# Linear change and minutes variability of solar wind velocity revealed by
FAST

Li-Jia Liu,1,2,3 Bo Peng,1,6 Lei Yu,1,3 Bin Liu,1,6 Ji-Guang Lu,1 Ye-Zhao Yu,4
Hong-Wei Xi,1,3 E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Ming Xiong,3,5 O. Chang,7

1CAS Key Laboratory of FAST, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, China
2State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing
100190, China
3University of Chinese Academy of Sciences, Beijing, 100049, China
4School of Physics and Electronics, Qiannan Normal University for
Nationalities, Duyun, 558000, China
5National Space Science Center, Chinese Academy of Sciences, Beijing 100190,
China
6Hebei Key Laboratory of Radio Astronomy Technology, Hebei 050081, China
7RAL Space, United Kingdom Research and Innovation - Science and Technology
Facilities Council - Rutherford Appleton Laboratory,
Harwell Campus, Oxfordshire, OX11 0QX, UK

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Observation of Interplanetary Scintillation (IPS) provides an important and
effective way to study the solar wind and the space weather. A series of IPS
observations were conducted by the Five-hundred-meter Aperture Spherical radio
Telescope (FAST). The extraordinary sensitivity and the wide frequency
coverage make FAST an ideal platform for IPS studies. In this paper we present
some first scientific results from FAST observations of IPS with the $L$-band
receiver. Based on the solar wind velocity fitting values of FAST observations
on September 26-28, 2020, we found that the velocity decreases with increasing
frequency linearly, which has not yet been reported in literature. And we have
also detected a variation of solar wind velocity on a timescale of 3-5
minutes, which imply the slow change of the background solar wind, a co-
existence of high- and low-speed streams, or a reflect of the quasi-periodic
electron-density fluctuations.

###### keywords:

scattering – methods: data analysis – sun: solar wind – sun: heliosphere

††pubyear: 2015††pagerange: Linear change and minutes variability of solar
wind velocity revealed by FAST–Linear change and minutes variability of solar
wind velocity revealed by FAST

##  1 Introduction

Interplanetary scintillation (IPS) is caused by the density irregularities in
the solar wind plasma(Hewish et al., 1964). Observations of IPS of compact
radio sources can be used to study the solar wind and the angular structures
of the scintillating radio sources (Hewish & Symonds, 1969; Armstrong & Coles,
1972; Manoharan & Ananthakrishnan, 1990). Compared with in-situ measurement
provided by spacecraft, ground based observations of IPS can give information
over all the heliographic latitudes and cover a wide range of solar
elongations from the Sun (Dennison & Hewish, 1967). Therefore, it provides an
efficient way to study the heliospheric physics.

After IPS phenomenon was discovered in the 1960s (Clarke, 1964; Hewish et al.,
1964), many instruments have been dedicated to the observations of IPS, such
as the Ooty Radio Telescope (Manoharan & Ananthakrishnan, 1990), the multi-
station system of the Institute for Space-Earth Environmental Research (Asai
et al., 1995), the Mexican Array Radio Telescope (Mejia-Ambriz et al., 2010)
and the Puschino radio telescope (Artyukh et al., 2003). After decades of
studies, the results deduced from the observations of IPS can be used to form
the three-dimensional (3D) distribution sky map for solar wind velocity and
electron density (Jackson et al., 1998, 2019; Kojima et al., 1998), and can
also be used to forecast the arrival time of coronal mass ejection to the
Earth environment (Tokumaru et al., 2003; Manoharan, 2006; Iwai et al., 2019,
2021).

The Five-hundred-meter Aperture Spherical radio Telescope (FAST), which is
located in Guizhou Province of China, was completely constructed in 2016. The
high sensitivity and wide frequency range of FAST make it powerful to carry
out observations of IPS.

This paper is laid out as follows: IPS observations with FAST are presented in
Section 2. The data reduction and observation results are described in Section
3. Sections 4 and 5 present the discussions and concluding remarks,
respectively.

##  2 Observation of IPS with FAST

FAST has an aperture diameter of 500 m and an illuminated aperture of 300 m.
As the largest single-dish radio telescope in the world, FAST has not only a
high sensitivity, but also a high time and a high frequency resolution, which
makes it the most powerful telescope to detect weak or short timescale signals
(Peng et al., 2000).

