16
The University of Manchester Research Anisotropic Radio-Wave Scattering and the Interpretation of Solar Radio Emission Observations DOI: 10.3847/1538-4357/ab40bb Document Version Final published version Link to publication record in Manchester Research Explorer Citation for published version (APA): Kontar, E. P., Chen, X., Chrysaphi, N., Jeffrey, N. L. S., Emslie, A. G., Krupar, V., Maksimovic, M., Gordovskyy, M., & Browning, P. (2019). Anisotropic Radio-Wave Scattering and the Interpretation of Solar Radio Emission Observations. The Astrophysical Journal, 884(2), [122]. https://doi.org/10.3847/1538-4357/ab40bb Published in: The Astrophysical Journal Citing this paper Please note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscript or Proof version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version. General rights Copyright and moral rights for the publications made accessible in the Research Explorer are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Takedown policy If you believe that this document breaches copyright please refer to the University of Manchester’s Takedown Procedures [http://man.ac.uk/04Y6Bo] or contact [email protected] providing relevant details, so we can investigate your claim. Download date:30. Jul. 2021

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Page 1: Anisotropic Radio-wave Scattering and the Interpretation of ......scattering is anisotropic, with the dominant effect being perpendicular to the heliospheric radial direction (Kontar

The University of Manchester Research

Anisotropic Radio-Wave Scattering and the Interpretationof Solar Radio Emission ObservationsDOI:10.3847/1538-4357/ab40bb

Document VersionFinal published version

Link to publication record in Manchester Research Explorer

Citation for published version (APA):Kontar, E. P., Chen, X., Chrysaphi, N., Jeffrey, N. L. S., Emslie, A. G., Krupar, V., Maksimovic, M., Gordovskyy,M., & Browning, P. (2019). Anisotropic Radio-Wave Scattering and the Interpretation of Solar Radio EmissionObservations. The Astrophysical Journal, 884(2), [122]. https://doi.org/10.3847/1538-4357/ab40bb

Published in:The Astrophysical Journal

Citing this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.

General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.

Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact [email protected] providingrelevant details, so we can investigate your claim.

Download date:30. Jul. 2021

Page 2: Anisotropic Radio-wave Scattering and the Interpretation of ......scattering is anisotropic, with the dominant effect being perpendicular to the heliospheric radial direction (Kontar

Anisotropic Radio-wave Scattering and the Interpretation of Solar Radio EmissionObservations

Eduard P. Kontar1 , Xingyao Chen1,2 , Nicolina Chrysaphi1 , Natasha L. S. Jeffrey1,3 , A. Gordon Emslie4 ,Vratislav Krupar5,6 , Milan Maksimovic7 , Mykola Gordovskyy8 , and Philippa K. Browning9

1 School of Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK2 Key Laboratory of Solar Activity, National Astronomical Observatories Chinese Academy of Sciences, Beijing 100101, People’s Republic of China

3 Northumbria University, Department of Mathematics, Physics and Electrical Engineering, Newcastle upon Tyne, NE1 8ST, UK4 Department of Physics & Astronomy, Western Kentucky University, KY 42101, USA

5 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA6 Institute of Atmospheric Physics CAS, Prague, Czech Republic

7 LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, F-92195 Meudon, France8 School of Physics & Astronomy, University of Manchester, Manchester M13 9PL, UK

9 Jodrell Bank Centre for Astrophysics, School of Physics & Astronomy, University of Manchester, Manchester M13 9PL, UKReceived 2019 July 16; revised 2019 August 21; accepted 2019 August 31; published 2019 October 17

Abstract

The observed properties (i.e., source size, source position, time duration, and decay time) of solar radio emissionproduced through plasma processes near the local plasma frequency, and hence the interpretation of solar radiobursts, are strongly influenced by propagation effects in the inhomogeneous turbulent solar corona. In this work, a3D stochastic description of the propagation process is presented, based on the Fokker–Planck and Langevinequations of radio-wave transport in a medium containing anisotropic electron density fluctuations. Using anumerical treatment based on this model, we investigate the characteristic source sizes and burst decay times forType III solar radio bursts. Comparison of the simulations with the observations of solar radio bursts shows thatpredominantly perpendicular density fluctuations in the solar corona are required, with an anisotropy factor of ∼0.3for sources observed at around 30MHz. The simulations also demonstrate that the photons are isotropized near theregion of primary emission, but the waves are then focused by large-scale refraction, leading to plasma radioemission directivity that is characterized by a half width at half maximum of about 40° near 30MHz. The resultsare applicable to various solar radio bursts produced via plasma emission.

Unified Astronomy Thesaurus concepts: Radio bursts (1339); Solar coronal radio emission (1993); Solar radioemission (1522); Solar radio flares (1342)

1. Introduction

Solar radio emission is produced in the turbulent medium ofthe solar atmosphere, and its observed properties (sourceposition, size, time profile, polarization, etc.) are significantlyaffected by the propagation of the radio waves from theemission site to the observer. Bright radio emission produced inthe outer solar corona during flares is mostly produced viaplasma emission mechanisms, so that the radiation is generatedclose to the plasma frequency or its harmonic (see, e.g., Suzuki& Dulk 1985; Pick & Vilmer 2008 for reviews). Since therefractive index of an unmagnetized plasma

w w= -n 1 peref2 2 1 2( ) is significantly different from unity

for ω close to ωpe, the effects of density inhomogeneity alongthe wave path play a particularly strong role in the propagationof solar radio bursts produced by plasma processes. Appreciat-ing this fact, even early observations (e.g., Wild et al. 1959;Smerd et al. 1962; Steinberg et al. 1971) considered radio-waveescape to be an important effect.

Scattering of radio waves on random density irregularitieshas long been recognized as an important process for theinterpretation of radio source sizes (e.g., Steinberg et al. 1971),positions (e.g., Fokker 1965; Stewart 1972), directivity (e.g.,Thejappa et al. 2007; Bonnin et al. 2008; Reiner et al. 2009),and intensity-time profiles (e.g., Krupar et al. 2018). In theparticularly strong scattering environment appropriate forelectromagnetic waves close to the plasma frequency, thewave direction is quickly randomized, and the waves quickly

become isotropic. As the waves propagate farther away fromthe source, large-scale refraction also produces a degree offocusing/defocusing. The observed properties of solar radioemission are therefore determined by an interconnectedcombination of scattering off small-scale inhomogeneities,which generally shifts the observed positions of sources awayfrom the solar disk center (Riddle 1972; Gordovskyy et al.2019), and refraction by relatively large-scale density inhomo-geneities, such as coronal mass ejection fronts (Afanasiev 2009)or coronal streamers and fibers (Bougeret & Steinberg 1977),which generally shifts the sources toward the disk center (Wildet al. 1959; Smerd et al. 1962; Steinberg et al. 1971).Subarcminute imaging observations of Type III solar radio

bursts have shown that intrinsic sources with sizes 0 1 resultin observed sources as large as ∼20′ at 30MHz (Kontar et al.2017; Sharykin et al. 2018), demonstrating that scatteringdominates the properties of observed source sizes. Moreover,the locations of the upper and lower subband sources of Type IIsolar radio bursts are observed to be spatially separated (e.g.,Zimovets et al. 2012; Chrysaphi et al. 2018), with the amountof separation being consistent with radio-wave scattering ofplasma radio emission from a single region (Chrysaphi et al.2018).The majority of both past (e.g., Steinberg et al. 1971) and

recent (e.g., Thejappa & MacDowall 2008; Krupar et al. 2018)ray-tracing simulations have assumed isotropic scattering bysmall-scale density fluctuations. However, there are observa-tions (McLean & Melrose 1985) that cannot be explained by

The Astrophysical Journal, 884:122 (15pp), 2019 October 20 https://doi.org/10.3847/1538-4357/ab40bb© 2019. The American Astronomical Society. All rights reserved.

