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

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

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

Vp

0.3 1 2 5Heliocentric distance

~235 000data points

~45 000data points

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

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

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

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)

𝑽𝑽 𝒓𝒓 = 𝑨𝑨 × 𝒓𝒓 + 𝑽𝑽𝟎𝟎

𝑻𝑻𝒑𝒑 𝒓𝒓 = 𝑻𝑻𝒑𝒑𝟎𝟎𝒓𝒓𝒓𝒓𝟎𝟎

𝜶𝜶

𝑻𝑻𝒑𝒑 ∝ 𝑽𝑽𝒑𝒑𝟐𝟐

Internal energy equation for protons in spherical symmetry

Cranmer et al., 2009Hellinger et al. 2013

Heating rate per mass unit

𝝏𝝏𝝏𝝏𝝏𝝏

= 𝟎𝟎 𝐚𝐚𝐚𝐚𝐚𝐚 𝒒𝒒~𝟎𝟎

If there is an equation of state

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

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

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

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

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

Are the protons in coronal holes cooler than in the quiet corona ?

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) !!

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)

Interplanetary radio scintillation, Grall (1996)

Sheeley et al., 1997

Other outflow speed diagnostics

𝑽𝑽 𝐤𝐤𝐜𝐜/𝒔𝒔 = 𝟏𝟏 − 𝐞𝐞𝐞𝐞𝐞𝐞 −

𝒓𝒓𝒓𝒓𝒔𝒔− 𝟐𝟐.𝟏𝟏

𝟏𝟏.𝟏𝟏