During the commissioning, the Ultra-Wideband (UWB) receiver was installed in
the feed cabin between September, 2016 and July, 2018. This receiver covers a
wide frequency range of 1.35 GHz from 270 to 1620 MHz. The huge frequency
coverage allows it to carry out the single station single frequency (SSSF) and
single station dual-frequency (SSDF) analysis modes observations of IPS
simultaneously, which makes it possible to compare the solar wind velocities
deduced from the two modes. Based on which, an optimized model-fitting method
for SSSF analysis mode was developed and the frequency difference of the SSDF
analysis mode were discussed. A pilot observation of IPS has been carried out,
which was presented in an earlier paper (Liu et al., 2021).

Since July 2018, the 19-beam receiver at $L$-band was equipped on FAST. The
frequency coverage of the receiver is from 1.05 to 1.45 GHz with a system
temperature of 20 K (Qian et al., 2020).

IPS phenomenon is strong in the area close to the Sun, but is weak in most of
the interplanetary space. Therefore, the IPS study requires high sensitive
observations made by large telescopes like FAST. The lower frequencies are
useful for studying the regions that are further away from the Sun, and the
higher frequencies are useful for those very close to the Sun (Cohen &
Gundermann, 1969). The frequency coverage of the $L$-band receiver on FAST is
capable to probe as close as 18 solar radius from the Sun (Zhang, 2007).

The FAST telescope has a digital acquisition back-end, which is the
reconfigurable open-architecture computing hardware-version 2 (ROACH2) board,
developed by the Collaboration for Astronomy Signal Processing and Electronics
Research (CASPER). When performing observations of IPS, the data recorded by
the central beam of the 19-beam $L$-band receiver was splitted into 4096
channels. The wide frequency coverage and the high frequency resolution of the
FAST $L$-band receiver allow it to perform a dynamic spectrum of the IPS
observations, which can be used to study the variations and microstructures of
the solar wind (Coles & Filice, 1984; Fallows et al., 2013).

Radio source 3C 286 was chosen as a target source, which has also been
observed during September 26-28, 2020. The solar elongation of 3C 286 in three
days was $37^{\circ}$, which was in the weak scattering regime.

The central-beam of the 19-beam receiver was used for the observations of IPS.
The original sampling interval recorded by the digital back-end was 0.2 ms.
When carrying out the IPS data reduction, a new time interval with integration
time of 20 ms was generated by performing summation over every 100 raw
sampling points along time. The telescope tracking on our target 3C 286 stayed
for 30, 60 and 30 minutes respectively, as summarized in Table 1,where
observing dates, on-source and off-source time are listed in columns 1 to 3.

Table 1: Summary of observations with FAST. $Dates$ | on-source time | off-source time |
---|---|---|---
| (min) | (min) |
Sept. 26, 2020 | 30 | 5 |
Sept. 27, 2020 | 60 | 5 |
Sept. 28, 2020 | 30 | 5 |

##  3 Data Reduction and Results

In this Section, the data reduction method and power spectrum results are
presented, including the mitigation of radio frequency interferences (RFI)
mitigation, forming and model-fitting the SSSF mode power spectrum.

###  3.1 Data Reduction

The raw data recorded by the $L$-band receiver is partially contaminated by
RFIs in some channels, which should be removed. The whole frequency band is
divided into multi-channels, which allows us to locate the RFI precisely. The
steps are briefly described below:

1\. The original data recorded by FAST back-end was integrated to 20 ms each
point by performing summation every 100 raw sampling data along time;
2\. RFI channels are identified and removed by the method of random sample
consensus (RANSAC), which has been described in Liu et al. (2021);
3\. The mean power spectrum is calculated for a certain time interval, usually
of 50 s;
4\. The mean power spectrum is subtracted to remove the systematic
instrumental response between different frequencies.

Fig. 1 presents an example of the RFI removing process. 3C 286 was chosen as a
target source for FAST observation on September 27, 2020. Fig. 1 (a), (b), (c)
and (d) show the spectrum of integrated raw data, the spectrum after removing
the RFI by RANSAC, the 50 s mean spectrum and the spectrum which has been
removed by subtracting Fig. 1 (c). It can be seen that after the RFI
mitigation, the interferences throughout the frequency bandpass had been
obviously removed.

Figure 1: The process of data reduction. The target source was 3C 286 which
was observed on September 27, 2020. (a)-(d) shows the power spectrum of raw
data, the spectrum after removing the RFI by RANSAC, the mean spectrum and the
spectrum after subtracting the mean of different frequencies respectively.