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the earlier models; for example, to provide a plausibleexplanation of the size and directivity of Type I solar radiobursts, a fibrous structure was invoked (Bougeret & Stein-berg 1977). Other recent observations also suggest that thescattering is anisotropic, with the dominant effect beingperpendicular to the heliospheric radial direction (Kontaret al. 2017).

A quantitative understanding of radio-wave propagation isparticularly timely in the view of the opportunities to be openedby the Square Kilometer Array (Dewdney et al. 2009; Nindoset al. 2019) and the observations with the Chinese SpectralRadioheliograph (Yan et al. 2009; Li et al. 2016). While therehave been a number of Monte Carlo simulations developed todescribe wave scattering (mostly for isotropic density fluctua-tions), these do not all agree. Therefore, the present workaddresses this important issue both by extending the isotropicplasma treatment of Bian et al. (2019) into the anisotropicscattering domain and by improving the previous descriptionsby Steinberg et al. (1971), Arzner & Magun (1999), andThejappa & MacDowall (2008). The description presentedcaptures both multiple scattering of radio waves in anisotropicsmall-scale turbulence and refraction of waves in the presenceof large-scale plasma inhomogeneity.

In Section 2, we present a general theoretical treatment of thescattering process, and apply it to both isotropic and (using adiagonalization scaling technique) axially symmetric anisotro-pic scattering. In Section 3, we derive the pertinent stochasticdifferential equations that allow for a numerical solution forboth isotropic and anisotropic turbulence. In Section 4, wereview the numerical Monte Carlo technique used to solveLangevin equations modeling both source sizes and timeprofiles. In Section 5, we review relevant observations of thevariation of radio source sizes and decay times with frequency,and we compare these observations with our numericalsolutions. This leads us to the conclusion that observations ofType III solar radio burst sizes and durations, over a broadrange of frequencies, require anisotropic scattering, in the entireheliosphere between the Sun and the Earth, with an anisotropyfactor of around 3–4 and with the density fluctuationspredominantly perpendicular to the radial direction. Asdiscussed in Section 6, these observations provide essentialdensity fluctuation anisotropy constraints for MHD turbulencemodels (Shaikh & Zank 2010; Zank et al. 2012) over a widerange of locations between the Sun and the Earth.

2. Radio-wave Scattering Equations

The propagation of radio waves in a turbulent medium canbe effectively described using a kinetic approach (e.g.,Mangeney & Veltri 1979; Arzner & Magun 1999; Bian et al.2019). This approach describes the evolution of radio waves inan inhomogeneous plasma with quasi-static density fluctuationsin the geometrical optics approximation (Tatarskii 1961;Ishimaru 1978), i.e., when the scale length for variation ofthe wavelength λ due to inhomogeneity is much smaller thanthe wavelength itself:

ld

dr1. 1( )

This description ignores diffraction effects and is generallyvalid only for small amplitude density fluctuations (e.g.,Pécseli 2012). Nevertheless, it adequately describes the multi-ple-scattering transport of radio waves with angular frequency

ω (s−1) near the local plasma frequencyw p=r re n m4pe e

2( ) ( ) (where e and me are, respectively,the electron charge [esu] and mass [g], and rn( ) [cm−3] is thelocal plasma density) in the turbulent plasma of the solaratmosphere. Similar to the weak turbulence theory of Langmuirwaves in a plasma (Tsytovich & ter Haar 1995), such adescription provides the basis for a statistical description ofdensity and electromagnetic wave interactions. Since the groupvelocity of density fluctuations is much less than the speed oflight, the density fluctuations can be treated as effectivelystatic. Therefore, only elastic scattering conserving wavevectork∣ ∣ of radio waves is considered. The description presented isalso limited to an unmagnetized plasma environment(Zheleznyakov 1996).The spectral number density (or photon number) k rN t, ,( )

(cm−3 [cm−1]−3) can be described in the geometric-opticapproximation using a Fokker–Planck equation

g¶¶

+¶¶

+¶¶

=¶¶

¶¶

-r

rk

kN

t

d

dt

N d

dt

N

kD

N

kN , 2

iij

j· · ( )

where ò =k r k rN d N, 30( ) ( ) [cm−3] is the number density of

photons, ki are Cartesian coordinates of wavevector k, and thesummation is understood for a repeated index, i, j=1, 2, 3.rd dt, kd dt are given by the Hamilton equations corresp-onding to the dispersion relation for electromagnetic waves inan unmagnetized plasma (Haselgrove 1963):

ww

= =¶¶

=r

vk

kd

dt

c, 3g

2( )

w ww

w= -

¶¶

= -¶

¶k

r rd

dt. 4

pe pe ( )

Here the photon packet frequency in Equation (3) is found fromthe dispersion relation w w= + c kpe

2 2 2 2 for electromagneticwaves in an unmagnetized plasma, and γ [s−1] is the collisional(free–free) absorption coefficient for radio waves in a plasma(e.g., Lifshitz & Pitaevskii 1981).The diffusion tensor Dij appropriate to scattering (see Arzner

& Magun 1999; Bian et al. 2019) is given by

ò

ò

pw

wd

ppw

wd

p

=

=

q q v

q q k

D q q Sd q

cq q S

d q

4 2

4 2, 5

ijpe

i j g

pei j

4

2

3

3

4

2

3

3

( ) ( · )( )

( ) ( · )( )

( )

where q is the wavevector of electron density fluctuations. qS ( )is the spectrum of the density fluctuation normalized to therelative density fluctuation variance:

òd

p=

á ñ= q

n

nS

d q

2, 62

2

2

3

3( )

( )( )

where = á ñn n is the average plasma density, taken to be aslowly varying function of position. Note that Equations (5)and(6) include a scaling of (2π)3 in the definition of thespectral density qS ( ), consistent with the treatment of Arzner &Magun (1999), but not with the scaling used by Bian et al.(2019).

2

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2.1. Isotropic Scattering

The bulk of radio-wave scattering research has assumed anisotropic spectrum of density fluctuations: =qS S q( ) ( ). Suchan assumption substantially simplifies the expression for thewavenumber diffusion tensor Dij (see Appendix A for details),so that Equation (5) becomes

òdp

w

w

p w

wd

dn

d

= -

=á ñ

- = -

¥D

k k

k c kq S q dq

c kq

n

n

k k

k

k k k

k

1

32

8 2,

7

ij iji j pe

peij

i jij

i j

2

4

2 0

3

4

2

2

2 2s

2

2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( )

¯

( )

where δij is the Kronecker delta and we have introduced thespectrum-averaged mean wavenumber

ò p=

q q S qd q1

28

2

3

3¯ ( )

( )( )

and the scattering frequency

np w

wd p w

w=

á ñ=

c kq

n

n c kq

4 4. 9

pe pes

4

2 3

2

2

4

2 32¯ ¯ ( )

Since the scattering frequency νs is proportional to thespectrum-weighted mean wavenumber q 2¯ , knowing thislatter quantity leads to a determination of the scatteringfrequency. Equivalently, observations of radio-wave scatteringin the solar corona provide a diagnostic of the level of densityfluctuations via the quantity q 2¯ . The assumption of isotropyof the scattering density fluctuations allows us to substantiallysimplify the diffusion operator, so that in spherical coordinates

mn

mm

¶¶

¶¶

=¶¶

-¶¶k

Dk 2

1 , 10i

ijj

s 2⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟( ) ( )

where m q= cos , θ being the polar angle for k.

