# Polarization in Low Frequency Radio Astronomy

Baptiste Cecconi

(2016)

## Editorial Notice

This content of this document has been initially written in 2016 as a chapter
of a book entitled “The Universe in polarised light”, after the “Ecole
Internationale de Polarimétrie en Astrophysique” (International School on
Polarimetry for Astrophysics) organized in June 2013, at Centre Paul-Langevin,
Aussois, France. This summer school has been funded by the EU funded COST
action MP1104111http://www.polarisation.eu and OSUG (Observatoire des Science
de l’Univers de Grenoble). This book has not been published yet due to delays
and lack of funding.

## Foreword

This chapter introduces the concepts of polarimetry in the case of low
frequency radio astronomy. In this regime radio waves are usually not the
signature of atomic or molecular transitions lines, but rather that of
unstable particle distribution functions releasing their free energy through
electromagnetic radiation. As the radio source region is usually magnetized,
the propagation medium (at least close to the source) is anisotropic, and the
polarization depends on the local magnetic field direction, the propagation
mode and the direction of propagation.

##  1 Introduction

Low frequency radio astronomy is defined by a frequency span ranging from
$\sim$1 kHz to $\sim$100 MHz (or $\sim$3 m to $\sim$300 km in terms of
wavelengths). The energy of a single photon at 100 kHz is 7$\times$10-29 J (or
4$\times$10-10 eV). The wave properties of light are used instead of its
corpuscular properties. The electric (and/or magnetic) field fluctuations of
the wave are sampled. The polarization of the radio waves is thus directly
measured by the sensors. Depending on the sensor physical shape different
polarization components are intrinsically measured: linear electric dipoles
measure linear polarization, loop or helicoidal antenna are measuring the
circular polarization.

The electromagnetic radiations that can be detected in the low frequency range
have various origins. In the vicinity of the Earth, the most intense sources
are the Sun and the magnetized planets, as shown in Figure (1). Human
technologies are strong sources of Radio Frequency Interferences (RFI).
Atmospheric electricity (i.e., Earth and planetary lightnings) emit wide band
electromagnetic pulses. Our Galaxy is radiating a continuum of emission
(called the Galactic Background), resulting from free-free interactions of
electrons in the interstellar medium. The Galactic Background is almost
isotropic and its brightness temperature exceeds 107 K at 1 MHz [6, 12, 37].
The brightness temperature of Solar radio bursts and of Jovian radio emissions
are respectively of the order of $\sim$1012 K and $\sim$1018 K. Those
brightness temperatures usually do not correspond to the actual black body
temperature of the radio source medium. Hence, these radio emissions are
called “non-thermal” radio emissions.

Figure 1: Compared planetary radio emission spectral flux densities. The radio
emissions from Jupiter are in black (_nKOM_ : narrowband Kilometric Radiation;
_bKOM_ : broadband Kilometric Radiation; _QP_ : Quasi-Periodic Bursts; _HOM_ :
Hectometric Radiation; _S-bursts_ : Short (or Millisecond) Bursts; _Io-DAM_ :
Io-controlled Decametric Radiation; _non-Io-DAM_ : non-Io-controlled
Decametric Radiation; _DIM_ : Decimetric Radiation; _Thermal_ : Black body
thermal radiation), from Saturn in green (_n-SMR_ : narrowband Saturn
Myriametric Radiation; _n-SKR_ : narrowband Saturn Kilometric Radiation; _SKR_
: Saturn Kilometric Radiation), from Earth in red (_LFb_ : Low-Frequency
Bursts; _TKR_ (or _AKR_): Terrestrial (or Auroral) Kilometric Radiation), and
from Uranus and Neptune in blue (_UKR_ /_NKR_ : Uranus/Neptune Kilometric
Radiation). The vertical grey dashed line represents the Earth ionospheric
cut-off. Figure Adapted from [55], updated with [53, 28, 30].

