Frank H. (Frank Hagar) Bigelow.

# A treatise on the sun's radiation and other solar phenomena, in continuation of the meteorological treatise on atmospheric circulation and radiation, 1915 online

. (page 27 of 29)
Font size 6.548 X lO-^y, m = 8.845 X 10-Â», e = 4.774 X 10-Â«> E. S. U, or
1.5913 X lO-Â«> ÂŁ. M. U. Thence, it follows that,

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(343) v^ ^_ (___),

T being an integer characteristic for the orbits, 1, 2, 3, 4 . . .
For T = 1, the inner orbit gives the convergence frequency.

The Series of Spectral Lines

The frequencies of the series of spectral lines can be expressed
by Balmer's Formula,

(344) V = .0 (l - ^,)

where m takes successive integral values, or by Bohr's Formula,

(345) , = ^__ (___),

where t2 and n are whole niunbers.

m = mass, e = charge, h = potential of the electron, such that

(346) â€” ^^ â€” = 3.235 X lO^S for ÂŁ = e.

Since the curves on Fig. 32 are fimctions of T and h, both
being variable, we may note the following fact: If the h â€”
ordinates be drawn for equal Tâ€” intervals, as Ti'-To=A degrees,
they are spaced in positions very similar to those of the- above
formulas. It is supposed that this is in harmony with the
computations which depend upon h and T, as they are related
to the thermodynamic conditions of the gas. In this case the
spacing depends upon the temperature of the live emissions in
succession, and they become a method of determining the
temperatures of the solar gases at different depths. Further-
more, variations in the positions of the lines of a series, as
hydrogen, indicate temporary changes in the local temperatures
of solar emission. The positions of the lines, as measured in
the spectrum, may be used inversely to establish the position
and the correct curve of the (A . T) function.

It is evident that A is a variable, and we shall show that the
spectrum lines can be produced by the vibrations imposed upon
the electronic orbits by reaction from the external collisions of the
atoms and molecules.

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352 A TREATISE ON THE SUN'S RADIATION

Derivation of 'the Orbital Formulas

Assume the following relations:
V = the orbital frequency = the vibration frequency.

e
F = â€” = the potential at the distance r.

9 F e

(347) ÂŁ = â€” TTâ€” = - = the central force.

dv r^

(348) W = â€” = F c = the work done between charges E, e.

2t c

(349) Â« = â€” = 2ti'=2t - - = the angular velocity.

(350) ^ = â€” = â€” = â€” = the periodic time.

(a V c

(351) ,.-y- â€” - =^-^-â€” .

X 2 T 2tc , , , , .

(352) ^ = w / = -â€” ./ = 2tj'./=â€” â€” ./,fortheelapsedtime/.

d A

Â« ÂŁ

(353) V = = , if the central charge is ÂŁ = n e.

ra ma Â°

From the orbital velocity v, the constant kinetic energy in the
orbit y2 m u^, and the central acceleration/, have the following
formulas:

c 2t^

(354) Velocity. z; = 2Ta.j' = aw = 2Ta

(355) Kinetic Energy. \mj^^hv^V e^

' X rh

Ee e e

ma ra

g (zi - Zo)

Wio(t^i â€” Wo)io

/or:^^ A 1 ^- ^i-^ ^^ ^Aj' 2F^ 2Ee
(356; Acceleration to Center. =

a a a m a^

2ee
ra^

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BIGELOW'S AND BOHB*S FREQUENCY 353

T A^

(357) Radius of Orbit, a = ;r-T 1. Log. 2 ^2 ^ e^ =

- 45.59979.

(358) Velocity. v = ^^. Log. 2^^^ = - 18.15594.

(359) Frequency. v= ^ " 2^2^e* = - 64.95755.

4 ?r^ f^ ^

(360) Angular Velocity, w = ^ ^ . Log. 4 x^ w e* =

- 63.75573.

2x2 w^

(361) Potential. F= ^^, . Log. 2T2weÂ» = - 54.27867.

(362) Nuclear charge. ÂŁ = â€”

(363) Kinetic Energy, ^mv^ = â€” ^-75â€”. Log. 2T^me* =

- 64.95755.
(364; Acceleration to Center. = â€” ^-tt â€” .

Log. 8 7r*w2 6Â« = - 108.85837.

These can be used in computing the elements in the Jf -series,
L-series, and the other constituents of the spectnun.

From the theory of electronic orbits we have directly the

centrifugal force = imv^ = hv = â€” , where E is the charge

on the positive nucleus, and n the nimiber of electrons; hydro-
gen n = 2. For the velocity in the orbit, z^ = 2 x a. y, where
the angular frequency and the nimiber of vibrations v are assimied
to be the same. Hence, we have,

(365) Frequency. " = -gX = aT^

h

niTV

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354

A TREATISE ON THE SUN S RADIATION

The central force of acceleration is,

^ mv^ 2Ee Ee , ^ ^
(367) / = â€” = ;j^ = â€” for hydrogen, n = 2.

