- Email: [email protected]
Clues to discretization on the cosmic scale
17 March 1997
PHYSICS ELSEVIER
Physics Letters A 227
LETTERS
A
(I 997) 16% 17 I
Clues to discretization on the cosmic scale A.G. Agnese ap’,R. Festa b*Z a Dipurtimenro di Fisica, Universith di Genovu. and INFN, S&me
dr Gmovu. Geneva. I& b Dipurrimento di Fisicu. Universirh di Genova, und INFM. Unitir di Geneva. Genova. Italy
Received 26 June 1996; revised manuscript
received I2 December 1996; accepted for publication Communicated by P.R. Holland
I8 December
1996
Abstract Starting from discretization rules analogous to those of Bohr and Sommerfeld, values for a few mechanical quantities associated to celestial periodic motions are obtained. A gravitational dimensionless structure constant (Yecan be singled out, whose value, determined from data relative to the solar system, allows us to quantitatively account for phenomena on a much wider scale. PACS: 96. IO. + i; 98.50. - v; 03.65.B~
1. Introduction
It is well known that the weak equivalence principle, i.e. the fact that the trajectory of a particle subjected to gravitational forces does not depend on its mass, is satisfied only in the classical limit by the axioms of both the old (Bohr-Sommerfeld) quantum mechanics $ Pj dq, = ‘1’
(1)
and the properly called quantum [p,,
qj]=
mechanics
-ih.
Some years ago, Greenberger
(2) [ 1I, in order to recover
the overall validity of the equivalence principle, formulated the hypothesis that the more general commutation rule holds [p.
-i(h+mcXf(-&.
$)),
where A (X = A/27r) is a fundamental length and f< . > a dimensionless function with the necessary symmetries with respect to its arguments. On the r.h.s. of Eq. (3) the first term would dominate on the microscopic scale (negligible gravitational fields) and the second on the “megascopic” or celestial one (predominating gravitational fields). Taking into account the necessary invariances, the author considered the simple commutation rule [P,
x]=
-i(h+y2-J
(4)
where y is a parameter, which can be variously defined provided that it is very small but in strong
’ E-mail: [email protected]. ’ E-mad: [email protected]. 0.375.9601/97/$17.00 Copyright Pfi SO375-9601(97)00007-8
xl=
0 1997 Published
by Elsevier Science B.V. All rights reserved.
166
A.G. Agnese, R. Festa/Phy.sics
gravitational approximated
fields. by
In this limit
Eq. (4) can be
Letters A 227 (1997) 165-171
from suitably tuned (Bohr-Sommerfeld).
action
discretization
rules
2. Orbital motions around the Sun so that the particle momentum corresponds to a non-local (integral) linear operator, in contrast with the local (differential) operator of quantum mechanics. The hypothesis that the celestial phenomena obey quantum rules is nowaday repugnant to the minds of almost all physicists. Nevertheless, considering that the gravitational forces represent the dominant forces only at large scales, i.e. at the celestial level, one can accept that some breakdown of the classical “macroscopic” mechanics at this “megascopic” level would not be astonishing, though in no way necessary. The Greenberger’s assumption (3) represents by no means the sole possible generalized commutation rule, as explicitly pointed out by the author himself. Moreover, it turns out to be Lorentz invariant only if fl p/me, x/A) takes a constant value. As a matter of fact, even from a purely mathematical point of view, the dependence of the commutator (4) on p* is by no means necessary. If we simply consider the constant term in the Taylor’s power development of f in Eq. (31, we obtain [p,
xl=
-i(h+iXmc),
(6)
where A is a suitably chosen length. In the case of a body gravitating in the force field of another body with mass M, A could be directly related to the gravitational source strength through the dimensionally determined relationship 1 GM x=_..-._, %
C2
Let us choose as the starting point the customary Bohr-Sommerfeld rules for (multiply) periodic motions
#
e2
pj dqj=nj2r--,
(8)
ffec
where the Planck’s constant is written, using the fine structure constant ff, = 1/ 137, in the form
fi=e’.
