Download deduction of the gravity law and quantum mechanical model of

1st International Congress of Serbian Society of Mechanics,
10-13th April, 2007, Kopaonik
DEDUCTION OF THE GRAVITY LAW AND QUANTUM MECHANICAL
MODEL OF DISCRETIZATION IN THE MACROSCOPIC GRAVITY
SYSTEM FROM SOLAR SYSTEM DATA
Aleksandar S. Tomić
VI Gymnasium, Milana Rakića 33, 11000 Belgrade
Peoples observatory, Kalemegdan, 11000 Belgrade
e-mail: [email protected]
Abstract:
The analysis of the historical process of coming to know the solar system from Kopernik’s measurement of the planetary distances and periods, and Kepler laws of planetary motion is applied for a reconstruction of the path for a direct deduction of the law of gravity interaction (different from Newton’s
procedure) and quantum mechanical model of the mass discretization in macroscopic gravity systems
(Bohr-Sommerfeld’s solution), too. We used in consideration the ideas of Borelli, Kepler, Huigens,
Boscovich, Titius and Bode and one’s own consideration of the second Kepler’s law as a possible discrete
distribution determined by natural numbers.
The conclusions were: gravity constants by nature is kinetic, kinetic concept demonstrated the
structural analogies between gravity and electrical force law with the same quantization rules. We extracted
the possibility of unification of gravity and electric force in kinetic formalism. Structural analogy of electric
and gravity interaction, and effors for unification of these physical fields, in this analysis obtained an
important support – unification is possible, but on kinetic level, instead used dynamic level.
Key words: Newton’s gravity law, Bohr’s quantization, Solar system, epistemology, unification of forces
1. Introduction
Kopernik published in the year 1543rd De revolutionibus orbium coelestium. His measurement
confirmed concept of world introduced by Aristarch approximately 18 century ago, in which
planets revolving around the Sun, instead around the Earth. Trigonometric method has been used
by solving the plane triangle for distance estimation, simultaneously planet to the Sun and planet
to the Earth. In the intervals bordered with accuracy of the measurement Kopernik obtained
permanent distances to the Sun and very variable distances to the Earth.
With regard to the consequences produced on the theological doctrines, usually were thought
on the Kopernik’s astronomical revolution [1], or on the overthrow by which the Earth removed
from the center of the world. Numerical data obtained by Kopernik presents also the same big
surprise. Difference in the measured data and adopted evaluations until Kopernik [2] were
remarkable. Solar system were much bigger than it were assumed. Kopernik introduced a new
measure for distance – astronomical unit, in absence of exact value for Earth’s radius. Also,
estimated sideral planetary periods from intervals between opposition (for external planets) and
maximal elongation (for internal planets), introducing implicit a hypothesis on the summing of
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the angular velocities as the vector, what is the most often ignored. As the result on this way were
obtained the possibility to calculate planetary circular velocities, with important detail - faster
decreasing of the velocity by increasing of the distance.
Kepler held his attention on this detail and on the fact that seasons (determined precisely by
solar position on the celestial sphere) were not equal in the duration. He is looking for cause, into
data for three decades permanently Brahe’s measurement of planetary positions, with the
accuracy better than each others. Result (on practically equally long calculation) appears as
Kepler’s three law of planetary motion around the Sun, published in De harmonices mundi-Liber
V , 1619 [3]. In Astronomia nova sive Physica coelestis [4] he proposed the mass of the celestial
body as the origin of the attraction between celestial bodies. So were Kepler the first man which
unified celestial physics (planetary motion) and terrestrial physics (Galilean kinetics). Both were
considered in the frames of kinetics, and Kepler attributed to the planets (freely moving through
the space, but the most not on the circles, and not uniformly) the same inertia. Looking for
explanation appears as imperative.
Galilei and its followers regard that transmission of the inertia principle to the planets were
just enough explanation, and ignored Kepler’s discovery. In these times Kepler marked not that
planetary motion must have centrifugal force as a compensation in equilibrium.
Borelli were the first which as scientist bring out the question a quo moventur planetae and
why planets stand on these distances.(A century latterly it repeat Charles Bonnet. [9]) He
considered [5],[6] this problem and gives attention to the missed compensation. Consequently
following Aristotelian method connection of the consequence and cause, Borelli concluded that in
the stable circular motion must be the equilibrium. He found geometrical explanation for the
existence of elliptical orbit (by conical sections), considered problem of the stability of the Solar
system as the equilibrium of the centrifugal and centripetal force (using Galilean method of
reduction celestial motion on the mechanical examples from terrestrial physics of fluid and quasi
–magnetic interaction). Explicit conclusion were that it what moving is internal principle, not
intelligent external while. Celestial mechanics were founded on the principle of conservation
the motion.
