·.:
GRAViTT.:KTION · . AND :COSMIC ROT . A'.TlON
Julian M. Avery
May 1954
. GRAVITATION-. AND - COSMIC -· ROTATION
PAGE.
·introduction......................... .. ... .... ........ .... ....... ...... ... .. .. .. .... .... .. .............. .. ....... ........ .......... .... ... .. 1
Explanatory .Foreword···· ···· ·············· ···· ··· ·· ··········· ·· ···· ······· ······· ··········· ··· ·· ·················· ····· ······· 2
Presentation of General Expressions··· ··········· ·· ·············· ·· ········ ········ ·····: ·············· 4
Application to ·spherical Bodies ······ ········ ··· ······················· ····· ·· ·· ·········:·········· ········· 5
Calculations for the ·sun··· ···· ···· ·· ············ ········ ··········· ·· ····· ········ ··· ········· ··· ········ ·············· 7
Effect. of Loss of Mass ···· ··· ··· ···· ········ ············ ·····: ················· ·········· ···· ······· ······· ················· 9
Calculations for the · Earth··· ···· ······ ········ ······ ·· ······· ····· ··· ······· ·· ········· ·· ····························· 11
Calculations for the Planets .... ... .................. .......................................................... :... 13
The System Earth-Moon........... .. ........ ............. ....... ... ....... ........... ... ... ......... ....... ... ... ....... ........ 15
The Solar System ............ ... ........... ....... .... ... ..................................................... ,........ .... .. ...... .... 19
Application to Main Sequence Stars....... ........................... .... .. ................ ............... 24
Application to Red Giant Stars .... ........................... ... .......... .... ..... .-.................... ........ 30
Deductions Regarding Stellar Evolution........... ... ........ ..... ................................. 32
Application to the Unive_rse··················· ··· ··················· ·· ··· ··· ·········· ········ ····· ·············:·... 37 ·
General Consideration&······ ········ ················· ····· ··· ···················· ············· ········· ·· ··· ··· 37
Comparison With Rel a tiv.i ty ·Theory...... ............. .... ......... .. ..... ...... ..... ... ..... .. 37
Curvature . of Space..... ..... ... ...................... .. ....... .... ......... ... ... ............... ... ...... ............. 39
The Bending of Light Grazing Cosmic Bodies .. ... ... .. .... ...... ...... .......... 39
The Expanding Universe ... ...... .. ............ ....................... ...... ... .... ... ........... ...... .......... · 41
The Radius of the Universe......... ... ....... .......................... ... ..... ...... ............ ....... 43
Concept of the Universe.............. ... .............. .. ........ ...... .... ................................... 45
Development of Basic Hypothesis········ ····· ···· ··· ······· ·· ····· ···· ···· ··· ·········:······ ······ ·· ········· 47
Origin and Nature of Gra.vitational .Fields.... ........ ..... ... ... ...... ........ .. ..... ........ 53
'
May 19M
·.I N · T R O D U C T 1 0 N
-Surely one · of the most · intriguing l.lllsolved mysteries of Nature is the
· cause .. of' the axial ·rotation · exhibited by cosmic · bodies and · systems. No theory
of cosmic evolution · can ·be · intellectually satisfying·: or · mathematically complete
unless ·- 1 t · includes a rational · explanation of such phenomena.
The only ·serious attempt to provide a solution appears to be the
"grazing stellar collision" .hypothesis, which was invented to explain the . angular
momentum of our solar · system. ·But this hypothesis : was found l.llltenable,
and science continues to beg the question by citing the law of the · conservation
of angular momentum for · isolated systems. : This merely states that a .cosmic body , _
· or system always possessed its present · angular · momentum, unless · it has been
acted upon .by external forces. · It offers no exp~anation· whatever as to what
caused -the . rotation, much less why the amount of angular momentum is that which
is . observed. ·
The purpose of this -paper is to present an hypothesis which attributes
the rotational · characteristics exhibited by cosmic bodies and - systems to physical
phenomena inherent in -the process · of the · central gravitational condensation of
· nebulous matter. These phenomena are shown to be · relativistic in char·acter, and to
involve hitherto unsuspected properties of ponderable matter and gravitational
fields. · It will in fact be shown that the axial rotation which · characterizes
cosmic , bodies and systems is closely akin to the relativistic precession · of orbi.
tal atomic ·, electrons.·
-1-
PRESENTATION OF BASIC · MATHEMATICAL , EXPRESSIONS
We ,begin .by ascribing . to a cosmic system of .constant . mass (M) and
progressively decreasing .radius (R) a gravitational energy (Eg) and a gravita:
tional ' ve_locity · (Vi) · such that:
M2 G
Eg .=---it .=; MVg2
V .=·12 MG g . R
where (G .= 6. 66 x 10- 8 ) is the gravitational ·. constant.
The ·velocity · (Vg) will at once · be recognized as the "parabolic" or
"escape" 0·ve~ocfty of astrophysics.· · It : is . the velocity -which a particle would
) : .·
a,cquire if it . fell freely · from outer · space to the distance ( I(. ) under the -. influence
of the central .gra:vi tational field of mass , (M). ·rn a · condensing • system
such as the creation of a star from · nebulous matter this theoretical radial
velocity, developed by myriads of particles, is converted into the random motion
of · thermal · energy.
This would seem to negate the possibility of any causitive connection ·
between the process of central .gravitational condensation, and the · development of
oriented -motion of the individual particles about an : axis of revolution, · such as
· is · required for the axial rotation actually · exhibited by the system as a whole.
Nevertheless, we · shall 0 show that such a relation does exist, and that it can be
expressed mathematically.
We now ascribe to the · system a rotational energy ( En.,) and an effective
average velocity-( V/,v) such that:
EA. = ; MVA. 2
-2-
The .basic premise of the present · hypothesis is that for a cosmic · system
of constant mass 0 undergoing the process of central gravitational condensation
free from external forces, ' the gravitational and · rotational energies of the systehn
are related by the expression:
E/t. .. = Eg x (~)
2
X tt .= i ~ 2
whence:
E - y20 M~ x·; ,.r, . - -- X -2 "~ . R Re
If we now · ascribe · to · the. system an angular momentum (p) such that:
Pi== M VA.. R
we find that:
p .= M X MG X R .= u:20
Re · C
This .is very gratifying, partly because of the extreme simplicity of the
result , and partly because sine~ the mass . of the system was assumed to be .constant,
its theoretical angular momentum is constant, and the law of the conservation of
angular momentum is honored. ·
If the · reader is sceptical, · he may at this point turn to the Appendix
· where he will find that these expressions ·for rotational energy and angular momen~
tum give . mathematically correct results for · the Sun · within the · limits of accuracy
of available data. · In : subsequent papers the application of these expressions to
the · rotational =characteristics of other cosmic bodies and systems will be developed,
and a relation with certain aspects of general Relati"vity Theory will be shown
to exist.
-3-
INTERPRETATION OF MATHEMATICAL .EXPRESSIONS
We have now . to interpret our math'#;ffilatical expressions in terms of an
acceptable theory representing physical phenomena inherent · to the process of
central · gravi.tatielnal condensation .
In approaching · this problem, it : seems · logical to treat the cosmic
system as composed · of discrete particles of ·mass · (m) :, which mi ght be related to
nuclear particles (neutrons or protons) or hydrogen atoms. Molecules and even
dust particles may then be looked upon as aggregates of particles of this unit
mass. · · The expression . for · rotational energy then becomes :
M MmG MG . .
E,z... -= m: X It X Re' X !
and the rotational energy · (E~ ·J.) developed. by a typic~l . particle is:
. l = Mni.G MG x 1. E/L .- R x Rc2 :a-
In astrophysics the velocity:
.Vo .= \MG .= !L
. R . l2
is the " e quilibrium" oi: "orbital" ·yeloci t y corresponding to the "escape " or
" parabolic" velocity which w~ _, hav.e .identified a.-_s ~the '·~ra:,,itati ~nal velo·cicy
(Vg) ; of t~e syste,n. It · can · be · shown on thermodynamic grounds that a cosmic
· system .can -condense · under the influence of its central gravitational field
only as it progressively loses half of its gravitational (thermal) energy.
This means that (Vo) is the average residual velocity of particles under conditions
for dynamic equilibrium of the system. ·
We therefore write
E 1
Yl,..
= MmG x -Vo2 x !
R ~
-4-
Now the expression :. (~ ; G) represents · the · gravitational energy of' · the
particle (m) with , respect to the central ' f'leld, and·is of course associated ' with
t,he gravitational·velocity (Vg). We · may therefore write:
1 Mm {V 2
EA, - -
Jl~2 X Vo X !
m V 2 , 2 = ! · ~ :. X Vo X ! .c
= ! b. mg ·
2
Vo
where (6 mg) is the . relativistic mass associated with the kinetic energy of' mass
· (m) ·moving at velocity (Vg).
The total rotational energy developed .by the system is of' course:
1 · 2
E,11.-= .L E,-2. .= ! b. Mg Vo
Interpreting . these ·.expressions:
1) A cosmic · system of' ·. constant mass undergoing · the process of' central
' gravitational · condensation develops -axial -rotational energy equivalent to the
· kinetic energy associated with a ·mass equal to the · relatlvistic mass corresponding
to the gravitational ' energr · of the -system, :moving at · the equilibrium veloc~ty
of' · the · sys tern..
2) A cosmic system of constant mass undergoing the process of central
gravitational condensation possesses · as an inherent property a constant angular
momentum proportional to the · square of its mass .
3) A · cosmic system of constant mass undergoing the process of' central
gravitational condensation develops axial · rotational · energy proportional to the
cube of its mass and the inverse square of its ef'fective radius; the effective
axial angular .velocity of' · the system is proportional to the · inverse square of its
ef'fective radius; · the rotational · energy developed by the · system therefore irt- .
· creases progressively as condensation proceeds whereas its axial angular momentum~
remains · constant.
, lfPLLANATORY HYPOTHESIS
If all 0 tn.is is true, forces must be at work which are not recognized
by present day theoretical . physics. · It ·: follows that any explanation · offered
, here to account · for the · relation . be tween the , axial. rotation of ·. cof!mic · systems
and·the process : of · central gra:vitational · condensation must .be · hypothetical. If
· the · hypothesis .about =to be pres.ented is : found · untenable or inadequate, it · is
hoped . that others may be · stimulated to determine the true nature of the phenomena
· which result in the relations which · have · been expressed.
We ·begin by ascribing to any ponderable . body or particle of mass · (M)
a gravitational pole · strength · (Pg) :
· i
Pg .= Mu._
which · has the · same · dimensions as electrostatic pole strength. Assuming a · Coulomb
type fieldt · the gravitational - field · strength (Hg) at distance · (L) is then:
H - M G1 g --L2
In -terms · of field action, the · gravitational energy (Eg 1)between a
particle . of mass· (m) -and the central mass -· (M) is then: :
i
Egl = ~ x m G'
L
and the rotational ' energy may be expressed · in terms;of' gravitational field action:
1 _ M . G' i
EA, .- ~ x m G
We now introduce the concept that a particle of mass · (m) moving at any
velocity · (V) develops a second-order relativistic "gravitational" pole strength
· (PA,) proportional to the relativistic -mass (6 -my) corresponding to its kinetic
energy' ~
( y_c )2 P/t- .= ! m \- Gl .= 6.m., G•
- f\ -
1
It is at once evident · that· the above expression · for · (E.,1- ) can be written in
terms · of.' this .hypothetical "relativistic" pole · strength-: ·
where (m 0) is the relativistic ·mass at · velocity (Y 0
).