(9)
%C
Our conjecture is that a simple correspondence holds between electric and gravitational force fields, which can be expressed by e2 -*a,
GMm ffg
(10)
’
where (Ye is a gravitational structure constant, whose value must be evaluated using available experimental data. Thus, for periodic plane motions of a particle of mass m in the gravitational field of a particle of mass M the discretization rules should hold #
#
p, dr = k2rr-
GMm ffgc
pv dq=12Ir-
’
(‘1)
GMm
(12) %c
(k
(7)
where (us is a dimensionless constant. This form takes implicitly into account the required relative weight of the gravitational field in comparison with other “microscopic” fields represented by h. In fact, we prefer to stop at this stage and not try for the moment the setting up of a megascopic analogous of the full quantum mechanics, as Greenberger and other authors [2-41 do. In this Letter we will only conjecture that periodic motions of celestial bodies can be described, approximately enough, starting
and 1 are the customary radial and azimuthal numbers). We will now examine the possibility that this rule accounts for a “gravitational atom” model of the solar system. Introducing the gravitational potential energy v=
GMom -__ r ’
(‘3)
where MO is the mass of the Sun, it follows almost immediately that the major semi-axis a, (where n = k + I is the principal number) of an allowed discretized elliptical orbit is given by a = &2 n I
’
(14)
A.G. Agnese. R. Festu/
Physics Letters A 227 (1997) 165-171
where
and the corresponding
167
period
GM,
represents the “Bohr radius” of the planetary system. The minimum and maximum distances from the Sun, and the corresponding maximim and minimum orbital velocities are then given by 4 n.l =a,(1
-&).
Up)= .
&/=a,(1
f&),
u;y’“‘=(l
(1 +,)F,
-$$
(16)
where l2 &=
(17)
/-- ‘-2
indicates the elliptical eccentricity, and, as customary. I- l,..., n. For circular orbits (I = n) one obtains the constant velocity v, = cygc
P, = 2li--n3 (‘9) 03c3 . B A first estimate of uy can be achieved through Eq. (14) by using the known mean distance of a given planet from the Sun and choosing a suitable (small) principal number n for its orbit. This first estimate can then be used with the same equation in order to single out the appropriate principal number for each of the remaining planets. A final estimate of 0”: (and consequently of LYEthrough Eq. (1.5)) is then obtained by an overall least-square fit on all planets. It turns out that the only consistent choice lies in assigning to Mercury the principal number n = 3. Then, to the other planets of the solar system the following quantum numbers almost automatically correspond: Venus -+ 4, Earth -+ 5, Mars + 6. (Ceres --) 81, Jupiter --, 11, Saturn -+ 15, Uranus -+ 21, Neptune -+ 26, Pluto + 30. The best fit using these quantum numbers gives uy = 0.0439 * 0.0004 a.u.,
(20)
(18)
n
Planets of the Sun
IO
Fig. I. Experimental
20
15 Pr~nupal
ndmbar
25
30
n
orbital radii of the planets compared with the values given by the formula u,, = ~l,n’. The correspondence between
planets and principal numbers is reported in the text.
A.G. Agnese, R. Festu / Physics Letters A 227 (1997) 165-171
168
In Fig. 1 the a, quadratic rule given by Eq. (14) is graphically compared with the actual orbital major semi-axes of the planets of the Sun. We remark that Eqs. (1 1), (12) and their consequences only determine the allowed orbits. Actually. only a few allowed orbits are occupied by the planets. The reason for this can only be the subject of guesses (in fact, the origin of the planets is yet today a subject of guesses). One can suppose for instance that the protoplanetary material was originally distributed to form rings in correspondence with every quantum number, and that gravitational instabilities or perturbations have brought a final state of aggregation of the material in planets over some orbits in correspondence with particular quantum numbers. Otherwise, one can guess that external gravitational perturbations acting on an almost continuum protoplanetary material in gravitational, centrifugal and thermodynamical equilibrium, led to spatial density resonances mainly corresponding to some transversal eigenfunctions (which, of course, must satisfy circular periodicity conditions). Anyway, and with a certain surprise, one can see from Fig. 1 that the interpretation of the planetary orbits with a gravitational atom model is in reasonable agreement with the observed data, even for small principal numbers. We acknowledge that it is still not enough to give validity to our conjecture, and that alternative hypotheses could be formulated. In order to extend the investigation about this topic, one would need some more information about other planetary systems. Till the end of 1995 no such information was available. Fortunately, in recent paper [5], the planetary motion of a newly discovered companion of the star 51 Pegasus, with mass ranging between one half and twice the mass of Jupiter and mean distance from 51 Pegasus of about 0.05 a.u., has been examined: it turns out that its orbit nearly corresponds to the
quantum number n = 1 of its planetary system. Some reported uncertainty on the mass of 51 Peg (a star “quite similar to the Sun”) could be removed if one guesses that the orbit of its large planet exactly corresponds to n = 1: on the basis of the observed period (P = 4.23 days) a distance a, = 0.055 a.u and a mass MSIPes = 1.26& can be obtained. Thus, some recent knowledge about a simple “gravitational atom” different from our solar system seems to support our guess. Very recently other extrasolar simple planetary systems have been discovered. For the moment, the available data [6] seem to validate our guess. In particular the orbits of the planets of HR5185 and HR458 correspond to II = 1.
