Radial E olu ion o Non-Maxwellian Elec on Popula ions De i ed om Quasi- he mal
Noise Spec oscopy: Pa ke Sola P obe Obse a ions
Xianming Zheng
1,2
, Mihailo M. Ma ino ić
3
, Vi iane Pie a d
4,5
, K is ophe G. Klein
3
, Mingzhe Liu
6,7
,
Joel B. Ab aham
8
, Yong Liu
1
, Jing ing Luo
1
, Xiaodong Lin
1
, Guoqing Liu
1
, and Jingchun Li
1
1
Shenzhen Key Labo a o y o Nuclea and Radia ion Sa e y, Ins i u e o Ad anced S udy in Nuclea Ene gy & Sa e y, College o Physics and Op oelec onic
Enginee ing, Shenzhen Uni e si y, Shenzhen 518060, Peopleʼs Republic o China; [email p o ec ed]
2
Depa men o Ea h and Space Sciences, Sou he n Uni e si y o Science and Technology, Shenzhen, Peopleʼs Republic o China
3
Luna and Plane a y Labo a o y, Uni e si y o A izona, Tucson, AZ 85721, USA; [email p o ec ed]
4
Royal Belgian Ins i u e o Space Ae onomy (BIRA-IASB), Space Physics, Sola -Te es ial Cen e o Excellence, B ussels, Belgium
5
Cen e o Space Radia ion (CSR), Ea h and Li e Ins i u e—Clima e Sciences (ELI-C), Uni e si é Ca holique de Lou ain, Lou ain–la-Neu e, Belgium
6
LESIA, Obse a oi e de Pa is, Meudon, F ance
7
Space Sciences Labo a o y, Uni e si y o Cali o nia, Be keley, CA 94720-7450, USA
8
Mulla d Space Science Labo a o y Uni e si y College London, Holmbu y S . Ma y, Do king RH5 6NT, UK
Recei ed 2024 July 8; e ised 2024 Sep embe 16; accep ed 2024 Sep embe 17; published 2024 No embe 29
Abs ac
Unde s anding he anspo o ene gy wi hin space plasmas, pa icula ly in he sola wind, emains a complex
challenge. Accu a e measu emen o elec on empe a u es and hei non he mal cha ac e is ics is c ucial o
comp ehending ene gy anspo p ope ies in plasmas. Quasi- he mal-noise (QTN)spec oscopy has eme ged as a
dependable ool o p ecise elec on pa ame e s assessmen as i is less suscep ible o spacec a e ec s han pa icle
de ec o s. In his s udy, we apply a QTN spec oscopy fi ing me hod o analyze da a om he Pa ke Sola P obe
FIELDS adio ins umen ob ained du ing Encoun e s 2 h ough 13, unde unbiased an enna condi ions. We use he
kappa unc ion o cha ac e ize he elec on eloci y dis ibu ion and employ a fi ing echnique o de i e he
changes in each pa ame e ac oss heliocen ic dis ances anging om 12 Rs o76Rs. Specifically, we find ha he
elec on densi y scales as n
e
∝
−2.09±0.04
and he T
e
∝
−0.65±0.02
. The dis ibu ion o he kappa index has h ee
dis inc egions as a unc ion o adial dis ance om he Sun. Fu he mo e, we conduc a s a is ical analysis o sola
wind ene gy flux which we finds ollows a powe -law ela ionship w
o al
∝
−1.92±0.04
.
Unified As onomy Thesau us concep s: Space p obes (1545)
1. In oduc ion
The sola wind, an in ica e and con inuous exodus o highly
ionized plasma om he sola co ona, cons i u es o a mix u e
o p o ons, α-pa icles, aces o hea y ions, and elec ons. The
majo i y o momen um flux wi hin he sola wind can be
a ibu ed o ions, owing o hei la ge mass, which plays an
impo an ole in shaping sola wind dynamics (D. Ve scha en
e al. 2019). Elec ons, being compa a i ely ligh e , assume he
p ima y ole as ca ie s o hea flux due o hei highe he mal
eloci ies (E. Ma sch 2006; G. Le Cha e al. 2009). The e o e,
elec ons a e expec ed o play a key ole in he he mally d i en
expansion o he sola wind. Thus, ha ing an accu a ely
measu ed elec on empe a u e adial p ofile is o p ime in e es
o quan i ying he ene gy anspo in he sola wind and
se es as an impo an ing edien o cons ain he mally d i en
sola wind models (E. N. Pa ke 1958; V. Pie a d & J. Lem-
ai e 1996; M. Maksimo ic e al. 1997a; I. Zouganelis e al.
2004).
To simpli y ma e s, he obse a ion o sola wind elec on
empe a u es can be desc ibed by a powe law as a unc ion o
he dis ance om he Sun. This powe law dependence a ies
be ween iso he mal and adiaba ic models (M. Maksimo ic e al.
2000); desc ibing he elec on empe a u e as T
e
=T
0
α
,αis
obse ed o ange be ween 0 (iso he mal)and −4/3(adiaba ic).
Fo he kine ic empe a u e, αis ound be ween −0.2 and −0.9
(E. Ma sch e al. 1989; W. G. Pilipp e al. 1990), whe eas o he
elec on co e empe a u e, αis ound be ween −0.3 and −1.1
(M. Maksimo ic e al. 1995;J.L.Phillipse al.1995; K. Issau ie
e al. 1998). The conside able a ia ion in he αmeasu emen s is
o be expec ed and a ises om mul iple ac o s (G. Le Cha e al.
2011; M. Liu e al. 2023).
(i)I is di ficul o sepa a e genuine adial a ia ions along
s eam flux ubes om hose ac oss hem.
(ii)T ansien s uc u es such as co onal mass ejec ions, co-
o a ing in e ac ion egions, and in e plane a y shocks
in oduce significan a ia ion.
(iii)The obse a ions ha e been ca ied ou o e di e en
la i udinal and adial anges du ing di e en phases o he
sola ac i i y.
(i )Many di e en da a acquisi ion, educ ion, and fi ing
echniques ha le e age di e en obse able quan i ies
ha e been used.
Quasi- he mal-noise (QTN)spec oscopy, heo e ically
desc ibed mo e han hal a cen u y ago (J. A. Feje &
J. R. Kan 1969), can yield accu a e elec on densi y and
empe a u e measu emen s using obse a ions o he plasma
peak in elec ic field spec a in he sola wind. I has been used
in many space missions, including he ISEE-3, Ulysses, Wind,
and STEREO missions (N. Meye -Ve ne 1979; S. Hoang e al.
1980; P. J. Kellogg 1981; K. Issau ie e al. 1999,2005;
M. Moncuque e al. 2005; I. Zouganelis e al. 2010;
M. M. Ma ino iće al. 2016,2020). Recen in es iga ions
(M. Maksimo ic e al. 2020; M. Moncuque e al. 2020;M.