The data set after RFI removal was then integrated on frequency. The original
sub-band frequency span was 0.12 MHz, when carrying out IPS data processing,
the sub-band will be integrated to 10 MHz to form a new data set. Then the new
time series with a bandwidth of 10 MHz and an integration time of 20 ms will
be used for further data reduction. The SSSF mode data processing is
illustrated below:

1\. The new time series is divided into blocks of length of 512 data points
corresponding to a duration time of about 10 s. Then each block is subtracted
by its own average;
2\. The data points of each block from step 1 is smoothed by using Hanning
window. The fast Fourier transform is then applied to get power spectra;
3\. The spectra of 8 sub-sets are averaged and normalized to form the SSSF
mode power spectrum.

###  3.2 Power Spectrum Results

In our IPS studies, the solar wind velocity was deduced from a model-fitting
method. The fitting parameters include 1) solar wind velocity ($V$); 2) the
anisotropic axial ratio (AR) (Yamauchi et al., 1998); 3) the spectral index of
electron density fluctuation ($\alpha$) and 4) the size of source ($\theta$).

For SSSF analysis mode, in the weak scattering regime, equation (1) gives the
temporal power spectrum $P(f)$ with $C=(2\pi r_{e}\lambda)^{2}$ which depends
on the observing wavelength $\lambda$, $Z$ is the distance to the scattering
screen, $V_{p}$ is the solar wind velocity projected on to the plane
perpendicular to the direction of line-of-sight $z$, and $r_{e}$ is the
electron radius. The spectrum of the electron-density fluctuations
$\Phi_{ne}\propto q^{-\alpha}$, where $q=\sqrt{(q_{x}^{2}+q_{y}^{2})}$ is the
wave number. $f=\frac{q_{x}V_{p}}{2\pi}$ is the temporal frequency,
$F_{diff}=4sin^{2}(\frac{q^{2}z\lambda}{4\pi})$ is the Fresnel propagation
filter and $F_{source}=\exp(-(\frac{qz\theta}{2.35})^{2})$ is the squared
modulus of the source visibility (Manoharan & Ananthakrishnan, 1990; Mejia-
Ambriz et al., 2015).

$\centering
P(f)=C\int^{Z}_{-Z}\frac{1}{V_{p}}dz\int^{+\infty}_{-\infty}\Phi_{ne}F_{diff}F_{source}dq_{y}.\@add@centering$
(1)

In our previous studies (Liu et al., 2021), an optimized model-fitting method
has been developed, which can obtain the fitting value of four solar wind
parameters (V, AR, $\alpha$, $\theta$) and the fitting errors simultaneously.
The parameters fitting results of our method are deterministic, which means a
set of initial parameters corresponding to an identical result. We applied
multi-initialization strategy (Ugray et al., 2007) to find the global minimum,
which guarantees the fitting convergence in the global minimization. Fig. 2
demonstrates one of the fitting results, as an example of radio source 3C 286
which are resulted from a continuous 10 MHz bandwidth sub-bands data using 80
s time length on September 28, 2020. The dots and solid lines show the fitting
values and the errors of the four parameters.

Figure 2: The solar wind parameters of source 3C 286 resulted from a
continuous 10 MHz bandwidth sub-band data using 80 s time length on September
28, 2020. The four fitting parameters velocity, AR, $\alpha$ and source size
are displayed from top to bottom panels. The dots and solid lines represent
the fitting values and the error bars (95$\%$ confidence intervals).

##  4 Discussions

In this section, a linear changing trend and a minutes scale oscillation of
the solar wind velocity are presented and discussed.

The velocities through the whole frequency band (Channels affected by strong
RFI are rejected) were fitted by a weighted linear function, which is given
below in equation (2)

$V=k\times x+b$ (2)

Where $x=1+\frac{f-1050}{10}$ is the serial number of each sub-set, $f$ is the
observing frequency of each sub-band in MHz, and $V$ is the solar wind
velocity in $km/s$. The dots and the vertical lines are the fitted solar wind
velocities and errors, and the dashed line shows the linear fit of the
velocities with error. The best fit parameters of the dashed line in Fig. 3
are $k=-2.8\pm 0.3$ and $b=655.8\pm 6.0$. The same procedure was also
performed on the data observed during the three days. Totally, we got 80 sets
of values for $k$ and $b$.

