# Automatic detection of solar flares observed at 45 GHz by the POEMAS
telescope

Vanessa Lessa  Adriana Valio

###### Abstract

Every 11 years, the Sun goes through periods of activity, with the occurrence
of many solar flares and mass ejections, both energetic phenomena of magnetic
origin. Due to its effects on Earth, the study of solar activity is of
paramount importance. POEMAS (Polarization of Millimeter Emission of Solar
Activity) is a system of two telescopes, installed at CASLEO (El Leoncito
Astronomical Complex) in Argentina, which monitors the Sun at two millimeter
wavelengths (corresponding frequencies of 45 and 90 GHz). The objective of
this work is to automatically detect solar flares observed by the polarimeter.
First it is necessary to eliminate the background noise, caused mainly by
instrumental problems, from the light curves of millimeter solar emission. The
methodology used to exclude the noise proposed in this work is to use the
tendency of time series. The subtraction of this model from the light curves
provides the input to automate the detection of solar flares using artificial
intelligence techniques. A Neural Network was trained to recognize patterns
and analyze a dataset in order to identify solar flares. Previously, a total
of 30 flares had been visually identified and analyzed in the POEMAS database
between 2011/11/22 and 2013/12/10. The methodology presented here confirmed
87% of these events, moreover the neural network was able to identify at least
9 new events. As the neural network was trained to detect impulsive events
(lasting less than 5 min), long duration bursts were not automatically
detected, nor were they detected visually due to the background noise of the
telescope. Visual inspection of the POEMAS data, when comparing with microwave
data from the RSTN, allowed the identification of an additional 10 long-
duration solar flares at 45 GHz. We discuss some problems encountered and
possible solutions for future work.

###### keywords:

Solar Flares , Millimeter Emission , Pattern Recognition , Neural Network ,
Artificial Intelligence

††journal: Astronomy $\&$ Computing

[first]organization=Center for Radio Astronomy and Astrophysics Mackenzie,
Mackenzie Presbyterian University,addressline=Rua da Consolacao, 930, city=São
Paulo, postcode=01302-907, state=SP, country=Brazil

###### Contents

  1. 1 Introduction
  2. 2 POEMAS Telescope
    1. 2.1 Data Acquisition
    2. 2.2 Data calibration
  3. 3 Neural Network
    1. 3.1 Neural Network Training (NNT)
    2. 3.2 Classification
  4. 4 Results and discussion
    1. 4.1 Events identified previously and confirmed by the Neural Network
    2. 4.2 Events previously reported and not identified by the Neural Network
    3. 4.3 New events identified by the Neural Network
    4. 4.4 New long duration events visually identified
  5. 5 Summary and conclusions

##  1 Introduction

The Sun is an active star with a magnetic cycle of about 11 years (Hathaway,
2015). In periods of maximum activity, an increase in the frequency of solar
flares and coronal mass ejections can be observed. Both particles and magnetic
fields thrown into interplanetary space by coronal mass from the Sun may
impact the Earth. Geomagnetic storms, disruption of telecommunications
signals, GPS malfunctions, and blackouts are some of the disruptions affecting
Earth.

Over the years, studies on solar activity and the Sun’s behavior have been
carried out trying to mitigate these effects on Earth (Pulkkinen, 2007). For
example, studies involving active regions, magnetic fields, solar flares,
coronal mass ejections, and others are all relevant. Since the emission from
solar activity is produced at all wavelengths of the eletromagnetic spectrum,
observations at different frequencies is crucial to understand solar phenomena
and the mechanisms involved (Dulk, 1985).

In 1859, English astronomers Richard C. Carrington and Richard Hodgson
identified the first solar flare (Tsurutani et al., 2003). The explosion was
quite intense, and a flash located in a small region was detected in images of
the Sun’s visible light. Just 17 hours later, a coronal mass ejection hit the
Earth, causing one of the largest magnetic storms ever recorded. If a similar
storm reaches our planet today, it would cause severe communication and
electrical energy problems, among others (Phillips, 2014).

Solar phenomena are usually associated with active regions of the solar
atmosphere. When a solar flare occurs, a large amount of energy is released
($10^{28}-10^{32}$ erg), this energy is used in accelerating particles and
heating the plasma, which generate radiation across the entire electromagnetic
spectrum (from X rays to radio waves) (Mann et al., 2009).

Observations of solar activity, both from ground and space observatories, have
generated a large amount of data. Thus it is necessary to apply artificial
intelligence techniques to analyze the data in search of the sudden increases
in the emission caused by solar flares. Here we have used a neural network to
find patterns and identify solar flares automatically.

This work involves the automatic detection of solar flares in the data of the
Polarization Emission of Millimeter Activity at the Sun (POEMAS) telescope
(Valio et al., 2013). POEMAS is a polarimeter that observed the Sun daily from
December 2011 to December 2013 at the rarely explored frequencies of 45 and 90
GHz.

The paper is organized as follows. In Section 2, we describe the data, and in
Section 3, the Neural Network methodology. In Section 4, the results of the
Neural Network experiments are detailed. Finally, we conclude, in Section 5,
and anticipate future research.

##  2 POEMAS Telescope

POEMAS is a system of two telescopes, installed at the CASLEO Observatory (El
Leoncito Astronomical Complex), in Argentina. The POEMAS telescope provides
solar left and right circular polarization measurements at two-millimeter
wavelengths (45 and 90 GHz) with a temporal resolution of 10ms. It operated
continually for two years (Dec 2011 - Dec 2013) observing the full disk of the
Sun, and detected several flares. The data collected from the Sun every day
were written to binary files, converted using the Python programming language,
and finally written to FITS files.

###  2.1 Data Acquisition

The antenna temperature data at both left and right circular polarized
emission at 45 and 90 GHz are recorded in the daily binary files (TRK
extension). Also the azimuth and elevation angles of the Sun are recorded in
the file. For the analyzes, we used the light curve resulting from the sum of
the two, right (RCP) and left (LCP), circular polarizations of the antenna
temperature at 45 GHz. Using the Python programming language version 3.6, the
TRK files were converted to FITS files using the following procedures:

  * 1.

FITS level 0 - conversion of data from the TRK to FITS file, using the 10ms
configuration of the original file.

  * 2.

FITS level 1 - integration of the temporal resolution of the FITS file level 0
from 1 ms to 1s using the median of the data within the 1s interval.

  * 3.

FITS level 2 - the merger of all FITS files level 1 of the same day into a
single new file

  * 4.

FITS level 3 - application of the time series (Trend) for all FITS file level
2

The first day of POEMAS observation used in this work was 12/01/2011, while
the last day used was 12/10/2013. In this period, we did not have data for 51
days, which resulted in a total of 690 days for analysis.

After converting the POEMAS binary files to Flexible Image Transport System
(FITS) files, the data is integrated into the database. Finally, the automatic
detection of solar flares is applied using Artificial Intelligence techniques,
especially pattern recognition and machine learning (Deep Learning).

###  2.2 Data calibration

Unfortunately, there is a misalignment of the telescope support structure due
to mechanical problems, causing the signal to abruptly decrease during local
noon. This decrease of the antenna temperature is clearly seen in Figure 1,
especially between 16 and 17 UT. The red curve on the same plot depicts the
expected light curve profile of the observations. Due to variations in the
solar emission caused by this telescope misalignment, it is difficult to
identify any increase caused by a solar event, except for the most intense
ones, which are usually rare.

Figure 1: Full-day observation of solar energy flux at 45 GHz, left circular
polarization (black curve) and attenuated flux (red curve).

