Quantisation of the Auxiliary Gravitational Field
in Astronomical Systems
R. Wayte
29 Audley Way, Ascot, Berks, SL5 8EE, England, UK
e-mail: rwayte@googlemail.com
Research article. Submitted to vixra.org
11 January 2011
Abstract: A quantisation-field model has been developed to explain the
general dependence of angular momentum on mass squared of astronomical
bodies. The gravito-cordic field is proposed as a real controlling force acting
azimuthally, in harmony with normal gravity acting radially, to encourage
long-term stability of astronomical systems. The quantisation of the field
involves a gravitational de Broglie wavelength and associated force, which
organises material into stable orbits. Optimum coupling between the field and
orbiting material occurs for a specific velocity and spatial dimensions, as
derived by way of electromagnetic theory. For every system, the atomic fine
structure constant (α ≈ 1 / 137) has appeared as the major factor.
Keywords: stellar dynamics, galaxy dynamics, quantum gravity
PACS Codes: 04.60.Bc, 98.62.Dm
2
1. Introduction
It will be shown that numerical commensurabilities can be generated in
astronomical systems if we assume that a quantising gravito-cordic field is emitted by
orbiting and rotating masses. The original derivation of this field (Wayte 2010,
hereinafter Paper 1) [1], described how it propagates around orbits, contributing to the
binding and long-term stability of rotating systems. The quantised field model is a
mathematical analogy to the Bohr model of the atom, and we shall see the frequent
appearance of the inverse atomic fine structure constant (α −1 ≈ 137) , which suggests a
deep connection between electromagnetism and gravity. Detailed explanation of
quantisation in terms of electromagnetic theory is to be given in Section 3, but practical
inferences will be applied first to the observations in Section 2. Several aspects have
already been given in Paper 1 for galactic spirals and rings. Logically, quantisation
itself implies that material was moved forcibly into certain orbits and held there against
the perturbing forces which must have existed during the creation period. Thus the
infinite variety of spins and orbits and classes available according to Newtonian theory
was reduced to those which obey a few rules extrapolated from atomic theory.
2. The J proportional to M2 law
Figure 1, taken from Paper 1, shows how galaxy-clusters, spiral galaxies, globular
clusters, binary stars, main sequence stars and planetary bodies fit the J proportional to
M2 law. Although all these individual classes do not lie on the main line, they each
have their own
line, signifying a special proportionality constant. We shall try
to account for this observed degree of order in each astronomical class by proposing
that they have a quantisation-field which governs their angular momentum. It also
accounts for the conspicuous absence of objects between the classes. The field is
therefore proposed as a real controlling force, acting in addition to normal gravity.
3
80
GALAXY CLUSTERS
Sb & Sc GALAXIES
70
logJ
OPEN CLUSTERS &
T-ASSOCIATIONS
60
GLOBULAR CLUSTERS
VISUAL BINARIES
50
STARS
40
PLANETS
30
Fig.1
logM
40
50
The angular momentum versus mass relationship for various astronomical
bodies, showing a theoretical average line for J = 2.00 x 10-15 M2 over 40 decades.
Data taken from Allen (1973) [3].
Within each of these classes the angular momentum is proportional to mass
squared, for no known classical reason. The constant of proportionality varies from
class to class, and will be found to correspond to a
quantisation wavelength in
each case. Then the number of wavelengths around a given body has a characteristic
value or increases monotonically with body size. That is,
2πr = Nλ = N(h / mjv) ,
(2.1)
where N is usually an integer, mj is an effective mass characteristic of the quantisation
species (electron, meson, proton, hydrogen-electron) to be derived, h is Planck's
constant; therefore λ is the gravitational de Broglie wavelength for mj travelling at
velocity v. This explanation is unique in establishing some control in the creation of
astronomical bodies as a whole. The quantisation force appears relatively weak and is
easily destroyed by turbulence, which results in a scatter range of sizes in each class.
However, for the larger systems, the ratio (2πrmax / λ) only varies by a factor of 3 from
4
the mean value. Definite gaps exist between the classes because suitable quantisation
rules cannot be established there. Again, no classical explanations exist for these gaps,
nor for the particular sizes of bodies. Characteristic features of galaxies and galaxyclusters have already been covered in Paper 1, so the remaining classes will be analysed
here.
