O N D E T E R M I N I N G T H E E L E C T R O N D E N S I T Y D I S T R I B U T I O N

O F T H E S O L A R C O R O N A F R O M K - C O R O N A M E T E R D A T A

MARTIN D. ALTSCHULER and R. MICHAEL PERRY

High Altitude Observatory, National Center fi)r Atmospheric Research*, Boulder, Colo. 80302, U.S.A.

(Received 7 October, 1971)

Abstract. The electron density distribution of the inner solar corona (r ~< 2 R| as a function of lati- tude, longitude, and radial distance is determined from K-coronameter polarization-brightness (pB) data. A Lcgendre polynomial is assumed for the electron density distribution, and the coefficients of the polynomial are determined by a least-mean-square regression analysis of several days of pB-data. The calculated electron density distribution is then mapped as a function of latitude and longitude. The method is particularly useful in determining the longitudinal extent of coronal streamers and enhancements and in resolving coronal features whose projections on the plane of the sky overlap.

1. Introduction

The K-coronameter of the High Altitude Observatory at Hawaii is used to measure

the distribution of the product pB of polarization and brightness in the plane of the

sky fi'om the solar limb (where r= R--solar radius) to r~<2 R. This (two-dimension-

al) pB distribution changes each day, primarily because the projection of the inhomo-

geneous corona onto the plane of the sky changes as the Sun rotates. Granted two

assumptions, we can use the coronameter pB data to calculate the three-dimensional

electron density distribution of the inner solar corona. The first assumption is that

the corona is optically thin (only one Thomson scattering per photon), which implies

that we can observe without hindrance a photon scattered (toward the observer) by

any electron in the corona above the limb. The second assumption is that the solar

corona does not change in density or geometry throughout a 14 day time period, so

that the two-dimensional pB distribution changes only because the Sun is rotating.

Both the first and second assumptions together imply that 14 consecutive days of

coronameter data (corresponding to half a solar rotation) are sufficient. No assump-

tions whatsoever are made about the symmetry of the electron density distribution.

Moreover, the method to be described is valid at all latitudes. In this paper we discuss the mathematical procedure for calculating the electron

density distribution of the inner solar corona given several days of K-coronameter

measurements.

2. Data

The data we use are provided by the HAO K-coronameter (Wlerick and A• 1957) presently located on Mauna Loa (altitude 3.3 kin) in Hawaii. The instrument is call-

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Solar l"hysies 23 1",1972) 410-428. All Rights Reserved Copyright @ 1972 by D. Reidel Publishhlg Company, Dordrecht-Holland

ON DFTERMINING THE ELECTRON DENSITY DISTRIBUTION OF THE SOLAR CORONA 41 I

brated at a wavelength of 520.0 mn with the brightness of the center of the solar disk Io=2.49 x 10 7 W n1-2 s r -z . The product pB of polarization and brightness is given

in units of 10 .8 I o. For the data corresponding to the November, 1966 eclipse, pB ranges from a maximum of 130 pB units at the lower scanning heights to only a few pB units at more distant scanning heights. A calibration for sky brightness is made at i"=3 R (about 32 arc min from the limb).

K-coronalneter data are referenced to polar coordinates in the plane of the sky; the origin is at the center of the solar disk. Measurements ofpB are made at tive scanning heights (in our particular example): 2, 4, 8, 12, and 16 arc rain from the limb and at 72 positions (every 5 ~ around the Sun beginning at heliocentric north. Thus there are 360 measured values ofpB in a complete daily K-coronameter map. Since 1 arc min corresponds to a distance of 43.5 Mm at the Sun, the coronameter provides data to 700 Mm above the limb (that is, to r = 1400 M m = 2 R).

The data we use have sky and instrumental contributions already removed. The mean random error of the data is estimated by Hansen (1971) to be within +_2pB units. The circular scanning aperture as projected on the plane of the sky is 1.3 arc min in diameter (or 56.5 Mm), which corresponds to 4.1 ~ of heliocentric angle at a scanning height of 2 arc min, and to 2.6 ~ at a scanning height of 12 arc min. Bohlin (1968) estimates the distortion of the true dimension by the scanning aperture to be 0.5%, which is negligible with respect to the uncertainty of the other parameters.

3. Geometry

There are two important coordinate systems: (1) the spherical coordinate system centered in the Sun and oriented with the solar rotation axis, and (2) the data coordi- nate system based on the plane of the sky and the line of sight. The first system is used to express our calculated electron density. The second system is used to express the coronameter data. In this section we find expressions relating these coordinate systems.

In the spherical coordinate system, each point in space is labelled by (r, 0, qS) where r is the radial distance from the solar center, 0 is the polar angle (the angle at the solar center between the rotation axis and the radial vector to the point), and q5 is the azimuthal angle in the Carrington longitude system. In this paper we are interest- ed in the domain R<.Nr<~2R, O<.%0<~zc, 0~b<2~r , where R is the solar radius. The spherical coordinate system is natural for our coronal calculation because the derived values of (r, O, (a) which locate a coronal feature do not depend on time (if we assume the corona rotates with the Carrington period essentially as a rigid body over a 14 day time scale).

