Coronal scattering
under strong regular refraction
Alexander Afanasiev
Institute of Solar-Terrestrial Physics
Irkutsk, Russia
The problem of accounting for the combined influence
exerted by the regular inhomogeneity of background corona
and by random coronal inhomogeneities upon the propagation
of radio emission has been studied insufficiently. It is quite
clear, however, that in some cases regular refraction that
leads to multipathing and focusing of radio waves, must
influence the scattering process and promote new effects
during the propagation of radio emission through a randomlyinhomogeneous corona.
Questions
• Coronal sounding with spacecraft radio signals at small
elongations
• Coronal sounding with a pulsar at small elongations
• Scattering of radio emission from a solar source in the
presence of large-scale electron density inhomogeneities in
the corona
Coronal sounding with spacecraft radio signals
at small elongations
Schematic plot of ray trajectories in
a regular (i.e. without random
inhomogeneities) spherically-symmetric
solar corona
When a spacecraft is at rather
small angular distance from the
Sun, the observer on the Earth
may be in the ‘illuminated’
zone, close to the caustic or
may get in the caustic ‘shadow’
zone.
spacecraft
The interference integral method
(proposed by Yu. I. Orlov in 1972)
The scalar wave field U (for example, a component of the electric vector)
at any given point r is represented as an integral over partial waves:
U r  

Ar, a  exp ik r, a da
a
is a parameter that characterizes a partial wave
k
is the wave number in a vacuum;
 (r, a)
is the complete integral of the eikonal equation:
( )2   (r),
 (r ) is dielectric permittivity of plasma, and
A(r, a ) is the solution of the transport equation:
2A  A  0.
where
(1)
In the presence of electron density inhomogeneities in the solar corona
the integral representation for the wave field can be written in the form:
~
U (r, t )   A(r, a) exp[ ik~ (r, a, t )]da
~    ~ ;
The perturbation theory: 
~  
0
1
1
(2)
0
 0 is the eikonal in the absence of statistical inhomogeneities;
~1 is an addition introduced into the eikonal by the inhomogeneities;
In the shadow zone near the caustic boundary, the following expression
for the energy spectrum R() can be obtained:
 (   0 ) 2 [ F (, d )]2 
R () 
D1/ 2 [ F (, d )] exp 


2
2
2 2
2
4
  ( a    at )
 2k  

R0
shadow zone
Sun
d
point of
caustic
Earth’s orbit
D (x) is the parabolic cylinder function;
d is the depth of entry into the shadow
zone;
  ,  a , and at are the statistical
trajectory characteristics depending on
turbulent inhomogeneity parameters.
Some numerical modeling results
Energy spectra for different depths of entry into the shadow zone
(for λ=3m) Turbulent inhomogeneity parameters:
the density perturbation σN=1%;
the outer scale l0=106 km;
the inner scale q0=104 km;
the radial velocity of inhomogeneities
Vr=300 km/s.
d  10 6 rad
d  20 10 6 rad
d  10 10 6 rad
d  60 10 6 rad
Vr=300 km/s (curve 1)
Vr=800 km/s (curve 2)
Distortion of
the energy spectrum
with a change of the
velocity Vr of travel of
coronal inhomogeneities
λ=3 m
σN=1%;
l0=106 km;
q0=104 km;
d  10 6 rad
Energy spectra for
different outer scales l0
l0=106 km (curve 1)
of turbulent
l0=5105 km (curve 2)
inhomogeneities
5
l0=10 km (curve 3)
λ=3 m
d  10 6 rad
σN=1%;
q0=104 km;
Vr=300 km/s
Conclusion
• The form of the energy spectrum in the caustic shadow zone differs
from a Gaussian and depends critically on the properties of the
turbulent inhomogeneities. Therefore measurements of the radio
energy spectrum in the neighborhood of the caustic can be used
for the coronal plasma diagnostics.
Coronal sounding with a pulsar
at small elongations
Qualitative ray picture of radio
emission propagation from
a pulsar through the corona
For investigating the properties of the
near-solar plasma, natural distant
radio sources when they are occulted
by the corona, are also used. In
particular, pulsars that are virtually
point pulsed sources are applied for
this purpose.
Coronal inhomogeneities cause the
temporal broadening of the pulses and
distort their shapes. Therefore the
mean time profile of the pulse is
a useful characteristic that contains
information on coronal turbulent
inhomogeneities.
Of interest is to calculate the mean pulse profile in
the neighborhood of the regular caustic to analyse
the possibilities for the coronal turbulence diagnostics.
If the radiated (initial) pulse from a pulsar is specified by the Gaussian form,
the following expression for the mean pulse profile in the neighborhood of the
caustic boundary can be obtained:

[ F1 (t , d )]2 
J (t )  J 0CD1/ 2 [ F1 (t , d )] exp F2 (t , d ) 

4


d is the depth of entry into the shadow zone
Mean pulse profiles for different points of observation in the caustic shadow zone
σN = 1%
The initial pulse parameters:
the carrier frequency f = 111 MHz;
the initial pulse half-width T = 1.510-3 s.
σN = 3%
Turbulent inhomogeneity parameters:
the outer scale l0=106 km;
the inner scale q0=103 km.
Using the asymptotic representation for the parabolic cylinder function
2


x
1/ 2

D1/ 2 ( x)  x exp    as x  ,
 4
one can obtain an expression for estimating the variance of relative
fluctuations in the electron density:
2
k 2 (d 22  d12 )
~
2

 N / N
N
 d 1/ 2 J (d ) 
1

2( a2 ) ln  1 
 d 2  J (d 2 ) 
where J(d1) and J(d2) are the values of the maxima of the mean pulse
profile in the caustic shadow zone at distances d1 and d2 from
the caustic point;
(σa2) is a calculated statistical trajectory characteristic.