Therefore, to minimize the daily variation of the telescope measurements at 45
GHz, a time series subtraction of the signal was performed. This time series
is the component trend of the antenna temperature of the same polarization on
the day under consideration. In Figure 2, we can see the 3 components of the
time series for the observations on 01/27/2012. In the top panel of Figure 2,
the light curve observed by POEMAS, that is the result of the sum of the RCP
and LCP polarizations, is shown. The second plot from the top is the signal
trend for different growth and decrease patterns. In the third plot, the
seasonality is presented; in this case, we consider an interval of 50 points
to analyze the behavior for every period of 1s. In the last panel, the
residuals after subtraction of the effects of seasonality and trend from the
data are presented. The residual fluctuations are attributed to random
components.

Figure 2: Decomposition of the time series of 2012/01/27

In Figure 3, the original observed data (blue curve), the trend of the time
series (red curve), and the result after subtracting the trend from the
observed signal (black curve) are shown. A value of 1000 was added to this
subtraction residual to place it on the same scale as the original signal. On
this day, a solar flare occurred at 18:15 UT, and the impulsive peak can be
observed in the original curve (blue). On the same day, between 21:00 UT and
22:00 UT, there was a drop in signal due to interference in the Earth’s
atmosphere, probably due to clouds in front of the Sun. These drops in signal
due to clouds are significant noise in the signal, and we have disregarded
them from the data analysis.

Figure 3: Analysis of the emission observed by POEMAS at 45 GHz all day
2012/01/27

To verify the existence of a solar event, we used data from the Radio Solar
Telescope Network (RSTN) operated by the Meteorological Agency of the United
States Air Force (Guidice, 1979). This network is composed of 4 radio stations
located around the globe. Considering the location of POEMAS in Argentina, we
will use data from observatories in Palehua, Hawaii (USA), Sagamore Hill in
Massachusetts (USA), and San Vito (Italy). The 3 observatories cover the
POEMAS observation window depending on the time of year. However, the antenna
with the highest time intersection is in Sagamore Hill, Massachusetts (USA).
Data from Sagamore Hill and Palehua stations for 01/27/2012 are shown in
Figure 4, in the bottom and top panels, respectively.

Figure 4: RSTN data for 2012/01/27 - Palehua (upper) and Sagamore Hill
(bottom)

The solar flare that happened on this day at 18:15 UT is clearly seen in the
RSTN data shown in the two panels of Figure 4. The impulsive phase of this
event was also detected in the POEMAS signal at 45 GHz (Figure 3). However, to
see the gradual phase of the flare, it is necessary to subtract the daily
instrumental variation from the signal due to the telescope’s misalignment.
This can be done by subtracting the signal observed on the day before (or
after) the event, since the variation in the signal does not vary
significantly in the period of a day or so. However, this is not the procedure
performed in this work due to its requirement of human supervision.

##  3 Neural Network

Based on biological neural networks, that is, on the biological neuron,
Artificial Neural Networks (ANNs) are mathematical models that have
computational capacity acquired through learning and generalization. This
structure attempts to mimic a human brain with connections between neurons
(synapses) and input and output signals.

Frank Rosenblatt at the Cornell Aeronautical Laboratory developed the first
multi-neuron network of the linear discriminator type and named this network
the perceptron. A perceptron is a network with neurons arranged in layers.
This proposed model learns concepts and can answer with true (1) or false (0).
In the early 1960s, Rosenblatt extended his work by publishing several
articles and a book (Rosenblatt, 1962; Tappert, 2019).

The resulting 45 GHz emission signal, after subtraction of the trend of the
time series (black curve of Figure 3) was input to a Multilayer Perceptron
Neural Network (NN). The temporal resolution of the light curve is 1 second,
which would generate a lot of data points for the network. Therefore, the
temporal resolution was modified to 10 seconds, using the median in each
10-second interval to reduce the number of data points. Then a 5-minute window
was considered to go through the resulting signal extracting chunks every 10
seconds.

To better exemplify the process, in Figure 5 we show the validation of time
intervals to be later considered as true signals for input to the neural
network. In the example, the supposed event starts at 1:02:00 UT and ends at
1:03:00 UT. The first true interval would be [0:58:00, 1:03:00] UT, the second
true interval would be [0:58:10, 1:03:10] UT, and so forth. Therefore, we
would have a sliding 5 min window traversing the final signal every 10s. The
last true interval of this supposed event would be [1:02:00, 1:07:00] UT. If
this was the only event in one day, there would be 25 true intervals.

Figure 5: Example of interval extraction

This was the most critical process in evaluating the signal due to the large
volume of data. There were a total of 690 days of solar observation by the
POEMAS with an average of 10 hours per day. Considering a temporal resolution
of 10 s, there are approximately 3600 points in a day; thus resulting in
$2.484\times 10^{6}$ data points.

###  3.1 Neural Network Training (NNT)

Previously, Hidalgo Ramírez et al. (2019) visually detected and analyzed 30
events observed by POEMAS telescope, which are listed in Table 1. First, we
separate these events into two classes: training and classification. The first
14 flares were separated into the classification group. Events 15 through 30
were used for training of the neural network. The model adopted was supervised
learning, where we submitted the 228 intervals referring to the 16 events and
classified them as positive.

N | Date | Time (UT) | NN
---|---|---|---
1 | 2011/12/25 | 18:14 | ✓
2 | 2011/12/25 | 20:27 | ✓
3 | 2011/12/26 | 20:16 | ✓
4 | 2012/01/27 | 18:15 | ✓
5 | 2012/03/05 | 19:13 | ✓
5 | 2012/03/06 | 21:05 | –
7 | 2012/03/13 | 17:24 | ✓
8 | 2012/05/07 | 17:23 | –
9 | 2012/05/09 | 12:30 | ✓
10 | 2012/06/03 | 17:53 | ✓
11 | 2012/07/03 | 17:02 | ✓
12 | 2012/07/04 | 16:36 | ✓
13 | 2012/07/05 | 20:13 | ✓
14 | 2012/07/08 | 16:30 | ✓
15 | 2012/07/10 | 15:30 | –
16 | 2012/11/27 | 15:56 | ✓
17 | 2013/02/17 | 15:47 | ✓
18 | 2013/05/13 | 16:03 | ✓
19 | 2013/10/15 | 19:00 | –
20 | 2013/10/25 | 15:00 | ✓
21 | 2013/10/26 | 19:25 | ✓
22 | 2013/10/28 | 15:10 | ✓
23 | 2013/10/29 | 21:48 | ✓
24 | 2013/11/05 | 18:10 | ✓
25 | 2013/11/05 | 22:11 | ✓
26 | 2013/11/06 | 13:43 | ✓
27 | 2013/11/07 | 12:26 | ✓
28 | 2013/11/07 | 14:18 | ✓
29 | 2013/11/07 | 14:23 | ✓
30 | 2013/11/07 | 14:29 | ✓
Table 1: Events identified in the work of Hidalgo Ramírez et al. (2019). The
last column lists the events identified in this work by the Neural Network
(NN).

We need a balanced training base, and for that, we use the Nearmiss data
balancing algorithm, a subsampling algorithm that randomly reduces most class
examples. In the case of negative intervals, however, it selects samples based
on distance (Mani and Zhang, 2003).

The next step is to define the NN structure after the training base is
balanced (228 positives and 228 negative intervals). The input layer has 30
nodes that will receive each interval of 5 minutes with a resolution of 10
seconds. The output layer has only 1 node, which reveals if the prediction for
the data input is true or false. Several configuration tests were performed
for the middle layer, using one and two layers. The best training results were
found using only 1 intermediate layer. Thus, we decided to use one layer and
varied the number of neurons to compare the results.