2.2. Anisotropic Scattering

As we shall see below, using numerical simulations based onthe isotropic scattering analysis above, isotropic scattering isinconsistent with the observations of solar radio source sizesand time profiles. We therefore now develop a model forscattering in an anisotropic spectrum of density fluctuations

qS ( ). Similar to previous investigations (e.g., Hollweg 1968),we assume that the anisotropic density fluctuations are axiallysymmetric, so that the spectrum can be parameterized as aspheroid in q-space:

a= +^-qS S q q , 112 2 2 1 2( ) ([ ] ) ( )

where α=h⊥/hPis the ratio of perpendicular and parallelcorrelation lengths (see also Appendix B). Whenh⊥?hP(i.e., α?1), the density fluctuations are mostly inthe perpendicular direction; conversely, whenh⊥=hP(i.e., α=1), the spectrum of density fluctuations isdominated by the fluctuations in the parallel direction. Forexample, the direction parallel to heliospheric radial directionfollows the guiding magnetic field in spherically symmetriccorona.

It is convenient to introduce the anisotropy matrix

a=

-

1 0 00 1 00 0

. 121

A⎛

⎝⎜⎜

⎠⎟⎟ ( )

Then, defining =q qA (so that = -q q1A ), we can write

a+ = = = =^- q q q qq q q A q q q , 13i ij j i i

2 2 2 2 2A · ( )

where q⊥ and qP are, respectively, the perpendicular andparallel components of the wavevector q.Using Equations (5), (11), and (13), the wave vector

diffusion coefficient can be written as

ò

ò

ò

pw

wd

pw

p wd

w

p wd

=

=

= a b a b

- - -

- -

q q k

q q q q k

q q q q k

Dc

q q Sd q

cS d q

cA A S d q

4 2

32

32det ,

14

ijpe

i j

pei j

pei i

4

2

3

3

4

2 21 1 1 3

4

2 21 1 3

A

A A A

J

(∣ ∣) ( · )( )

( ) ( ) ( ) ( · )

( ) ( · ) ( )

( )

where aº =-det det 1J A( ) ( ) is the determinant of theJacobian matrix J transforming coordinates from q to q.Equation (14) can be written as

ò

w

p wd

pa= -

´

a b aba b- -

¥

Dc

A Ak k

k k

q S q dq

32

, 15

ijpe

i j

4

2 21 1

2

0

3

⎛⎝⎜

⎞⎠⎟

( ) ( )

where we introduced = -k k1A .We can now write the diffusion tensor components Dij in

terms of the original quantities k:

ò

w

pwa= -

´

-

-

- -

-

¥

k k

k k

k kD

A

c

q S q dq

32

. 16

ijij i j pe

2

2 1 2

2 2

2 3 2

4

2

0

3

AA AA

⎡⎣⎢⎢

⎤⎦⎥⎥( )

( ) ( )( )

( ) ( )

For isotropic scattering, the anisotropy matrix A reduces to theidentity matrix and Equation (16) correspondingly reduces toEquation (7). Equation (16) coincides with Equation (B10) ofArzner & Magun (1999; note the equation sign misprint in theirpaper’s appendix).

3. Stochastic Differential Equations

We now proceed to cast the Fokker–Planck Equation (2) in aform suitable for numerical computation. The scattering term inEquation (2) can be written as

=¶¶

¶¶

=¶¶

¶+

¶¶

=¶¶

¶+

¶¶

dN

dt kD

N

k kN

D

k kD N

kN

D

k kB B N

1

2, 17

iij

j i

ij

j jij

i

ij

j jim jm

T

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ ( )

3

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whereB is a positive-semi-definite matrix with matrix elementsdetermined by matrix D, so that

=D B B1

2. 18ij im jm

T ( )

The nonlinear Langevin equation for k t( ) corresponding to theFokker–Planck Equation (17) is

x=¶

¶+

dk

dt

D

kB , 19i ij

jij j ( )

where x t( ) is a Gaussian white noise with the propertiesxá ñ =t 0( ) and x x d dá ñ =t t0i j ij( ) ( ) ( ), where á ñ... denotes anensemble average, δ (t) is the Dirac delta function, and thek-dependent deterministic vectors ∂Dij/∂kj and Bij correspondto the diffusion tensor Dij. These are analogous to the equationsdescribing binary collisions in a plasma (see, e.g., Ivanov &Shvets 1978; Shvets 1979; Rosin et al. 2014). Equation (19) issimilar to the equation by Arzner & Magun (1999); it is thedefinition of the stochastic integral in Itôʼs sense, adopted in thetheory of random processes. Itô’s approach considerablysimplifies its numerical integration and requires the knowledgeof function kDij ( ) at the beginning of the time step rather thanhalf-step in Stratonovich form (Ivanov & Shvets 1978). Thefirst term on the right-hand side describes the so-called Itô drift,a systematic decrease of ki due to elastic scattering, while thesecond term represents diffusion. The presence of the Itô driftimproves the stochastic differential equations used in the past(e.g., Steinberg et al. 1971; Riddle 1974; Thejappa et al. 2007)and conserves the value of k∣ ∣ for elastic scattering.

If we apply Itôʼs formula to the square of the wavevector=k k k ki i· , one finds

= + =d

dtk k k

dk

dtB B2 0, 20i i i

iij ij ( )

where we have used ki dki/dt=−ki ∂Dij/∂kj=−νsk2 and

n=B B k2ij ij s2. One can see that the presence of the so-called

Itô drift is necessary to ensure conservation of = kk ∣ ∣ inscattering events, similar to pitch angle scattering in a Lorentzgas (e.g., Ivanov & Shvets 1978).

Including large-scale refraction due to gradual variation ofthe ambient density rn( ) of the solar corona, the equation forthe components of wavevector k becomes

ww

wx= -

¶+

¶+

dk

dt r

r

r

D

kB , 21i pe pe i ij

jij j ( )

which, in combination with the radio-wave transport equation

w=

dr

dt

ck , 22i

i

2( )

describes the propagation, refraction, and scattering of radio-wave packets in an inhomogeneous plasma.

3.1. Numerical Solution of the Langevin Equations

Following the conceptually similar description of plasmacollisions, we modify the transport code of Jeffrey et al. (2014),giving the wave vector and position of photons at the next time

step from the stepping equations

ww

w

x

w

+ D = -¶

¶D

¶D + D

+ D = + D

k t t k tr t

rt

r t

r tt

D

kt B t

r t t r tc

k t t

,

, 23

i ipe pe i

ij

jij j

i i i

2

( ) ( )( ( ))

( ) ( )( )

( ) ( ) ( ) ( )

where the ξi are random numbers drawn from the normaldistribution N(0, 1) with zero mean and unit variance.

3.1.1. Isotropic Scattering

In the case of isotropic density fluctuations (and henceisotropic scattering), the Langevin equations take on aparticularly simple form. With Dij now given byEquation (7), one finds that

p w

wd

¶= -

á ñ= -

D

k c kq

n

nk2

824

ij

j

pei

4

2

2

2 s¯ ( )

and

p w

wd

d

n d

=á ñ

-

= -

Bc k

qn

n

k k

k

kk k

k

4

, 25

ijpe

iji j

s iji j

4

2

2

2

1 2

2

22

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

¯

( )

so that Equation (23) becomes

x

ww

w

n n

x

+ D = -¶

¶D

- D +

´ - Dk

k t t k tr t

rt

r t

r tt

kk

kt k

k

kt 26

i ipe pe i

i

ii

s2

2 s2 1 2

21 2⎜ ⎟⎛

⎝⎞⎠

( ) ( )( ( ))

( ) ( )( )

( )

( · ) ( ) ( )

and

w+ D = + Dr t t r t

ck t t, 27i i i

2( ) ( ) ( ) ( )

where, again, x is a vector with components ξi being randomnumbers drawn from the normal distribution N(0, 1).Equations (26) and(27) are the Euler–Maruyama approx-imation to the Langevin Equations (21) and(22); they are in aform particularly useful for solving initial value problems. Thetime step Δt is chosen to be much smaller than thecharacteristic times due to scattering and refraction. The meanscattering time 1/νs is normally the smaller time, and so wechoose Δt=0.1/νs. Since νs (r) is a decreasing function of r,the time step is shortest near the radio emission source andquickly increases with distance.