Interpreting ' this expression:
4) The rotational energy developed by a cosmic · system of.' constant ·mass
-undergoing the process of · central gravitational condensation · is equal to the
· theoretical gravitational · energy between the actuai · mass of the system and a ·hypo
· thetical · relativistic :mass corresponding to the "equilibrium" energy of the systemj
at a distance equal to the -effective radius of the system.
Comparison of this result · with the pre'vious result in terms of the
"equilibrium" · velocity will show that - the two are -precisely equivalent.
·What, · then, 1s the mechanism or physical process which produces this
· result?
· We now introduce the radical concept that the hypothetical "relativistic"
pole ' strength of particle (m) reacts with the central gravitational f.'ield to produce
a tangential · instead . of a radial · (centripetal) - force-:
Ft 1 - ~
L~
M Gi
= X; m G* X
L2
at
E./i-
1
.= f. .Ft
1
d L
-; R
Mm G M G = ! X
R Rc 2
MG* X ;
y~2 = m -::-r
R C
-7-
2 M G
---r
Le
= MG'-
--
R
G*
X ; m Vo 2 G!
c2
whence:
1 M Gi
E/1... = .~ EA, .= -R- X !::, Mo a*
.= M2G x (!_g \2 x tr
R ·c-)
The i last 1lpression · is precisely the empirical expression presented as our basic
"··~
.premise - at ,the beginning of this paper., but we are now able to interpret it in
terms of a piausible hypothesis.
-5) The particles of a cosmic · system -undergoing the process of central
gravitational condensation experience a tangential force proportional to the
relativistic mass associated with their kinetic energy. · This force is produced
by the action of the central gravitational field upon the relativistic "gravitational
" pole : strength -of the particle as it moves radially inward under the influence
· of the . central · field. The · result of this tangential force is to . prod~ce
an oriented motion of rotation of the discrete particles about an axis of the
· central mass , · superimposed upon · the random motion of thermal .energy of the part-
· i e les .
Statistically, the result is as though the discrete particles -fell freely
from space to the effective radius of the system. · The selection of the axis of
rotation, and of the direction of rotation, are probably a matter of chance in an
isolated system. · But · in a s y stem which is not effectively isolated, as for
example the planets of our · solar · system, the axis of rotation, a.n,d the direction
of rotation are clearly influenced by the orientation established by the central
body.
-8-
ANALOGY WITH ELECTRONIC ·PRECESSION IN ATOMS
Using the Bohr model of the hydrogen atom in its lowest · energy · state
as our - example, the · energy associated with · the · re;I.at±vistic precession of the
orbital · electron may be · expressed:
-~ = ~ me Vu
2 x~ 2 xJ
where -~ is the energy of precession
Me is the rest mass of the · electron
I
·vu is · the orbital velocity of the · electron
rJ:.. .Yk. · = ' 2 . 'Tre.2 is the fine~structure constant = o •he of spectroscopy.
e is the electronic · charge · in e.;lsc~u.
h is Planck's constant · of · action.
y = ~ for the .case in question.
We also · recall -that · the electrostatic field .energy (E) between the electron and
the · nucleus, with respective electric charges or pole strengths (e-) and (B+) is :
E =
where~ is the orbital radius.
·E0 is the orbital kinetic energy
The energy of precession may therefore be expressed either in terms · of
the relativistic -mass of the electron due to its orbital motion:
· 2
-~ 62 - ~ =
=
-9-
· Or,in · terms of fiel.d . energy between the ·nuclear · pole and · a ' second-order or relativistic
.pole strength of the e1e·ctron · due to -its . orbital velocity:
C )2
- e VH .- ! - . -X e X --
RH . c
The corresponding expressions for the rotational·energy of a cosmic
.· system · were:
M
2
G ( )' J<;R = __:____x ~ X 1t
R .C
= ! I:). M X Vo 2
-g
= ! MG~
R
X MG' (:g )' X ~
Dimensionally, the expressions in terms of relativistic mass on the one
· hand, .. and in terms . of relativistic pole .strengt;h · on the other, are precisely equiva-
1lent. This extraordinary result lends · substance to the reality of the present
hypothesis, and to the concepts upon which it is based.
CONCLUSION
We conclude tnat the rotational characteristics exhibited by cosmic
bodies and · systems are in fact · the result of natural phenomena · inherent in the
process . of . central gravitational condensation of · nebulous matter. It is suggested
that these phenomena , have - their origin · in hitherto unrecognized properties of
matter and of gravitational · fields. The · interpretation of these properties appears
to involve an extension . of Relativity Theory which, because of its nature, may be
of assistance in the development of a Unified .Field Theory.
- 10 -
CALCULATIONS .FOR THE -.EARTH-Calculating
· from accepted data for ·mass, · ra4ius and peripheral angular
velocity, we have · for the axial angular momentum of the Earth:
M
2 'a = 9 ~00 ·x 10 37 .
C
.For the correction factors (n) . (k) we use · the gene:rally accepted values
(Blackett):
nk = O .258 k = 0.88
w,hence:
2 p = 6 nk"-} MR. 2 = 1.84 X 1040
= M2 G I_!: k tf
C 6 ..
where Cf is a function to be evaluated.
Numeri·cally., from the above date:
. r _ P __ 1.84 . X 1040 3
":1 = 0.388Xl0 =388
. 8.oox10 37 xOo594
dimensionless parameter.,and it was therefore logical to
see whether it had any connection with
inent a role in the expression already
the parameter cv~) which plays so prompresented.
· In this way we find, purely
empirically-:
V 0 (Earth) f = 0.795 x 106
1 0
= 2 \3. 77Xl0 4 =388
-11-
I
From the previous section we .conclude · tentatively:
!i_(EarthJ=i4 .c =388
M Vo
In ·view of the uncertainties involved in m~thods used · for estimating this · ratia,
the .check with the ·values estimated by Kuip·er and Urey would seem to be acceptable
- it m~y even be that the present estimate is nearer the truth.
Turning to axial rotational energy:
.M
2
G =3.76 x 10 39
R
ER(theoretic(ll)
2 ( l,2 . (iA.;i 2 = -JI.~~ x -Y:) x~ X
It thus · appea-rs that the expressions · for both angular momentum and.
rotational energy, as modified for the effect of mass · lost from the system, give
correct results for t he Earth.
We have at present no explanation to offer for the fact that the corl
rection factor for mass appears to be given by the parameter ,
but this ·may well . be ·left · for future analysis.
: -12-
.CALCULATIONS ·.FOR: THE : PLANETS ·
.For the other planets a different procedure , must be followed, since
values for (n) (k) can only be estimated from the physical ·state of each planet •
. Instead of checking the actual ·value . of (PJ or · (ER) agai~st · the theoretical
·value, . as . was .done .in the .case of the Earth, we · therefore reverse the ·procedure, .
and use the theoretical expressions to -.cal<!ulate ·values , for (n) (k) an\~lj· If
reasonable results are obtained, we shall . have further reason to believe -that
our thepretical expressions have mer.it, and may .in fact be · correct; within the
accuracy of available data.. ,
We · therefore wr.ite tentatively, as . for the ·Earth:
P= _! nkw .MR.2'= M2G [ .! k [4· c
6 · · c · 5 V 0
y2ri I .c fF I - 4,.___ -
nk~= . C ~ 5
.! w MR·2 ··
. 6 . .
Unfortunately .it .is impossible to separate (n) a~d (kiJ, because we do
·not knQ\W1 the value of either one alone, But we know from the apparent densities .;
of' the major planets that they must have very thick atmospheric shells which give
a greatly exaggerated .idea ·of the true radius of the quasi-solid planet. It
follows that (kiJ should be much smaller for these planets than for the Earth.
Somewhat similarly, Mars is commonly believed to have a very thick earthy. enve~ope,
I which accounts . for its low apparent density. · It too shpuld therefore show a low
value for (fe}!t.J,. · The , comparative , values for (n) are difficult to estimate even to
a rough approximation, but · from the evidence .in the cases of Jupiter and Saturn
one would expect. values of (n) considerably less than unity for the major planets,
and especially for Saturn,
·-13-
The actual data are :
Earth Mars Ju:eiter Saturn Uranus Ne:etune
:~(.t) MR.2 . 701·0X10 4 0 2 .• 12xl0 39 6.88x10H 1.42x1046 ' 3.63xl043 3 .llx10 4 3
6
· M:¥ G 8.0ox1037 9.30xlo 86 8.oox10 42 7.24xlo 41 l.85Xl0 40 2.;35Xl0 4 O
C
·vo 7.99Xl0 6 3 ~56xl0 6 4~25xl0 6 2 ~53xl0 6 L58Xl0 6 L62xl0 6
,~o =:l · 388 ;596 168 216 274 276
nkl 0.278 0.165 0 . 123 ·0 . 070 0.088 0.132
I These values for (nk2) are without exception evidently .in correct rela-tion
with the ·value for the Earth, and with respect to each other. We · therefore
ac~ept them as e v idence supporting the validity of the expressions from which
they were derived.
The values for(:~ likewise are in correct relative relation with one
another. This lends support 9 but by .no means .constitutes .proof;for the possibility
that · the parameter 14 ~o is a .correct measure of the ratio of the planetismal mass
to the final mass 9 or at least of the effect of the ·planetismal mass upon the
final rotational characteristics of the planet.
We now turn to the system Earth-Moon.
· -14- -
THE SYSTEM :EART&.-MOO'N .
The system Earth-Moon may · well , be .considered to be a binary system, and
as such .it·. is possible that a :further modification of the expressions thus far de,.
veloped :may : be ' required. The data · for angular momentum and rotational energy for
· the · system are:
p · (Earth}=: M2
G x ,~ k x 388 ==0.184X10 41
.c
2
J:> 0 (Koon~ ==Ct.JoMRo
" :L p (System)
M2 G == -~X
R
ER (Earth)
E · ( Noun) ==W · 2MR. 2
0 0 0
2 ER (System)
=
==
==
2~J.ct.C) , 1_
3 : 11Xl041
(ass) 2 = L 95x10 35
~as~10.s&
-5."84X10 Sb
where the · subscripts (o) ·.indicate that the symbols relate · to : the orbital .character.
is tics of t he Moon. ·
The : orbital angular momentum of · the Moon .is thus (15~9) times the axial
angular momentum o~ the Earth, while 1 the orbital ikinetic energy of the Moon .is
only (1~98) times the axial rotational energy of the ·Earth.
:rt ·.is ; thus at once .clear t:qat the · relation . between angular -momentum and
rotational energy · for the system is not the same · as for the Earth alone, .in · terms
. of a .co,rrection -for .change of Mass. An obvious possibility •.is - that the evolution
of the System may 1 ha.ve taken place -in · two ' more or l~ss distinct phases: Phase .I l .in which an .initial Mass · (M 1 ) .condensed to an- .intermediate Mass (M2) and a radius
(R
2
) : , at, t_he · end of which the Moon was "sloughed off" ; and Phase II ·_in which Mass
(M 2 ) .condensed to the present Mass (M) · and radius (R) of the Earth, during which
the axial rotational energy of the Earth was developed. Another tentative assumpti-
on .is · that (R 2 ) , may be approximately the orbital radius (R0 ) of' the Moon.
:-1'5-
Now, .if the factor (388) :_ in the mathe~atical -expressions for the Earth
.is taken to represent a mass ratio (¥j, we have to determine whether (Mx) relates
to (Ml) or (Mz) •
Trial calculation indicates that · if the assumption is made that (Mx=M 1):,
the ·numerical results are .irrational.
But .if we assume that (Mx =M 2
) is the mass at the end of Phase I and the
beginning of Phase II of the evolutionary process , of the System, we · -ha:ve :
p = ~ l_! k Mz =
c 6 M
= M VRl_! k R . 5
VR= ¥2G
- ~
E MV z x- - R= R
- -
= M2G ~ (MM2)2 X
R Rc2 ~
-16-
the Moon :
Then from previous data, if (E 2
) is the orbital rotational energy of
~J:,ca ~ 2X3600 (:9 2 ~ 2. 79 X 10 6
M . a
_ :..L_ = l.,67Xl0
M
M .