Because we are using supervised learning, when the NNT predicts a given input
and the result is not expected, the network must adjust the weights to reduce
the error and repeat the process until the error rate is 0. The activation
function used was ReLu, as it is not linear and has better results compared to
Tanh.

ReLU is the most commonly used activation function when designing neural
networks today. The function is non-linear, and it does not activate all
neurons at the same time, because if input is negative, it will be converted
to zero and the neuron will not be activated. The Sigmoid function is a
sensitive function and is continuously differentiable. This is not a linear
function, and this is an interesting feature because it essentially means that
when there are several neurons with sigmoid function as activation function
the output too is nonlinear. This function varies between values [0,1]. The
Tanh function is very similar to the sigmoid function. In fact, it’s just an
improved version of the sigmoid function as it varies between values [-1,1]
(Burns, 2019).

To assess the quality of training and classification, we consider accuracy as
a figure of merit, as it defines the proximity of an experimental result to
its actual value. The greater the accuracy, the closer it is to the actual
result. For all NNT configurations performed, the training had an accuracy of
100%. That is, they hit 228 positives (VP) and 228 negatives (VN) and had no
false positives (FP) and no false negatives (FN).

###  3.2 Classification

The network efficiency and learning quality depend on its architecture
specification, that is, the function of neuronal activation, learning rule,
initial values, and training data. We consider a network with 3 layers: one
input, one intermediate, and one output. This configuration showed the best
performance and results in the training phase. For the primary classification,
there were 14 events and 141 actual intervals. The 16 events used in the
training were labeled in the classification as negative.

For each experiment, we varied the number of intermediate layer neurons. We
started with 30 neurons and added 30 neurons in each experiment afterwards. In
Table 2, we present the results of the 8 experiments. In the first column of
the Table, the number of the experiment is given, in the second the number of
neurons used in the intermediate layer, in the third column the number of
positive intervals identified by the network as positive, and in the fourth
column the number of positive intervals specified as negative. In the fifth
column, the number of negative intervals identified as positive by the NNT are
seen. Finally, in the sixth column, the number of negative intervals that are
real negative, and in the seventh column, we have the accuracy of each of the
experiments.

| # Neurons | True | False | False | True | Acuracy
---|---|---|---|---|---|---
| Occult layer | Positive | Negative | Positive | Negative |
Exp. 1 | 30 | 126 | 15 | 1522161 | 747732 | 33%
Exp. 2 | 60 | 126 | 15 | 1334829 | 935064 | 41%
Exp. 3 | 90 | 127 | 14 | 1338661 | 931232 | 42%
Exp. 4 | 120 | 127 | 14 | 1246748 | 1023145 | 45%
Exp. 5 | 150 | 126 | 15 | 1522170 | 747732 | 33%
Exp. 6 | 160 | 126 | 15 | 1196951 | 1072942 | 47%
Exp. 7 | 170 | 131 | 10 | 1355206 | 914687 | 40%
Exp. 8 | 180 | 113 | 28 | 1764115 | 505778 | 22%
Table 2: Results of the Neural Network.

We started with 33% accuracy in experiment 1 with 30 neurons in the middle
layer. In experiments 2, 3, and 4 we had a gradual increase in accuracy,
reaching 45%. In experiment 5, with 150 neurons, we had a drop in accuracy to
33%, which is equivalent to the results in experiment 1. As the results were
not inferior to any previous investigation, we decided to continue increasing
the neurons to check the results. In experiment 6 we achieved an accuracy of
47% with 160 neurons in the hidden layer. In experiment 7, the accuracy
decreases to 40%, and experiment 8 reached the lowest accuracy found, 22%.
Experiment 7 had the highest number of true positives but a high number of
false positives, so we focused on experiment 6 to analyze the results.
Experiment 6 with a configuration of 160 neurons had an accuracy of 47%, the
best among all the configurations.

To improve the results, we note that each minute has 6 points (10 seconds time
resolution) evaluated several times by the neural network. For experiment 9,
if the network assessed less than 3 points within a minute as positive, we
would consider that minute as negative. For experiment 10, if less than 4
points are evaluated as positive within one minute, this minute is assumed to
be negative. The proposed method improved the accuracy of the neural network
to 60%, as seen in Table 3. There are still many false positives and a
considerable decrease in true positives, from 126 to 13, when comparing
experiments 6 and 10. Also, the false negative increased from 15 to 128 cases.

| # Positive | True | False | False | True | Acuracy
---|---|---|---|---|---|---
| Points | Positive | Negative | Positive | Negative |
Exp. 9 | $<3$ | 25 | 116 | 990033 | 1279860 | 56%
Exp. 10 | $<4$ | 13 | 128 | 906998 | 1362895 | 60%
Table 3: Results of the Neural Network for experiments 9 and 10.

Since the results of the neural network presented many false positives, we
then compared the neural network results with the RSTN light curves, as well
as the POEMAS observations, by visual inspection. First, the daily light
curves from December 2011 to December 2013 (years of POEMAS observation)
provided by RSTN were checked for events at frequencies $>4$ GHz (microwaves).
When identifying a microwave event in the RSTN data, we check if the Neural
Network identified this event at the same day and time in the 45 GHz data.

##  4 Results and discussion

In this work, we analyzed the 45 GHz light curves observed by POEMAS
telescopes for a total of 690 days, from December 2011 to December 2013. For
the analysis, we considered the sum of the signal involving the two circular
polarizations, RCP plus LCP. Moreover, visual inspection was performed on the
daily microwave light curves observed by the RSTN telescope network during the
same period.

The Neural Network (NN) application detected patterns in the 45 GHz light
curve of the POEMAs and identified already known events and new ones. We
compared the results of the Network with the work of Hidalgo Ramírez et al.
(2019), who identified 30 events, as listed in Table 1. In addition, some
events identified in the RSTN microwave data were not detected visually in the
POEMAS light curves nor by the NN, given its temporal characteristic of long
duration. Below we discuss each of these results in more detail.

###  4.1 Events identified previously and confirmed by the Neural Network

The Neural Network identified 26 of the 30 events from the work of Hidalgo
Ramírez et al. (2019), thus the NN was able to retrieve 87% of the events
(last column of Table 1). An example of such an event is shown in Figure 6
that occurred on 2013 May 13th. In the top panel, the POEMAS light curve is
shown in blue, whereas the points identified by the NN as positive are
depicted in red. In the bottom panel, the microwave light curves of the RSTN
data from the Sagamore-Hill telescope (USA) are presented, where the event at
16:03 UT is clearly seen at all frequencies. Thus, the NN correctly identified
the event that peaked at approximately 16:03 UT. This was a large event, of
GOES X-ray class X2.8, that occurred on the East limb of the Sun.

Figure 6: Light curves from the event on 2013/05/13 at 16:03 UT. Upper: 45 GHz
data from POEMAS (blue) and Neural Network result (red) and Lower: data from
RSTN.

###  4.2 Events previously reported and not identified by the Neural Network

In Table 1 there are four events identified visually by Hidalgo Ramírez et al.
(2019) but not recognized by the NN, which make up only 13% of the events. The
plots in Figure 7 represent the light curves of 2012 May 7th observed at 45
GHz by the POEMAS and in microwaves by RSTN. The event is clearly identified
in the RSTN at approximately 17:23 UT. This event is not easily recognized in
the POEMAS data and was neither identified by the NN, probably because of a
duration longer than 5 min.

Figure 7: Light curves from the event on 2012/05/07 at 17:23 UT. Upper: 45 GHz
data from POEMAS (blue) and Neural Network result (red) and Lower: data from
RSTN.

