5 Appendix: Coordinate Transformation

In this Appendix we describe the coordinate transformation that allows us to solve the 180°-ambiguity in the azimuth φ of the magnetic field vector (see Section 1.3.2). To that end let us define three different reference frames: {ex, ey,ez}, {eα,e β,eρ}, and ∗ ∗ ∗ {el,ex,ey }. The first frame is a Cartesian frame centered at the Sun’s center. The second frame is a curvilinear frame located at the point of observation P, where eρ is the unit vector that is perpendicular to the plane tangential to the solar surface at this point. This plane contains the unit vectors e α and e β. This is the so-called local reference frame. The third reference frame is the observer’s reference frame. It is centered at the point of observation P, with the unit vector e∗l referring to the line-of-sight. As mentioned in Section 1.3 the inversion of the radiative transfer equation (1View Equation) yields the magnetic field vector in the observer’s reference frame:

B = B cosγe∗ + B sinγ cosφe ∗ + B sinγ sin φe ∗= l x y B cosγ 0 0 e∗l e∗l 0 B sin γ cos φ 0 e∗x = ˆℬ e∗x . (27 ) 0 0 B sin γ sin φ e∗ e∗ y y
The key point to solve the 180°-ambiguity is to find the coordinates of the magnetic field vector into the local reference frame, where we will apply the condition that the magnetic field vector in a sunspot must spread radially outwards. In order to do so, we will obtain the coordinates of {e ,e ,e } α β ρ and ∗ ∗ ∗ {e l,e x,ey} into the reference frame at the Sun’s center: {ex,ey, ez}.

  1. Let us focus first on {ex,ey, ez}: the unit vectors defining a coordinate system at the Sun’s center. The unit vector ez connects the Earth with the Sun’s center, while ey corresponds to the proyection of the Sun’s rotation axis on the plane perpendicular to ez (see Figure 42View Image). In this coordinate system, the vector connecting the observer’s and the Sun’s center is OE (E means Earth):
    OE = Aez, (28)
    where A = 1.496 × 1011 m is the Astronomical Unit. Let us now suppose that we observe a point P on the solar surface. The coordinates of this point are usually given in the reference frame of the observer with the values Xc and Yc (usually in arcsec). These two values refer to the angular distances of the point P measured from the center of the solar disk as seen from the Earth (see Figure 42View Image). From this figure we can extract two triangles: OEP1 and OEP2 (see Figure 42View Image) that can be employed to obtain the values of α and β given Xc and Yc. From the latter triangle we obtain:
    sin α = A--tan Y , (29) R⊙ c

    where R ⊙ = 6.955 × 108 m is the Sun’s radius. Once α has been obtained, we can employ the cosine theorem in the triangle OEP1 to determine β as:

    − b ± √b2-−-4ac- cos β = ----------------, (30) 2a R2⊙-cos2α- a = sin2Xc , (31) b = − 2R ⊙A cosα, (32) 2 R2⊙-cos2α-- c = A − sin2 Xc . (33)
    Note that the obtained values for α and β will have to be modified depending upon the signs of Xc and Yc (which determine the quadrant on the solar disk). Equation (30View Equation) shows that there are two possible values for β, however, one of them always corresponds to an angle |β| > π∕2 and, therefore, can be neglected. It is also important to bear in mind that in Figure 42View Image the points labeled as P1 and P2 are the projections of the observed point on the solar surface, P, onto the planes y = 0 and z = 0 respectively. In fact, once that β and α are known, the coordinates of P in the reference frame of {ex,ey,ez} (the vector OP) can be written as:
    OP = R ⊙ cosα sin βex + R ⊙ sin αey + R⊙ cosα cos βez. (34)
    Another vector that will be useful later is the unit vector from the Earth to the observation point P. This can be written as follows:
    PE = − R cosα sin βe − R sin αe + [A − R cosα cos β]e . (35) ⊙ x ⊙ y ⊙ z
    View Image

    Figure 42: Sketch showing the geometry of the problem and the different reference frames employed in Section 1.3.2. The reference frame {ex,ey, ez} is centered at the Sun’s center ‘O’. The observed point at the Sun’s surface is denoted by ’P’. The observer is located at the point ‘E’, denoting the Earth. The vector OP is parallel to eρ, while EP is parallel to e∗ l.
  2. The next step is to realize that OP (Equation (34View Equation)) is parallel to the radial vector in the solar surface and, therefore, it is perpendicular to the tangential plane on the solar surface at the point of observation. This means that the vector eρ (see Figure 42View Image) belonging to the local frame can be obtained as:
    OP e ρ = -----= cosα sin βex + sin αey + cosα cos βez |OP | = eρxex + eρyey + eρzez. (36)
    With this data we can now define the local reference frame as {eα,eβ,eρ}. This coordinate system is defined on the plane that is tangential to the solar surface at the point P, with eρ being perpendicular to this plane, and eα and e β being contained in this plane. eα and eβ are the tangential vectors to the β =cons and α =cons curves, respectively. The relation between the local reference frame and the one located at the Sun’s center can be easily derived from Figure 42View Image:
    eα − sinα sinβ cos α − sin α cosβ ex ex eβ = cosβ 0 − sin β ey = ℳˆ ey . (37) eρ cosα sinβ sin α cosα cosβ ez ez
  3. The third reference frame we have mentioned is the observer’s reference frame: {e∗,e∗,e∗ } l x y. Note that the first of the unit vectors can be directly obtained (see Figure 42View Image and Equation (35View Equation)) as:
    ∗ -PE-- ∗ ∗ ∗ el = |P E| = elxex + elyey + elzez. (38)