A little preliminary theory, common to these classes, will be developed first. We
wish to relate the angular momentum to the square of the body mass through some
quantisation law. If angular momentum and mass are approximately conserved during
the formation of the body, any law derived for present day bodies must reflect some
conditions in the original gas clouds. Let those conditions be:
GM = V 2 R , and J = (3 / 5)MVR = p ′M 2 ,
(2.2)
where V is the rotational velocity at radius R, and p′ is a constant. Immediately, it
follows that V must have been a
ideally constant velocity in each class. This
will be interpreted to imply that there was a particular gravitational de Broglie
wavelength, defined as:
1/ 2
e2
h
λG =
×
m s V Gm 2
=
where mj = ms(e2/Gm2)-1/2 is a
emitting the gravito-cordic field.
h
m jV
,
(2.3)
mass of the real source particle mass
is Planck’s constant and (e2/Gm2) is the electronic
ratio of electric/gravitational force. Then an expression which describes the practical
influence of λG on the orbit is:
GMm j
2πR
= K
,
λG
c
(2.4)
where (GMmj /ћc) is the effective gravitational strength constant, analogous to the
electromagnetic strength constant (e2/ћc = 1/137); and K = c/V is a constant for each
particular class of object, when V is the preferred velocity. On Fig.1, the main line
corresponds to a preferred velocity of [201kms-1 = (4π/1372)c] as described in Paper 1.
The analogous equation in the first Bohr orbit of hydrogen is of course:
e2
2πr1
= 137 = 1
c
λ1
.
(2.5)
5
Let us consider normal dwarf and giant stars as a function of spectral class with
regard to mass M, radius R and rotation as listed by [3]. For the best fit, the proton mass
will be introduced for mj in Eq.(2.3), where:
mjp = mp(e2/Gm2)-1/2 .
(2.6)
Then Table 1 lists (2πR/ λGp) spanning unity, and (GMmjp /ћc) spanning 1/137. The
cross-over points are at spectral type A8.5 for dwarfs and F7.5 for giants, whereupon:
GMm jp
2πR
≈ 1 ≈ 137
λ Gp
c
.
(2.7)
The fact that (GMmjp /ћc ≈ 1/137) for the proton/star entity, is exceptional evidence for
the electromagnetic nature of gravitation. Equation (2.7) is directly analogous to the
first Bohr orbit in the hydrogen atom; although stars are now in
equilibrium.
Table 1. Quantisation values for dwarf and giant stars.
Dwarf
stars
Giant
stars
Type
Ratio
2πR/λGp
O5
B0
B5
A0
A5
F0
F5
G0
G5
K0
K5
18.26
8.20
4.52
2.70
1.54
0.69
0.16
0.068
0.061
0.055
0.048
Strength
factor
GMmjp/ћc
18.74/137
7.99 /137
3.33 /137
1.67 /137
1.03 /137
0.84 /137
0.665/137
0.505/137
0.439/137
0.38 /137
0.33 /137
Ratio
2πR/λGp
B0
B5
A0
A5
F0
F5
G0
G5
K0
K5
8.14
6.49
4.78
3.31
2.34
1.29
0.68
0.65
1.03
1.63
Strength
factor
GMmjp/ћc
7.99/137
3.33/137
1.67/137
1.03/137
0.84/137
0.66/137
1.18/137
1.49/137
1.87/137
2.36/137
It is interesting to note that if there was an original cloud of uniform density
which condensed into many stars, then the many protostellar cloud circumferences
would have varied either side of the mean value by less than a factor 2, from stellar
type O5 to K5. This could signify effective control that the quantisation field had
during the star formation period.
6
Given that a star’s equatorial orbit is being considered stabilised in the above
analysis, it is possible that the star’s bulk may also be stabilised. Bulk mixing would
probably be impeded to some extent, so evolution of the star might be slower.
In the case of planetary spins, π-mesons of approximately 250 electron masses
will be taken as the source of the quantisation field, so mass mj in Eq.(2.3) is to be
given by:
mjπ ≈ mπ(e2/Gm2)-1/2 .
(2.8)
Table 2. Quantisation values for the planets.
Planet
Ratio
2πR/λCGπ
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
0.77≈3/4
1.93≈2
2.02≈2
1.08≈1
22.6
19.1
7.80
8.00
1.02≈1
Strength
factor
GMmjπ/ћVs
1.05/137
26.5/137
0.13/137
0.026/137
1.47/137
0.55/137
0.21/137
0.36/137
0.24/137
However, because the gravitational de Broglie wavelength, ( λ Gπ = h / m jπ Vs ), is large
compared with the planetary circumference, the gravitational Compton wavelength
( λ CGπ = h / m jπ c ) will be used instead; see 2πR/λCGπ in Table 2. For rotational stability,
this quotient should ideally be an integer or simple fraction, and follows from
modifying Eq.(2.4):
GMm jπ
2πR
= K π
Vs
λ CGπ
.
