double-dash - dotted line
corresponds to $\theta^{\prime}=155$ degrees, the black solid line corresponds
to $\theta^{\prime}=160$ degrees, and the brown long-dashed line corresponds
to $\theta^{\prime}=172$ degrees. The corresponding energies of the cyclotron
absorption features are given in the text. The shock height is $H=10$ km.

Two additional things are worth noticing in Fig. 6. 1) The absorption features
are weaker when the photons have a grazing incidence in the atmosphere
($\theta^{\prime}=147$ degrees) than when they have a nearly perpendicular one
($\theta^{\prime}=172$ degrees). This is understood, because at grazing
incidence the photons see a smaller absorption depth for escape than at normal
incidence. 2) The absorption feature at grazing incidence
($\theta^{\prime}=147$ degrees) falls on top of the emission feature at
incidence at $\theta^{\prime}=149$ degrees, and similarly for subsequent
incidence angles. This has the important implication that when a beam falls on
the neutron star, the addition of the reflected spectra, properly weighted
with the angular distribution of the beam, will wash out the imprints of
weaker magnetic fields from the low-magnetic-latitude part of the polar cap.
We have found that this is indeed the case and this is the main point of our
paper.

In Fig. 7, we show the reflected spectrum (red dot-dashed line) from an
isotropic beam ($m=1$ in eq. 1) and similarly (blue solid line) from a
strongly beamed distribution ($m=3$ in eq. 1). In both cases, the input
spectrum is given by the black dashed line. The shock height is $H=5$ km,
which implies that, on the illuminated surface of the neutron star, $E_{c}$
ranges from 20.0 to 25.0 keV. In both cases, the absorption feature from a
broad beam is naturally wider than the corresponding one from a pencil beam
(see Fig. 6). The minima, though, of the absorption features are at 24.7 keV
for the $m=1$ case and at 24.5 keV for the $m=3$ case, which are very close to
the cyclotron energy at the pole. The contribution of the lower magnetic
latitudes is nearly washed out. The strongly beamed distribution ($m=3$)
produces an absorption feature (blue solid line in Fig. 7), which has slightly
lower energy than that of the isotropic distribution (red line in Fig. 7,
$m=1$). This is because, in the $m=3$ case, most of the photons hit the
neutron-star surface at low magnetic latitudes and very few near the pole. We
note that Fig. 7 shows only the reflected spectrum. The observed spectrum,
however, is a combination of direct and reflected spectra, which makes the
absorption features shallower (see Fig. 5).

Figure 7:  Reflected spectra in all directions in $E^{2}dN/dE$ form, with
arbitrary units, as a function of energy $E$ in keV. The red dot-dashed line
corresponds to a beam with $m=1$, while the blue solid line to a beam with
$m=3$. The shock height is $H=5$ km and the input spectrum is given by the
black dashed line.

A shock height $H=5$ km is rather large. We used it only for reasons of
demonstration. In Fig. 8, we show the reflected spectrum of a beam with $m=1$
and a reasonable shock height $H=2$ km. A beam with $m=3$ produces nearly
identical results. The absorption feature occurs at the cyclotron energy
corresponding to the magnetic field at the pole and, as expected, it is
narrower than that of Fig. 7, because a narrower range of magnetic field
strengths is sampled ($22.74\leq E_{c}\leq 25.0$ keV). In the mixed spectra
(stars and pluses), however, the absorption feature is less prominent.

Figure 8:  Reflected spectrum (crosses) in all directions in $E^{2}dN/dE$
form, with arbitrary units, as a function of energy $E$ in keV. The photons
are emitted in a beam with $m=1$. The shock height is $H=2$ km. The other two
lines show mixed spectra: 50% direct and 50% reflected (stars) and 70% direct
and 30% reflected (pluses). The input spectrum is given by the black dashed
line.

##  5 Comparison with observations

With regard to the change of the cyclotron-line energy with luminosity, the
most remarkable X-ray pulsar to date is V 0332+53. It was demonstrated by
Doroshenko et al. (2017) that, in its 2015 and 2016 outbursts, the cyclotron-
line energy exhibited both a correlation (at low luminosities) and an
anticorrelation (at high luminosities) with luminosity. Both, the correlation
and the anticorrelation have been seen in other sources (for a review see
Staubert et al. 2019), but todate this is the only source that exhibits both.