###  4.3 New events identified by the Neural Network

A total of 9 events were identified by the Neural Network, but went unnoticed
in the visual inspection of Hidalgo Ramírez et al. (2019). These flares are
listed in Table 4. An example of such is the burst that occurred on 2012 July
28th, identified by the NN at approximately 21:00 UT. The time profile of the
flare is shown in the top right panel of Figure 8, where the light curve of
the previous day was subtracted to eliminate the variation due to the
misalignment of the telescope, and better show the event.

N | Date | Time (UT)
---|---|---
1 | 2012/01/28 | 11:50
2 | 2012/05/08 | 13:00
3 | 2012/07/28 | 21:00
4 | 2012/09/02 | 18:10
5 | 2012/10/20 | 18:15
6 | 2012/10/21 | 20:00
7 | 2013/05/03 | 16:30
8 | 2013/05/03 | 17:30
9 | 2013/07/02 | 17:50
Table 4: Events identified only by the Neural Network

Figure 8: Top: Poemas light curve on 2012 July 28th, with the event at 20:50
UT identified by the NN depicted in red. Middle: Highlight of the 45 GHz event
after subtraction of the emission from the previous day. Bottom: Microwave
light curve of the same day observed by RSTN.

###  4.4 New long duration events visually identified

From the visual inspection of the RSTN microwave light curves, we identified
11 events not identified by Hidalgo Ramírez et al. (2019) nor by the Neural
Network, which are listed in Table 5. An example of such event is shown in
Figure 9 for the 2012 July 12th flare, a long-duration event that lasted for
more than 2 hours. The top panel shows the POEMAS 45 GHz light curve in blue,
withe the light curve of the day before depicted in green. The subtraction of
the emission from the previous day is shown in black on the top panel and
highlighted in the middle panel of Figure 9. The flare that started at
approximately 16:00 UT, peaked just before 17 UT and ended after 18:30 UT, was
not readily visually identified in the POEMAS light curve. In the bottom panel
of the figure, the microwave light curves observed by RSTN clearly show the
event at all frequencies, where the temporal profile of the 15 GHz emission
closely resembles that of the 45 GHz from POEMAS. The non-identification of
this and the other 9 events probably occurred due to their gradual temporal
profile, lasting from 30 minutes to more than an hour. We point out that the
data input for NNT consisted of 5 min intervals.

N | Date | Time (UT)
---|---|---
1 | 2012/03/02 | 17:40
2 | 2012/03/03 | 18:00
3 | 2012/03/04 | 11:00
4 | 2012/03/17 | 20:50
5 | 2012/06/06 | 20:00
6 | 2012/06/14 | 13:30-15:00
7 | 2012/07/12 | 16-17:00
8 | 2012/07/27 | 17:15
9 | 2013/07/03 | 20:00
10 | 2013/08/17 | 18:20-19:30
Table 5: Long-term events not identified in the work of Hidalgo Ramírez et al.
(2019) nor by the Neural Network

Figure 9: Solar flare of 2012/07/12. Top: Light curve mission at 45 GHz from
POEMAS for the day of the event, 2012/07/12 (blue curve) and the previous day,
2012/07/11 (green curve). The result of the subtraction of the emission on the
12th by the 11th of July 2012 is shown by the black curve (shifted by 900 to
fit the scale of the figure). Middle: Blow up of the subtracted light curve to
better identify the solar flare detected at 45 GHz. Bottom: Microwave light
curves observed by RSTN ($1-15$ GHz) for the whole day.

##  5 Summary and conclusions

In this work, we analyzed two years of light curves from POEMAS telescopes at
45 GHz, from December 2011 through December 2013. The main objective was to
automatically detect solar events in these light curves. The detection of
solar flares in the light curves of POEMAS was hindered due to problems with
the telescope’s pointing, causing daily variations in the signal. Therefore,
it was necessary to apply initial computational techniques to calibrate and
reduce the POEMAS data. We created and used a Neural Network (NN) to identify
solar flares in the data automatically. The application of this Neural Network
was later compared with the microwave emission ($1-15$ GHz) detected by the
RSTN radio-telescope network.

The first challenge of this work was the data transformation, due to noise and
augmented by the misalignment of the telescope. Moreover, clouds obstructed
the observation of the Sun and interference in the Earth’s atmosphere even in
the absence of clouds, such as increased water vapor or ice crystals, also
precluded the detection of the flare signal by creating several spurious peaks
in the light curves.

Using a Neural Network with supervised learning, we reached an accuracy of
47%. This value is low, however it is due to the few samples used for training
and the intrinsic noise of POEMAS’ light curves. Later the accuracy was
improved to 60% by using a constraint on the False Positives. Nevertheless,
despite the problems in the data mentioned above, we confirmed 26 previously
known events and could identify 9 new events detected in the light curves of
POEMAS, thanks to the NN.

Comparing the RSTN data for the two years of 2012 and 2013, we visually
identified 10 long-term events not previously identified in the POEMAS light
curve by visual inspection nor by the NN. As the Neural Network was not
supplied with any long-term events for training, it is not capable of
detecting this type of event. Thus the Neural Network constructed here can
detect only short-term impulsive events, with duration less than 5 minutes.

Event | Total | Old∗ | Neural Network | Table
---|---|---|---|---
Old∗ | 30 | 30 (100%) | 26 (87%) | 1
New | 9 | 0 (0%) | 9 (100%) | 4
Long duration | 10 | 0 (0%) | 0 (0%) | 5
Total | 49 | 30 (61%) | 35 (71%) |

Table 6: Summary of the events detected, or not, by the Neural Network.

∗ refers to the events identified visually by Hidalgo Ramírez et al. (2019).

In summary, with the aid of artificial intelligence, in this work we have
identified 35 solar events in the 45 GHz emission from 2012 to 2013 from a
total of 49 bursts, or 71%. If we consider, that the NN was not trained to
detect events with duration longer than 5 min, then the accuracy of the NN
increases to 90%. From the total of 49 events, 19 solar flares are
unprecedented at 45 GHz, not having been identified in previous works that
analyzed these data (Hidalgo Ramírez et al., 2019). The statistics of the NN
is summarized in Table 6.

The use of artificial intelligence is innovating Astronomy. Especially in
Solar Physics, we can mention the works by Hou et al. (2020); Ishikawa et al.
(2021); Neira et al. (2020), who obtained an accuracy of approximately 90%. To
leverage the study of solar flares, such techniques must be explored.

The search for solar flares using Neural Network was a first step to
automating the process. Several challenges were encountered, such as clouds,
periodic signal variations, and non-detection of long-term events, and their
solution will be proposed in future work.

## Acknowledgements

A.V acknowledges partial financial support from FAPESP grant #2013/10559-5.
V.L. thanks the fellowship from MackPesquisa funding agency.

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11institutetext: Yunnan Observatories, Chinese Academy of Sciences, Kunming
650011, PR China
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext:
University of the Chinese Academy of Sciences, Beijing 100049, PR China
33institutetext: International Centre of Supernovae, Yunnan Key Laboratory,
Kunming 650216, PR China

# Irradiation effect and binary radio pulsars with giant companions

Shunyi Lan 1122 Xiangcun Meng 1133

###### Abstract

Context. Pulsars are neutron stars that rotate rapidly. Most of the pulsars in
binary systems tend to spin faster than those in isolation. According to
binary evolution theory, radio pulsars in binary systems can have various
types of companion stars. However, binary radio pulsars with giant stars,
helium stars, and black holes as companions are lacking so far.