M. Ma ino iće al. 2022; M. Liu e al. 2023)ha e applied his
The As ophysical Jou nal, 977:39 (9pp), 2024 Decembe 10 h ps://doi.o g/10.3847/1538-4357/ad7d05
© 2024. The Au ho (s). Published by he Ame ican As onomical Socie y.
O iginal con en om his wo k may be used unde he e ms
o he C ea i e Commons A ibu ion 4.0 licence. Any u he
dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)and he i le
o he wo k, jou nal ci a ion and DOI.
1
echnique o Pa ke Sola P obe (PSP)elec ic ol age spec a
acqui ed by he Radio F equency Spec ome e (RFS/FIELDS;
S. D. Bale e al. 2016; M. Pulupa e al. 2017). Typical pa icle
analyze s a e a ec ed by spacec a pho oelec ons and
cha ging e ec s. Since he QTN elec on densi y is deduced
om a peak in he elec ic po en ial- equency spec um, his
measu emen is independen o gain calib a ion and spacec a
cha ging. Due o i s eliabili y and accu acy, he elec on
numbe densi y de i ed om he QTN spec oscopy is called
he “gold s anda d” o o al densi y and is ou inely used o
calib a e o he ins umen s (M. Maksimo ic e al. 1995;
C. Salem e al. 2001; K. Issau ie e al. 2001; P. L. Whi lesey
e al. 2020). The elec on numbe densi y de e mined using he
QTN echnique on PSP has se ed as a benchma k calib a ion
s anda d o scien ific analyses.
In his pape , we p esen ou me hodology o calcula ing
QTN spec oscopy assuming he elec on eloci y dis ibu ions
a e well desc ibed by kappa unc ions, which is elabo a ed in
Sec ion 2. We employ his me hod o analyze a subse o he
da a collec ed by he PSP/FIELDS ins umen , de i ing plasma
densi y and o al elec on empe a u e, as expounded in
Sec ion 3. Ou p ima y aim is o un eil he adial a ia ions
obse ed in he kappa index and sola wind ene gy flux using
PSP measu emen s om Encoun e s 2 h ough 13, co e ing
heliocen ic dis ances anging om 12 Rs o76Rs(whe e Rs
deno es he Sola adius), as discussed in Sec ion 4.In
Sec ion 5, we w ap up wi h a comp ehensi e discussion
encompassing u u e model p ospec s and an assessmen o
po en ial limi a ions o he me hodology.
2. Me hodology
2.1. FIELDS Ins umen Obse a ion
We u ilized he RFS componen o he FIELDS (S. D. Bale
e al. 2016)sui e onboa d PSP o collec elec ic field
fluc ua ion da a wi hin a specified equency ange. The RFS
comp ises bo h low- equency ecei e s and high- equency
ecei e s, co e ing equency anges o 10 kHz–1.7 MHz and
1.3–19.2 MHz (M. Pulupa e al. 2017), espec i ely. Each
ecei e is equipped wi h 64 loga i hmically spaced equency
bins, p o iding an app oxima e esolu ion o 4.5%, and
main aining he s a is ical unce ain y o he powe in each
a e aged spec um below 0.3 dB. Du ing ce ain segmen s o
each Encoun e , a bias cu en was applied o he an ennas o
main ain hei po en ial close o ha o he undis u bed plasma.
As his biasing impac s he elec ic field spec a, all signals
collec ed du ing compensa ion bias in e als a e no included in
his s udy. Consequen ly, ou analysis concen a es exclusi ely
on he in e als when he FIELDS an enna emains unbiased.
This op ion emained iable h oughou he en i e pe iod om
PSP E2 o E13 ha we a e in e es ed in. Unlike p e ious
me hods, ou fi ing ange includes he peak equency in e al,
allowing o he calcula ion o sola wind densi y and he
Kappa index.
To s eamline he calcula ions, we began by de e mining
plasma densi y h ough peak acking and compu ed i as
n
e
∼
2
p
, as desc ibed in D. A. Gu ne (1998)and S. D. Bale
e al. (2019). Subsequen ly, we conduc ed a fi ing p ocess o
de e mine he o al elec on empe a u e, u ilizing signals
anging om 3
p
o abo e 1.6 MHz, while excluding he
esis i ely coupled an enna mechanism, ollowing he me hod
ou lined in he pape (M. Maksimo ic e al. 2020). Las ly, we
ca ied ou a fi ing p ocedu e o he κindex using he
spec um spanning om 0.8
p
o 3
p
. This spec um was de i ed
om 1 minu e median fil e o elimina e signal con amina ion
om a ious high- equency sou ces, including Langmui
wa e bu s s and pollu ion om uniden ified sou ces. Because
hese fluc ua ions a e jus ansien dis u bances and as
fluc ua ions, and due o limi ed a ailabili y o QTN da a, we
a e using manual emo al o elimina e he adio emission
signal.
2.2. Quasi- he mal Noise Spec oscopy
A passi e elec ic field an enna egis e s a ia ions in he
elec ic po en ial gene a ed by he mo emen s o su ounding
elec ons and ions. The syn he ic QTN spec um, deno ed as
V
2
( ), is compu ed h ough he summa ion o indi idual
con ibu ions om elec ons, p o ons, impac (sho )noise,
ins umen noise, and galaxy adia ion as ollows:
()()=G + + + +VVVVVV,1
22
q n
2
pn
2
sn
2
gal
2
noise
2
whe e V
2
q n
,V
2
pn
,V
2
sn
,V
2
gal
,V
2
noise
ep esen he elec on QTN, he
Dopple -shi ed p o on he mal noise, he sho noise, he galac ic
adio backg ound noise, and ins umen noise, espec i ely. The
ins umen noise is es ima ed »-
*
V
V2.3 10 H
z
noise
217 2
(S. D. Bale e al. 2016;M.Maksimo ice al.2020).Γis he
an enna gain o he ecei e , which is exp essed as
()
G
=+
C
CC
2a
as
2
2.He e,C
a
and C
s
a e he dipole an enna
capaci ance and he s ay capaci ance. Fo he equency anges
conside ed, ()
=p
-
C
a
L
ln 1
L
a
0(N. Meye -Ve ne e al. 2017)and
C
B
≈18 pF (M. Moncuque e al. 2020),whe eL=2 m is he
leng h o an enna, a=1.5 ×10
−3
m is he wi e adius, and ò
0
is
he acuum pe mi i i y. Fo an iso opic Maxwellian, he
elec on con ibu ion is (N. Meye -Ve ne & C. Pe che 1989)
() ()()
∣( )∣ ()
ò
ww
p
w
w
=¥
VmBk Fk
kk dk
16 ,
,,2
ep
L
q n
2
2
0022
() () ()
ò
wp
=¥
w
Bk k d ,2.3
k
He e, ( ) ep esen s he elec on Veloci y Dis ibu ion
Func ion (eVDF)and ò
L
(k,ω)deno es he plasma longi udinal
unc ion, whe e kand ωdeno e he wa enumbe and angula
equency, espec i ely. Mo e de ail a e desc ibed in he pape
(M. Ma ino ić2016), such as he B(k)and ω
p
( he angula
equency o plasma oscilla ion). The F(k)An enna Response
Func ion is based on model esul s (M. M. Ma ino iće al.