For our purposes, we detine the plane of the sky to be that plane through the solar center which is perpendicular to a line from the center of the Earth to the center of the Sun. At any instant of time, a point in the corona can be referenced to the plane of the sky by a particular (x, y, z), where x is the distance along the line of sight (so that x = 0 is the plane of the sky as just defined, x > 0 is toward the observer), z is the distance along the projection of the solar rotation axis onto the plane of the sky ( z>0 is

412 MARTIN D. ALTSCHULER AND R. MICHAEL PERRY

toward the north), and y is perpendicular to both x and z (and is in the plane of the sky with y > 0 toward the west limb). The K-coronameter data, however, are given in terms of ~ and q, which are polar coordinates in the y - z plane. Thus it is more convenient to replace (x, y, z) with ((.~, q, ~) defined by

z=~ocosq , y = o s i n q , x = ~ o t a n z (1)

with domains R ~< (2 ~< 2R, 0 ~< q < 2re, - 7r/2 < r < 7t/2, where z is the angle between the radius vector to the coronal point and the radius vector in the plane of the sky to the line-of-sight projection of the point.

The data coordinate systems (x, y, z) or (~o, q, ~) are convenient for recording corona- meter data but have the disadvantage of being time dependent. That is, a coronal feature with fixed values of (r, 0, 4,) will have time-changing values of (x, y, z) and (0, r/, z). For our calculation it is necessary to express the spherical coordinates in terms of data coordinates.

Because the plane of the ecliptic is inclined 7 ~ to the plane of the solar equator, the z-direction of the data coordinates does not always coincide with the solar rotation axis. Let fl be the angle between the solar rotation axis and the plane of the sky. During the year, [1 oscillates in the domain - 7 ~ < ] / ~ 7 ~ For clarity, we treat the case f l=0 in detail in this and the next section, and describe the necessary modifications for /34:0 in Section 5.

For a point P(r , 0, qS) fixed with respect to the corona, the time-independent (Carrington) azimuthal angle ~/~ can be expressed as the sum of two time-changing angles/~ and 4. Referring to Figure 1, we define/~ as the Carrington longitude of the central meridian at the time of observation and 4 as the azimuthal angle in the plane of the solar equator from the meridian of P to either the meridian p + re/2 (if P is seen on the west limb) or to the meridian/~- z/2 ( ifP is seen on the east limb) with the con- vention that 4>0 is toward the Earth. This means that the data from each limb are treated separately and that 4 is restricted to the domain -7r/2< 4 < re/2. With these definitions of/~ and 4, the Carrington longitude of a coronal point P can be written succinctly as

4, = ~ + =~./2 - , ~ (2)

where I for 0~<q~<= (west limb)

=sgn(sinr/)= _1 for r r < t / < 2 r c (east limb). (3)

It is convenient for later use to define

q, - ~ + ~ / 2 (14)

and write Equation (2) as

The line of sight to (~.~, r/) in the plane of the sky, with central meridian/1, passes through a number of coronal particles. Hall" a solar rotation later, the line of sight to (o, Dr-q) , with central meridian /~-~, will contain the same coronal particles

ON DETERMINING TItE ELECTRON DENSITY. DISTRIBLITION OF THE SOLAR CORONA 413

~ ~ Pola__~r Vie.__w_! for ~ = O)

Q

"r = Z PCQ

if, q,, 4 are in plane

of solar equator

x t o ~

E y

SP

Fig. I. Geometry of the calculation is shown in polar and observer (plane of sky) views for the case ,8 -= 0. Coronarneter data is given in polar coordinates (Q, q) in the plane of the sky. When/7 ~ 0, observer's view would still show the same geometry, but polar view would be quite different. When

> 0, then r is the complement of the scattering angle.

(assuming rigid rotat ion about the solar axis), and have the same pB value, if and

only i f f l = 0 . I f fl-r the line o f coronal particles initially coincident with the line o f sight to (~o, q), central meridian/1, will be inclined by an angle 2[3 to the line o f sight to (~o, 2n-q), central meridian tl--n. Thus when/~-r we must treat data from each

solar limb separately; but when f l=0 , we can treat the data f rom the east and west limbs together and greatly simplify our procedures. When /~=0, coronameter data

(for a particular date corresponding to central meridian/ l ) which appear on the east

414 MARTIN D. ALTSCHULER AND R. MICHAEL PERRY

limb ( n < r / < 2 n ) are brought to the west limb (0~<r/~<n) by adding (or subtracting) n to the central meridian angle/~; the intersection of the line of sight through P with the plane of the sky has spherical coordinates (Q, q, p + n / 2 ) on the west limb and (~o, 2 n - ~ l , l ~ - n / 2 ) on the east limb. Since all data are now referred to the west limb, the domain of i ' /may be restricted to 0 ~< q ~ n.

The distance between P and Q is x. The relations between the spherical coordinate system (r, 0, q~) and the plane of the sky coordinate systems (x, y, z) and (2, 11, z) when the solar rotation axis lies in the plane of the sky, so that [ t=0 , are given by

x -= o tan 1" = r sin 0 sin ~ (6)

y - ~ sin q = r sin 0 cos ~ (7)

z - ~o cos rl = 1" cos 0 (8)

r = Q sec z (9)

0 = cos - j (cosq cos~) (10)

= t an - x (tan r/sin ~1) (11)

#5 = n/2 + ~, - { (12)

where we have restricted r / to the domain 0~<~l~<n and thus have transferred all p B

data to the west limb. Other useful relations are

sin ~ : sin v (sin 2 z + COS 2 1. sin 2 q)- 1/2

cos ~ : sinq cosz (sin 2 l" + COS 2 1" sin 2 q ) - 1 1 2

(13)

(14)

The leftmost equalities of Equations (6), (7), (8) are the definitions of T, ~, q given in Equation (1). The rightmost equalities of Equations (6), (7), (8) as well as Equations (10) through (14) are not valid if fl:~0.