Figure 3: An example of the solar wind velocities through sub-bands. The radio
source was 3C 286 observed by FAST on September 27, 2020. The dots and the
vertical lines are the fitted solar wind velocities and errors, and the dashed
line shows the linear fit of the velocities with error.(Channels affected by
strong RFI are rejected.)

The framework of classical IPS theory is based on some standard assumptions,
including the temporal intensity variation of a scintillating source is mainly
due to the bulk motion of the solar wind across the line of sight (Oberoi,
2000). According to the definition of IPS, the velocity derived from IPS
phenomenon reflects the bulk speed of the solar wind, therefore over the whole
observing frequency range, the velocities at each frequency should be the same
only with a small fluctuation. However, Fig. 3 clearly shows a linear trend
over the whole observing frequency, which has not yet been reported in
literature, and this represents one of the studies that one can make using the
capabilities of FAST. The histogram in Fig. 4 displayed the distribution of
$k$ during September 26-28, 2020. The statistics of $k$ through the three days
show that over $80\%$ of $k$ are negative. A similar phenomenon has also been
seen from the observations of IPS by the Arecibo Telescope also using $L$-band
(Arecibo team, private communications).

There are three possible origins causing the linear trend: 1) the observing
facility, 2) the model-fitting pipeline, 3) the intrinsic properties of the
solar wind plasma.

Figure 4: The distribution of $k$ during September 26-28, 2020.

In order to figure out the real cause of this trend, the three possibilities
were investigated one by one. After the RFI removal and the baseline flatten
by the mean subtract process, the difference caused by the instrument system
has been eliminated, as shown in Fig. 1. So, the first possibility can be
excluded. For the purpose to investigate the influence of the model-fitting
pipeline, a simulation was carried out. We adopted a characteristic solar wind
velocity value from our observations as the simulated one. The different S/N
ratios in different frequency channels were achieved by inserting the scaled
data from our off-source observation. The simulated data was processed by the
same model-fitting method, which was applied to our real data. Then we can
inspect the influence of the method on our result by comparing the recovered
value with input value. The simulation procedure is described below:

1\. Generate a series of theoretical power spectra according to equation (1)
with a same solar wind velocity value over the observing frequency range from
1.05 to 1.45 GHz.
2\. Carry out inverse FFT to the spectra to create the time domain data.
3\. Add the real off-source data observed by FAST to each time domain data
with a linear change from 14 dB at the lowest frequency 1.05 GHz to 11 dB at
the highest frequency 1.45 GHz.
4\. Perform the model-fitting pipeline with the time domain data to obtain the
fitting values and errors.

Fig. 5 displays the model-fitting results of the simulated data. The dots and
solid lines represent the fitting values and the errors. The dashed line in
the middle shows the linear fit with the error of the velocities. The $V$ used
for simulation is $650\,km/s$ and $k=-0.04\pm 0.2$. The solar wind velocities
through the whole frequency range demonstrate an excellent consistency, which
means the velocity trend shown in Fig. 3 is not caused by the model-fitting
method.

Figure 5: The linear fit result of the simulated data. The dots and solid line
represent the fitting values and the errors. The dashed line in the middle
shows the weighted linear fit of the velocities.

Both of the first two possible causes are unable to result in this velocity
changing trend shown in Fig. 3. So the different solar wind velocities
measured in different FAST frequency channels could be caused by some
intrinsic characteristics of the solar wind, requiring a theoretical work to
further study the radiative transfer mechanism with a physics-based turbulent
velocity field.

The solar wind magnetohydrodynamic turbulence, though overall balanced at a
macro scale, is always locally imbalanced in creating patches of positive and
negative cross-helicities (Perez & Boldyrev, 2008). The solar wind turbulence
cascade could be intermittently decaying towards these Fresnel-scale
irregularities, which might be ascribed to a gradual misalignment of electron
density irregularities at different spatial Fresnel scales ($\sqrt{Z\lambda}$,
where $Z\approx 1.2\times{10}^{11}\,m$, and $0.21\,m<\lambda<0.29\,m$) of
about 150 to 190 km. A big turbulence eddy associated with a large density
irregularity tends to be a bit faster than its nested small eddies, resulting
in higher velocities at lower frequencies, which causes linear change trend of
the solar wind velocity. Such an inconsistency in cross-scale density
irregularities is likely caused by a cascade of solar wind turbulence with a
minutes scale.

