Validating Onepoint Inversion Solution of the Elliptic Cone Model for FullHalo CMEs

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Validating One-point Inversion Solution of the
Elliptic Cone Model for Full-Halo CMEs

• X. P. Zhao
• Stanford University
• H. Cremades,
• UTN-FRM / CONICET
• SSH31A -0227
• AGU Fall Meeting, December 12, 2007

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• Abstract
• By using the elliptic cone model as a
  geometrical proxy of CME flux ropes, we have
  established an one-point approach to invert
  geometrical properties of front side, disk
  full-halo CMEs, such as the CME propagation
  direction, shape and angular widths. This work
  presents an algorithm for determining the radial
  speed and acceleration of full-halo CMEs on the
  basis of apparent speed and acceleration measured
  on the plane of the sky.
• It is found that by using the one-point
  approach more than one set of model parameters
  can be inverted, and all sets can be used to well
  reproduce observed CME halos. This work attempts
  to find a way to obtain the valid solution among
  all inverted sets of model parameters.

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1. Introduction

• Recent study shows evidence that all single CMEs
  are hollow flux ropes with two ends anchored at
  solar surface Krall, Ap. J., 2007.
• The geometry of the CME flux ropes can be
  approximated by cones with elliptic cone bases.
  Most of halo CMEs can be reproduced by projecting
  the elliptic bases onto the sky-plane Cremades
  Bothmer, 2005 Zhao, 2005.
• We have developed an one-point approach for
  inverting elliptic cone model parameters Zhao,
  2007. This poster studies how to find out the
  optimum inversion solution among many possible
  solutions.

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2. Characteristics of full halo CMEs
Zh
Zh
The center, semi-axes and orientation of 2-D CME
halos may be characterized by halo parameters
(Dse, a) (Saxh, Sayh) ?. These 5 parameters
contain the information of 3-D CME flux ropes. In
addition, the apparent speed acceleration
measured on the sky-plane provide information of
the kinematic properties of CMEs. How to invert
the CME propagation direction and flux rope
parameters from the 5 observed halo
parameters? How to invert the radial propagation
speed and acceleration from the apparent ones?
Yc
Yc
SAyh
SAyh
?
?
Yh
Dse
Dse
Yh
a
a
Xc
SAxh
Xc
SAxh
Fig1. Definition of five halo parameters, Dse, a,
Saxh, Sayh, ?. 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).
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• 3. The elliptic cone model
• 3.1 Model parameters, (Rc, ß, a), (?y, ?z), ?,
  denote, respectively, the position, size and
  shape, and the orientation of the cone base.

Ze
Zc
Xe, Xc
Xe, Xc
(a, ß)
(f, ?)
Semi-minor axis
?
Semi-major axis
Rc
Yc
Rc
Cone base
?y
Ye
Xe,Xc
?z
Cone apex
Suns center
Ye
Ze
Fig. 2 Definition of model parameters, Rc, ?y,
?z, ? in the elliptic base and cone coor. systems
XeYeZe and XcYcZc
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3.2 Parameters (a, ß) or (f, ?) denote the
direction of central axis (propagation) or the
source location of CMEs in the Heliocentric
Ecliptic Coordinate System XhYhZh. Here the model
parameter a is the same as the halo parameter a.
Zh
Xc
Xc
Yh
Rc
a
Relationship between ß, a and ?, f
sin? cosß sina (1.1) tanf cosa / tanß
(1.2) sinß cos? cosf (1.3) tana tan?
/ sinf (1.4)
ß
?
f
Xh
Fig. 3 Definition of propagation direction Xc,
i.e., ß, a or ?, f in the Heliocentric Ecliptic
coordinate system XhYhZh.
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• 3.3. Relationship between model and halo
  parameters
• Rc Dse / cos ß
  (2.1)
• tan ?y -(a - c sinß )(a c sinß )2
• 4 sinß b2)0.5 / 2Rc sinß
  (2.2)
• tan? (Rc tan ?y - c) / b
  (2.3)
• tan ?z -(a b tan? ) / Rc sinß
  (2.4)
• where a SAxh cos2 ? - SAyh sin2 ? (3.1)
• b (SAhx SAyh)sin? cos?
  (3.2)
• c -SAxh sin2 ? SAyh cos2 ?
  (3.3)
• Rc, ?y, ?, ?z can be calculated using Dse, ?,
• SAxh, Sayh if ß can be specified.
• How to determine ß using halo parameter a ?