3.2. Anisotropic Scattering

Now let us find the Langevin equation functions for theanisotropic scattering tensor given by Equation (16). For the

4

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anisotropy matrix A given by Equation (12),

¶¶

= =- - -k k kk

A k2 2 , 28j

j ji i2 2 2A A( ) ( ) ( )

¶¶

= =- - - -k k k k kk

kA k3 3 , 29j

j ji i2 3 2 2 1 2 2 2A A A( ) ( ) ( ) ( )

and

¶¶

= - = --

-

-

-

k k

k

k kk

A k

k

1, 30

j

j ji i

2 1 2

2

2 3 2

2

3AAA( )( )

( )( )

where the -Aji2 are elements of the diagonal matrix

=- - -2 1 1A A A and summation over repeated indices is implicit.Using the definitions = -k k1A and = = -k k kk 2 1 2A∣ ∣ ∣ ∣ , onefinds the explicit expressions for Langevin equations in case ofanisotropic scattering:

a

¶¶

= -

-+

+

= - -+

+

= - +

- +

- -

-

- - - -

-

- - -

-

- - - -

- -

- - -

k

k k

k k

k k

k k k

k k

k k k

k k k

k k k k

kD D

A

A A

Dk k

kD

kk

k

3

tr

3

2 3

2 , 31

jij A

ij j

ij j jj i

i j j

Ai i i

i

Ai i

2 2

2 3 2

2 2 2 2

2 3 2

2 2 2

2 5 2

4

3

4 2 2

3

2 4

5

52 4 2 4

2 2

AA

A AA

A A AA

A A A A

A A

A A A

⎡⎣⎢⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

( )( )

( ) ( )( )

( ) ( ) ( )( )

( ) ( ) ( )( )

( ) ( )

[ ( ) ( ) ( ( )

( ) )] ( )

where a= +-tr 22 2A( ) is the trace of matrix -2A for theanisotropy matrix A given by Equation (12), and

d= -

= -

a aa

--

- -

-- -

k k

k k

k k

BD

Ak

D

kA

k

2

2, 32

ijA

i jj

Aij

i j

2 1 21

1 1

2

12 1

2

AA A

A A

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

( )( ) ( )

( ) ( )( )

where

òw

pwa=

¥D

cq S q dq

3233A

pe4

2 0

3 ( ) ( )

is the k-independent coefficient in the diffusion tensor(16).The Langevin Equation (21), together with the vector functions(31) and(32), can be solved numerically for an arbitraryspectrum of density fluctuations. For isotropic scattering, i.e., inthe limit α=1, the functions (31) and(32) reduce toEquations (24) and(25), respectively.

Due to the choice of anisotropy matrix (Equation (12)), it isuseful to introduce the perpendicular

a a a

a a a

¶¶

= - + - + -

= - + + -

^^

^

kD

D k

kk k k

D k

kk k

2 1 3 1

1 3 1

34

jj

A

A

52 2 2 2 2 2

52 2 2 2 2

[ ( ) ( ) ]

[ ( ) ( ) ]

( )

and parallel

a a a

a a

a a a a

¶¶

= - + -

+ -

= - + + -

kD

D k

kk k

k

D k

kk k

2 1

3 1

3 3 1 35

jj

A

A

52 4 2 2 2

2 2 2

54 2 2 4 2 2

[ [( )

( ) ]]

[( ) ( ) ] ( )

components of ∂Dij/∂kj in the differential Equation (21). Theseequations differ from Equation (44) in Arzner & Magun (1999).For isotropic scattering (α=1), Equations (34) and(35)reduce to the isotropic case. The expressions (34) and(35)remain finite for the limiting cases of quasi-perpendiculardensity fluctuations, i.e., a ¥, as well as in the quasi-longitudinal case a 0.One can also readily verify the result (32) a posteriori:

d

d

d

´ = -

´ -

= -

- +

= -

=

- -- -

- -

- -- -

- - - -

-- -

k k

k k

k k

k k k k

k k

B B D A Ak

k

D A Ak

k

k

k

D Ak

D

1

2

, 36

ik jkT

A ik kj iki k

kjk j

A ik kj iji j

i j i j

A iji j

ij

1 11 1

2

1 1

2

1 11 1

2

1 1

2

2 1 1

4

22 2

2

A A

A A

A A

A A A A

A A

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥⎥

⎡⎣⎢

⎤⎦⎥

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )

as required. We note that the “square root” of a matrix is notunique, and so, to simplify the numerical solution of theequations, we follow Schmidt et al. (2011) in the choice for Bij.

3.3. Collisional Absorption of Radio Waves

The plasma of the solar corona is a collisional medium,which leads to free–free absorption of propagating electro-magnetic waves, with a characteristic rate γ. For binarycollisions in a plasma (e.g., Melrose 1980; Lifshitz &Pitaevskii 1981),

gw

wg= , 37

pe2

2 c ( )

where

gp

=Lre n

m v

4

3

2 ln. 38c

4

Te3

( ) ( )

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Here the thermal speed =v T me eTe , with Te the electrontemperature in energy units. A constant Coulomb logarithm lnΛ;20 is assumed, per Ratcliffe & Kontar (2014). We alsoassume an isothermal solar corona with temperatureT=86 eV.

The effects of collisional absorption are stronger in higherdensity plasmas. The attenuation of the signal due to absorptionis given by

= t-N t N e , 390 a( ) ( )

where the Coulomb collisional depth

òt g= r t dt. 40a ( ( )) ( )

Absorption is in general always important at higher frequencies50MHz and noticeably affects the time profiles at higherfrequencies. The effect of absorption is also noticeable whenthe scattering is so strong that the photons are trapped near thesource for the time longer than free–free absorption time 1/γ.

4. Monte Carlo Ray-tracing Simulations

4.1. Methodology

We have simulated the propagation of radio waves in thepresence of background density fluctuations, using the MonteCarlo ray-tracing method presented in Section 3(Equations (23)). Simulations were performed in the solarcentered coordinate system (x, y, z) as shown in Figure 1, withthe z-axis directed toward the observer; x and y are heliocentric-Cartesian coordinates in the plane of the sky, used in solarimaging observations (Thompson 2006).

The solar corona is assumed to be spherically symmetric andthe density fluctuations are assumed to be aligned with respectto the local radial direction, so that qP is parallel to r for a givenphoton location. Similar to Kontar & Jeffrey (2010) and Jeffrey

& Kontar (2011), before advancing the stochastic differentialEquations (31) and(32) corresponding to the LangevinEquation (21), the wavevector k is first rotated to a local (x′,y′, z′) coordinate system where z′ is radially aligned (seeFigure 2). In the paper, we only consider spherically symmetricsolar corona.10 The stochastic differential equations are then

Figure 1. Cartoon showing the Sun-centered Cartesian coordinate system (x, y, z), where the z-axis is directed toward the observer. The initial location of a pointsource of radio emission is given by the radial coordinate Rs and the polar angle θs; the azimuth angle in the plane of the sky is not relevant to our study. The photonsscatter until they cross a sphere at a distance large enough that scattering is no longer important, resulting in an apparent source size and position indicated by the redregion.