Solar and Solar Wind radio sources are produced by populations of relativistic
electrons in the Solar Corona or escaping the Sun in the interplanetary
medium. The two main low frequency Solar radio sources are called type II and
type III solar radio bursts. Type II bursts are emitted by electrons
accelerated in front of coronal mass ejections and interplanetary shocks (see
[17] and references therein). Type III bursts are produced by beams of
electrons ejected from the Sun and travelling outward along magnetic field
lines. The radio waves are believed to be produced by wave conversion from
strong Langmuir waves excited along the beam path (see [20, 45] and references
therein). Radio waves are scattered in the inner heliosphere by solar wind
inhomogeneities, resulting in an apparent spatially extended source (up to 90∘
at $\sim$100 kHz [49]).

In planetary magnetospheres, the two main low frequency radio emissions
families are the auroral radio emissions and the radiation belts. Planetary
aurorae result from particle precipitations towards the magnetic poles of the
planet. When those particles reach the planetary ionosphere, they transfer
their kinetic energy through collisions with the atmospheric populations of
atoms and molecules, which are releasing this energy by emitting photons in
the infrared to ultraviolet range, depending on the chemical species present
in the medium. A fraction of the particles does not reach the planetary
ionosphere thanks to the magnetic mirror effect. The up-going particles are
the source of the auroral radio emissions. The typical energy of particles
responsible for auroral radio emissions is of the order of 10 keV. The
electrons are the main driver for electromagnetic auroral phenomena and the
emission mechanism is the Cyclotron MASER Instability (CMI), see [35, 55, 51]
for more details. This non-thermal process has been observed in-situ at Earth
by various instruments onboard the Viking [24] and FAST [13] space missions.
Remote observations at Jupiter and Saturn are consistent with the CMI
phenomenology [55, 23, 9]. Furthermore, radio observations by Cassini near a
SKR radio source are showing results that are fully consistent with the cold
plasma theory and the CMI emission process [30, 31]. The apparent brightness
temperature of such radio source can be higher than 1015 K. The radio waves
are emitted at a frequency close to the local cyclotron frequency. The radio
source is strongly anisotropic. Its theoretical beaming pattern is a hollow
cone, aligned with the ambiant magnetic field, with an opening half angle
between 45∘ and 90∘, and a thickness of $\sim$1∘.

The planetary radiation belts radio emissions are produced by synchrotron
emission of highly relativistic electrons trapped in the planetary magnetic
field near the magnetic equator. The radio source is the radiation belts by
itself and spans up to several planetary radii in the equatorial plane. The
Jovian radiation belts are the most intense of the kind in the solar system,
and span from about 40 MHz up to $\sim$10 GHz [55, 48, 16]. The terrestrial
radiation belts are emitting a radio emission around 1 MHz, but have still
been poorly studied.

Astrophysical objects such as pulsars are also emitting in this frequency
range. Magnetized exoplanets are believed to host low frequency radio sources,
similarly the Solar System planets [56, 22, 38, 54]. Finally, the Dark Ages
and Cosmic Dawn radiation signatures is also predicted to appear in this
frequency range (see [5] and references therein).

##  2 Magneto-ionic theory

This section presents the theoretical background required to understand the
emission and propagation of low frequency radio emissions, and thus their
polarization.

###  2.1 Presentation of the assumptions and conditions

A series of preliminary assumptions are made:

  * •

The propagation medium is a plasma, which is globally neutral.

  * •

The plasma is cold, which implies: (i) without perturbations, the particles
are not moving; (ii) particles have no thermal velocity; (iii) the pressure
gradient is negligible.

  * •

The effect of the ions is negligible, as the radio frequency range is much
higher than that of the ion scales.

  * •

The medium is magnetized and thus anisotropic.

###  2.2 Basic Equations

Using the Fourier notation, the Maxwell-Faraday and Maxwell-Ampère laws
respectively write:

$\displaystyle i\bm{k}\times\bm{E}$ $\displaystyle=i\omega\bm{B}$ (1)
$\displaystyle i\bm{k}\times\bm{B}$
$\displaystyle=-i\frac{\omega}{c^{2}}\overline{\overline{\bm{\kappa}}}.\bm{B}$
(2)

The equation driving the electric field can thus be derived as:

$\displaystyle\bm{n}\times(\bm{n}\times\bm{E})+\overline{\overline{\bm{\kappa}}}.\bm{E}=0$
(3)

using the refractive index $\bm{n}$ vector defined as:

