Improvement of Inversion Solutions for Type C Halo CMEs Using the Elliptic Cone Model

1.1 Three types of halo CMEs The direction from solar disk center to elliptic halo center is defined as Xc’ axis (the green axis in Fig. 1). It is the projection of CME propagation direction onto the sky-plane. Halo CMEs may be characterized by 5 halo params: SAxo, SAyo (shape & size), ψ (orientation), and Dse, α (location) Type A: The minor axis is nearly parallel to Xc’ axis, ψ≈0 (top left) Type B: The major axis is nearly parallel to Xc’ axis, ψ≈0 (top right) Type C: The semi-axes have an angle with Xc’ axis, ψ≠0 (other 4) Fig 1. Three types of halo CMEs. 1. Validity of inversion solution for Type C halo CMEs?

1.2. Inversion solutions? We have established an inversion equation system for halo CMEs in Zhao [2008] (or Zhao08) and obtained inversion solutions for 4 Type C halo CMEs, as shown in Fig 2. Except the event with β > 70° (lower-left panel), all modeled halos (green ellipses) cannot match observed ones (white ellipses). In Zhao08, we concluded that the inversion equation system is valid only for disk halo CMEs of which β > 70°. Fig. 2 Comparison of modeled halos with observed ones. This work try to improve the inversion equation system and to obtain inversion solutions that can be used to reproduce all observed Type C halo CMEs.

2. Relationship between δh & δb 2.1 The inversion equation system in Zhao08 was established based on the following equations: Rc cos β = Dse (1.1a) Rc tanωy sinβ sinχ=SAxo cosψ sin∆δ+SAyo sinψ cos ∆δ (1.1b) -Rc tanωz sinβ cosχ=SAxo cosψ cos∆δ-SAyo sinψ sin ∆δ (1.1c) Rc tanωy cosχ=-SAxo sinψ sin∆δ+SAyo cosψ cos ∆δ (1.1d) where Rc, ωy, ωz, χ and β in left side are model params, and Dse, SAxo, SAyo, ψ in right side are observed halo params; ∆δ=δh-δb, and δh and δb are the phase angles of elliptic cone bases and CME halos, respectively, as shown in the following expressions yeb=Rc tanωy cosδb (1.2a) yeo=SAy cosδh (1.3a) zeb=-Rc tanωz sinδb (1.2b) xeo=SAx sinδh (1.3b) Here the projection angle β may be obtained from one-point approach, i.e., using observed α and the location of associated flares.

2.2 By assuming ∆δ = δh-δb ≈ ψ-χ (2.1) the inversion equations are, as shown in Zhao08, Rc cosβ = Dse (2.2a) [Rc tanωy sinβ+a]tanχ=b (2.2b) -Rc tanωz sinβ-b tanχ=a (2.2c) Rc tanω-b tanχ=c (2.2d) where a=SAxo cos²ψ-SAyosin²ψ (2.3a) b=(SAxo+SAyo)sinψcosψ (2.3b) c’=-SAxo sin²ψ+SAyo cos²ψ (2.3c)

2.3 Reexamination of the effect of projection on ∆δ when χ≠0 and β>70° By given 6 model params, we calculate cone bases (left coloum, the propagation direction view) and the projection of the bases onto the plane of the sky (POS) (right, the Earth view ). The left three panels show the XcYcZc coordinate system and the Xc view of cone bases, corresponding to SAyb >, =,,=,< ωz), respectively, from top to bottom. The small dots near symbol SAyb denote the starting phase angle of bases, δb, increasing counter-clockwise from 0° to 360°, with an angular distance from Yc axis, χ, measured clockwise. The right panels show the Xh view. Small dots here are the projection of small dots in left panels onto the POS, with a slight shift toward Yc axis (see χp). Open circles located at the semi axes near Yc axis are the starting phase angle of CME halos, δh, increasing counter- clockwise, with an angular distance from Yc axis, ψ, measured clockwise. Fig 3a. Xc and Xh View of coronal bases with χ=30° and β=70°.