Borelli had attended that starting force in the solar system is constant. On the oscillating
pendulum he demonstrated that the motion by action of the constant central force at bigger
distance were slowly, how it has found Kopernik for planetary motion. According to Galileo the
gravity between planets and the Sun were active by origin of the planets, and planets followed to
fall to the Sun up to the moment in which achieved its actual velocities. (Freely fall to the Sun is
possible – for quiet planet, and in the any sense Galileo is right.) He considered in terms of his
kinetics only linear gravity action, other words – make the difference between this and circular
motion in the cause, too. Hooke and Newton generalized gravity action to all bodies in the
universe. Method of consideration used by Newton is well known, and we wish it repeat not. We
can now ask the answer to the question: Is possible the riches of contents in Kopernik’s data,
Kepler’s laws and Borelli’s ideas utilize, and obtain somewhat moreover? Specially, we bare on
mind explanation on the necessary transit from consequently continuity to discontinuity, in
consideration by Boscovich of stable orbit in system of few (three) body with gravity force as
interaction [7],[8], and Titius - Bode’s rule for discrete planetary distance [9]. With a small
appendix we think that it were possible, on following way.
2. Initial data and assumed knowledge
Our concept is very simple. In the measurement were found the source of motile force – by
Galileo in the Earth, by Kopernik and Kepler in the Sun, i.e. in the material bodies. Before than it
were made Newton, Kepler brought out the hypothesis on the mass of central body as the cause of
planetary motion. For stable planetary motion the necessity of velocity and impulse conservation,
permanent presence of the central force as the cause of motion, and the equilibrium of opposite
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oriented forces as the condition of the stability introduced Borelli. Newton accepted the inertial
principle of Galileo, introduced hypothesis of the equivalence of the gravity acceleration on the
Earth’s surface and the centripetal acceleration to the Moon at the smallest possible distance - the
Earth’s surface as the consequence of the same origin. He defined the acceleration as the ratio
acting force /mass of exposed body and the equilibrium of action and reaction implicit as vectors.
On the examples of two orbital motion, first the Earth around the Sun, and second the Moon
around the Earth, supposed the same origin of attraction in both cases (or equivalent gravity
constants, as alternative), using Picard’s [10] measurement of the Earth’s radius (1669-1671), the
third Kepler’s law and definition of centripetal acceleration, Newton derived the gravity force as
function mass of bodies and distance. By retroactive consideration obtained also the third
Kepler’s law in the form which determines the sense of the Kepler’s constants as the multiple of
gravity constants and the mass of central body - if angular velocity is used: ω 2 r 3 = γ ⋅ M . Just it
is contents hypothesis of Kepler, for which Kepler were not have had the time to make test of
statement. How big task it were, shown Newton in his Principia. Keplerian and Galilean physics
stay kinetic, other words defined for mass unit.
The conservations laws for isolated systems were formulated latter, as a generalization of
Newton’s mechanics Our consideration of the second Kepler’s law as possible discrete
distribution determined by natural number (Table 1) for the next consideration were needy too. A
simple procedure in connection of these facts gave very interesting results.
3. Deduction of quantum mechanical discretization
We start from the third Kepler’s law in the form:
v 2 r = const ,
(1)
also from the second Kepler’s law in the evidently discrete formulation, following from data
given in Table 1:
vr = (v1r1 )n .
(2)
By introduction of functional multiplier f n for the velocity discrete transformation by distance
change, which is unknown, if Eq.1 divides by Eq.2:
v 2 r (v1r1 )n ⋅ (v1 ⋅ f n )
=
= v = v1 ⋅ f n .
vr
(v1r1 )n
(3)
and dividing Eq.1 by squared Eq.2:
(v r )n ⋅ (v1 ⋅ f n ) 1 1 f n
v 2r
= 11
= = ⋅
.
(vr ) 2
(v1r1 ) 2 n 2
r r1 n
Now insert v = v1 ⋅ f n and r = r1 ⋅ n / f n in the third Kepler’s law:
n
v 2 r = const = v12 r1 = (v1 f n ) 2 (r1 ) = v12 r1 ⋅ n ⋅ f n .
fn
(4)
(5)
It is satisfied only for:
f n = 1/ n ,
(6)
i.e. velocity and stable distance transformation possess the form:
v = v1 / n ,
r = r1 ⋅ n 2 .