During the decay of the luminosity, the cyclotron-line energy increased from
27.9 keV to 30.4 keV. This is approximately a 9% increase.

To compare with these observations, we have run models with the shock height
at $H=10,8,5,3$, and $1$ km to mimic the decrease in the observed luminosity.
We have also used a beam with $m=3$, which is the most favorable to exhibit a
variation of the cyclotron-line energy with shock height or equivalently
luminosity. In Fig. 9 we show our results.

Figure 9:  Reflected spectra in all directions in $E^{2}dN/dE$ form, with
arbitrary units, as a function of energy $E$ in keV for different shock
heights: $H=10$ km (black solid line), $H=8$ km (blue dotted line), $H=5$ km
(red dashed line), $H=3$ km (magenta dot-dashed line), and $H=1$ km (green
dash - dotted line). The dashed straight line is the input spectrum. The
photons are emitted in a beam with $m=3$. In the inset, we zoom into the
absorption feature, with the horizontal axis linear.

In varying the shock height by an order of magnitude, the cyclotron line
varies from $24.5\pm 0.1$ keV (the uncertainty is due to binning) to 25 keV,
i.e., only about 0.2%. This is practically unobservable.

We remark that had we used a downward beam, most of the photons would hit the
neutron-star surface near the magnetic pole and the variation of the
cyclotron-line energy would be even smaller.

##  6 Discussion and conclusions

We have studied reflection of continuum X-ray photons from the surface of a
magnetic neutron star. With a Monte Carlo code, we have simulated photons
emitted from a radiative shock at height $H$ above the surface of the neutron
star, on the magnetic axis. The photons are either unidirectional or are in
beams with specific distributions: isotropic, fan beam, or downward beam.

When the photons have a specific direction, they impinge in a polar ring on
the neutron-star surface, that has a specific magnetic field. The lower the
magnetic latitude of the ring, the lower the strength of the local dipole
magnetic field. In such a case, the reflected spectrum exhibits a prominent
cyclotron absorption line at the local cyclotron energy. Naturally, if the
photons are emitted in a beam, a range of magnetic fields is sampled and the
absorption line is wider.

If we could see only the reflected spectrum, these absorption features would
be easily detectable. However, the observed X-ray spectra are a mixture of
direct photons from the radiative shock and reflected ones. We have found
that, for reasonable mixtures of direct and reflected photons, which depend on
the inclination angle of the spin axis as well as on the angle between the
magnetic and the spin axes (both unknown), the cyclotron features become much
weaker than the ones observed (see, e.g., Fig. 3 of both Tsygankov et al. 2006
and Doroshenko et al. 2017).

Another more serious finding, that speaks against the idea of cyclotron line
formation at the surface of magnetic neutron stars, is that the reflected
spectra are essentially unaffected by the height of the radiative shock. This
is contrary to intuition, where one would expect an anticorrelation between
the luminosity of the source, and therefore the height of the shock, and the
cyclotron absorption feature (Poutanen et al 2013). The reason for this is the
following.

Consider photons at grazing incidence to the neutron-star surface. These
photons impinge on the furthermost ring of the polar cap, which has the lowest
magnetic field in the polar cap. Reflection of these photons creates a) an
absorption feature at the local cyclotron energy and b) an emission feature
below this. Consider now photons impinging on the adjacent ring, inside the
first one. The strength of the magnetic field here is larger than that of the
first ring. Thus, the emission feature at the local cyclotron energy falls on
top of the absorption feature of the first ring, and the absorption feature is
nearly washed out. The same thing happens with all inner rings. Since, in the
illuminated polar cap, the magnetic field strength varies continuously from
the circumference to the magnetic pole, illumination of the polar cap results
in a broad cyclotron line, like the one shown in Fig. 7. The magnetic field
inferred from such a line corresponds to that of the magnetic pole. We remark
that the distribution of the photons in the beam is practically irrelevant. An
isotropic beam and a strong fan beam give practically the same reflected
spectrum. For this reason, we think that the use of gravitational bending will
not alter the above picture.