The charge on the central nucleus of positive electricity is,
Charge (+),ÂŁ= = . = â€” = e. Hence,

2hv
m

(368) v^ = ^4r" = ^- Thence,

(369) Frequency, v = -y^ = â€” J^\â€”t)y hydrogen.
We compute the two formulas for v and X = â€” :

^ g (ft - a) 1^ J.
(n â€” m)yÂ» ' niÂ» ' h
Bigdow

Log.f(2i-so) 13.13805

Log. (viâ€” Â»b)io
Log. Â«io
Log. h

Log. if,â€ž
Log. c

5.15619
17.11295
-25.96720

- 2.23634

14.90171
10.47712

Log. \n - 5.57541

Wave length

{cm)\n 0.00003760

2*^ me* / 1 \

Bohr

Log. X*
Log. w
Log.<5*

Log. 2
Log. AÂ«

Log. V
Log. c

Log. X

X

0.99430
-28.94670
-38.71552

- 64.65652

0.30103
-79.44770

-79.74873

14.90779
10.47712

- 5.56933

0.00003710

Lyman

as observed,

also

Millikan

Science

April 6,

1917

0.00003650

The Bigelow formula seems to be equivalent to the Bohr
formula for the case of the hydrogen series, since the convergence
wave length is practically the same. It is evident that since in
the Bigelow formula the A is an external potential, while in the
Bohr formula A is a fimction of the kinetic energy,

Digitized by

moseley's law 355

it will be necessary to examine the formulas in their other

relations.

Our thermodynamic data refer in all cases to the maximiun

.2891
Pmy Xm, in the Wien Displacement Law, Xm = -yi â€” , so that

these results are practically identical. It should be noted that
the Bigelow form uses the hm of the external potential, while
Bohr uses h = the constant or adiabatic value. We may,
therefore, conclude that the Bohr computation refers to the
adiabatic case, generally, and the Bigelow computation to
the nonadiabatic case near the bottom of the solar reversing
layer.

Moseley's equation for the relation between the frequency
and the atomic niunbers,

(370)' vn==A(N- by OT vn= {a + b N)^

or Uhler's more accurate equation, which is hyperbolic,

(371) vn = A+BN + j)^j^ {Physical Review. April, 1917),

shows that the ultimate relations between the atomic nimibers
in the Kay K^y La, L^, Ly, types of the X-ray series are really
very complicated. It remains to be seen what form further
experiments will assign to these structural relations. We shall,
therefore, reserve this subject for further examination, though
it has seemed proper in this chapter to indicate some of the
interesting developments which come from the non-adiabatic
thermodynamics.

Sanford applies the Bohr theory to the K and L series with
much success in Physical Review, May, 1917. These subjects
will be resinned in a further publication on the structure of
matter,

Moseley^s Law

Moseley foimd an important relation between the atomic
numbers of the elements in certain series, by assuming that the

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356 A TREATISE ON THE SUN's RADIATION

addition of the b unit charges, 6 . e, to the nucleus suffices to pass
from one element N to the next in order N+b, For r = 1
(convergence).

(372) / = â€” = â€” .4TÂ«a2i^=4TÂ»wai^

/Â«Â«Â«x r 2ÂŁie Eie ^^ Etc , ^ ,

^^^^^ ^' " IT^ ""^ ^"^^^ " lij' ^""'^ ^^^' ^^"^""^^

(374) /iai* = ÂŁie = 4T*wpiÂ«ainÂŁj ^ 1^ Oi^

(375) /,a2Â« = ÂŁ,e = 4T*wir2Â«a,Â»jÂŁi"" I'l^'fli*

Moseley finds that the frequency is proportional to the
square of the nuclear charge.

(376) â€” = ^r; = Moseley's Law. Hence,
V2 -or

(377) f^ = f^, . ^ and ÂŁiÂ» aiÂ» = ÂŁ2* fli*, so that

(378) (Millikan) Ei Ei = a^ E,.

This is derived from the central acceleration. On the other
hand, from equation (330), we have, by assuming that the kinetic
energy in the orbit is a constant, a different relation.

(379) (Bigelow) ai Ej = as Ei

There are, therefore, two distinct solutions by (346) and
(347), the former depending upon assuming that the frequency
in the orbits of the electrons is the same as the frequency in the
spectral lines; the latter assimies that the kinetic energy in the
orbit is a constant for all electrons, the radial distance conform-
ing thereto. The former assimies that h is constant, and the
latter that A is a variable; the former supposes that the funda-
mental relations are internal to the atom, the latter that the
primary relations are external to the atom, and that the varia-
tions in the position of the lines of the spectrum are due to
perturbations impressed by interpenetration of the active orbits
of the electric charges during the collisions. The Bohr theory
makes the variation of energy which causes radiation to depend

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EVALUATION OF SERIES FACTORS

357

TABLE 112

Evaluation of ( â€”z â€” â€” : ) in v = =- ( â€” ; 1

1 1

006^

*

if[m>

Tl= 1

Tl = 2

0.750

0.760

4r

25'

\^

3

0.889

0.889

27

16

X.

