Sola Wind Densi y and Co e Tempe a u e De i ed om he PSP Quasi- he mal Noise
Measu emen s
Xianming Zheng
1
, Kaijun Liu
1
, Mihailo M. Ma ino ić
2,3
, Vi iane Pie a d
4,5
, Mingzhe Liu
3,6
, Qingbao He
1
,
Kun Cheng
1
, Yuqi Liu
1
, and Yan Wang
1
1
Depa men o Ea h and Space Sciences, Sou he n Uni e si y o Science and Technology, Shenzhen, Peopleʼs Republic o China; [email p o ec ed]
2
Luna and Plane a y Labo a o y, Uni e si y o A izona, Tucson, AZ, USA; [email p o ec ed]
3
LESIA, Obse a oi e de Pa is, Meudon, F ance
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), Geo ges Lemaî e Cen e o Ea h and Clima e Resea ch (TECLIM), 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
Space Sciences Labo a o y, Uni e si y o Cali o nia, Be keley, CA 94720-7450, USA
Recei ed 2023 Decembe 5; e ised 2024 Janua y 25; accep ed 2024 Janua y 26; published 2024 Ma ch 8
Abs ac
Quasi- he mal noise (QTN)spec oscopy is a aluable me hod o deduce impo an pa ame e s in space plasma,
such as plasma densi y and empe a u e, especially when di ec pa icle measu emen s a e no a ailable. The
p esen s udy de elops a new fi ing me hod o fi he QTN spec a obse ed by he Pa ke Sola P obe (PSP)wi h a
comp ehensi e heo e ical QTN spec al model. By combining he s eepes descen and Le enbe g–Ma qua d
algo i hms, he new me hod is mo e flexible wi h ini ial guess alues bu s ill yields eliable sola wind elec on
densi y and empe a u e alues. The new me hod is applied o de i e he sola wind densi y and co e empe a u e
om he QTN measu emen s du ing 10 encoun e s o PSP. The elec on densi y and empe a u e alues ob ained
a y wi h he adial dis ance om he Sun as n
e
∝
−2.12
and T
e
∝
−0.71
, bo h o which a e consis en wi h exis ing
models and p e ious esul s.
Unified As onomy Thesau us concep s: Sola wind (1534);Space p obes (1545);Space plasmas (1544);Plasma
physics (2089)
1. In oduc ion
Quasi- he mal noise (QTN)spec oscopy p o ides a alu-
able me hod o ob ain plasma densi y and empe a u e
in o ma ion, especially when di ec measu emen s a e no
a ailable. Ea ly heo e ical s udies ha e sugges ed ha a QTN
spec um measu ed by elec ic field an ennas on boa d a
spacec a in a space plasma depends on he eloci y
dis ibu ions o he plasma pa icles (Feje & Kan 1969).
Subsequen ly, plasma pa ame e s, such as densi y and emp-
e a u e, can be deduced by fi ing he QTN spec a eco ded
wi h a heo e ical QTN model (Meye -Ve ne e al. 2017). The
echnique was fi s used in he ISEE-3 mission (Knoll e al.
1978; Meye -Ve ne 1979)and has also been ou inely used o
in e in si u elec on densi ies and empe a u es in a ious sola
wind missions (e.g., Ulysses, Maksimo ic e al. 1995; Issau ie
e al. 1996,1999; Ma ino iće al. 2017; Wind, Maksimo ic
e al. 1998; Issau ie e al. 2005; Ma ino iće al. 2020,2022;
STEREO, Zouganelis e al. 2010; Ma ino iće al. 2016)and
plane a y missions (such as Cassini, Moncuque e al. 2005).
In heo y, a QTN spec um is de e mined by bo h he
an enna geome y and he pa icle (mos ly elec on) eloci y
dis ibu ions (Cou u ie e al. 1981; Meye -Ve ne &
Pe che 1989). The influence o he o me is cha ac e ized by
he an enna esponse unc ion (ARF), which a ies ac oss
di e en missions. Abou he pa icle eloci y dis ibu ions,
heo e ical QTN spec a ha e been de i ed o plasmas wi h
one-Maxwellian, wo-Maxwellian, kappa, and fla - op elec on
dis ibu ions. In he sola wind plasma, he wo-Maxwellian
dis ibu ion composed o he co e and halo elec ons is o en
assumed. In he case o a wi e dipole an enna wi h L
an
>L
D
,
whe e L
an
is he an enna leng h o each wi e and L
D
is he
plasma Debye leng h, he QTN spec um de i ed o he wo-
Maxwellian elec on dis ibu ion ollows a s anda d pa e n
ac oss h ee dis inc equency ( ) egions: a spec al pla eau a
equencies below he elec on plasma equency (
p
), a spec al
peak jus abo e
p
, and a powe -law spec al a ia ion o
−3
a
>
p
.