$\displaystyle\bm{n}=\frac{n}{k}\bm{k}=\frac{c}{\omega}\bm{k}$ (4)

The permittivity tensor is computed from the equation of motion of the
particles in the plasma. After the assumptions presented in the previous
paragraph, it takes the following form:

$\displaystyle\bm{\kappa}=\left(\begin{array}[]{ccc}S&-iD&0\\\ iD&S&0\\\
0&0&P\end{array}\right)$ (8)

with $S=1-X/(1-Y^{2})$, $D=XY/(1-Y^{2})$, $P=1-X$, and defining
$X=(\omega_{p}/\omega)^{2}$ and $Y=\omega_{c}/\omega$. The characteristic
frequencies $\omega_{p}$ and $\omega_{c}$ are respectively the plasma
frequency and the electron cyclotron frequency.

###  2.3 Refraction index: Appleton-Hartree equation

As the medium is magnetized, we choose to represent the various vectors in the
reference frame defined as: $\bm{z}$ is along the main ambiant magnetic field
direction $\bm{B}_{0}$, and the wave vector $\bm{k}$ is in the
$(\bm{x},\bm{z})$ plane. Hence the refractive index vector is
$\bm{n}=(n\sin\theta,0,n\cos\theta)$, with $\theta$ the angular separation
between $\bm{B}_{0}$ and $\bm{k}$.

The dispersion equation derives from the equation of the electric field of the
wave, by searching for the eigenmodes of propagation, i.e., solving the
following linear system:

$\displaystyle\left(\begin{array}[]{ccc}S-n^{2}\cos^{2}\theta&-iD&n^{2}\sin\theta\cos\theta\\\
iD&S-n^{2}&0\\\
n^{2}\sin\theta\cos\theta&0&P-n^{2}\sin^{2}\theta\end{array}\right)\left(\begin{array}[]{c}E_{x}\\\
E_{y}\\\ E_{z}\end{array}\right)=0$ (15)

The roots of the determinant of the previous matrix provides the eigenmodes of
propagation. The following system has to be solved:

$\displaystyle\left|\begin{array}[]{ccc}S-n^{2}\cos^{2}\theta&-iD&n^{2}\sin\theta\cos\theta\\\
iD&S-n^{2}&0\\\
n^{2}\sin\theta\cos\theta&0&P-n^{2}\sin^{2}\theta\end{array}\right|=0$ (19)

The general solution of this equation is known as the Appleton-Hartree
equation:

$\displaystyle
n^{2}=1-\frac{2X(1-X)}{2(1-X)-Y^{2}\sin^{2}\theta\pm\sqrt{Y^{4}\sin^{4}\theta+4(1-X)^{2}Y^{2}\cos^{2}\theta}}$
(20)

This equation links the scalar refractive index $n$ with $X(\omega)$,
$Y(\omega)$ and $\theta$.

Figure 2: Dispersion diagram with following plasma and cyclotron frequency
values: $f_{p}=500$ kHz and $f_{c}=1500$ kHz. Vertical lines, from left to
right are: $f_{L}$, L-mode cutoff frequency; $f_{p}$, plasma cutoff frequency;
$f_{c}$, cyclotron resonance frequency; $f_{UH}$ upper hybrid resonance
frequency; and $f_{R}$, R-mode cutoff frequency.

###  2.4 Propagation Modes, propagation angle

The $\pm$ signs in the Appleton-Hartree equation correspond to two eigenmodes
and thus two propagation modes. We hereafter refer to them as the $\oplus$ and
$\ominus$ modes, respectively. Figure (2) shows the various propagations
modes. Each curve on the figure correspond to a propagation angle with respect
to the ambiant magnetic field $\bm{B}_{0}$: from parallel propagation in bold
plain line; to perpendicular propagation in bold dashed line. Each region of
the diagram is a specific propagation mode and is named as shown on the
figure. The $\oplus$ modes are the whistler and LO modes, whereas the
$\ominus$ modes are the Z, LX and RX modes. The vertical lines are the
resonnance (when $n\rightarrow\infty$) and cutoff (when $n\rightarrow 0$)
frequencies.