Figure 2. Coordinate systems (x, y, z) and (x′, y′, z′) with the Sun center in theorigin, where z-axis is directed to an observer and the z′-axis is parallel to r,and the y′-axis is tangent to the circle created by the intersection of the planeformed by the z and z′ axes and a spherical surface of radius r.

10 The approach can include arbitrary alignment and hence trace the localdensity anisotropy given by, e.g., a magnetic field.

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advanced one time step and then the wavevector k is rotatedback to the fixed (x, y, z) coordinate system for propagation tothe next scattering event. Figure 2 shows the correspondinggeometry; the z′-axis is parallel to r, and the y′-axis is tangentto the circle created by the intersection of the plane formed bythe z and z′ axes and a spherical surface of radius r. Therelationships between the wavevector components are

f q q ff q q f

q q

=- + -= + -= +

^ ^

^ ^

^

k k k k

k k k k

k k k

sin sin cos cos

cos sin cos sin

cos sin , 41

x x y

y x y

z y

( )( )

( )

where (kx, ky, kz) are the components in the (x, y, z) coordinatesystem, (k⊥x, k⊥y, kP) are the components in the (x′, y′, z′)coordinate system, and the rotation angles are given by thephoton position in the (x, y, z) coordinate system,

f q q= = - =y x z r z rtan , sin 1 , cos . 422 2 ( )

In all simulations, the initial radio source was modeled as apoint source with an isotropic distribution of wavevector k andwith a frequency ω=1.1 ωpe(Rs) corresponding to the near-fundamental plasma emission at a distance Rs from the solarcenter, determined using a spherically symmetric Parkerdensity model (Parker 1960) with constant temperature andconstants chosen to agree with satellite measurements adaptedfrom Mann et al. (1999). The absolute value of the wavevector

w w= -k cpe2 2 1 2( ) is therefore the same for all photons.

Although this density model is relatively simple and has beenused successfully for the simulations of Type III bursts in thepast (e.g., Kontar 2001), it does not have a simple analyticalform, which is needed for the solution of the differentialEquation (23). To simplify the density model, we fit thenumerical solution with three power-law functions (see, e.g.,Alcock 2018), giving

= ´ + ´

+ ´

n rR

r

R

r

R

r

4.8 10 3 10

1.4 10 , 43

914

86

62.3

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )

( )

which can be easily differentiated to find the derivatives usefulto solve the ray-tracing Equation (23).

The simulations begin with approximately 104 photons withdifferent initial positions given by Rs from 1.05 to 57 Re. Usingthe coronal density model given by Equation (43), thesecorrespond to plasma frequencies from 460 to 0.1 MHz,respectively. The photon transport was simulated until adistance where both refraction and scattering become negli-gible, or until the photon frequency ω (which is conserved inthe simulations) became much larger than the local plasmafrequency. In each simulation run, a photon was traced until itcrossed a sphere where scattering becomes negligible or to 1 au(whatever is less) and the arrival time and photon properties atthis sphere were recorded. The locations of the photons on thissphere directed toward the observer (i.e., those with0.9<kz/k<1) were then back-projected to the source plane,thus defining the apparent source intensity map I(x, y) (Kontar& Jeffrey 2010, red region in Figure 1). Similarly, the spread ofarrival times on this sphere determines the observed burstintensity-time profile. In order to calculate the decay time atdifferent frequencies, we first select the peak time of the flux(maximum in the histogram of the arrival times); times greater

than the peak time are regarded as defining the decay phase,which was fitted with a Gaussian form. The delay time isdefined as the half width at half maximum (HWHM) of theGaussian fit.The total flux was evaluated by performing an integral

ò I x y dx dy,( ) over the corresponding source area. Also, usingsolar disk-centered coordinates, the centroid position of thesource (x, y) was found by calculating the first normalizedmoments (means) of the distribution:

ò

ò

ò

ò= =-¥

¥

¥-¥

¥

¥xx I x y dx dy

I x y dx dyy

y I x y dx dy

I x y dx dy

,

,,

,

,44¯

( )

( )¯

( )

( )( )

and the variances (s x2, sy

2) calculated using the secondnormalized moments:

ò

ò

ò

ò

s

s

=-

=-

¥

¥

¥

¥

x x I x y dx dy

I x y dx dy

y y I x y dx dy

I x y dx dy

,

,,

,

,. 45

x

y

22

22

( ¯) ( )

( )

( ¯) ( )

( )( )

The full width at half maximum (FWHM) in each direction canthen be calculated using

s=FWHM 2 2 ln 2 , 46x y x y, , ( )

based on the assumption that the distribution I(x, y) isGaussian. To evaluate the FWHM source sizes we also fittedI(x, y) with a 2D Gaussian and determined the sizes using thebest-fit parameters. Typical images I(x, y) are shown inFigures 3 and 4.Because of the finite number of photons in the sample, the

source centroids (Equation (44)) and sizes (Equations (45))have associated statistical errors (see, e.g., Rao 1973). Theuncertainties in the mean values can be estimated as

ds

ds

xN

yN

, , 47x y¯ ¯ ( )

and the uncertainty in the FWHM sizes as

ds

NFWHM 2 2 ln 2

2, 48x y

x y,

, ( )

where N?1 is the number of photons used to determine themeans x y,( ¯ ¯) and the standard deviations σx, σy. Theseuncertainties are used in all numerical results presented in thispaper.Krupar et al. (2018) have recently investigated the effects of

isotropic scattering on time profiles generated in the inter-planetary medium using Monte Carlo simulations. Theyassumed a power-law spectrum of electron density fluctuations(see Appendix C for the derivation) and also used expressionsfor the diffusion coefficient from Thejappa et al. (2007) andThejappa & MacDowall (2008) to describe the scatteringeffects. We adopt the same density fluctuations model here.Krupar et al. (2018) used11 Equation (64), viz.

p - - q l l4 , 49i2

02 3 1 3 2¯ ( )

11 Note a missing factor of π/2 in their equation.

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where li=(r/Re) [km] is the inner scale of the electrondensity fluctuations (Manoharan et al. 1987; Coles &Harmon 1989), R is the heliocentric distance, lo=0.25 Re

(R/Re)0.82 is an empirical formula for the outer scale

(Wohlmuth et al. 2001), and d= á ñ n n2 2 is the level ofdensity fluctuations with the spectrum given by Equation (62).ò was taken as a quantity independent of radial distance.

We stress that for the density fluctuations spectrum (62), thescattering rate is determined by the density fluctuations atscales near li. Since both the density fluctuations variance ò2

and the outer scale lo(r) determine the level of densityfluctuations in Equation (49), ò(r) cannot be determinedwithout knowledge of l0(r), and different models for l0(r)result in different values for ò(r). Hence the ò values taken forthe simulations in the next section should be viewed as thestandard deviation of density fluctuations for a given outerscale model lo(r), and may not be suitable for direct comparisonwith density fluctuation measurements in the corona.