2022), which we will e e o as “M22”in he ollowing
sec ion. In he analysis o QTN, eVDFs ha e o en been
modeled as a combina ion o wo Maxwellian unc ions
(K. Issau ie e al. 2004,2008). Howe e , his ype o eloci y
dis ibu ion does no adequa ely cap u e he beha io o
supe he mal elec ons (Y. F. Cha eau & N. Meye -Ve -
ne 1991; V. Pie a d & M. Laza 2010). The e o e, in his
pape , we employ a heo e ical app oach o calcula e QTN,
using a kappa unc ion o desc ibe he elec on eloci y
dis ibu ion (G. Le Cha e al. 2009,2010). The iso opic kappa
2
The As ophysical Jou nal, 977:39 (9pp), 2024 Decembe 10 Zheng e al.
eloci y dis ibu ion is defined as ollows:
()
() ()
()
()
⎜⎟
⎛
⎝
⎞
⎠
k
pk k k
G
G
=+
-+
k--
11,4
k
0
31
2
2
0
2
1
3
2
whe e Γdeno es he gamma unc ion,
0
is he he mal speed
ela ed o he kine ic empe a u e
()
()
=k
k
-
kT
m
0
23 e
e
B,k
B
is
he Bol zmann cons an , and m
e
is he elec on mass.
To de e mine he e ec i e elec on QTN spec oscopy, we
ely on model esul s (J. C. No aco & L. W. B own 1978;
A. Zasla sky e al. 2011) o calcula e he con ibu ions o galaxy
adia ion powe . The backg ound adio galac ic noise is modeled
as =G
p
V
ZLB
Rgalaxy
24
302e
2model,whe eZ
0
=120 πis he impe-
dance o acuum, Γ
R
L
e
=1.17 is he educed e ec i e leng h,
and B
model
is he empi ical model o he iso opic sky
backg ound b igh ness, which is =
-
B
B e
model 0 MHz
0.76 .He e
=*
-
B
1.38
0
10 W
mHzs
19
2, =-
3.28 MHz
0.64 a e bes -fi pa ame e s.
Figu e 1depic s s anda d powe spec al densi y RFS
spanning om 100 kHz o 1.7 MHz, measu ed by he FIELDS
V1–V2 an enna dipole. The ed da a poin s signi y obse a-
ions ob ained om PSP, which we e p ocessed using
1 minu e median alues; i.e., each poin in equency is being
a e aged o e a minu e. In e e y in e sion calcula ion, i is
impe a i e o ini ially deduc he influences o galaxy
adia ion and ins umen noise (bo h o hem a e cons an s,
which we ha e no plo ed in he Figu e 1). This necessi a es
acqui ing a p is ine QTN spec um, ensu ing i s composi ion
solely comp ises elec on, p o on, and impac noise. As
you can see he black line is he elec on QTN, he ed and
blue lines ep esen impac /sho noise and p o on noise,
espec i ely. Panels (c)and (d)display he esul s o
compa ing he kappa eVDF wi h he wo-Maxwellian
eVDF esul s. I is impo an o no e ha , wi hin his con ex ,
we ha e cons ained ou kappa index o posi i e in ege s less
han 15, wi h lowe bound o 1.5 due o he defini ion o
he eVDF.
Figu e 1. Example QTN fi ing esul s using di e en eVDF models a e shown, wi h he ed do s ep esen ing he obse a ions made by PSP. Panel (a)displays he
esul s p esen ed in M. Maksimo ic e al. (2020), while (b)showcases ou fi ed esul s. The fi ing p ocedu e is de ailed in Sec ion 2.1. Bo h se s o esul s
demons a e a high deg ee o concu ence. Panels (c)and (d)compa e esul s o a di e en in e al be ween he wo Maxwellian fi and kappa fi .
3
The As ophysical Jou nal, 977:39 (9pp), 2024 Decembe 10 Zheng e al.
3. Deducing he Elec on Densi y and Tempe a u e
3.1. Elec on Densi y
In his sec ion, we u ilize he kappa eloci y dis ibu ion,
Equa ion (4), along wi h he M22 an enna esponse model o
de i e eVDF pa ame e s om QTN spec a acqui ed om PSP
measu emen s du ing encoun e s 2 h ough 13. This calcula ion
fi s hese spec a o he 1 minu e median alues o RFS da a,
which we e sampled using he unbiased V1–V2 dipole. Gi en he
subs an ial numbe o spec a sampled du ing each o hese
in e als, i is wo h no ing ha each panel in Figu e 1co es-
ponds o a median o app oxima ely 10 o 15 downloaded
spec a. Fo each o he spec a, we employ he SL algo i hm
(X. Zheng e al. 2024; combining he s eepes descen and
Le enbe g–Ma qua d me hods) o fi he spec a a hei ull
esolu ion.
The da a was binned in o 14 adial dis ance in e als o equal
wid h in linea space, each sepa a ed by a dis ance o 5 Rs. In
each o hese adial bins, we compu ed he a e age (mean alue)
and s anda d de ia ion o he ele an pa ame e s. The la e we
use as ou es ima e o he unce ain y in each adial bin.
Figu e 2depic s he median alues o n
e
, which we e
de e mined by acking he plasma line a
p
, and we ha e
emo ed any ques ionable measu emen s (especial he Lang-
mui wa e and adio emission as desc ibed in M. Liu 2022).
This algo i hm d aws inspi a ion om he Spec oscopie Ondes
and B ui Elec ons a ique The mique adio ecei e (M. Mon-
cuque e al. 2006)u ilized du ing he BepiColombo mission.
The co e p inciple o his algo i hm e ol es a ound iden i ying
he change in he ampli ude wi hin each aw spec um. The
esul s show ha he elec on densi y a ies as
()=´ -
n
6.2 10
e5 2.09, which is consis en wi h ou p io
wo k whe e we calcula ed ()=´ -
n
5.54 10
e5 2.08. The g ay
squa es ep esen he SPAN-I (R. Li i e al. 2022)measu e-
men s o p o ons compa ed o he QTN esul s. The SPAN-I
ins umen om he SWEAP sui e is designed o measu e he
eloci y dis ibu ion unc ions o sola wind p o ons and
alphas. Densi ies measu ed by SPAN-I a e lowe han he QTN
esul s, which may be due o he cha ge densi y o o he ions,
such as alpha pa icles, in he sola wind, o a limi ed
ins umen field o iew (L. Woodham e al. 2021).