###  2.5 Polarization

When studying wave propagation in a cold plasma, the sense of polarization is
always given considering the wave propagating in the direction of the magnetic
field.

In case of parallel propagation, the polarization of the wave is always
circular. The $\oplus$ and $\ominus$ modes are respectively fully LH and RH
polarized.

In case of perpendicular propagation, the $\oplus$ mode is linearly polarized
along the ambiant magnetic field direction, and the $\ominus$ mode is
elliptically polarized perpendicular to the ambiant magnetic field direction
(this means that the wave propagation is not transverse).

In case of oblique propagation, the $\oplus$ and $\ominus$ modes are
elliptically polarized.

##  3 Radio Waves Propagation

The propagation of radio waves is usually studied with ray tracing methods.
They can include scattering effects. The main ray tracing algorithm for radio-
astronomy is the Haselgrove algorithm [18, 19]. Other algorithms are available
(such as Poeverlein [43]), but are mainly adapted to horizontally stratified
propagation media, such as the Earth’s ionosphere.

Figure 3: Ray tracing in the inner Saturn magnetosphere. a) Dynamic spectrum
of circular polarization degree of SKR. b Ray tracing performed by ARTEMIS-P
through the Kronian magnetosphere at $f=50\,\textrm{kHz}$. Green lines are ray
trajectories, red arrows are wave vectors, black arrows are directions of the
magnetic field, blue line are electron density contours at
$[0.001,0.01,0.5,1,10,50]\textrm{cm}^{-3}$, dotted black line is the iso-
surface where $f_{ce}=50\,\textrm{kHz}$, long dashed black lines are dipolar
magnetic field lines. The magnetospheric plasma density is taken from the
MDISC model [1]. Figure adapted from [15]

The two main propagation effects are the refraction and the scattering of
radio waves. While the former is directly related to the evolution of the
refractive index along the ray path, the latter includes random changes in ray
propagation, due to interaction is particle and inhomogeneities in the
propagation medium. Refraction is the core of the ray tracing techniques.
Scattering can be included [50]. Further non-linear effects, such as caustics
can not be handled by such simulation codes. Full electromagnetic code (such
as Finite Difference Time Domain (FDTD) methods) should then be used.

Figure (3) is showing an examples the effect of radio wave propagation. It
displays the trace of several radio rays in the inner magnetosphere of Saturn,
at a frequency of $50\,\textrm{kHz}$. The apparent location of the radio
source as seen from a remote location highly depends on the initial ray
inclination with the source ambient magnetic field vector.

###  3.1 Propagation of polarization

The propagation of polarization can be treated in ray tracing codes. Two kind
of propagation regions shall be separated, where (1) the medium is imposing
the polarization on the wave, or (2) the wave is freely propagating with its
own polarization. These regions are called “weak” and “strong” mode coupling,
respectively. The “coupling” is here referring to that between the wave modes,
not between the waves and the medium. The limit between the two regimes is
given by the following relation [3, 4]:

$\displaystyle|n_{\oplus}-n_{\ominus}|\sim\frac{1}{k}\frac{\mathrm{d}}{\mathrm{d}k}n_{(\oplus\textrm{
or }\ominus)}$ (21)

In the strong mode coupling region, the polarization is frozen and is
projected on the two modes of the medium, as in a birefringent medium in
classic optics. If the two modes have different phase velocities, Faraday
rotation occurs (see [41] and references therein).

##  4 Detection

The Earth ionosphere is reflecting radio waves below 10 MHz whether the wave
is coming from space or from ground. This is used for telecommunication on
ground in the so-called long wavelengths radio bands. Space based observations
are required below 10 MHz. Between 10 MHz and $\sim$80 MHz (i.e., the between
ionospheric cutoff and the FM broadcasting band), several ground based
instruments are available. Figure (4) is showing several types of radio
antenna used for ground based observations.

The angular resolution of a telescope is defined by the ratio between the
observed wavelength and the telescope aperture diameter. Hence, in order to
get a few arcseconds resolution at 30 MHz (i.e., a wavelength of 10 m), an
aperture of about 2500 km in needed. As single piece focussing devices of this
size cannot be built on ground, phased array are used together with
interferometric techniques.

