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

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Improvement of Inversion Solutions for Type C
Halo CMEs Using the Elliptic Cone Model
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1. Validity of inversion solution for Type
C halo CMEs?

• 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), ? (orientation), and
  Dse, a (location)
• Type A The minor axis is nearly
• parallel to Xc axis, ?0 (top
  left)
• Type B The major axis is nearly
• parallel to Xc axis , ?0 (top
  right)
• Type C The semi-axes have an
• angle with Xc axis, ??0 (other 4)

Fig 1. Three types of halo CMEs.
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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
• ß gt 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 ß gt 70 .

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.
Fig. 2 Comparison of modeled halos with observed
ones.
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2. Relationship between dh db

• 2.1 The inversion equation system in Zhao08 was
  established based on the following equations
• Rc cos ß Dse
  
  (1.1a)
• Rc tan?y sinß sin?SAxo cos?
  sin?dSAyo sin? cos ?d (1.1b)
• -Rc tan?z sinß cos?SAxo cos?
  cos?d-SAyo sin? sin ?d (1.1c)
• Rc tan?y cos?-SAxo sin? sin?dSAyo
  cos? cos ?d (1.1d)
• where Rc, ?y, ?z, ? and ß in left side are model
  params, and Dse, SAxo, SAyo, ? in right
• side are observed halo params ?ddh-db, and dh
  and db are the phase angles of
• elliptic cone bases and CME halos, respectively,
  as shown in the following expressions
• yebRc tan?y cosdb
  (1.2a) yeoSAy cosdh (1.3a)
  
• zeb-Rc tan?z sindb
  (1.2b) xeoSAx sindh (1.3b)
• Here the projection angle ß may be obtained from
  one-point approach, i.e., using
• observed a and the location of associated flares.

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• 2.2 By assuming
• ?d dh-db ?-?
  (2.1)
• the inversion equations are, as shown in Zhao08,
• Rc cosß Dse
  (2.2a)
• Rc tan?y sinßatan?b
  (2.2b)
• -Rc tan?z sinß-b tan?a
  (2.2c)
• Rc tan?-b tan?c
  (2.2d)
• where aSAxo cos²?-SAyosin²?
  (2.3a)
• b(SAxoSAyo)sin?cos?
  (2.3b)
• c-SAxo sin²?SAyo cos²?
  (2.3c)

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2.3 Reexamination of the effect of projection on
?d when ??0 and ßgt70

• 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 gt, , lt Sazb (or ?y gt,,lt
  ?z), respectively, from top to bottom. The small
  dots near symbol SAyb denote the starting phase
  angle of bases, db, increasing counter-clockwise
  from 0 to 360, with an angular distance from Yc
  axis, ? , 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 ?p).
  Open circles located at the semi axes near Yc
  axis are the starting phase angle of CME halos,
  dh, increasing counter-clockwise , with an
  angular distance from Yc axis, ?, measured
  clockwise.

Fig 3a. Xc and Xh View of coronal bases with
?30 and ß70.
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• Fig 3b is the same as Fig 3a except
• ? -30.
• Since ? and ? are measured clockwise, and db and
  dh are counter-clockwise, the Expres. for ?d
  should be
• ?d dh-db-??
  
• differ from Expres (2.1), i.e.,
• ?d ?-?.
• However, when ß70, , ??, ?d0
• regardless ?y gt ?z or ?y lt ?z, and ?gt0 or ?lt0.
  That is why the inversion equation system (2)
  can be used to approximately invert model params
  for disk halo CMEs with big value of ß, and the
  modeled halos match the observed ones very well.
  

Fig 3b. The same as Fig 3a, but ?-30
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• As shown in Fig 3c, ?0when ?0, thus we have
• ?d0 (3.1)
• Rc cosß Dse (3.2a)
• Rc tan?ySAyo (3.2b)
• -Rc tan?z sinßSAxo (3.2c)
• If ?y ?z, the inversion equation system
  becomes for the circular cone model
• Rc cosß Dse (3.3a)
• Rc tan?ySAyo (3.3b)
• -Rc tan?y sinßSAxo (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, ß. 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 ?0. Note the right 3 halo are identical
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2.4 Reexamination of the effect of projection on
?d when ??0 and ß80,70,60,50,40 (1)
Fig 4a. ?y/?z lt 1 and ?30 (left) and ?-30
(right). The separation between the small dot and
open circle increases clockwise (left) and
counter-clockwise (right) as ß decreases.
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2.4 Reexamining the effect of projection on ?d
when ??0 and ß80,70,60,50,40(2)
Fig 4b. ?y/?z gt 1 and ?30 (left) and ?-30
(right). The seperation between the small dot and
open circle increases counter-clockwise (left)
and clockwise (right) as ß decreases, and the
seperation for ?y/?z gt 1 is much less than for
?y/?z lt 1 .
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• The reexamination further confirms that
• ?ddh-db-??
  (4.1)
• the inversion equations become
• Rc cosß Dse
  (4.2a)
• Rc tan?y sinß-atan?-b
  (4.2b)
• -Rc tan?z sinß-b tan?a
  (4.2c)
• Rctan?btan?c
  (4.2d)
• where aSAxo cos²?SAyosin²? (4.3a)
• b(SAxo-SAyo)sin?cos?
  (4.3b)
• cSAxo sin²?SAyocos²? (4.3c)

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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 ß60.
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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 ß60.
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4. Comparison of inverted with observed Types A
B full halo CMEs
Type A
Type 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 ß others are significant
different each other, showing the validity of
solution needs to be further confirmed.
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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 ß lt 70.
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5. Comparison of inverted with observed Type C
full halo CMEs (2)
Type C
Type C
Fig 8. The green modeled halos match the observed
white ones much better than the red and blue
ones.
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5. Summary Discussion

• By reexamining the effect of projection on ?d,
• we find that the correct expression for ?d,
  (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, ß.
• 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.