3.2. To al Elec on Tempe a u e
Figu e 3illus a es he o al elec on empe a u e, T
e
, de i ed
om he QTN spec oscopy using a kappa unc ion wi h he
selec ed da a se , as a unc ion o heliocen ic dis ance. The
esul s om M20 (M. Moncuque e al. 2020)deno ed by ed
line, show ()»-
T eV 418
c0.74, while he esul s om L23
(M. Liu e al. 2023)deno ed by blue line, depic
()=
-
T 491.7 61.0
e0.66 0.09. Acco ding o he ela ionship
T
e
≈T
c
(1+a
h
h
;a
h
=n
h
/n
c
,
h
=T
h
/T
c
), he exp ession
a
h
h
=1.17
0.07
is de i ed om p e ious esul s. Howe e ,
in ou fi ing esul s, we assumed a
h
=0.05, and
h
=2
( esul ing in a
h
h
<0.17), ollowing p e ious esul s (Š.Š e-
ák e al. 2009). This assump ion implies ha he M20 esul
will be smalle han he esul s o ou fi ing. This disc epancy
is e iden in he figu e. Consequen ly, based on he di e ences
in T
c
alues be ween ou esul s and M20, i is appa en ha we
a e unde es ima ing he supe hea ed elec on pa ame e s.
Howe e , i emains unce ain whe he his unde es ima ion
is ela ed o he supe he mal elec on densi y o empe a u e.
In he u u e, we plan o ea supe he mal elec ons as an inpu
pa ame e in he fi ed model and conduc u he adjus men s
o accu a ely fi he eal supe he mal elec on pa ame e s.
The compa ison o he esul s demons a ed a ema kable
le el o consis ency wi h findings om p e ious esul
(M. Moncuque e al. 2020; M. Liu e al. 2023). Specifically,
he o al elec on empe a u e, fi ed wi h he powe -law model
T
e
∝
−0.65±0.02
, exhibi ed a modes ly fla e p ofile compa ed o
he co e empe a u e, which ollowed T
c
∝
−0.71±0.03
.Acco d-
ing o SPANe obse a ions (L. Be čiče al. 2020), he e appea s
o be no significan end in he a ia ion o s ahl elec on
empe a u e wi h adial dis ance. Addi ionally, s ahl elec ons
Figu e 2. Radial a ia ion o elec on densi y wi hin he selec ed da a ange. The g ay boxes ep esen p o on densi y da a ob ained om SPAN-I measu emen s
(momen s), while he da k g ay do s ep esen he fi ing esul s de i ed h ough QTN spec oscopy. The elec on densi y (n
e
)is fi ed using a powe -law exp ession:
()=h-
n
n
e0(depic ed by he blue line). No ably, bo h se s o esul s exhibi a consis en end. The ed and black do s (p o ons and elec ons espec i ely)show he
densi y a e ages a a dis ance o 5 Rs o each bin.
4
The As ophysical Jou nal, 977:39 (9pp), 2024 Decembe 10 Zheng e al.
a e mo e p ominen as one app oaches he Sun, while he densi y
a io be ween halo and s ahl elec ons inc eases wi h inc easing
adial dis ance (M. Maksimo ic e al. 2005;Š.Š e ák e al.
2009), implying a sca e ing o some s ahl elec ons in o halo
elec ons. The e o e, he fla e T
e
p ofile compa ed o T
c
may
p ima ily esul om he fla ening o he s ahl elec on
empe a u e p ofile. I is wo h no ing ha p e ious esul s om
he PSP mission (J. B. Ab aham e al. 2022)sugges ha he
physical pic u e di e s somewha om he simple con e sion o
s ahl o halo elec ons seen in Helios obse a ions. PSP findings
indica e ha he o e all sup a he mal elec on ac ion
(halo +s ahl)inc eases wi h heliocen ic dis ance below
0.25 au (whe e au ep esen s an as onomical uni ). Fu he mo e,
he ela i e densi ies o halo and s ahl elec ons a e become
inc easingly small as a unc ion o adial dis ance nea
pe ihelion. Howe e , when in close p oximi y o he Sun, he e
a e ins umen al limi a ions in measu ing bo h halo and s ahl
elec ons using SPAN-E. Thus, he o e all sup a he mal ac ion
(halo+s ahl)close o he Sun may be unde es ima ed, and such
ac o s should be handled wi h g ea e cau ion in such scena ios.
4. Supe he mal Popula ions and Ene gy Flux
4.1. Kappa Index and SPAN-E Da a Fi ing
In Figu e 4, we p esen he kappa index ob ained om ou
da a se as a unc ion o he heliocen ic dis ance. The g ay do s
ep esen he esul s o ou QTN fi ing; ecall ha ha he
kappa index mus be a posi i e in ege . As he kappa index
Figu e 3. The adial a ia ion o elec on empe a u e wi hin he chosen da a se . The le panel illus a es he o al elec on empe a u e de i ed using he kappa
unc ion. The igh panel displays he co e empe a u e, which is fi ed using a wo-Maxwellian dis ibu ion. The bes -fi powe laws o hese empe a u es a e as
ollows: Fo he o al elec on empe a u e, we ha e T
e
=T
e0
−0.65∼−0.67
; Fo he co e empe a u e, he powe law is T
c
=T
c0
−0.71∼−0.74
.
Figu e 4. The adial a ia ion o he kappa index in he selec ed QTN da a is shown in g ay. The black do s ep esen he s a is ical esul s om each 5 Rs. The blue
line shows he esul s o di ec fi ing om SPAN-E da a, compa ed o kappa-fi ing esul s om p e ious s udies ep esen ed by he ed, g een, and pink lines. No e
ha he o ange dashed line ep esen s he u bulen dissipa ion ange index as a unc ion o sola adial dis ance (S. Lo z e al. 2023). Du ing he fi ing p ocess, we
cons ain he kappa index wi hin he ange o [2, 15]. Be ween 30 Rs and 60 Rs, he kappa index expe iences an inc ease, and i s end is essen ially consis en wi h
ha o he u bulen dissipa ion ange index, indica ing a dec ease in supe he mal elec ons wi hin his ange. Con e sely, beyond 60Rs, he kappa index dec eases,
signi ying an inc ease in supe he mal elec ons wi hin ha specific ange.
5
The As ophysical Jou nal, 977:39 (9pp), 2024 Decembe 10 Zheng e al.
app oaches infini y, he dis ibu ion becomes mo e akin o a
Maxwellian dis ibu ion. The ed do s co espond o he fi ing
esul s om J. B. Ab aham e al. (2022), wi h he fi ing ange
spanning om 0.13 o 0.5 au. Addi ionally, he g een and pink
do s deno e he kappa indices o he s ahl and halo
componen s, espec i ely, as epo ed in he pape by Š.Š e-
ák e al. (2009), which was om Helios I and II. The mean
alue encompassed by he s anda d de ia ion o he kappa
index is depic ed by he black do s, while he eVDF fi ing
esul s a e ep esen ed by he blue do s. De ails o his me hod
a e shown in Figu e 5, In he con ex o his s udy, he kappa
index is a c i ical pa ame e o accu a ely modeling he
exosphe e, especially wi hin he amewo k o he kine ic
exosphe ic model o he sola wind. The kappa index, which
alls wi hin he ange o 1.5–15, is pa icula ly significan
because i de e mines he dis ibu ion o pa icle eloci ies,
influencing he beha io o sup a he mal pa icles and he
o e all dynamics o he sola wind. This is especially impo an
o low kappa alues, which a e obse ed in he as sola wind
and di ec ly a ec he wind’s accele a ion o supe sonic speeds.