Aims. We aim to investigate the possible parameter space of binary radio
pulsars with giant companions.

Methods. We used the MESA stellar evolution code to consider the effect of
irradiation in binary evolution and evolved a series of binary models.

Results. We present the potential physical properties of binary radio pulsars
with giant star companions in the framework of the classical recycling
scenario and the irradiation model. We found that the parameter space of
binary radio pulsars with giant companions can be greatly expanded by the
effect of irradiation. Moreover, when irradiation is strong enough,
submillisecond pulsars might be accompanied by giant companion stars. We also
demonstrate a correlation between the timescale of the binary system as a low-
mass X-ray binary and the irradiation efficiency.

Conclusions. The birthrate problem between millisecond pulsars and low-mass
X-ray binaries might not be resolved by the irradiation effect. A more
detailed binary population synthesis is necessary. Our findings may offer
guidance for observers that wish to locate binary radio pulsars with giant
companions or submillisecond pulsars.

###### Key Words.:

binaries: close – stars: neutron – pulsars: general – X-rays: binaries

##  1 Introduction

A pulsar is a type of neutron star (NS) that spins rapidly. The first pulsar
was discovered in 1967 (Hewish et al., 1968). Since then, over 3,300 pulsars
have been discovered, and the count continuously increases (Manchester et al.,
2005). Most millisecond pulsars (MSPs), which have a spin period $P_{\rm
spin}$ shorter than 30 milliseconds, are found in binary systems. It is well
established that these MSPs have increased their spin rate by accreting mass
from their low-mass companion star, causing them to become recycled pulsars.
During the mass-transfer phase, these systems can be observed as low-mass
X-ray binaries (LMXBs). The discoveries of transitional millisecond pulsars
and accreting millisecond X-ray pulsars (AMXPs) have strongly supported this
scenario (e.g., Papitto et al., 2010, 2013).

The general picture of binary evolution is well established. Starting from a
binary of two zero-age main-sequence (ZAMS) stars, the initially more massive
star first dies in a supernova event and leaves an NS+MS or black hole (BH) +
MS binary system if the binary system remains bound (e.g., Brandt &
Podsiadlowski, 1995; Renzo et al., 2019). When the orbital period is short
enough, mass transfer and mass loss can occur. The form of the mass transfer
varies depending on the orbital period and evolutionary stage of the companion
stars. For a companion mass $M_{2}>10\leavevmode\nobreak\ M_{\odot}$, the mass
transfer is dominated by the stellar wind. These systems can be observed as
high-mass X-ray binaries (HMXBs), and they may be the progenitors of double
neutron stars (DNSs) or NS+BH (e.g., Tauris et al., 2017). When the mass-
transfer process of NS+MS or BH+MS is dynamically unstable, the envelope of
the companion star may swallow the NS or BH and enter the so-called common-
envelope (CE) phase (Ivanova et al., 2013). The CE can be ejected under
certain conditions and leave an NS+helium star system or a BH+helium star
system. For $M_{2}<10\leavevmode\nobreak\ M_{\odot}$, the mass transfer is
normally dominated by Roche-lobe overflow (RLO), and these systems can be
observed as intermediate-mass X-ray binaries (IMXBs) or LMXBs. NS IMXBs lead
to the formation of MSPs with carbon-oxygen WD (CO WD) or oxygen-neon-
magnesium WD (ONeMg WD) companions, and NS LMXBs lead to the formation of MSPs
with helium WD (He WD) companions (e.g., Tauris & Savonije, 1999; Tauris et
al., 2012). For other formation channels and a more detailed binary evolution
review, we refer to Han et al. (2020) and Tauris & van den Heuvel (2023).

The main framework has been established, but certain physical details still
need to be addressed. Podsiadlowski (1991) found that the evolution of LMXBs
can be significantly altered by the irradiation effect. Harpaz & Rappaport
(1994) showed that when the irradiation effect is considered, episodic mass
transfer occurs. Further research on the detailed mechanism of the irradiation
effect was conducted by Ritter et al. (2000) and Büning & Ritter (2004),
revealing that the irradiation effect is vital not only for LMXBs, but also
for accreting WD binaries (e.g., Zhao et al., 2022) . By applying this effect,
a series of works by Benvenuto et al. (2012, 2015, 2014, 2017) showed that the
irradiation effect could play a critical role in the formation of black widow
and redback pulsars. Black widows might be evolving from redbacks. The
irradiation effect may also account for the formation of AMXPs, as stated in
Goodwin & Woods (2020).

The irradiation effect on a binary system is strong because it causes mass
transfer to occur in a cyclic manner (e.g., Benvenuto et al., 2014). This
enables the system to accommodate a radio pulsar before the mass transfer
ceases completely. According to theory, binary radio pulsars can have
companions of various types, such as MSs, giant stars, helium stars, He WDs,
CO WDs, ONeMg WDs, NSs, and BHs. However, binary pulsars with companions that
are giant stars, helium stars, and BHs are yet to be discovered (Manchester et
al., 2005). This paper aims to provide the possible parameter space for binary
radio pulsars with giant companion stars, with and without the irradiation
effect.

In Sect. 2 we introduce our methods and the input physics for numerical
calculations. In Sect. 3 we present our results. The discussions are given in
Sect. 4, and we conclude in Sect. 5.

##  2 Input physics and methods

We used the stellar evolution code Modules for Experiments in Stellar
Astrophysics (MESA, version 10398, Paxton et al., 2011, 2013, 2015, 2018,
2019) to run evolutions of LMXBs. In all of our calculations, we assumed an
initial NS mass of $1.4\leavevmode\nobreak\ M_{\odot}$. The companion stars
were all ZAMS stars, and the masses ranged from $1.0$ to
$2.0\leavevmode\nobreak\ M_{\odot}$ with a step of $0.1\leavevmode\nobreak\
M_{\odot}$. The metallicity of the companion star was $Z=0.02$. We considered
the prescription of Reimers (1975) for the stellar wind, and we set the
scaling factor to 0.1. The initial orbital periods of binary systems, $\log
P_{\rm orb}/{\rm days}$, range from $0.5$ to $2.0$. We adopted the Kolb &
Ritter (1990) scheme to calculate the mass-transfer rate of RLO. We adopted
the isotropic reemission model (Tauris & van den Heuvel, 2023) to compute the
mass loss from the binary system with the following parameters:
$\alpha=0,\leavevmode\nobreak\ \beta=0.7,and\leavevmode\nobreak\ \delta=0$,
which correspond to the fraction of the mass that is lost in the vicinity of
the companion star, in the vicinity of the NS, and in the circumbinary coplane
toroid, respectively. The mass-accretion rate onto the NS is $\dot{M}_{\rm
NS}=(1-\alpha-\beta-\delta)|\dot{M_{2}}|$, where $\dot{M}_{2}$ is the mass-
loss rate of the companion star. We did not consider mass accretion of the
stellar wind of the companion star, and $\dot{M}_{\rm NS}$ was limited by the
Eddington accretion rate (Tauris et al., 2012),

$\dot{M}_{\rm Edd}\simeq 3.0\times
10^{-8}M_{\odot}\mathrm{yr}^{-1}\leavevmode\nobreak\ \left(\frac{R_{\rm
NS}}{13\leavevmode\nobreak\ {\rm km}}\right)\left(\frac{1.3}{1+X}\right),$ (1)

where the $R_{\rm NS}=15\leavevmode\nobreak\ (M_{\rm
NS}/M_{\odot})^{1/3}\leavevmode\nobreak\ {\rm km}$ is the radius of the NS,
and $X$ is the fraction of hydrogen in the transferred materials.