Figure 4: Various types of ground based radio antenna used in the low
frequency range (10 MHz to 80 MHz): (a) Array of helicoidal antenna, Nançay
Decameter Array (Nançay, France); (b) Array of thick linear dipoles, UTR-2
(Kharkov, Ukraine); (c) Array of thick folded crossed dipoles, LWA (New
Mexico, USA) and Nenufar (Nançay, France); (d) Array of thin folded crossed
dipoles, LOFAR-LBA (EU); (e) Log Periodic Yagi antenna, Iitate HF Radio
Monitor (Iitate, Japan); (f) thin dipole, RadioJOVE (world wide).

###  4.1 Space based sensors

In the vicinity of Earth, the accessible spectral range includes frequencies
as low as a few kHz for observations of radio waves above the local
propagation cut-off. The constraints on space based instrumentation, such as
reliability and cost imply that only simple sensors can be used on a
spacecraft. Space-based electric and magnetic sensors are very simple.
Electric sensors are either wire boom monopoles or dipoles, or a pair of probe
antenna. Magnetic sensors are either magnetic loops (single loop) and search
coils (many loops). We will focus in the following on wire dipole electric
sensors.

Space based low frequency radio interferometry has not been implemented yet,
for obvious reasons of cost for building and operating a fleet of several tens
or hundreds of spacecraft. Nanosatellite concepts may help, and several
studies are ongoing [40, 44, 5].

###  4.2 Presentation of Goniopolarimetry

At frequencies much lower than the resonance frequency of wire dipole electric
sensors (i.e., at wavelengths longer than 20 times the antenna length, so-
called short antenna or quasistatic range), the gain properties of the antenna
is very simple:

$\displaystyle G(\theta)\propto\sin^{2}\theta,$ (22)

where $\theta$ is the angle between the antenna axis and the wave vector
direction $\bm{k}$. Figure (5) is showing the antenna beaming pattern at
different wavelengths. In the quasistatic range (upper left corner), the
beaming pattern is single lobed, and its gain figure varies as presented in
Eq. (22). When the wavelength is getting shorter, the beaming pattern is
getting narrower, up to the resonance frequency ($L/\lambda=1/2$) and then
becomes multi-lobed.

Figure 5: Linear dipole antenna beaming pattern for various wavelengths to
antenna lengths ratio.

Goniopolarimetric (i.e., simultaneous derivation of direction of arrival and
polarization) inversions are making use of the simple form of the beaming
pattern in the quasistatic range. Figure (6) is showing a simplified case of
linearly polarized wave observed with a set of orthogonal crossed dipoles, in
a fully planar configuration. The direction of arrival $\theta$ can simply be
determined after the ratio of the $P_{1}$ and $P_{2}$ measured power. The
solution is not unique in this case.

Figure 6: Principle of goniopolarimetry illustrated in a simple planar case.
The wave arrives with an angle $\theta$ from the blue antenna. The ratio of
the radio power measured on each antenna is directly linked to $\theta$.

For a three-dimensional analysis, either three quasi-orthogonal dipoles, or
dipoles on a spinning spacecraft (1 aligned with the spin axis, the other(s)
in the spin plane) shall be used. In the quasistatic range, the electric
signal induced by the wave on an antenna is:

$\displaystyle V_{a}=\bm{h}_{a}.\bm{E}_{\omega}(t),$ (23)

where $\bm{h}_{a}$ is the antenna vector and $\bm{E}_{\omega}(t)$ the wave
instantaneous electric field. The radio wave is assumed to be in a transverse
propagation regime. Hence, the instantaneous electric field is:

$\displaystyle\bm{E}_{\omega}(t)=e^{i(\omega
t+\phi_{0})}\left|\begin{array}[]{l}a_{x}\\\ a_{y}e^{i\delta}\\\
0\end{array}\right.$ (27)

in the wave polarization plane (i.e., perpendicular to $\bm{k}$).