3. Orbital motions around the planets We wiH now shortly examine some consequences of our guess on the allowable orbits around the various planets. Of course, the major semi-axes of these orbits are obtained by substituting in Eqs. (13) and (15) the mass of the Sun MB with the mass Mp of the concerned planet. Assuming the universality of the gravitational structure constant CY~,one immediately gets from Eq. (15)
(21) where u\P) indicates the “Bohr radius” for orbitating motions around the planet. By using the value of uy given by Eq. (20) one can obtain some interesting results. For instance, considering the principal number of the first “free” (i.e. out of the planet) allowable orbits, one can note that for Mercury, Venus, the Earth, Mars and Pluto these orbits correspond to relatively high (> 19)
Table I Structure of the main Sacurn rings Rings [IO’ km] n o,, [IO3 km1 Au,, [IO’ km] uobs
DIE
67.0 6 67.6 + 0.6
c ,E
74.5
BIE
92.0 7 92.0 IO.8
Cassini 119.8
AI,
F
G
E IE
E OE
122.2 8 120.1 f 1.1
140.4 9 152.1 f 1.6
170
180 IO 188.0 * 1.7
480 16 481.0 + 4.4
A.G. Apese,
principal numbers. The opposite, i.e. small (< 11) “free” quantum numbers, is verified for the big planets. As regards the orbits of the satellites one really obtains both good and bad results. This disappointing ambiguity could be partly attributed to the important role that other forces, besides the gravitational one, play in this case (see below). More interesting coincidences are found if one considers the Saturn ring “powder” structure. In Table 1 we show the main features of this structure and the corresponding discretized radii. One can note that the inner edges of the main rings (D. B and the Cassini division together with the neighbouring A ring) are almost exactly singled out by the corresponding calculated radii. An exception is represented by the inner edge of the C ring. On the other hand this ring does not originate from purely gravitational forces, since, as explicitly remarked in the Landolt-Bbmstein handbook [71, the boundary between the B and C rings is presumably related to electrodynamical interaction between the charged dust particles of the ring system and the planetary magnetic field. One must finally note the remarkable calculated singling out of the (somewhat conventional) limits of the broad rarefied E ring. Of course, we agree on the fact that the previous remarks about the orbital motions around the planets are in no way conclusive in order to check our guess, and in particular the “universality” of the gravitational structure constant (Y , which, we recall, has been estimated by calculat;ons concerning only orbital motions around the Sun.
4. Angular momenta
169
R. Festu / Physics Letters A 227 (1997) 165-171
of celestial bodies
As already said, very few planetary systems are presently known. Thus, still with the purpose of validating (or disproving) the universality of the calculated LYE,we are compelled to take into consideration a seemingly different empirical evidence, i.e. the relationship between the angular momentum J of a gravitationally bounded celestial body and its mass M. As Wesson [8] reported in a work about the self-similarity of many astronomical systems, the validity of a rule of the type J a Mk has been remarked by several authors both in planetary physics
and in stellar and galactic astrophysics, with k ranging in the interval 1.67 < k < 2. For systems held together by purely gravitational forces Wesson points out that self-similarity demands k = 2. In a subsequent article [9] the same author, looking for a universal constant p to be added to the gravitational constant G and the light speed c in order to get a dimensionless gravitational parameter, fits the law J = PM* for a set of celestial bodies with masses ranging on about twenty orders of magnitude. He finds log ,0 p = - 15.1 f. 0.9 and chooses, for consistency with his general guess. a mean value p = 8 X 10-‘h cm2 gg’ s-‘. In the spirit of the present paper. recalling that in quantum mechanics the electron spin is given by (221 we want to assume that the spin of a purely gravitationally bound celestial body is related to its mass through the relationship 1 Gm’ J=-----.
(23)
2 cYygc’
One can compare the values obtained through this equation with those found in the literature or, as in the case of most planets, with the “observed” values Jobs =
$fR2;,
(24)
obs
which are calculated from the observed periods Pobs by considering spherically shaped homogeneous rigid bodies. These quantities represent in fact upper limits for the actual values. We want to note that expression (231, which has been inferred by pure analogy with (221, could be obtained on suitable conditions by applying our guess (101 to a flat continuous axially symmetric purely gravitating system. In fact, if one considers the total angular momentum of such a system as the sum of the collinear angular momenta of its constituting infinitesimal rings, each of them orbiting on the “Bohr radius” determined by the already accounted internal part of the system. the simple expression J=
/
dJ=
“jm a,c
dm
holds, which just equals Eq. (231.