The impac o he kappa index has been ex ensi ely s udied and
alida ed in he li e a u e, including wo ks (H. Lamy e al.
2003; V. Pie a d & J. Lemai e 1996; V. Pie a d e al. 2023),
unde sco ing i s impo ance in bo h heo e ical models and
obse a ional da a.
Analyzing he esul s o he kappa index s a is ical s udy as
depic ed in Figu e 4, we no e a ela i ely s able a ia ion in he
kappa index wi hin he ange o 12 o 30 sola adii. Howe e ,
as we ex end he dis ance o he ange o 30 o 60 Rs, he kappa
index shows an inc ease wi h dis ance om he Sun, and i s
end is essen ially consis en wi h ha o he u bulen
dissipa ion ange index, he me hodology desc ibed in he
Lo z pape p ima ily in ol es calcula ing and alida ing bo h
he u bulen ine ia egion index and he u bulen dissipa ion
egion index (iden ified as he s eepes powe spec um index).
This is achie ed using high- esolu ion PSP magne ic field
powe spec um da a, which is hen ollowed by s a is ical
analysis. Tha means as u bulence in ensifies, he eVDF
inc easingly esembles a Maxwellian dis ibu ion due o
enhanced pa icle sca e ing and ene gy homogeniza ion
mechanisms. This occu s because s ong u bulence acili a es
he uni o m dis ibu ion o ene gy and iso opy in he elec on
eloci y space, he eby d i ing he eVDFs owa d he mal
equilib ium. Beyond a dis ance o 60 sola adii, he kappa
index exhibi s a con inuous dec ease. I is no ewo hy ha
beyond 200 sola adii, bo h κ_s ahl ≈4 and κ_halo ≈2
emain cons an (Š.Š e ák e al. 2009). This obse ed
ela ionship be ween he kappa index and dis ance sugges s a
co ela ion wi h he sola wind pa icles eloci y dis ibu ion
unc ion.
Fo he s a is ical s udy and compa ing he QTN me hod
esul s o sola wind p ope ies, we ha e ga he ed a la ge
numbe o measu ed eVDFs combining obse a ions om
SPAN-E da a. A de ailed desc ip ion o he ull da a se and he
associa ed ins umen is gi en in he Appendix. The fi ing
p ocedu es ha we ha e used o he analysis o he measu ed
eVDFs a e use he Global Kappa (M. Maksimo ic e al. 1997b;
M. Laza e al. 2017; L. Be čiče al. 2020). Figu e 5is he
example o he global kappa-fi ing p ocedu es me hod.
4.2. Sola Wind Ene gy Flux
The sola wind ene gy flux (w), which includes he kine ic
ene gy (w
kine ic
), he en halpy ene gy (w
en halpy
)and he flux
equi alen o he ene gy equi ed o o e come he sola
g a i a ion (w
g
), and he wa e ene gy flux (w
w
), is exp essed
as:
()=+wnmV
Vnm V V
22
,5
ppp
p
aaa a
kine ic
22
()
⎛
⎝⎞
⎠⎛
⎝⎞
⎠
=++wnV
kT nV kT nV kT5
2
5
2
5
2,
6
ep epp
p
aa a
en halpy BBB
Figu e 5. Example o eVDF fi ing p ocedu e. The blue do s ep esen he SPAN-E obse a ional esul s. The fi ing ange is indica ed by he black squa e do s, while
he ed and pu ple do s depic he esul s o he kappa and Maxwellian fi ings, espec i ely. Tha shows in his case he n
e
=522.02 ±0.2 cm
−3
,
T
c
=29.05 ±0.01 eV o Maxwellian and o kappa eVDF he n
e
=542.02 ±0.5 cm
−3
,T
e
=32.23 e±0.01 V, kappa =5.79 ±0.1. I is impo an o highligh ha
measu emen s become un eliable o channels wi h ene gies below 30 eV, a ibu ed o seconda y con amina ion o he da a by ins umen ( he 30 eV ene gy line is
delinea ed by a e ical ed line in his g aph).
6
The As ophysical Jou nal, 977:39 (9pp), 2024 Decembe 10 Zheng e al.
()
⎜⎟
⎜⎟ ⎜⎟
⎛
⎝⎛
⎝⎞
⎠⎛
⎝⎞
⎠⎞
⎠⎛
⎝⎞
⎠
=+ -
aaa
wnmV
GM
RnmV GM
R
R
1,7
gppp
s
s
s
s
s
∣∣ ()
⎛
⎝⎞
⎠
=+ áñ
WVVnm z
3
24
,8
w
p
App
2
wi h he Elsässe a iables z
±
=δ mδb(W. M. Elsasse 1950),
and he ±chosen o cap u e ou wa d-p opaga ing fluc ua ions
(i.e., he minus sign o adially ou wa d magne ic fields, and
he plus sign o adially inwa d magne ic fields). Fo a mo e
comp ehensi e analysis o wa e ene gy flux calcula ions,
please e e o he pape by J. S. Halekas e al. (2023). The hea
flux (Q)is exp essed as:
() ()
ò
=Qm d
1
2.9
ee
hea , 23
So, he o al sola wind ene gy flux is exp essed as:
()=+ +++ww w w Q w.10
gewkine ic en halpy hea ,
In he abo e exp ession, o all he ene gy fluxes excep o
w
en halpy
, we ha e neglec ed he e ec o he elec on due o i s
mass being negligible compa ed o ha o he p o on. Howe e ,
o he hea flux, we ha e only conside ed he con ibu ion o
elec ons, as p e ious epo s e.g (J. S. Halekas e al. 2021)
indica e ha he elec on hea flux is significan ly la ge han
he p o on hea flux.