4. Determination of the Electron Density

The K-coronameter gives the distribution of the product of polarization and bright- ness in the plane of the sky at the time of observation.

Suppose the corona had zero density everywherc except in a small volume located at the point P ( r , 0, c/~) where the electron density was N e and the thickness in the line of sight direction was 3x . The coronagraph would then record the product of polari- zation

p = (I, -- I,.)/(I, + 1,) (15)

and brightness

B = It + I r ([6)

where I, and lr are the intensities of the polarized light tangential and radial to the solar limb. Thus (Minnaert, 1930; Billings, 1966)

pB = I, - I~ = ~-naloN,~ cosZz [(1 - u ) . J . / + u:~] A x (17)

ON I)ETERMINING THE ELECTRON DENSITY DISTRIBUTION OF THE SOLAR CORONA 415

where ,~;/= cosQ sin 2 Q (18)

,~ . . . . . . . . . . . . . . . . . . . (I 9) 8 sin Q cos_O, fl_] "

and a is the square of the electron radius and equals 7.95 x 10 -26 cm2; f2 is the angle at the scattering point P between the radius vector and a tangent to the limb of the

Sun; u is a function of wavelength which arises f rom limb darkening; I 0 = 2.49 x l 0 a~ erg cm -2 s -1 sr -1 is the light intensity at the point where the radius vector to P

intersects the photosphere (that is, the central disk intensity). The angle f2 is given in terms of spherical and data coordinates by

sin f2 = R/r = (R cos r,)/9. (20)

Thus f2, , J , and ,~ are functions only of r in the spherical coordinate system.

For our data, we take u=0 .63 , a constant value corresponding to 2=520 .0 nm (in the green). It is also convenient to mult iply and divide by R, so that Equat ion (17) can be written

pB (0, q, ,u) = a o I-0.37 ~J (r) + 0.63 (~ (r)] ?4,. (r, O, qS) cos 2 r, Ax/R =

= C(r) N,, (,', 0, ~b) cos~ ~ A~/R (21)

where the constant is given by

ao -~ ~-~loR. (22)

Since the coronamete r data are given in units o f 10-8 io ' we convert the expression for pB to the same units by replacing a o with

a = {n 108oR = lOSao/Io = 8.69 x 10 -7 cnl 3 . (23)

In the plane of the sky, we see not the scattered light f rom one small volume element, but the scattered light f rom all the volume elements along a given line of sight. Therefore,

pB(~,,7,~) I c(,.)N~(,.,o, ' = ( p ) C 0 S 2 r, dx/R =

. - Qt 2

,7/2

Ne (r, O, t[~) C (r) t2 dr.jR (24)

- h i 2

where by Equat ions (9) through (12), r, 0, ~/~ are given as ffmctions of(o, q, r,, I~. In the last step we have used d x = o sec2r dr, since ~ = c o n s t along the line of sight.

If the electron density N,, (r, 0, 40 of the solar corona were known, we could expand this distribution in terms of or thogonal Legendre polynomials

2 X 2(',:)' t,r (,., o, 4,) = P2 (0) (g.,., cos,,ub + h,,,,,, sin m4,) (25) n = O m = O l = 1

416 MARTIN D. ALTSCHULER AND R. MICHAEL PERRY

where the coefficients gn,a and hn,a have dimensions (cm -3) forpB in units of 10 -s Io. (In the limited domain of interest R<~r~2 R, Equation (25) allows for an increase of coronal density with height. Negative or zero values for the integer l are not necessary.) In general, the distribution N~(r, O, 4)) is not known. Our problem is to obtain fl'om K-coronameter data a set of g~,a and h,,,t which when inserted into Equation (25) will adequately approximate Ne (r, O, ~/~).

Putting Equation (25) into Equation (24), we can write

~/2

n m l - z~/2

• (g,~l cosm~b + hn,a sin mqS) dz (26) R

where the unknowns are now the discrete set of numbers gn,,a and h,,mv Using Equation (2) or Equation (5), we can expand cosmt/~ and sin mc/~ and write

Equation (26) in the form

p n ( o, P~, ].1) = Z E E [ ( gnml COS Wl l/I "3 t- hnm I s in Wl l/g ) ~lml ( ~) , ti) "t- n m I

+ (g.,,a sin n+~p - hnm I cos n+l/I) o+Vnn,i (~..+, 17)- l (27)

where m and t~ are defined by Equations (3) and (4) and

+r/2

=- [" (R/r)' C(r) I~'(0)cosm~(.o/R)dz (28) U,.nt (0, //) i i /

- n / 2 7r!2

~1) = I" (R/")* C (r) P~ (0) sin m~ (e/R) dr. (29) Vnml t /

-7t/2

Since r, 0, ~ are functions only oleo, I1, z (cf. Equations (9), (10), (11)), the dependence ofpB on the central meridian I~ (or ~) now appears explicitly in Equation (27). With the present restriction to fi = 0, we employ the following simplification. In performing the integrals of Equations (28) and (29), we note from Equations (9), (10), (I 1) that r, C(r), P,7(0), and cosine do not depend on the sign of r, but that sin m~ changes sign when T changes sign. Consequently,

n/2

2 I" (R/r)' C (r) P,~ (0) cos m~ (o/R) dz (30) U,,m, (,.,, ~)

0

v;,m, ,1) = 0 (31)

so that finally (for fl=O)

pB (o, 'l, F') = ~ X ~ U..a (~,, ~) (g,,.a cos re,l, + h,,,,,, sin toO). (32) I1 II1 l