The p esen s udy analyzes he QTN spec a eco ded by he
Pa ke Sola P obe (PSP) o de i e he sola wind elec on densi y
and empe a u e. PSP is o bi ing he Sun on highly ellip ical
ajec o ies (Fox e al. 2016). I s pe ihelion g adually dec eased
om 35.7 R
s
(R
s
is he sola adius) o 9.86 R
s
(es ima ed o 2024
Decembe ), a e mul iple g a i y assis s om Venus. The
FIELDS ins umen on boa d PSP was designed o measu e DC
and fluc ua ion magne ic and elec ic fields (Bale e al. 2016).I
con ains wo wi e dipoles wi h L
an
=2 m and a 1.98 m gap
be ween he woa mso eachdipole.Thela gegapmakesi
challenging o fi he obse ed QTN spec a wi h heo e ical QTN
models, as mos o he models we e de i ed o he case o dipoles
wi h a negligible gap. Ne e heless, ea ly s udies ha e shown ha
he sola wind densi y could be in e ed by co ela ing he peak
equency in a QTN spec um wi h
p
(Gu ne 1998;Balee al.
2019). In addi ion, as he disc epancies due o he an enna gap
p ima ily appea in he icini y o he spec al peak a ound
p
in a
QTN spec um, some s udies es ima ed he elec on co e
empe a u e T
c
(Moncuque e al. 2020)and o al empe a u e T
e
(Maksimo ic e al. 2020;Liue al.2021)by sepa a ely fi ing he
QTN spec a below o abo e
p
. Recen ly, Ma ino iće al. (2022;
he ea e M22)calcula ed he ARF o PSP using he Analysis o
The As ophysical Jou nal, 963:154 (6pp), 2024 Ma ch 10 h ps://doi.o g/10.3847/1538-4357/ad236d
© 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
Wi e An ennas and Sca e e s (AWAS)so wa e and achie ed
be e ag eemen be ween he obse ed and he model QTN
spec a a ound he spec al peak. The elec on densi y (n
e
)and
co e empe a u e we e hen ound ia s anda d Le enbe g–
Ma qua d (LM)leas squa e fi o he obse ed QTN spec a
(du ing Encoun e 7)using he new model. While he esul s
o M22 a e consis en wi h he PSP pa icle measu emen and he
p e ious s udies, he LM algo i hm adop ed o in e ing plasma
pa ame e s su e s om i s sensi i i y o ini ial pa ame e guesses.
Following M22, he p esen s udy uses he new ARF o
calcula e he heo e ical QTN spec um. The QTN measu e-
men s du ing 10 encoun e s o PSP (Encoun e s 3 o 13, excep
Encoun e 11)a e hen fi ed o he heo e ical QTN spec um
in he equency ange a ound
p
o de i e he sola wind
elec on densi y and empe a u e. A new fi ing me hod has
been de eloped by combining he s eepes descen and
Le enbe g–Ma qua d algo i hms. The new me hod, e e ed
o as he “s eepes descen and Le enbe g—Ma qua d”
(SL)me hod in he es o he pape , is mo e flexible wi h
ini ial guess alues bu s ill yields eliable sola wind elec on
densi y and empe a u e.
The es o he pape is o ganized as ollows. Sec ion 2
in oduces he PSP QTN measu emen s and he heo e ical
QTN model used. Sec ion 3desc ibes he new SL fi ing
app oach. Sec ion 4p esen s he elec on densi y and co e
empe a u e calcula ed and s a is ically examines hei a ia-
ions wi h he heliocen ic dis ance. Finally, Sec ion 5
summa izes he s udy and discusses he limi a ions and possible
u u e wo k.
