Vol.00 (2022) No.0, 000–000

11institutetext: CAS Key Laboratory of Solar Activity, National Astronomical
Observatories of Chinese Academy of Sciences, Beijing 100012, China,
<EMAIL_ADDRESS>
22institutetext: School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100049, China

# Diagnostic functions of solar coronal magnetic fields from radio
observations

Baolin Tan 11 2 2

(Received 2022 month day; accepted 2022 month day)

###### Abstract

In solar physics, it is a big challenge to measure the magnetic fields
directly from observations in the upper solar atmosphere, including the
chromosphere and corona. Radio observations are regarded as the most feasible
approach to diagnose the magnetic field in solar chromosphere and corona.
However, because of the complexity and diversity of the emission mechanisms,
the previous studies have only presented the implicit diagnostic functions of
the magnetic field for specific mechanism from solar radio observations. This
work collected and sorted out all methods for diagnosing coronal magnetic
field from solar radio observations, which are expressed as a set of explicit
diagnostic functions. In particular, this work supplemented some important
diagnostic methods missed in other reviews. This set of diagnostic functions
can completely cover all regions of the solar chromosphere and corona,
including the quiet region, active region and flaring source regions. At the
same time, it also includes incoherent radiation such as bremsstrahlung
emission of thermal plasma above the quiet region, cyclotron and gyro-
synchrotron emissions of magnetized hot plasma and mildly relativistic
nonthermal electrons above the active regions, as well as coherently plasma
emission around flaring source regions. Using this set of diagnostic functions
and the related broadband spectral solar radio imaging observations, we can
derive the magnetic fields of almost all regions in the solar atmosphere,which
may help us to make full use of the spectral imaging observations of the new
generation solar radio telescopes (such as MUSER, EVOSA and the future FASR,
etc.) to study the solar activities, and provide a reliable basis for the
prediction of disastrous space weather events.

###### keywords:

Sun: magnetic fields – Sun: corona – Sun: radio emission

##  1 Introduction

In solar physics, there are many important unsolved mysteries, including the
coronal heating, the origin of solar eruptions, and the structure formation of
solar atmosphere, etc. As for the coronal heating mechanism, so far, more than
tens of models are proposed to explain the formation of the hot upper
chromosphere and very hot corona, which can be summarized into three classes:
(1) Wave-heating (AC mechanism, Davila 1987, De Pontieu et al. 2007, etc.),
(2) Magnetic reconnection heating (DC mechanism, Parker 1988, Sturrock 1999,
Rappazzo et al. 2008, etc.), and (3) Magnetic-gradient pumping mechanism (Tan
2014, Tan et al. 2020, Tan 2021). Although these models vary widely, one thing
they have in common is that they are so closely linked to the solar magnetic
fields that many people say there could be no hot corona without magnetic
fields. As for the solar eruptions, almost all of the source regions of solar
flares, coronal mass ejections (CMEs) and even the plasma jets are located in
the solar atmosphere above the photospheric regions with relative strong
magnetic fields, they are intrinsically rapid release of magnetic energy in
the chromosphere and corona (Hudson 2011, Fletcher 2011, etc.). In fact,
magnetic field dominates all the dynamics, structures and evolutionary
phenomena of the solar atmosphere, and even particle accelerations and
propagations. We may say that magnetic field ultimately controls the evolution
of everything in solar system, including the Sun itself and the disastrous
space weather events in the solar-terrestrial and the interplanetary space
environment. It is because of the magnetic field that the Sun becomes so
colorful and so interesting. Therefore, it is a high priority goal for
observational solar physics to measure or diagnose the magnetic fields in the
solar atmosphere.

Since coronal magnetic fields are so important in solar physics, then, how can
we get accurate information about them? So far, there are several possible
methods to obtain the information of coronal magnetic fields which can be
listed in following:

(1) Zeeman effect

It is well-known that we can directly measure the magnetic fields by using
Zeemen splitting effect of spectral lines (Hale 1908).
$\triangle\lambda_{B}\approx 4.67\times 10^{-13}g_{eff}\lambda^{2}B$, here,
$\triangle\lambda_{B}$ is split of the spectral line, $g_{eff}$ is the
effective Lande factor, $\lambda$ is the wavelength at Å, $B$ is the magnetic
field at Gauss. Despite the great successes of magnetic field measurements in
the solar photosphere and low chromosphere by applying this method at optical
and infrared wavelengths, because of the very high temperature and relatively
weak magnetic field, the reliable measurements of magnetic fields in the high
chromosphere and corona have still remained elusive (Casini, White & Judge
2017). Zeeman splitting observations of infrared coronal forbidden emission
lines Fe XIII 1075 nm have been successfully used to deduce the coronal
magnetic field with a sensitivity of about 1 G with 20$"$ spatial resolution
after 70 minutes of integration (Lin et al. 2004, Liu & Lin 2008), but this
method only works for measuring coronal magnetic fields outside the solar
limb. Due to high temperature and low density, the well-developed magnetic
field measurements of using Zeeman effect are hardly applicable to the upper
chromosphere and corona.

(2) Hanle effect

The Hanle effect (Hanle, 1924) is the modification of the linear polarization
of a spectral line scattered by a local magnetic field, may provide strong
diagnostics of regions of weak magnetic fields such as the quiet Sun and the
corona, where a number of spectral lines with different but complementary
sensitivity ranges are present (Ignace et al. 1997). It has been used to
determine the field strength and distribution of magnetic structures of the
solar prominence (Raouafi et al. 2016). However, the observation of the
associated coronal linear polarization is always very difficult and the
interpretation depends on very complicated physical processes.

(3) Faraday effect

The magnetic field in the high corona and the solar wind can be obtained from
the Faraday effect of linearly polarized celestial radio sources (Spangler
2005): the angle changes of polarization can be expressed as
$\triangle\varphi=C\lambda^{2}\int N_{e}B\cdot ds$, here, $C$ is a constant,
$N_{e}$ is the plasma density along the line of sight which need to be
obtained from other approach. The Faraday effect can be applied to diagnose
the magnetic fields in the high corona and interplanetary space, but it meets
many difficulties in the chromosphere and low corona (Hatanaka 1956).

(4) Magnetic-field-induced transition method

The magnetic-field-induced radiation originates from atomic transitions where
the lifetime of the upper energy level is sensitive to the local, external
magnetic field (Beiersdorfer et al. 2003, Li et al. 2015). Recently, based on
the theory of magnetic-field-induced transition (MIT), some people proposed to
apply the EUV spectral observations to derive the coronal magnetic field and
showed very good prospects (Li et al. 2021, Chen et al. 2021).

(5) Coronal seismology

Coronal seismology is mainly concerning the oscillations in coronal loops,
interpretation of the oscillations observed in terms of global MHD waves
allows to connect the period of the oscillations and the loops length with the
magnetic field strength in the loops. Therefore, the coronal seismology can be
applied to diagnose coronal magnetic fields (Nakariakov et al. 1999,
Nakariakov & Ofman 2001, De Moortel & Pascoe 2009, Andries et al. 2009).
Recently, many people applied the coronal seismology to obtain the global
magnetogram from 1.05 $R_{sun}$ to 1.35 $R_{sun}$ outside the solar disk (Yang
et al. 2020, Jess et al. 2016, etc.).

(6) Modeling extrapolations

Dulk & McLean (1978) proposed an estimation of the averaged magnetic field
distribution in corona. The most economical approach is to derive the coronal
magnetic fields by using the modeling extrapolations from the observed
photospheric magnetographs (Yan & Sakurai 2000, Metcalf et al. 2008, Jiang &
Feng 2013, Chifu et al. 2015, Zhu & Wiegelmann 2018, etc.). But the modeling
extrapolation can not provide the dynamical and fast variations of the coronal
magnetic fields.