4

0.938

0.938

20

17

X

5

0.960

0.960

16

16

!

\^

6

0.972

0.972

13

36

\

7

0.980

0.980

11

16

\

8

0.984

0.984

10

16

5"

\

9

0.988

0.988

8

52

Â§

\

10

0.990

0.990

8

8

\

11

0.992

0.992

7

16

\

Conver

gencei'

1.000

\

11 7 5 4 3 2 ^

r"^iTT>

s-

Tl = 2

Tl = 3

0.139

0.666

56'

12'

^

Tin

4

0.188

0.752

41

13

X

5

0.210

0.840

32

51

be]

\

6

0.222

0.888

27

23

?

>s^

7

0.230

0.920

23

5

p

V

8

0.234

0.986

20

46

o

\

9

0.238

0.962

17

50

â€˘^

\

10

0.240

0.960

16

7

\

11

0.242

0.968

14

31

\

Conver

gence V

0.250

11 8

est 3 â€˘Â«

p-^

TlÂ»3

TÂ«-4

0.049

0.441

65*

50'

|T^

\

6

0.071

0.640

60

8

X

V

6

0.083

0.748

41

36

1^

\

7

0.091

0.820

34

28

n

\

8

0.096

0.866

30

8

\

9

0.099

0.892

26

52

^

10

0.101

0.910

24

30

\

11

0.103

0.928

21

52

\

Conver

gence v

0.111

]

11 9 7 6 5 \ *p

o

Fig. 38. Orbital Distributions.

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358 A TREATISE ON THE SUN's RADIATION

upon the electrons jumping from one stable non-radiating orbit
to another, but this would make the structure of atoms wholly
unstable and precarious. The flat Satumian systems of orbits
should probably be superseded by a series of orbits which are
arranged upon the surface of a sphere, in order to produce the
polarizations and magnetizations that exist in molecules.

The evaluation of the term f â€” jjin the expression for

the frequency v gives the relative position of the spectrum lines
in different series. The fimction cos ^ is formed from successive
divisions by the convergence value, computed for t% = Â», and ip
is the angle from the equator. If the electrons should revolve
on these planes, as indicated, the frequencies would correspond
with an Amperean polarized sphere. If the several atoms occur
in different environments of A, depths, densities, temperatures,
the corresponding radiations conform to those observed in the
spectrum. This research will be continued

The Electronic Orbits in the K and L Series of Radiation Lines

for h'Variable,

We have computed the values of the several terms according
to formulas (322) to (339) for the Ka and LÂ« series, quoting
Uhler's and Sanford's wave length values X, as given in the
American Physical Review for April and May, 1917. From these
values of X the frequency v is computed, and with this v the
corresponding variable h by the formula,

(380) h=( j.

\ p /

This variable h is then applied to a, t;, F, J m t^, , E, in

succession. Finally, we have,

(381) A = â€” = a constant by Moseley's Law.

It is noted that in the K^ radiation the value of ^ is somewhat
smaller in the middle than at the ends of the series; the same is
true of the La series. In each case the divergence is small, and

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EVALUATION OF SERIES FACTORS

369

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EVALUATION OF SERIES FACTORS

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362

A TREATISE ON THE SUN S RADIATION

U proves that Mosdey's Law conforms to the variable h in, the series.
Since the potential energy h represents the resistance to the kinetic
energy in the medium, because we have assumed the kinetic
energy constant, and the central force variable from one element
to another, it is seen that the theory of orbits takes on very
much greater flexibility, and really represents the complex of
the thermodynamic conditions which produce radiation. These
vary with the depth in gases, temperature, density, pressure,

KlneUc

Fig. 39. Interpenetration of the Orbits of the Electrons in the Atoms
During Collisions.

gas efficiency, surroimding each atom as represented by n and N.
Hence, different subordinate series of the same element originate
in different environments on the otUside, while the several
chemical elements are derived from the different structural con-
figurations of the electrons on the spherical surface of the atoms.

The Interpenetration of the Electron Orbits at the Contact of

Collision

The complex systems of atoms in collision must cause inter-
penetration of the individual orbits of the electrons, and this

Digitized by

VARIABLE INTENSITY OF THE SOLAR RADIATION 363

must communicate a series of internal vibrations to the radiat-
ing particles or electric charges.