(25)
170
A.G. Agnese, R. Festa/Physics
qr-----l
Table 2 Angular momenta for the planets of the Sun
YIRRL 70
GRLRXIES
-
z=
$‘/
/
/
7,” “s
/ / 5TRR
CLUSTERS w
6fl-
F 2
ss’ q
_
SOLRR
5V5TEfl
Letters A 227 (1997) 165-171
imE
5TRR5
sBody
Jca~c[J ~1
Jobs [J ~1
Mercury Venus EaTth Mars Jupiter Saturn uramls Neptune Pluto
2.56 x 5.57 x 8.40X 9.68 x
< 1.02x 10’” <2.14X 103’ 5.88 x 1033 <2.10x lo32 < 6.83 x lO38 8.16X 103’ < 2.50 x 1036 < 2.30 x lO36 < 1.5x 10’9
10” lo= lo33 10” 8.48 x 1038 7.55 x 10” 1.78X 1036 2.49 x lO36 5.3 x 1028
/ i
+%
40 -
+< /C
PLANETS
/ / 30
’ xl
I
’ 25
I
I
I
30
35
40
I
45
a
50
Loqo M Is1 22 3
Fig. 2. Law .I =(G/2a,c)m* on logarithmic scale, superimposed on a data plot from Wesson f91.
In Fig. 2 (whose background experimental points have been reproduced from a figure of Wesson’s paper [9]) we draw the graph on logarithmic scale of Eq. (23) with the choice G/2 crgc = 2.35 X lo-l5 cm2 g- ’ s- ’ deduced from our previous result. If one considers that no free parameter has been used to fit the data, one can conclude that a quite satisfactory agreement is shown over more than twenty orders of mass magnitude. The less satisfactory part of the comparison regards the solar system planets, which on the other hand are held together by additional forces besides the gravitational one, as Wesson explicitly remarks. In Table 2 we report in detail the calculated and “observed” (upper limits) angular momenta for some planets. Apart from the Earth, only for Saturn, owing to its oblate shape, the angular momentum J = 8.16 X 103’ J s can be directly inferred. The discrepancy regarding the Earth, in fact the Earth-Moon system, is the subject of a somewhat long discussion which we prefer to leave out from this Letter. A further and, for the present, last clue to the universality of cyg can be inferred by considering the independence of the orbital velocities given by Eq. (16) from the mass of the gravitational source. The
maximum given by Eq. (23), would be results in [IO,1 11.
orbital velocity, obtained for n = I = 1, is U, = 142 km s- ‘. For a galaxy, recalling the upper bound for rotational velocity 284 km s-‘. This bound v,,, = 2a,c= a fair agreement with observational data
5. Conclusions To sum up, if one agrees that the simple analogies (IO), (1 I), (12) with the old quantum mechanics hold, one can observe a number of remarkable coincidences between calculated and measured celestial mechanical quantities. We have no idea, in fact no indisputable idea, about the why of these coincidences. Interpretations which call into question stochastic mechanics [12] with a diffusion coefficient given, consistently with our guess, by v = GM/2 ergc could be plausible, though by no means unique. Finally, we explicitly acknowledge that for the moment we do not have much belief in the validity on a large scale of something similar to the customary quantum mechanics. If one would take seriously a complete quantum mechanical analogy (as Greenberger does in his theory), a de Broglie general wavelength could be defined as
where Q,,(g) is the gravitational field flux throughout the surface connected with an experiment (for orbital motions, i.e. for a solid angle R = 47~, one of course obtains Eq. (7) again).
A.G. Agnew.
R. Fe&to/ Physics Letters A 227 (1997) 165-171
However, in spite of the intriguing aspect of the previous formulae, we believe that more research and speculation are needed and that one must proceed with extreme caution along the tricky road of “celestial quantum mechanics”.
References [I] D.M. Greenberger, Found. Phys. I3 (1983) 903. [z] M. Dersarkissian, Lett. Nuovo Cimento 40 (1984) 390. [3] J.C. Carvalho, Lett. Nuovo Cimento 44 (1985) 337.
[4] [5] [6] [7] [8] [9] [IO] [I I] [l2]
171
W.J. Cock, Astrophys. J. 288 (1985) 22. M. Mayor and D. Queloz, Nature 378 (1995) 355. J. Schneider, Internet, http://www.obspm.fr/ (1996). Landolt-Biimstein, Handbook V1/3A. 3.2. The planets and their satellites (Springer. Berlin, 1993). P.S. Wesson. Astron. Astrophys. 80 (1979) 296. P.S. Wesson, Phys. Rev. D 23 ( I98 1) 1730. C. de Vaucouleurs. L‘exploration des galaxies voisines (Masson. Paris, 1958). Laudolt-BBmstein, Handbook VI/2C. 9.2 Internal structure and dynamics of galaxies (Springer, Berlin. 1982). Ph. Combe, Ph. Blanchard and W. Zheng, Mathematical and physical aspects of stochastic mechanics (Springer. Berlin, 19871.