Since most of the solar wind velocities show a linear trend, the velocities at
some certain frequencies can characterize the change in velocity over time.
Fig. 6 presents the solar wind velocity change during September 26-28, 2020.
The black dots, red circles, blue squares, and solid lines represent the
velocities and their fitting errors obtained at 1.06 GHz, 1.25 GHz and 1.35
GHz, respectively (the data affected by strong RFI are rejected). The
velocities deduced from the three different frequencies have also revealed the
phenomenon demonstrated in Fig. 3, which is the velocity decreases with the
increasing observing frequency. Furthermore, the structure function of the
velocities show that there appears a 3-5 minute-scale oscillation in the solar
wind velocity. The velocity oscillation may imply that, during our observation
with FAST, a slow change of the background solar wind, an evidence of high-
and low-speed streams, the quasi-periodic electron-density fluctuations
(Efimov et al., 2012), or the presence of slow-mode wave in the solar wind
which is caused by the parametric decay instability of Alfvén wave (Tenerani &
Velli, 2013; Shoda et al., 2018; Réville et al., 2018; Shoda et al., 2019).

Figure 6: The solar wind velocity variations through September 26-28, 2020.
The black dots, red circles, blue squares and solid lines represent the
velocities and the fitting errors obtained at 1.06 GHz, 1.25 GHz and 1.35 GHz,
respectively (the data affected by strong RFI are rejected).

##  5 Concluding remarks

In conclusion, the Five-hundred-meter Aperture Spherical radio Telescope
(FAST) has outstanding advantages to carry out observations of IPS. The high
sensitivity allows it to discover weak signals that can not be detected by
other telescopes. The wide coverage of observing frequency makes it capable of
effectively removing the RFI and performing dynamic spectrum analysis.

Based on the observations of IPS by FAST, we observed a linear trend of solar
wind velocity change in the entire frequency band of FAST 19-beam receiver,
and an oscillation in minutes scale. The linear trend represents a new result,
which cannot be done by most of the existent radio telescopes. According to
the FAST observation, the interplanetary solar wind flow is variable on
timescale of some minutes, and the solar wind density irregularities are not
strictly co-moving with the speed of its bulk flow. These observations of the
radio source 3C 286 during September 26-28, 2020 show a very exciting IPS
prospect, but also revoke the necessity for more studies.

## Acknowledgements

This work is supported by the National Key $R\&D$ Program of China under grant
number 2018YFA0404703, the Open Project Program of the CAS Key Laboratory of
FAST, NAOC, Chinese Academy of Sciences, the Specialized Research Fund for
State Key Laboratories, the basic research program and project of Yunnan
province of China (2019FB009) as well as the National Natural Science
Foundation of China under grant number 41874205. The authors thank M.M. Bisi
(RAL Space, United Kingdom Research and Innovation - Science and Technology
Facilities Council), for his conversations about the Arecibo comparisons. We
are grateful to M. Tokumaru (Institute for Space-Earth Environmental Research,
Nagoya, Japan) and Jian-Sen He (School of earth and space sciences, Peking
University), for their helpful discussions. The authors also thank Dr. Juan
Arratia, for his help in polishing the English. This work made use of the data
from FAST (Five-hundred-meter Aperture Spherical radio Telescope). FAST is a
Chinese national mega-science facility, operated by the National Astronomical
Observatories, Chinese Academy of Sciences.

## Data Availability

According to the FAST data policy, the data underling this article has been
released. Please contact FAST Data Center<EMAIL_ADDRESS>for the data.