1 =4·.3
~
M
--.!x-1-=2810
M w· _k
5
These results appear to be acceptable. The indicated ratios of Masses :
M1 =1670 M
M2 =388 M
bracket Kuiper I s estimate for the initial Mass -of the pro to planet, ·and the mean
:of the two (1029) is almost exactly Ureys estimate of (1050).
It is possible that the factor (1 k) does not - belong .in the calculation
for the ratio ( : 2). In that case :
.M2 G
p (Earth) = .c x
2
~ -2 X 3600
251 M2 -
251 : ;
M
~251
M
M
-..L=251
M
: ;
The result (M 1 =1290 M) .is very close to Kuipers estimate of (M 1 =1200 M) •
We accept the results as supporting the following hypothesis :
The evolution of the Earth-Moon system 0 from a nebulous mass occurred in
in two fairly ·distinct successive phases. In the first phase an initial mass of
-17-
something dver (1000) times the mass of the Earth .condensed to about the orbital
radius of the Moon, losing a major . portion of .its mass .in the process.
At the end of Phase .I the Moon was "sloughed off," · taking with .it a
disproportionately large _· share · (in ·te nns . of . i"t,s mass) of the rotational energy
and angular momentum developed . by the System during Phase .I. This -probably was
· the . result , of the development . of an equatoriai · bulge or beit . of matter which
devel0ped a peripheral : velocity high enough to -balance the gra:v:-itational attraction
. of the residual .central mass, and .consequently established .itself .in dynamic
equilibrium at the orbit . of the . Moon.
Dur.ing . Phase .II the remaining mass shrank to the radius of the Earth,
losing most . of .its substance .in the process. Dur.ing this period the present axial
rotational .characteristics of the Earth were _developed.
The hypothesis , thus explains both the rotational .characteristics . of the
Earth and of the Earth-Moon System, as the .inherent result of an orderly natural
process of .central gravitational .condensation, during which most . of the original
mass was lost. No .cataclysm . of nature .is required, nor .is .it necessary to speculate
as to whether the Moon was torn loose from the Earth, leaving a pit which we know
as the Pacific Ocean • . It .is to be hoped that the present hypothesis, or something
akin to .it, may be found acceptable. ·
. -18-
THE SOLAR SYSTEM
The data for the Solar System are:
Angular momentum.
· 2
p (Sun) =6 nliw MR.2 = . o.023x1q50
1: Po"fPlanetJs) = 2 WO MR.02 = 3.18 ; x10
50
:L··p(System)=3.20 · x 1060
Rotational · energy
2
ER (Sun)- 6 n .2 k cu~ MR. 2 X = 4.08 x 1042
E 0 (Plu.netJs):2 w 0
2 MR.
0
2 X =2.00 ··~ 10 4 ·2
The total orbital angular momentum is thus (140). times the actual
2 ,·
angular momentum of the Sun, and (35.6) times the theoretical ·value (~) for
C
the Sun. This brings us squarely up against the rock upon which so many nebu-lar
hypothesis have struck and foundered!how to explain the very large angular
momentum of the Solar System.
But we are now armed with an hypothesis which offers the prospect of
a -logical solution; a step-wise evolution in which, during successive phases,
a .nebulous mass initially much greater than that of the Sun., condensed and
"sloughed off" successive planetismals which subsequently in turn condensed to
form the planets.
It might be possible to calculate the changes in mass and rotational
characteristics of the system for each successive phase resulting in the formation
of a planet. But since nearly (2/3) of the orbital angular momentum and
over (3/4) of the rotational energy of the planets reside · in the orbital motion
of Jupiter, and since the orbital angular momenta and rotational ene·rgies of the
minor ·planets are negligibly · small, it seems reasonable to treat the · system
initially as a binary system and then make any necessary modification.
Previous calculations · for the Sun alone gave nb\ ;i.ndica. ti.on of any
effect from a larger initial mass upon the rotational characteristics of the
Sun. There is therefore an important difference between the expressions developed
for the system Earth-Moon, and those · to be developed for · the Solar System.
We begin with the assumption that an initial mass (M 1
) condensed during
Phase I to radius (R :2 ~ and mass (M 2 ) : , and that during this phase the major
planets were 11 sl9ughed offn and given their orbital characteristics. Since the
to:tal mass of the planets is only about (_1· ) part of' the mass of the Sun, we
10 0 0
may without .important error assume that (M 2 ~M).
The .inherent angular momentum of the initial mass was accordingly:
M 2 G 1
. C
and at the end of Phase I:
M1 .2G x ~ _ M1MG
.c Ml --c-
If (p
2
) .is the total orbital angular mom~ntum of the ·planets:
( \f k= 0.253)
We then have two possible values for the mass ratio·:
Ml : =140 (if factorli k is applied)
M
M ·,--..2
_ 1 =35.5 (if factorl 6 k is not applied)
M
- .20!...
The residual rotational ' energy at the end of Phase I is then:
E2 M1 2M R 2 _____ x __
ER R2.2 Ms
If M
_..!. = 140 R2 ::: 197 R
M
Ml
M
= 35.5 R2 = 50 R
The hypothetical radii are then:
R
2 = 197 R - 6.97Xl0lO X 197 ::= 1.37 X 101 3
R 2 ::: 50 R = 6.97xlo 10 ·x 50 =0.35 x 1018
Comparing these results with orbital radii of the planets :
R 0 (Earth) ·- 1.50xlo 13
R 0 . (Hercury) = o.5sx1ols
·R0 tAsteroids) = .4.19xlol 3
We · conclude from this that during Phase I a large degree of differential
central condensation developed, and that the radius (R 2
) is not the
peripheral radius of the system, but the effective radius of the central mass.
Thus the major planets were "sloughed off" - during a . period when the effective
radius of the central mass shrank from about the zone of the asteroids to
-21-
about the orbit of Mercury. · During this phase the ·loss :gf mass from the system
occurred, and thereafter the loss of mass was negligibly small.
Phase II therefore-. represents the condensation of the Sun proper from
an effective radius of about the orbit of Mercury to - the present radius of the
Sun, ·and the rotational energy of the Sun was developed. , Whether the minor
planetesimals were created during Phase. I 9r phase II is problematic, but .it
seems plaμsible that this may have happened during Phase II.
This concept of the evolutionary process of the Solar System is rather
satisfying, because .it prov.ides a basis for the obvious major change .in the
characteristics of the system inside and outside of the - region -of the asteroids,
·. in addition to providing a logical solution of the problem of angular momentum
without invoking a cataclysm of Nature such as the grazing stellar collision
hypothesis.
The present hypothesis of .cosmic evolution has now been applied with
some apparent success to the Sun, the Earth, the planets 1 the system Earth-Moon,
and the Solar System as a whole. This encourages us to see whether .it also
applies to the stars, or at least to those of the Main Sequence of which · the Sun
is an average · example •
. NOTE:
As a matter of further interest, we examine the possibility that _ the
ratio of masses for the Solar System may, as in the case of the Earth, be indica-ted
. C
by the parameter (~). We then find, · for the Sun:
0
~36.8 I Vo
as compared with the mass ratio :
just calculated.
M1=35.5
Ai
The corresponding parameter for the Earth wasj4c and it
Vo
. .
-22-
therefore seems probable that something more than a mere numerical coincidence
is indicated. This problem may well be ·left for future study, in the hope that
the true significance of this parameter may ·be ·discovered.
We ·note in· passing, ·however., that the present ·value of (V 0
) for the
Sun is not definitive. As the ·sun shrinks further the ' value of (V 0
) will in.
crease somewhat and it is .certainly possible that, after · the Sun has run its
course, the final or definitive yalue of (V 0 ) will be such that the parameter
~~ applies, as in the case of the Earth.
-23-
APPLICATION .TO MAIN . SEQUENCE STARS
We •found that in the · case · of the Sun per se no .correction for change
·of mass was required in calculating its rotational characteristics, whereas for
the Solar System as a whole, a correction for change -of mass was required. A
logical question, then, ._is whether ou.r mathematical expressions may -be expected
to give · approximately ·. correct results for the rotational characteristics of
stars in general, or at least fpr those of the Main Sequence.
Earth~bound observers have at present no means of determining whether
or not a star posse_sses a planetary system. : Bli.t if the hypthesis presented
·here has substance, it .is .clear · that the development of planetary ·systems is a
-normal result - of the process of stellar evolution.
In the .case of our Solar System the effect of change · of mass was found
to be confined to the orbital motion of the planets. We ·therefore make ;the
assumption that the same is probably true of stars in general, and that the Sun
may be used as a "yard-stick" in .calculating their · rotational .characteristics,
ignoring both the effect of change ·of mass and the probabl~ existence of a planet-ary
system.
Turning to existing ' data on the subject, Strtive gives the following
table for stars of various types .in the Main Sequence:
Type
Oe-Be
0 - B
A
Fo-Fz
.F.6-.Fil
d G
dK
d M
Peripheral Velocity (w R) of Stars
Relative
15
15
4
2.5
1.5
1..2
1.0
0.5
Mass ·w R(Average)
3:50 X 107
0.94 X 10 7
1.12 X 107
0.51 X 10 7
9.20 X 107
0
0
0
where (M) is the mass relative to the Sun.
-24-
w R (Maximum)
5.00 X 10 7
2.50 X 10 7
2.50 X 107
2.50 X 107
0.50 X 10 7
0
0
0
These data are derived from spectroscopic observation of the Doppler
effect due to rotation across the line of vision. Stellar masses, as Struve
points out, are estimated from data obtained on binary systems but, as Struve
also points out, it is by no means sure that · this is an accurate method for
single stars, We ·shall question here both the interpretation of the spectroscopic
data as related to peripheral velocity, and · the estimated stellar masses.
The data in the · above table are most extr.aordinary, The equatorial
ivelocity of the Sun is
w R(Sun) = 2 x 10 5
1
or only about (250) of the maximum value in the table, and (_!_) of the value
10
for stars of nearly the same mass as the Sun. But more - extraordinary still is
the fact that the stars of the · same mass as the Sun, or smaller, show no rota-tion
at all, This has been remarked upon by others, and it seems obvious that
.it is due to a limitation in the spectroscopic method as applied to smaller stars.
The present method will exhibit no such limitation.
The mathematical expl'.essions with which we are here concerned are :
p - M:G ,~k = ! nkw MR2= MVR
MG
VR= Re
=
If k R
V =~
0 fR
w R = peripheral velocity
If (M) :, (R), · (w) are expressed in terms of the corresponding values
for the Sun, we ·have:
w R(Star) =
w R(Sun)
M (Star)
M (Sun)
X R(Sun)
R(Star)
:-25-
x n~ (Sun)
n \~ k (Star)
We have no method for determining (n) (k) for a star except purely
theoretical considerations:. But if (k) is . relatively small, (n) is relatively
large, and conversely. It .is therefore · reasonable to assume that the ratio
~rtvolving ( n H · k) may be taken as unity without · introducing an important ·error.
This simplifies our .expression to:
M
WR':'-'R.
where all values are measured in terms of the Sun.
Now ( R) . can be determined from the · temperature-luminQsity, relation:
where (L) is the luminosity in terms of the Sun
(T) is the surface temperature in terms of the Sun.
The accepted method for estimating · the stellar mass is the mass -
luminosity relation
L"'M3. 5
determined for binary systems and applied, ' with some question as . to validity, to
!
stars .in general.