(2.9)
Here the gravitational strength factor normalised for spin velocity (GMmjπ /ћVs) is seen
to be clustered around 1/137, for no known classical reason given that the planets are
now either solid or in hydrostatic equilibrium. It is possible that quantisation fields
helped produce the present Solar System from a gaseous nebula and continue to
7
stabilise the orbiting planets, see Wayte (1982) [2]. Binary stars and clusters also obey
the equations of quantisation as follows.
Many stars are members of binary systems, so it is interesting to know how
these are stabilised in orbit by a quantised gravito-cordic field. Table 3 lists the relevant
parameters derived from [3] for visual and eclipsing binaries. For orbiting bodies, the
quantisation law changes from Eq.(2.4) to:
GMm j
M1 2πR 1 M 2 2πR 2
,
= K ′
+
λ
λ
c
M
M
G 2
G1
(2.10)
where M=(M1 + M2 ), λG1,2 = h / mjV1,2 , and K′ = p′c /G. The meson and proton
masses have been used for mj in eclipsing and visual binaries respectively, as they give
the best values for the strength factor (GMmj /ћc ~ 1/137), and orbit fitting (2πR/λG).
Average radii in elliptical orbits have been used.
Table 3. Quantisation values for eclipsing and visual binaries.
Eclipsing
binary
σAq1
WW Aur
AR Aur
β Aur
yz Cas
AR Cas
AH Cep
αCrB
AR Lac
U Oph
VV Ori
AF Per
ξ Phe
RS Sgr
R CMa
RZ Cas
U Cep
U Her
δ Lib
β Per
V Pup
U Sge
V356 Sgr
V505 Sgr
µ´ Sco
λ Tau
TX UMa
RS Vul
Averages
Ratio
2πR1
Ratio
2πR2
Strength
factor
λGπ1
λGπ2
GMmjπ
ћc
0.864
0.515
0.703
0.704
0.339
0.305
1.593
0.263
0.394
0.803
0.415
0.811
0.383
0.256
0.016
0.085
0.255
0.283
0.192
0.086
0.940
0.224
0.710
0.194
0.957
0.202
0.125
0.200
0.458
1.369
0.526
0.864
0.755
1.440
4.800
2.151
2.074
0.400
1.043
3.613
1.042
0.781
0.569
0.073
0.695
1.093
2.256
1.075
2.270
2.700
2.514
4.625
0.718
2.217
1.263
1.361
2.160
1.659
0.79/137
0.25 "
0.31 "
0.30 "
0.32 "
0.97 "
2.00 "
0.22 "
0.17 "
0.65 "
1.56 "
0.62 "
0.33 "
0.15 "
0.039 "
0.16 "
0.28 "
0.69 "
0.24 "
0.40 "
1.71 "
0.57 "
1.08 "
0.23 "
1.50 "
0.21 "
0.24 "
0.39 "
0.58/137
Visual
binary
ηCas
O2 EriBc
ξBoo
70 Oph
αCenAB
Sirius
Kru60
Procyon
ζHer
85Peg
Ross614AB
Fu46
Averages
Ratio
2πR1
λGp1
1.52x137
0.485 "
1.61 "
1.06 "
1.37 "
0.739 "
0.234 "
0.455 "
0.880 "
1.01 "
0.123 "
0.322 "
0.817x137
Ratio
2πR2
λGp2
Strength
factor
GMmjp
ћc
4.00x137
2.22 "
2.06 "
2.03 "
2.07 "
4.00 "
0.811 "
3.33 "
1.65 "
1.06 "
0.378 "
0.496 "
2.01x137
0.715/137
0.311/137
0.753/137
0.730/137
0.923/137
1.53 /137
0.202/137
1.13 /137
0.871/137
0.763/137
0.104/137
0.264/137
0.692/137
8
It can be seen that the visual binaries differ as a group from the eclipsing
binaries in that the orbit length relative to the gravitational de Broglie wavelength is
consistently around 137 times larger. On Fig.1, the short line through the visual binaries
corresponds to a preferred nominal velocity of c/(4x1372) in Eq.(2.2). Several of the
ratios 2πR/λGπ for the eclipsing binaries are small, such that a proton mass instead of a
meson mass for mj would be much better suited (x7) to bring the ratio near to unity.
Both mesons and protons could assist in the quantisation process, at the same time.