(7)
Direct application of formulae (7) on planets in solar system, by analogy with atom, is not
effective – for Mercury order number n = 1 is not adequate. But, for numbers obtained from
quantization of angular moment (Table1) it is just adequate.
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A.S. Tomić: Deduction Of The Gravity Law And Quantum Mechanical Model Of Discretization…From Solar System Data
km
)
s
planet
r (AU )
v(
Pluto
Neptune
Uran
Saturn
Jupiter
Asteroids
Mars
Earth
Venus
Mercury
**
*
39,53
30,110
19,218
9,555
5,203
2,709
1,523
1,000
0,723
0,389
0,1655
0,0414
4,73
5,43
6,80
9,65
13,06
18,11
24,13
29,79
35,02
47,87
73,22
146,44
r ⋅v (
AU ⋅ km
)
s
186,98
163,50
130,68
92,21
67,95
49,06
36,75
29,79
25,32
18,52
12,12
6,06
v 2 km 2
( )
2 s
n
∆
11,19
14,74
23,12
46,56
85,28
163,99
291,13
443,72
613,20
1145,77
2680,58
10722,34
31
27
22
15
11
8
6
5
4
3
2
1
0,178
0,048
0,458
0,200
0,201
0,086
0,058
0,089
0,174
0,055
E1
E2
30,955
26,972
21,536
15,175
11,213
8,086
6,069
4,916
4,182
3,059
Table 1: Quantization of angular moment and energy of mass unit in the Solar system
Here we have additional argument. If planetary distances express via geometric progression,
known as Titius-Bode rule [9] the average distance ratio of successively planets is equal [11]:
rn / rn −1 = 1.695 .
(8)
From quantization of angular moment for mass unit and third Kepler’s law can be calculated
distance and velocity of the first two (missing) planets, as:
v 2 ⋅ r = γ ⋅ M = const
v2 = (v 2 ⋅ r ) / 2(v ⋅ r )1 ,
v1 = (v 2 ⋅ r ) / 1 ⋅ (v ⋅ r )1 ,
(9.1)
r2 = (v ⋅ r ) 2 / v2 ,
(9.2)
r1 = (v ⋅ r )1 / v1 .
(9.3)
Obtained values are included in Table 1. It is obvious that distance ratio increasing if planets
coming nearer to the Sun:
rn / rn −1 = 1.695 , r4 / r3 = 1.859 , r3 / r2 = 2.350 , r2 / r1 = 3.998 .
(10)
Expressed with first possible planet distance r1 , determined by quantization of angular moment,
instead of real nearest planet, we have beautiful agreement:
r2 = 3.998 ⋅ r1 = 1.9992 ⋅ r1
(11.1)
r3 = 2.350 ⋅ r2 = 9.4 ⋅ r1 = 3.0552 ⋅ r1
(11.2)
r4 = 1.859 ⋅ r3 = 17.9 ⋅ r2 = 4.179 ⋅ r1
(11.3)
2
r5 = 5.089 ⋅ r1 , etc.
(11.4)
We can conclude that the same formulas are really valid for electrical force in the atoms and
gravitational force in planetary system, with only difference that here exist vacant positions.
The formulas (7) are well known, but here obtained from kinetic concept, while Bohr in his
atomic model derived from dynamical concept applied to the electric force. Electrical force law,
analogous in form with gravity force law, were obtained experimentally a century after appeared
Principia. For difference related to Galileo and Kepler which start from measured data, Bohr used
much more – mathematical model known as Newton’s concept of mechanics, and conservation
laws for system in weak interaction with environment.
Applying formulae for kinetic and potential energy and conservation of energy, valid because
of big distance to nearest stars, i.e. good isolation of solar system, it can be obtained formula for
specific energy (energy per mass unit) of each planet in kinetic concept, too:
2
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A.S. Tomić: Deduction Of The Gravity Law And Quantum Mechanical Model Of Discretization…From Solar System Data
E
E 1
= ( 1) 2 ,
m
m1 n
(12)
and it’s obvious from last column of Table 1. We obtained formulas for velocity, distance and
energy (pro mass unit) in known form, but here all quantity were kinetic, not dynamic. Other
words all can stay kinetic. It is original procedure, dynamical procedure is upgraded annex.