The potential energy h in the radiation fimction is related to
the kinetic energy k in much the same way that has just been
described. If ^T is a constant then Ai^is a variable, and a very
complex variable whose mean values alone can appear in the
thermodynamic formulas. The important point to recognize is
that the electrons may remain on their stationary orbits accord-
ing to the structure of the chemical elements, and acquire the
perturbations producing the radiation during the confusion of
interpenetration. This avoids the difficulty in Bohr's theory
which requires the electrons to pass froin one orbit to another
during radiation. The amperean spherical distribution of the
interacting elementary charges, E at the center, and e in the
several orbits, probably present the basis for the structural
permanences which are inherent in the chemical elements.
Fig. 39 represents the collision of two polarized atoms whose
orbits are projected on the equatorial plane.

The Electromagnetic Waves Due to the Sudden Motion and Stoppage
of an Electric Charge in Collisions

Following Heaviside's exposition of the effect of suddenly
starting or stopping the motion of an electric charge, as in col-
lisions, we have the distribution of the electric disturbance D
and the magnetic induction B, in producing the plane elec-
tromagnetic waves of Fig. 40.

Let p take on suddenly the velocity v, so that in the time
/ it reaches the polar position of the sphere +p. The outside
radial displacement changes into a current along the meridians
from the positive to the negative pole, with its magnetic induction
on the parallels in the vector sense, so that the polar field be-
comes the plane wave [D . B], as the sphere enlarges. Let p
be suddenly stopped, then the displacement reverses from the
negative to the positive pole along the meridians, with induction
on the parallels in the opposite direction, and renewed inner
radial displacement to the center. This, also, releases a plane

Digitized by LjOOQ IC

364 A TREATISE ON THE SUN's RADIATION

wave [D , B] at the positive pole. Similarly, during collisions
of two orbit-atoms, there are sudden motions and stoppages,
with reversed displacements and inductions, sending plane

Sudden Stopping. Sudden Starting.

Fig, 40. The Sudden Motion and Stoppage of an Electric Charge P,

electromagnetic waves into space. When the orbits of atoms
in collision interpenetrate, these waves are complicated in their
frequencies, in accordance with the results seen in their charac-
teristic spectrum lines.

Furthermore, during successive collisions at the end of the
free path C, there exist the potential and the kinetic energies,
which may be analyzed as potential along the free path with
h as the average value, and 2 A as the maximum value, the time

of one period being â€” second of time, while the kinetic energy

is expressed by the motion in the circle, such that kT = ^ mv^.

(382) divW=-eoJ-hoG + Q + j+H

(383) W = V {E - Co) {H ^ ho) = V EiHi =^ V + H).

All these forces, expressed and implied, are in action during
the collisions of complex atoms and molecules.

Digitized by LjOOQ IC

VARIABLE INTENSITY OF THE SOLAR RADIATION

365

The Variable Intensity of the Sun's Radiation in the 26.68-Day
Period of the Synodic Rotation

Besides showing that the intensity of the solar radiation is
variable in the ii-year period to the amount of 1% to 2%, it
appears that a similar variation occurs as the sun turns on its
axis. Fig. 41 gives the normal direct curves of the solar varia-
tion, as registered in the terrestrial magnetic field and the
meteorological elements, according to the author's papers of
1893, 1895, 1898, and the Meteorological Treatise, 1915. The
Cordoba-Pilar curve of the pyrheliometric mean intensities,
1912-1916, results in a nearly identical curve, and this proves
that the solar radiation is variable in solar longitude and aflfects
all the terrestrial elements in the 26.68-day period. Similarly,
the La Quiaca pyrheliometric data produce the small curve in the

12 8 4 6

6 7 8 9 10 Bia IS 14 1516 17 lÂ«t9 SO 21 22 28 94261627

so

10

Ha

Agnetio

^

H

'\_/

\-A"

Direct

-10

-20
4.000

vA

r

\J

\j

\j *

8.950

CordoU

Pilar

^

\:A

^r

r\

Direct

4.000

1 \j

LaQD

jusa

h

A

f]

if]

r

InverBe

4.060

H3

Xf

\J

n

'

t^

\f^

Fig. 41. â€” The variable intensity of the solar radiation in the 26.68-day
synodic period.

inverse foi?n, the amplitudes being much larger. It seems that
there are inversion and damping of the variations in the intensity
of the radiation in the lower atmosphere^ a subject of importance
for further research.

Digitized by

366 A TREATISE ON THE SUN'S RADIATION

International Character Numbers

The character numbers of the amplitude of the disturbance
mission for 35 stations during 191 5, produces nearly the same
inverse curve. This curve distributes its minor crests on four

Fig. 42 â€” Magnetic Character Numbers, 1915.
International Commission, 35 Stations.

axes, whose center is eccentric to the axis of rotation, as if the