By measuring the maxima J(d1) and J(d2) at the points d1 and d2, and
calculating (σa2), one can estimate the value of the variance of
coronal plasma electron density fluctuations.
Conclusion
• The variation of the pulsar’s pulse energy in the caustic shadow
zone can be treated as an indicator for turbulent inhomogeneity
intensity in the solar corona
Scattering of radio emission from a solar source in the
presence of large-scale electron density
inhomogeneities in the corona
Of special interest is the combined
influence of scattering and strong
regular refraction on characteristics
of radio emission from coronal
sources. It is known that around such
sources there can exist different
large-scale regular electron density
structures (coronal arches,
streamers, and others). These
structures may give rise to regular
caustics and multipathing of radio
emission.
The appearing refraction effects should be taken into account in
the analysis of the emission structure of solar radio bursts and
their generation mechanisms.
Part I. Solar millisecond spike bursts
Among the great variety of solar radio bursts, millisecond
spike bursts represent one of the less understood solar
phenomena. Spike bursts are intense narrowband
(Δf / f < 1%) flashes of subsecond duration, which
accompany solar flares. They are observed in different
wavelength ranges from centimetric to decametric.
Examples of spikes
There are currently a number of models for the
mechanism of radio spike generation. Nevertheless,
the question regarding the origin of spikes remains
to be conclusively answered. One difficulty is that it is not fully clear as to the
particular influence exerted by an inhomogeneous propagation medium upon
observed characteristics of radio spikes.
Consideration of the propagation effects usually assumes that the solar corona
is spherically-symmetric in general, the influence of regular refraction is
negligible, and the spike characteristics are determined by the scattering and
diffraction of the waves by turbulent coronal inhomogeneities.
On the other hand, radio spikes can be generated by sources located in high
coronal arches. Not only the scattering but also strong regular refraction of radio
emission in the arch structure can be important in this case.
A point radio source is located at the
coronal arch top and emits a δ-pulse
Geometry of the problem
Ray pattern of the field (f = 100 MHz)
Mean profile of the pulse after its passing through the corona
for different values of the density perturbation σN
(numerical modeling results)
σN = 1%
Intensity
Intensity
σN = 0.2%
Time, s
Time, s
σN = 2%
Intensity
σN = 4%
Time, s
Time, s
Time profile at 408 MHz of the radio burst observed with the Trieste
Solar Radio System (Trieste Astronomical Observatory) on 15 April 2000 b
J. Magdalenic, B. Vrsnak, P. Zlobec, and H. Aurass.
Flux density (SFU)
Time (UT)
The analysis of the event has shown that it is associated with
a coronal mass ejection which could be responsible for the
complex temporal structure of the pulses.
Conclusions
• When the spike emission is propagating in the solar corona, strong
regular refraction due to large-scale regular electron density
structures such as coronal arches can lead to the formation of
multipathing and regular caustics. These phenomena promote
formation of a multi-component mean time profile of the radio spikes.
• To understand the causes of formation of the complex time
profiles of spikes associated with the propagation effects, it is
necessary to consider the data concerning the large-scale structure
of the solar corona (CME, arches, etc). This would create the
conditions for a more correct investigation of spike generation caused
by physical processes occurring within the solar radio source.
Part II. Type IIId solar decameter radio bursts with
echo-components
One important feature of type IIId radio bursts
DB
(which was revealed by Abranin, Baselyan, and
Tsybko [Astron. Rep. 1996, V. 40, 853] during the
solar observations with the UTR-2 radio telescope)
is that as the burst source approaches the central
solar meridian, a temporal splitting of the bursts is
developed and thus an additional burst component
is produced.
Time profiles of some type IIId burst events
contain several additional components.
Examples of time profiles
of type IIId bursts
f=25 MHz
additional
component
DB
What is more, position observations showed that
the positions of visible sources of the initial burst
and its additional component usually do not
coincide and can be spaced by a distance
comparable to the Sun’s optical diameter.
To explain this observational evidence, Abranin, Baselyan, and Tsybko
suggested that the additional components represent echoes of the original burst
in the corona, which are produced due to strong regular refraction of radio
emission on large-scale structures lying at heights of the middle corona.
The observed radio burst echo-components with long delays can be
explained by the production of additional radio emission propagation
modes within a ‘transverse’ refraction waveguide arising between the
localized electron density nonuniformity and deeper layers in the corona.
As these additional modes are reflected from the streamers, they can
reach the Earth.
Ray pattern illustrating formation of the refraction waveguide in the corona
(f = 25 MHz)
direct
signal
source
photosphere
Mean time profiles for the radio pulse for different values of intensity of
the large-scale localized nonuniformity forming the refraction waveguide
f = 25 MHz
direct signal
echo-components
Conclusions
The results of the theoretical analysis and numerical
modeling, presented here reveal the importance of the
regular refraction phenomenon in the solar corona.
On the one hand, analysing statistical characteristics of
radio emission from distant non-solar sources in the
neighborhood of the regular caustic is useful for coronal
turbulence diagnostics. On the other hand, strong regular
refraction due to large-scale coronal structures can
influence substantially the emission from Sun’s own radio
sources.