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Empirical model of the Solar Wind protons :from the Helios & Ulysses
observations back to the corona
M. Milan, A. Zaslavsky, O. Alexandrova, L. Matteini & F. Pantellini
LESIA & CNRS, Paris Observatory
Parker Solar Probe & Solar Orbiter joint meeting, APL USA, October 2017
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Helios & Ulysses proton data
Vp Np Tp
0.3 1 2 5Heliocentric distance
0.3 1 2 5Heliocentric distance
0.3 1 2 5Heliocentric distance
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Helios & Ulysses proton data
Vp Np Tp
0.3 1 2 5Heliocentric distance
0.3 1 2 5Heliocentric distance
0.3 1 2 5Heliocentric distance
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Vp
0.3 1 2 5Heliocentric distance
~235 000data points
~45 000data points
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𝑵𝑵𝒑𝒑𝑽𝑽𝑹𝑹𝟐𝟐 = 𝑭𝑭𝟎𝟎
And a density model 𝑵𝑵𝑷𝑷 it is possible to deduce a velocity model
𝑽𝑽 =𝑭𝑭𝟎𝟎 × 𝑹𝑹−𝟐𝟐
𝑵𝑵𝒑𝒑
Assuming
𝑵𝑵𝒑𝒑𝑽𝑽𝑹𝑹𝟐𝟐 = 𝑭𝑭𝟎𝟎~𝟐𝟐𝟐𝟐𝟎𝟎𝟎𝟎 𝐜𝐜𝐜𝐜−𝟑𝟑𝐤𝐤𝐜𝐜/𝐬𝐬
0.3 1 2 5Heliocentric distance
0.3 1 2 5Heliocentric distance
𝑵𝑵𝒑𝒑𝑽𝑽𝑹𝑹𝟐𝟐 = 𝑭𝑭𝟎𝟎~𝟏𝟏𝟏𝟏𝟐𝟐𝟎𝟎 𝐜𝐜𝐜𝐜−𝟑𝟑𝐤𝐤𝐜𝐜/𝐬𝐬
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Semi empirical density models which fit both the coronal observation and solar wind data
𝑵𝑵𝒆𝒆 𝐜𝐜𝐜𝐜−𝟑𝟑 = 𝟑𝟑.𝟐𝟐 𝟏𝟏𝟎𝟎𝟐𝟐𝐞𝐞𝐞𝐞𝐞𝐞 −𝟑𝟑.𝟕𝟕𝒓𝒓𝑺𝑺𝒓𝒓
𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐+ 𝟒𝟒.𝟗𝟗
𝒓𝒓𝑺𝑺𝒓𝒓
𝟑𝟑+ 𝟕𝟕.𝟔𝟔
𝒓𝒓𝑺𝑺𝒓𝒓
𝟒𝟒+ 𝟔𝟔
𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐
𝑵𝑵𝒆𝒆 𝐜𝐜𝐜𝐜−𝟑𝟑 = 𝟏𝟏.𝟑𝟑 𝟏𝟏𝟎𝟎𝟐𝟐𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐+ 𝟏𝟏𝟐𝟐.𝟑𝟑𝟒𝟒
𝒓𝒓𝑺𝑺𝒓𝒓
𝟒𝟒.𝟏𝟏+ 𝟏𝟏𝟑𝟑𝟑𝟑𝟐𝟐.𝟒𝟒
𝒓𝒓𝑺𝑺𝒓𝒓
𝟏𝟏𝟑𝟑.𝟕𝟕
Equatorial Streamers (slow wind); Sittler & Guhatakurta, ApJ 1999; Guhatakurta et al. 1998
Coronal holes (fast wind); Guhatakurta et al. 1999
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𝑵𝑵𝒆𝒆 𝐜𝐜𝐜𝐜−𝟑𝟑 = 𝟑𝟑.𝟐𝟐 𝟏𝟏𝟎𝟎𝟐𝟐𝐞𝐞𝐞𝐞𝐞𝐞 −𝟑𝟑.𝟕𝟕𝒓𝒓𝑺𝑺𝒓𝒓
𝟎𝟎.𝟗𝟗𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐+ 𝟒𝟒.𝟗𝟗
𝒓𝒓𝑺𝑺𝒓𝒓
𝟑𝟑+ 𝟕𝟕.𝟔𝟔
𝒓𝒓𝑺𝑺𝒓𝒓
𝟒𝟒+ 𝟔𝟔
𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐
