691
THE SOLAR CORONA,
DISCUSSED BY SPHEKICAL HARMONICS.
BY
Professor FRANK H. BIGELOW.
CITY OF WASHINGTON:
PUBLISHED BY THE SMITHSONIAN INSTITUTION.
1889.
691
THE SOLAR CORONA,
DISCUSSED BY SPHEKIGAL HARMONICS.
BY
Professor FRANK H. BIGrELOW.
CITY OF WASHINGTON;
PUBLISHED BY THE SMITHSONIAN INSTITUTION.
1889.
JUDD & DETWEILER, PRINTERS,
WASHINGTON, D C.
ADVERTISEMENT.
The following mathematical study of the solar corona, as shown in the photographs
taken by Messrs. Pickering and Barnard during the total eclipse of January 1, 1889, is
submitted to astronomers and physicists as a possible clue to the explanation of the corona,
and as suggesting the direction to be taken in future observations and investigations.
The paper has been recommended for publication by Professors Asaph Hall and
Cleveland Abbe, to whom it was referred in accordance with the usage of the Smith¬
sonian Institution.
S. P. LANGLEY,
Secretary.
Smithsonian Institution,
Washington, October 1, 1889.
(3)
THE SOLAR CORONA,
DISCUSSED BY SPHERICAL HARMONICS.
Bv Professor Frank H. Bigelow.
t/
The difficulty of analyzing the structure of the solar corona is increased by
the superposition of individual rays in projection on a plane perpendicular to the
line of sight. The polar streamers and the outline of the equatorial wings are
relatively free from this overlapping, and the body of the moon, in transit, cuts
off such rays over the disk as are most distorted, so that the problem ought to
be soluble by some theory applicable to the case of the rays specified
The structure to be accounted for consists : (1) of polar rays nearly vertical
at the coronal poles or axis of reference for the symmetrical figure, but inclining
more from this axis than a radius vector to any point as the vectoral angle in¬
creases ; (2) four wings disposed upon two axes, each inclined at an angle of
about 40° from the vertical; and (3) extensive equatorial wings seen more dis¬
tinctly at periods of solar quiescence. This appearance upon a meridian section
must be translated into corresponding zones and sectors on the figure of revolu¬
tion of the sun.
We propose to treat this subject by the theory of spherical harmonics, on
the supposition that we see a phenomenon similar to that of free electricity,
the rays being lines of force and the coronal matter being discharged from the
body of the sun, or arranged and controlled by these forces. In order to give the
solution a general foundation the important points of the theory of harmonics
specially relating to the case will be recapitulated, and the corresponding geo¬
metrical solution will be given in a notation adapted to the sun. My references
are to Maxwell, Mascart and Joubert, Ferrer, Todhunter, Thomson and Tait
in their treatises on harmonics.
(5)
6
THE SOLAR CORONA.
THE HARMONIC THEORY.
Assume the centre of the sun's corona as the origin, the co-ordinate axes,
X, Y, Z, at any instant being the radius vector to the observer, that at right
angles, and the polar axis respectively. Take any set of secondary polar axes
distributed at will over the spherical surface, each axis, A, etc., being defined
as a definite direction from the origin, the cosines of the angles between these
axes being cos mi2, cos ^¿3, etc., in all combinations. Let any point in space
be defined as (r, ö)i from axis Äi, (r, from axis etc. Then suppose there
are n axes, and that s is the number of cosines between them. Assume that a is
the number of poles of the 71 axes distributed uniformly on the equator from X at
distances —. From the point (r, 0) draw planes pei-pendicular one to each axis
(7 *
and successively differentiate the equation v = ^ relatively to each pole.
It is known that La Place's equation, = 0, is satisfied by a solid
harmonic of the degree i of the form M¿ y. = ^
In order that for a spherical closed surface the potential may satisfy the
equation continuously without becoming infinite at the origin or at infinity it is
converted into three terms :
= 1_^M¿ r' Yi within the sphere,
(T == C on the spherical surface,
V. = Y i without the sphere.
These become, when expressed in terms of C :
_ fYC r'
2i+l • '
_ 4/7C R»+ 2
2Í + 1 • fr~r- i '
wherein C is a constant and is a surface harmonic. Y^ when expressed in the
general trigonometrical form is :
f f 21 — 2 S /
Y i = sum I ( — 1)" 2^ - s [i 11 -Eg 0'-^® cos m'
In case s = 0, and consequently no poles are assumed symmetrically disposed
around the equator nor at random over the surface, but all are collected into one
pole, the surface harmonic becomes a zonal harmonic, whose form is :
Qi = sum„ I ( — 1)"
{ 2-" — 2n
where n receives the values 0,1, 2, 3, etc., for summation.