Figu e 6displays w
o al
,w
kine ic
,w
en halpy
, and w
g
as unc ions
o heliocen ic dis ance in sola adius uni s (Rs). Ou s a is ical
analysis indica es ha w
o al
≈w
g
>w
kine ic
>w
en halpy
om he
QTN esul s. Howe e , he o al ene gy flux alues dec ease as
he dis ance om he Sun inc eases. Fu he mo e, we obse e
ha (w
g
∝
−2.0
)due o he ela ionship
()
µ-
nmV
ppp 2.0 .We
excluded he alpha pa icles om ou analysis due o
inconsis en a ailabili y o adial ends o he alpha popula-
ions da a. None heless, we calcula ed he indi idual pa ial
ene gies by combining SPAN-E da a wi h a fi ing algo i hm
o he eVDF. The fi ing esul s indica e ha he dis ibu ion o
ene gy in he ene ge ic componen s dec eases om
(
)w
g o he
hea flux (Q). S a is ical esul s show ha he kine ic ene gy
(w
kine ic
)is compa able o he en halpy (w
en halpy
)in a small
dis ance ange, and bo h a e app oxima ely an o de o
magni ude g ea e han he wa e ene gy and 2 o de s o
magni ude g ea e han he hea flux ene gy. In ou analysis o
he QTN fi ing esul s, we ha e omi ed he conside a ion o
wa e ene gy flux. The o al hea flux, deno ed as Q,
comp ises he sum o elec on hea flux, q
e
, and p o on
hea flux, q
p
. Simila findings we e epo ed in he s udies by
Figu e 6. The a ia ion o he sola wind ene gy flux wand i s cons i uen componen s wi h heliocen ic dis ance. The ene gy alues a e ep esen ed by dis inc do s
wi h he squa es ep esen ing median alues su ounded by hei s anda d de ia ions The ed line ep esen s he fi ed p ofile, which aligns wi h he model and
measu emen esul s p oposed in he s udy by M. Liu e al. (2021),
()
()= -
w52.1 1.4
R
1.92 0.007
s,(a)is he ene gy flux as an unc ion o he heliocen ic dis ance,
and he do s ep esen he QTN esul s and he s ep line ep esen he SPAN-E and SPAN-I (wa e ene gy flux)da a fi ing esul s, (b)is he ac ional ene gy flux o
he w
g
,w
en halpy
, and w
kine ic
.
7
The As ophysical Jou nal, 977:39 (9pp), 2024 Decembe 10 Zheng e al.
M. Liu e al. (2021), J. S. Halekas e al. (2020,2021), and
J. B. Ab aham e al. (2022), which indica ed ha q_e ypically
alls wi hin he ange o 10
−4
o 10
−3
Wm
−2
, while q
p
anges
om app oxima ely 10
−7
o 10
−5
Wm
−2
(P. Hellinge e al.
2011). Consequen ly, hese ene gy flux alues a e significan ly
lowe compa ed o o he ene gy flux componen s. Howe e , in
his s udy, we calcula ed wa e ene gy and hea flux (Q)
u ilizing SPAN-E da a. The esul s a e p esen ed in Figu e 6.
To o m a ac ional ene gy flux in Figu e 6(b), we u ilized
he QTN esul s o calcula e he pe cen age o o al ene gy
con ibu ed by each componen . In his calcula ion, we
delibe a ely excluded he hea flux ene gy (Q)and wa e
ene gy flux, ocusing solely on he ene gy equi ed o
o e come he sola g a i a ional po en ial ene gy (w
g
), kine ic
ene gy (w
kine ic
), and en halpy ene gy (w
en halpy
). F om ou
analysis, he esul s indica e ha (w
g
)cons i u es app oxima ely
70% o he o al ene gy (w
o al
), which is sligh ly lowe han
ha epo ed by M. Liu e al. (2021;∼75% o he heliocen ic
dis ance conside ed). In con as , he con ibu ions o bo h
w
kine ic
(20%–25%)and w
en halpy
(5%–10%) o w
o al
in ou
analysis a e sligh ly highe compa ed o p e ious esul s (eg.,
M. Liu e al. 2021). The e a e wo possible easons accoun ing
o his disc epancy. Fi s o all, he e ec s o alpha pa icles
a e neglec ed in his wo k, which may be he main eason
causing he unde es ima ion o w
g
. Second, he esul s he ein
co e a la ge ime ange ( om 2018 o 2023)bu limi ed o he
pe iodical unbiased ime in e al, while M. Liu e al. (2021)
makes use o he da a se s mainly when he an enna a e
nominally biased om 2018 o 2020. Bo h ac o s may help
explain he mino disc epancy, and he e o e hese esul s a e in
a he good ag eemen s conside ing he unce ain y. Fu he -
mo e, hese pe cen ages highligh he undamen al oles o
g a i a ional po en ial ene gy, kine ic ene gy, and en halpy
ene gy in he dynamics and he modynamic beha io o space
plasmas. G a i a ional po en ial ene gy domina es, making up
he majo i y o he o al ene gy, which is c i ical o
unde s anding he la ge-scale beha io and s uc u e o he
plasma. Kine ic ene gy, on he o he hand, is essen ial o
desc ibing he mo ion o pa icles and he anspo o ene gy
wi hin he plasma. Meanwhile, en halpy ene gy p o ides a
comp ehensi e iew o he he modynamic s a e and ene gy
ansi ions, o e ing insigh s in o he he mal ene gy and hea
con en o he plasma.
5. Conclusions
In his pape , we explo e he a ia ions in he kappa index
and sola wind ene gy flux wi hin he inne heliosphe e using
obse a ions om PSP. Ou analysis encompasses helio-
cen ic dis ances anging om 12 Rs o76Rs, inco po a ing
da a collec ed du ing encoun e s E02 o E13. This comp e-
hensi e da a se allows us o in es iga e he beha io o he
elec on densi y, empe a u e, kappa index and sola wind
ene gy flux in he inne heliosphe e, p o iding aluable
insigh s in o he mechanisms behind sola wind hea ing. We
ha e obse ed ha he elec on densi y ollows a powe law,
()µ-
n
e2.09, and he o al empe a u e exhibi s a fla ening
end, ()µ-~-
T
kappa 0.65 0.67, which con as s wi h he co e
empe a u e, ()µ-~-
T
co e 0.71 0.74. Ou s a is ical analysis o
he kappa index o sola wind elec ons e eals in e es ing
pa e ns: wi hin he ange o 12 o 30 sola adii om he Sun,
he kappa index emains ela i ely cons an , app oxima ely
kappa ≈5±1. As we ex end o he ange o 30 o 60 sola
adii, he kappa index inc eases, his phenomenon aligns wi h
changes in he dissipa ion ange index o u bulence in he
egion, sugges ing ha u bulence enhances ene gy ans e
e ficiency h oughimp o edene gymixingandinc eased
in e -pa icle in e ac ions (H. Che & M. L. Golds ein 2014;
K. Ho ai es e al. 2019). Consequen ly, he eVDF mo es
p og essi ely close o he he mal equilib ium Maxwell
dis ibu ion. This p ocess demons a es how a u bulence-
d i en sys em app oaches a he modynamic equilib ium s a e,
and beyond 60 sola adii, i dec eases wi h dis ance.