## References

  * Armstrong & Coles (1972) Armstrong J. W., Coles W. A., 1972, J. Geophys. Res., 77, 4602
  * Artyukh et al. (2003) Artyukh V. S., Kopylov A. I., Kopylova F. G., 2003, A&A, 403, 555
  * Asai et al. (1995) Asai K., Ishida Y., Kojma M., Maruyama K., Misawa H., Yoshimi N., 1995, Journal of geomagnetism and geoelectricity, 47, 1107
  * Clarke (1964) Clarke M., 1964, PhD thesis, Cambridge University
  * Cohen & Gundermann (1969) Cohen M. H., Gundermann E. J., 1969, ApJ, 155, 645
  * Coles & Filice (1984) Coles W. A., Filice J. P., 1984, Nature, 312, 251
  * Dennison & Hewish (1967) Dennison P. A., Hewish A., 1967, Nature, 213, 343
  * Efimov et al. (2012) Efimov A. I., Lukanina L. A., Samoznaev L. N., Rudash V. K., Chashei I. V., Bird M. K., Pätzold M., MEX VEX R. R. S. T., 2012, Advances in Space Research, 49, 500
  * Fallows et al. (2013) Fallows R. A., Asgekar A., Bisi M. M., Breen A. R., ter-Veen S., 2013, Sol. Phys., 285, 127
  * Hewish & Symonds (1969) Hewish A., Symonds M., 1969, Planetary and Space Science, 17, 313
  * Hewish et al. (1964) Hewish A., Scott P., Wills D., 1964, Nature, 203, 1214
  * Iwai et al. (2019) Iwai K., Shiota D., Tokumaru M., Fujiki K., Den M., Kubo Y., 2019, Earth, Planets, and Space, 71, 39
  * Iwai et al. (2021) Iwai K., Shiota D., Tokumaru M., Fujiki K., Den M., Kubo Y., 2021, Earth, Planets, and Space, 73, 9
  * Jackson et al. (1998) Jackson B. V., Hick P. L., Kojima M., Yokobe A., 1998, J. Geophys. Res., 103, 12049
  * Jackson et al. (2019) Jackson B. V., Odstrcil D., Hick P. P., Buffington A., Tokumaru M., Bisi M. M., 2019, in AGU Fall Meeting Abstracts. pp SH43D–3362
  * Kojima et al. (1998) Kojima M., Tokumaru M., Watanabe H., Yokobe A., Asai K., Jackson B. V., Hick P. L., 1998, J. Geophys. Res., 103, 1981
  * Liu et al. (2021) Liu L.-J., et al., 2021, MNRAS, 504, 5437
  * Manoharan (2006) Manoharan P. K., 2006, Sol. Phys., 235, 345
  * Manoharan & Ananthakrishnan (1990) Manoharan P., Ananthakrishnan S., 1990, Monthly Notices of the Royal Astronomical Society, 244, 691
  * Mejia-Ambriz et al. (2010) Mejia-Ambriz J. C., Villanueva-Hernandez P., Gonzalez-Esparza J. A., Aguilar-Rodriguez E., Jeyakumar S., 2010, Sol. Phys., 265, 309
  * Mejia-Ambriz et al. (2015) Mejia-Ambriz J. C., Jackson B. V., Gonzalez-Esparza J. A., Buffington A., Tokumaru M., Aguilar-Rodriguez E., 2015, Sol. Phys., 290, 2539
  * Oberoi (2000) Oberoi D., 2000, PhD thesis, Department of Physics Indian Institute of Science
  * Peng et al. (2000) Peng B., Strom R. G., Nan R., Ma E., Ping J., Zhu L., Zhu W., 2000, in van Haarlem M. P., ed., Perspectives on Radio Astronomy: Science with Large Antenna Arrays. p. 25
  * Perez & Boldyrev (2008) Perez J. C., Boldyrev S., 2008, ApJ, 672, L61
  * Qian et al. (2020) Qian L., Yao R., Sun J., Xu J., Pan Z., Jiang P., 2020, The Innovation, 1, 100053
  * Réville et al. (2018) Réville V., Tenerani A., Velli M., 2018, ApJ, 866, 38
  * Shoda et al. (2018) Shoda M., Yokoyama T., Suzuki T. K., 2018, ApJ, 860, 17
  * Shoda et al. (2019) Shoda M., Suzuki T. K., Asgari-Targhi M., Yokoyama T., 2019, ApJ, 880, L2
  * Tenerani & Velli (2013) Tenerani A., Velli M., 2013, Journal of Geophysical Research (Space Physics), 118, 7507
  * Tokumaru et al. (2003) Tokumaru M., Kojima M., Fujiki K., Yamashita M., 2003, in AGU Fall Meeting Abstracts. pp SH42C–0548
  * Ugray et al. (2007) Ugray Z., Lasdon L., Plummer J., Glover F., Kelly J., Marti R., 2007, INFORMS Journal on Computing, 19, 328
  * Yamauchi et al. (1998) Yamauchi Y., Tokumaru M., Kojima M., Manoharan P. K., Esser R., 1998, J. Geophys. Res., 103, 6571
  * Zhang (2007) Zhang X.-Z., 2007, Chinese J. Astron. Astrophys., 7, 712