The spin evolution of the NS was implanted and calculated self-consistently
with the MESA code. We calculated the magnetospheric radius $r_{\rm mag}$, the
corotation radius $r_{\rm co}$, and the light cylinder radius $r_{\rm lc}$ as
(Frank et al., 2002)

$r_{\rm mag}=\xi\left(\frac{\mu^{4}}{2GM_{\rm NS}\dot{M}_{\rm
NS}^{2}}\right)^{1/7},$ (2)

$r_{\rm co}=\left(\frac{GM_{\rm NS}P_{\rm
spin}^{2}}{4\pi^{2}}\right)^{\frac{1}{3}},$ (3)

$r_{\rm lc}=\frac{cP_{\rm spin}}{2\pi},$ (4)

where $\xi$ is a dimensionless constant depending on the details of the disk-
magnetosphere interactions, which is unknown for now. We took $\xi=0.5$ in
this paper. $G$ is the gravitational constant, and $\mu$ is the magnetic
moment of the NS. $M_{\rm NS}$ and $\dot{M}_{\rm NS}$ are the mass and
accretion rate of the NS, respectively. $P_{\rm spin}$ is the spin period of
the NS, and $c$ is the speed of light in vacuum.

We implemented the irradiation effect by injecting additional energy into the
outer envelope of the companion star. The amount of this energy was determined
by the irradiation luminosity $L_{\rm irr}$, and we followed Harpaz &
Rappaport (1994),

$\\\ L_{\rm irr}=\left\\{\begin{array}[]{l l}{{\eta L_{\rm
X}\left(\frac{R_{2}}{2a}\right)^{2}}}&{{\dot{M}<\dot{M}_{\rm Edd}}}\\\ {{\eta
L_{\rm
X}\left(\frac{R_{2}}{2a}\right)^{2}\exp\left(1-\frac{|\dot{M}|}{\dot{M}_{\rm
Edd}}\right)}}&{{\dot{M}\geq\dot{M}_{\rm Edd}}}\end{array}\right.,$ (5)

where $\eta$ is the irradiation efficiency, $L_{\rm X}$ is the accretion
luminosity, $R_{2}$ is the radius of the companion star, $a$ is the binary
separation, $\dot{M}$ is the mass-transfer rate, and $\dot{M}_{\rm Edd}$ is
the Eddington limit.

The maximum age of the system was set to 14 Gyr. For the other details of
physics that are not mentioned here, we followed Lan & Meng (2023).

##  3 Results

Figure 1: Binary evolution results in a $M_{2}-P_{\rm orb}$ diagram. The
observed binary pulsars with He WDs and MSs are plotted as cyan upward-
pointing triangles and magenta squares, and the masses are the lowest possible
masses of the companions of observed binary pulsars. The red dots indicate the
initial grid. The color bar indicates the mass-transfer rate. The different
panels show different irradiation efficiencies.

Figure 1 shows our binary evolution results in a $M_{2}-P_{\rm orb}$ diagram
for $\mu_{\rm i}=1.0\times 10^{30}\leavevmode\nobreak\ {\rm
G\leavevmode\nobreak\ cm^{3}}$. The dots that represent observation indicate
the minimum mass of the companion star, and we did not plot error bars for
clarity. The different panels show different irradiation efficiencies $\eta$
of $0,\leavevmode\nobreak\ 0.001,\leavevmode\nobreak\
0.01,and\leavevmode\nobreak\ 0.1$. The deep blue segments indicate that the
mass-transfer rate is very low, that is, the system is a detached binary. The
figure shows that when the irradiation is weak, for example, $\eta=0.001$, the
overall evolution is similar to the results of $\eta=0$. For
$\eta=0.01and\leavevmode\nobreak\ 0.1$, the mass-transfer cycles are more
evident and the average mass-transfer rate is higher than the results of
$\eta=0,\leavevmode\nobreak\ 0.001$, as shown by some previous studies (e.g.,
Benvenuto et al., 2014, 2015). Some evolutionary tracks for converging systems
were stopped manually. We only focused on systems that are able to have giant
companions.

###  3.1 Irradiation-induced mass-transfer cycles

Figure 2: Evolution of the mass-transfer rate for different $\eta$. The
initial companion mass is $M_{2}=1.0\leavevmode\nobreak\ M_{\odot}$, and the
initial orbital period is $\log\leavevmode\nobreak\ P_{\rm orb}/{\rm
days}=0.5$. The initial magnetic moment of the NS is $\mu_{\rm i}=1.0\times
10^{30}\leavevmode\nobreak\ {\rm G\leavevmode\nobreak\ cm^{3}}$. The blue,
dashed orange, and dash-dotted green lines show the mass-transfer rate, the
mass-loss rate of the companion, and the Eddington limit, respectively.
Figure 3: Similar to Fig. 2, but for
$(M_{2}/M_{\odot},\log\leavevmode\nobreak\ P_{\rm orb}/{\rm days})=(1.3,0.5)$.
Many mass-transfer cycles lie in the orange region for the case of
$\eta=0.001$. Figure 4: Similar to Fig. 2, but for
$(M_{2}/M_{\odot},\log\leavevmode\nobreak\ P_{\rm orb}/{\rm days})=(1.3,1.5)$.

In Fig. 2 we show an example for the evolution of the mass-transfer rate as a
function of time for different $\eta$ . The initial companion mass is
$M_{2}=1.0\leavevmode\nobreak\ M_{\odot}$ and initial orbital period is
$\log\leavevmode\nobreak\ P_{\rm orb}/{\rm days}=0.5$. The initial value of
the NS magnetic moment is $\mu_{\rm i}=1.0\times 10^{30}\leavevmode\nobreak\
{\rm G\leavevmode\nobreak\ cm^{3}}$. For $\eta=0$, the mass transfer is
continuous. The mass transfer was only halted once in the middle, when the
hydrogen-burning shell encountered the layer in which the outer convective
zone reached its deepest penetration, creating a molecular weight
discontinuity. This reduction in molecular weight caused the companion star to
contract, and this interrupted the mass-transfer process (Kippenhahn et al.,
2013). For the case with weak irradiation, that is, $\eta=0.001$, the
irradiation effect on the companion star can be easily offset by thermal
relaxation. The mass-transfer rate of $\eta=0.001$ therefore evolves in a
similar way to that of the nonirradiated star. For $\eta=0.01$, the thermal
equilibrium is disrupted by the stronger irradiation, causing the companion
star to expand and the mass-transfer rate to exceed the Eddington limit. When
the expansion due to irradiation is outweighed by the thermal relaxation of
the companion star, the companion star almost immediately detaches from its
RL. The system remains a binary radio pulsar until the companion star refills
its RL. For more details about irradiation-induced mass-transfer cycles, we
refer to Ritter et al. (2000). For $\eta=0.1$, the irradiation is even
stronger, and it is quite difficult to reach thermal equilibrium for the
companion stars. The mass-transfer rate can reach a higher peak, and the
system looses more mass within a single cycle. The reattachment timescale is
therefore longer and the number of mass-transfer cycles is smaller.