(7) Radio diagnostics

Besides the above direct or indirect methods, the solar radio broadband
observations are regarded to be the most important approach for diagnosing the
magnetic field in the high chromosphere and corona. Essentially, we can see
almost everything occurred in solar chromosphere and corona by using radio
observations, including the magnetic fields (Dulk 1985, Bastian, Benz, & Gary
1998) from the face-on to the limb of the solar disk. Radio emissions are very
sensitive to both thermal and nonthermal processes in solar atmosphere,
including magnetic field and its variations, plasma instabilities, particle
accelerations and propagations. Additionally, solar radio observations span
frequencies of more than five orders of magnitude, from sub-millimeter waves
to hectometer waves, and the emission source region locates from the low
chromosphere to the very high corona, and even the interplanetary space,
including solar quiet regions, active regions, and flaring eruptive source
regions. The emission source involves a wide variety of plasmas, including the
dense partial ionized chromospheric plasma and the fully ionized hot dilute
coronal plasmas, as well as the energetic nonthermal electrons. Additionally,
radio imaging observation can obtain the solar full-disk images with sub-
second cadence, much faster than any other approaches and is very useful for
providing fast dynamic images of the source regions. For example, the Mingantu
Spectral Radioheliograph (MUSER) can capture nearly 600 solar radio full-disk
images every 0.2 seconds and every 25 MHz in the frequency range of 0.4-15.0
GHz at frequency intervals of 25 MHz (Yan et al. 2021). Another important
instrument is the Expanded Owens Valley Solar Array (EVOSA) which can also
provide high-cadence, spatially-resolved spectral imaging observations in the
frequency range of 1-18 GHz (Gary et al. 2018, Fleishman et al. 2020, Chen et
al. 2020).

The radio emission mechanism depends on the energy of emitting electrons,
which may be different in different frequency ranges and in different source
regions. In the solar quiet region, the emitting electrons are mainly thermal
equilibrium and the radio emission comes from the bremsstrahlung process. In
active regions, the radio emission may produce from cyclotron of thermal
electrons at low harmonics of the electron gyrofrequency. And in flaring
regions, the emission may be contributed from multiple emission mechanisms,
including the gyro-synchrotron emission of mildly relativistic electrons,
synchrotron emission of nonthermal high-energy electrons, and coherent plasma
emission triggered by some plasma instabilities and nonthermal energetic
electrons. Due to the diversity of emission mechanisms and the variability in
different regions and at different frequency range, the radio observations
with high temporal, spatial and spectral resolutions are necessary for
obtaining the maps of magnetic fields in chromosphere and corona,
respectively.

Many people have already tried to derive the magnetic fields of solar
chromosphere and corona from radio observations (Dulk 1985, Gelfreikh et al.
1987, Kundu 1990, Zhou & Karlicky 1994, Huang 2006, Huang et al. 2013,
Wiegelmann et al. 2014, Yan et al. 2020, etc.). There are several review
papers elaborated on this topic (Dulk & McLean 1978, Kundu 1990, Kruger &
Hildebrandt 1993, White 2005, Casini, White, Judge 2017, Alissandrakis & Gary
2021). However, so far, most of them are fragmented and mainly based on the
gyro-synchrotron mechanism, and presented implicit functions of the magnetic
field with respect to the parameters of solar radio observations (Dulk & Marsh
1982, Huang 2006). These implicit functions are not convenient in practice to
derive magnetic fields from the solar lower chromosphere to the higher corona
from the observations. Even the latest review article has still missed some
very important diagnostic methods. As the new generation solar radio
telescopes come into operation, such as MUSER, EVOSA and the future FASR cover
a very large space from the solar chromosphere to the corona, and has very
high temporal, spatial and spectral resolutions (Yan et al. 2009, Yan et al.
2021), it is urgently needed to provide a complete set of explicit diagnosing
functions of magnetic field from the radio observations which may cover all
regions above solar disk and most frequency range of the radio emissions.

This paper attempts to collect and compile all the methods of radio coronal
magnetic field diagnosis so far, and summarizes them in the form of explicit
functions, which may cover the full solar disk, including the solar quiet
region, active region, and flaring source regions, so that they can be easily
applied to the magnetic field diagnosis of all regions of the solar
atmosphere. Section 2 will address the criterion of emission mechanism in a
given radio frequency range and target region. Section 3 presents the explicit
diagnosing functions of magnetic fields from solar radio observations.
Finally, some conclusions and discussion of the diagnostic procedure are
summarized in Section 4.

##  2 Emission mechanisms and the criterions in solar chromosphere and corona

As we know, solar radio emission comes from a variety of physical processes,
including bremsstrahlung, cyclotron, gyro-synchrotron, synchrotron, and
coherent plasma emissions. The emission may produce at different frequency
range in different altitudes above the same region of the sun’s surface, and
the emission mechanism may generate from different radiation processes. At the
same time, the emission mechanism may be different even at the same frequency
range but in different locations in the solar atmosphere (Dulk 1985, Bastian,
Benz, & Gary 1998). Therefore, it is very important to distinguish the solar
radio emission mechanism for diagnosing the magnetic fields at the given
frequency range and location. According to the physical conditions (including
the possible magnetic field strength, plasma density and temperature, thermal
or nonthermal electrons, etc.), the diagnostic target area can be divided into
three kinds of distinct regions: quiet region, active region, and flaring
region. Here, we try to present the main emission mechanism in different
regions and their criterions from radio observations.

###  2.1 Emission in and above the solar quiet regions

The basic feature of the atmosphere in and above the Sun’s quiet regions is
that it has only a weak magnetic field and thermal plasma without nonthermal
electrons. This nature implies that the emission is contributed mainly from
the bremsstrahlung mechanism (Dulk, 1985). The bremsstrahlung emission is
produced when individual electrons are deflected in the coulomb fields of
ions. The emission frequency ($f$) is from the plasma cutoff frequency
($f_{pe}\approx 9.0n_{e}^{1/2}$) up to $\frac{v}{d}$,

$f_{pe}<f<\frac{v}{d}.$ (1)

Here, $v$ is the velocity of electron and $d$ is the impact parameter (the
distance from ion). In thermal plasma, $d$ is in the range from Laudau length
($\lambda_{L}\approx 1.67\times 10^{-5}T_{e}^{-1}$) to Debye length
($\lambda_{D}\approx 69(\frac{T_{e}}{n_{e}})^{1/2}$), and the velocity
$v\approx 3.89\times 10^{3}T_{e}^{1/2}$, $n_{e}$ is the plasma density, and
$T_{e}$ is the temperature. The collision is far-encounter when the impact
parameter is in the range of $b_{0}<d<\lambda_{D}$ (here, $b_{0}\approx
n_{e}^{-1/3}$ is the averaged distance between electrons), and it is close-
encounter when the impact parameter is in the range of $\lambda_{L}<d<b_{0}$.
Generally, the far-encounters are much more important than the rarely close-
encounters, and $\frac{v}{b_{0}}\approx 3.89\times
10^{3}n_{e}^{1/3}T_{e}^{1/2}$. Therefore, the solar bremsstrahlung emission
mainly occurs in the following frequency range,

$9.0n_{e}^{1/2}<f<3.89\times 10^{3}n_{e}^{1/3}T_{e}^{1/2}.$ (2)

For example, for the typical parameters in coronal plasmas, $n_{e}\sim
10^{16}$ m-3, $T_{e}\sim 10^{6}$ K, then the bremsstrahlung emission will
occur in a very wide frequency range from 900 MHz to 1.2 THz. However, in hot
plasma or with nonthermal energetic electrons, the close-encounter will become
considerable, the upper limit of frequency of the bremsstrahlung emission may
approach to $\frac{v}{\lambda_{L}}$ and even close to $\frac{v}{\lambda_{d}}$
($\lambda_{d}$ is the de Broglie wavelength of the electron), and this may
produce X-ray emission and even hard X-ray emissions. These facts indicate
that bremsstrahlung emission has a very wide frequency range. The
bremsstrahlung is an incoherent emission process whose brightness temperature
($T_{b}$) does not exceed the thermal temperature of the radiating background
plasma ($T_{e}$) (Benz 2009, Nindos 2020).