O N D E F E R M J N T N G T I l E E L E C T R O N D E N S I T Y D I S T R I B U T I O N O F T I l E S O L A R C O R O N A 417

Equation (32) contains (1) the coronameter data p.B(o, q, ~), (2) the quantities ~;,.l (O, r/) which can be evaluated .for any (0, ~7) in the plane of the sky, and (3) the unknown real numbers g.mt and h,,,. v

Our purpose now is to obtain the values of g,,mt and h.~ which minimize F, the sum of the squares of the differences between the observed pB values (which we now denote by D(ll, O, ~1)) and the calculated pB values:

[ F = E E Y. D (/~, o, '1) - E E E U.,,:, (0, r/) (g,,m, COS m,/, + I,,,,,,, sin ,,,,/, /* o I/ n m 1

(33) The equations

(~F/C"g,,.,,) = 0, (~F/eh,,,.,) = 0

for each (n, m, l) are linear in g,,,.t and h.,nt and can be written as

and

(34)

E E Yl G,., (0, '1) cos m,/, E E E U,~ (o, ,7) (,q,.~,, coss,p + h,,. sin sIp) = l * O ~/ t s I.'

= Z Y', Z D (tt, 0, q)/d,~t (0, ,'I) cos t,, o (35) I1 ~d r/

~ E U,,m' (0, q) sin mO E E E Ut~r (0, q) (g,~r cossl/J + h,,p sin .s'l/*) = it o q t s p

= ~ ~ E D (t~, 0, q) U~,., (o, ,i) sin m~p (36) I 1 s I/

where t, s, p are alternative indices for n, m, I respectively. The unknowns are g,,mt

and h,~t. The quantities U,,.~ (0, ~7) are evaRmted from Equation (28) and thepB(/4 0, q) are given data.

Since m~<n and we have restricted l to a positive integer, we can uniquely define the column vector

~9,,,.,; when i = ( n 2 + m ) K + k ( l ) x, = ~ th. , . , ; when i = (,,-' + ,, + ,,,) K + / , (0; "* # 0 .

(37)

The integer k (1) is an index for the radial component, and K is the total number of values for k. For example, if / is chosen to take all values f r o m / = 1 to I=L, then k ( l )= l and K= L. But if l is chosen to take only the values 3, 5, and 8 (corresponding to radial dependences r -a, r -5, and r -8 for the electron density), then K=3, and l=3 gives k (3 )= l , and l=5 gives k(5)=2, and I=8 gives k(8)=3.

Similarly, we define the column vector

i o(l~.o,,fl G.~ cos,,.,,/,. when i = ( n z + m ) K + k ( 1 ) ,

Yl = ~ ~ DOt, O, '1) U,,,,,,(O, q) sin ,n,//,

when i = ( n 2 + n + m ) K + k ( / ) ; Let

•UtsP , , , , , - Z Z G,,,, (e, ,7) G,, (~, ,7)

(38)

m r O.

(39)

418 MARTIN D. ALTSCHULER AND R. MICIIAEL PERRY

and define the symmetr ic matrix

Ai. i : A j i =

{j : + ,,,) K + k (0 ~,,,~r Imp coss,/J; when 0 2 + s) K + k(p) )'~nml ~ COS

it �9

,,,I Y~ sin m~/J sin s~p; when , . ( t 2 + t + s ) K + k ( p ) ; s~aO

Wt ,p m,/,sins,/J when ~i = ( n z + m ) K - l - k ( 1 ) t"_

nml Z COS u ... = ( t 2 + t + s ) K + k ( p ) ; s~:O + k ( 0 ;

I L , , , , Z s i n m ~ c o s s , p ; , when . = ( P +s) K + k ( p )

(40)

where k (l) and k (p) are the indices for l and p respectively. In the actual calculation, we compute only a finite series to principal index N = max (n). Thus the column vectors x~ and j.,~ each have .~4 =K(N + 1 ) z components , and the square matrix A~.i has 11.1z

components of which at most ~M(M + I) are distinct. Then Equat ions (35) and (36) can be written

M

Aijx.i = y; (41) j= I.

or in matrix notat ion

Ax = y . (42)

The solution to this equat ion

x = A - i y (43)

gives the required values of the unknowns g,,mt and h,mt for the electron density d istribu- tion of the solar corona.

5. Modifications for 16 r 0

W h e n / / r Equat ions (6), (7), (8) become

x ' = x cos fi - z sin [3 = i- sin 0 sin

z ' = x s i n / / + z c o s / / = r cos 0

Y = 3' = ~r sin0 cos

(44)

(45)

(46)

where we have rotated the data coordinates (x, y, z) o f the plane of the sky through an angle / / a b o u t the ),-axis so that the z ' direction coincides with the actual solar rota t ion axis (Figure 2). The quanti ty :~ is detined in Equat ion (3). Our convention is that positive fl tilts the north pole of the Sun toward the Earth. Equat ions (9), (10), and (I 1), are now replaced with

r = 9 sec z (47)

0 = cos --~ ( s in t s i n / / + c o s t cos t / cos / / ) (48)

= t a n - J [( tan t c o s / / - cos q sin//)/Isin 111]. (49)

ON DETERMINING THE ELECTRON DENSITY DISTRIBUTION OF THE SOLAR CORONA 4l 9

Equation (47) is the same as Equation (9). Equation (48) is derived from Equations (45), (1), (47). Equation (49) is derived by dividing Equation (44) by Equation (46) and proceeding as before.