It is obvious, from the inspection of the symmetrical disposition of the corona,
that we deal with only one axis, and that therefore our harmonics are of the first
degTee, i = 1. Hence :
cos O'-ä« gin 02
THE SOLAR CORONA.
7
Yi = Qi = 1 . cos d ,
Hi = . r cos 6 , Inside sphere.
(7=0 . cos 6, Upon "
V = ' Outside "
1 3
which upon differentiation satisfies the equation :
^ + 4 n a = 0.
d r dr
THE GEOMETRICAL THEORY.
Let us now pass to the corresponding geometrical conditions. If we sup¬
pose equal masses of potential of opposite signs, + m and — m, to be located on
4
the extremities of the polar axis, the moment is M = 2 R m = -g- H R'* C. The
equation for equipotential surfaces is v = M ^ ) where r and are the
distances from any point to the positive and negative poles respectively. (Fig. 1.)
8 THE SOLAR CORONA.
N
The equation of the lines of force is (cos 6 — cos 6') = 1 — cos u = 2 uM.
where 6 and are the angdes between r and Ti and the positive direction of the
axis, and u is the angle between the tangent to the line of force at the given point
and the polar axis. If ß and /î' are the angles that this tangent makes with
r and n then ^ and the force itself is F = M 1 + ^'4- 1
If we suppose these potentials to be distributed over the surrounding hemi-
spheres by the law of cosines, the surface density at any point is (j = ; and if
these two masses instead of being distributed are brought infinitely near together
at the centre of the sphere, so that 0 = 0' and r = r^, then the equation of an
equipotential surface is y = M . > and the equation of a line of force is
-vr Q // T\/r siu ^ d /-j -V N . 2 // siu"^ 0
JN == 2 // M . QY (1 — COS u)—' ~
/y» ^ ^3
where N is the order of the line taken, being proportional to the square root of
natural numbers. (Fig. 2.)
THE SOLAR CORONA
9
To obtain the interior force, Fj, let r cos 0 = 1 in the expression for Hi ;
4
hence r¡ = — fl C, being directed opposite to the positive direction.
4
The moment, M = + 11 C . r cos 0
4
The interior potential, Hi = + -g- H C . r cos 0
rr^i . , ... 4 ^ "T* Q COS 0
i he exterior potential, Vi = + -g- IT C . R® .
Resolving the exterior potential tangentially and normally to the circle
whose radius is r :
4 SIH 0
The tangential component, F^ = + 11 C . R® . ^ •
The normal component, F,„ = + -g- 11 C . R'' .
Also resolving along the polar and equator axes ;
The polar component, Fp = — 4" n C . R® —^3°^ — •
mi . , T-i , 4 „ ^ -1-.0 3 sin 0 cos 0
i he equator component, F^ = + qy II C . R®
The whole mass of potential is m = H R^ . C, and the total flow of force, or
the quantity Q, = (4 n R^) 11 C.
Now construct a diagram convenient for our purpose, in reference to the
corona, representing lines of equipotential and of force.
4 4 cos 0
In H] = g n C . r cos 0, and Vi =-g- 11 C R® ^2 , we may regard the con-
COS 0
stants as unity ; hence Hi = r cos 0 and Vj = In the interior of the sphere
the equipotential lines are parallel to the equator. Draw a spherical meridian
and divide the vertical radius into ten equal parts, each of which will represent
an equal diminution of potential in passing from the maximum at the poles to
/cos 0 •
zero at the equator. Outside the sphere compute r= Conveniently we
assume for Vi the successive values 1.0, .9, .8, .7, .6, 0. and for cos 0 the
same in succession. A double-entry table will give us the values of r at the
angles 0 corresponding to the potential Vj. *
Plotting points on radii extended through the angles of equal différence of
cosines and connecting all points for same V^, we have a diagram of ovals sur¬
rounding the poles becoming tangent to the equator at the centime of the sphere.