    The heliocentric angle Θ is defined as the angle between the normal vector to the solar surface at the point of observation P (aka eρ) and the line-of-sight (aka ∗ el). Therefore, the scalar product between these two already known vectors (Equations (35View Equation) and (36View Equation)) yields the heliocentric angle cosΘ = eρ ⋅ e∗l:

    cos Θ = A-cosα-cosβ-−-R-⊙-. (39) |P E|

    The other two vectors of this coordinate system, i.e., e∗x and e∗y must be perpendicular to the line-of-sight. The first one, e∗ x, can be obtained from Equations (35View Equation), (36View Equation), and (38View Equation) as:

    ∗ eρ × e∗l 1 ∗ ∗ ∗ ∗ ∗ ∗ ex = -------∗-= -------∗- [eρyelz − eρzely]ex + [eρzelx − eρxelz]ey + [eρxely − eρyelx]ex |eρ × el| |e∗ρ × el|∗ ∗ = exxex + exyey + exzez. (40)
    Finally, the vector ∗ e y can be obtained employing the following conditions:
    8> e∗⋅ e∗ = 0 < x∗ y∗ >: el ⋅ e∗ y = 0 |ey| = 1 e∗ = e∗ ex + e∗ey + e∗ ez. (41) y yx yy yz
    This represents a system of three equations with three unknowns: e∗yx,e∗yy,e∗yz. These represent the proyections of the e∗y unit vector on the base located at the Sun’s center: {e ,e ,e } x y z. Solving the previous system of equations yields the following solutions:
    2 3 1 e∗ e∗ − e∗ e∗ !2 e∗ e∗ − e∗ e∗ !2 − 2 e∗yz = 4 -x∗z-ly∗----x∗y-l∗z + -x∗z-lx∗----x∗x-lz∗- + 15 , (42) exyelx − exxely exyelx − exxely
    ! ∗ ∗ e∗xze∗ly − e∗xye∗lz eyx = eyz -∗--∗----∗--∗- , (43) exyelx − exxely
    e∗ e∗ − e∗ e∗ ! e∗yy = − e∗yz -x∗z-lx∗----x∗x-lz∗- . (44) exyelx − exxely
    All the three components of e∗ y – Equations (42View Equation), (43View Equation), and (44View Equation) – can be obtained through Equations (38View Equation) and (40View Equation). We are, therefore, able to construct the matrix ˆ𝒩, which transforms the observer’s reference frame into the reference frame at the Sun’s center:
    ∗ ∗ ∗ ∗ el elx ely e lz ex ex e∗x = e∗xx e∗xy e∗xz ey = 𝒩ˆ ey . (45) e∗y e∗yx e∗yy e∗yz ez ez
    An important point to consider is that the local reference frame, {e α,eβ,eρ}, and the observer’s reference frame, {e∗l,e∗x, e∗y}, vary with the observed point P on the solar surface. This means that each of these coordinates systems must be recalculated for each point of an observed 2-dimensional map.
  4. We can now express the magnetic field vector in Equation (27View Equation), in the local reference frame:
    e∗l ex eα B = ℬˆ e∗x = ℬˆ𝒩ˆ ey = ℬˆˆ𝒩 ℳˆ −1 eβ = B αeα + B βeβ + Bρeρ. (46) e∗ e e |---{zy-} |---{z-z-} |-----{z-ρ--} Eq.(27) Eq.(45) Eq.(37)
    Note that because of the 180°-ambiguity in the determination of φ, the two possible solutions in the magnetic field vector also exist (in an intricate manner) in the local reference frame {e α,eβ,eρ}: B (φ) and B (φ + ϕ ) (Equation (46View Equation)). In order to distinguish which one of the two is the correct one, we will consider that the magnetic field in a sunspot is mostly radial from the center of the sunspot. This means that we will take the solution that minimizes Equation (9View Equation). To evaluate that equation we already know the coordinates of the magnetic field vector B in the local reference frame. However, we must still find the coordinates, in this same reference frame, of the vector r which, as already mentioned in Section 1.3.2, is the vector that connects the center of the umbra (denoted by U) with the point P of observation:
    r = OP − OU = R⊙ [(cosαp sinβp − cos αusin βu)ex + (sin αp − sin αu)ey + (cos αpcos βp − cosαu cosβu )ez] = rxex + ryey + rzez, (47)
    where we have distinguished between (αp,βp) and (αu,βu) to differentiate the coordinates of the point of observation P and the umbral center U, respectively. For convenience, we re-write Equation (47View Equation) as follows:
    rz 0 0 ex ex ˆ r = 0 ry 0 ey = ℛ ey . (48) |--0-0{z-rz-} ez ez ℛˆ
    Note that Equations (47View Equation) and (48View Equation) refer to the reference frame at the Sun’s center. However, in order to calculate its scalar product with the magnetic field vector (Equation (9View Equation)) we need to express it in the local reference frame {e α,eβ,eρ} at the point of observation P:
    ex eα r = ℛˆ e = ˆℛ ℳˆ −1 e , (49) y β ez |-----{z-eρ--} Eq. (37)
    where the inverse matrix ˆ −1 ℳ (Equation (37View Equation)) must be obtained employing αp and βp (coordinates on the solar surface for the observed point P). Once B and r are known in the local reference frame (Equations (46View Equation) and (49View Equation), respectively), it is now possible to evaluate Equation (9View Equation). This allows us to determine which solution for the azimuth of the magnetic field, φ or φ + π, yields a magnetic field vector which is closer to be radially aligned.

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