The procedure of referring east limb data to the west limb by adding ~ to/~ is not valid when [3~0. Instead east and west limb data must be calculated separately.

Because Equation (47) is the same as Equation (9), Section 4 is applicable without change until Equations (30) and (31). These equations are not valid because in general

Z

z' I

NP

to ~9

C

X

SP

Fig. 2. When fl g: 0, we define the (x', z') axes by rotating the (x, z) axes through an angle fl about the y-axis so that z' coincides with the solar rotation axis.

~-~0 when z = 0 (cf. Equation (49)) and hence sinm~ does not change sign when changes sign (except in the planes where q=n/2 or 3n/2). Thus Equation (32) is no longer valid and the :full expression of Equation (27) must be used. Equations (33), (35), (36), (38), (39), and (40) are now more complicated. We write down the modifica- tions for Equations (38), (39), and (40). Equation (38) is now replaced with

= w h e n i = 0 , 2 + , , ) K + k ( 0 , , (SO)

I when i = ( 2 + , , + m ) ~ + k ( 0 ; m # 0

where ~. and ~[J are defined in Equtations (3), (4). The definitions of U,,,~(~, 17) and V,,,~(o, q) are still given by Equations (28) and (29). Equation (40) must be replaced.

420 M A R T I N D . A L T S C I I U L E R A N D R, M I C H A E L P E R R Y

Instead of Equation (39), we now let

~q - Z Y, u.,., (~, ,7) <,,, (e, ,7)

~vl - Y 52 v,,,,, (~, ,I) v;,p (e, ,7)

W3 ~ Z 520~CTntn' ((2, tl) gtsp (~), t])

tr - 52 ~ ~v;,,., (~,, ,1) u;,, (~.,, ,) O t l

where l,t,) is indexed by (n, m, I; t, s, p), and

c , - }2 cos m,/J c o s s 0 #

G2 - Y~ sin m~p sin s,/J It

G 3 = 52 cos trt@ sill sI/I .a

G, -= ~' sin m~p coss~/s //

where G i is indexed by (m; s). Then Equation (40) becomes

WtGI + I"FzG 2 + W3G 3 + W4.G 4 { i =(n2 + m) K + lc(I)

when . (t 2 + s) K -t-/c (p)

I'VIG 2 + B%GI - 14%G4- W4Gs { i = ( n 2 + n + m ) K + lc(l);

when . (t 2 + t + s ) K + k(p) ;

A~i = Aji = [tqG3 - W2G4 - W3G1 + W4G2

{ i =(nZ + ln) K + k( / ) when . (t 2 + t + s ) K + k(p) ;

W1G4.- W2G3 q- I '1/~3G2 - I,'I/~.G~ {~ = ("2 + n + m) K + k( / ) ;

when (t 2 + s) K + k(p)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

m # 0 s" # 0

(59)

s 4 : 0

m r 0

With Equations (50) through (59), and Equation (37), we can carry through Equa- tions (41), (42), and (43) for the matrix solution as before and obtain the values for g,,,a and h,,,, t. The modifications discussed in this section, although tedious to write out, arc in practice only a minor modification of the computer program.

6. Computer Techniques and Practical Limitations

Relating to the present procedure there are a number of practical considerations. Most important are (I) the quality and quantity of the data, (2) the size of the matrix A, and (3) the conditioning of the matrix A. We discuss these in turn.

The data from the High Altitude Observatory coronameter on Mauna Loa, Hawaii arc extremely good and have a large signal to noise ratio, particularly at the inner

ON DETERMINING THE ELECTRON DENSITY DISTRIBUTION OF THE SOLAR CORONA 421

scanning heights. Although a coronameter scan of the Sun is obtained daily, the vicissitudes of atmospheric turbidity often preclude obtaining data from the upper scanning heights. With a simple data format devised in collaboration with D. Trotter,

missing data do not enter into or affect our calculation (except that less data means less resolution). No attempt is made to compensate for (or interpolate for) missing data, since our derived regression equation is itself an interpolation.

In our computer calculation, we have so far only one value of l (that is, K = I) for

the radial variation of the electron density distribution (see Equation (25) or Equa- tions (50) and (59)). The best value for l was determined from the axisymmetric coronal model of Saito (1970) in the domain R<~r<~2R, and was found to be between I = 7 a n d / = 8 . We chose l=8 . This restriction to a simple radial dependence is the worst defect of our calculated results at present. There are two primary reasons why, in this initial attempt, we have chosen a simple radial dependence.

First, the expansion given by Equation (25) is unwieldy, even as a truncated series, if a high surface resolution in (0, c/~) and an accurate radial (or r) dependence are both desired. F'or example, if n runs from 0 to N = max(n) and l runs from 1 to L, there are L ( N + l ) 2 Legendre coefficients that must be evaluated. With N = 9 and L = 5 we must evaluate 500 coefficients and invert a 500 x 500 symmetric matrix (with at most 1.25250 distinct elements). I f we choose only two values for l (for example, l = 3 and I--7), and N = 7 , then we require the inversion of a 128 x 128 matrix (with

at raost 8256 distinct elements), which is feasible. Thus the computation (particularly the time and computer storage required to set up the matrix) would become excessive with present facilities if more than one or two values o f l were chosen and a high surface resolution were not sacriticed.