Fu he mo e, ou s udy o sola wind ene gy flux has shown
ha w
g
>w
en halpy
>w
kine ic
>Q,wi h heQhea flux con ibu -
ing abou 1.0% o he o al ene gy flux and w
g
≈70%w
o al
,
w
kine ic
≈20 ∼25%w
o al
,w
en halpy
≈5∼10%w
o al
.Thew
kine ic
/
w
o al
(w
en halpy
/w
o al
)sligh ly inc eases (dec eases)wi h espec
o he heliocen ic dis ance, while w
g
/w
o al
is almos conse ed
o he heliocen ic dis ance conside ed. This is in ag eemen
wi h he ac ha he p is ine sola wind de ec ed by PSP is s ill
unde accele a ion and ge s coole du ing he expansion. These
findings align wi h p e ious esul s ob ained om long- e m
obse a ions a g ea e dis ances and a ious la i udes (R. Sch-
wenn & E. Ma sch 1990;G.LeCha e al.2009,2012;
D. J. McComas e al. 2014). Specifically, bo h he ac ual
pe cen ages o di e en ene gy flux ypes and hei adial end
a e in a he good ag eemen s wi h he ecen esul s epo ed by
M. Liu e al. (2021), conside ing he di e en ways o deal wi h
alpha pa icle e ec s and he sligh ly di e en ime in e als used
o analysis. We awai mo e da a ha a e o come in he u u e
PSP encoun e s, wi h he eco e y o he well calib a ed alpha
pa ame e s. And i is widely employed in global hemisphe ic
s udies and modeling o deduce densi y om sola wind speed
(and ice e sa), as suppo ed by a ious s udies (D. J. McCo-
mas e al. 2017,2020;F.Shene al.2018; S. M. K imigis e al.
2019;Y.Wange al.2020).
In he u u e, we plan o employ exosphe ic models
alongside he measu ed kappa index o compu e a ia ions in
plasma pa ame e s and sola wind along he adial o he Sun.
Tha will be enhancing he conside a ion o sola wind
accele a ion. Fu he mo e, ou esea ch eam aims o b oaden
he s udy’s scope by in es iga ing sola wind eloci y s a is ics
a a ious la i udes, he eby ad ancing ou comp ehension o
sola wind dynamics.
Acknowledgmen s
This wo k was suppo ed by he Na ional Key R&D
P og am o China unde g an 22022YFE03070004,
S a egic P io i y Resea ch P og am o Chinese Academy o
Sciences (g an XDB 41000000), he Na ional Na u al
Science Founda ion o China (NSFC)g an s 41974168
and 42174203, he Guangdong Pea l Ri e Talen P og am
(2019QN01G838), Pionee p ojec o China Na ional
Nuclea Powe Co po a ion, and Shenzhen Science and
Technology P og am (g an JCYJ20210324104810027).
V.P. acknowledges he p ojec 21GRD02 BIOSPHERE om
he Eu opean Pa ne ship on Me ology, co-financed by he
Eu opean Union’s Ho izon Eu ope Resea ch and Inno a ion
P og amme and by he pa icipa ing s a es. We also hank he
PSPmission o heuseo FIELDSandSPANda ain his
s udy.
8
The As ophysical Jou nal, 977:39 (9pp), 2024 Decembe 10 Zheng e al.
Appendix
Compa a i e Analysis o Measu emen Resul s Using QTN
Fi and eVDF Fi
To acili a e he compa ison o measu emen esul s, we
employed bo h QTN fi and eVDF fi me hodologies (G. Le
Cha e al. 2009; M. Laza e al. 2017; J. B. Ab aham e al.
2022). I is impo an o no e ha he quan i y o da a and he
du a ion o he measu emen s we e no iden ical o bo h
app oaches. Table A1 p esen s he espec i e da a se s o QTN
fi and eVDF fi , allowing o a comp ehensi e examina ion o
he ob ained esul s.
In his da a se , accu a ely calcula ing he ac ual kappa index
alue is no easible when he FIELDS is biased due o
compu a ional cons ain s. The kappa alue is gene ally
influenced by he ela i e magni udes o he peaks in he
powe spec um. Specifically, when a cu en bias is applied,
he powe spec um signal a low equencies becomes
ele a ed, con amina ing he ue peak signal. Consequen ly,
his app oach is no sui able o accu a ely de e mining he
ac ual kappa index when cu en bias is p esen . Fu he de ails
ega ding he a ia ion o QTN unde bias condi ions will be
p esen ed in ou u u e wo k.
ORCID iDs
Xianming Zheng h ps://o cid.o g/0009-0005-7089-9749
Vi iane Pie a d h ps://o cid.o g/0000-0001-5014-7682
K is ophe G. Klein h ps://o cid.o g/0000-0001-6038-1923
Mingzhe Liu h ps://o cid.o g/0000-0003-2981-0544
Joel B. Ab aham h ps://o cid.o g/0000-0002-6305-3252
Re e ences
Ab aham, J. B., Owen, C. J., Ve scha en, D., e al. 2022, ApJ,931, 118
Bale, S. D., Badman, S. T., Bonnell, J. W., e al. 2019, Na u ,576, 237
Bale, S. D., Goe z, K., Ha ey, P. R., e al. 2016, SSR ,204, 49
Be čič, L., La son, D., Whi lesey, P., e al. 2020, ApJ,892, 88
Cha eau, Y. F., & Meye -Ve ne , N. 1991, JGR,96, 5825
Che, H., & Golds ein, M. L. 2014, ApJL,795, L38
Elsasse , W. M. 1950, R MP,22, 1
Feje , J. A., & Kan, J. R. 1969, RaSc,4, 721
Gu ne , D. A. 1998, in Measu emen Techniques in Space Plasmas: Fields.
Geophysical Monog aph, ed. R. F. P a , J. E. Bo o sky, & D. T. Young,
Vol. 103 (Washing on, DC: AGU),121
Halekas, J. S., Bale, S. D., Be homie , M., e al. 2023, ApJ,952, 26
Halekas, J. S., Whi lesey, P. L., La son, D. E., e al. 2021, A&A,650, A15
Halekas, J. S., Whi lesey, P., La son, D. E., e al. 2020, ApJS,246, 22
Hellinge , P., Ma eini, L., Š e ák, Š., T á níček, P. M., & Ma sch, E. 2011,
JGRA,116, A09105
Hoang, S., S einbe g, J. L., Eps ein, G., e al. 1980, JGR,85, 3419
Ho ai es, K., Boldy e , S., & Med ede , M. V. 2019, MNRAS,484, 2474
Issau ie , K., Le Cha , G., Meye -Ve ne , N., e al. 2008, GeoRL,35, L19101
Issau ie , K., Meye -Ve ne , N., Moncuque , M., & Hoang, S. 1998, JGR,
103, 1969
Issau ie , K., Meye -Ve ne , N., Moncuque , M., Hoang, S., & McComas, D. J.