Last, but not least, we remark that a shock height $H=10$ km is unreasonably
high. It was used only for the purpose of demonstration. For more reasonable
heights of a few kilometers, the variation of the magnetic field in the polar
cap is small and the spectra exhibit practically a broad absorption feature,
which, in addition, is very shallow if the reflected spectra are mixed with
direct ones. Thus, reflection at the surface of a neutron star is not the
mechanism for cyclotron line formation.

Due to the overlap of the CRSFs at different magnetic-field strengths,
reflection at the surface of a neutron star cannot explain the anticorrelation
of cyclotron line energy and luminosity. Several alternative scenarios have
been mentioned by Becker et al. (2012) Doroshenko et al. (2017), and Staubert
et al. (2019), but a quantitative explanation of the effect is still pending.

###### Acknowledgements.

N.D.K. would like to thank Sterl Phinney for his notes on the classical
magnetic scattering cross sections.

## References

  * (1) Araya, R., & Harding, A. 1999, ApJ, 517, 334
  * (2) Araya-Góchez, R., & Harding, A. 2000, ApJ, 544, 1067
  * (3) Basko, M. M., & Sunyaev, R. A. 1976, MNRAS, 175, 395
  * (4) Becker, P. A., Klochkov, D., Schönherr, G., et al. 2012, A&A, 544, 123
  * (5) Bethe, H. A., & Salpeter, E. E. 1957, Quantum Mechanics of One- and two-electron Systems (New York: Academic Press)
  * (6) Cashwell, E. D, & Everett, C. J. 1959, A Practical Manual on the Monte Carlo Method for Random Walk Problems, Pergamon Press
  * (7) Cusumano, G., La Parola, V., D’Ai, A., et al. 2016, MNRAS, 460, L99
  * (8) Daugherty, J. K., & Harding, A. K. 1986, ApJ, 309, 362
  * (9) Daugherty, J. K., & Ventura, J. 1978, Phys. Rev. D, 18, 1053
  * (10) Doroshenko, V, Tsygankov, S.S., Mushtukov, A. A. et al. 2017, MNRAS, 466, 2143
  * (11) Fürst, F., Pottschmidt, K., Wilms, J., et al. 2014, ApJ, 780, 133
  * (12) Gonthier, P. L., Baring, M. G., Eiles, M. T., Wadiasingh, Z., Taylor, C. A., & Fitch, C. J. 2014, PhysRevD, 90, 043014
  * (13) Harding, A. K., & Daugherty, J. K. 1991, ApJ, 374, 687
  * (14) Herold, H. 1979, Phys. Rev. D, 19, 2868
  * (15) Herold H., Ruder H., & Wunner G., 1982, A&A, 115, 90
  * (16) Kaminker, A. D., Fedorenko, V. N, & Tsygan, A. I. 1976, SvA, 20, 436
  * (17) Klochkov, D., Doroshenko, V., Santangelo, A., et al. 2012, A&A, 542, L28
  * (18) Klochkov, D., Staubert, R., Santangelo, A., et al. 2011, A&A, 532, 126
  * (19) Kylafis, N. D., Trümper, J. E., & Ertan, Ü. 2014, A&A, 562, 62
  * (20) La Parola, V., Cusumano, G., Segreto, A., & D’Ai, A. 2016, MNRAS, 463, 185
  * (21) Loudas, N. A., Kylafis, N. D., & Truemper, J. E. 2021, A&A, in press
  * (22) Lyubarskii, Y. E., & Sunyayev, R. A. 1988, SvAL, 14, 390
  * (23) Malacaria, C., Klochkov, D., Santangelo, A., & Staubert, R. 2015, A&A, 581, 121
  * (24) Mitrofanov, I. G., & Tsygan, A. I. 1978, A&A, 70, 133
  * (25) Mowlavi, N., Kreykenbohm, I., Shaw, S. E., et al. 2006, A&A, 451, 187
  * (26) Mukherjee, D., & Bhattacharya, D. 2012, MNRAS, 420, 720
  * (27) Mushtukov, A. A., Nagirner, D. I., & Poutanen, J. 2016, PhysRevD, 93, 105003
  * (28) Mushtukov, A. A., Tsygankov, S. S., Serber, A. V., Suleimanov, V. F., & Poutanen, J. 2015, MNRAS, 454, 2714
  * (29) Nagel, W. 1981, ApJ, 251, 288
  * (30) Nishimura, O. 2008, ApJ, 672, 1127
  * (31) Nobili, L., Turolla, R., & Zane, S. 2008, MNRAS, 386, 1527
  * (32) Poutanen, J., Mushtukov, A. A., Suleimanov, V. F., et al. 2013, ApJ, 777, 115
  * (33) Pozdnyakov, L. A., Sobol, I. M., & Synyaev, R. A. 1983, ASPRv, 2, 189
  * (34) Rothschild, R. E., Kühnel, M., Pottschmidt, K., et al. 2017, MNRAS, 466, 2752
  * (35) Sartore, N., Jourdain, E., & Roques, J. P. 2015, ApJ, 806, 193
  * (36) Schönherr, G., Wilms, J., Kretschmar, P. et al. 2007, A&A, 472, 353
  * (37) Schwarm, F. -W. 2017, PhD Thesis, Friedrich-Alexander Universität
  * (38) Sina, R. 1996, PhD Thesis, University of Maryland (unpublished)
  * (39) Staubert, R., Shakura, N. I., Postnov, K. et al. 2007, A&A, 465, L25
  * (40) Staubert, R., Trümper, J. E., Kendziorra, E. et al. 2019, A&A, 622, 61
  * (41) Trümper, J. E., Dennerl, K., Kylafis, N. D., et al. 2013, ApJ, 764, 49
  * (42) Trümper, J., Pietsch, W., Reppin, C., et al. 1977, Ann.N.Y.Acad.Sci, 302, 538
  * (43) Trümper, J., Pietsch, W., Reppin, C., et al. 1978, ApJ, 219, L105
  * (44) Tsygankov, S. S., Lutovinov, A. A., Churazov, E. M., & Sunyaev, R. A. 2006, MNRAS, 371, 19
  * (45) Tsygankov, S. S., Lutovinov, A. A., & Serber, A. V. 2010, MNRAS, 401, 1628
  * (46) Ventura, J. 1979, Phys. Rev. D., 19, 1684
  * (47) Ventura, J, Nagel, W., & Meszaros, P. 1979, ApJ, 233, L125
  * (48) Vybornov, V., Doroshenko, V., Staubert, R., & Santangelo, A. 2018, A&A, 610, 88
  * (49) Vybornov, V., Klochkov, D., Gornostaev, M., et al. 2017, A&A, 601, 126
  * (50) Zampieri, L., Turolla, R., Zane, S., & Treves, A. 1995, ApJ, 439, 849