PHYSICS ELSEVIER
Physics Letters A 227
LETTERS
A
(I 997) 16% 17 I
Clues to discretization on the cosmic scale A.G. Agnese ap’,R. Festa b*Z a Dipurtimenro di Fisica, Universith di Genovu. and INFN, S&me
dr Gmovu. Geneva. I& b Dipurrimento di Fisicu. Universirh di Genova, und INFM. Unitir di Geneva. Genova. Italy
Received 26 June 1996; revised manuscript
received I2 December 1996; accepted for publication Communicated by P.R. Holland
I8 December
1996
Abstract Starting from discretization rules analogous to those of Bohr and Sommerfeld, values for a few mechanical quantities associated to celestial periodic motions are obtained. A gravitational dimensionless structure constant (Yecan be singled out, whose value, determined from data relative to the solar system, allows us to quantitatively account for phenomena on a much wider scale. PACS: 96. IO. + i; 98.50. - v; 03.65.B~
1. Introduction
It is well known that the weak equivalence principle, i.e. the fact that the trajectory of a particle subjected to gravitational forces does not depend on its mass, is satisfied only in the classical limit by the axioms of both the old (Bohr-Sommerfeld) quantum mechanics $ Pj dq, = ‘1’
(1)
and the properly called quantum [p,,
qj]=
mechanics
-ih.
Some years ago, Greenberger
(2) [ 1I, in order to recover
the overall validity of the equivalence principle, formulated the hypothesis that the more general commutation rule holds [p.
-i(h+mcXf(-&.
$)),
where A (X = A/27r) is a fundamental length and f< . > a dimensionless function with the necessary symmetries with respect to its arguments. On the r.h.s. of Eq. (3) the first term would dominate on the microscopic scale (negligible gravitational fields) and the second on the “megascopic” or celestial one (predominating gravitational fields). Taking into account the necessary invariances, the author considered the simple commutation rule [P,
x]=
-i(h+y2-J
(4)
where y is a parameter, which can be variously defined provided that it is very small but in strong
’ E-mail: [email protected]. ’ E-mad: [email protected]. 0.375.9601/97/$17.00 Copyright Pfi SO375-9601(97)00007-8
xl=
0 1997 Published
by Elsevier Science B.V. All rights reserved.
166
A.G. Agnese, R. Festa/Phy.sics
gravitational approximated
fields. by
In this limit
Eq. (4) can be
Letters A 227 (1997) 165-171
from suitably tuned (Bohr-Sommerfeld).
action
discretization
rules
2. Orbital motions around the Sun so that the particle momentum corresponds to a non-local (integral) linear operator, in contrast with the local (differential) operator of quantum mechanics. The hypothesis that the celestial phenomena obey quantum rules is nowaday repugnant to the minds of almost all physicists. Nevertheless, considering that the gravitational forces represent the dominant forces only at large scales, i.e. at the celestial level, one can accept that some breakdown of the classical “macroscopic” mechanics at this “megascopic” level would not be astonishing, though in no way necessary. The Greenberger’s assumption (3) represents by no means the sole possible generalized commutation rule, as explicitly pointed out by the author himself. Moreover, it turns out to be Lorentz invariant only if fl p/me, x/A) takes a constant value. As a matter of fact, even from a purely mathematical point of view, the dependence of the commutator (4) on p* is by no means necessary. If we simply consider the constant term in the Taylor’s power development of f in Eq. (31, we obtain [p,
xl=
-i(h+iXmc),
(6)
where A is a suitably chosen length. In the case of a body gravitating in the force field of another body with mass M, A could be directly related to the gravitational source strength through the dimensionally determined relationship 1 GM x=_..-._, %
C2
Let us choose as the starting point the customary Bohr-Sommerfeld rules for (multiply) periodic motions
#
e2
pj dqj=nj2r--,
(8)
ffec
where the Planck’s constant is written, using the fine structure constant ff, = 1/ 137, in the form
fi=e’.