From Table 3 it is plain that the various ratios of orbit length to gravitational de
Broglie wavelength are around unity or 137, and the strength factors around 1/137; so
the quantisation phenomenon must have selected these from the continuum of classical
dimensions.
.
Globular cluster dimensions fit the gravitational de Broglie wavelength emitted
by the bound electron within the hydrogen atom. This wavelength is defined as:
h
,
λ GH = 137
m je V
(2.11)
where mje = m(e2/Gm2)-1/2 , for electron mass m. Table 4 lists the relevant parameters of
several globular clusters derived from [3]. The ratio of the cluster circumference 2πR to
the wavelength λGH is scattered around 137, and the gravitational strength factor is of
the order 1/137. It therefore looks as though the quantisation field from hydrogen
governed the size of the globular clusters at some stage. On Fig.1, the short line for
globular clusters corresponds to a preferred velocity of c/(1372) in Eq.(2.2).
The long-term stability of globular clusters may be assisted by detailed control of
stars throughout the body. Given the average result (2πR/λGH ~137), then around 137
quantised interior orbits will also exist due to hydrogen-electrons.
9
Table 4. Quantisation values for some globular clusters,
open clusters and T-associations.
Globular
clusters
Ratio
2πR
λGH
Strength
factor
GMmje/137
ћc
Open
clusters
Ratio
2πR
λGH
Strength
factor
GMmje/137
ћc
M3
M5
M4
M13
M92
M22
M15
N104
Averages
0.53x137
0.27 "
0.24 "
0.58 "
0.38 "
2.56 "
2.62 "
0.74 "
0.99x137
0.39/137
0.11 "
0.11 "
0.56 "
0.26 "
13.1 "
11.2 "
0.99 "
3.34/137
T-
Ratio
0.54
0.76
3.06
2.56
1.08
0.68
1.37
0.97
0.99
1.17
0.76
1.76
0.84
0.42
1.05/1373
2.10 "
10.5 "
8.43 "
4.21 "
2.10 "
2.81 "
4.21 "
3.51 "
3.51 "
1.76 "
7.01 "
2.10 "
1.05 "
associations
2πR
Praesepe
0.88
3.51 "
λGH
Strength
factor
GMmje/137
ћc
M103
N752
hPer
XPer
Stock2
M34
Perseus
Pleiades
Hyades
M38
M36
M37
SMon
τCMa
0.46
0.56
0.71
1.62
4.83
13.38
2.21
1.41
1.58
0.31
2.71
0.42/1373
0.35 "
0.45 "
1.40 "
14.0 "
4.90 "
3.15 "
1.05 "
0.87 "
0.56 "
2.72/1373
oVel
M67
θCar
N3532
Sco-Cen
Coma
KCru
Ursa Maj
M21
M16
M11
M39
Averages
0.24
0.79
0.38
1.33
4.64
0.74
0.48
1.17
0.56
0.62
0.97
0.28
1.11
0.52 "
2.81 "
0.88 "
4.54 "
3.86 "
1.40 "
1.05 "
3.51 "
1.40 "
1.40 "
2.81 "
0.70 "
3.06/1373
Tau T1
Tau T2
Aur T1
Ori T1
Ori T2
Mon T
Ori T3
Sco T1
Del T1
Per T2
Averages
Open clusters and T-associations have diameters about half that of globular
clusters, with masses 1372 times less on average. Nevertheless, the gravitational de
Broglie wavelengths of the hydrogen-electron give very good fits to cluster
circumferences, see 2πR/λGH in Table 4. The corresponding gravitational strength
factor is 1372 times less typically. It is possible that the definite step in (2πR/ λGH),
from 1 in open clusters to 137 in globular clusters, accounts for the dearth of
intermediate-sized bodies. According to Newtonian theory, there is no reason for two
such definite star cluster species to exist within an infinite continuum of sizes. On
Fig.1, the short line for open clusters corresponds to a preferred velocity of c(4/1373) in
Eq.(2.2).
10
3. Theory of quantisation of the gravito-cordic field
The previous section has shown that some astronomical phenomena may be
explained by proposing the existence of a quantising gravito-cordic field. This field is
emitted around orbits by the orbiting material, serving to increase the overall radial
binding force and also to organise material and encourage long-term stability in
preferred orbits. Astronomical features explained so far are: flat rotation curves in disc
galaxies, creation and maintenance of bar and spiral structures, rings of stars within the
discs, the universal angular momentum / mass squared relationship, and the general
masses and segregation of objects.