Second, the basis functions (R/r) z are not very good for representing radial de- pendence.

The use of only one value for l means that all coronal features are given the same radial dependence. As a consequence, we can not at present use our calculation to

distinguish coronal condensations from the bases of streamers (although according to Hansen and Hansen (1971.) this can be done by examining the shapes of the pB polar plots as a function of height). In our next effort we will try to obtain a more accurate radial dependence for the electron density distribution.

During 1964 to 1967 pB coronameter scans were taken only at daily intervals. Ideally, 14 consecutive days of data are sufficient for this analysis. When such data are not available because of atmospheric turbidity or bad weather, we use data collected over a longer time period (say 20 days) together with the implicit assumption that the corona does not change significantly during this longer period. Our com- putational scheme does not require consecutive days of data. Each day of data is referred to the central meridian (Carrington) longitude.

If a polynomial with too large an N = m a x ( n ) is fitted to too few days of corona- meter data, there is an 'overfitting' of the data, and incorrect oscillations appear. Over fitting means that some of the Equations (35) and (36) for larger n add little or no new infbrmation, or equivalently, are nearly linearly dependent upon the other

4 2 2 MARTIN D. ALTSCH ULER AND R. MICHAEL PERRY

~ - , . j

o~ o ~

C_. .~ ~--~

u l too , ( l o , I ~ tn r lN ~

tit ~r~0~,~o~ r')r.- om')~0c~Jur)~r o l r l n ~ l ~ , e a 0 ~ ( = , , D o ~ o ~ ~ ~...~(~ ~.

�9 - IN t~MFr ) N N N N , - ~ , - ~ , - I ~" r "

. . . . . . . . . . . ~ N

~'.__.

m ,.. "~ ' ~ "

~ o . . . . g

I ,',, I ~ ~ ~~" ~ _ , ~

. . . . . o ~

ON DETERMINING THE ELECTRON DENSITY DISTRIBUTION OF THE SOLAR CORONA 423

equat ions. Thus the symmetr ic mat r ix A may become i l l -condi t ioned and the solut ion

for the co lumn vector x is ei ther no longer unique or is very sensitive to small changes

in y. To avoid an i l l -condi t ioned matr ix , we use the p rocedure o f G o l u b and Reinsch

(1970) as modif ied by A. Cline o f N C A R . The matr ix A is first decomposed into a

p roduc t of o r thogona l and d iagonal matr ices ( A = UrDV). The componen t s of the

d iagonal mat r ix D are called the s ingular values of A, and are equal to the square

roots o f the eigenvalues o f ArA. W e then de te rmine the condi t ion number , which is

the ra t io o f the largest to the smallest (nonzero) s ingular value. I f this number ex-

ceeds 1014, the solut ion vector x may not be accurate to any significant figure (because

our compute r has only 14 significant figures in single precision). Consequent ly , when

the condi t ion number exceeds 1014, the lowest s ingular values are e l iminated until

the condi t ion number falls be low 101r thus the d iagonal matr ix is modif ied to a

new matr ix of lower rank. This p rocedure gives the best poss ible solut ion for the

co lumn vector x ( that is for g , ~ and h,,,~). Thus, for example, if we t r ied to fit a

Legendre po lynomia l of N = m a x ( n ) = 7 to only one day o f data , then the cont r ibu-

t ions o f the non-ax i symmet r ic modes would be severely modif ied to provide con-

sistency with the data. In pract ice, however, our condi t ion number rarely exceeds 3 x 104.

The two-d imens iona l m a p o f relat ive electron densi ty at some given radia l d is tance

is shown in Figure 3 for the per iod abou t the 12 N o v e m b e r 1966 eclipse. Uni ts are

relat ive and run between 0 and 100. When the abso lu te value o f densi ty co r respond ing

to 100 is known, the other densit ies can be found by tak ing the direct p r o p o r t i o n

(or percent ) ; for example , the region label led 45 is 45 ~ of the densi ty of the region

label led 100. Because we have chosen only one value for l, the m a p of F igure 3 in

relat ive units is the same for all heights, a l though the absolute value o f e lectron densi ty

at any given (0, qS) changes according to r -t. The m a x i m u m values o f e lectron densi ty

(cor responding to the box label led 100) are given in Table I for several heights; also

TABLE I

Data for 12 November 1966

max(Ne(r, O, r (cm -s)

r/R = 1.0 = 1.33 = 1.67 = 2.0

Condition number

N = 6 1.2 x 109 1=8

N = 7 1.3 x 109 l = 8

N = 8 1.3 x 109 l = 8

(Map of Figure 3)

N = 7 1.0 x 10 9 I = 7

N = 7 1.5 X 109 1=9

1.2 • 10 8 2.1 • 10 7 4.8 x 10 6 3.6 • 10 2

1.3 x 10 8 2.2 • 10 7 5.0 x 10 6 2.3 x lO s

1.3 x 10 8 2.3 x 10 7 5.3 x 10 6 2.4 x 10 4

1.4 x 10 8 2.9 x 10 7 8.2 x 10 6 4.9 x lO s

1.1 x 10 s 1.5 x 10 7 2.9 x 10 6 1.1 x lO s

424 MARTIN D. A L T S C H U L E R A N D R. M I C H A E L PERRY

shown is the variation in calculated density when different values of l and different values of N = max (n) are chosen (that is, when the radial dependence and the length of the truncated Legendre expansion are both varied).