2. PSP QTN Measu emen s and Theo e ical Model
The Radio F equency Spec ome e (RFS)o FIELDS on
boa d PSP collec s he elec ic field fluc ua ions up o
19.2 MHz (Pulupa e al. 2017). The QTN measu emen s
analyzed in he p esen s udy we e made by he Low F equency
Recei e (LFR)o RFS. I co e s he equency ange o
10.54 kHz o 1.69 MHz wi h 64 loga i hmically spaced
equencies, p o iding ∼4.5% spec al esolu ion. As
in M22, he p esen s udy ocuses on he QTN spec a
measu ed when he FIELDS an ennas we e no biased because
he bias cu en p oduces an inc eased impac noise signal jus
below
p
. The QTN measu emen s du ing 10 encoun e s
(Encoun e 3 o Encoun e 13 excep Encoun e 11, as
Encoun e 11 does no con ain unbiased in e als)ha e been
analyzed, bu only he V1–V2 channel da a a e conside ed o
simplici y. In addi ion, he s udy uses he 1 minu e median da a
in o de o emo e sho - e m signal pollu ion.
The PSP QTN measu emen s a e fi ed o he heo e ical
QTN model o M22 assuming ha he sola wind elec ons
consis o wo iso opic Maxwellians: a he mal co e and a
sup a he mal halo. As desc ibed in Sec ion 2.2 o M22, he
syn he ic QTN spec um gene ally con ains con ibu ions om
he elec ons, p o ons, impac (sho )noise, galaxy adia ion,
and ins umen noise. Howe e , a ound
p
, he syn he ic QTN
spec um is domina ed by he con ibu ion om he elec ons,
i.e., he elec on QTN. Co espondingly, he p esen s udy
di ec ly fi s he obse ed QTN spec a in he ange o 0.9–5
p
o
he heo e ical elec on QTN. No e ha he ARF in ol ed in he
calcula ion o he elec on QTN has been accu a ely de e mined
using he AWAS so wa e o PSP in M22. In addi ion, he
an enna gain ac o Γ=0.33 has been adop ed as in Moncuque
e al. (2020)and Liu e al. (2023).
3. S eepes Descen and Le enbe g–Ma qua d Me hod
In QTN spec oscopy, ele an plasma pa ame e s a e
de e mined by minimizing he di e ence be ween obse ed
and model QTN spec a. The di e ence is o en quan ified as
he chi-squa e: i
NOE
O
2
1
ii
i
2
c=å=
-()
, whe e O
i
and E
i
a e he
alues o he obse ed and model QTN spec a, and Nis he
numbe o da a poin s. As a widely used op imiza ion
algo i hm, he LM me hod was o en employed in QTN
spec oscopy o find he plasma pa ame e s ha minimize χ
2
.
The pa ame e upda e ule in he LM algo i hm is
xxJJIg,1
kk
k
Tkk
11
l=- +
+-
() ()
whe e he subsc ip s kand k+1 ep esen he numbe s o i e a ion
s eps, x ep esen s he pa ame e ec o , Jis he Jacobian ma ix
( he supe sc ip “T”deno es i s anspose),λis he LM pa ame e ,
Iis an iden i y ma ix, and g
k
is he g adien o χ
2
(wi h espec o
he pa ame e ec o ). In compa ison, he upda e ules in he
simple s eepes descen (SD)and Gauss–New on me hods a e
x
k+1
=x
k
−αg
k
and xxJJg
kk
k
Tkk
11
=-
+-
()
, espec i ely. He e
αis a posi i e s ep size ac o . I is clea om Equa ion (1) ha he
LM me hod alls be ween he SD me hod and he Gauss–New on
me hod. I app oaches he SD me hod when λis la ge bu e e s
o he Gauss–New on me hod o small alues o λ(As e e al.
2018).
The LM me hod was also used in M22 o ob ain he sola
wind elec on pa ame e s by fi ing he PSP obse ed QTN
spec a wi h he imp o ed heo e ical QTN model. As an
example, he do s in Figu e 1display he PSP QTN spec um
(1 minu e median alues)obse ed be ween 17:50 and 17:51
UT on 2021 Janua y 15. The obse ed spec um in he
equency ange o 0.9–5
p
(highligh ed in ed)is fi ed o he
heo e ical elec on QTN model o M22 using he LM me hod,
and he fi ed model QTN spec um is shown as he dashed
cu e. Following M22, he ini ial pa ame e guesses a e:
n
e
=584.03 cm
−3
,T
c
=30.29 eV, he halo- o-co e elec on
densi y a io n
h
/n
c
=0.01, and he halo- o-co e elec on
Figu e 1. PSP QTN spec um (1 minu e median alues)obse ed by he V1–
V2 channel o LFR be ween 17:50 and 17:51 UT on 2021 Janua y 15 and fi ed
heo e ical elec on QTN using he LM me hod. The do ed and dashed cu es
display he obse ed and heo e ical QTN spec a, espec i ely. The ed egion
ma ks he equency ange o 0.9–5
p
, whe e he obse ed spec um is fi ed o
he heo e ical elec on QTN model.
