Determination of Geometrical and Kinematical Properties of Disk Halo CMEs Using the Elliptic Cone Model X. P. Zhao, H. Cremades, J. T. Hoeksema, Y. Liu CISM All-hand Meeting, Boston September 17, 2007

1. Introduction White-light coronagraph images are formed by Thomson scattering of free electrons along the line-of-sight. The morphology of 2-D halo CMEs depends on the 3-D CME magnetic geometry that confines CME plasma, and on the CME propagation direction. The 3-D magnetic geometry of most, if not all, single CMEs is expected to be flux rope-like because the free magnetic energy that generates CMEs is stored in the field-aligned electric currents in the low-beta corona, leading to a force-free field.

The geometry of CME flux ropes with two ends anchored on the solar surface can be approximated by cones with elliptic bases, and different from the circular cone model [Zhao et al., 2002; Xie et al., 2004], most halo CMEs can be reproduced by projecting the cone bases onto the sky-plane [Cremades & Bothmer, 2005; Zhao, 2005]. Such geometrical properties of CME flux ropes as the CME propagation direction, the size and orientation have been inverted using observed halo parameters [Zhao, JGR, 2007]. This work present an algorithm to determine CME propagation speed and acceleration, and to further validate the inversion solution of the elliptic cone model.

Fig1. Definition of five halo parameters, Dse, α, Saxh, Sayh, ψ. The white ellipse is determined using 5 points (‘+’) method (Cremades, 2005). Shown at the top are the values of the five halo parameters (Dse, SAxh & SAyh are in solar radii). 2. Observed halo parameters Xc’ Yc’ SAxh SAyh Yh Zh α Ψ Dse The five halo parameters include the information of the CME magnetic geometry and propagation direction. In Addition, the apparent speed & acceleration on the sky-plane provide information of the kinematic properties of CMEs. How to invert the elliptic model parameters from the halo parameters? How to invert the radial propagation speed and acceleration from the apparent ones? Zh Yh Xc’ Yc’ SAxh SAyh α ψ Dse

3. Six model parameters (Rc, α, β) or (Rc, φ, λ) characterize the location of the cone base center in XhYhZh system, (Rc, ωy, ωz) the size and shape, and χ the orientation of the cone base. Rc Ye Xe, Xc ωyωy Ze Xe, Xc ωzωz Rc Ye Ze Yc Zc χ (α, β)(φ, λ) Fig. 2 Definition of model parameters, Rc, ωy, ωz, χ in XeYeZe and XcYcZc coordinate systems Cone base Cone apex Semi-major axis Semi-minor axis

Xh Yh Zh Xc Xc’ φ λ α β Rc Fig. 3 Definition of Xc direction, i.e., β, α or λ, φ in XhYhZh sinλ = cosβ sinα (1.1) tanφ = cosα / tanβ (1.2) sinβ = cosλ cosφ (1.3) tanα = tanλ / sinφ (1.4) Relationship between β, α and λ, φ

4. Invesion solution of cone parameters Rc = Dse / cos β (2.1) tan ωy = {-(a - c sinβ )+[(a + c sinβ )^2+ 4 sinβ b^2)]^0.5} / 2Rc sinβ (2.2) tanχ = (Rc tan ωy - c) / b (2.3) tan ωz = -(a + b tanχ ) / Rc sinβ (2.4) Rc, ωy, ωz, χ in left side, Dse, SAxh, SAyh, ψ & β in right side. Here a = SAxh cos^2 ψ - SAyh sin^2 ψ (3.1) b = (SAxh + SAyh)sinψ cosψ (3.2) c = -SAxh sin^2 ψ + SAyh cos^2 ψ (3.3) How to determine β using halo parameter α ?