The problem is to see how the gravito-cordic field generates a real ponderomotive
force, which is physically capable of coercing material into specific orbits and
velocities. Rules covering the quantisation aspect of the field will be based upon the
Bohr atom and de Broglie hypothesis. The effective wavelength of the field is actually
to be the gravitational equivalent of the atomic de Broglie wavelength:
1/2
h e2
×
λ G =
Gm 2
m
v
s
(3.1)
.
Here, ms is the mass of the emission source particle which couples best to the size of
the astronomical object; for example, proton for stars, pion for binaries, electron for
globular clusters and galaxies.
is Planck’s constant, v is the orbital velocity, and
(e2/Gm2) is the electronic ratio of electromagnetic to gravitational force.
The
quantisation field can be optimised around galactic orbits for a particular material
velocity. This will appear to be equivalent to tuning a waveguide system to match a
microwave source. Various properties of the field, which were arbitrarily introduced
into the Paper 1, will now be derived in detail.
By analogy with the Compton wavelength in electromagnetic theory it is proposed
that the gravito-cordic field quanta emitted from particles of mass mo have a
gravitational wavelength:
1/ 2
λ CGo
e2
h
=
×
m o c Gm 2
=
h
,
m jo c
(3.2)
11
where mjo = mo(Gm2/e2)1/2 acts like a characteristic mass, but is not an actual particle
mass. The wave amplitude at the particle, as seen by a local observer may be expressed
as:
y = y o exp[− i 2πν o t o ]
,
(3.3)
where frequency ν o = c / λ CGo . If, however, the particle has a velocity v in the x
direction relative to a coordinate observer, the wave amplitude at the particle as seen by
the coordinate observer is, by application of the Lorentz transformation:
y ′ = y ′o exp
(
− i 2πν o t − vx / c 2
(1 − v
2
/c
)
2 1/ 2
)
.
(3.4a)
,
(3.4b)
This may be written as:
~ x )]
y′ = y′o exp[− i 2π(ν ′t − υ
(
where ν ′ = ν o 1 − v 2 / c 2
)−1 / 2
and ~
υ = ν′v / c 2 .
A Compton gravitational quantum emitted by a particle in the direction of motion in
a circular orbit would be measured by a stationary coordinate observer as having a
Doppler-frequency ν′(1 + v / c ) . Alternatively, a quantum emitted backwards would
have a coordinate Doppler-frequency ν′(1− v / c ) . Since these two quanta travel around
the circular orbit and may interfere, the net amplitude at distance x from any designated
origin on the orbit could be given as:
y = yo exp[− i 2π{(ν′t − x / λ′)(1 + v / c )}] + yo exp[− i 2π{(ν′t + x / λ′)(1 − v / c )}]
= 2 y o cos[2π{ν ′(v / c )t − x / λ ′}]exp[− i 2π{ν ′t − ~
υx}]
.
(3.5)
Here λ ′ = c / ν ′ , and t starts from zero as the particle crosses the origin. This is a
circularly polarised wave of fundamental frequency in the exponential term:
−1 / 2 2
c / h = m jc 2 / h
ν ′ = m jo 1 − v 2 / c 2
(
)
(
)
,
(3.6)
where mj = mjo(1-v2/c2)-1/2 is the increased relativistic mass. In the cosine term, the beat
frequency is ν ′(v / c ) , and the beat wavelength is therefore:
λ = c / (ν ′v / c ) = 1 / ~
υ = h / m jv = λ G
which is the
!
,
(3.7a)
wavelength. Hence, by superimposing or
interfering two Doppler-shifted Compton quanta in a circular orbit, we get a standing
wave pattern rotating around the orbit with the particle, and a simple interpretation of
12
the de Broglie wavelength λ G = 1 / ~
υ . The individual Compton quanta propagate around
the orbit at the velocity of light but we can show that the beats naturally stay fixed
relative to the orbiting particle source, as follows. From Eq.(3.5) the condition for a
beat maximum, for a given beat number s, is:
2π{ν′(v / c )t − x / λ′} = sπ .
(3.7b)
Hence by differentiation, the beat envelope velocity is:
dx / dt = λ′ν′(v / c ) = v
.
(3.8)
At an anti-node, a particle receives the two Compton quanta in phase with its
emission so there is resonance at the particle. All particles around the orbit would like
to align themselves so as to emit coherently because this is a lowest energy state.
Furthermore, the amplitude Eq.(3.5) will be single-valued when c / v = λ G / λ ′ = integer
n, if the orbit circumference is an integral number of de Broglie wavelengths. Then the
amplitude repeats itself spatially every distance λ G .