The emissivity of bremsstrahlung process is proportional to
$n_{e}^{2}T_{e}^{-1/2}$, so the plasmas with low temperature and high density
are favorable for bremsstrahlung emission (Raulin & Pacini 2005). Because both
impact parameters and electron velocities are distributed continuously over a
wide range, the solar bremsstrahlung emission generally appears as broad band
continuum with long duration on the observed spectrum, and the peak frequency
depends on the emission measure ($\propto n_{e}^{2}$).

In brief, the criteria of bremsstrahlung can be concluded as: $T_{b}\leq
T_{e}$, broadband continuum, long duration, weakly or even non-polarization,
and relatively weak magnetic fields.

###  2.2 Emission in and above the active regions

Active regions (ARs) are the most prominent manifestation of the large-scale
solar magnetic field in the photosphere of the Sun (Shibasaki et al. 2011).
The active regions have relatively strong magnetic field (200 – 2000 G) and
large number of hot plasma loops which may occupied an area extended from
$10^{4}$ km to more than $10^{5}$ km (Seehafer 1986). As magnetic cyclotron
emission is approximately proportional to $n_{e}T_{e}^{\alpha}B^{\beta}$ ($B$
is magnetic field strength, $\alpha>1$, $\beta>1$), therefore, in and above
the solar active region, the bremsstrahlung emission tends to be negligible
and the emission is dominated by cyclotron of thermal electrons (Bastian et
al. 1998, Nindos et al. 2008) which is produced in thin layers around iso-
Gauss surfaces where the magnetic field strength is just fitted to the
observing frequency equal to a harmonic of the local gyrofrequency (Lee 2007).
The emission frequency can be simply expressed as the resonance condition
(Melrose 1980, Robinson & Melrose 1984, Dulk 1985),

$f=\frac{n(1+\mu\beta cos\alpha cos\theta)}{\gamma}f_{B}=sf_{B}.$ (3)

Here, $f_{B}\approx 2.8\times 10^{6}B$ is the electron gyrofrequency at unit
of Hz, here the unit of magnetic field strength $B$ is Gauss (G). The typical
value of $B$ is about 100 - 1000 G in solar active region, and the typical
frequency $f_{B}$ is in the range from 280 MHz to 2.8 GHz. $n$ is an integer
related to the energy of the radiated electrons. $\mu$ is the index of
refraction, and $\beta=v/c$. $\gamma$ is the Lorentz factor of electrons,
$\alpha$ is the angle between electron velocity and the magnetic field line
(pitch angle) and $\theta$ is the angle between the magnetic field line and
the line-of-sight. $s=\frac{n(1+\mu\beta cos\alpha cos\theta)}{\gamma}$ is
called harmonic number and not an integer. During the period without solar
flares in the active region, the radiating electrons are nonrelativistic
thermal electrons, for example in the typical coronal plasma in and above the
active region, $T_{e}\sim 10^{6}$ K, $\beta\sim 10^{-2}$, $\gamma\sim 1.0$,
and $\mu\sim 1.0$, then we may have $n\approx s$. The emission mainly occurs
at the fundamental frequency ($s=1$) or low harmonics ($s=$ 2 or 3) which is
called as cyclotron emission. The emission at fundamental frequency is mainly
along the direction of magnetic field, and the low harmonics emission is at
the moderate angle with respect to the magnetic field.

Generally, the integer $n$ is approximated as $n\sim\gamma^{3}sin\theta$. Then
the harmonic number ($s$) can be expressed as:

$s=\gamma^{2}sin\theta(1+\mu\beta cos\alpha
cos\theta)\approx\gamma^{2}sin\theta.$ (4)

With the increase of electron’s kinetic energy ($\propto\gamma^{2}$), the
harmonic number $s$ will increase rapidly. As for the mildly relativistic
electrons, $s>10$, which is called gyrosysnchrotron emission. Although
cyclotron and gyrosysnchrotron radiation are emitted at discrete frequencies,
they generally give rise to a broadband continuum due to the variations of
magnetic field with height (Kakinuma & Swarup 1962, Fleishman & Melnikov
2003). Obviously, both cyclotron and gyrosysnchrotron are incoherent emission,
which criteria may include: $T_{b}\leq T_{e}$, and long duration continuum
(for the continuously varying of $\gamma^{2}sin\theta$) with strong circular
polarization and localized only in and above the strong magnetic field
regions.

In addition to solar active regions, the solar internetwork magnetic fields in
the quiet regions sometimes may reach to kG magnitude (e.g. Wang et al. 1995,
Orozco et al. 2007, Zhou et al. 2013, etc.), and magnetic cyclotron radiation
also provides considerable contribution to the emission. However, as the
internetwork always occupies only a very small fraction area of the solar
quiet region, and the existing telescopes have not been able to distinguish
them clearly. With the development of the future next generation of solar
radio telescopes, such as the SKA (Nindos et al. 2019), it will be possible to
identify the magnetic cyclotron emission of the solar network in the same way
that we observe in active regions.

###  2.3 Emission around the flaring source regions

Generally, the flaring source region is only a small patch embedded within the
solar active region where flare occurs, usually much smaller than the active
region. It is always composed of a series of loops with very hot plasma and
relatively strong magnetic fields, and containing a great number of
nonthermal, collisionless, energetic electrons with power-law distributions.
Fast variation, rapid energy-releasing, and nonthermal process are the
prominent features of the flaring source region. With such conditions, the
emission should have very high brightness temperature, very short lifetime,
and rapidly variations. The emission mechanism may include incoherent emission
of nonthermal energetic electrons and coherent emission triggered by some
plasma instabilities.

(1) Incoherent emission in flaring source regions

The incoherent emission around the flaring source regions should be
gyrosysnchrotron emission of the mildly relativistic electrons and synchrotron
of the relativistic electrons ($\gamma\gg 1$) over a broad continuum mainly at
the instantaneous direction of velocity. The distribution of radiating
electrons is power-law. The emission frequency approximates (Dulk & Marsh
1982),

$f\approx f_{B}\gamma^{2}sin\theta$ (5)

The synchrotron emission is directed strongly at the direction of the
instantaneous electron motion and perpendicular to the magnetic field. On the
broadband spectral observations, it appears as a broadband continuum spectrum
with relatively short lifetime, such as patches, type V bursts, etc. Because
the radiating electrons have high energy which is much more energetic than the
ambient plasma, their brightness temperature will be higher than the local
thermal temperature but usually not exceed 1010 K in solar regime.

(2) Coherent emission in flaring source regions

There are mainly two kinds of coherent emissions in flaring source regions:
plasma emission (PE) and electron cyclotron maser emission (ECME).

The first coherent emission around the flaring source regions is PE, which is
triggered by nonthermal energetic electrons and the related Langmuir
turbulence and plasma instabilities. Intrinsically, the plasma emission is a
strong wave coupling process between the Langmuir wave and other low-frequency
waves. The frequency of plasma emission can be expressed as (Ginzburg &
Zheleznyakov 1958, Zheleznyakov & Zlotnik 1975, Robinson & Benz 2000),

$f=f_{pe}+f_{3}\approx sf_{pe}.$ (6)

Here, $f_{pe}\approx 9.0n_{e}^{1/2}$ is the plasma frequency. The unit of
plasma density $n_{e}$ is $m^{-3}$ and the typical value is from
$10^{15}m^{-3}$ to $10^{17}m^{-3}$ in the flaring source regions, and the
corresponding typical plasma frequency is from 280 MHz to 2.80 GHz. $s$ is the
harmonic number. $f_{3}$ is the frequency of low-frequency waves, such as
magnetosonic wave, electron cyclotron wave, whistler wave, and even scattering
Langmuir wave. Generally, $f_{3}\ll f_{pe}$ for magnetosonic wave, electron
cyclotron wave, and whistler wave, then $s\approx 1$, $f\approx f_{pe}$, which
is called the fundamental PE. When $f_{3}$ is the scattering Langmuir wave,
$f_{3}\approx f_{pe}+f_{B}$, here, $f_{B}\ll f_{pe}$, then
$f=2f_{pe}+f_{B}\approx 2f_{pe}$, $s\approx 2$, the emission is the second
harmonic PE. The fundamental PE is always strong polarization in the sense of
O-mode in a magnetized plasma, while the second harmonic PE is usually weak
polarization.