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4. Determination of ß using one-point ..
Equations (1.1), (1.2) show that a contains
information of (?,f) . The dotted curve in Fig. 4
corresponds to a 56.46. All possible ß are
located on the curve. CME source is believed to
be located near the associated flare or filament
disappearance. The dot fe is the location of
associated flare. The red dot, me, with
shortest distance from fe is one possible ß. The
blue and green dots, le, re, denote two
extreme ß. Which one is the optimum ß ?
Flare
ß
a56.45
Fig. 4 Find out the optimum ß on the basis of the
location of the flare (the black dot) associated
with the 12/13/2006 halo-CME
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Blue Red Green a
56.4 56.4 56.4 ß
83.6 72.3 52.5 ?y
19.8 44.5 63.1 ?z 17.6
42.1 65.3 ? 6.6 6.8
7.4 Rc(025404) 12.28 4.50 2.25 Vc in
km/s 4210 1538 757 Ac in km/s2 -0.146
-0.053 -0.026 Rc(030606) 16.60 6.08
3.30 Rc(034204) 28.85 10.56 5.23
Rc(041804) 40.13 14.68 7.26 ?y, ?z are
too small and Rc is too large for Blue cone to
produce such bright halo CMEs
Fig. 5 The three sets of inverted model
parameters are shown on the right (Rc in solar
radii). The blue, red, and green dotted ellipses
are reproduced using the 3 sets of model
parameters. All three ellipses agree well with
observed white ellipse, though the model
parameters for blue case may not be reasonable.
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• 5. Determination of kinematical properties
• The sky-plane speed acceleration, Vsp, and
  Asp, are measured at a given position angle, PA.
  For the 2006.12.13_025404 halo, Vsp1773.7 km/s
  and Asp-0.0614 km/s2 at PA193. It may be used
  to get the radial speed and acceleration at that
  position angle, Ve, Ae for base edge and Vc, Ac
  for base center as follows
• VcVsp / v(d2 e2) AcAsp / v(d2 e2)
  (4)
• VeVc f AeAc f
  (5)
• Where
• dcosßcosa(sinßsin?cosacos?sina)tan?ycosd -
  
• - (sinßcos?cosa-sin?sina)tan?zsind
  (6.1)

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• e- cosßsina-(sinßsin?sina-cos?cosa)tan?ycosd
• (sinßcos?sinasin?cosa)tan?zsind
  (6.2)
• f v(1tan2 ?ycos2 dtan2 ?zsin2 d) (6.3)
• dPAa-??.
  (6.4)
• Using calculated Vc, Ac, the time difference
  between a later halo and the halo at 025404,
  the value of Rc can be calculated (see Fig. 5).
  The blue, red and green ellipses in Figures 6, 7,
  8 show the reproduced halos at 030606,
  034204, 041804 . All ellipses agree with
  observed halos very well. Therefore, it is not
  possible to find out the optimum solution by
  comparing calculated with observed halo.

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Fig.6 Based on Vc and Ac, and time difference,
we obtain three Rc for the 2006.12.13_030606
halo. The blue, red, and green ellipses are
obtained using these values of Rc and other model
parameters. Although all agree with observed halo
pretty well, the parameter Vc Rc are too
large for blue and too small for green .
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Fig. 7 The same as Fig. 6 but for
2006.12.13_034204.
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Fig. 8 The same as Fig. 6 but for 2006.12.
13_041804 .
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6. Comparison with near-surface features
The three-part structure in cone-like limb CMEs
implies that all low-corona features associated
with CMEs, such as flare, SFD, dimming,
post-eruption arcade, should occur within cones.
The solar disk projection of a cone base that
intersects with photosphere should cover all
associated features. This inference may be used
to determine which ß should be selected among
many possible ß values. The three solid
ellipses in Fig. 9 are calculated using the
three sets of a, ß, ?y, ?z, ? for 025404 halo
shown in Fig. 4 and Rccos ?y. The blue one is
too small to cover bright features, and the red
green ones cover coronal holes that should
occur outside cones. The different mean
angular widths (37, 47 and 72) among data sets
of MK3, SMM and LASCO suggests that some CMEs
originated in and near active regions may expand
in the early stages of their formation and
propagation (Dere et al., 1997 Hudson et al.,
2007). Thus, angular widths in low corona may be
less than in corona. The dashed ellipses in
Fig. 9 are obtained by assuming that the angular
width at solar surface is 0.75 of its coronal
value. Fig. 9 shows that all bright features are
located within only the red dashed ellipse, and
all coronal holes are located outside only the
red dashed ellipse. The red solution may be the
optimum one.
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Fig. 9 EIT_195 image showing near-surface
features, say, flare and coronal holes. The blue,
red and green solid (dashed) ellipses are
calculated using ?y (0.75 ?y) and other cone
parameters. The dashed red ellipse appears to be
the optimum solution.
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7. Summary Discussion

• By combining the halo parameter a and the
  location of associated flare, more than one ß
  value, and more than one set of model parameters,
  can be inverted. Although some of model
  parameters are obviously not reasonable, all
  three sets we tested can still be used to
  reproduce observed CME halo pretty well (Fig. 5).
• By determining the radial CME propagation speed
  and acceleration on the basis of the measured
  apparent (sky-plane) speed and acceleration, the
  observed CME halos at later times can also be
  well reproduced using these sets of model
  parameters (Fig. 6, 7, 8).
• It shows that the capability of reproducing CME
  halos is not a strong argument for the validation
  of the inverted 6 model parameters.

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• The three-part structure occurred in cone-like
  limb CMEs implies that all CME-associated
  low-corona features should be located within an
  ellipse that is the solar disk projection of the
  cone base at solar surface. For the event tested
  here, it can be used to determine the optimum ß
  and other model parameters. More samples are
  needed to be tested. The two-point approach can
  be used to further confirm the result.
• It should be noted that the inversion solution is
  valid only for disk frontside full halo CMEs,
  i.e., for ß greater than 45 Zhao, 2007.
• Reliable determination of halo parameters,
  especially a, is the key for successfully using
  the elliptic cone model.