Goniopolarimetric radio receivers are measuring the power on each sensor, as
well as the cross-correlation of the signals measured on each pair of
antennas. This provides us with the full spectral matrix $P_{ij}$ of the wave
[27, 8, 7]:

$\displaystyle P_{ij}=\frac{Z_{0}Gh_{i}h_{j}S}{2}$
$\displaystyle\left[(1+Q)A_{i}A_{j}+(U-iV)A_{i}B_{j}\right.$
$\displaystyle\left.+(U+iV)A_{j}B_{i}+(1-Q)B_{i}B_{j}\right]$ (28)

where: $S$, $Q$, $U$ and $V$ are the four Stokes parameters of the wave;
$h_{i}$ and $h_{j}$ the antenna electrical lengths; $G$, the gain of the
receiving chain; $Z_{0}$ the impedance of free space; and $A_{n}$ and $B_{n}$
(with $n=i$ or $n=j$) defined as:

$\displaystyle A_{n}$
$\displaystyle=-\sin\theta_{n}\cos\theta\cos(\phi-\phi_{n})+\cos\theta_{n}\cos\theta$
(29) $\displaystyle B_{n}$ $\displaystyle=-\sin\theta_{n}\sin(\phi-\phi_{n})$
(30)

where: $\theta$ and $\phi$ defines the direction of the wave vector $\bm{k}$
in a spherical frame; and $\theta_{n}$ and $\phi_{n}$ defines the direction of
the $n$-th antenna $\bm{h}_{n}$ in the same frame. All the spectral matrix
terms $P_{ij}$ are not always measured, depending on the receiver
capabilities. Goniopolarimetric techniques are inverting the set of
measurements $P_{ij}$ into the wave parameters. The Stokes parameters can be
related to the wave parameters as follows:

$\displaystyle S$
$\displaystyle=\frac{<E_{x}.E_{x}^{*}>+<E_{y}.E_{y}^{*}>}{2Z_{0}}=\frac{<a^{2}_{x}>+<a^{2}_{y}>}{2Z_{0}}$
(31) $\displaystyle Q$
$\displaystyle=\frac{<E_{x}.E_{x}^{*}>-<E_{y}.E_{y}^{*}>}{<E_{x}.E_{x}^{*}>+<E_{y}.E_{y}^{*}>}=\frac{<a^{2}_{x}>-<a^{2}_{y}>}{2SZ_{0}}$
(32) $\displaystyle U$
$\displaystyle=\frac{<E_{x}.E_{y}^{*}>+<E_{y}.E_{x}^{*}>}{<E_{x}.E_{x}^{*}>+<E_{y}.E_{y}^{*}>}=\frac{<a_{x}a_{y}\cos\delta>}{SZ_{0}}$
(33) $\displaystyle V$
$\displaystyle=-\frac{<E_{x}.E_{y}^{*}>-<E_{y}.E_{x}^{*}>}{i(<E_{x}.E_{x}^{*}>+<E_{y}.E_{y}^{*}>)}=\frac{<a_{x}a_{y}\sin\delta>}{SZ_{0}}$
(34)

The radio convention [25] of the sign of $U$ and $V$ is the opposite of the
optical convention. Radio astronomers consider they measure the polarization
following the radio wave instead of looking at the source.

The goniopolarimetric inversions are used to derive the wave polarization
state. The plane of polarization is also determined. The direction of
propagation is then derived as the normal direction to the polarization plane,
with the transverse propagation assumption. This assumption is valid most of
the time, except close to the wave propagation resonances and cutoffs. In that
case, determining the polarization of the magnetic component is directly
providing the direction of $\bm{k}$. However, in such a case, the direction of
propagation (the Poynting vector) is not that of the wave vector. Hence,
simultaneous electric and magnetic component sensing is required.

In case of planetary radio emissions, the radio source location (or at least
the magnetic hemisphere containing its location) can be derived from the
direction of arrival, see Figure (8), and the sense of polarization indicates
the emission mode. In weak mode coupling conditions (the medium is setting the
polarization), the sign of the Stokes parameter $Q$ is providing the
propagation mode:

  * •

$Q>0$ means $\oplus$ mode (L-O branch of the dispersion diagram)

  * •

$Q<0$ means $\ominus$ mode (R-X branch of the dispersion diagram)

###  4.3 Various goniopolarimetric inversions

Several goniopolarimetric inversions are available in the literature. The
following list is grouping them with their assumptions:

  * •

Point sources. The output parameters are $S$, $Q$, $U$, $V$, $\theta$ and
$\phi$. They are used for auroral radio emissions (Earth, Jupiter, Saturn).
They have been used on Cassini/RPWS (with 2 or 3 antenna modes) and
INTERBAL/Polrad (with 3 antennas) [8, 27, 32, 42].