4.2. Simulation Results for a Single Frequency

Using the assumptions presented in the previous section, wecan choose ò so that the characteristic size of the radio source isabout 19′ for fpe=32MHz (observing frequency ∼35MHz),as typically observed for fundamental plasma emission (Kontar

et al. 2017). Figures 3–8 plot the main results of the ray-tracingsimulations. Figures 3 and 4 show the results for a point sourcelocated above the solar disk center at a height 0.75 Re abovethe photosphere, where fpe=32MHz according to the densitymodel(43). The simulations presented in Figure 3 use the samelevel of density fluctuations ò but different values of theanisotropy parameter (α=0.3 and α=0.5, respectively). Forboth cases, the FWHM source size is about 1.15 Re (consistentwith 19′ FWHM size observations), but the time profile for thesimulation with α=0.5 (Figure 3) is significantly broader thanthat for α=0.3 (Figure 4). Turbulent density fluctuationswhich have a power that is weaker in the parallel directioncompared to the perpendicular to radial direction result in areduced time-broadening effect (i.e., radio-wave cloud broad-ening along the z direction); consequently, the results withanisotropy factor α=0.3 give a characteristic decay time∼0.6 s, exactly as observed (see Figure 4 in Sharykin et al.2018).Figures 5 and 6 demonstrate how the observed source sizes

and the decay times vary with the value of the anisotropyparameter α. Low-level density fluctuations (e.g., ò=0.2;Figure 5) are too weak to provide sufficient scattering toexplain FWHM sizes as large as 1.15 Re. At the same time,nearly isotropic scattering (Figure 6) with ò=0.8 provides theobserved sizes, but the decay time appears to be larger than

Figure 3. Simulations for a point source located at RS=1.75 Re ( fpe=32 MHz), and using ò=0.8, α=0.5. Left: time profile of the observed photons—blue withabsorption, red without absorption, and dashed line indicates the location of the time-profile maximum. Center: observed radio image in Sun-centered coordinates. Theorange circle denotes the Sun, the dashed line denotes the radius where the plasma frequency is 32 MHz, and the blue circle is the FWHM source size. Right:directivity of the observed radio emission. The red dashed line shows the width at half maximum.

Figure 4. Simulation results as in Figure 3 but with stronger anisotropy, α=0.3.

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observed. Reduced scattering along the radial direction (e.g.,density fluctuations that are predominantly in perpendiculardirections) decreases the characteristic decay time and aniso-tropy, and a value α=0.3 provides the best match to theobservations. Indeed, comparing Figures 5 and 6, we find that adensity fluctuation level of ò;0.8 and an anisotropyparameter of α=0.3 are the parameters that best explain

recent LOFAR observations by Kontar et al. (2017) andconsistent with the source sizes reported by Dulk &Suzuki (1980).Scattering of photons close to the intrinsic source contributes

substantially to the free–free absorption of radio waves.Photons experiencing strong scattering stay longer in thecollisional medium and hence are absorbed. Indeed, Figure 3

Figure 5. FWHM sizes and decay time (HWHM) with ò=0.2 as a function of anisotropy α. The black symbols are from fitting the simulation data with a 2DGaussian function to determine the size and centroid position, the blue sizes are using Equation (46). One standard deviation uncertainty is calculated usingEquation (48).

Figure 6. Same as Figure 5, but for ò=0.8.

Figure 7. Radio images for a point source located at RS=1.75 Re ( fpe=32 MHz), and for three different source locations θs=0°, 10°, 30° from the disk center. Allimages are for anisotropic turbulence with anisotropy parameter α=0.3 and a level of turbulence ò=0.8. The projected positions of the source and the imagecentroid are shown by red and blue crosses respectively. The orange circle denotes the Sun, the dashed line denotes the radius where the plasma frequency is 32 MHz,and the blue circle is the FWHM source size.

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demonstrates that the time profile is significantly extendedwhen absorption is switched off. This difference is smaller forthe stronger anisotropy case presented in Figure 3.

The main effects on source location and size are shown inFigure 7. Because of projection effects along the radialdirection, the FWHM source size along the x-directiondecreases with heliocentric angle (Figure 8), while the FWHMin the y-direction (perpendicular to the radial direction) changesonly weakly, remaining 1–1.2 Re. Sources located away fromthe disk center are shifted radially (along the x-direction in oursimulations), and the near-linear dependence of the sourceposition on sin θs can be clearly seen from Figure 8. Theobserver sees an apparent position that is shifted radially awayfrom the disk center, with the shift projected onto the skyplaneproportional to sin θs. While sources near the disk centerθs=0 are radially shifted toward the observer, the true andapparent sources coincide in the (x, y) plane of the sky. Thecase of more isotropic scattering (Figure 9) suggests that thedegree of anisotropy only weakly affects source sizes andpositions close to the disk center, but has a stronger effect closeto the limb. Thus, the radial size (along the X-axis) for α=0.5(nearly isotropic scattering) does not decrease toward the limbas fast as in the case with stronger anisotropy α=0.3(Figure 8). This is consistent with the observations of angularbroadening of the Crab Nebula (e.g., Dennison & Blesing 1972)

by coronal turbulence, which show a preferential elongationalong the tangential direction.Similarly, the interplay between scattering and the focusing

effects determine the directivity of the escaping emission. Thesimulated directivity patterns show that although radio-wavescattering effects lead to large source sizes, the directivity (rightpanel in Figures 3–4) is predominately in the radial directionwith half widths at half maximum ;47° and ;40° foranisotropy α=0.5 and α=0.3 correspondingly. Theseresults are different from early results suggesting isotropicdirectivity due to scattering as reviewed by McLean &Melrose (1985).

5. Observations of Type III Solar Radio Bursts in theHeliosphere: Source Sizes and Decay Times

It is instructive to review observations of solar radio burstsource sizes and decay times for comparison with the ray-tracing results. Solar radio bursts are observed over a widerange of frequencies from about ∼500MHz down to ∼20 kHznear 1 au. Therefore, the variation of burst parameters withfrequency allows us to diagnose the scattering over a widerange of heliocentric distances.

Figure 8. Left: shift of the centroid position x as a function of the source heliocentric angle θs. The shifts are calculated for anisotropic scattering with α=0.3 andturbulence level ò=0.8 as in Figures 7. Center: FWHM X-size given by Equation (46). Right: FWHM Y-size given by Equation (46). The error bars show onestandard deviation given by Equation (47) and (48). The number of detected photons in z-direction is decreasing, so the uncertainties are large for angles closeto θs;90°.

Figure 9. Same as Figure 8, but for α=0.5.

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Figure 10 combines measurements12 by several differentauthors (Bougeret et al. 1970; Abranin et al. 1976;Alvarez 1976; Abranin et al. 1978; Chen & Shawhan 1978;Dulk & Suzuki 1980; Steinberg et al. 1985; Saint-Hilaire et al.2013; Krupar et al. 2014; Kontar et al. 2017) over the last 50years. The Type III source sizes (FWHM; degrees) are forfrequencies ranging from ∼0.05 to 500MHz. Using a weightedlinear fit in log-space, the FWHM depends on the observingfrequency ( f; MHz) as (see Figure 10)

= ´ - fFWHM 11.8 0.06 . 500.98 0.05( ) ( )

Similarly, a collection of Type III burst decay timemeasurements (Alexander et al. 1969; Aubier & Boischot 1972;Elgaroy & Lyngstad 1972; Alvarez & Haddock 1973; Barrow& Achong 1975; Krupar et al. 2018; Reid & Kontar 2018),over the frequency range from ∼0.1 to 100MHz, is presentedin Figure 10. The best-fit power-law dependence of the decay

time τ (s) on frequency f (MHz) is

t = ´ - f72.2 0.3 . 510.97 0.03( ) ( )

For comparison, Wild (1950) derived an expressionτ=100×f−1 for the decay time, based on observations inthe frequency range 80–120MHz, while Alvarez & Haddock(1973) obtained τ=51.29×f−0.95 based on observations inthe frequency range 50 kHz–3.5 MHz, and Evans et al. (1973)obtained τ=(2.0±1.2)×100×f−(1.09±0.05) based onobservations in the frequency range 67 kHz–2.8MHz for 1/edecay.Figure 11 shows the results of our simulations, assuming

isotropic scattering. The decay time agrees within a factor of 2with that by Krupar et al. (2018); this difference is likely due tothe different numerical schemes used (see the discussionaround Equation (20)). While a detailed comparison for variousanisotropies would require substantial computation effortoutside the scope of this work, it is nevertheless clear thatisotropic scattering cannot explain the observations. Forexample, if the level of density fluctuations ò is chosen toexplain the decay times, the predicted source sizes are far toosmall to explain the observations. Similarly, if the level ofisotropic density fluctuations is chosen to match the sourcesizes, the decay times are too long. Evidently, anisotropicscattering, with a reduced level of scattering along the radialdirection, is needed to account for both observed source sizesand decay times.