##  Appendix A Geometry

We consider a point source on the magnetic axis $z$, at height $H$, above the
surface of a neutron star. The source emits a pencil beam of photons with
direction

$\hat{n}=\cos\theta^{\prime}\hat{k}+\sin\theta^{\prime}\hat{i}$

in the $xz$ plane, with the polar angle $\theta^{\prime}>\pi/2$ such that
$\sin\theta^{\prime}\leq R/(R+H)$. Under these conditions, the pencil beam
hits the neutron-star surface at a point $(x,z)$, which is determined from the
intersection of the straight line

$z=\cot\theta^{\prime}x+R+H$

and the circumference of the circle

$x^{2}+z^{2}=R^{2}.$

Eliminating $z$ from these two equations we obtain

$(1+\cot^{2}\theta^{\prime})x^{2}+2\cot\theta^{\prime}(R+H)x+(R+H)^{2}-R^{2}=0,$

from which we get

$x={{-b-\sqrt{b^{2}-4ac}}\over{2a}}>0$

and

$z=\sqrt{R^{2}-x^{2}}>0,$

where

$a=1+\cot^{2}\theta^{\prime}>0,$ $b=2\cot\theta^{\prime}(R+H)<0,$

and

$c=(R+H)^{2}-R^{2}.$

The polar angle $\theta$ of the point $(x,z)$ is determined from

$\cos\theta=z/R,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ 0\leq\theta<\pi/2.$