(A table of Equipotential Surfaces is given on the following page.)
2
o
EQUIPOÏENTIAL SURFACES.
COS 0, log. cos 0.
0
1.00
0.00000
0° 0^
.99
9.99564
8 7
.98
9.99123
11 29
.96
9.98227
16 16
.94
9.97313
19 57
.92
9.96379
23 5
.90
9.95424
25 51
.85
9.92942
31 47
.80
9.90309
36 52
.75
9.87506
41 25
.70
9.84510
45 34
.60
9.77815
53 8
.50
9.69897
60 0
.40
9.60206
66 25
.30
9.47712
72 33
.20
9.30103
78 28
.10
9.00000
84 16
.00
90° 0^
log.V. V.
0.00000 1.00
9.95424 .90
9.90309 .80
9.84510 .70
9.77815 .60
9.69897 .50
9.60206 .40
9.47712 .30
9.30103 .20
9.00000 .10
0° 0^
GO
O
j
11°29^16°16^
!
19°57^
23° 5^
25°51^
31°47/
36° 52^
41°25^
45°34^
V
00
o
CO
lO
60° 0^
66°25/72°33^
;
V
00
(M
0
00
84°16/
90°.0/
1
1
1
1
1.000
:
i
1.054
1.049
1.044
1.033
1.022
1.011
1.000
1.118
1.112
1.107
1.095
1.084
1.072
1.061
1.031
1.000
1.195
1.189
1.183
1.171
1.159
1.146
1.134
1.102
1.069
1.035
1.000
1.291
1.285
1.277
1.265
1.252
1.239
1.225
1.190
1.155
1.118
1.080
1.000
1.414
1.407
1.400
1.386
1.371
1.357
1.342
1.304
1.265
1.222
1.183
1.095
1.000
1.581
1.573
1.565
1.549
1.533
1.517
1.500
1.458
1.414
1.369
1.323
1.225
1.118
1.000
1.826
1.817
1.807
1.789
1.770
1.751
1.732
1.683
1 633
1.581
1.528
1.414
1.290
1.155
1.000
1
i
,
2.236
2.225
2.214
2.191
2.168
2.145
2.121
2.062
2.000
1.936
1.871
1.732
1.581
1.414
1.225
1.000
3.162
3.146
3.130
3.098
3.066
3.033
3.000
2.916^
2.828
2.739
2.646
2.450
2.236
2.000
1.732
1.414
1.000 i
1
r
Icos 6
\'~V7-
H
w
O
w
o
o
td
b
THE SOLAE CORONA.
11
The lines of force are constructed by the equation:
N = 2 lie R'
or, calling the constant 11 R"' C unity,
sin^ 0 . S II
N =
r ó
The successive integral numbers mav be given
CJ mj O
to
N, 0. 1
. 2.
corresponding values of 0 computed. They are :
11
o
II
o
II
0
o
II
1 20° 12^7
5
o
o
to
35.0
2 29 15.0
6
57
48.6
3 36 4.5
7
66
4.7
4 43 42.5
8
77
44.7
when we assume in the formula that r = 1. These give us the points at which
the lines of force of integral orders depart from the surface of the sphere. But
more conveniently for our purposes we may assign values to the angle 6, such
that the cosine of the successive angles differ by one-tenth radius, and compute
the values of N under this case :
If 0 = 25° 51' N = 1.593 If 0 = 72 38 N = 7.624
36 52 3.016 78 28 8.043
45 34 4.272 84 15 8.294
53 8 5.362 87 8 8.357
60 0 6.283 90 0 8.878
66 25 7.037
3 r N
To trace out the path of a line of force of any order N, take sin^ 0
8 U
assume the required hi, assign successive values to r at convenient distances, and
compute 0 ; e. g. :
If N = 1.593 and r = 1 0=25.51 If N = 1.593 and r = 4 0=60.42
2 38. 4 5 77.10
3 49. 3 6
or assign values to 0 and compute r.
(A table of Lines of Force is given on the following page.)
To find where the lines of any order N cut the equator axis, take
3rN 1 8 //
_ J or 1'
8 // ~ 3 N
assign the values to N and compute r.
11= 1.539 frrr 5.248 7,624 r = 1.099
3.016 2.779 8.043 1.042
4.272 1.961 8.294 1.010
5.362 1.562 8.357 1.002
6.283 1.333 8.378 1.000
7.037 1.191
LINES OF FORCE.