But let us assume that the surface temperature of a star is proportion ,a.l
to its gravitational energy per unit mass:
2 MG
·T "'VG =2 R
Then, in terms of the Sun:
! T
-1
·y "-' TR "'L
M
CJ) R "-' -R "-' T
-26-
Applying these expressions to data for stars of the Main Sequence, the
tabulated results are:
L!T- 1 L!T- 2 .. M Vo
Type T . L it3 (juR.) (ei..R) (L) (M) (R)· 'C..6)
· 05 5.22 106 60~5 11.6 0.039 · 2.2!;!
Bo 3.64 6xl0 8 21.2 5.83 ·0.107 L9i
Ao 1~84 70 4.'55 2.47 0.30 f.3q
.Fo 1.23 4 1.62 1.32 0.11 16·10
· Go 1 ·1 1 1 1 1
Ko 0.85 0.32 0.66 0.77 . 1.19 0.92
Mo 0.59 0.032 0.31 0.52 2.22 0.77
where all data are in relation to unity for the Sun:
The column (:
8
) is introduced to show the relative densities ~) based
on the estimated mass and radius. The results seem reasonable.
Comparison of the stellar masses thus calculated with those given in
the preceeding table, -show relatively larger masses for the more massive stars,
and relatively smaller masses for the less massive stars ~ than those obtained from
the application of binary data. But the differences are not order-of magnitude,
and it is believed that the data provide ·strong support for validity of the method
used, and its possible superiority over present · reliance on data derived from
binary systems.
The results for peripheral velocity are extremely interesting, since
they eliminate the fantastically high velocities presently accepted for massive
star.s,_ and fill in the blanks for stars the size of the Sun and smaller. These
results seem to speak for themselves in support of the method by which they were
derived.
-27-
But if they · are the true peripheral -stellar velocities 9 how are the
spectroscopic data to be explained? It is suggested -that they actually represent
the optical effect of matter rotating about the star in gravitational equilibrium,
_. but at some distance - possibly a considerable dista.nee - from the actual
surface -of the star. The 11 spectroscopic 11 velocities are · thus not "peripheral"
velocities, but 11 orbi ta.1 11 ·velocities for which ·we -have used · the symbol "
We ; found earlier that:
M
w R "'· R. :
· from which, in terms of the ·Sun:
The data in the right hand column of the previous table · were calculated
from this relation.
In a previous Section we found that
·v 0 (Sun) = 4.40 ·x 10 7
The .corresponding ·value for an (05) type ·star is accordingly:
Vo (05)= 'Vo(Sun) x 2,29 =10,1 x 107
as .compared with a 11 spectroscopic 11 ·velocity ranging from about (lxl07 ) to (5xl0 7 ),
Remembering that (V 09 Sun) represents the equilibrium velocity at the surface of
the Suri., and that (V 0
) decreases with distance · from the surface, it is evident
that the . results strongly support the belief that the spectroscopic velocities
have in fact been misinterpreted and that they do ·actually represent the optical
· effect of matter (probably proto-planets or the - equivalent) rotating in gravita-tional
equilibrium at some distance from the true surface of the star.
-28-
We ·.conclude that all stars of the Main Sequence possess axial rotatiorr,
that the development ·of such rotation is an .inherent result of phenomena
· corrected with the process -of central gravitational condensation, and that ·the
amount of rotation can be .calculated at least approximately., by means of the
mathematical . expressions developed here.
We ·now turn to data for the Red Giant Sequence of · stars, to determine
if possible whether the method is equally_useful for . such stars.
-29-
APPLICATION TO RED GIANT STARS
'The spe_ctroscap_ic method gives no ·detectable result for · the axial
rotation of stars of the Red Giant Sequence. _ But if we apply to such stars
the ·methods just used -for the·Main Sequence, the following tabulated results
are obtained:
L~T'."' 1
M
Type T L -L~T- 2 Ra Vo
( Ct) R) (L) {M) - (R) . (,P) (Fa)
.Fo 1.30 25 3 .-75 2 . 90 0.154 1.14
Go 0.92 20 4.9 5 .-35 -3 .20x1C)- 2 0 . 96
Gs 0~78 20 5.8 7~4 1.43x10- 2 0.89
Ko 0.70 31 8.1 1L7 5.lxlo- 3 0.84
Ks 0.56 90 17.1 30.8 · 5.85Xl0-( 0.75
Mo 0.52 160 24.1 46.2 2.45x10- 4 or.72
Ms 0 . 36 2000 127 358 2 .'11x10- 6 0 . 60
The trend of estimated mass is the same as · that shown on the customary
Herzsprung - Russell diagram : the "redder" the star the more massive it is,
which of course is the reverse of the Main Sequence relation . But as in the
.case of our results for the Main Sequence, the mass estimated for the more massive
stars is considerably larger than accepted ·values.
Again, we find reasonable values for peripheral velocity, instead of no
apparent rotation according to spectroscopic data.
A very important point is that the estimated radius of the largest (M5)
Red Giant is (358) times the radius of the Sun, and its mass is (127) times that of
of the Sun. This radius is somehwat larger than the orbital radius of Mars. In
-30-
a previous section we found that the Sun apparently lost no appreciable mass
· after the radius shrank to about the zone of the Asteroids. It follows that
a star (127) · times as massive as the Sun would also lose no appreciable mass
·during the further process of condensation, from a radius of similar size.
Thus all of the Red Giant ·stars ·presumably have reached the second phase ·of
their evolutionary process as postulated for our own Sun.
We are now ready to . discuss the · results obtained for the two classes
of stars, and to present some · rather sweeping deductions regarding the ·process
of stellar evolution.
-31-
DEDUCTIONS REGARDING STELLAR .EVOLUTION
·Aside -from spectroscopic · data relating · to rotation, . the only
observational data we have for estimating the physical characteristics of stars,
are measurements of sul'face temperature and luminosity. · It is . customary to plot
thes~ data according to spectral types, and thus ·derive the familiar Herzspl'ungRussell
or (H-R) ,diagram. · In such diagrams stal's of the · Ma:in Sequence · form an
S-shaped ,band with brighter, hotter stars at the upper ,left, the Sun ·in appro~imately
the center, and ,red dwarfs at the lower right. · ·The Red Giant ,Sequence
also forms a band, ,running .at first horizontally from left to right from an intersection
with the Main Sequence band .at about ·Type . (A5) . to about .Type (Ko), and
.from thence sloping ·Sharply upward to the ·right to Type . (M5) · . But ,the Red Giants
differ from the Main Sequence in that the mass-temperatul'e relation is reversed,
:the ,cooler stars being larger and more massive than those .exhibiting ·higher temp.
eratures.
It has become the fashion .to interpret this diagram as indicating
that the ,evolutionary path of Main Sequence stars constitutes a "sliding ·down"
·t~e Main Sequence curve, with consequent loss of mass, into a "sinkn of Red
Dwarfs. · Speculation as to the ,evolutionary path of Red Giants is more vague; some
apparently belleve thatthesestars also lose mass and ,wind ,up in the Red Dwarf
"sink", while others suggest that they may traverse the chart horizontally from
right to left, with more or less loss of mass, and thus :wind up ·in the Main Sequence. ·
There is also much argument as to whether the Red Giants as a class
are relatively older or younger than stars of the Main Sequence , or whether all
stars are of approximately the same age.
-32-
. ... . · ·· ·•·
I .
. . 1 ..
' . '
\\ .... ;_-··· ! . :: I I . ' :
-- i . ... L. - .. ! ... ,... : T . i . I -· r ·--- +--- --- --~--- -1--- -- ·1 · ·
I • · • • • • • • f . . ' . i . I I
···· ·- --' - ...... !. .-- -·· ---r···-- ·i .. . .
j • I •
i . .•. • .• l . ... .. .. : .. _~- -~ -+--..
' . . I • t
i
i ·1 . ·:
~ •
i , .
I • ..
I - •
, . . . 1
' •
i
. I
I
::J
' ~ - .. r -1 ' .. l D ....
~ z
~
Q • • ..
i • • ...
•
..
' C . -. '
0
e
,-
In the hope -of throwing -some light · on these speculative matters, data
set forth in the previous sections ,were plotted on the accompanying · logarithmic
Mass Radius diagram. · As -will .be seen, ·data for the -various Main Sequence types
fall without exception on a straight -line passing -through . the -Sun -as origin, and
-having a slope of' (1. 75) · , indicating the -relation :
M "'Rl.75
Data for the Red Giants · fall very nearly on a straight line with a
slope -indicating the relation
M,..., R0;86
But as the Red Giants approach the Main Sequence, the .actual data create a curve
which intersects the Main Sequence line at about Type · (A-.F), as in the ·H~R diagram
from -which the basic data were drawn.
In a previous -section it was shown -that stars of both the Main and Red
Giant sequences probably have already passed through · the phase where large loss of
mass occurs, and that their subsequent evolution will therefore take plaoe with-out
, important f'urther loss of mass.
-If this is so, the evolutionary path of a Red Giant ,should be horizontally
from right to left, until the Main Sequence is reached. Thus an (M 0
) Red Giant
would become approximately a (B 0
) Main Sequenoe ·star.
It follows that Red Giants are a younger ·class of stars than those of the
Main Sequence, at least in the -evolutionary sense. Possibly, as has been suggested,
they represent the result of super-nova "explosions", and are beginning once more a
renewed process of central gravitational condensation. · But if this were so, the
very large rotational energy developed -before the "explosion" -would presumably result
in excess or at least observational rotation, and ·-we therefore rate this as
improbable.
The .fact · that the Red -Oiant line bends toward · the Main Se~uence is suggestive
of .the fact that less massive stars will pass through . the process 0 of gravitational
condensation more · rapidly than ,larger stars, and thus reach the Main Sequence more
rapidly. Are we then to conclude that ·Red Giant ·. stars are all of approximately the
,same .age, and much younger than Main Sequence stars? On the ·basis of available · evidence
this would appear ·to ·be so, but logic - suggests that new ·Stars are constantly
being formed, . by condensation of matter which ·escaped from the ,initial condensation. ·
If this is so, there should be · stars between the ,lines representing ·the Main and Red
Giant .sequences, as well as to the right of the Red Giants, but ·We still have to explain
the preponderance of Red Giants in the state of ·evolution which creates the
,line on the charts unless they .are approximately of the same age.
How .then does one .explain the Main Sequence? With the advent of the
nuclear energy theory we have only .to assume that ,when a star reaches, by central
·gravitational condensation, conditions of temperature and .pressure such that the =release
of nuclear energy becomes important, the gravitational condensation process is
at first slowed down, .and ·then possibly stopped or even reversed. It can be shown
on thermodynamic grounds ·that central gravitational .condensation can .proceed only ,if
a star progressively loses thermal energy corresponding to half of its gravitational
energy. The .superimposition of nuclear energy upon gravitational .energy thus inevitably
slows down the process of condensation • .And :since the ·nuclear .energy available
is so enormous, we hav:e adequate reason to assume that :the Ma:in Sequence represents
a long pause in the process of gravitational condensation.· The result is of
course an accumulation of stars in the process: . of .. developing .and ,radiating nuclear
energy into space, with trivial loss of mass corresponding ,to this energy.
But ·What happens when -this evolutionary phase -is over if it does not re-sult
in an "explosion" ? ·on the ·Cha:rt presented here , the White Dwarfs, always
-34-
represented as being anomalous on the H-R diagram, · form a vague -pattern at the -lower
left corner of the diagram, between . extensions ·. of the lines -for the Main Sequence
and , the Red Giants. · It seems possible that these stars, -being ·-much less massive,
have merely run their course -m·ore rapidly, have -lost :most -of their nuclear energy,
and -are now ,in a stage where final readjustments must occur. · Their tremendously . high
densities indicate a high degree . of ionization ;-as they -cool ,down, -they ,may ·be -expected
to expand -to give ultimate densities corresponding -to those of the known ,elements.