In practice, the density maps for N = 7, 8, 9 (and fixed l) are vcry similar. Although better resolution is achieved for higher values of ,V, the condition number of the matrix A also increases, and presumably increases catastrophically if N is increased much beyond N = 10. Thus how much is actually gained in resolution by increasing N beyond about N = 7 depends on the amount and the quality of the data. Related to this point is the convergence of the Legendre series.

The Legendre series we obtain converges in the sense that the values of O,,,,t and h,,mt do not change very much when the length of the series changes from N = 7 to N = 8 or N = 9 ; in particular, the lower the integer n compared with N=max07), the less is the change in 9,,,,~ and hn,,, t when N is incremented. The power spectrum (that is, _ 2 • 2 S,,,,~-9,,,,t , h,,,~) for given N, however, decreases fairly slowly with increasing n for those modes which involve mostly longitudinal dependence (m~n) (note that always m~<n), but decreases fairly rapidly for those modes which involve mostly latitudinal dependence (0~<m<<n). This occurs because there is less longitudinal data (i.e. fewer observations with different central meridians) than latitudinal data (where there are 37 points to cover the limb at each ccntral meridian). Consequently, for N~>7 (depending on the amount and quality of the data) the modes for m ' ~ n become nearly linearly dependent, so that the magnitudes of their 9,,,t and h,,mt do not have a sharp power spectrum (i.e. do not decrease rapidly); this also explains the increase in the condition number of the matrix A as N is increased. This linear dependence effect can be mitigated in two ways. First, we can change the matrix A to include more latitudinal information and less longitudinal information by going to higher N but omitting modes with m ~ n when n > 7 or 8. Second, we can increase the frequency of K-coronameter measurements beyond once per day; this can be done with several calibrated coronameters located at different longitudes around the Earth, or by a coronameter in Earth orbit, as is planned for the Skylab project.

We have varied N=max(n) from 1 to 9 and lind that when 7~<N~9 (depending on the data) the location of high and low density regions is rather insensitive to the variation of I between 7 and 10.

To test the 'reality' of our procedure, we compared our maps with those of 9 cm radio emission, coronal green line emission, and plage location for three independent time periods, and found excellent agreement between our high density regions and the regions of high activity.

The calculated electron density is plotted on a two-dimensional contour map (Figure 3) as a function of latitude and longitude at some specified radial distance. 'Three-dimensional' computer maps have also been constructed in which regions of density enhancement are shown by a cloud of points in spherical coordinates. The three-dimensional effect can then be obtained by examining two different projections through a stereo viewer or by watching a movie of the rotating map. The 3-D map can be used to compare the electron density (projected on the plane of the sky) with

ON DETERMINING THE ELECTRON DENSITY DISTRIBUTION OF TIIE SOLAR CORONA 425

the observed pB intensity data, and thus to check the validity of the calculation. In practice, the 3-D map for coronal density is qualitative, whereas the detailed 2-D map (Figure 3) gives quantitative values of electron density as a function of latitude and longitude at any specified radial distance. The computer can also draw the density distribution as it would appear looking down at the poles of the Sun, and this view gives at a glance the longitudinal distribution of coronal density.

As a point of information, the determination of the coronal electron density dis- tribution from 21 days of K-coronameter data with four scanning heights and N = = m a x ( n ) = 7 requires 2 min 17 s of computation on the CDC 7600 computer.

7. Discussion

We have used the formulas of Schuster (1880) and Minnaert (1930) together with a regression analysis procedure to determine without ambiguity the inhomogeneous electron density distribution of the solar corona using only pB coronameter data. No assumptions of coronal symmetry were made. When the kegeudre coefficients 9,m~ and k,,,,~ (obtained from Equation (43) via Section 5) are inserted into Equation (25) (and the electron density Ne(r, O, ~D) is plotted as for example in Figure 3), we can accurately locate coronal features in latitude and longitude, and (in principle though not yet attempted) determine how they change in dimension and orientation as a function of radial distance.

The absolute values for the electron density are as accurate as the coronameter calibration to the resolution determined by either the polynomial (Equation (25)) or the periodicity of the data (whichever is worse). Our choice of only one radial dependence is probably the most serious source of error at present.

Other methods which use K-eoronameter measurements to determine the inhomo- geneous structure of the solar corona are merely different ways of plotting pB data to enhance interpretation (see for example Dollfus, 1968; Hansen et al., 1969). A day of data can be displayed as a numerical table ofpB values distributed as on the sky plane, or as a polar plot ofpB for each scanning height, or as a plo! of the isophote contours which connect regions of equal pB.

Over a period of time (roughly half a solar rotation), the pB amplitudes observed on the limb can be mapped onto a cylindrical projection (latitude vs longitude, with the longitude corresponding to the longitude of the limb at the time of observa- tion) and plotted as contours. Because a coronal feature extends beyond the limb and is seen in the plane of the sky Ibr several days before and after its base transits the limb, these contours for pB are elongated in longitude compared with the actual dimensions of the coronal feature. Nevertheless, when an isolated coronal feature is at maximum brightness and elongation at the limb, its longitude can be identified with the longitude of the limb. In fact, however, none of the methods of displaying pB data can resolve the coronal density distribution if the Sun is relatively active and several coronal features overlap in projection on the plane of the sky; to surmount this difficulty was a motivation for the present paper. We believe that the method de-

426 M A R T I N D . A L T S C H U L E R A N D R . M I C H A E L P E R R Y

scribed in this paper makes the best use of coronameter data and in fact is the only self-consistent procedure presently available which determines the electron density distribution of the entire corona from pB data alone.