8-
sin^ 0. ^ = T ■ sí"'
N.
0.000
8.29960
9.22272
0.167
8.59806
9.52118
0.332
8.89466
9.81778
0.657
9.06602
9.98914
0.975
9.18672
0.10984
1.288
9.27900
0.20212
1.593
9.44314
0.36626
2.324
9.55624
0.47936
3.015
9.64110
0.56422
3.666
9.70748
0.63060
4.272
9.80622
0.72934
5 362
9.87506
0.79818
6.283
9.92424
0.84736
7.037
9.95908
0.88220
7.624
9.98228
0.90540
8.043
9.99564
0.91876
8.294
0.00000
0.92312
8.378
0° 0^
1.000
80 7/
CO
1.000
11°29^
16°16^
CO
CO
1.988
3.936
1.000
1.980
1.000
19°57^
CO
23° 5^25°51^ 31°47^
CO
2.937
¡.878
.J 1.000
1.320
1.000
CO
9.537
36° 52^
CO
CO
4.797 Í 7.000
I
2.423 ¡ 3.536
1.633 ' 2 383
13.916 ! 18.057
9.082
4.588
3.092
41°25^
1.237
1.805
1.000 1.459
1.000
2.342
1.893
1.298
1.000
CO
21.953
11.042
45° 34^
53° 8^
CO
CO
25.579 32.10:
12 865
J6.149
60° 0^
66° 25^ 72° 33^
CO
37.624
5.577
6.499
8.157
3.759
4.380
5.498
2.847
3.317
4.164
2.302
2.682
3.367
1.577
1.838
2.307
1.216
1.417
1.778
1.000
1.165
1.463
1.000
1.255
1.000
CO
42.135
21.192
18 924 ^
' 1
7.215
2.703
2.084 2.333
I
1.714 1.919
1.000 1.120
1.000
5.464
4.418
3.028
2.528
2.080
1.471 ¡ 1.647 1.785
1.312
CO
78° 28^
CO
45.654 48.159
22.963 24.222
11.599
7.817
5.934
12.235
8.246
6.245
4.787 j 5.050
3.280 I 3.460
2.667
2.194
1.883
1.422 1.500
1.214 1.280
1.083 1.143
1.000 1.055
1.000
84° 16^
90° 0^
CO
CO
49.663
50.165
24.980
25.231
12 618
12.745
8.504
8.590
6.440
6.505
5.208
5.260
3.569
3.605
2.750
2.778
2.262
2.285
1.941
1.961
1.547
1.562
1.320
1.333
1.179
1.191
1.088
1.099
1.031
1.042
1.000
1.010
1.000
8-
N = -o- sin ^ 0.
o
r
Stt
r>
O
sin' 0
log. ^ 0.92312.
THE SOLAR COROFA«
13
To find the order of line of force at the earth's inean distance from the sun^
take the mean semi-diameter of the siin^ 962'', and the mean parallax of the
earth, and the earth is 108.7 radii of the sun distant from it. For
T == 109, K 0.07686. The angular distance from the pole at which this line of
force leaves the sun is 6° 29' 47".
Graphically the lines of force cut the epuipotential lines orthogonally, and
may be so drawn, starting at the points of the surface heretofore marked by
the equipotentials. These lines are ovals cutting the equator perpendicularl}^ and
becoming tangent to the polar axis at the centre of the sphere. A test of the
accuracy of the drawing is found by taking the sides of any of the quadrilateral
figures, wherein the ratio of the mean distance between consecutive equipotential
surfaces is to the mean distance betv/een consecutive lines of force as the half
the distance of the centre of the figure from the polar axis is to the unit of
measure.
APPLICATION TO THE COEONA.
An analysis of these lines of force appears to be a description of the visible
solar corona, and this analogy first suggested the explanation of the phenomena
now given. The concentration of potential at each pole throws lines vertical at
the polar region, bending gradually each side, and at a distance of 26° losing
oneAenth of the force, — the angle of the line of force to the polar axis bring
nearly 45° ; this curve closes on the equa.tor at 5.25 radii from the centre. The
next decimal line leaves the sphere at an angle of 67° to the vertical axis, and
haviiig a potential of eightAenths closes on the equator at 2.8 radii. The third
line of force is inclined at 76° to the axis, and having potential sevenAunths closes
on equator at 1.96 radii. The fourth line starts perpendicular to the vertical axis,
leaves the sun at polar distance 53°, closing on equator at 1.66 radii. The other
lines rapidly become more nearly parallel with the surface and close in as they
X (L/' ey X ^
lose potential.