Now, even if the emission from several particles around an orbit is totally
incoherent, the resultant amplitude from q particles will, from Eq.(3.5) be:
y q ≈ q1 / 2 2 y o cos[2π{ν ′(v / c )t − x / λ ′ + ϕ}]exp[− i 2π{ν ′t − ~
υx}] , (3.9)
where ϕ is some arbitrary constant phase, for constant velocity of all the particles. By
convenient choice of origin, ϕ may be set to zero. The average intensity Iq of the net
quantisation-field due to q incoherent particle emitters is proportional to q, but there
will be a superimposed intensity modulation at frequency (2ν G = 2c / λ G ) proportional
to q1/2. That is, from Eq.(3.9) we get:
[ {
I q = ∑ 2 y o cos 2π ν ′(v / c )t − x / λ ′ + ϕ q
(
}] ) 2
q
(
)
(
) [ {
≈ q 2 y o 2 + q1 / 2 2 y o 2 cos 2 2π ν ′(v / c )t − x / λ ′ + ϕ q
}]
.
(3.10)
This intensity profile normally travels round the orbit at velocity v. It appears from the
galactic bar and spiral pattern data that there are two stable nodes per λ G so this
implies that incoherent matter is guided towards the two intensity minima per λ G by
some longitudinal radiation pressure gradient of Eq.(3.10). Since the intensity minima
13
are λ G / 2 apart, and the radiation pressure gradient must change direction across a
minimum or across a maximum, the resultant longitudinal force probably has the form:
FRP = F0 cos 4[2π{ν ′(v / c )t − x / λ ′ + ϕ f }]
,
(3.11)
which exhibits two stable points and metastable points per λ G along the orbit. An
estimate of force magnitude could be made by arbitrarily letting it be
to
the gravito-cordic field strength given in Paper 1, Eq.(3.6). The directional nature of
quantisation-field emission (see Section 4.4) helps to increase the quantisation force,
whatever the degree of coherence.
It is to be noted that the wave phenomenon expressed by Eqs.(3.5) to (3.11) is
entirely due to
of real circularly polarised quanta. There is no actual de
Broglie-type quantum, although the interference is characterised by the gravitational de
Broglie wavelength and propagates at the velocity of light. Gravito-cordic field quanta
are emitted by orbiting particles (electrons, for example), and exist as loops of material
attached continuously to their particles.
4. Matching the gravito-cordic field to orbital parameters
It will be shown that the quantised gravito-cordic field may be optimised in its
propagation around galactic orbits for a certain material velocity. This is equivalent to
tuning a waveguide system to match a microwave source. In Paper 1 it was shown how
orbit lengths in disc galaxies are stable when they satisfy the de Broglie condition for
hydrogen-electrons, (2πr = Nλ GH ) ; but the actual orbital velocity is also important.
The average rotational velocity V for flat rotation curves in Sa,b,c galaxies is of the
order of 201kms-1. In addition, the universal J proportional to M2 law for astronomical
bodies has a slope linked to this velocity; see Paper 1, Section 7. Consequently, this
velocity is probably not random, especially as it has a special relationship to the
velocity of light:
V201
4π
=
= 4πα 2 ,
2
c
137
(4.1)
14
where (α = e 2 / c = 1 / 137) is the atomic fine structure constant. Given that hydrogen-
electrons emit the controlling gravito-cordic field in galaxies, it is appropriate to
operate initially in atomic units. Thus, every unit is referred to the hydrogen 1st Bohr
orbit such that (e = m = ћ = v1 = r1 = 1 and c = 137v1). Then Eq.(4.1) may be expressed:
V201
= 4πα .
v1
(4.2)
Now, there is no established interpretation for this formula, but electric
is
inversely proportional to velocity, so this may represent a ratio of impedances. Then the
4πα term on the right turns out equal to the value in
of the characteristic
impedance of electromagnetic radiation, (alias impedance of vacuum Z0 = 376.73Ω ; see
Glazier and Lamont 1958, p133; [4]). For this interpretation, an atomic unit of
resistance is defined as the unit potential (e/r1), divided by the unit of current e/(r1/v1),
and will be named an
. That is:
(e / r1 )
(e /(r1 / v1 )
=
1
= 1.0 atohm ,
v1
(4.3a)
or in SI units:
e/4πε 0 r1
= 4108 ohms .
e /(r1 / v1 )
(4.3b)
It follows that:
4πα atohms = 376.73 ohms = Z 0
,
(4.4)
All electromagnetic radiation exhibits electric and magnetic fields in the ratio E/H = Z0
ohms. By
then, the controlling gravito-cordic field from hydrogen-electrons is
an electromagnetic phenomenon. So Eq.(4.2) is able to relate the de Broglie wavelength
of a galactic electron moving at velocity 201kms-1 to this most fundamental impedance.