(9)
%C
Our conjecture is that a simple correspondence holds between electric and gravitational force fields, which can be expressed by e2 -*a,
GMm ffg
(10)
’
where (Ye is a gravitational structure constant, whose value must be evaluated using available experimental data. Thus, for periodic plane motions of a particle of mass m in the gravitational field of a particle of mass M the discretization rules should hold #
#
p, dr = k2rr-
GMm ffgc
pv dq=12Ir-
’
(‘1)
GMm
(12) %c
(k
(7)
where (us is a dimensionless constant. This form takes implicitly into account the required relative weight of the gravitational field in comparison with other “microscopic” fields represented by h. In fact, we prefer to stop at this stage and not try for the moment the setting up of a megascopic analogous of the full quantum mechanics, as Greenberger and other authors [2-41 do. In this Letter we will only conjecture that periodic motions of celestial bodies can be described, approximately enough, starting
and 1 are the customary radial and azimuthal numbers). We will now examine the possibility that this rule accounts for a “gravitational atom” model of the solar system. Introducing the gravitational potential energy v=
GMom -__ r ’
(‘3)
where MO is the mass of the Sun, it follows almost immediately that the major semi-axis a, (where n = k + I is the principal number) of an allowed discretized elliptical orbit is given by a = &2 n I
’
(14)
A.G. Agnese. R. Festu/
Physics Letters A 227 (1997) 165-171
where
and the corresponding
167
period
GM,
represents the “Bohr radius” of the planetary system. The minimum and maximum distances from the Sun, and the corresponding maximim and minimum orbital velocities are then given by 4 n.l =a,(1
-&).
Up)= .
&/=a,(1
f&),
u;y’“‘=(l
(1 +,)F,
-$$
(16)
where l2 &=
(17)
/-- ‘-2
indicates the elliptical eccentricity, and, as customary. I- l,..., n. For circular orbits (I = n) one obtains the constant velocity v, = cygc
P, = 2li--n3 (‘9) 03c3 . B A first estimate of uy can be achieved through Eq. (14) by using the known mean distance of a given planet from the Sun and choosing a suitable (small) principal number n for its orbit. This first estimate can then be used with the same equation in order to single out the appropriate principal number for each of the remaining planets. A final estimate of 0”: (and consequently of LYEthrough Eq. (1.5)) is then obtained by an overall least-square fit on all planets. It turns out that the only consistent choice lies in assigning to Mercury the principal number n = 3. Then, to the other planets of the solar system the following quantum numbers almost automatically correspond: Venus -+ 4, Earth -+ 5, Mars + 6. (Ceres --) 81, Jupiter --, 11, Saturn -+ 15, Uranus -+ 21, Neptune -+ 26, Pluto + 30. The best fit using these quantum numbers gives uy = 0.0439 * 0.0004 a.u.,
(20)
(18)
n
Planets of the Sun
IO
Fig. I. Experimental
20
15 Pr~nupal
ndmbar
25
30
n
orbital radii of the planets compared with the values given by the formula u,, = ~l,n’. The correspondence between
planets and principal numbers is reported in the text.
A.G. Agnese, R. Festu / Physics Letters A 227 (1997) 165-171
168
In Fig. 1 the a, quadratic rule given by Eq. (14) is graphically compared with the actual orbital major semi-axes of the planets of the Sun. We remark that Eqs. (1 1), (12) and their consequences only determine the allowed orbits. Actually. only a few allowed orbits are occupied by the planets. The reason for this can only be the subject of guesses (in fact, the origin of the planets is yet today a subject of guesses). One can suppose for instance that the protoplanetary material was originally distributed to form rings in correspondence with every quantum number, and that gravitational instabilities or perturbations have brought a final state of aggregation of the material in planets over some orbits in correspondence with particular quantum numbers. Otherwise, one can guess that external gravitational perturbations acting on an almost continuum protoplanetary material in gravitational, centrifugal and thermodynamical equilibrium, led to spatial density resonances mainly corresponding to some transversal eigenfunctions (which, of course, must satisfy circular periodicity conditions). Anyway, and with a certain surprise, one can see from Fig. 1 that the interpretation of the planetary orbits with a gravitational atom model is in reasonable agreement with the observed data, even for small principal numbers. We acknowledge that it is still not enough to give validity to our conjecture, and that alternative hypotheses could be formulated. In order to extend the investigation about this topic, one would need some more information about other planetary systems. Till the end of 1995 no such information was available. Fortunately, in recent paper [5], the planetary motion of a newly discovered companion of the star 51 Pegasus, with mass ranging between one half and twice the mass of Jupiter and mean distance from 51 Pegasus of about 0.05 a.u., has been examined: it turns out that its orbit nearly corresponds to the
quantum number n = 1 of its planetary system. Some reported uncertainty on the mass of 51 Peg (a star “quite similar to the Sun”) could be removed if one guesses that the orbit of its large planet exactly corresponds to n = 1: on the basis of the observed period (P = 4.23 days) a distance a, = 0.055 a.u and a mass MSIPes = 1.26& can be obtained. Thus, some recent knowledge about a simple “gravitational atom” different from our solar system seems to support our guess. Very recently other extrasolar simple planetary systems have been discovered. For the moment, the available data [6] seem to validate our guess. In particular the orbits of the planets of HR5185 and HR458 correspond to II = 1.