2
The As ophysical Jou nal, 963:154 (6pp), 2024 Ma ch 10 Zheng e al.
empe a u e a io T
h
/T
c
=2.10. In addi ion, he LM pa ame e
λis se o 0.1 in he beginning bu is mul iplied o di ided by 5
depending on whe he he con e gence a io kk1
22
cc
+
(
) alls
below 0.25 o exceeds 1.1. A e 100 i e a ions, he p ocedu e
e en ually yields n
e
=517.0 cm
−3
,T
c
=27.0 eV, n
h
/n
c
=0.02,
and T
h
/T
c
=2.44. These alues a e in good ag eemen wi h he
esul s shown in he op-le panel o Figu e 2 in M22, despi e
ha in M22, he obse ed QTN spec um in a wide equency
ange was fi ed o a syn he ic QTN spec um including
con ibu ions om he elec ons, p o ons, impac noise, galaxy
adia ion, and ins umen noise.
The LM me hod desc ibed abo e is sensi i e o he ini ial
pa ame e guesses used. Fo he example QTN spec um shown
in Figu e 1, he black, blue, and ed do s in Figu es 2(a)and (b)
display, espec i ely, how n
e
and T
c
a y wi h he numbe o
i e a ion s eps when di e en ini ial alues a e used in he LM
me hod. The h ee se s o ini ial pa ame e alues es ed a e
(n
e
,T
c
,n
h
/n
c
,T
h
/T
c
)=(600.0 cm
−3
, 40.0 eV, 0.05, 2.00),
(600.0 cm
−3
, 40.0 eV, 0.02, 2.44), and (500.0 cm
−3
, 20.0 eV,
0.05, 2.00), as lis ed in he legend o Figu e 2(b). The
calcula ed n
e
alues a e 100 i e a ions a e 448.3, 460.6, and
498.3 cm
−3
, espec i ely. Fo compa ison, he ho izon al
dashed line in Figu e 2(a)ma ks he “ ue” alue o
n
e
=517.0 cm
−3
which is ob ained wi h he ca e ully selec ed
ini ial pa ame e alues as discussed abo e. Clea ly, he n
e
alues calcula ed in he LM me hod o di e en ini ial
pa ame e alues a e qui e di e en . Simila ly, Figu e 2(b)
demons a es ha he T
c
alues ob ained also a y wi h he
ini ial pa ame e alues. In compa ison wi h he “ ue” alue o
T
c
=27.0 eV deno ed by he ho izon al dashed line, he
calcula ed T
c
alues a e 100 i e a ions a e 29.7, 30.5, and
26.8 eV o he h ee es se s o ini ial pa ame e s, espec i ely.
On he o he hand, i should be men ioned ha he alues o
n
h
/n
c
and T
h
/T
c
ha e been fixed du ing he i e a ions shown in
Figu e 2 o simplici y. This is because ou es calcula ions
(no shown)ha e shown ha he heo e ical elec on QTN
spec um is no sensi i e o he alues o n
h
/n
c
and T
h
/T
c
. This
is indeed confi med by he esul s indica ed by he black and
Figu e 2. Va ia ions in he ob ained (a)elec on densi ies and (b)co e empe a u es wi h he numbe o i e a ion s eps when di e en ini ial guess alues and fi ing
me hods a e used (as labeled in he legend). The ho izon al line in each panel ma ks he “ ue” alue o he co esponding pa ame e which is gi en by he LM me hod
using he ca e ully selec ed ini ial guess alues (see ex o de ails). Fo he combined SL me hod, he esul s gi en by he ea ly SD i e a ions a e shown as emp y
ci cles o be dis inguished om he alues yielded by he la e LM i e a ions.
3
The As ophysical Jou nal, 963:154 (6pp), 2024 Ma ch 10 Zheng e al.
blue do s being close o each o he in Figu es 2(a)and (b). The
wo co esponding se s o ini ial pa ame e s ha e he same n
e
and T
c
alues bu di e en alues o n
h
/n
c
and T
h
/T
c
.