1999, JGR,104, 6691
Issau ie , K., Moncuque , M., & Hoang, S. 2004, SoPh,221, 351
Issau ie , K., Moncuque , M., Meye -Ve ne , N., Hoang, S., & Manning, R.
2001, Ap&SS,277, 309
Issau ie , K., Pe che, C., Hoang, S., e al. 2005, AdSpR,35, 2141
Kellogg, P. J. 1981, PlPh,23, 735
K imigis, S. M., Decke , R. B., Roelo , E. C., e al. 2019, Na As,3, 997
Laza , M., Pie a d, V., Shaaban, S., Fich ne , H., & Poed s, S. 2017, A&A,
602, A44
Lamy, H., Pie a d, V., & Maksimo ic, M. 2003, JGR,108, 1047
Le Cha , G., Issau ie , K., & Meye -Ve ne , N. 2012, SoPh,279, 197
Le Cha , G., Issau ie , K., Meye -Ve ne , N., & Hoang, S. 2011, SoPh,
271, 141
Le Cha , G., Issau ie , K., Meye -Ve ne , N., e al. 2009, PhPl,16, 102903
Le Cha , G., Issau ie , K., Meye -Ve ne , N., e al. 2010, in AIP Con . P oc.
1216, 12 h In . Sola Wind Con . (Mel ille, NY: AIP),316
Liu, M. 2022, PhD hesis, Uni e si é Pa is Sciences e Le es
Liu, M., Issau ie , K., Meye -Ve ne , N., e al. 2021, A&A,650, A14
Liu, M., Issau ie , K., Moncuque , M., e al. 2023, A&A,674, A49
Li i, R., La son, D. E., Kaspe , J. C., e al. 2022, ApJ,938, 138
Lo z, S., Nel, A. E., Wicks, R. T., e al. 2023, ApJ,942, 93
Maksimo ic, M., Bale, S. D., Be cic, L., e al. 2020, ApJS,246, 62
Maksimo ic, M., Ga y, S. P., & Skoug, R. M. 2000, JGR,105, 18337
Maksimo ic, M., Hoang, S., Meye -Ve ne , N., e al. 1995, JGR,100, 19881
Maksimo ic, M., Pie a d, V., & Lemai e, J. F. 1997a, A&A, 324, 725
Maksimo ic, M., Pie a d, V., & Riley, P. 1997b, GeoRL,24, 1151
Maksimo ic, M., Zouganelis, I., Chau ay, J.-Y., e al. 2005, JGRA,110,
A09104
Ma sch, E. 2006, LRSP,3, 1
Ma sch, E., Pilipp, W., Thieme, K., & Rosenbaue , H. 1989, JGRA,94, 6893
Ma ino ić, M. 2016, PhD hesis, Uni . Belg ade
Ma ino ić, M. M., Do de ic, A. R., Klein, K.G., e al. 2022, JGRA,127,
e30182
Ma ino ić, M. M., Klein, K. G., G amze, S. R., e al. 2020, JGRA,125,
e28113
Ma ino ić, M. M., Zasla sky, A., Maksimo ic, M., e al. 2016, JGRA,
121, 129
McComas, D. J., Alleg ini, F., Bzowski, M., e al. 2014, ApJS,213, 20
McComas, D. J., Bzowski, M., Dayeh, M. A., e al. 2020, ApJS,248, 26
McComas, D. J., Zi ns ein, E. J., Bzowski, M., e al. 2017, ApJS,229, 41
Meye -Ve ne , N. 1979, JGR,84, 5373
Meye -Ve ne , N., Issau ie , K., & Moncuque , M. 2017, JGRA,122, 7925
Meye -Ve ne , N., & Pe che, C. 1989, JGR,94, 2405
Moncuque , M., Lecacheux, A., Meye -Ve ne , N., Cecconi, B., &
Ku h, W. S. 2005, GeoRL,32, L20S02
Moncuque , M., Ma sumo o, H., Bouge e , J.-L., e al. 2006, AdSpR,38, 680
Moncuque , M., Meye -Ve ne , N., Issau ie , K., e al. 2020, ApJS,246, 44
No aco, J. C., & B own, L. W. 1978, ApJ,221, 114
Pa ke , E. N. 1958, ApJ,128, 664
Phillips, J. L., Bame, S., Ga y, S., e al. 1995, in P oc. 28 h ESLAB Symp., The
High La i ude Heliosphe e (Do d ech : Kluwe ),109
Pie a d, V., & Laza , M. 2010, SoPh,267, 153
Pie a d, V., & Lemai e, J. 1996, JGR,101, 7923
Pie a d, V., Pé e s de Bonhome, M., Halekas, J., e al. 2023, Plasma, 6, 518
Pilipp, W. G., Miggen iede , H., Mühläuse , K. H., Rosenbaue , H., &
Schwenn, R. 1990, JGRA,95, 6305
Pulupa, M., Bale, S. D., Bonnell, J. W., e al. 2017, JGRA,122, 2836
Salem, C., Bosqued, J. M., La son, D. E., e al. 2001, JGR,106, 21701
Schwenn, R., & Ma sch, E. 1990, Physics o he Inne Heliosphe e I: La ge
Scale Phenomena (Be lin: Sp inge ),99
Shen, F., Yang, Z., Zhang, J., Wei, W., & Feng, X. 2018, ApJ,866, 18
Š e ák, Š., Maksimo ic, M., T á níček, P. M., e al. 2009, JGRA,114,
A05104
Ve scha en, D., Klein, K. G., & Ma uca, B. A. 2019, LRSP,16, 5
Wang, Y., Guo, X., Wang, C., e al. 2020, SpWea,18, e02262
Whi lesey, P. L., La son, D. E., Kaspe , J. C., e al. 2020, ApJS,246, 74
Woodham, L., Ho bu y, T. S., Ma eini, L., e al. 2021, A&A,650, L1
Zasla sky, A., Meye -Ve ne , N., Hoang, S., Maksimo ic, M., & Bale, S. D.
2011, RaSc,46, RS2008
Zheng, X., Liu, K., Ma ino ic, M. M., e al. 2024, ApJ,963, 154
Zouganelis, I., Maksimo ic, M., Meye -Ve ne , N., Lamy, H., & Issau ie , K.
2004, ApJ,606, 542
Zouganelis, I., Maksimo ic, M., Meye -Ve ne , N., e al. 2010, RaSc,45,
RS1005
Table A1
The Kappa Values Calcula ed in his S udy a e P esen ed Below, Ob ained
Using Two Di e en Me hods: QTN and eVDF
Me hod Ins umen Biased/Unbiased Pe iod Samples
QTN FIELDS Unbiased E2 ∼E13 ≈1300
eVDF SPAN-E LE1 ∼E13 ≈400,000
No e. The ollowing able p o ides he s a is ical s a us o he ins umen s used
in hese me hods and hei co esponding s a is ical quan i ies.
9
The As ophysical Jou nal, 977:39 (9pp), 2024 Decembe 10 Zheng e al.