The coherent PE may produce various kinds of spectral fine structures with
very short lifetime, very narrow frequency bandwidth, and superhigh brightness
temperature ($T_{b}\gg 10^{10}$ K), such as type II burst, type III burst, and
the superfine structures overlapping on type IV continuum including Zebra
pattern (ZP) structures (Chernov 2010, Tan et al. 2014), fiber bursts
(Karlicky et al. 2013, Wan et al. 2021), spike group bursts(Wang et al. 2008,
Chernov et al. 2010, Tan 2013, Tang et al. 2021), etc. These spectral fine
structures are intrinsically related to the electron gyrofrequency $f_{B}$
which contains the information of magnetic fields (Chernov 2006, 2011).

Generally, the plasma emission is usually only effective with relatively weak
magnetic field ($f_{B}<f_{pe}$).

The second coherent emission in flaring source regions is electron cyclotron
maser emission (ECME) which is occurred when the magnetic field is relatively
strong or the plasma density is relatively low: $f_{B}>f_{pe}$. The second
condition for ECME is a population inversion in the electron distribution as
compared with equilibrium, such as the loss-cone distribution in magnetic flux
tubes (Twiss 1958, Schneider 1959, Wu & Lee 1979). Many spectral fine
structures, such as the spike bursts are possibly generated from ECME
mechanism (Fleishman et al. 2003, Tang et al. 2016).

In a word, different emission mechanisms will dominate in different regions of
the solar atmosphere, so that radio emission has different characteristics,
including emission intensity ($I_{f}$), brightness temperature ($T_{b}$), peak
frequency ($f_{peak}$), bandwidth ($\triangle f$), lifetime ($\tau$), degree
of polarization ($d_{p}$), frequency drift rate ($\frac{df}{dt}$) and so on.
Through the measurement of these parameters, we can identify the emission
mechanism in the diagnosing area, and through the physical relationship
between the above parameters and the magnetic field, it is possible to
establish a definite coronal magnetic field diagnostic method.

##  3 Radio diagnosing functions of magnetic fields

Generally, most solar radio telescopes are designed to receive two components
of left- and right-handed circular polarizations (LCP and RCP). According to
the magnetoionic theory (Hartree 1931, Appleton 1932), when a beam of
electromagnetic wave propagates parallel to the magnetic field, or the X-mode
when a beam of electromagnetic wave propagates perpendicular to the magnetic
field, both of LCP and RCP have the cutoff frequencies:

$f_{Rc}=\sqrt{f_{pe}^{2}+\frac{f_{ce}^{2}}{4}}+\frac{f_{ce}}{2}$ (7)

$f_{Lc}=\sqrt{f_{pe}^{2}+\frac{f_{ce}^{2}}{4}}-\frac{f_{ce}}{2}$ (8)

By solving the above two equations, we can get: $f_{ce}=f_{Rc}-f_{Lc}$, and
$f_{pe}=(f_{Rc}\cdot f_{Lc})^{1/2}$. Then, if we can measure the cut-off
frequencies of LCP and RCP from the solar radio observation, we may obtain
simultaneously the magnetic field strength and plasma density in the medium of
the magnetized plasma:

$B=357.1(f_{Rc}-f_{Lc})$ (9)

$n_{e}=1.24\times 10^{16}f_{Rc}\cdot f_{Lc}.$ (10)

Here, the unit of $f_{Rc}$ and $f_{Lc}$ is GHz, $B$ is at Gauss (G), and
$n_{e}$ is at $m^{-3}$. This method can be called as low-frequency cut-off
diagnostics, and the error of the diagnosing result of this method depends on
the measurement accuracy of the cut-off frequency.

Because the cut-off effect is only produced by the propagation process of
electromagnetic waves in magnetized plasma, they have nothing to do with the
emission mechanisms. In the solar atmosphere, the magnetic field and plasma
density of magnetized plasma decrease rapidly with distance from the solar
disk. Therefore, the outer atmosphere can almost be regarded as a transparent
medium. The diagnostic results given by the above formula almost reflect the
real information near the emission source regions. However, so far, no one has
seen this method to diagnose the coronal magnetic field, and this method is
always ignored by other reviews. The main reason is that this method requires
that the radio telescope should have sufficient frequency bandwidth and
frequency resolution, and can be extended beyond the cut-off frequency to
ensure the accurate $f_{Rc}$ and $f_{Lc}$. With the new generation of solar
radio telescopes, such as MUSER and the future FASR, it is possible to apply
the low-frequency cut-off diagnostics to obtain the information of magnetic
fields in solar atmosphere.

At the same time, the emission mechanism should be different from different
source region, and the corresponding observational results will have different
characteristics, which may provide different diagnostic functions to derive
the magnetic fields in the solar emission source regions. In the following
sections, we will discuss the diagnosing functions of magnetic fields in
different region with different emission mechanism.

###  3.1 Diagnosing functions of magnetic fields in and above the quiet
regions

The quiet regions cover most of the Sun’s surface. In and above the Sun’s
quiet regions, there is only a weak magnetic field, and the bremsstrahlung
emission of thermal electrons is dominated. From the emission mechanism
itself, bremsstrahlung has nothing to do with magnetic field. However, if the
source region has magnetic field, then the emission is composed of ordinary
mode (O-mode) and extraordinary mode (X-mode), the absorption coefficients of
O-mode and X-mode are a bit of different, which can be expressed in the
following form (Ratcliffe 1959),

$\kappa_{\sigma}=0.2\frac{n_{e}^{2}}{T^{3/2}(f+\sigma f_{B}cos\theta)^{2}}.$
(11)

Here, $\sigma=1$ means the O-mode, and $\sigma=-1$ means the X-mode. $f_{B}$
is the electron gyrofrequency. $\theta$ is the angle between the magnetic
field line and the line-of-sight. Supposing the emission source region is
optically thin, and $f\gg f_{B},f_{p}$, ($f_{p}$ is the plasma Langmuir
frequency), then the polarization degree ($d_{p}$) can be approximated as,

$d_{p}=\frac{\kappa_{x}-\kappa_{o}}{\kappa_{x}+\kappa_{o}}\approx
2\frac{f_{B}}{f}cos\theta.$ (12)

Because $f_{B}\propto B$, therefore, we may obtain the diagnosing functions of
the longitudinal component of magnetic fields from the measurement of
polarization degree,

$B_{l}=Bcos\theta\approx 178.6fd_{p}.$ (13)

Here, the unit of observing frequency $f$ is GHz, and the unit of magnetic
field $B_{l}$ is Gauss (G). For example, when $f=1.0$ GHz, and the observed
polarization degree $r=10\%$, then the longitudinal component magnetic field
is about 18 G. At the same time, if we can obtain the azimuth information of
the magnetic field with the help of imaging observations at other wavelengths,
such as EUV images, we may obtain the total magnetic field strength ($B$).