Fig 3b is the same as Fig 3a except χ = -30°. Since χ and ψ are measured clockwise, and δb and δh are counter- clockwise, the Expres. for ∆δ should be ∆δ = δh-δb=-ψ+χ differ from Expres (2.1), i.e., ∆δ = ψ-χ. However, when β=70°,, ψ≈χ, ∆δ≈0 regardless ωy > ωz or ωy 0 or χ<0. That is why the inversion equation system (2) can be used to approximately invert model params for disk halo CMEs with big value of β, and the modeled halos match the observed ones very well. Fig 3b. The same as Fig 3a, but χ=-30°

As shown in Fig 3c, ψ≈0°when χ≈0°, thus we have ∆δ=0° (3.1) Rc cosβ = Dse (3.2a) Rc tanωy=SAyo (3.2b) -Rc tanωz sinβ=SAxo (3.2c) If ωy = ωz, the inversion equation system becomes for the circular cone model Rc cosβ = Dse (3.3a) Rc tanωy=SAyo (3.3b) -Rc tanωy sinβ=SAxo (3.3c) Note: the halo params for right three CME halos are exactly the same, though the left cone bases are significantly different each other. It implies that the circular cone model is only one of various possibilities, and correct inversion solutions depend on the correct determination of the projection angle, β. Reproduction of observed halo is only a necessary but not sufficient condition for the validity of the solutions Fig 3c. The Xc and Xh views of the cone bases with χ=0. Note: the right 3 halo are identical

2.4 Reexamination of the effect of projection on ∆δ when χ≠0 and β=80°,70°,60°,50°,40° (1) Fig 4a. ωy/ωz < 1 and χ=30° (left) and χ=-30° (right). The separation between the small dot and open circle increases clockwise (left) and counter-clockwise (right) as β decreases.

2.4 Reexamining the effect of projection on ∆δ when χ≠0 and β=80°,70°,60°,50°,40°(2) Fig 4b. ωy/ωz > 1 and χ=30° (left) and χ=-30° (right). The seperation between the small dot and open circle increases counter-clockwise (left) and clockwise (right) as β decreases, and the seperation for ωy/ωz > 1 is much less than for ωy/ωz < 1.

The reexamination further confirms that ∆δ=δh-δb≈-ψ+χ (4.1) the inversion equations become Rc cosβ = Dse (4.2a) [Rc tanωy sinβ-a’]tanχ=-b’ (4.2b) -Rc tanωz sinβ-b’ tanχ=a’ (4.2c) Rctanω+b’tanχ=c’ (4.2d) where a’=SAxo cos²ψ+SAyosin²ψ (4.3a) b’=(SAxo-SAyo)sinψcosψ (4.3b) c’=SAxo sin²ψ+SAyocos²ψ (4.3c)

3. Comparison of inverted with given model parameters (1) Fig 5a. The same as Fig 4a but with three sets of inverted model params from three inversion equation systems, as shown by red, blue and green. The inverted green params match white ones better than others, especially when β≤60°.

3. Comparison of inverted with given model parameters (2) Fig 5b. The same as Fig 4b but with three sets of inverted model params from three inversion equation systems, as shown by red, blue and green. The inverted green params match white ones better than others, especially when β≤60°.

4. Comparison of inverted with observed Types A & B full halo CMEs Type AType B Fig 6. All three kinds of inversion equation systems (red, blue and green) cab be used to reproduce observed Type A & B halo CMEs, but inverted model params β & others are significant different each other, showing the validity of solution needs to be further confirmed.

5. Comparison of inverted with observed Type C full halo CMEs (1) Type C Type c Fig 7. the green modeled halos match the observed white ones better than the red and blue ones, especially when β < 70°.

5. Comparison of inverted with observed Type C full halo CMEs (2) Type C Fig 8. The green modeled halos match the observed white ones much better than the red and blue ones.

5. Summary & Discussion By reexamining the effect of projection on ∆δ, we find that the correct expression for ∆δ, (4.1), and establish the correct inversion equations, (4.2), (4.3). The inversion equations are valid for all three types, especially Type C, halo CMEs in a wide range of the projection angle, β. Note: Reproducing observed CME halos is only a necessary but not sufficient condition for the validity of inversion solutions. Further confirmation is necessary for the validity of the inversion solutions. In addition to the inversion equations, a correct inversion solution depends also on the correct identification of CME halos and correct determination of the projection angle.