  * •

Spatially extended sources. The output parameters are $S$, $Q$, $U$, $V$,
$\theta$, $\phi$ and $\gamma$, where $\gamma$ is the angular size of the
source (assuming a circular shaped source region). They are used for solar
radio emissions. They have been used on STEREO/Waves (with 3 antennas) and
WIND/Waves (spinning spacecraft) [36, 7, 26].

  * •

Radio sources along a spatial profile. The output parameters are $S(a)$,
$Q(a)$, $U(a)$, $V(a)$, with $a$ the curvilinear coordinate along the spatial
profile. Such inversion can be used for auroral radio sources [21].

  * •

All sky source. The output parameters are the Stokes parameters distribution
on the sky $S(\theta,\phi)$, $Q(\theta,\phi)$, $U(\theta,\phi)$,
$V(\theta,\phi)$. Such inversions can be used for mapping the galactic
background emission, but are still to be developed. The only Galactic emission
mapping below $10\,\textrm{Mhz}$ was obtained with Lunar occultations measured
by Radio Astronomy Explorer 2 (RAE 2) [39].

###  4.4 Limitations

Although very powerful compared to the use of a single dipole,
goniopolarimetric system have several intrinsic limitations:

  * •

Electric versus magnetic sensors. Magnetic component intensity is $c$ times
fainter than the electric component. Furthermore, the magnetic sensor have an
intrinsic band pass input filter that forbids wide band applications.

  * •

Sense of propagation. Simple goniopolarimetry provides the wave direction of
propagation, but not its sense of propagation. Coupled electric and magnetic
components sensing are required to do so.

  * •

Wave vector direction. Goniopolarimetric inversions with electric sensors
assume transverse propagation. It is not true when $n\neq 1$. As discussed
earlier, observing the magnetic components provides directly the wave vector
direction. However, the direction of propagation is not that of the wave
vector.

  * •

Accuracy. The goniopolarimetric accuracy depends on two factors: the accuracy
of the receiver system calibration (effective antenna directions and lengths,
phase miss-match between sensing channels); and the receiver and onboard
processing noise. The main source of noise is usually the digitization noise,
which quantifies the output signal. This results in the following typical
accuracies: $\sim$2∘ on direction of arrival, $\sim$10% on polarization
degrees and $\sim$3 dB on flux densities.

  * •

Effective sensor directions. Electric and magnetic sensors must be accurately
calibrated to obtain their effective parameters (length and direction). This
calibration process is more critical for electric sensors, as the conductive
spacecraft body can strongly alter the antenna beaming pattern.

  * •

Propagation mode. The propagation mode can not be derived unambiguously
without simultaneous and coupled measurements on electric and magnetic
sensors.

  * •

Direction of propagation. The wave vector is aligned with the Poynting vector,
except when $n\neq 1$.

  * •

Radio source location. The goniopolarimetric systems are measuring the wave
parameter at the place of the spacecraft. Hence the radio source location can
not be retrieved without specific assumptions, such as: straight line
propagation, emission process.

##  5 Example of results and open questions

According to the magneto-ionic theory, there is a direct link between the
propagation angle and the polarization. As seen in the previous section,
perpendicular emission leads to linear or elliptical polarization, parallel
emission to circular polarization, and oblique emission to elliptical
polarization. The sense of polarization depends on the direction of the
magnetic field in the source (hence on the hemisphere of emission), as well as
on the propagation mode.

Solar radio emissions are showing a limited degree of polarization. In the
lower frequency range (below $\sim$10 MHz), radio observations are not showing
evidences of significant degree of polarization (excepty for a limited series
of event [47]), whereas at higher frequency, up to 35% of polarization can be
observed [11]. Contrarily planetary radio emissions are fully polarized,
either circularly or elliptically [55]. The planetary radiation belts are
elliptically or linearly polarized [10, 52, 33, 34].