In general observations, most solar radio telescopes are designed to receive
signals of left-handed (L) and right-handed (R) circular polarizations. In
such case, the polarization degree can be expressed as,

$d_{p}=\frac{R-L}{R+L}.$ (14)

In practice, the actual emission source tends to be partially optically thick,
and the polarization degree depends not only on the magnetic field strength
but also on the temperature gradient of the source region (Bogod & Gelfreikh
1980, Gelfreikh 1990),

$d_{p}\approx\delta\frac{f_{B}cos\theta}{f}.$ (15)

Here, $\delta=-\frac{dlnT}{dlnf}$ is the spectral index of emission electrons.
The diagnosing functions of the longitudinal component of magnetic fields is
then rewritten into the following form,

$B_{l}=357.1\frac{f}{\delta}d_{p}.$ (16)

As for the optically thin thermal source region, the energy spectral index
$\delta=2$ and the above formula reverts to Equation (13). Obviously, the
diagnosing accuracy of magnetic fields mainly depends on the measurements of
spectral index $\delta$ and the degree of polarization ($d_{p}$).

Additionally, the observed sense of polarizations may provide the information
of direction of magnetic field: the right-handed circular polarization
($d_{p}>0$) indicates that the magnetic field is at an acute angle from the
line of sight, while the left-handed circular polarization ($d_{p}<0$)
indicates an obtuse angle between the magnetic field line and the line of
sight.

So far, there is no appropriate method to obtain measurements of the
transverse component of magnetic fields in quiet regions. We can only get
these information through some indirect approaches, such as modeling
extrapolation.

###  3.2 Diagnosing functions of magnetic fields in and above the active
regions

As we mentioned in Section 2.2, the emission is dominated by cyclotron and
gyro-synchrotron in and above the active regions. In this case, the opacity of
gyroresonance emission from hot plasma allows the measurement of magnetic
field strength, and the polarization may provide the information of the
direction of magnetic fields. When the magnetic field is at the direction
pointing towards the observer, the circular polarization is dominantly right-
hand circular, and when the field is at the direction pointing away from the
observer, the polarization is dominantly left-hand circular (White 2005).
Actually, the opacity of gyroresonance emission is a complicated function of
the temperature, density, magnetic field strength, and even the wave mode
(X-mode or O-mode) and the direction of the magnetic field with respect to the
line of sight (Akhmedov et al. 1982, Alissandrakis et al. 2019). Many people
derived the spatial resolved magnetic field strength by fitting the microwave
spectrum with gyrosynchrotron model at given spatial locations (Chen et al.
2020, etc.). However, these fitting methods are very complicated and have no
implicit diagnostic functions for deriving the magnetic fields.

There are several methods which may present implicit diagnostic functions for
deriving the magnetic fields in and above the active regions.

(1) Peak frequency diagnostics

For the typical coronal hot plasma in and above the active region, the gyro-
synchrotron emission of thermal electrons becomes optically thick at 3
harmonics for O-mode and at 2 harmonics for X-mode. That indicates that the
peak frequency ($f_{pk}$) is $3f_{B}$ for O-mode and $2f_{B}$ for X-mode. This
may give a diagnostic function of the magnetic field in the source region,

$B=357.1\frac{f_{pk}}{s}$ (17)

The unit of $f_{pk}$ is GHz, and $B$ is Gauss. $s$ is the harmonic number,
$s=2$ for X-mode and $s=3$ for O-mode. Here, we can partition the active
region into many small grids and find out the peak frequency in each grid, and
then derive the magnetic fields by using the above diagnosing functions (Equ.
17). Because the magnetic field will be different at different height above
the solar surface, the peak frequency changes accordingly. For example, we may
obtain the peak frequencies at foot-point, loop-leg, and loop-top,
respectively, and derive the magnetic field accordingly (White 2005). The
shortcoming of this method is that, at some cases, both the second and the
third harmonics may contribute to the emission, and the result should be
somewhere derived between the second and the third harmonic emission
(Alissandrakis et al. 1980).

(2) Spectral diagnostics

In high temperature coronal plasma, it is often difficult to determine the
harmonic number of cyclotron emission directly, and it is also difficult to
extract the exact peak frequency from observations. Therefore, it is hard to
use Equation (17) to derive the magnetic field in the active region. Zhou &
Karlicky (1994) obtained another expression,

$B=(\frac{c^{2}}{k_{B}T_{b}A_{1}}f_{p}^{1.3+0.98\delta}f^{-0.78-0.9\delta}A_{2}^{-2.52-0.08\delta})^{1/(0.52+0.08\delta)}.$
(18)

In this expression, $A_{1}=4.24\times
10^{1.4+0.3\delta}(sin\theta)^{0.34+0.07\delta}$, $A_{2}=2.8\times 10^{6}$.
$k_{B}$ is the Boltzman constant. $T_{b}$ is the emission brightness
temperature, which can be derived from the observed value emission flux
intensity ($F$) and the scale length of the source region. The electron energy
spectral index $\delta$ can be derived from the observed emission spectral
index ($\alpha$), $\delta=-1.1(\alpha-1.23)$. The scope of application of the
above function is $\delta=2-7$, and the corresponding emission spectral index
($\alpha$) is in the range from -0.6 to -5.1. By using the measurement of
polarization degree, the value of $sin\theta$ and then the direction of
magnetic field can be obtained (Huang & Nakajima 2002).

Additionally, Equation (18) also contain a variable of $f_{p}$ which depends
of the local plasma density, and this should be derived from other methods or
some models.

Huang (2008) applied the imaging observations of NoRH at 17 GHz and 34 GHz,
and obtained a vector magnetograph. Wang et al. (2015) used this method and
the imaging observation of EVOSA and got the whole map of magnetic field in an
active region. Zhu et al. (2021) adopted the above method and obtained the
evolution of magnetic fields of the coronal loop with respect to the height
above the solar surface.

(3) QPP diagnostics

Quasi-periodic pulsations (QPP) are frequently occurred in the solar
observations at optical, UV and EUV, soft X-ray, hard X-ray, and radio
wavelengths in the solar active regions with and without flares (Aschwanden
1987, Nakariakov et al. 2010, Li et al. 2015, Inglis et al. 2016, Zimovets et
al. 2021). As the radio observations may have very high temporal resolutions
(down to millisecond of cadence), they may provide the evidence of multiple
periods of QPP with period from decades of minutes to milliseconds (Tan et al.
2010). Similar as seismic waves can be used to detect the internal structure
of the Earth, QPP can be used to diagnose the physical parameters in the solar
source regions, such as the magnetic field, electric currents, plasma density,
etc. In solar active regions, radio emissions are always modulated by some MHD
oscillations, such as the fast sausage mode or fast kink mode (Abrami 1970,
Aschwanden 1987, Nakariakov & Melnikov 2009, Tan et al. 2010). These QPPs can
be used to derive the magnetic field related to the coronal loops in active
regions. For example, when the QPP is associated with fast sausage modes, then
the magnetic field can be derived,

$B\approx 2.02\times 10^{-16}\frac{an_{e}^{1/2}}{P}.$ (19)

Here, $P$ is the period of the QPP, $n_{e}$ is the plasma density, $a$ is the
radius of the section of the coronal loop. When the radio QPP is associated
with fast kink modes, then the magnetic field can be derived,

$B\approx 6.48\times 10^{-17}\frac{Ln_{e}^{1/2}}{P}.$ (20)

$L$ is the length of the coronal loop. The parameters $a$ and $L$ can be
measured from the imaging observations.

Equation (19) and Equation (20) indicate that the QPP diagnostics depends not
only on the measurement of oscillation period ($P$), but also on the
measurement of the geometric parameters ($a$ and $L$) and plasma density
($n_{e}$) of the coronal loops. The geometric parameters depends on the
imaging observations with broadband frequency and high resolutions, and the
determination of plasma density is related to the judgment of the emission
mechanism and the supplement of imaging observations at other wavelengths,
such as multi-wavelength EUV imaging observations, etc.

It is worth noting that radio QPP doesn’t always show up in observations. Its
diagnosing results can be used to verify the results derived by peak frequency
diagnostics and spectral diagnostics. In fact, the results of the above three
diagnostic methods can be verified with each other.