We show another two examples of the mass-transfer evolution in Figs. 3 and 4
to illustrate the impact of the initial companion mass and orbital period on
the evolution. In Fig. 3, the initial companion mass is
$M_{2}=1.3\leavevmode\nobreak\ M_{\odot}$ and the initial orbital period is
$\log\leavevmode\nobreak\ P_{\rm orb}/{\rm days}=0.5$. The mass transfer
initiates at the Hertzsprung gap (HG), and for this system, even for weak
irradiation, that is, $\eta=0.001$, the irradiation-induced mass-transfer
cycles may also occur at an early stage of the mass transfer because the
companion star has a thicker convective envelope than a $1\leavevmode\nobreak\
M_{\odot}$ star (see the panel of $\eta=0.001$). The later evolution of the
mass-transfer rate is similar to that of $\eta=0$ because the companion star
is more evolved and the irradiation can easily be relaxed. For $\eta=0.01$ and
0.1, the evolution of the mass transfer is similar to that of 2. In Fig. 4,
the initial companion mass is $M_{2}=1.3\leavevmode\nobreak\ M_{\odot}$ and
the initial orbital period is $\log\leavevmode\nobreak\ P_{\rm orb}/{\rm
days}=1.5$. The irradiation cannot significantly affect a highly evolved giant
star as a companion because the thermal timescale of the envelope is very
short and the irradiation effect can easily be relaxed by the companion star.

###  3.2 Binary radio pulsars with giant companions

Figure 5: Corresponding evolution of Fig. 2 in the HR diagram. The different
panels represent various irradiation efficiencies. The red dots indicate
$r_{\rm mag}>r_{\rm lc}$, i.e., the system can be considered as a binary radio
pulsar.

Figure 6: Corresponding evolution of Fig. 2 in the $M_{2}-P_{\rm orb}$
diagram. The differently colored lines indicate different irradiation
efficiencies. The red dots indicate $r_{\rm mag}>r_{\rm lc}$, i.e., the system
can be considered as a binary radio pulsar.

In order to intuitively present the evolutionary state of a binary radio
pulsar with a giant companion star, we show in Figs. 5 and 6 the corresponding
evolution of the system in Fig. 2 in a Hertzsprung-Russell (HR) diagram and a
$M_{2}-P_{\rm orb}$ diagram, respectively. In Fig. 5, the cases with different
$\eta$ are shown in different panels, while in Fig. 6, differently colored
lines correspond to distinct irradiation efficiencies. In both figure the red
dots indicate $r_{\rm mag}>r_{\rm lc}$, i.e., the system can be considered as
a binary radio pulsar. Compared to the evolution of weak irradiated
$\eta=0.001$ or non-irradiated model, for $\eta=0.01,\leavevmode\nobreak\
0.1$, when mass transfer turns on, the position of a companion star in the HR
diagram deviates from red giant branch (RGB) to become brighter and hotter.
The additional energy provided by irradiation that is stored on the surface
can travel inward or radiate off the surface of the companion star and causes
the stellar luminosity to rise. As the irradiation pushes the companion star
out of thermal equilibrium, it expands, which causes its effective temperature
to rise at the same time. The stronger the irradiation, the larger the
deviation from the RGB for the companion star. In Fig. 6 we show for a given
evolutionary stage that the orbital period of a binary system that is strongly
irradiated is longer than that of a system that is weakly irradiated or not
irradiated at all. The reason is that when the irradiation is strong, the
mass-transfer rate can exceed the Eddington limit, which results in the
ejection of more mass from the binary system, as shown in Fig. 2. Moreover,
the final mass of the WD is slightly higher because the final orbital period
is longer because the system still follows the relation between the mass of
the WD and the binary orbital period (e.g., Tauris & Savonije, 1999).

Figure 7: HR diagram showing the parameter space for giant stars, which serve
as the companions of radio pulsars. The different panels show different
$\eta$. Figure 8: Possible parameter space of binary radio pulsars in a
$M_{2}-P_{\rm orb}$ diagram for different $\eta$. The upperward-pointing red
and downward-pointin green triangles, blue squares, and cyan stars indicate
observed binary pulsars with He WDs, CO WDs, MSs, and NSs. The data are taken
from the ANTF pulsar catalog (Manchester et al., 2005). The black dots are the
results we calculated for binary radio pulsars with giant companions. Figure
9: Possible parameter space of binary radio pulsars with giant companions in a
Corbet diagram. The different panels show different $\eta$. The horizontal red
line indicates a spin period of 1 ms.

Figure 7 shows the possible parameter space of giant star companions of binary
radio pulsars in the HR diagram for different $\eta$ . As expected, the
parameter space of binary radio pulsars with giant companion stars is expanded
through the irradiation effect, that is, the parameter space increases with
$\eta$. In our models, the mass transfer can be initiated by a companion star
in the HG or on the RGB, which means that the companion stars can be a
subgiant or a red giant. In the case of $\eta=0.001$, it is clear that the
parameter space is divided into two parts. In Fig. 8, we show the possible
parameter space of binary radio pulsars in a $M_{2}-P_{\rm orb}$ diagram for
different $\eta$ . Observed binary pulsars with different types of companion
are also shown in the figure. The parameter space is smallest for $\eta=0$.
For $\eta=0.001$, part of its dots coincide with that of $\eta=0$ because most
of the evolution with weak irradiation is basically not affected. Some other
systems initiate mass transfer on the subgiant branch, and the evolution can
be significantly altered by even weak irradiation in the early stage of mass
transfer (see Fig. 3). For $\eta=0.01$ and $\eta=0.1$, the parameter space is
clearly enlarged due to irradiation-induced mass-transfer cycles.

In Fig. 9, we show the possible parameter space of binary radio pulsars with
giant companions in a Corbet diagram, that is, a $P_{\rm orb}-P_{\rm spin}$
diagram. Similar to Figs. 7 and 8, the different panels correspond to
different $\eta$. The red horizontal line indicates the spin period of 1 ms.
Our results show that when irradiation is strong enough, the pulsar can be
accelerated to the submillisecond level. Although our model is simplified (see
Sect. 4 for a more detailed discussion), these results imply that the sub-MSPs
might be accompanied by giant stars.

###  3.3 Mass-transfer timescale

Figure 10: Relation between $\mathcal{R}$ and the final mass of a WD $M_{\rm
WD}$. The dot shows our calculated results, and different colors represent
different irradiation efficiencies. See text for the definition of
$\mathcal{R}$.

Kulkarni & Narayan (1988) first reported that the birth rates of MSPs and
LMXBs do not match. There are two ways to resolve this birthrate problem: The
number of LMXBs can be increased through observations, or the system timescale
can be reduced for an LMXB in theory. In the context of the classical
recycling scenario, that is, $\eta=0$, the system timescale for LMXBs,
$\tau_{\rm LMXB}$, typically ranges from $10^{7}$ to $10^{10}$ yr, which
depends on the initial parameters of a binary (e.g., mass of companion or
orbital period, see also Podsiadlowski et al., 2002).

As previously demonstrated, the irradiation effect can significantly reduce
$\tau_{\rm LMXB}$, as shown in Figs. 2, 3, and 4. The system timescale of
LMXBs after considering the irradiation effect, $\tau_{\rm LMXB,irrad}$, not
only depends on the initial parameters of the binary (mass of companion and
orbital period), but also on the efficiency of the irradiation. We define the
parameter $\mathcal{R}$ to characterize the degree to which $\tau_{\rm LMXB}$
is reduced,

$\mathcal{R}=\frac{\tau_{\rm LMXB,irrad}}{\tau_{\rm LMXB}}.$ (6)

Since the initial companion mass and orbital period may be related to the
final mass of the WD $M_{\rm WD}$ (e.g., Tauris & Savonije, 1999), we relate
$\mathcal{R}$ to $M_{\rm WD}$ in Fig. 10. For $\eta=0$, $\mathcal{R}$ is
always 1, which corresponds to the canonical recycling scenario. For weak
irradiation, $\eta=0.001$, most systems are not affected and $\mathcal{R}$ is
very close to 1. For stronger irradiation, $\mathcal{R}$ depends on $M_{\rm
WD}$. For a given $\eta$, there is a threshold of $M_{\rm WD}$, that is, when
$M_{\rm WD}$ exceeds the threshold, $\mathcal{R}$ is equal to 1, and
$\mathcal{R}$ increases with $M_{\rm WD}$ when $M_{\rm WD}$ is lower than the
threshold. For $\eta=0.01$, the threshold is $\sim 0.31\leavevmode\nobreak\
M_{\odot}$, while for $\eta=0.1$, it is $\sim 0.34\leavevmode\nobreak\
M_{\odot}$. The smallest $\mathcal{R}$ can be $\sim 0.05$. The reason for
these results is that for a given companion star, the irradiation effect can
only be significant when the companion star is in an earlier evolution state,
which depends on $\eta$. For a more evolved companion star, the thermal
timescale of the envelope is very short, and the irradiation effect can easily
be relaxed by the companion star (see Fig. 4).