Why is it important? Gravity quantization can be realized only in this case. For electrical
quantization it is preferable, not critical. It is one from reasons why through over eight decades all
attempts by unification of electrical and gravity force were not gave result.
A remark. The first useful attempt of connecting formulas for electric and gravity interaction
made Fernando Sanford, just after Bohr’s model were accepted [12]. He attempt derive Bohr’s
equations from data for planets using only second and third Kepler’s law, and found for planetary
distances (r) and periods (T) relations:
r2 /T = Q ⋅ n ,
(13)
r = Q 2 ⋅ n 2 = 0.0428[ AU ] ⋅ n 2 ,
(14)
starting with n=3 for Mercury. This were equivalent to our formulas (2) and (7) for angular
moment of mass unit discrete distribution (what Sanford were not recognized), and for distances.
For each planet Sanford looking for formula giving energy, but not for mass unit. How planetary
masses were different, obtained result were thrown of, where editor and reviewer were not
pointed out that division by planetary mass make all regular, but now as energy of mass unit
( 1 / T = ν − frequency, k = const ):
E / m = k ⋅ν .
(15)
This attempt Sanford made in the best moment. But, conceptual ommission by author and not
enough attention by reviewers resulted as failure. (Sanford's paper author obtained by kindness of
dr Siniša Ignjatović, in this moment in Toronto, Canada, after investigation were finished.)
4. Gravity law
Introduced hypothesis on the mass of central body as the cause of planetary orbital motion
showed as right. Only this quantity is the same in the interaction with each planet. In the third
Kepler's law, the constants which as multiplyer of the mass equalize quantities and units, is
gravity constants, introduced by linear proportionality:
v 2 r = const = γ ⋅ M .
(16)
Centripetal acceleration is equal to the ratio of squared velocity and distance. If simple divide by
r 2 obtaines acceleration to the central body:
v 2 / r = γ ⋅ M / r 2 = ac ,
(17)
enough for description of planetary motion. Connection of centripetal acceleration and planetary
mass into gravity force made Newton. Using here only Newton's the second principle it follows
directly the gravity law:
F = m ⋅ ac = γ ⋅ M ⋅ m / r 2 .
(18)
The previous consideration gave a possibility to define gravity constante in solar system as
Kepler's constante on mass unit of central body (the most correct - at minimal possible distance
from the Sun) as follows from Eq.12:
v 2 r / M = v12 r1 / M = γ .
(19)
How we see, it presents also kinetic quantity, and gravity constants realy belongs to Kepler's
concept. Mass stay as a cause of motion, but determined (or harmonized) by kinetic parameters.
In the pairs of body Earth-Moon and other system planet-satelite, the same numerical value were
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A.S. Tomić: Deduction Of The Gravity Law And Quantum Mechanical Model Of Discretization…From Solar System Data
obtained, so that can be word on the universality of this constants. Dimenzionaly:
[γ ] = kg −1s −2 m3 , what can be expressed also as claim that a multiple of mass and squared time
is inverse equivalent (on the unit circle) to the space, i.e. to volume, if the gravity constants
treated as dimensionless quantity.
In the same status is Coulomb's constants, defined pro unit of electricity of the central body.
On this way it can be word on the mass (or electricity) as origin of the space, and the time as the
measure of the change (or evolution) in the distribution of the mass / electricity. These categories
were not so clean defined in published papers.
5. Conclusions
The historical process of how to know the solar system from Kopernik, Kepler, Huigens and
Newton is permanent present part of scientific heritage in contemporary education. With addition
of Borelli idea's (end of 16.century), Kepler's hypothesis on the cause of orbital motion, and
Boskovic's deduction of the transformation of the continuity into discontinuity in gravity systems
[7],[8], it appears as consequently logic connected with our consideration of the 2nd Kepler's law
as the discrete distribution determined by natural numbers - other words, with fractal structure of
solar system [13]. By using methaphysic most strong arms - logical deduction, we shown that in
the last etape existence of methaphysics in science were omited obvious possibility for short
deduction of (Newton's) gravity law and Bor- Somerfeld's solution for discretization in the
macroscopic gravity systems.
The first atempt of connection obtained formulas for electrical and gravity field, made by
Sanford in the year 1921 were estimated as failure. Structural analogy of electric and gravity
interaction, and effors for unification of these physical fields, in this analysis obtained an
important support – unification is possible, but on cinetic level, instead on used dynamic level.
References
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1591/1971.
[3] Kepler J., De harmonices mundi, Liber V, Wuetemberg, 1619.
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