The solar corona can now be analyzed. The straight polar rays of high
tension carry the lightest substances, as hydrogen, meteoric matter, débris of
comets, and other coronal m^aterial, away from the sun, and they become soon
invisible by dispersion. INext we come to the strong quadrilateral rays of poten-
tial .9 .8 .7 .6, which united form the appendages conspicuously seen at periods
of great solar activity. They rapidl}^ diminish in intensity, and at the distance
of one radius have generally a potential of one to two tenths. The explanation
of the long equatorial wings, wi(h absence of well-marked quadrilaterals, seen at
periods of minimum activity, is that they are due to the closing of the lines of
force about the equator. The re-entrance of these lines forms along the equator,
14
THE SOLAR CORONA,
the place of zero potential, a sort of pocket or receptacle wherein the coronal
matter is gradually carried hy the forces, accumulated and retained as a solar
accompaniment. During periods of inactivity or low maximum potential the
streams along the region 40° — 60° polar distance diminish in intensity, so that
huge volumes are not carried away from the surface, but none the less what does
leave the sun is persistently transported to the equatorial plane of the corona.
In fact, the zodiacal light may be the accumulation at great distances from the
sun along this equator of such like material, being carried by forces, all of
which approach the equator perpendicularly, but there become zero. Here the
zodiacal coronal material has no way of escape, being once deposited.
We have a test of the accuracy of our theory which may be applied to
any portion of the coronal rays, using the caution that we deal with true rays
undisturbed by perspective and diffraction, and notably the polar and the outer
boundaries of the quadrilaterals are best available. From the centre of the sun,
with a radius vector r, draw a circle at any chosen point of such ray where the
curvature is well marked, and a tangent to the circle, prolonged to intercept the
polar axis with which it makes an angle. (See Fig. 2.) Let/" equal the angle at
which the line of force crosses this tangent ; draw another tangent to the line of
force and prolong it to the polar axis, then tan / = 2 cot 6 = 3 tan k. The inter¬
cept cut off from the centre of the sun by the force-tangent is one-third the inter¬
cept cut of by the circle-tangent. I believe that this criterion holds good on
the photographs taken during recent eclipses, as the following readings show :
(A table of Readings is given on the opposite page.)
These readings were taken from Professor Holden's diagram (Monthly
Notices R. A. S., April, 1889) by centering one side of a right triangle on the
sun with radius 2 rotating to the.several angles 6, previously selected to mark
the prominent rays, and reading the other side of the triangle on the axis ex¬
tended and marked on a scale with the radius as unit. Finallv an edge was laid
on the local lines at (r6), and the reading on the axis again taken. It must be
clearly kept in mind that it is not the direction of the whole ray from its base on
the sun to the point (r 6), but the direction tangent to the ray at the point (r 6).
Besides the local inaccuracies there may be a slight error in placing the direc¬
tion of the axes, and the readings in the S. E. quadrant suggest this presump¬
tion. Still the approximation of the ratio to 3.0 is so evident as to show the
application of this theory to the solar corona and also to witness the fidelity of
Professor Holden's drawing'.
I have just had the pleasure of seeing one of the photographs of the inner
corona and one of the outer taken by the Harvard College party January 1,
1889, and the details are shown so clearly that our theory is at once able to be
THE SOLAR CORONA,
15
READINGS ON HOLDEN'S DIAGRAM,
Kadins ¡Anille |
r
2
o W, Quadrant.
N. E, Quadrant
S, W. Quadrant-
S. E. Quadrant.
Intercepts by—
CircIe-Tano-ent. ' Eorce-Tano-ent,
Eatio.