The above three kinds of diagnosing methods, peak frequency diagnostics,
spectral diagnostics and QPP diagnostics, any of the above three methods is
difficult to give the magnetic field information of the whole solar active
region, but they need to complement each other to give a complete magnetograph
of the active region.

###  3.3 Diagnosing functions of magnetic fields around the flaring source
regions

As we mentioned in Section 2.3, the flaring source region is smaller than the
active region and much smaller than the quiet region but with rapid
variations. The emission around the flaring source region should be including
incoherent synchrotron of the relativistic electrons and the coherent plasma
emission. Therefore the diagnostics can also be classified into two kinds.

(1) Spectral diagnostics

When the emission is generated from the incoherent synchrotron mechanism and
form various kinds of broadband continuum bursts, such as the bright patches,
etc, the magnetic field can be derived from the following diagnosing function
(Zhou & Karlicky 1994),

$B\approx(\frac{c^{2}}{k_{B}T_{b}f^{2}A_{2}^{5/2}A_{3}})^{2}.$ (21)

Here, $A_{3}=2.245\times
10^{15}(sin\theta)^{1/2}(\frac{2f}{f_{pk}})^{\frac{\delta-1}{2}}f_{pk}^{-\frac{5}{2}}$.
The above function is applicable when the index $\delta$ is in the range of 2
- 6. The diagnosing results depends on the measurements of several spectral
parameters, including the peak frequency $f_{pk}$, plasma frequency $f_{p}$,
index $\delta$, and flux intensity $F$. Some of these parameters can be
extracted only from the broadband spectral observations with high temporal and
spectral resolutions.

(2) Diagnostics of spectral fine structures

Many spectral fine structures are associated with solar flares, their
formations are related to the magnetic fields in the source region and
therefore can be used to derive magnetic fields.

Zebra Pattern

The solar radio Zebra patterns are a kind of spectral fine structures on the
solar radio broadband continuum spectrogram (Elgaroy 1959, Slottje 1972,
Chernov et al. 2003, Altyntsev et al. 2005, etc.), which consist of several or
decades of almost parallel and equidistant stripes, and frequently observed in
all phases of solar flares, see the statistics of Tan et al. (2014). Because
the structure characteristics of ZP contains the information of magnetic
fields in the source region, it is an important tool to diagnose the magnetic
fields in the flaring source regions.

When the emission is generated from the coherent plasma emission, then we can
apply various spectral fine structures to diagnose the magnetic field in the
flaring source region, among which the radio ZP structure is the most suitable
for diagnosing the magnetic field for their frequency separation between the
adjacent ZP stripes is proportional to the magnetic field strength in the
source region. We may approximately express the relationship between frequency
separations of ZP stripes and the magnetic fields in the ZP source region,

$B\approx 357.1M_{\alpha}\cdot\triangle f.$ (22)

Here, $\triangle f$ is the frequency separation between the adjacent Zebra
stripes at unit of GHz. $M_{\alpha}$ is a factor which depends on the
formation mechanism of the ZP structure. According to the physical
classification (Tan et al. 2014), different type of ZP may generates from
different coupling processes of the plasma emission and therefore have
different value of $M_{\alpha}$. It is necessary to identify the coupling
processes related to the ZP structure before deriving the magnetic fields in
the source regions:

As for the Bernstein wave coupling processes, all stripes in a ZP structure
are proposed to produce from a small compact source region, the emission
originates from nonlinear coupling processes either between two Bernstein
waves or a Bernstein wave and and the plasma electrostatic upper hybrid wave.
The energetic electrons with non-equilibrium distribution excite longitudinal
electrostatic waves at frequency of the sum of Bernstein mode frequency
$sf_{ce}$ and the upper hybrid frequency $f_{uh}$: $f=f_{uh}+sf_{ce}$. Here,
$s$ is the harmonic number. This model predicts that the frequency separation
between the adjacent Zebra stripes should be equal to the electron
gyrofrequency $\triangle f=f_{ce}$, and then $M_{\alpha}=1$ (Rosenberg 1972,
Zheleznyakov & Zlotnik 1975, Zaitsev & Stepanov 1983). If the source region is
small enough, the $\triangle f$ should be a constant, and this point is
consistent with the class of Equidistant ZP (EZP) (Tan et al. 2014).

As for the whistler wave coupling processes of ZP formation, the whistler
waves are generated at the anomalous Doppler cyclotron resonance
($f_{w}-\frac{k_{\|}}{2\pi}+f_{ce}=0$) under larger angles to the magnetic
field, they may form standing wave packets in front of the shock wave, and
when the group velocity of whistler waves is equal to the shock velocity, a ZP
structure with slow oscillating frequency drift will produce. Here, each Zebra
stripe corresponds to a propagating whistler wave packet. The emission
frequency is $f=f_{pe}+sf_{w}$, here $f_{w}\approx 0.1-0.5f_{ce}$ is the
whistler frequency peaking at $\frac{1}{4}f_{ce}$. The frequency separation
between the centers of stripes in the emission and absorption spectrum of the
ZP structure approximates to whistler frequency (Kuijpers 1975, Chernov 1976,
Maltseva & Chernov 1989), then the frequency separation between the adjacent
ZP stripes should be around two times of whistler frequency, and therefore,
$\triangle f\approx\frac{1}{2}f_{ce}$ with peak whistler wave group velocity,
then $M_{\alpha}\approx 2$.

As for the coupling of double plasma resonance (DPR) model, the Zebra stripes
are proposed to produce at some resonance levels where the upper hybrid
frequency coincides with the harmonics of electron gyrofrequency in
inhomogeneous magnetic flux tubes $f_{uh}=sf_{ce}$ (Berney & Benz 1978,
Winglee & Dulk 1986, Labelle et al. 2003, Yasnov & Karlicky 2004, Kuznetsov &
Tsap 2007). Under DPR model, the frequency separation between the adjacent ZP
stripes depend on not only the magnetic field strength, but also on the scale
heights of magnetic fields and plasma density in the source region (Chen et
al. 2011, Karlicky & Yasnov 2018, 2019). For example, if the emission is
triggered by the coalescence of an excited plasma wave and a low frequency
electrostatic wave, the emission frequency should be $f\approx f_{uh}\approx
sf_{ce}$ which is called the fundamental plasma emission, and the strip
frequency separation $\triangle f=\frac{H_{b}}{|sH_{b}-(s+1)H_{n}|}sf_{ce}$,
$H_{b}$ and $H_{n}$ are the scale heights of magnetic field and plasma density
in the source region. Obviously, here the factor
$M_{\alpha}=\frac{|sH_{b}-(s+1)H_{n}|}{sH_{b}}$. When the emission is
triggered by the coalescence of two plasma waves which is called as the second
harmonic plasma emission, the frequency of the Zebra stripes should be
$f\approx 2f_{uh}\approx sf_{ce}$ and the stripe frequency separation will be
$\triangle f=\frac{H_{b}}{|sH_{b}-(s+1)H_{n}|}2sf_{ce}$, the factor
$M_{\alpha}=\frac{|sH_{b}-(s+1)H_{n}|}{2sH_{b}}$. Here, we can see that the
stripe frequency separation is proportional to the number of harmonics,
$\triangle f\propto s$, that is, increases with emission frequency. And this
is consistent with the class of growing-distant ZP (GZP) (Tan et al. 2014).
Here, the scale heights of $H_{b}$ and $H_{n}$ need to be derived from other
approaches, for example, the Newkirk model for plasma density (Newkirk 1961),
the Dulk & McLean model for the coronal magnetic field above the solar active
region (Dulk & McLean 1978), etc. Just because the above diagnostic functions
also depends on the spatial variations of plasma density and magnetic field,
this function is not an explicit function in the strict sense. Kuznetsov &
Tsap (2007) and Yasnov & Karlicky (2020) proposed an improved method to
determine the gyro-harmonic numbers of the zebra stripes that are essential in
obtaining the electron density and magnetic field strength in ZP sources.