Support for this special velocity-impedance relationship is illustrated in Figure 2
where the hydrogen-electron's orbit around the proton p+ is aligned in the direction of
galactic velocity V201 so that the electron describes a helix of pitch V201/v1 . Now,
is by definition the inverse of
and is equivalent to velocity in
Eq.(4.3a); therefore, the admittance of electromagnetic radiation (Y0 = 1/Z0) is
equivalent to a velocity V0 as shown in Figure 2b. It follows that along the helical
trajectory we have a rather special arrangement:
15
tan θ =
V201 v1
=
v1
V0
.
(4.5a)
Then, in terms of the impedances involved in the real electromagnetic interactions
(where z1 = 1/v1 ), we get:
Z
z1
= 0 = 4πα .
Z 201 z1
(4.5b)
θ
θ
Figure 2 (a) The electron in the first Bohr orbit of a hydrogen atom travelling around
the galactic orbit at velocity V201 describes a helical trajectory. (b) The instantaneous
electron velocity is ve , with V201 and V0 occupying congruent triangles.
To get this result, V0 had to be parallel to V201 , which is in the direction of net current
flow and gravito-cordic field emission around the galactic orbit. Equation (4.5) is
therefore taken to mean that at an orbit velocity of 201kms-1, the impedance of
electromagnetic radiation is coupled to the hydrogen-electron impedance z1, and this
optimises the emission and reception of gravito-cordic field quanta,
"
#
The next step is to apply this optimum propagation equation (4.5) directly to
galactic structure. Practical units for the gravitational domain must be reinstated by
putting mje = m(e2/Gm2)-1/2: then the gravitational de Broglie wavelength for hydrogenelectrons (see Section 4.5) becomes
λ GH = (h / mV201 )[ 137 × (e 2 / Gm 2 )1/ 2 ] , and r1
becomes rGH = r1 [ 137 × (e 2 / Gm 2 )1 / 2 ] . Equation (4.2) with (4.4) may now be written:
16
1
r
Z 0 = GH = 4πα atohms.
λ GH / 2π v1
(4.6)
Classical waveguide theory enables us to interpret this equation, (see [4], pp194,
228). For a waveguide of dimensions
transmitting an H10 wave of guidewavelength
λ g and free space wavelength λ , the waveguide impedance is:
b λ g
Zg = 120π .
a λ
(4.7)
Thus, for optimum coupling of a gravito-cordic field to such a galactic waveguide, Zg
should probably be equal to Z0 . In the simplest case, let λ g = λ GH / 2 for intensity
tuning, and λ = 2π rGH . As the radial separation of stable galactic orbits is λ GH / 2π ,
(Eq.(7.5) in Paper 1), let this define the waveguide width . It follows that
= 2rGH is to
be the effective waveguide height in Eq.(4.7). That is, each stable galactic orbit acts
like a waveguide with dimensions defined by the hydrogen first Bohr radius and the
gravitational de Broglie wavelength such that the waveguide impedance is optimally
matched to Z0 for V = 201kms-1. The scatter of velocities around 201kms-1 for Sa,b,c
galaxies indicates that the quantisation force is not strong enough to coerce the disc
matter into perfect agreement with Eq.(4.6), but nevertheless there is no known
classical reason why this particular average velocity should exist at all.
$
% !
It has now been shown that the quantisation wavelength is matched to the
dimensions of the stable orbits, as far as characteristic impedance is concerned.
However, it was seen in Section 3 that there are
de Broglie-type quanta, so the
question arises as to why the orbital dimensions should match the
wavelength λ GH . According to Eq.(3.9) the net field amplitude is a circularly polarised
wave of frequency ν ′ modulated at frequency ν ′(v / c ) . In order to extract this
modulation frequency it is necessary for the wave to interact with a body.
For example, in the static coordinate reference frame there is a field amplitude yq in
Eq.(3.9) acting on particles of internal field Eq.(3.4), so by Lenz’s law the interaction
force is:
17
Fc = − yq exp[+ i 2π(ν′t − ~
υx )]
= q1 / 2 (2 yoi )sin[2π{ν′(v / c )t − x / λ′}]
.