3. Orbital motions around the planets We wiH now shortly examine some consequences of our guess on the allowable orbits around the various planets. Of course, the major semi-axes of these orbits are obtained by substituting in Eqs. (13) and (15) the mass of the Sun MB with the mass Mp of the concerned planet. Assuming the universality of the gravitational structure constant CY~,one immediately gets from Eq. (15)
(21) where u\P) indicates the “Bohr radius” for orbitating motions around the planet. By using the value of uy given by Eq. (20) one can obtain some interesting results. For instance, considering the principal number of the first “free” (i.e. out of the planet) allowable orbits, one can note that for Mercury, Venus, the Earth, Mars and Pluto these orbits correspond to relatively high (> 19)
Table I Structure of the main Sacurn rings Rings [IO’ km] n o,, [IO3 km1 Au,, [IO’ km] uobs
DIE
67.0 6 67.6 + 0.6
c ,E
74.5
BIE
92.0 7 92.0 IO.8
Cassini 119.8
AI,
F
G
E IE
E OE
122.2 8 120.1 f 1.1
140.4 9 152.1 f 1.6
170
180 IO 188.0 * 1.7
480 16 481.0 + 4.4
A.G. Apese,
principal numbers. The opposite, i.e. small (< 11) “free” quantum numbers, is verified for the big planets. As regards the orbits of the satellites one really obtains both good and bad results. This disappointing ambiguity could be partly attributed to the important role that other forces, besides the gravitational one, play in this case (see below). More interesting coincidences are found if one considers the Saturn ring “powder” structure. In Table 1 we show the main features of this structure and the corresponding discretized radii. One can note that the inner edges of the main rings (D. B and the Cassini division together with the neighbouring A ring) are almost exactly singled out by the corresponding calculated radii. An exception is represented by the inner edge of the C ring. On the other hand this ring does not originate from purely gravitational forces, since, as explicitly remarked in the Landolt-Bbmstein handbook [71, the boundary between the B and C rings is presumably related to electrodynamical interaction between the charged dust particles of the ring system and the planetary magnetic field. One must finally note the remarkable calculated singling out of the (somewhat conventional) limits of the broad rarefied E ring. Of course, we agree on the fact that the previous remarks about the orbital motions around the planets are in no way conclusive in order to check our guess, and in particular the “universality” of the gravitational structure constant (Y , which, we recall, has been estimated by calculat;ons concerning only orbital motions around the Sun.
4. Angular momenta
169
R. Festu / Physics Letters A 227 (1997) 165-171
of celestial bodies
As already said, very few planetary systems are presently known. Thus, still with the purpose of validating (or disproving) the universality of the calculated LYE,we are compelled to take into consideration a seemingly different empirical evidence, i.e. the relationship between the angular momentum J of a gravitationally bounded celestial body and its mass M. As Wesson [8] reported in a work about the self-similarity of many astronomical systems, the validity of a rule of the type J a Mk has been remarked by several authors both in planetary physics
and in stellar and galactic astrophysics, with k ranging in the interval 1.67 < k < 2. For systems held together by purely gravitational forces Wesson points out that self-similarity demands k = 2. In a subsequent article [9] the same author, looking for a universal constant p to be added to the gravitational constant G and the light speed c in order to get a dimensionless gravitational parameter, fits the law J = PM* for a set of celestial bodies with masses ranging on about twenty orders of magnitude. He finds log ,0 p = - 15.1 f. 0.9 and chooses, for consistency with his general guess. a mean value p = 8 X 10-‘h cm2 gg’ s-‘. In the spirit of the present paper. recalling that in quantum mechanics the electron spin is given by (221 we want to assume that the spin of a purely gravitationally bound celestial body is related to its mass through the relationship 1 Gm’ J=-----.
(23)
2 cYygc’
One can compare the values obtained through this equation with those found in the literature or, as in the case of most planets, with the “observed” values Jobs =
$fR2;,
(24)
obs
which are calculated from the observed periods Pobs by considering spherically shaped homogeneous rigid bodies. These quantities represent in fact upper limits for the actual values. We want to note that expression (231, which has been inferred by pure analogy with (221, could be obtained on suitable conditions by applying our guess (101 to a flat continuous axially symmetric purely gravitating system. In fact, if one considers the total angular momentum of such a system as the sum of the collinear angular momenta of its constituting infinitesimal rings, each of them orbiting on the “Bohr radius” determined by the already accounted internal part of the system. the simple expression J=
/
dJ=
“jm a,c
dm
holds, which just equals Eq. (231.