Perhaps -such a readJustment occurs through · the , steps -of an "explosion" -followed by a
final gravitational condensation.
-It seems pertinent to speak here of the end ·-result of -stellar evolution in
terms of the type of matter produced. · · In recent years it has become fashionable ·to
speak of -the -cosmic distribution of' matter in terms -of proportions .-of' various -known
-elements, and to assume that cosmic bodies are created by the gravitational condensation
of matter of such composition.
-The suggestion is offered here, that the primordial matter ·may have been
hydrogen, and that heavier elements are the result of nuclear reactions -common -to all
stars as the evolutionary process progresses to the ,point -where the necessary -conditions
-of temperature and pressure are reached.
Support for this view is found ,in the ·.following ·-data for nuclear binding
·-energies :
Element
He
.Fe
u
Per Atom
28.18 m.e.v.
483 . m.e,v,
1782 m,e.v. ·
Per Nucleon
7;04 :m.:e,v. ·
8.62 m,-e.v. ·
·7 .49 m.-e .-v.
On · thermodynamic grounds the principle product of nuclear . reactions, -beginning
with hydrogen, should therefore be iron and companion elements. The Earth itself
is perhaps the best evidence (supported by the composition of meteorites) that .this
supposition is sound.
We also see that most of . the available nuclear energy is already released
when conversion of hydrogen into helium is completed. · ·Thus gravitational condansation
can proceed f'airly rapidly after this evolutionary stage has -been passed, · to form a
body ·- consisting, :like our -Earth, of a metallic core within a rocky shell.
-36-
" ) APPLICATION - TO ·THE UNIVERSE
General Consideration
We have seen that a nebulous .cosmic system of mass (M) and radius (R)
has associated with .it a · gravitational energy (EG):, a gravitational ·velocity
(VG) :, and a hypothetical angular momentum (p) and rotational ·velocity (VR) such
that :
·v _ MG R-
-Re
Now .if, as .is generally ·believed, the average density of matter .in the
Universe .is uniform, it seems possible that the Universe .itself may represent the
limiting .case of these relations. This -possibility -will -now be explored.
If we begin with a big enough yardstick · (the civera{.'e dvstance ·between
galax i es, for examp l e) and ".carve out" .concentric spheres o_f .increasing radius
from any point of origin · (the center of our own galaxy, for .i nstance) the mass
.included w.i thin the · spherical space will :·vary as the .cube of the radius, whence:
The limiting .case .is obviously the ·v elocity of light :
VG = .C= ~ (l= 1)
M2 G. 2
E · = -- = MV X = Mc 2
G R G
Comparison With Relativ ity Theory
.In general Relativity Theo,ry the radius (Re) of the Universe .is expressed
by Einstein :
Mk
R=e
4 7T2
Re = _2_ MCL
1T .c 2
87TG
k=c2
~ I Now · in · the -four -dimensional continuum of Relativity Theory the ·d:tstance a.round ·- the
·Universe is (2.7TRe) · , as in Euclidean geometry, ·but all .physical action must take
place along geodesic lines - -i.e. within the ·"-sur-:face" of the -spher-e. The ·,equ:tva-
.lent o:f the -Euclidean diameter is there:fore (7T Re) and the ·.equivalent · "radius" -is
(.!!...Re). · · Since our ·,calculations have been based on Euclidean geometry, the ·-rela.tion-
2 -
.- ship o:f ·"radii · of action" is therefore:
. 7T
whence:
R =2-lie
7T MG
-R --= R
2 . e- :c 2 -
Then, :for the -gravitational energy of the ·Universe:
It follows that according to both the ,general ·Theory of Relativity and : the
present hypothesis:
· a) ·The ·gravitational energy of the Universe is equal . to the ·,energy associated
with the total mass of the -Universe. ·
b) The =gravitational velocity of the Universe is the -velocity of -light.·
c) The radius of the ·Universe is that distance at which the -total gravitational
energy o:f the cosmic .system, defined by the -radius, -is . equal . to the ·. energy associated
with the rest mass of the ·.system, starting ·from any point . of origin. ·
d) The radius of the ·.universe is the distance at which a particle starting
from any point of origin with radial velocity (c) will have lost all of its kinetic
energy due to the effect of the general ·gravitational field.
-38-
CURVATURE · OF SPACE
According to the present hypothesis, · there is associated with a cosmic
system of' mass (M) -an _inherent angular momentum (p)
-whence:
M2G .
P . = -c- = ~RR= uJ MR2
We · have just -found that for the -Universe:
MG
VR(Universe) = WR = ,Re= ;c
If' we -now assume that a ray .of' light ·emanating f'rom any point . of' origin
· is · susceptible to this hypothetical · rotation, then ·, in traversing · the ·:rad:tus -of ·- the
Universe the 0 elapsed time (t) is:
R
t - C
a.nd the angle through -which the ray of light -is def'lected is:
(.u R C
wt=~= c=1
Since · (w)is measured in radian, we conclude that -in traversing ,the :Universe from any
point of' origin a ray or light -is deflected through an angle ·of one radian.·
We thus explain the -curvature of the -path or a ray .of' light traversing
"emptyn , :-spaoe as due ·not to a physical curvature of' -space, as in realtivity theory,
but ·to a.n inherent rotational .property or the 0 general gravitational rield of the
Universe, ·
THE BENDING OF LIGHT GRAZING COSMIC BODIES
One of the -first triumphs .of Einstein's General Relativity Theory -was his
prediction that a ray of light grazing the 0 surface of the Sun must .be 11 derlected11 -by
" )
\
l
a certain . amount. · The ·,extent of the :deflection of path is given by Einstein as:
kM
a=27TR
(radian)
.where - (R) -is the -radius of the -Sun. · -Of , this ' deflection one --half represents the
ordinary effect of the -gravitational field of -the ·sun ,upon a photon of mass . (M)
where (E) · is the energy of the photon
(h) .is Planck's .constant of action
. .
Cf) Hi · the ,frequency · of vibration of the photon. -
The present hypothesis associates with the ·-gravitational field of the -Sun
a hypothetical .rotational velocity ( VR):
where (w) is a . hypothetical angular ·velocity associated with the '. gravitational field, ·
If a ray of light approaching · the -Sun -is affected by this rotational · property of the
·gravitational , field, we .-have for the d eflection, · first approaching and · then ·-receding
· from the -Sun :
= 2 MG
. Rc 2
R
t=2.c
which is precisely the amount of -deflection attributed by Einstein to the -relativistic
effect alone. ,, .
Again we find that the "bending" . of .-light · traversing a gravitational field
is attributable to a rotational property of the ·- gravitational field, rather than -to
a physical "curvature of space".
-40-
It will not have escaped · the .attention of' · the ·reader that .-the ·relation · thus
,expressed -lends ·substance to the concept that (VR)~ as applied to the ·rotationai
characteristics of' cosmic bodies by the mathematical :expression -developed in earlier
sections .of' this paper, has a real physical meaning.
THE"EXPANDING" :UNIVERSE
We .have identif'ied the gravitational or "escapen . ;velooity of' the Universe •
. with the -velocity of' light, and ·we now interpret this · to · mean that a particle of' any
mass ,leaving any .point of' origin with · radial .velocity · (c) · will lose all of' ,its kinetic
energy in traversing the ,radius of' -the Universe. · If' · the · loss of' energy -is a linear
-f'unction .of' the -distance · traveled:
where:
L
x-
R
6 Ek is the ·loss of energy
L is the distance traveled
R is the -radius of' the , urtiverse
·Let us now assume that the "·particle·" is a photon of initial energy:
·A photon -dif'f'ers f'rom an ordinary particle in that instead of' losing energy by slowing ;
· down, it continues to move at the ·velocity (c) and the 0 loss of energy is ref'lected by
a decrease in frequency . (f) ~
6 f == f
The change in · frequency • (6 f) at distance (L) is thus:
L xR
Since (6!) ·represents a decrease in energy and frequency, the result is a shif't in the
.color of the original photon toward the red or inf'ra-'red end of' the · spectrum. ·It
-41..:.
follows that a ray of light from a distant star . or gala.xy :will suffer a red shift in
traversing the distance to the Earth, and : the -extent of · the ·- shift will be porportional
to the -distance . travelled. ·
Such a red shift of light · from distant galaxies is known -to exist, and ·is
of course the -accepted basis for the concept of the · "Expanding ·-Universe . " According
-to this · theory, the red shift is attributed to the Doppler effect caused by an
assumed mutual . recessional velocity of the ·-galaxy and the ·,Earth:
V
·b. f = f X C""' L
where Vis the velocity of recession
Lis the distance.
The accepted conclusion is that every galaxy in the Universe is moving away : from every
other galaxy, and that the nutual .speed of recession · is directly proportioned · to the
·distance between the galaxies.
The present hypothesis is in direct conflict ·with such a concept; it postu-lates
an essentially static Universe in the sense that -in the universal scheme . of
things, galaxies represent "local".condensations of matter, which presumably retain
their original relative positions ·in space. The sred shift then ·becomes both a yard-
0stick by means of which the · radius .of the Universe can be measured, and at · the ·- same
time a means .for checking the plausibility of the ·present hypothesis. For if the
·radius thus -calculated is in keeping with the ·radius ,calculated by general Relativity
Theory, ·we shall be standing · on reasonably :solid · ground. ·
-42-
THE RADIUS OF THE UNIVERSE
where
:Einsteins · expression for the ·-radius of the , universe -is :
8 7TG · ·
K = - 2- = l.86x10-: 27
· c
(.('O) is the average ·density of · matter in the · universe. ·
·This reduces to :
2 - MG
R=-;; :~
as set forth in a previous section. ·
But to calculate (R) it is necessary to set a . value ·, for either · (/0) or · (M) ' •
Einstein originally solved the problem by ·introducing ·, a.n · nexpansive force " related
to the density(~) and ,related this in turn to Hubbles expansion by the -factor:
h = 4.71 -X 10-28
,,-0 = 3 o 5 X 10- 28
3h 2 = k,,<J
whence :
·R= ~ 3
h =1.73 · X 1027 ~
:1.94 .x 109 light years
Eddington, using a somewhat -different approach, ,calculated :
·2 . 1T Re = 6 ." 7 x 109 light years
= 1. 67 · x 10 9 light years
where · (Re) is the Einstein . or " initial " radius of the Universe.
-43-
-..
More recently, McVittie , has estimated the average density Qf matter .in the
Universe as approximately :
_/D = 10-27 · (Nax i mum)
/0 = 10-29 ( '1a more likely ·value·")
These : figUres give for the radius respectively :
R= 2·.02 x 109 light years
or 2o02 x 10 1 0 light years
Using the pres·_ent hypothesis, we _ ha:ve for one. set or data the ·value given
by Eddington for the Hubble . expansion velocity (V h) of frqm (5 x 107 ) to (1 x 10 8 )
per megaparsec of distance (L). Since a megaparse·c .is . eq~ivalent to (3 .26 x . 10.6)
light years, the result .is :
V =_!:___
.c R
a.oo:x1010
R= 3o26 X 10 6 X 5 ~QOX:J,.()'t - 1•95 X 10 9 light years.
To .check this, the estimated _average distance between two galaxies .is
(L = 1 . 1-x106 light years):, and the .corresponding "velocity of -mutual recession" .is
3 oOOxlo.io
R=l.50xlo8 x ..
9
=1.80 x 109 light years.