To o e come he limi a ion o he LM me hod being
sensi i e o ini ial guess alues, he p esen s udy combines
he SD and LM algo i hms o cons uc a new fi ing app oach
(named he SL me hod), which is mo e flexible o ini ial
pa ame e guesses. As he SD algo i hm o en p o ides a slow
bu ce ain con e gence o a local minimum, he new fi ing
app oach fi s employs he SD algo i hm wi h a gene ic ini ial
pa ame e se (n
e
,T
c
,n
h
/n
c
,T
h
/T
c
)=(600.0 cm
−3
, 40.0 eV,
0.05, 2.00). The s ep size ac o αin he SD i e a ions is se o
0.01 a e lo s o es s (wi h α a ying om 0.001 o 1). The SD
i e a ions would be s opped when he con e gence a io
consecu i ely exceeds 1 wice. Subsequen ly, he LM me hod,
as desc ibed abo e, akes o e wi h he pa ame e alues
yielded by he SD i e a ions as ini ial alues. The LM i e a ions
a e finally e mina ed when he o al i e a ion numbe
(including he p eceding SD i e a ions) eaches 100. The g ay
emp y ci cles and solid do s in Figu e 2display he esul s om
he combined SL me hod using he gene ic ini ial pa ame e
combina ion. The a ia ions o n
e
and T
c
wi h he i e a ion
numbe a e shown in Figu es 2(a)and (b), espec i ely. The e
a e 19 SD i e a ions and 81 LM i e a ions. The esul s gi en by
he ea ly SD i e a ions a e shown as emp y ci cles o be
dis inguished om he alues yielded by he subsequen LM
i e a ions (solid do s). The p ocedu e e en ually p oduces
n
e
=515.1 cm
−3
and T
c
=26.9 eV. Clea ly, he combined SL
app oach yields mo e accu a e n
e
and T
c
han he LM me hod
(see he black do s in Figu e 2), wi h he gene ic ini ial
pa ame e alues. In addi ion, he esul s om he SL me hod
using he ini ial pa ame e se o (n
e
,T
c
,n
h
/n
c
,
T
h
/T
c
)=(500.0 cm
−3
, 20.0 eV, 0.05, 2.00)a e shown as he
magen a emp y ci cles and solid do s in Figu e 2. As he g ay
ones, he magen a emp y ci cles and solid do s deno e he
alues yielded by he ea ly SD and la e LM i e a ions,
espec i ely. Compa ed wi h he LM me hod esul s ( he black
and ed do s in Figu e 2), he esul s om he new SL app oach
wi h he di e en se s o ini ial pa ame e s a e close o each
o he , indica ing ha he SL app oach is less sensi i e o he
ini ial alues used.
4. Da a Analysis
The new SL me hod desc ibed in Sec ion 3is applied o
analyze he PSP QTN spec a wi h he gene ic ini ial pa ame e
se (n
e
,T
c
,n
h
/n
c
,T
h
/T
c
)=(600.0 cm
−3
, 40.0 eV, 0.05, 2.00).
As a o emen ioned, he heo e ical elec on QTN spec um is
no sensi i e o he alues o n
h
/n
c
and T
h
/T
c
, so hei alues
ha e been fixed du ing he i e a ions o sa e compu a ion ime.
The calcula ed n
e
and T
c
du ing he an enna-unbiased in e als
o Encoun e 7 a e shown as he ed do s in Figu es 3(a)and
(b), espec i ely. In compa ison, he n
e
and T
c
alues gi en by
he LM me hod in M22 a e displayed as black diamonds. These
alues we e ob ained using specific ini ial pa ame e alues as
lis ed in Table 1. In addi ion, he blue do s u he p esen he
Figu e 3. Compa ison o (a) he elec on densi y and (b)co e empe a u e calcula ed using he new SL me hod ( ed do s)and he LM me hod in M22 (black diamonds)
o PSP Encoun e 7. The p o on densi y ob ained om SPAN-I measu emen s and he elec on co e empe a u e om SPAN-E da a a e shown as blue do s in (a)and
(b), espec i ely. (c)The adial dis ance o PSP om he Sun.