(25)
170
A.G. Agnese, R. Festa/Physics
qr-----l
Table 2 Angular momenta for the planets of the Sun
YIRRL 70
GRLRXIES
-
z=
$‘/
/
/
7,” “s
/ / 5TRR
CLUSTERS w
6fl-
F 2
ss’ q
_
SOLRR
5V5TEfl
Letters A 227 (1997) 165-171
imE
5TRR5
sBody
Jca~c[J ~1
Jobs [J ~1
Mercury Venus EaTth Mars Jupiter Saturn uramls Neptune Pluto
2.56 x 5.57 x 8.40X 9.68 x
< 1.02x 10’” <2.14X 103’ 5.88 x 1033 <2.10x lo32 < 6.83 x lO38 8.16X 103’ < 2.50 x 1036 < 2.30 x lO36 < 1.5x 10’9
10” lo= lo33 10” 8.48 x 1038 7.55 x 10” 1.78X 1036 2.49 x lO36 5.3 x 1028
/ i
+%
40 -
+< /C
PLANETS
/ / 30
’ xl
I
’ 25
I
I
I
30
35
40
I
45
a
50
Loqo M Is1 22 3
Fig. 2. Law .I =(G/2a,c)m* on logarithmic scale, superimposed on a data plot from Wesson f91.
In Fig. 2 (whose background experimental points have been reproduced from a figure of Wesson’s paper [9]) we draw the graph on logarithmic scale of Eq. (23) with the choice G/2 crgc = 2.35 X lo-l5 cm2 g- ’ s- ’ deduced from our previous result. If one considers that no free parameter has been used to fit the data, one can conclude that a quite satisfactory agreement is shown over more than twenty orders of mass magnitude. The less satisfactory part of the comparison regards the solar system planets, which on the other hand are held together by additional forces besides the gravitational one, as Wesson explicitly remarks. In Table 2 we report in detail the calculated and “observed” (upper limits) angular momenta for some planets. Apart from the Earth, only for Saturn, owing to its oblate shape, the angular momentum J = 8.16 X 103’ J s can be directly inferred. The discrepancy regarding the Earth, in fact the Earth-Moon system, is the subject of a somewhat long discussion which we prefer to leave out from this Letter. A further and, for the present, last clue to the universality of cyg can be inferred by considering the independence of the orbital velocities given by Eq. (16) from the mass of the gravitational source. The
maximum given by Eq. (23), would be results in [IO,1 11.
orbital velocity, obtained for n = I = 1, is U, = 142 km s- ‘. For a galaxy, recalling the upper bound for rotational velocity 284 km s-‘. This bound v,,, = 2a,c= a fair agreement with observational data
5. Conclusions To sum up, if one agrees that the simple analogies (IO), (1 I), (12) with the old quantum mechanics hold, one can observe a number of remarkable coincidences between calculated and measured celestial mechanical quantities. We have no idea, in fact no indisputable idea, about the why of these coincidences. Interpretations which call into question stochastic mechanics [12] with a diffusion coefficient given, consistently with our guess, by v = GM/2 ergc could be plausible, though by no means unique. Finally, we explicitly acknowledge that for the moment we do not have much belief in the validity on a large scale of something similar to the customary quantum mechanics. If one would take seriously a complete quantum mechanical analogy (as Greenberger does in his theory), a de Broglie general wavelength could be defined as
where Q,,(g) is the gravitational field flux throughout the surface connected with an experiment (for orbital motions, i.e. for a solid angle R = 47~, one of course obtains Eq. (7) again).
A.G. Agnew.
R. Fe&to/ Physics Letters A 227 (1997) 165-171
However, in spite of the intriguing aspect of the previous formulae, we believe that more research and speculation are needed and that one must proceed with extreme caution along the tricky road of “celestial quantum mechanics”.
References [I] D.M. Greenberger, Found. Phys. I3 (1983) 903. [z] M. Dersarkissian, Lett. Nuovo Cimento 40 (1984) 390. [3] J.C. Carvalho, Lett. Nuovo Cimento 44 (1985) 337.
[4] [5] [6] [7] [8] [9] [IO] [I I] [l2]
171
W.J. Cock, Astrophys. J. 288 (1985) 22. M. Mayor and D. Queloz, Nature 378 (1995) 355. J. Schneider, Internet, http://www.obspm.fr/ (1996). Landolt-Biimstein, Handbook V1/3A. 3.2. The planets and their satellites (Springer. Berlin, 1993). P.S. Wesson. Astron. Astrophys. 80 (1979) 296. P.S. Wesson, Phys. Rev. D 23 ( I98 1) 1730. C. de Vaucouleurs. L‘exploration des galaxies voisines (Masson. Paris, 1958). Laudolt-BBmstein, Handbook VI/2C. 9.2 Internal structure and dynamics of galaxies (Springer, Berlin. 1982). Ph. Combe, Ph. Blanchard and W. Zheng, Mathematical and physical aspects of stochastic mechanics (Springer. Berlin, 19871.