5°
2.01
.65
3.09
■
16
2.07
.66
3.14
:
22
2.15
.71
1 3.03
i
29
2.26
.50
4.51
35
2.42
.59 :
:
4.10
47
2.83
.85
3 32
3.20
3°
2.00
.65
GO
O
CO
10
2.03
72
2.82
16
2.07
73
2.83
18
2.09
.72
2.90
23
2.13
.75
2.83
28
2.26
.74
3.01
36
2.45
. / /
3.18
53
3.32
1.07
8.10
64
4.62
1 35
3.45
72
6.70
1.20
i
!
5 58
2.97
3°
2.00
.50
4.00
6
2.02
.55
3.67 '
11
2.04
.58
3.53 ■
19
2.11
.70
3.01
41
2.62
.84
3 12
50
3.07
J.02
3.01
56
3.60
1.21
2.97
3 03
6°
2.02
.61
CO
CO
12
2.04
.57
3,52
19
2.10
.60
3.50
41
2.59
.72
3.59
51
3.08
.90
3.42
56
3.60
1 20
3.00
3 39
Local curvature too strais^bt« i
O {
Mean,
Local curvature faulty.
Mean»
Local lines in error
Mean.
Whole Quadrant may need
adjustment as to its axis.
i Mean
The necessity is obvious of rejecting freely such lines of force as are not naturalj and the difficulty of obtaining
true lines is at present great.
16
THE SOLAR CORONA.
tested. I give a table of the measures when our rule of polar intercepts is ap¬
plied to the ray structure. They can be verified by any one possessing a Pickering
photograph on celluloid.
THE PICKERING PHOTOGRAPHS.
THE INNER CORONA.
m
c3
P5
o
C) OJJ
G
I
Ph-Y
CD <V
CD Ö
-hJ CD
G ^
M
>- I
S bfi
C
CD CD
Ü CD .
ÍTÍ ^
S P ^
"tí ^ CD
Pi
-l-i
PH
CD
Ü
CD
P
o3
Pi
1.00 5°
10°
15°
20°
25°
30°
85°
40°
45°
50°
55°
60°
1.00
1.01
1.08
1.05
1.10
1.16
1.22
1.81
1.42
1.56
1.74
2.00
N. E. Quadrant.
N. W. Quadrant.
0.88
0.84
0.88
0.40
0.89
0.42
0.42
0.46
0.48
0.58
0.58
0.68
8.08
2.97
2.70
2.68
2.84
2 76
2.90
2.85
2.96
2.94
8.00
2.94
2.88
0.88
0.84
0.88
0 71 :
I
0.67
0.58 i
0.54 I
0 51 !
0.50
0.56
0.58
8.08
2.97
2.70
1.50
1.64
2.00
2.26
2.57
2.84
2.79
8 00
2.89
Ph
o;
O
a>
P
i—i
o3
Pi
S. W. Quadrant.
0.31
0.88
0.44
0.49
0.51
0.5Ö
0.52
0.48
0.49
0.51
0.58
8.02
8.01
2.84
2.14
2.16
2.82
2.85
2.78
2.89
8.01
8.00
2 94
-(-3
Ph
a>
Ü
u
(d
-H-3
c3
Pi
S. E. Quadrant.
0.82
0.84
0.40
0.45
0.42
0.42
0.48
0.45
0.48
0 52
0.56
THE OUTEE COKONA.
5°
1.21
0.40
8.00
10°
1.22
0.42
2.90
15°
1.24
0.48
2.87
20°
1.28
0.44
2.90
25°
1.82
0.45
2.98
80°
1.89
0.47
2.96
85°
1.47
0.50
2.94
0
0
1.57
0.53
2.96
45°
1.70
0.56
8 00
50°
1.87
i 0.60
8.01
55°
2.09
■ 0.67
3.01
1
2.95
8.01
2.97
2.58
2.88
2.62
2.79
2.84
2.91
2.96
8.00
3.01
2.94
0.41
2.95
0.48
2.81
0.41
2.95
0.42
2.90
0.67
1.97
0.42
2.90
0.44
2.84
0.62
2.00
0.42
2.95
0.63
2.00
0.68
2.00
0.45
2.84
0.64
2.01
0.58
2.28
0.66
2.00
0.65 :
2.29
0.54
2.89
0.68 ;
2.00
0.68
2.33
0.55
2.67
0.68
2.88
0.54
2.91
0.65
2 41
0.57
2.98
0.60
2.83
—
0.61
■
;
8.07
-
2.90
2.94
2.89
Bracketed intercepts omitted in taking means ; 55 readings retained ; 27 show distortion.