𝑵𝑵𝒆𝒆 𝐜𝐜𝐜𝐜−𝟑𝟑 = 𝟏𝟏.𝟑𝟑 𝟏𝟏𝟎𝟎𝟐𝟐 (𝟏𝟏.𝟏𝟏,𝟎𝟎.𝟏𝟏)𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐+ 𝟏𝟏𝟐𝟐.𝟑𝟑𝟒𝟒
𝒓𝒓𝑺𝑺𝒓𝒓
𝟒𝟒.𝟏𝟏+ 𝟏𝟏𝟑𝟑𝟑𝟑𝟐𝟐.𝟒𝟒
𝒓𝒓𝑺𝑺𝒓𝒓
𝟏𝟏𝟑𝟑.𝟕𝟕
Correction in order to fit the Helios/Ulysses data
Saito, 1970
Koutchmy, 1977
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Empirical solar wind speed profiles
(3) Grall, 1996 (Interplanetary radio scintillation)(1) Gabriel et al., 2005 (SOHO SUMMER Doppler Diming)
(2) Antonucci et al., A&A 2004 (SOHO/UVCS) (4) Sheeley et al., 1997 (slow wind blobs)
(1)
(2)
(3)
(4)
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𝑽𝑽 𝒓𝒓 = 𝑨𝑨 × 𝒓𝒓 + 𝑽𝑽𝟎𝟎
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𝑻𝑻𝒑𝒑 𝒓𝒓 = 𝑻𝑻𝒑𝒑𝟎𝟎𝒓𝒓𝒓𝒓𝟎𝟎
𝜶𝜶
𝑻𝑻𝒑𝒑 ∝ 𝑽𝑽𝒑𝒑𝟐𝟐
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Internal energy equation for protons in spherical symmetry
Cranmer et al., 2009Hellinger et al. 2013
Heating rate per mass unit
𝝏𝝏𝝏𝝏𝝏𝝏
= 𝟎𝟎 𝐚𝐚𝐚𝐚𝐚𝐚 𝒒𝒒~𝟎𝟎
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If there is an equation of state
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Estimations of the turbulent cascade rates by MacBride et al. 2008 (ACE data) in the slow solar wind (Vr < 400 km/s) 𝜺𝜺𝑻𝑻~ 𝟐𝟐𝟏𝟏𝟎𝟎𝟑𝟑 W/kg in the fast solar wind,
𝜺𝜺𝑻𝑻~ 𝟏𝟏𝟎𝟎𝟒𝟒 W/kg the direction ⊥ to B 𝜺𝜺𝑻𝑻~ 𝟏𝟏𝟎𝟎𝟑𝟑 W/kg the direction ∥ to B
If there is an equation of state
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𝒗𝒗𝒅𝒅𝒗𝒗𝒅𝒅𝒓𝒓
= −𝟏𝟏𝝆𝝆𝒅𝒅𝑷𝑷𝒅𝒅𝒓𝒓
−𝑮𝑮𝑴𝑴⊙
𝒓𝒓𝟐𝟐
𝒗𝒗𝒅𝒅𝒗𝒗𝒅𝒅𝒓𝒓
+𝟏𝟏𝝆𝝆𝒅𝒅𝑷𝑷𝒅𝒅𝒓𝒓
+𝑮𝑮𝑴𝑴⊙
𝒓𝒓𝟐𝟐= Ϝ𝑫𝑫(𝒓𝒓)
Ϝ𝑫𝑫 𝒓𝒓 = 𝟎𝟎Ϝ𝑫𝑫 𝒓𝒓 > 𝟎𝟎Ϝ𝑫𝑫 𝒓𝒓 < 𝟎𝟎
No extra force needed to explain the observed profiles
an extra force needed to explain the observed profiles
extra force in the system, extra energy available
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𝒗𝒗𝒅𝒅𝒗𝒗𝒅𝒅𝒓𝒓
= −𝟏𝟏𝝆𝝆𝒅𝒅𝑷𝑷𝒅𝒅𝒓𝒓
−𝑮𝑮𝑴𝑴⊙
𝒓𝒓𝟐𝟐
𝒗𝒗𝒅𝒅𝒗𝒗𝒅𝒅𝒓𝒓
+𝟏𝟏𝝆𝝆𝒅𝒅𝑷𝑷𝒅𝒅𝒓𝒓
+𝑮𝑮𝑴𝑴⊙
𝒓𝒓𝟐𝟐= Ϝ𝑫𝑫(𝒓𝒓)
Ϝ𝑫𝑫 𝒓𝒓 = 𝟎𝟎Ϝ𝑫𝑫 𝒓𝒓 > 𝟎𝟎Ϝ𝑫𝑫 𝒓𝒓 < 𝟎𝟎
No extra force needed to explain the observed profiles
an extra force needed to explain the observed profiles
extra force in the system, extra energy available