##  4 Discussion

Figure 11: Isochrones of three hypothetical pulsars. The gray dots show the
currently observed population of pulsars. The data are taken from the ANTF
pulsar catalog (Manchester et al., 2005). The red stars indicate the birth
positions of the hypothetical pulsars. The colored lines indicate isochrones
and were calculated for different values of the braking index $2\leq n\leq 5$.
The dashed black lines from top to bottom indicate the evolution of
$n=2,\leavevmode\nobreak\ 3$, and 5. The shaded orange area indicates the so-
called death valley (Rea et al., 2024).

We calculated a series of binary evolution models to show the possible
parameter space of binary radio pulsars with giant companions. The results we
show in Figs. 7, 8, and 9 ignore the evolution before the first RLO, even
though the conditions $r_{\rm mag}>r_{\rm lc}$ and the presence of a giant
companion star are both met. In our models, binary evolution starts with
NS+ZAMS. The range of time needed to initiate the first RLO is $\sim
0.9-12\leavevmode\nobreak\ {\rm Gyr}$. The lifetime of an MS star is a
function of mass. For a $2.0\leavevmode\nobreak\ M_{\odot}$ ZAMS star, it is
$\sim 1.0\leavevmode\nobreak\ {\rm Gyr}$, and for a $1.0\leavevmode\nobreak\
M_{\odot}$ ZAMS star, it is $\sim 10\leavevmode\nobreak\ {\rm Gyr}$
(Kippenhahn et al., 2013). Following Tauris et al. (2012), we analyzed the
spin evolution of an NS before the first RLO. The isochrones of three
hypothetical pulsars are plotted in Fig. 11. We assumed that the red stars are
the birth positions of pulsars in the $P_{\rm spin}-\dot{P}_{\rm spin}$
diagram. The radio emission is quenched when pulsars are within or below a
death valley. For the top hypothetical pulsar, it takes less than 0.3 Gyr to
reach the death valley, and for the middle hypothetical pulsar, this is $\sim
2$ Gyr. For MSPs, the corresponding timescale is longer than 12 Gyr. Although
the birth state of an NS remains unclear, if the birth state of the NS is
similar to the states of normal single pulsars, the part of the parameter
space of binary pulsars with giant companions that contribute to the binary
evolution before the first RLO can be neglected.

In our simplification, we omitted the accretion of stellar wind onto the NS.
Although the stellar wind from a giant companion may approach
$10^{-8}\leavevmode\nobreak\ M_{\odot}/{\rm yr}$, the accretion efficiency
onto an NS is relatively low; it is estimated to be about $10^{-4}$ to
$10^{-3}$. Therefore, the accretion rate of the NS by the stellar wind of the
companion is $\lesssim 10^{-11}\leavevmode\nobreak\ M_{\odot}/{\rm yr}$.
Instead of spinning up, the NS may instead spin down due to the propeller
effect. In our numerical calculation, we also assumed a perfect dipole for a
radio pulsar, that is, n=3. As mentioned before, there are uncertainties in
the spin-down process (see Fig. 11). The parameter space of binary radio
pulsars shown in Fig. 9 might therefore be slightly shifted upward (or
downward). There are still parts of the parameter space crossing the 1 ms
line. This indicates that a radio sub-MSP might be accompanied by a giant star
if $\eta$ were high enough. If researchers are consistently unable to detect
binary radio pulsars with giant companions or binary radio sub-MSP with giant
companions, this suggests that the irradiation effect might not a significant
factor in the evolution of LMXBs. Furthermore, this implies that this effect
might not resolve the birthrate problem. A more detailed binary population
synthesis is necessary to determine whether the irradiation effect may resolve
the birthrate problem in the future. We have only explored a limited range of
companion masses (1.0-2.0$\leavevmode\nobreak\ M_{\odot}$), and the parameter
space we showed might therefore be biased. A more comprehensive range of
companion masses should also be investigated in the future.

Theoretical models predict the existence of binary radio pulsars with giant
companions, but these objects remain undetected. According to our model, even
when the irradiation effect is accounted for, which expands the parameter
space, the timescale for the observation of such a system as a binary radio
pulsar remains brief, $\lesssim 0.3$ Gyr. This can make these objects
difficult to detect. Another reason why binary radio pulsars with giant
companions remain undetected may be the viewing geometry. The mass transfer
can cause the spin axis of the pulsar to align with the angular momentum
vector of the binary system, and thus, the pulsar may not be detected.
Although the parameter space becomes larger, the mass-transfer cycles
constantly cause the spin axis of the pulsar to align with the angular
momentum vector of the binary, which may make it difficult to observe (e.g.,
Guillemot & Tauris, 2014).

##  5 Conclusions

We studied the irradiation effect and calculated a series of binary models to
present the potential parameter space for binary radio pulsars from a purely
binary evolutionary perspective. These models were compared with those that
evolve without the irradiation effect. The inclusion of irradiation-induced
mass-transfer cycles results in an expanded parameter space compared to models
without irradiation, as illustrated in Figs. 7, 8, and 9. The parameter space
is very limited when the irradiation effect is not considered or when the
irradiation is weak. This may be one reason why binary radio pulsars with
giant companions are so difficult to find. Our results suggest that radio sub-
MSPs might be accompanied by giant stars when the irradiation is indeed strong
in the evolution of LMXBs. We also found a relation between the final mass of
WDs and the irradiation efficiencies (see Fig. 10).

The discovery of binary radio pulsars with giant companions can significantly
constrain theories of binary evolution. Our findings may be verified by future
observations or guide observers in the search for these objects.

###### Acknowledgements.

S.L. sincerely thanks Hai-Liang Chen, Zheng-Wei Liu, Hong-Wei Ge and Wei-Tao
Zhao for helpful discussions. This work is supported by the National Natural
Science Foundation of China (Nos. 12288102 and 12333008) and National Key R&D
Program of China (No. 2021YFA1600403). X.M. acknowledges support from the
Yunnan Ten Thousand Talents Plan - Young & Elite Talents Project, and the CAS
‘Light of West China’ Program. X.M. acknowledges support from International
Centre of Supernovae, Yunnan Key Laboratory (No. 202302AN360001), the Yunnan
Revitalization Talent Support Program-Science & Technology Champion Project
(NO. 202305AB350003), Yunnan Fundamental Research Projects (Nos.
202401BC070007 and 202201BC070003) and the science research grants from the
China Manned Space Project. The authors also acknowledge the “PHOENIX
Supercomputing Platform” jointly operated by the Binary Population Synthesis
Group and the Stellar Astrophysics Group at Yunnan Observatories, Chinese
Academy of Sciences.

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