Note.—It is evident that the effect of projection of the lines of spherical harmonics on a plane is to flatten them, so that the
force-tangent becomes elevated at its intercept on the polar axis. Hence the readings of this factor are too large, and the value of
the ratio too small, by an amount depending upon the error of the curves in projection.
THE SOLAR CORONA, 17
A scale was constructed as follows to facilitate the measurement of the lines on
the photographs : A positive on glass, showing fine lines on a transparent field, was
made from a drawing, which consists of concentric circles, the first coinciding with
the sun's disk, the others expanding by tenths of a radius to the distance of three
radii ; also a series of radii at five degrees aj)art. The polar axis was subdivided and
marked in figures, and the radii were numbered. This was reduced to the size
of the picture to be discussed, and the celluloid photograph being laid against
the scale and backed by a plate of glass formed a transparency which, viewed
against bright sky light, rendered the direction of the rays vexy distinct. (See
Plate I.) The circle-tangent iixtercept readings wei'e taken from a table of secants ;
the foi'ce-tan gent intei'cepts xvei*e l'eacl fi'om the pictui'e by laying an ivoiy scale on
the i*ay in question. Soixxe practice and judgment were I'equii'ed to distinguish
true and false directions, but considerable uniforixiit}^ was acquired in the way
of independent measui-es.
An inspection of the table for the inner and the outer c(xi'ona shows a decided
determination of the constant ratio 3.00. In the N. E. quadrant for both coronas
there is hardly a divei'geixce from it ; in the N. W. quadrant between 20° and
40° thei'e is a shai'p change iixdicating some distui'bance at this place ; the S. W.
quadrant shows a similar confusion ; and the S. E. is again quite regular. Some
solar currents seem to have swept the poles and the rays on the western side of
the sun.
It may be ixientioned that these readings are for iixdividual pictui'es and
with poles selected by best judgment. A compaifison of many pictui'es taken at
différent times and with various kinds of apparatus under the assuixxption that
our constant 3.00 is a fundaixientxil ratio may lead to valuable deductions as to
ft/
the coronal foi'ces. In this connection the solar prominences and the fibres of
the chx'omosphei'e should be coixipai'ed with the dmection of the lines of foi-ce as
they leave the solar surface.
It is hopexl that future eclipses may furnish us with pictures of the corona
so clear that the measures may be made with certainty.
It is plain that the accuracy of the results depends upon our ability to locate
the polar axes. The genei^al radiation at the poles shows the appi*oximate posi¬
tion, and the radial ray is probably near the vertex, but if our rule is granted as
trixe for the corona it becomes a means of fixing the pole precisely, referred to
the whole structui'e of a hemisphere, ivxther than leaving us to depend upon ap¬
pearances of rays, xvhich pi'obably undergo a certain amount of local Amriation.
The assumption regarding the poles of the corona has usually been that they
are in a diametr-al line passing tlirough the centime of the sun. Upon applyiixg*
the pxfinciple just stated to the southern vertex, at first assuming that it lay on
o
O
18 THE SOLAR CORONA.
SOLAR CORONA.—Bigelow.
Plate I.
Diagram of Scale for
Intercept Readings.
THE SOLAR CORONA. 19
the same diameter as the northern, I found that my intercept ratios were untrue.
However, on taking the vertex in the southwestern quadrant at 169° from the
northern, the readings were rectified.
We have avoided speaking of the apparent coronal structure as a phenom¬
enon of electricity in deference to the doubt that free electricity can exist at such
high temperatures as prevail on the sun's surface, but have shown that some
force is jaresent acting ujDon the corona according to the laws of electric potential.
An inverse argument might at once be drawn from this applicability of the for¬
mulae of statical electricity to the coronal structure that a form of energy analo¬
gous to electricity exists on the surface of the sun, but we need not insist upon
the name of the active repulsive force whose potential we are discussing.
The value of the potential at any point of a line of force can be easily com¬
puted, but a diagram plotted in tenths-potentials renders the work very simple.
Referring again to the Holden drawing, and for the present calling C equal to
unity, we may estimate the value of the potential at the edges of the corona as
recorded by the photograph.