2;5 xlO
As a still · further check we . t ake · the actual o.ata for the galaxy .in Gemini,
where the di·stance from the Earth .is (L = L50x108) . and the irveloci ty of recession"
.is (Vh=2 0 50xlo 9):, whence :
a.oox1010
R = L50xlo 8 x . 9 = L sox109 light years .•
2 o50Xl0 ,
These results l ·end solid· support to the hypothesis -which provided the basis
from which they ~.ere calculatedo We thus have a method for .calculating the radius . of
I
the Universe which does not .irtvo1ve . the average density o·f' ~atter., although the re-sults
.indicate . that the estimates used by others are .close to the tr-uth_,
·CONCEPT OF THE UNIVERSE
:We . thus picture the Universe not as a four-dimensional surface in which
galaxies are embedded, ·but as a three-dimensional space throughout -which ·galaxies
are more or less uniformly distributed, 'Each galaxy . is the -center of .its own
,special Universe, ·whose size or radius is determined -by the ·-distance to which · light
·-can travel from the given point of origin. These .are then an infinite .number .of
such11(Jniverses/1 overlapping each other -endlessly , throughout -limitless .spaces each
with its own "radius" , of about two .billion light-years,
For the -concept of a physicially .curved sp~ce we substitute the -concept
of a .rotational property inherent in the Universal gravitational field, ·which causes
light to pursue a curved path as it traverses "empty" space, or as it "grazes" a
star, · · The ,four-dimensional "surface" of relativity theory thus becomes -merely the
locu:p of the paths of rays of light -emanating from any point of origin in all direc-
· tions. ·
The ,General Theory of Relativity pictures the , four :-dimensions .of the -continuum
as rotating about each other, In his .book "The ·-Expanding Universe"-, Eddington
has this to say on the ·: subject:
"Although · the ·mathematician visualizes four dimensions, his picture is
wrong ·in essential .particulars - at least mine is. -I see our physical Universe like
a ·bubble in four dimensions; length, breadth and , thickness all lie in the slkin of
the ·- bubble, · Can I picture this bubble rotating? Why, of course I can, · I fix on one
direction in the ,four -dimensions .as axis, and -I see the other thI'ee dime}lsions ,whirling
round ,it. Perhaps I never see more than two at a time; but thought , flits rapidly
.from one pair to another, so that all three seem hard at it, · Can you picture it like
that? If you fail, it is just as well. · For we know by analysis that a bubble in
, four dimensions does not rotate that way at all, · Three dimenlS~ons cannot spin around
-45-
~) a fourth. They must rotate two around two; that .is to say, the bubble does not rotate
about a line axis but about a plane. · .I know that .is true, but :1 .cannot ·visual-
:.ize .it."
.It .is hoped that the present hypothesis may help to resolve this dilemma
by attributing the rotational .characteristics of the Universe to the .inherent . properties
of gravitational fields. Surely it .is posslble - to picture a gravitational
field as having at any given point .components .corresponding to a three-dimensional
coordinate system with any desired orientation, each .component representing a
Poynt-ing ·vector (or .its equivalent) as .in .current physical theory. We thus as-cr.
ibe to each ·vector, attributes which ar~ .in keeping with the rotational .characteristics
exhibited by the field .in that area. The difficulty of trying to picture
space itself as rotating simultaneously about three (or four} axies disappears, and
we have field equations which .can be applied without misgiving as to their physical
.interpretation.
-46-
DEVELOPMENT . OF BASIC HYPOTHESIS
The ,basic ·mathematical expressions ;linking ·- the ·-rotational characteristics
o:f cosmic .bodies and ·system · to the ·process -o:f ·-central gravitational · condensation
are :
2 . M -G
V =G
R
2
= MV X G
~
2 x= MVR X
A correction :factor :for the ,e:ffect of loss of mass du~ing the process of
·x
central condensation ·was revealed in the parameter (Y.Vo) where (Y) appears to
have the· values (Y = 2 . - or Y = 4) . and (x) -has the •value (x= !) ,:for angular momentum
and · (x = 1) :for rotational ·energy:
ER(corrected) _M:G ~:o)
2
Xx 2 -~:) · (Y=2)
M
2
G~ p · (corrected)~
1
2y
0
(Y=2)
(Y=2)
A further correction factor for the effect of dif:ferential central con-densation
upon angular momentum ·was found to be:
p (corrected) =M~G 11 k or
'Thus far the · fundamental : expression :for (ER) .has been presented as purely
. empirical in derivation, . It .remains · to interpret ·this expression in terms of the
properties of matter and ·of ·gravitational -fields, and · thus provide theoretical basis
for an hypothesis which relates the ·,rotational characteristics developed ·by cosmic
bodies and system to actual .physical .phenomena ,inherent in the process of central
gravitational condensation,
We -begin by rearranging terms :
M2 G 2
ER=~ x Vo X
={), M V 2 X G o
where (/), MG) -is the relativistic increment of mass associated with the -gravitational
energy of the -s ystem.
· rt ·will be recalled that (V 0 ) was indentified as the "gravitational equilibrium"
velocity of the ·,system, and · that it was stated -that gravitational condensa-tion
can proceed only as the :difference between the · kinetic energy corresponding · to
(VG) 2 and · (V 0 ) 2 is dissipated from the system. We ,may therefore state :
The -rotational .energy · developed by a cosmic body or -system of constant
mass, during · the process of c entral gravitational condensation, is equal to the ·kinetic
energy of a mass equivalent to the relativistic mass associated with the -gravitational
energy of · the ·,system, ·moving at the " equilibrium ·veloci ty n, of the ·system, ·
This is very satisfying, and · it indicates that · we are dealing · with relativistic
phenomena , ·but it does · not provide a theoretical explanation of the r ota-tional
phenomenon .
Th e ·next step is to introduce the ,concept of gravitational .pole strength
and field strength as associated with ponderabler matter, thus permitting gravitational
.phenomena to be treated mathematically in a manner ·Corresponding to the :classical
-48-
treatment of electrostatic and · magnetic phenomena. It is a strange · circumstance
that in , the classical · treatment of gravitational phenomena, this seems seldom if
ever to be done. · Instead, . it is customary to use the · Newtonian expression for · the
grav.ttational force and field energy between two masses · (M 1
) (M 2 ) at distance (L):
FG =MlM2G
L2
MMG
EG= 1 2
L
where (G= 5.·55x10-s )is the "gravitational .constant" .whose .value -is determined empii::ically
by laboratory measurements,
·But if we write the · expression for gravitational force in the ·form:
we -see at once . that this is precisely of the -form of the 0 expression for the .force
between two electrostatic charges · (Q 1) (Q 2h
F
0
=Q1xQ
2
=
L2
where ( :
2
) .is the strength of the Coulomb-type field of .charge (Q) at distance (L) :,
and ( ~).-is the electrostatic potential of the field,
.In like manner we ascribe to a particle of ponderable matter of any mass
(M) a gravitational pole strength :
G- = MG!
and a Coulomb-type gravitational field whose potential at any distance (L) .is:
GL
= MG 1
L
-49-
It seems hardly necessary to note in passing that (MG!) has the same dimensions
as the unit o:f electrost_atic ·. charge or pole · strength,
.In order to .introduce the concept o:f gravitational pole strength and
field · action .into our equation , we -. consider any individual . particle o:f the system ,
and picture . it as :falling freely :from space toward the .center of mass under the· .
. in:fluence o:f the .central gra:v.i tational field, If, .. :for example, the . particle .is
a hydrogen a tom, .its mass is essentially that of a proton. .If -.it .is a hea:v.ier
atom, . or a molecule, .its mass .is essentially · the sum of the masses of - the protons ·
and neutrons .compassing_ the atomic -nucleus or nuclei ·, Since the masses of the
protons and. neutrons are practically. identical, : we may say without importa_nt
error that the mass of any particle whatever is the sum of the - masses of the
nucleons .comprised -.in the atomic -nuclei of the particle. Then .if (M) is the
mass of the particle -or body:
MG! =:.Z: ~G!"' ~
Mn
where (Mn) .is the mass of· a · nucleon,
The expression for (ER) ma.y now be wr.i tten _:
_M x M X ~nG
X (:o)
2 .x ER --
~ ' R
= 2 MG! x Mn ai t? )2 X
R . C
:~ im! x M~Vo 2 ·x a! =
R
. C 2
= 1.: M.G!
It
x 6 Mn G;
where~ Mn .is the relativistic mass .corresponding to the kinetic energy o:f a
nucleon moving at ·velocity (V 0
). It follows that each nucleon develops rotational
energy equal to the gravitational energy .corresponding to the action of
-50-
l
the .central gravitational pole : strength upon a . hypothetical gr~vitational pole
strength which .is related to the relativistic mass of the nucleon due to .its
velocity.
Since the rotational motion thus developed .is perpendicular to the
normal gravitational ·velocity, we hypothesize:
A particle falling freely under the .influence of a .central gravita-tional
field experience a transverse force which .causes .it to develop a rot~tional
motion about the - main .center of mass of the system to whicp .it .is related.
A particle moving at any velocity - (V) develops a second-order grav:....
itational pole strength .corresponding to .its relativistic .increment of mass at
velocity (V).
The nature of this relativistic gravitational pole strength is such
that .in moving radially , through a .central gravitational field it reacts with the
field to produce a tangential .instead of a .centripetal force.
The phenomenon .is therefore somewhat akin to the action of a magnetic
field upon an electrically .changed particle moving through .it. .In this .case, the
force .is due to the action of the applied magnetic field upon the magnetic pole
strength developed by the electric .charge because of .its motion, in accordance
with .classical physics •
. In support of the hypothesis thus presented, let us .consider the
relativistic precessi_on of an electron about an atomic -nucleus. The quantuum
mechanics expression for the · energy of such precession, taking the simplest .case
of the hydrogenatoiμ, .can be simplified to the form:
where
E/Precession)= «_:_(V•)' xY
'lH ~ - C
(-t...) .is the electronic charge .in e.s.u.
('l..H) .is the orbital radius
(V H) .is the orbital ·velocity
(Y) .is a numerical factor of the order of unity., whos·e precise ·value ·varies
with .circumstances.
This expression .is of .course analagous to our original expression for
(ER) : , and .it can be written:
E(Precession)= ...e.. x ~(vH \
2
x Y
/1Ji ~ -;
which .is .completely analagous to our expression for (ER) .in terms of the potential
of the central field and the second-order pole strength developed by .individual
particles.
We thus arrive at the extraordinary .conclusion that the phenomenon of
cosmic rotation .is akin to the phenomenon of relativistic precession .in atomic
physics.
Support .is also -provided for the .concept that a "gravitational pole" .in
motion .creates a second order pole strength. quite akin to the second order, pole
strength produced by the motion of an electric .charge:
e~e(;) 2
MG
1
~MnG
1
(~)
2
We have thus achieved our purpose of providing an hypothesis which .interprets
the development of axial rotation .in .cosmic systems .in terms of understandable
natural phenomena, and .in so doing have introduced the .concept of a new property of
matter and of gravitational fields.
The reader will not be surprised .if we now undertake to link gravitational
phenomena with the properties of fundamental particles. ·
-52-
THE ORIGIN AND .· NATURE OF - GRAVITATIONAL .FIELDS .
. It -was pointed out .in the previous section· that the mass ; of an atom
or of a molecule .is essentially -the sum of the masses of the .individual nucleons
(neutrons and proton) ·.contained within the particle, and also that the masses
of the neutrons and protons are so nearly .identical that the difference may be
ignored here. The gravitational properties of a ponderable body are a function
of .its mass regardless of size, shape, density. or physical state. That this
statement . applies to .ionized as well as "normal 11 matter is evidenced by the fact
that the Sun and Stars, which .consist primarily of .ionized atoms, possess normal
gravitational properties.
:rt seems .clear that the gravitational field of a ponderable body is
the sumatioh .in space of the gravitational fields of the nucleons .comprised
within. the atomic nuclei of the matter of which the body or particle .is .composed.
It follows that the gravitational pole strength of a particle or body may be
expressed:
where (Mn) is the mass of the neutron (or proton).