###  5.1 Latitudinal variability of polarization

All magnetized planets are producing radio emissions in their auroral regions.
The main emission is produced in the R-X mode (i.e., RH polarized in the
Northern magnetic hemisphere). The radio waves are 100% polarized. At Earth,
the observations show circularly polarized waves. At Jupiter and Saturn,
auroral radio waves are circularly polarized when observed from near
equatorial regions, while they are elliptically polarized at high latitudes.
The limit latitude is about $\pm$30∘ as shown in Figure (7) [46, 14].

Figure 7: Observed polarization state at Jupiter and Saturn, with Ulysses/URAP
and Cassini/RPWS respectively. The left-hand panel shows a meridian plane
projection of the trajectory of the Ulysses spacecraft during its Jovian flyby
[46]. Axes are number in units of Jovian Radii. Bold segments of the
trajectory correspond to the location where elliptical polarization was
observed. The right-hand panel shows a similar figure measured at Saturn [14].
Axes are number in units of Kronian Radii. The displayed fraction of orbit
correspond to elliptical polarization waves. On both cases, circularly
polarized waves are observed elsewhere.

###  5.2 Saturn Kilometric Radiation

Cassini/RPWS/HFR (Radio and Plasma Waves Science/High Frequency receiver) is a
goniopolarimetric radio receiver. Cecconi et al. [9] proposed a simple way to
derive the radio sources locations assuming straight line propagation and CMI
emission, as shown in figure (8). Figure (9) shows a comparison of active
radio source magnetic footprints with ultraviolet (UV) aurora observed with
the Hubble Space Telescope (HST), at Saturn [29]. Both datasets were observed
at the same time, and with a very similar observing geometry. The figure shows
an very good conjugacy between the two phenomena, which comforts the idea that
the same electron populations are producing radio and UV emissions.

Figure 8: Derivation of the radio source location at Saturn, from the
goniopolarimetric products provided by Cassini/RPWS/HFR. Straight line
propagation is assumed, together with CMI emission process, which implies an
emission at the local electron cyclotron frequency ($f_{ce}$)in the source.
The three-dimensional location of the radio source is derived, as well as the
location of the magnetic footprint of the source (for comparison with
atmospheric aurora). The error on the wave propagation direction is also used
to derive the radio source footprint error ellipse. Figure extracted from [9].
Figure 9: Comparison of radio source location with UV aurora at Saturn. The
top panel shows radio source locations derived from Cassini/RPWS/HFR
measurements. The bottom panel shows the UV aurora as observed from HST. The
left-hand column is displaying the radio and UV sources as seen from the
observer, whereas the right-hand column shows the same data projected on the
southern polar ionosphere. Figure extracted from [29].

On Oct. 17th 2008, the Cassini spacecraft flew through the auroral radio
source region for a few minutes [31]. Figure (10) shows the polarization
parameters with respect to the distance to the radio source. It confirms that
the wave has a quasi perpendicular propagation, with elliptical polarization
consistently with the magnetoionic theory.

Figure 10: Polarization parameters of Saturn Kilometric Radiation in the
vicinity of a source region: (a) Circular polarization degree plotted versus
the reduced frequency Figure extracted from [31].

###  5.3 Future prospects

Goniopolarimetric methods are using measurements acquired on a single
location. Although implying several assumptions they are very efficient for
auroral planetary radio emissions, which are point sources with a high degree
of intrinsic polarization. Solar radio bursts can also be measured, but a
limited view of their spatial structure can be derived.

The two main limitations of goniopolarimetric inversions are the assumptions
of (i) a plane transverse wave propagating on a straight line from the source
to the observer and (ii) a point or circularly-shaped single source observed
at a time. The first limitation can be fixed using sensors measuring the full
electric and magnetic components of the wave. This set up is proposed for the
Alfvén space mission [2], which is dedicated to the study of the Terrestrial
auroral cavities. Solving the second limitation implies space-based
radioastronomy interferometric instrumentation [40, 5].

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