At the north pole the rays extend to potential .... 0.35
At the south pole the rays extend to potential .... 0.50
At the northwest quadrilateral the rays extend to potential . 0.15
At the southwest quadrilateral the rays extend to potential . 0.10
At the northeast quadrilateral the rays extend to potential . 0.20
At the southeast quadrilateral the rays extend to potential . 0.15
Remembering that the smaller the potential the greater the distance seen
from the edge of the sun, we note that the western quadrilaterals are generally
longer, extending about one and one-third radii from the sun. They are sym¬
metrically disposed to the poles assigned by our formula, the axis of symmetry
lying 85° from the north to the west. The northeast quadrilateral is shorter for
the same reason, and the tendency is to make the larger amount of matter visible
in the 169° side of the axis. The diminution of matter along the axis of sym¬
metry is very obvious. At certain parts of the quadrilateral the curvature of
the rays is marked and in the right direction.
I had the diagram transferred to a transparent positive photograph scale
reduced to the solar diameter, upon which the mounting was made, as above
described, for the measure. It was seen at a glance, by counting the value of the
lines, to what potential the matter attaching to any line of force was visible. On
the same scale were produced the lines of force at the decimal potentials, and an
inspection of the curvature of the computed lines and the coronal lines, when
superposed, is sufficient to substantiate the truth of the theory. (See Plate II.)
It is seen also that a field of accurate and intelligent study of the solar forces
is now opened, and that the coronal pictures which show true structure become
valuable.
20 THE SOLAK COEONA.
SOLAE CORONA.—Bigelöw.
Plate II.
Diagram of Lines of Force.
(Upper part.)
Diagram of Equipotential Surfaces.
(Lower part.)
THE SOLAR CORONA.
21
TERRESTRIAL MAGNETISM.
In treating the problem of the earth's magnetism it has been generally sup¬
posed that the forces of induction from the sun to the earth are in straight lines
following the vector joining these bodies. We now see that the earth lies in a mag¬
netic field, uniform by reason of its distance from the sun, the lines of force
being directed nearly perpendicular to the plane of the earth's orbit instead of
parallel to it, and of low potential, as the formula shows that the earth lies near
the plane of the equator of the corona. It is not yet known exactly what rela¬
tion the polar axis of the corona holds to the axis of revolution of the sun, or to
the true N. and S., but it may be determined by a study of the coronal lines.
If it should appear that the angle is considerable between the plane of the coronal
equator and the ecliptic, even supposing the corona does not oscillate, yet the
earth in its orbit must be passing through fields variable in potential and direc¬
tion, which will condition some of the periodic changes of the terrestrial mag.
netism. Knowing the potential of the earth's magnetism and its variations the
data ought to be accessible for obtaining the solar constant of maximum super¬
ficial density of electricity, and thus give a clue to the forces acting within the
sun.
With the data at present available it is diificult to assign the position of the
coronal pole to its true place on the solar surface, and the pictures heretofore ob¬
tained, which deal almost exclusively with the general outline of the corona instead
of with the direction of the rifts and structural lines, afford little ground for
deduction from symmetrical forms. If we suppose the poles of the ecliptic, the
sun, and the corona to be in the plane of vision the relative places are from
pole of ecliptic to the pole of the sun and probably 12°.5 to the pole of the
corona. One rotation of the sun on its axis will then cause the coronal equator
to range from about 13° north to 0° on the ecliptic. If Vi is the maximum po¬
tential at the pole of the corona on the surface of the sun the corresponding-
potentials at the mean distance of the earth, 108 solar radii, are:
r = 108, e = 75°, Vk
80°
85°
90°
If the corona follows the rotation of the pole the effect is to draw the lines
of force up and down, north and south, in the region of the earth, so that it lies
= 0.0000222 — 45 000 •
67.000 •
V,
133.000-
0.0000149
0.0000075
0.0000000
00
22
THE SOLAR CORONA.
in potentials continuonsly changing from zero to qqa in a period of 26.33 days.
\
At the same time the earth's orbital motion causes it to cut them into forces of
induction, which would tend to make variations in the earth's magnetism. It is
obvious that the general view is sustained that the direct magnetic influences
from the sun are very slight, yet Hornstein's period should show variations con-
flrming these coronal changes, and if the final residual of the earth's magnetic
variations can be completely assorted we should have from the coronal period a
means of fixing the polar positions, and also the ratio of the solar potential to
that of the earth's magnetism.
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