We thus assume that the gravitational field of the neutron, which .is
electrically . neutral, · is .identical with that of the proton, which is positively
charged. In -other words we assume that the electric .charge of the proton has
no appreciable effect upon .its gravitational properties. This is .in accord with
. .
the fact that gravitational phenomena are not electrostatic .in .character.
The next question . is whether the neutron and proton possess in .common
a property which might be related to their gravitational properties. We seize
at once upon the fact that both particles have magnetic moments which are of
nearly the same magnitude, but of opposite sign:
-53-
- h e
=-1.93~ -
477~c
wher~i(P .is - the magnetic moment of the proton
_..,t{n is · the magnetic moment of the neutron
.is . the "nuclear magn eton II of atomic theory.
It seems reasonable to assume that the difference between the two .is due
to the electric .charge of the proton. Since the neutron .is electrically -neutral,
we shall therefore treat the proton as an electrically _charged proton (or the
neutron as a neutralized proton) and assume -either that each possesses a ".core"
which has the Magnetic properties of the neutron, or that .it .is the neutron .itself
which set'ves as the 11 core 11 •
Now a magnetic moment is produced by a magnetic dipole, and the magnetic
- field associated with .it .is therefore · not of Coulomb - type. We therefore know in
advance that the magnetic field of the neutron per se .cannot be .its gravitational
field; .in the first place the gravitational field .is _not magnetic .in .character., .in
the second place we must seek for the sour.ce of a Coulomb type field.
One supposes therefore that the magnetic dipole field of the neutron,
though not .itself the gravitational field, .is .closely associated with .it. Since
we have hypothesi~ed that the gravitational field is a spec·ial kind of electremagnetic
field, we therefore surmise that the same physical .characteristics of
the neutron which produce .its magnetic dipole field, also produce an electromagnetic
field which is in fact .its gravitational field,
Quantum Theory ascribes to the electron a magnetic moment (,t(s) due to
inherent spin about .its own axis:
-54-
If we ascribe to the electron the .classical radius:
../?e = ~
M c 2
e
the spin-magnetic moment may -be written:
where:
// _ _e_! ~x ~
.......-t.,( s -~cz z,re
4ex~e x;
(,#fe)
h C
27Te
=6 .5sx10-s
has the dimensions of a pole strength.
This expression for (.-??/e) has two remarkable attributes:
1) Its numerical ·value is almost .identical with that of the gravitational
.constant. (G =6.66Xl0-8)
2) . . It is related to the fine structure .constant of spectroscopy by the
e·xpression:
where
/1,; e =-=- = ~
e . C
VH
!'..?< ) = :z,re_2 = ·v H . = 1
137 nc .c
(V 8 ) .is the orbital ·velocity of the hydrogen electron
(°'j is the fine struc tu-i·e constant
Whether these attributes ha:ve any deep physical significance remains to
be seen 9 but · they do support the possibility . that V/lfe) .is a hitherto unrecognized · l natural cons-tant, and that .it represents the .inherent magnetic pole strength of the.
electron due to axial spin.
Turning now to the neutron, in like manner we ascribe to it a .classical
radius:
-55-
and express .its magnetic moment:
-where
1.935
= /( n /J,rn X -2-
h .c
z,re
The possibility is thus presented that the neutron, like the electron,
also possess an .inherent magnetic pole strength which has the same ·value as that
of the electron, and which may also be due to axial spin.
Since the neutron · is electrically . neutral, the possibility suggested
.is that the neutron .is somehow .created by the union of an electron and a positron,
revolving about their .common .center of attraction or of mass. Like Bohr•s mech-anical
model of the hydrogen atom this .is doubtless an oversimplification. But is
inthe case of Bohr•s model such a .concept may provide a basis from which progress
.can be made . in developing an acceptable .interpretation, thus permitting an understanding
as to what the neutron really .is.
However., our objective is to find, .if possible, a meaningful relation
between the gravitational pole strength of the neutron and the hypothetical magnetic
pole strength, which will -hereafter be written (41() since .it apparently relates
to both the electron and the neutron. Thus, by .calculation:
=4.33Xl0-28 = 0.656X10-20
5.5sx10-B
This at once suggests that the numerical factor .is related to the.·velocity of
light, and to preserve dimensional integrity we write:
X -5.92
X 2:fT
1.06
-56-
.I
1
Treating the small numerical factor ("""i:o""~ as of no .immediate significance,
we recall Eddington 1 s complaint that he found such factors as (!) (~1_) more
2 7T
troublesome than the formulation of basic theory. The same . difficulty plagues us
here, because .it .is not clear whether the factor (27T) should be applied to the
· r.ight or left hand side of the equation, or even why .it appears at all. Presumably
.it is related to the . choice of units when tubes of force were .in ·vogue and the
factors (477)
1
(
27r) etc. were .introduced .into magnetic and electrostatic expressions.
Be that as .it may, it seems .clear that there is actually a meaningful
relation between the hypothetical magnetic pole . strength (/J?/) and the gravitational
pole strength (MnG!) of the neutron (or proton) • . It is nothing new to us to find a
second order pole strength related to velocity . - we have seen .it .in the .case of the
relativistic precession of the hydrogen electron, and in the rotational .character.
is tics developed by .cosmic . bodies, where it appli·es to gravitational pole . strength.
But there are two .important points : still · to be .cleared up.
The · hypothetical radius of the neutron (l.5x10- 16 ) .involves scientific
heresey, because .it is a dogma of present-day Quantum Theory that no length smaller
than the wave length :
has any physical meaning. The writer believes that this .is fallacious, and that
eventually .it will . be found that the radius of the neutron actually .is of the
order of (lo-1 6) cm.
The second point .is that the presence of the factor (~) .in the expression
.c
for (MnG!~ .implies that the gra:vi tational pole strength of the neutron varies with
·velocity, and that .it is zero if the neutron is at rest. This is inadmissable,
-57-
-because we have assumed that the gravitational pole :_strength of the neutron .is an
inherent property not dependent upon ·velocity.
A logical solution of this dilemma, appears to be the .introduction from
classical physics of the relation :
where
EA_ ; =_!_ (Numerically)
.c2
( E) .is the dielectric .constant .in ·vacuum
(.l<) .is the magnetic permeability in ·vacuum
( EPJ is a pure number.
We have assumed that the field . in question is electromagnetic in char-acter.,
and .it therefore seems not unreas.onable that · the factor (EA<) :, which .is . the
ratio of electrostatic or magnetic units to electromagnetic units, for either the
dielectric constant (E) or the magnetic perIIEabili ty (A.():, should appear .in the
present .case as a pure number :
or:
he
:27Te
It .can be shown by ·vector analysis that a particle .composed of an elect-ran
and a proton r~volv.ing about a .com.mon . . center of attraction wolllld .in fact produce
both a magnetic dipole · moment, and an external electromagnetic field which
has at least some of the .character.is tics required for the gra:v.i ta tional field.
The electric and magnetic ·vectors are tangential, not radial, and the
assumed rotation of the particle offers a possible explanation of the rotational
.characteristics which we have ascribed to gravitational fields. It .can be shown
-58-:-
that the orbital ·velocity· of the electron and positron .comprising this hypothetical
particle .is numerically .indistinguishable from the ·velocity of light (V 0,999999 .c) :,
and the strengths of the magnetic and electric field vectors are equal,
The gravitational field of a ponderable body .is thus tentatively pictured
as the summation .in space of myriads of .individual fields emanating from the nucleons
.comprised within the matter of which the body .is .composed. presumably the axis of
each neutron .is by no means fixed .in space, but precesses in such a way that .it
assumes all possible oreintations, probably within a very short period of time.
Moreover., the axes of .individual nucleons at any given moment -may be assumed to have
random orientation.
At any given exterior point, at any given moment, the local field there-fore
.is the ·vector sum of the magnetic and electric .components of the electromagnetic
fields of the .individual particles, with each Poynting ·vector possessing rotational
charac ~er.is tics involving a frequency of high order of magnitude. .It seems possible
that statis~ical treatment of such a field would yield fruit:f'u.l results.
It .is not our purpose to do more here than to offer the suggestion that
if the mystery of the nature of the neutron were attacked with some such .concept
as a starting point, an explanation of the gravitational properties of nucleons
might be found, and the true nature of gravitational fields might be revealed.
Clearly .it seems likely that .if this concept , or something akin to .it, has a basis
in physical reality there .is -some prospect that progress toward developing a
Unified .Field Theory might be made.
####
-59-
APPENDIX I
In support of the hypothesis presented in this paper, linking the
rotational characteristics of cosmic :bodies and systems with phenomena inherent
in the process of central gravitational condensation, the following calculations
are given for the · Sun :
Classically, the expression for the rotational energy of a solid
sphere rotating about a central axis is:
= 1 w 2 M R2
5
2 = 1 X 2 M VR
2 5
where · (w) is · the angular velocity, (M) is the mass, · (R) is the radius, and · (YR)
is the peripheral·velocity.
The Sun, however, is a gaseous sphere which is centrally condensed,
and the angular ··velocit y about the axis is not uniform. · It is therefore necessary
to apply . correction factors ' for · variable densitf and angular velocity.
Blackett, · in his paper on "The ·Magnetic Properties of Massive Rotating
Bodies" suggests the correction factors :
·k .= Ool6 · (for density)_
n .= ·> 0.7. '< 2.5 (for angular velocity)
where :
Using accepted values for (w,M,R) we find :
E/l... (Sun) .= n 2k x L63 x 10 4 3
On the basis of the present hypothesis :
Efi.- .= l, M~G tco) 2 .= 4.08 X 10"
- 11-
whence:
4~(l8 _ X 10 42 _
}6~3 X 104 2
k .= 0.16
0.25
n .= 1.26
This · calculated value f'or fn) · is not only . within the range indicated
by · Bjl.ackett; it · is close : to · Blackett•s ·mean · value of' (1.6). We - therefore write:
EA,.. (Sun) .= .! n2 k eu 2 M R2 ,= 1
5 2
and accept · this as strong support f'or the hypothesis on which the · calculations
· were -based.
· In · calculating the · angular momentum we proceed as follows :
· Inserting the correction factors (11, k) · in the classical expression
·f'or the angular momentum of' a rotating · sphere, we : f'ind by calculation:
p (actual) .= .! n k. eu M R 2 .= 2.27 · x 1048
5
This ·. compares with our theoretical expression:
p (theoretical) = M
2
G = 8.94 .x 10 48
· ·--· --c- -
This is rather -goo·d · support f'or the theory, but the calculated value is neverthe
· less - f'our times too large.
The necessary corre·ction f'actor., -due to the ef'f'ect of' central condensa-tion,
is f'ound readily:
E./l.. = l n2 k eu2 M R2 .=
6
- l2 -
M:G rco)2
'-
x ! .= ! M V./L 2
,r
Now (-y;,J ·is not the peripheral · velocity of the · sphere, but the · effective
average velocity corrected · for both · (n) and (k). It may therefore be · expressed :
~ -= n w x effective -radius
from · which it · is clear that the effective radius is
Reff.= 1 ~ k R
The correct theoretical · expression for the angular momentum of the Sun
is therefore :
p = M "'0, x Reff
=MxMG x
Re
Thus for a centrally condensed sphere th~ basic theoretical expression
· for angular momentum must · be · corrected for the effect or central condensation, by .
application of the factor 1~ k
Returning to the case of the Sun ~
If k .= ,0 . 40 x 0 . 16 .= 0.25
whence :
p(actual) = 'n kw M R 2 = M'G ,_
5
2 k
5 .c
We accept this as additional support for the present hypothesis.