6. Summary and Discussion

Radio emission from solar sources is strongly affected byscattering on small-scale density fluctuations. In general, theobserved source sizes and positions, time profiles, anddirectivity patterns are determined mainly by propagationeffects and not by intrinsic properties of the primary source. Wehave constructed a new model that allows quantitative analysisof radio-wave propagation in a medium that contains an axiallysymmetric, but anisotropic, scattering component. We havecompared the results of numerical simulations using this modelwith observations of source sizes and time profiles over a widerange of frequencies. Since plasma emission sources with smallintrinsic size are observed in type III bursts (Kontar et al. 2017;Sharykin et al. 2018), the observed radio sources are dominatedby the scattering, at least at these frequencies. Hence their sizescan be used as diagnostics of radio-wave propagation effects.In general, a typical source of plasma emission (e.g., Type I,

II, III, IV, or V solar radio bursts) might have a finite sizeFWHMsource defined by the intrinsic size of the regionproducing the radio emission. The observed FWHM size forsuch a source is given by (FWHMsource

2 +FWHMscat2 )1/2,

where FWHMscat is calculated in this paper. Thus forfrequencies around 35MHz, FWHMscat;1.1 Re, so if thesource is substantially smaller than this value, the observedsource sizes are dominated by scattering effects. For largesources 1.1 Re (i.e., 18′), the source sizes due to scatteringcalculated in this paper can be subtracted in quadrature fromthe observed source size to give the dimensions of the intrinsicsource, corrected for wave propagation effects. However, thesize of density fluctuations, and hence the scattering efficiency,can vary appreciably from event to event and from one solaratmosphere region to another, consistent with the considerable

Figure 10. Top: source sizes (FWHM; degrees) of typeIII solar radioobservations vs. frequency f(MHz). A combination of observations is plottedas indicated by the legend, and a weighted linear fit was applied to the data.The dashed line shows the fit given by Equation (50). Bottom: decay times τ(defined as the e-folding time in seconds) of Type III solar radio observationsvs. frequency f (MHz). A combination of observations is plotted as indicated bythe legend, and a weighted linear fit was applied to the data. The dashed lineshows the fit given by Equation (51). The standard deviation error bars werecalculated from the statistical distribution of the data and measurement errors ifreported.

12 The source sizes reported by Dulk & Suzuki (1980) and Steinberg et al.(1985) were given as the full width at 1/e of the distribution, so the values wererecalculated into FWHM values by multiplying by a factor of ln 2 .Measurements above 1 MHz from Krupar et al. (2014) were not plotted as“the analysis above 1 MHz is perhaps distorted by background signals resultingin increased source sizes” and thus, the results were deemed unreliable.

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variability of the density fluctuation spectrum observed in thesolar wind (e.g., Celnikier et al. 1983; Marsch & Tu 1990).

The main result of our work comes from the comparison ofthe simulation results with combined imaging and time-delayobservations. For a given density fluctuation magnitude ò andouter and inner scales lo, li, changing the anisotropy parameterα only weakly affects the source size over a broad range ofangles near the disk center. (These effects are most noticeableclose to the limb, where the anisotropy direction corresponds tothe line of sight.) However, the time profiles (or, equivalently,the radio pulse expansion along the line of sight) are stronglyaffected by the value of α. Comparison of the simulation resultswith observations of source size and time delay, both as afunction of frequency, suggests that anisotropic densityturbulence, with preferential scattering perpendicular to thesolar radial direction (α;0.3) is required to account for boththe source size and time-delay variations at frequencies close to30MHz. In order to explain the Type III observations in theheliosphere between 0.1 and 1MHz, additional simulations arerequired. Indeed, the simulations by Krupar et al. (2018)demonstrate that although isotropic scattering withò;0.06–0.07, lo and li given by Equation (49) can explainthe decay time, the anisotropy of density fluctuations isinadequate to explain the typical source sizes (e.g., Figure 11).The numerical model developed in Section 3 suggests that theanisotropic density fluctuations (lower power in the paralleldirection) are required to account for the source sizes and decaytimes simultaneously. This result requires further computation-ally intensive investigations using the method outlined in thepaper.

The other interesting result is that the directivity of solarradio bursts is determined by a combination of wave focusingdue to large-scale refraction and scattering on small-scaledensity fluctuations. At the same time, the intrinsic directivityof the source, e.g., the dipole pattern associated with radioemission near the plasma frequency (Zheleznyakov &Zaitsev 1970) is quickly lost due to scattering and thus is notevident in observations. Contrary to the results of earlysimulations (e.g., McLean & Melrose 1985, for a review), theresulting directivity appears to have a width of approximately40° near 30MHz. The observed directivity pattern is acombination of the focusing due to large-scale refraction andthe scattering. The anisotropy of the density fluctuationspectrum plays an important role in governing the emission

pattern of solar radio bursts. Therefore, efficient isotropizationof radio waves near the emission source does not automaticallyimply an isotropic emission pattern as sometimes assumed.Free–free absorption appears to have a small or negligible

effect for frequencies below 30–50MHz. However, thecollisions are important for higher frequencies and candetermine the time profile. It is also important to note thatthe stronger the scattering of radio waves, the more pronouncedthe effect of the free–free absorption. Photons that are stronglyscattered are also absorbed stronger and hence produce aweaker contribution to the observed properties.The effect of radio-wave scattering depends on the radial

profiles of the quantities q r2( ¯ )( ) and α (r), representing thesize and anisotropy of density fluctuations, respectively. For adecreasing spectrum of electron density fluctuationsS(q)∝q−5/3, scattering is most sensitive to the largest q (i.e., the smallest scales) in the inertial range spectrum—thescale of energy dissipation—and so provides key diagnosticsfor the inner scale li(r) (Equation (49)). At the same time,conclusions regarding the level of density fluctuations ò arealso dependent on, and so require knowledge of, the outerdensity scales l0. For example, to explain the observations near30MHz, a high level of density fluctuations ò=0.8 is requiredfor the model of lo (r) adopted, and it is possible that the modello(r) is not valid at these frequencies. Comparison betweenobservations and simulations therefore provides a powerful toolwith which to infer the radial variation of density fluctuationsfrom the Sun to the Earth, which will be the subject offurther work.

E.P.K. and N.L.S.J. acknowledge the financial support fromthe STFC Consolidated Grant ST/P000533/1. A.G.E. wassupported by grant NNX17AI16G from NASA’s HeliophysicsSupporting Research program. V.K. acknowledges support byan appointment to the NASA postdoctoral program at theNASA Goddard Space Flight Center administered by Uni-versities Space Research Association under contract withNASA and the Czech Science Foundation grant 17-06818Y.We also acknowledge support from the International SpaceScience Institute for the LOFARhttp://www.issibern.ch/teams/lofar/ and solar flarehttp://www.issibern.ch/teams/solflareconnectsolenerg/ teams.

Figure 11. FWHM size (left) and decay time (HWHM; right) calculated at various frequencies for isotropic scattering and for disk center source(FWHMx=FWHMy) for frequencies 0.1–1 MHz. The red dashed line indicates the best fit to the observations from Figure 10.