Table 1
Ini ial Pa ame e Values Used in he LM Me hod o M22
Time n
e
T
c
n
h
/n
c
T
h
/T
c
2021-01-15 05:50 ∼51 402.63 30.72 0.02 1.90
2021-01-15 17:50 ∼51 584.03 30.29 0.01 2.10
2021-01-16 05:50 ∼51 896.47 40.37 0.005 6.00
2021-01-16 17:50 ∼51 897.57 42.43 0.02 1.90
2021-01-17 05:50 ∼51 596.69 44.46 0.02 2.00
2021-01-17 17:50 ∼51 1337.55 50.04 0.005 2.00
2021-01-18 05:50 ∼51 697.91 41.85 0.02 4.00
2021-01-18 17:50 ∼51 1699.54 37.17 0.02 1.90
2021-01-19 05:50 ∼51 1066.82 48.36 0.02 1.50
2021-01-19 17:50 ∼51 881.81 49.47 0.04 1.90
2021-01-20 05:50 ∼51 983.89 41.03 0.02 2.10
2021-01-20 17:50 ∼51 673.13 35.04 0.25 2.40
2021-01-21 05:50 ∼51 424.75 37.16 0.25 1.40
2021-01-21 17:50 ∼51 456.04 34.63 0.025 1.40
2021-01-22 05:50 ∼51 358.18 29.19 0.04 2.10
2021-01-22 17:50 ∼51 387.21 26.34 0.01 2.10
4
The As ophysical Jou nal, 963:154 (6pp), 2024 Ma ch 10 Zheng e al.
p o on densi y om he Sola P obe Analyze o Ions (SPAN-
I; Kaspe e al. 2016)in Figu e 3(a)and he co e elec on
empe a u e om he Sola P obe Analyze o Elec ons
(SPAN-E; Whi lesey e al. 2020)in Figu e 3(b). Besides he
gene al consis ency be ween he M22 esul s and he measu e-
men s o SPAN-I and SPAN-E (which has al eady been
demons a ed in Figu e 3 o M22), Figu es 3(a)and (b)he e
show nice ag eemen be ween he esul s calcula ed using he
new SL me hod and hose in M22. This demons a es ha he
new SL me hod is ai ly eliable, e en wi h he gene ic ini ial
pa ame e alues. Finally, Figu e 3(c)p esen s he adial
dis ance o PSP om he Sun. The pe ihelion is a ∼20 R
s
du ing Encoun e 7. Figu es 3(a)–(c) oge he sugges ha bo h
he elec on densi y and co e empe a u e end o inc ease as he
adial dis ance dec eases, consis en wi h he findings in
Maksimo ic e al. (2020).
To u he examine how n
e
and T
c
a y wi h he adial dis ance
om he Sun ( ), hen
e
and T
c
alues calcula ed using he SL
me hod (wi h he gene ic ini ial pa ame e alues)du ing he
an enna-unbiased in e als o 10 encoun e s (Encoun e s 3 o 13,
excep Encoun e 11)a e p esen ed in Figu es 4(a)and (b)as a
unc ion o . As men ioned in Sec ion 2, he p esen s udy deals
wi h he 1 minu e median da a. The e a e 744 da a poin s in o al
du ing he 10 encoun e s analyzed. They encompass a adial
dis ance ange be ween 12 and 80 R
s
. S a is ically, bo h n
e
and T
c
dec ease wi h inc easing . The powe -law fi ing o he calcula ed
elec on pa ame e alues yields n
e
=n
0
−2.12±0.03
wi h
n
0
=(6.20 ±0.01)×10
5
cm
−3
and T
c
=T
0
(−0.71±0.03)
wi h
T
0
=(3.2 ±0.2)×10
2
eV. They a e shown as he blue cu es in
Figu es 4(a)and (b), espec i ely. In addi ion, he ed do s in
Figu e 4(a)a e om he heo e ical sola -wind model o Whang &
Chang (1965). They a e in good ag eemen wi h he blue cu e.
On he o he hand, he ed do s in Figu e 4(b) ep esen he
empi ical model esul s o C anme e al. (2009)basedon he
Helios and Ulysses plasma ins umen da a be ween 0.29 and
5.4 au. They also ollow he blue cu e closely. No e ha
Moncuque e al. (2020)de i ed om he PSP QTN spec al
measu emen s du ing Encoun e s 1 and 2 ha T
e
∝
−0.74±0.03
.