Fig. 4 Find out the optimum β on the curve determined by α on the basis of the location of the CME associated flare (the dot) β Equations (1.4), (1.3) show that there are a series of (λ,φ) pairs, or a series of possible β, corresponding to a specific value of α, as shown by the dotted curve in Fig. 4. The angular shift between the CME propagation direction and the associated flare location may be caused by the interaction between CME and adjacent high-speed streams [Cremades, 2005]. Thus the possible angle β may be occurred on the thick segment of the curve. 4.1 Determining β using α and the location of CME- associated flare

4.2 CMEs are large-scale dynamic phenomenan and believed to originate in large-scale closed field regions [e.g., Zhao & Webb, 2003]. The CME propagation (cone’s central axis) direction is expected to be located near the center of closed regions, though the closed regions contain often more than one single arcade. However, CME-associated flares may be located near the center or edge of closed regions, depending on the location of the arcade where CMEs origin. If it is the case, the optimum β should be determined using the minimum distance between the dot and the curve (see the symbol β in Fig 4) Figure 5 shows the inversion solution obtained using the β value corresponding to minimum distance between the dot and the curve.

Fig. 5 The inverted model parameters are shown at the top. Rc in solar radii. The red dotted ellipse is the projection of the cone base on the sky-plane, which agrees with the white ellipse very well.

5. Determining the kinematical properties The sky-plane speed & acceleration, Vsp, and Asp, are measured at a given position angle, PA. For the _02:54:04 halo, Vsp= km/s and Asp=-61.4 m/s^2 at PA=193 degs. It may be used to get the radial speed and acceleration of CMEs,Vc, Ac for base center and Ve, Ae for base edge at that position angle as follows: Vc=Vsp / √(d^2 + e^2) (4.1) Ac=Asp / √(d^2 + e^2) (4.2) Ve=Vc f and Ae=Ac f (5) Where f = √(1 + tan^2 ωy cos^2 δ + tan^2 ωz sin^2 δ) (6.1)

d=cosβcosα+(sinβsinχcosα+cosχsinα)tanωycosδ- - (sinβcosχcosα-sinχsinα)tanωzsinδ (6.2) e=- cosβsinα-(sinβsinχsinα-cosχcosα)tanωycosδ+ + (sinβcosχsinα+sinχcosα)tanωzsinδ (6.3) δ=PA+α-ψ+χ. (6.4) Using Vc, Ac, & the time difference of a halo relative to the first one, the value of Rc can be calculated, the halo at the time can be reproduced if α value is correctly determined and if CMEs propagate radially & keeping angular widths unchanging. Figures 6, 7, & 8 show the result for the 13 December 2006 event.

Fig.6 Based on Vc=1541 km/s, Ac= m/s^2 (and Ve=2084 km/s, Ae= m/s^2), Rc=6.07 Rs for the _03:06:06 halo. The red ellipse is obtained using calculated Rc and other model parameters. It agrees with observed halo pretty well taking consideration that the elliptic cone model is just a proxy of a CME rope.

Fig. 7 The same as Fig. 6 but for _03:42:04 with Rc = Rs.

Fig. 8 The same as Fig. 6 but for _04:18:04 with Rc=14.69 Rs.

6. Summary & Discussion By combining the halo parameter α and the location of CME-associated flare, the CME propagation direction and the other four model parameters can be inverted. The CME propagation speed and acceleration can also be determined on the basis of the measured apparent (sky- plane) speed, acceleration and the model parameters.

Observations show that limb CMEs propagate often radially & keeping angular widths constant, i.e., CMEs keep a self-similarity movement. The halos observed at later time can be calculated since parameter Rc at the time can be calculated. If such calculated halos agree with observed ones, the inversion solution is valid & acceptable. If it is not the case, other β value on the thick curve segment of Fig. 4 should be tested to get a better agreement. It should be noted that the inversion solution is valid only for disk frontside full halo CMEs, i.e., for β greater than 45 degs [Zhao, 2007].

Whether or not an inversion solution is acceptable depends strongly on the measurements of halo parameters, especially the parameter α. Recognizing the elliptic halo from obseved halo CMEs is not a easy task since the elliptic cone model is only a simplified approximation to flux-rope-like CME structures. Any attempt to further improve the recognition technique would be desired and should be encouraged.