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Appendix AIsotropic Density Fluctuations

For isotropic density fluctuations, due to spherical symmetry,

d= -D Dk k

k.ij ij

i j0 2

⎛⎝⎜

⎞⎠⎟

Taking the projection of Dij with δij gives

d d= - =D Dk k

kD2 ,ij ij ii

i i0 2 0⎜ ⎟⎛⎝

⎞⎠

where the ki are components of k and the summation overrepeated indices is implicit. Using the wavenumber diffusiontensor given by Equation (7),

òpw

wd

p= q kD

cq q S q

d q

4 2, 52ij

pei j

4

2

3

3( ) ( · )

( )( )

or, in polar coordinates,

ò

ò

d q p q

p

=

D q S q qk q d dq

kq S q dq

1

2cos 2 cos

.

02 2

0

3

( ) ( )

( )

Hence one finds that Equation (52) can be written as

òdp

w

w= -

¥D

k k

k c kq S q dq

1

32. 53ij ij

i j pe

2

4

2 0

3⎛⎝⎜

⎞⎠⎟ ( ) ( )

This wave vector diffusion tensor has the same structure as thatfor Langmuir waves (e.g., Goldman & Dubois 1982;Muschietti & Dum 1991; Ratcliffe et al. 2012).

Appendix BGaussian Spectrum of Density Fluctuations

Following early works by Hollweg (1970) and Steinberget al. (1971) we assume that the density fluctuations have aGaussian correlation, so the Gaussian autocorrelation functionof the density fluctuations is

d=

á ñ=

á ñ- -^

^

rC r

n n

n

n

n

r

h

r

h

0exp , 54

2

2

2

2

2

2

2

⎛⎝⎜⎜

⎞⎠⎟⎟( ) ( ) ( ) ( )

where h2 and hPare the perpendicular and parallel correlationlengths, respectively, and dá ñn2 is the variance of densityfluctuations. For isotropic fluctuations,

d=

á ñá ñ

=á ñ

-r

C rn n

n

n

n

r

h

0exp , 55

2

2

2

2

2

⎛⎝⎜

⎞⎠⎟( ) ( ) ( ) ( )

where h=h⊥=hPis the correlation length. The spectrum S(q), defined as

ò= -S q C r e d r,k ri 3( ) ( ) ·

also has a Gaussian form

dp=

á ñ-S q

n

nh

q hexp

4, 56

2

22 3 2

2 2⎛⎝⎜

⎞⎠⎟( ) ( ) ( )

so that the variance of density fluctuations is

ò òd

pp

pá ñ

= = =¥

n

nS q

d qS q q

dq

24

2. 57

2

22

3

3 0

23

( )( )

( )( )

( )

Substituting the isotropic Gaussian spectrum(56) into the wavevector diffusion tensor(53), one finds

òdp

w

w

dp d w

w

= -

= -á ñ

¥D

k k

k c kq S q dq

k k

k

n

hn c k

1

32

4.

ij iji j pe

iji j pe

2

4

2 0

3

2

2

2

4

2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( )

The average wavenumber vector q, given by Equation (8), forthe density fluctuation spectrum of Equation (56), is

ò òpp

p p

= =

´ - =

¥

q q S q

d qh q

q h dq

h

1

2

exp4 2

4, 58

2

3

32 3 2

0

3

2 2

3

⎛⎝⎜

⎞⎠⎟

¯ ( )( )

( )

( )( )

so that the diffusion coefficient Dθθ becomes

òp w

w p

p d w

wp w

w

=

=á ñ

=

qq¥

Dc k

q S q dq

h

n

n c k

qc k

4

1

2

4

16, 59

pe

pe

pe

2 4

2 3 3 0

3

2

2

4

2 3

24

2 3

( )( )

¯ ( )

where d= á ñ n n2 2 2. The angular scattering rate per unit timebecomes

q p w

wá ñ

= =qq d

dtD q

c k2

860

pe2

24

2 3¯ ( )

or, per unit distance, for a photon with group speedvgr=c2k/ω,

q q p w

p w

w w

p w

w w

á ñ=

á ñ=

=-

=-

d

dx v

d

dtq

c k

q

h

1

8

8

2, 61

gr

pe

pe

pe

pe

pe

2 22

4

4 4

24

2 2 2

2 4

2 2 2

¯

¯( )

( )( )

an expression widely used (e.g., Chandrasekhar 1952; Holl-weg 1968, 1970; Steinberg et al. 1971; Lacombe et al. 1997;Chrysaphi et al. 2018; Krupar et al. 2018; Gordovskyy et al.2019) and identical to the expression (27) in Arzner & Magun(1999) (noting that =h l2 needs to be redefined toobtain h q= á ñd dt2 2* ).

Appendix CPower-law Spectrum of Density Fluctuations

In situ observations of density fluctuations suggest aninverse power-law spectrum of density fluctuationsS(q)∝q−( p+2), with the exponent p close to 5/3 as observed(Alexandrova et al. 2013). This power-law normally holds over

13

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a broad inertial range from outer scales l0=2π/q0 to innerscales li=2π/qi (see, e.g., Alexandrova et al. 2013, for areview):

=>

´ < <<

- +S q

q q

q q q q

q q

0,

const , ,

0,

62pi

i

02

0

⎧⎨⎪⎩⎪

( ) ( )( )

where the constant follows by normalizing the integratedspectrum to the level of density fluctuations dá ñn2 . Then thespectrum-weighted average wavenumber q (Equation (8))becomes (Lacombe et al. 1997; Arzner & Magun 1999)

=--

-

-

- -

- -qp

p

q q

q q

1

2. 63i

p p

ip p

202

101

¯ ( )( )

( )

This is often simplified further by assuming a large range ofwave numbers, so that qo=qi. For example, Thejappa et al.(2007) and Krupar et al. (2018) used Equation (63) withp=5/3 in the limit qo=qi, giving the particularly simpleform

p= - -q q q l l2 4 , 64i i02 3 1 3

02 3 1 3¯ ( )

for which the variance of density fluctuations is

ò òpp

p= =

¥ S q

d qS q q

dq

24

2. 652

3

3 0

23

( )( )

( )( )

( )

Then the scattering rate with q given by (64) becomes

q p w

wp w

wp w

w w w

á ñ= =

=-

- -

- -

d

dtq

c kl l

c k

l lc

216 4

2. 66

pei

pe

ipe

pe

22

4

2 3

2

02 3 1 3 2

4

2 3

2

02 3 1 3 2

4

2 2 3 2

¯

( )( )

Expressed as a scattering per unit of length x, Equation (66) is

q p w

w wá ñ

=-

- - d

dxl l

2. 67i

pe

pe

2 2

02 3 1 3 2

4

2 2 2( )( )

This is the expression used by Thejappa & MacDowall (2008)and Krupar et al. (2018), but includes an additional factor of π/2. It also coincides with Equation (30) from Arzner &Magun (1999).

ORCID iDs

Eduard P. Kontar https://orcid.org/0000-0002-8078-0902Xingyao Chen https://orcid.org/0000-0002-1810-6706Nicolina Chrysaphi https://orcid.org/0000-0002-4389-5540Natasha L. S. Jeffrey https://orcid.org/0000-0001-6583-1989A. Gordon Emslie https://orcid.org/0000-0001-8720-0723Vratislav Krupar https://orcid.org/0000-0001-6185-3945Milan Maksimovic https://orcid.org/0000-0001-6172-5062Mykola Gordovskyy https://orcid.org/0000-0003-2291-4922Philippa K. Browning https://orcid.org/0000-0002-7089-5562

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