Since the · Sun is a typical : star ·.of' · the Main Sequence, it seems r-easonaJ,le
to suppose that the present hypothesis also applies to such stars. In a subsequent
paper this matter will .be developed further. , and the hypothesis extended to cover
stars in general 9 our Solar Sy stem as a whole, the axial rotation of planets, and
the · s y stem Earth-Moon.
- 13 -
'r
'
May 1954
i N T . R·. O D. rr .c Y .I . 0 N
The purpose,. of this paper . is to present mathematical . expressions which
link . the rotational .characteristics of .cosmic bodies and systems with the process
_, of :. central gravit_atiohal .condensation; and to discuss :- t.he phys-i:cal significance
·of :th~se expressions. ·
The writer : has long . believed that . the axial . rotation :exhibitedby our
Solar system and its .component parts, by stars, and even ·by galaxies, .mu·s ~t have
.its . or.igin .in some .hitherto unrecognized natural law. It will .be shown· that.such
a law apparently exists and that the rotational .characteristics of .cosmic .bodies
,and systems are the result of .physical phenomena .. inherent in the process of .cen-tral.
gravitational :.condensation of .cosmic nebulous masses. Interpretation of.'. the
expressions .leads to the .conclusion that the ph~rronre.n ·a .refle,ct,. ~.he _existence of
hitherto .unrecognized .properties of matter, of gravitational :fields, and of.' the
action of a central g _rav.i tational field upon matter in ·motion within .its sphere. of
.influence.
It will .be found that the expressions set forth .in this .connection bear
a .close relation to certain aspects of general Relativity Theory. · ·These . include
the "bending" . of a ray of light grazing the Sun; · the ". curvature" of spac~:9 _ the
"radius" .of the Universe, and the hypothesis of . the "expanding" Universe. The
fact that such relations were obtained without the awesome trappings of relativistic
·m11thematics is rather startling, -but it was a not al together unexpected re-:sult
, of the present studies.
The simplicity of the expressions presented here, and of the methods by
- Which they were derived, plead strongly for their acceptability. The physic al
interpretation of these expressions .clearly . touches upon the absorbing.problem of
a Unified .Field Theory. It is therefore hoped that the subject matter of .this
papermay contribute in some small way to the development of such a theory.
EXPLANATORY 'FOREWORD
We · begin with the assumption that one may ascribe to a .cosmic . body of
·mass (M) and radius (R): , undergoing the process of central gra:vi tational..condensation
from a nebulous state, . the development of .. gra:vi tational . energy (EG) according
to the expression :
E _ M2G
G- -.
R
where G =6.66xl0- 8 is the gravitational c -onstant.
We may then ascribe to such a body a gra:vitational ·velo . .city(Vt,) such
that :
where
X=
V Jim
G -IR
. V . . ,/3~
. ' ~
.c = the ·velocity of light
The source of the parameter (X) .is a general expression for the kinetic
energy (Ek) of a body or particle of mass (M) ·. in motion at any ve.locity (V }: , de-rivedfrom
the relativistic expression for kinetic energy in terms of rest~mass
Ek~MoC' (11~ -1)
Whence :
Ek =MV
2
~-~) =MV
2
X
Now the function (X} has the interesting property that as (V) approaches
(V:o), the value of · (XJ approaches (X==!) as a lower limit; and as (V) approaches
(V=c) as an upper limit, (X) approaches (X=l) as an upper limit. It .is then a
kind .of numerical bridge between the classical expression
-2-
and the more ·. cumbersome relativistic expression . embodying the effect of .change
,of ·mass.
The justification for introducing (X)in the development of the present
expressions, is that its use avoids numerical . confusion which otherwise
resultsJ'rom·variations in . the ·numerical value of . (XJ •.
1
·.For ·velocities .up . to about · ( -yu) of the ·velocity of light, the ·value
of . (I) •departs very little from : (X=:iJ• ' It · follows that for most .cases we may
take: ·
V """'l2 MG G - R (X==!l
which.will at . once be recognized as the "escape" ·velocity of astrophysics. It
. is. the velocity which a body would acquire if .it fell freely from outer sp a.c...e
to distance . (R) .under the influence of the central gravitational field of a
body. of .mass (M) •
We -shall find especially useful a related velocity:
which .is the orbital or gravitational equilibrium velocity for a particle revolving
about . mass (M) at radius · (R). We -shall . for convenience identify this as the
"equilibrium" velocity, since it will .in general be related to a hypothetical,
rather than an actual, . orbital velocity.
· -3-
PRESENTA.TION .. OF. GENERAL .EXPRESSIONS
The basic concept of the present hypothesis .. is that there is a definite
mathematical relation . between the gravitational energy (EG) and the axial rotational
-energy (ER) developed.by a .cosmic body .or system undergoing the process of
central gravitational · cnndensation. If . the mass of the system is constant, the
relation. is:
~) 2 ·x
we -shall see that within . the accuracy of available data this gives the c a-r l"'e:-c t
numerical . result for the Sun •.
We may now ascribe to the. system a hypothetical average rotational velocity
(YR) · such that :
1.'he axial angular momentum (1>) of'·· the system is then :
M2 G
P =MVRR, =--c-
This extraordinarily interesting result may .be expressed:
·A cosmic body or system of constant mass, ,undergoing the process - Of
central gravitational , condensation, possesses an inherent constant angular momen-tum
whose numerical value is dependent only on the ~ass of the body or . system.
Wespoke . earlier of the beautiful simplicity of the expressions which
would be presented here ; these are excellent . examples . of what was meant.
-4-
APPLICATION . TO SPHERICAL BODIES ..
Before proceeding with : theoretical discussion of the physical significance
of these · expressions, ·. it seems . desirable . to prove their ·validity by
actual .calculation. To apply them to spherical .cosmic -bodies requires .certain
modifications, which will , now .be introduced.
The axial rotational energy of a solid sphere of uniform density .is
.claSsd.cally:
Where ~ .is the angular ·velocity
w R ,,,:vp .is . the peripheral ·velocity 1~R is the effective radius of revolution about the axis
(~) .is . the Newtonian .counterpart of the parameter · { X)
But .in general, ·. cosmic bodies are not solid spheres of uniform density;
they show differential .condensation . toward the .center, and the angular ·velocity
is not uniform· throughout the body. .It .is .customary to apply two factors· {n)
and {k) to account. for these .conditions. Thus for a spherical .cosmic body:
ER a~ n
2 kw 2 MR 2 la M"'1_
2
Xa Mn' "'' ~~ k R) 2
X
nu .is the effectiv~ average angular ·velocity
R .is the .. effective- radius of revolution about the axis, ·
taking .cent_ral .condensation·. into ·account ~·
n w \~ : k R =·v~ .is . the effect.tve average rotational ·velocity.
X replaces the cl~ssical factor (~).
-5-
.In-: like -manner., the axial angular momentum of' a solid rotatiing sphere
is classically:
Whence, for a .cosmic spherical body· . pa.! nkw MR ;a MV 'i.2. kRa Mnj r .! k R)
2
6 P Is .. ~ 6
In - the previous section we f'ound that · the .inherent angular momentum of'
a .cosmic system· of .constant ·mass ·.is:
M2 G
p (theoretical)=M.VRR = .c ·
It follows that we must apply a .correction for the ef'fect of different~
.ial .condensation on the · radius of gyration, and the angular momentum becomes:
. ~ .M_~ ~ P. (corrected) =M.V - k R= -- - k R 6 .c 6
We are now ready to .consider an actual .case, for which purpose we
select the Sun.
-6-
CALCULATIONS FOR ·.TaE . SUN
Using accept.ed data for mass, radius and per.ipheral angular ·velocity,
we have t'or the Sun:
· ex= o
°l,f'f"'- I 48
-=--u. -=3.83 X l0
R;
v~ = ~ -a :;:4~40 x 107
V ' 1 · 10-. 3 _:_a__:_ .= , 46· x
c ·
~ 2
=2.15 X 10-&
. ..
,.-6. tH 2 · . 43 ~ V · X=0.408 x 10
R c
The ratio of the two · results is about (4):, ·but the .correction factors
(n) · ( k) have still to be applied. The accepted ·values (Blackett) appear to be:
k = 0016
n= '> 0.7 '< 2 -.5
In view of the uncertainty in (n) we accept the value for (k) and calculate the
corresponding ·value for (n);
0 .408Xl04 S .
n 2 k = . 1.63 X 1043 = 0.25 n =1.26
The value thus .calculated - for (n) · is not only within the accepted
range; .it is very nearly the mean ·value and therefore would seem to be accept-able.
·
We then have, for rotational energy :
ER =1 n2kw~MR2X = M;·at:OJ 2 X
2 4 2
=MVR X = 4.08 x 10
-7-
For axial angular momentum ·the -cal.culations are:
2
p(actual) = 1- nkw MR. 2 = 2o27 x 104 8
5
,~ · G= 8.94 X 1048
l! ~ -- 2 ' 2G rr; · p{.theoretical) = MVR h· k R= T t k =2.27 X 1048
-We may therefore say .with .confidence that .in the case of the Sun, at
least, the validity of the mathematical expression presented here .is completely
supportedo
Since the Sun is a typical star of the Main Sequence, and these re-sults
were obtained without reference to the fact . that the Sun possesses a planetary
system, .it is to be expected that the same expression will apply at least to
stars of the Main Sequence. It will be shown .in a subsequent section of this
paper that this appears to be true, and that the data thus obtained for ·various
classes of stars throw .considerable light on ,the ·subjeq.t of stellar evolution,
including the relation between the Red Giant Sequence and the Main Sequence.
\
Before turning · to the problem of the Solf!:r S_ys tern, it .is necessary to
consider theoretically the effect of loss of mass during the evolutionary process
upon our mathematical expressionso
--8-
EFFECT OF LOSS OF MASS
Comte~porary -~osmogonical theory, as applied to our Solar system as a
whole, and more especially to the planets individually, teaches that during the
process of central gravitational condensation from the nebulous state, a very
large ·fraction of the original mass must have been lost. In the .case of the
Earth, Kuiper. estimated that the ratio of the .initial mass (M0) of the planetismal
to the present mas~ (M) .is about:
M1 =1200M
Urey, using a different method,arrived at the result:
.In our own trial calculation for the Solar system, and especially for
the planets, the necessity for a substantial correction factor was obvious, base·d
on the expressions given previously. On a purely superficial bas}s: it-appeared
that application of the parameter (~o) in inverse form solved the numerical problem,
but this was a mere rationalization until a reason for its use could be sug-
•gested. ·The reason, as will be shown., is that such a parameter appear&~J;o .·,be
measure of the effect of the larger original mass upon the rotational . . characteristics
of the residual body or system, and that it may even provide a measure of
.t.he ratj..ci of.· the .initial t:o the final mass •
. ·· - ·- _, i - - · •• ·.,! .
If the mass of the planetesimal is (M
1
) :, the corresponding .initial in- l . herent angular momentum of the system .is:
P - M1 2 G
1--c
Tentatively, we assume that mass lost from the system as ·.central gravitational
condensation proceeds, takes with .it a proportionate share of angular
momentum. The ratio of final to initial angular momentum is then :
-9-
- '\
J and the angular momentum lost from the ·system is:
M1-M M1
2 G M 1~M
pf last) p 1 -- = -- x--
Ml c momMen
1
tum to the quant1· ty · ,
2
.G)
The ratio of the final angular ~ is · there-fore
a measure of the ratio of initial to final mass.
We ,have seen however that · when differential ,central condensation .is significant,
we must write for a spherical body.
M2-G I.! M~ ~ p(residual) == .c 5 k x M = MVR 16 k R
And the residual rotational energy should be:
"d l =MR2_ G (vco) 2 X 0_MM·1.)2 X = ,MVR2 X
ER(resi ua ) ~ J ~
We are now ready to consider actual -cases, to determine ·whether these
assump.tions have meri,t.
--10-