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Going back to the extrapolation towards the Sun
𝑵𝑵𝒆𝒆 𝐜𝐜𝐜𝐜−𝟑𝟑 = 𝑨𝑨𝟎𝟎𝐞𝐞𝐞𝐞𝐞𝐞 −𝟐𝟐.𝟗𝟗𝒓𝒓𝑺𝑺𝒓𝒓
𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐+ 𝑨𝑨𝟑𝟑
𝒓𝒓𝑺𝑺𝒓𝒓
𝟑𝟑+ 𝑨𝑨𝟒𝟒
𝒓𝒓𝑺𝑺𝒓𝒓
𝟒𝟒+ 𝑨𝑨𝟐𝟐
𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐
Diamonds : Saito, 1970Stars : Koutchmy, 1977
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Going back to the extrapolation towards the Sun
𝑵𝑵𝒆𝒆 𝐜𝐜𝐜𝐜−𝟑𝟑 = 𝑨𝑨𝟎𝟎𝐞𝐞𝐞𝐞𝐞𝐞 −𝟐𝟐.𝟗𝟗𝒓𝒓𝑺𝑺𝒓𝒓
𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐+ 𝑨𝑨𝟑𝟑
𝒓𝒓𝑺𝑺𝒓𝒓
𝟑𝟑+ 𝑨𝑨𝟒𝟒
𝒓𝒓𝑺𝑺𝒓𝒓
𝟒𝟒+ 𝑨𝑨𝟐𝟐
𝒓𝒓𝑺𝑺𝒓𝒓
𝟐𝟐
Diamonds : Saito, 1970Stars : Koutchmy, 1977
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Are the protons in coronal holes cooler than in the quiet corona ?
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Conclusions
1. There is a quite good radial continuity between HELIOS & Ulysses for the 𝑵𝑵𝒑𝒑,𝑻𝑻𝒑𝒑𝐚𝐚𝐚𝐚𝐚𝐚 𝑽𝑽𝒑𝒑 measurements
2. The fast wind is accelerated very close to the corona (confirmation) and reaches its terminal speed at a few tens 𝒓𝒓𝑺𝑺
3. The slower winds are still accelerated between 0.3 & 1 AU (and beyond)4. Heating rates with respect to the adiabatic case have been estimated5. The extrapolation back to the corona of the bulk speed assuming a
constant mass flux fits quite well with the SOHO estimates of 𝑽𝑽𝒑𝒑6. And the electrons ??7. Go PSP and SO (hopefully not too late) !!
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Proton outflow speeds & temperature diagnostics in the corona← Lyα and O VI emission lines with SOHO/UVCS, Protons assumed to be in collisional equilibrium with the neutral Hydrogen, Kohl et al., ApJ 1998
Proxy for Tp
Plume & interplumeoutflow speeds by Doppler dimmingobservations on SOHO Summer, Gabriel et al., 2005 →
Similar techniques stillwith SOHO/UVCSAntonucci et al., A&A 2004 →
1 2 3 4Heliocentric distance (Rs)
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Interplanetary radio scintillation, Grall (1996)
Sheeley et al., 1997
Other outflow speed diagnostics
𝑽𝑽 𝐤𝐤𝐜𝐜/𝒔𝒔 = 𝟏𝟏 − 𝐞𝐞𝐞𝐞𝐞𝐞 −
𝒓𝒓𝒓𝒓𝒔𝒔− 𝟐𝟐.𝟏𝟏
𝟏𝟏.𝟏𝟏