1. Flux Rope from Eruption Data (FRED) and its Interplanetary Counterpart
Nat Gopalswamy
NASA Goddard Space Flight Center
S. Yashiro, S. Akiyama, H. Xie
The Catholic University of America
ISEST/MiniMax24 Jeju Sep 18-22, 2017

2. Motivation
All CMEs reaching 1-AU seem to have flux rope structure (irrespective of observed MC or non- MC structure of ICMEs)
All ICMEs are associated with post-eruption arcades at the Sun; similar in MC and non-MC sources (Yashiro et al. 2013)
near-Sun and 1- AU flux rope fits (Marubashi et al. 2015) to MC and non-MC events
Similar heavy-ion charge states requiring flare plasma injection into flux ropes (Gopalswamy et al. 2013)
On the other hand CME arrival models use a pressure pulse or ad hoc magnetic structures as input
Needed: a realistic flux rope input – Flux Rope from Eruption Data (FRED)
Mouschovias and Poland, 1978
Both geometrical and magnetic properties of a coronal flux rope can be obtained from eruption data
Post-eruption arcade, LOS magnetogram, white-light CME data under the assumption of cylindrical force-free flux rope

3. Simultaneous Onset of Flare & CME
van Ballegooijen &
Martens, 1989
Gosling 1990
Longcope et al. 2007
Qiu et al. 2007
Two magnetic structures during eruption:
Post eruption arcade
Flux rope (CME)
Reconnection at * * * * *
Origin of Mass and EM emissions
Gopalswamy 2009
Gopalswamy 2009

4. Flux Rope Fit to White-light CMEs
Krall & St Cyr (2006) single view (SOHO/LASCO)
Thernisien (2011) Multiview (STEREO/SECCHI)
Geometrical properties: flux rope radius, leading-edge height, aspect ratio
Main parameter we use: R0/Rtip
Related to the kappa parameter in the GCS model (Thernsien 2011)
Λ = 2R1/d = R1/R0 R0
Rtip
Krall & St Cyr (2006); Krall (2007)

5. Magnetic Properties P r
Coronal flux rope can be constructed using Φr and CME flux rope fit
CME and flare properties closely related to Φr
Coronal flux ropes can be tracked and compared with 1-au flux ropes
If CME flux ropes are formed during eruption, then the total reconnected (RC) flux (Φr) is same as the poloidal flux (Φp) of the resulting FR: Φr ≈ Φp (Longcope et al. 2007; Qiu et al ; Hu et al. 2014; Gopalswamy et al )
Φr can be computed using the photospheric B underlying cumulative ribbon area (Longcope et al. 2007; Qiu et al. 2007; Kazachenko et al. 2017)
Φr can also be computed in a simpler way as half the flux underlying post eruption arcades (Gopalswamy et al. 2017)
ribbons r P
Longcope et al. 2007

6. Reconnected Flux fro PEA
 Photospheric flux in the area under the post eruption arcade in the flare decay phase gives 2ΦrA
GOES C4.1 flare
There is good agreement between RC fluxes obtained from Ribbon (ΦrR) and Arcade (ΦrA) methods:
ΦrA = 1.24(ΦrR)0.99
ΦP = 1.29(Φr)0.85
Cumulative flare-ribbon area (A) is projected on photospheric magnetogram to get the reconnected (RC) flux ΦrR =BA, where B is the average magnetic field (Qiu et al. 2007)
The RC flux (Φr) is also significantly correlated with the poloidal flux (ΦP) of the associated magnetic clouds
Gopalswamy et al. 2017, Solar Phys. 292, 65

7. Soft X-ray Flare Size and Fluence
Kazachenko et al. 2017
Kazachenko et al. 2017
The RC flux is well correlated with flare size and flare fluence
The flare fluence has a better correlation
True for both MCs and Ejecta
Gopalswamy et al JASTP

8. CME Speed and Kinetic Energy
Both speed and kinetic energy of the CMEs are well correlated with the RC Flux (r = 0.60, 0.69)
The higher the RC Flux, the faster are the CMEs
KE has the second highest correlation with RC flux (the highest was with soft X-ray flare fluence)
True when the CMEs ended up as MCs or ejecta
(a)
(a)
One event (2000 October 2) was excluded from the correlations because the arcade was ill-defined.
Diamonds – no mass data, so 1016 g assumed
Gopalswamy et al JASTP

9. Force-free Flux Ropes
Φp = (L/x01)B0R0 L-FR length; R-radius; Ba-axial field strength
Φt = (B0R02/x01) 2πJ1(x01) - 1st order Bessel function; J1 (x01) =0.5192
Bp = HB0J1(αr) with H = ±1 (helicity sign)
Bt = B0J0(αr); J0 – zeroth order Bessel, x01- its zero; α = x01/R0
Hr/L = 0.7B02R03
Φt /Φp = (R/L) 2πJ1(x01) = 1.63R/Rtip
Φp from Φr
L, R0 from FR fit to white-light data
H from PEA skew or hemisphere rule
Get flux rope B0 in the corona or any other heliospheric location

10. Example: 2012 July 12 Helicity sign: positive (RH) tilt angle -47 deg
Area =  7.2×1019  cm2, <B> = 392 G; RC Flux = 1.42×1022 Mx
Gopalswamy et al IAU 335

11. Flux Rope Fit to white-light CME
Direction: S15W01
tilt = -53 deg
Aspect ratio = 0.31
R0/Rtip = 0.26
At Rtip = 10 Rs, R0 = 2.6 Rs
At Rtip = 1 AU R0 = 0.26 AU
B0 = Φrx01/R0L = 0.13 G axial field strength (L = 2 Rtip =20 Rs)
Self-similar expansion 
B01au = B0 (10/214)2 = 28.3 nT
Hess & Jang 2014

12. Consistency Check V = 298φr0.75 (Gopalswamy et al. 2017 JASTP)
V = 1548 km/s
φr = 9.0×1021 Mx vs. observed 1.42×1022 Mx
Hr/L = 0.7B02R03
Coronal FR: Hr = 9.98 ×1043 Mx2

13. 1- AU Flux Rope
Bt ~ 30 nT, not too different from the estimate from coronal value
MC size at 1 AU =0.26 AU
(self similar expansion)
MC duration = 0.26 AU/500 km/s = 22 h
(average MC speed 500 km/s)
Similar to GS size (Hu et al. 2016)
Hess & Zhang 2014

14. Coronal Flux Rope B0 vs. 1-AU Bt
Cycle 23 Eruptions (CDAW)
Krall & St Cyr flux rope
MCs and non-MCs considered
Correlation significant with
p = 0.005
Btot = 21.9 logφr−453.2 nT
For φr = 1.42×1022 Mx
Btot = 31.9 nT
- another consistency check
Gopalswamy et al IAU 335

15. Summary
Flux ropes can be completely defined near the Sun using eruption data (photospheric B, PEA, and ME in white-light or EUV), RC flux being the fundamental quantity
RC flux is highly correlated with Flare fluence and CME kinetic energy
Possibility of flux rope formation during eruption: the flux of pre-existing flux rope if any is very small
Axial field magnitude and direction determine asymptotic flux rope properties