Recently the work of Withbroe et al. (1971) has provided another systematic procedure to determine the electron density of the entire solar corona. They analyze the Mgx fine (262.5 nm) generated at the base of the corona (and observed with the OSO-6 satellite). The detailed comparison of the Mgx maps with our maps derived from pB data should prove informative; for the period around the 7 March 1970 eclipse, a cursory comparison indicates a reasonable agreement.

For special time periods, particularly around eclipses, it is possible to use plots of raw pB data together with white light eclipse (or balloon coronagraph) photo- graphs and data from other sources (X-ray, UV, radio, magnetic field, etc.) to eliminate ambiguities in the positions of coronal features along the line-of-sight and to accurately determine the inhomogeneous global electron density distribution (see Bohlin, 1968, 1970a, b). Such work is extremely valuable because it interrelates different kinds of data. However, it is also useful to quickly and routinely determine the coronal electron density distribution from the data of a single instrument.

Because accurate maps of tile large-scale inhomogeneous electron density distribu- tion have heretofore not been available, researchers calculated an average spherical or axisymmetric density model for the corona of quiet and active years (Baumbach, 1937, 1939; Allen, 1947; van de H ulst, 1950; Saito, 1959, 1970; see Newkirk, 1967; Billings, 1966; and Shklovskii, 1965 for reviews), usingpB and eclipse data. A glance at our computed density map (Figure 3), however, indicates an extremely inhomo- gencous corona, with peak densities 3 to 5 times the median density (which is 19',;~i of maximum for the map of Figure 3). The extremely inhomogeneous corona which we tind is in complete agreerncnt with the results of Ney et al. ( 1961 ), Bohlin (1970a, b), Newkirk (1967), Leblanc (1970), Hansen et al. (1971), and others. Thus the significance and utility of average density models for the corona (particularly when they are com- pared with radio data from active regions) seems to us to be questionable, and now no longer necessary. Similar suspicions were also raiscd by Mustel (1963).

In fact, a preliminary examination of our calculated density maps for several different time periods shows large coronal regions of negligible electron density. We do not believe these coronal 'holes' are mathematical effects bccause for the period around the 7 March 1970 eclipse the holes we find coincide with those detected by Withbroe et al. (1971) in Mgx at tile base of the corona. Thus there are holes of relatively negligible electron density which extend out from the base of the corona to at least 1.5 Re. Orrall (1971) suggests these holes are structures as important as enhancements and streamers, and are caused by magnctic properties of the chromo- sphere. Perhaps, however, there is no 'homogeneous corona', but only a corona associated with active regions and coronal magnctic lields (see also Newkirk and Altschuler, 1970 and N ewkirk, 1971).

Coronal enhancements and streamers have generally been treated as local regions of high density superimposed on a coronal background of average (and constant)

ON I)ETERMININ(} TIlE ELECTRON DENSITY D1STRIBU2-1ON OF TIlE SOLAR CO:RONA 427

dcnsity. Often radio, eclipse, magnetic, and p B data are used together with geo- metrical assumptions of symmetry. (See Waldmeier, 1956, 1963; Waldmeier and Muller, 1950; Saito, 1959; Kawabata, 1960; Newkirk, 196l; Hiei, 1962; Saito and Billings, 1964; Nishi and Nakagomi, 1963; Dollfus, 1968; Leblanc et a/., 1970; Hanscn et al., 1971, and the reviews of Billings (1966) and Newkirk (1967).)

Thc method of Leblanc et al. (1970) uses only K-coronametcr data to determine the density structure of coronal enhancements; it is mathematically rigorous when applied to cnhancemcnts at low solar latitudes, but requires an assumption for the geometric form of the density distribution. Nevertheless, this is the best method at present for determining the radial variation of density in corona~ enhancements.

Since the method we have described in the present paper eliminates the ambiguity due to overlapping prqjections of coronal features on the plane of the sky, we can compare our coronal density maps with fcatures of the solar disk observed over the entire electromagnetic spectrum, and with magnetic maps (Altschuler and Newkirk, 1969). -I'he improvement and application of our coronal density maps are currently in progress.

Acknowledgements

We would like to thank R. Hansen, S. Flansen, and C. Garcia for their excellent coronamcter data and for helpful comments, G. Newkirk, Jr. for stimulating and focusing this work in its formative stages, D. Trotter for reducing all the data to a convenient computer format, A. Cline (NCAR) for helpful discussions on matrix analysis and singular value decomposition and for the use of his SVD subroutine, P. Swarztrauber (NCAR) for developing an extremely fast subroutine for generating associated Legendre polynomials, G. Dulk (Univ. of Colo.) for computational assis- tance which sped up our program, and F. Orrall (Univ. of Hawaii), C. Querfeld, A. Skumanich, and M. Raadu for many helpful comments. One of us (RMP) would like to thank NCA R for support on their Summer Student Program,

Note added in proof: Because we used the Ephemeris of 0 UT rather than that corre- sponding to the Hawaiian observations (1800 UT), the Carrington longitudes given in Figure 3 arc 10: too large. Thus 0 ~ is actually 350". and 50" is actually 40:', etc.

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428 MARTIN D. ALTSCHULER AND R. MICHAEL PERRY

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