Tan et al. (2012) applied the ZP structures to derive the magnetic fields in
the source region before and after the peak time of an X2.2 flare, and the
results show that the magnetic field strength is about 230-345 G in the flare
impulsive phase at frequency range of 6.4-7.0 GHz, about 126-147 G after the
flare peak phase at frequency range of 2.60 - 2.75 GHz, and about 23-26 G in
the postflare phase at frequency range of 1.04-1.13 GHz. These diagnostic
results are basically consistent with our understanding of the flaring source
regions.

Fiber Burst

Solar radio fiber bursts have intermediate frequency drift rate which is
between type II and type III bursts. They are frequently observed in solar
flares (Young et al. 1961, Aurass et al. 1987, Chernov 1990,Wang et al. 2006,
Wan et al. 2021). Fiber bursts may allow one to measure the strength of the
coronal magnetic field in the radio burst source region. However, this is
depending on the theoretical interpretations. One interpretation of the
formation of fiber bursts is based upon the model of Alfven solitons which
establishes the relationship between magnetic field and frequency drift rate.
Therefore, radio fiber bursts can also be applied to derive the magnetic
fields of the flaring source region (Kuijpers 1975, Bernold & Treuman 1983,
Wang & Zhong 2006),

$B\approx 10.15\times 10^{-14}H_{n}\cdot\frac{df}{dt}.$ (23)

$\frac{df}{dt}$ is the frequency drifting rate of the fiber burst at unit of
Hz$\cdot s^{-1}$. The above function contains two variables: the scale height
of plasma density ($H_{n}$) and the frequency drifting rate ($\frac{df}{dt}$).
Here, $\frac{df}{dt}$ can be extracted directly from the dynamic spectrum of
the source region, and $H_{n}$ should be derived indirectly from models or
other approaches.

Wang & Zhong (2006) obtained the magnetic field strength of the fiber source
related to an X1.5 flare in the range of 130 - 270 G around the frequency of
2.6 - 3.8 GHz.

Another interpretation of radio fiber bursts is the whistler wave model, which
presumes that the fiber bursts are caused by packets of low frequency whistler
waves propagating along the magnetic field lines of a coronal loop (Benz &
Mann 1998). This model gives the magnetic field as,

$B\approx 17.8\times
10^{-12}\frac{H_{w}}{\sqrt{x}cos\theta}\cdot\frac{df}{dt}.$ (24)

$H_{w}$ is the scale height in whistler wave model.
$x=\frac{\omega_{w}}{\omega_{ce}}$, $\omega_{w}$ is the whistler wave
frequency, $\theta$ is the propagation angle of the fiber burst and its
guiding field line relative to the vertical. Generally, $x\ll 0.1$. Wang &
Zhong (2006) adopted the parameter values as $x=0.01$, $H_{w}=2\times 10^{5}$
km, and $\theta=0$, then obtained the magnetic field in the range 50 - 100 G
for the source region of fiber bursts at frequency of 2.6-3.8 GHz, which is a
bit weaker than the results derived from the above Alfven soliton model.
Obviously, Equation (24) is not a complete explicit function for the parameter
$x$ also containing the information of magnetic field $B$. Additionally, this
equation contains several variables, $H_{w}$, $x$, $\theta$ and
$\frac{df}{dt}$. Therefore, it is changeable and depending on the supplements
of other methods.

Type II Burst

The solar radio Type II bursts are usually regarded as exciting by MHD shock
waves associated with fast CMEs traveling through the solar corona with super-
Alfvenic speed along the shock trajectory (Vrsnak & Cliver 2008). They often
expose the fundamental and harmonic emission band, both frequently being split
in two parallel lanes that show a similar frequency drift and intensity
behavior (Vrsnak et al. 2002, etc.). The frequency drifting rate of type II
burst is proportional to the speed $v$ of the CME-associated shock waves and
therefore linked with Alfvenic speed $v_{A}$, such relationships provides a
method to estimate the magnetic field along the path of CME,

$B=4\pi\frac{\sqrt{m_{e}m_{p}}}{e}\frac{H_{n}}{M_{A}}\frac{df}{dt}.$ (25)

Here, $M_{A}=\frac{v}{v_{A}}$ is the Alfven Mach number which could be
estimated from other approaches, and generally $1.2<M_{A}<1.5$ in the regime
of CMEs (Smerd et al. 1974, Vrsnak et al. 2002). $H_{n}$ is the scale height
of the local plasma density which depends on certain models of plasma density,
such as the Newkirk model (Newkirk 1961).

The frequency separation of splitting type II radio bursts also depends on the
magnetic field strength along the path of CMEs, therefore, the splitting of
type II radio bursts might be an useful tool to diagnose the magnetic fields
in the related source regions. One of the interpretations of splitting type II
bursts is magneto-ionic resonance which gives the frequency separation
(Sturrock 1961), $\triangle f=\frac{1}{2}\frac{f_{B}^{2}}{f_{p}}$. From this
we can obtain a diagnostic function of the coronal magnetic field,

$B\approx 5.05\times 10^{-7}\sqrt{f_{p}\cdot\triangle f}.$ (26)

Here, the unit of $\triangle f$ and $f_{p}$ is Hz, and magnetic field $B$ is
in Gauss. Many people obtained similar results from different interpretations
of the formation of type II radio bursts (Vrsnak et al. 2002, Cho et al. 2007,
Kishore et al. 2015, Lv et al. 2017). For example, Cho et al. (2007) obtained
the coronal magnetic field strength is from 1.3 G decreasing to 0.4 G while a
CME-related shock passes from 1.6 to 2.1 $R_{sun}$, and this result is agree
with the potential field source surface modeling. Actually, as the type II
radio bursts only occurred in the meter-wavelength and more lower frequencies
($f<300$ MHz), and Equation (20) can derive the magnetic fields only in the
high corona and interplanetary space where CMEs pass through.

Type III pair bursts

The solar radio Type III bursts are frequently occurred on the spectrogram
with fast frequency drifting rates around the period of solar flares. They are
interpreted as being caused by energetic electron beams streaming through the
background coronal plasma at speed of 0.1-0.9 c (c is the speed of light) (Lin
& Hudson 1971, Lin et al. 1981, Huang et al. 2007, and a review in Reid &
Ratcliffe 2014). Occasionally, we can observe type III pair bursts which are
simultaneously composed of normal type III burst with negative fast frequency
drifting rates and reversed slope type III bursts with positive fast frequency
drifting rates (Aschwanden et al. 1993, Tan et al. 2016a). Type III pair
bursts occurred mainly in the decimetric wavelengths and are interpreted as
being triggered by bi-directional electron beams produced from the site of
magnetic reconnection, the normal type III bursts show the propagation of
upward electron beams and the reversed slope type III bursts show the downward
electron beams, and therefore the separation frequency between the normal and
reversed slope type III bursts might pinpoint the site of magnetic
reconnection and electron accelerations. Based on the framework of plasma
emission, Tan et al. (2016b) proposed a method to estimate the magnetic field
around the reconnection site by using the observations of type III pair
bursts,

$3.40\times 10^{-19}(n_{st}T\bar{D}R_{c})^{1/2}<B<3.29\times
10^{-16}[\frac{n_{st}T\bar{D}R_{c}}{(n_{st}\tau)^{1/3}}]^{1/2}.$ (27)

In above equation, $n_{st}$ indicates the plasma density near the start site
of the electron beam which can be derived from:
$n_{st}=\frac{f_{st}^{2}}{81s^{2}}$. $f_{st}$ is the start frequency of the
type III pair bursts. $T$ is the temperature. $\bar{D}=\frac{df}{fdt}$ is the
relative frequency drifting rate. $R_{c}$ is the curvature radius of the
magnetic field lines. $\tau$ is the lifetime of type III bursts. Obviously,
the equation (27) can only give the range of magnetic field strength around
the magnetic reconnection site. By using this method, Tan et al. (2016b) got
the estimation of the magnetic field around the reconnection site in the range
from about 50 G to 80 G for the X3.4 flare on 2006 December 13.