Also, he s a is ical s udies o Maksimo ic e al. (2020),
Maksimo ic e al. (2000)sugges ed ha T
e
∝
−0.6∼−0.8
.Allo
hese esul s a e consis en wi h he powe -law fi o T
c
in
Figu e 4(b). In con as , Š e ák e al. (2015)concluded om
Helios da a be ween 0.3 and 1 au ha T
e
∝
−0.59
o slow sola
wind and T
e
∝
−0.31
o as sola wind, espec i ely. The
disc epancies could be a ibu ed o se e al ac o s, such as he
di e en sola condi ions when measu emen s we e made, he
di e en adial dis ance anges o he measu emen s, and he
impac o nonlocal hea ing and hea ans e on he elec on
empe a u e, e en in he p esence o he same sola -wind s eam.
5. Conclusions
The p esen s udy combines he SD and LM algo i hms o
cons uc an imp o ed me hod o fi he PSP QTN measu e-
men s o calcula e he sola wind elec on densi y and co e
empe a u e. Compa ed wi h he adi ional LM me hod, he
new SL me hod is mo e flexible wi h ini ial guess alues and
yields eliable sola wind elec on densi y and empe a u e wi h
a gene ic se o ini ial pa ame e alues. The fi ing is ocused
on he obse ed QTN spec a in he equency ange o 0.9–5
p
,
whe e he elec on QTN domina es he QTN con ibu ions om
o he sou ces. Subsequen ly, he obse ed QTN spec a a e
di ec ly fi ed o he heo e ical elec on QTN wi h he ARF
gi en in M22. In addi ion, o simplici y, he fi ing has been
pe o med wi h he alues o n
h
/n
c
and T
h
/T
c
fixed du ing
i e a ions, because ou calcula ions demons a e ha he esul s
a e no sensi i e o hese wo pa ame e s.
The new SL fi ing app oach has been applied o analyze he
PSP QTN measu emen s du ing he an enna-unbiased in e als
o 10 encoun e s (Encoun e s 3 o 13, excep Encoun e 11).
The analysis gi es he sola wind elec on densi y and co e
empe a u e be ween 12 and 80 R
s
o heliocen ic dis ance. The
esul s indica e ha n
e
∝
−2.12
and T
e
∝
−0.71
, in ag eemen
wi h exis ing models and p e ious esul s. On he o he hand,
he heo e ical elec on QTN model adop ed in he p esen
s udy assumes ha he sola wind elec ons consis o a he mal
co e and a sup a he mal halo ollowing iso opic Maxwellian
dis ibu ions. This is ce ainly no a ealis ic ep esen a ion o
he sola wind elec ons. How o be e model he sola wind
elec on QTN spec um and, mo e impo an ly, expand he
QTN analysis o he PSP an enna-biased in e als emains an
impo an ask o u u e esea ch.
Acknowledgmen s
This wo k was suppo ed by he S a egic P io i y Resea ch
P og am o he Chinese Academy o Sciences (g an XDB
41000000), he Na ional Na u al Science Founda ion o China
(NSFC)g an 42174203, he Guangdong Pea l Ri e Talen
P og am (2019QN01G838), 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, cofinanced by he
Figu e 4. (a)Elec on densi y and (b)co e empe a u e as a unc ion o he
adial dis ance om he Sun. In each panel, he black squa es a e he alues o
he co esponding elec on pa ame e calcula ed using he new SL me hod om
he PSP QTN spec a o 10 encoun e s, he blue cu e ep esen s hei powe -
law fi , and he ed do s a e om a heo e ical o empi ical sola wind model as
desc ibed in he ex .
5
The As ophysical Jou nal, 963:154 (6pp), 2024 Ma ch 10 Zheng e al.
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
PSP mission o he use o FIELDS and SPAN da a in his
s udy.
ORCID iDs
Xianming Zheng h ps://o cid.o g/0009-0005-7089-9749
Kaijun Liu h ps://o cid.o g/0000-0001-5882-1328
Mihailo M. Ma ino ićh ps://o cid.o g/0000-0002-
7365-0472
Vi iane Pie a d h ps://o cid.o g/0000-0001-5014-7682
Mingzhe Liu h ps://o cid.o g/0000-0003-2981-0544
Qingbao He h ps://o cid.o g/0000-0003-2605-7083
Kun Cheng h ps://o cid.o g/0000-0003-1019-2598
Yan Wang h ps://o cid.o g/0000-0001-6783-7591
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