(4.8)
Thus at any position # the matter is excited by a field oscillating at the de Broglie
frequency ν ′(v / c ) . Now, from scattering theory, (van der Hulst 1957) [5] and radar
theory ([4], p.273) it is known that Mie resonance is optimum when the particle is
around ( λ / 2π ) in radius; this actually corresponds with the spacing of the galactic
orbits, ( λ GH / 2π ). It is suggested therefore that the force Eq.(4.8) causes resonance at
the gravitational de Broglie frequency in any orbiting gas clouds with this radius. Given
that reception and radiation characteristics of antenae are identical, according to the
Reciprocity Theorem, ([4], p.269), then the reception of the interference wave Eq (3.9)
is also optimised by matching radiation impedance to the orbital dimensions. Thus, the
field Eq.(3.9) really is capable of exciting resonance through radiation
reaction forces, (see Panofsky & Phillips, 1962) [6].
It is interesting to calculate the effective polar diagram of the ponderomotive force
due to the gravito-cordic field around the orbit. The transverse force field Eq (4.8),
vibrating at frequency ν ′(v / c ) may be viewed as an equivalent dipole. If there are
( N = 2πR / λ GH ) (typically several million) gravitational de Broglie wavelengths
around an orbit, then there are effectively N dipoles properly phased to enhance one
another in the forward direction. Such an assembly of dipoles constitutes an “end-fire”
aerial array ([4], p.314).
The main lobe total angular width is given by
( β = 2 cos −1 (1 − 4 / N) ≈ (32 / N)1 / 2 ), for large N. Therefore the beam width after one
revolution is ( ∆R = 2πRβ = λ GH (32 N)1 / 2 ), and this corresponds to a spread over
∆R /(λ GH / 2π) orbits. The total number of nodes in these orbits encompassed by this
beam spread is then 2π(32 N 3 )1/ 2 . Thus, the quantisation force of each orbit extends
laterally across a range of orbits and tends to synchronise the de Broglie frequency,
thereby encouraging production of
! . Since, the overall
synchronisation force increases with N3/2, the larger disc galaxies should show flatter
rotation curves, except when turbulence destroys the coherence.
18
&
!
According to Eq.(3.7a),
% !
electrons would carry a gravitational de Broglie
wavelength ( λ Ge = h / m je V ). However, in galaxies the quantisation field appears to be
emitted from the atomic hydrogen-electron, because the replacement of λ Ge by
( λ GH = 137λ Ge ) fits the galactic parameters better and may be explained as follows.
The hydrogen-electron’s motion around the proton in the 1st Bohr orbit of length
2πr1 = 137λ C will modulate the gravito-cordic field from the electron itself at 137 times
the Compton wavelength. This modulation introduces a factor 137 into Eqs.(3.1) and
(3.7a), as was employed in Section 4.2. Orientation of the hydrogen atoms in the
galactic orbits will not affect this process, although it was assumed that circularly
polarised quanta would be optimum for the analysis in Section 4.1.
5. Conclusion
Published astronomical data from a variety of sources [3] have been analysed to
reveal significant quantisation of angular momenta and body parameters. Analogy with
the Bohr atom is striking and the regular appearance of the fine structure constant
implies that gravity is electromagnetic by nature. The gaps between classes may he
attributed to a lack of quantisation rules to operate there. One aspect of the results is
that the quantisation forces are strong enough to cause approximate quantisation but not
so strong as to eliminate variety in each class. A great deal of further evidence for
quantisation has already been presented in Paper 1 on the characteristic features of disc
galaxies.
The gravitational de Broglie wavelength of the gravito-cordic field emitted by a
particle has been calculated from first principles, and then the corresponding
ponderomotive force derived. Emission of the gravito-cordic field has been found to be
optimum for a particular velocity of 201kms-1 because of its dependence upon the
characteristic impedance of electromagnetic radiation. This impedance was also found
to be matched to galactic orbit dimensions in terms of classical waveguide theory.
19
Finally, a polar diagram for the gravito-cordic field radiation was calculated, which
confirmed that larger disc galaxies should possess flatter rotation curves.
Acknowledgements
I would like to thank Imperial College Libraries and A. Rutledge for typing.
References
1. Wayte R: 2010, viXra:1002.0018, (Paper 1).
2. Wayte R: '
1982, 26:11.
3. Allen CW: "
(Athlone Press: London); 1973.
(
4. Glazier EVD, Lamont HRL:
!
)
$
*
&
+
HMSO, London; 1958.
5. Van de Hulst HC: ,
6. Panofsky WKH, Phillips M: Addison-Wesley, London; 1962.
Wiley, NY; 1957.
.
'
.