Here it should be noted that the above diagnostic functions of solar radio
spectral fine structures (Equations 22 - 27) are very immature which contain
many free parameters and depend on the theoretical models of the formation of
the spectral fine structures. So far, the diagnostic methods of spectral fine
structures can only present the magnetic field strength and even only give a
wide range of the strength without information of direction of the magnetic
fields. The relevant theoretical models need to be further improved and the
diagnostic methods based on them also need to be optimized. Figure 1 presents
a brief summary of the diagnostic functions of the magnetic field from radio
observations in solar chromosphere and corona.

Figure 1: Brief summary of the diagnostic functions of the magnetic field from
radio observations in solar chromosphere and corona.

##  4 Conclusions and discussions

Solar radio emissions are very sensitive to magnetic fields in the very wide
space of solar atmosphere, therefore, radio observations are one of the most
reliable diagnostic methods of magnetic fields in the upper chromosphere and
corona. This work gives 14 equations, including Equation (9), (13) and
Equations (16)-(27) and forming a set of implicit diagnosing functions for
deriving the chromospheric and coronal magnetic fields. Some of them do not
appear in the previous reviews, such as the low-frequency cut-off diagnostics,
etc. This set of diagnosing functions cover all quiet regions, active regions,
and flaring source regions from the face-on to the limb of the solar full-
disk. By using this set of functions and the broadband spectral solar radio
imaging observations, we can derive the magnetic fields of almost any region
in the solar atmosphere.

Obviously, different diagnostic functions are applicable to different physical
conditions. A complete coronal magnetic map needs to consider a variety of
emission mechanisms and adopt a variety of diagnostic methods. Therefore, it
is necessary to partition the diagnostic area into many small diagnostic
grids, and each grid has only a single emission mechanism at work. Such
partitioning requires high spatial-, temporal-, and spectral-resolutions of
radio observations, such as the observations obtained by MUSER (Yan et al.
2009, 2021), LOFAR (van Haarlem et al. 2013), VLA (Perley et al. 2011), and
the future FASR, etc. Then we can choose an appropriate diagnostic function
for each grid to derive the magnetic field strength. When we obtain the
magnetic field at each grid, we can get a complete coronal magnetic map.

It is reasonable to discuss the diagnostic procedure. The diagnostics of high
chromospheric and coronal magnetic fields can be divided into the several
steps,

(1) Solar radio imaging observations.

It is required that the observations have high spatial-, temporal-, and
spectral-resolutions, and have dual circular polarizations in a broadband
frequency range, complete data calibrations (including phase and amplitude
calibration, the emission intensity is in unit of sfu at each pixel), and
obtain images at every frequency channel. At present, the observations can be
obtained by MUSER, LOFAR, EVOSA, and even VLA are suitable for our
diagnostics.

(2) Partitioning the diagnostic area.

The diagnostic source region should be divided into small grids. The scale of
grids should be small as soon as possible, so that there is only one emission
mechanism working in each grid. The grid partitioning also affects the
accuracy of magnetic field diagnosis. Generally, the scale of grids depends on
the resolutions. The higher the resolutions, the smaller the grids.

(3) Extracting observational parameters at each grid.

Generate a spectrum diagram at each grid, and extract the observational
parameters for diagnostics of magnetic fields from the spectrum diagram. These
observational parameters include,

a. Flux density $F$ (or brightness temperature $T_{b}$).

b. Peak frequency $f_{pk}$.

c. Cut-off frequency, $f_{Rc}$ and $f_{Lc}$.

d. Spectral index $\alpha$.

e. Degree of polarization $d_{p}$.

f. Bandwidth $\triangle f$.

g. Lifetime $\tau$.

h. Frequency drifting rate $\frac{df}{dt}$.

i. Pulsating period $P$.

j. Frequency separation $\triangle f$ for ZP.

k. Splitting frequency separation $\triangle f$ of type II burst.

And so on.

Accordingly, make the judgement of the emission mechanism at each grid. At
this step, the multi-wavelength EUV images with high spatial resolutions (such
as AIA images) and the vector magnetograph (such as HMI images) are the very
good supplementary materials which may help us to identify the mechanism.

(4) Deriving magnetic fields.

After we obtained all the observational parameters and identified the emission
mechanism at each grids, we may derive the magnetic fields for every grid.
When the grid is located in the quiet region, the magnetic field can be
calculated from Equation (13) or (16). When the grid is located in the active
region, the magnetic field can be calculated from Equations (17) and (18). At
the same time, if we can extract the signal of QPP, we can also use the
Equation (19) and (20) to give the magnetic field. When the grid is located in
the flaring source region during the nonthermal eruptions, the magnetic field
can be calculated from Equation (21), and at the same time, if we can identify
the kinds of spectral fine structures (such as ZPs, fiber bursts, splitting
type II bursts, etc.) related to the grid, we may derive the magnetic field
from Equation (22), or Equation (23), or Equation (24).

After we derived the magnetic fields at each grid, then a full coronal
magnetic field map will be completed. Here, we meet another question: Are
these diagnostic results credible? We may apply several methods to verify
these results.

The first simple method is to compare the diagnosing result with the ones
deriving from a simple modeling function proposed by Dulk & McLean (1978),

$B\approx 0.5(\frac{R_{sun}}{h})^{3/2}.$ (28)

The unit of $B$ is G, $R_{sun}$ is the solar photospheric radius and $h$ is
the height of the source region above the solar photosphere. The comparison
between the diagnosing result and the calculating ones from Equation (26) can
preliminarily verify the reliability of the diagnosis results. In the work of
Tan et al. (2012), the diagnosing results of microwave ZPs were verified by
above methods and they are consistent with each other.

Another method is to compare the diagnosing result with MHD modeling and
extrapolations (such as Yan & Sakurai 2000, Yan & Li 2006, Metcalf et al.
2008, Wiegelmann 2008, Jiang & Feng 2013, Chifu et al. 2015, Zhu & Wiegelman
2018, etc.) from direct measurements in the photosphere and, to a lesser
degree, in the chromosphere, which are based mainly on the Zeeman effect. Such
comparison can not only verify the reliability of the diagnosis results, but
also optimize and improve the modeling extrapolation methods.

When the source region locates at the solar limb region, we may apply the
simultaneous observations of radio imaging observations and Zeeman splitting
observations of infrared coronal forbidden emission lines (Lin et al. 2004,
Liu & Lin 2008) and the results of coronal MHD waves diagnostics (Nakariakov &
Verwichte 2005, Yang et al. 2020) to cross verify the above diagnosing
results.

Generally speaking, the coronal magnetic field diagnostics is a key problem in
solar physics. However, the diagnosing functions presented in this work are
approximation at present, some of them have limitations, for example, so far
we have no effective method to obtain the transverse magnetic field component
of the quiet region, and the diagnostics of spectral fine structures can not
provide the information of direction of magnetic fields. There are still many
aspects of these functions that deserve modification and improvement in the
future works. In the next works, we will attempt to apply the MUSER solar
radio imaging observations and the above diagnosing functions to produce the
solar full-disk coronal magnetic maps in different frequencies, which may help
us to understand a series of important problems of solar physics, such as the
mechanism of coronal heating, the origin of solar flares and their precursors,
and their possible relationship with space weather, etc.

###### Acknowledgements.

This work is supported by NSFC Grants 11790301, 11973057, 12003048 and
11941003, National Key R&D Program of China 2021YFA1600500, 2021YFA1600503,
and the International Partnership Program of Chinese Academy of Sciences
(183311KYSB20200003).

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