aa r X i v : . [ a s t r o - ph ] A ug On the Planetary acceleration and the Rotation of theEarth
Arbab I. Arbab Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321, Khartoum11115, Sudan
Abstract
We have developed a model for the Earth rotation that gives a good account (data) of theEarth astronomical parameters. These data can be compared with the ones obtained usingspace-base telescopes. The expansion of the universe has an impact on the rotation of planets,and in particular, the Earth. The expansion of the universe causes an acceleration that isexhibited by all planets.
It has been understood that the impact of the universe expansion on our solar system is negligible.This is however not very true. The consequences of the expansion on the earth - moon system is inthe measurable limit. The evolution of the earth-moon system was understood to be mainly due totidal evolution. We have recently shown (Arbab, 2003) that the present acceleration of the universeis due to the ever increasing gravity strength. Very recently, we have found that the evolution ofangular momenta and energy of the earth-moon system can be accounted as due to cosmic expansion(Arbab, 2005). This system is affected by the perturbation due to other planets or the sun. Thecosmic expansion may show up in raising tides in this system. The influence of the expansion iscontained in changing the value of the gravitational constant appearing in Kepler’s third law andNewton’s law of gravitation. At any rate, the total cosmic effect is embedded in the an effectivegravitational constant ( G eff . ) that takes care of any gravitational interactions with thee system. Fora flat universe if gravity strengthens, expansion has to increase, in order to maintain a flatnesscondition. If gravity increases, then its effect on rotation of the earth - moon system will show upin its evolution. Astronomical investigations show that the preset earth’s rotation is decreasing, so E-mail: [email protected]
We have recently developed a model that accounts for the present cosmic acceleration (Arbab, 2003).We have shown that, in the present epoch, the gravitational constant ( G ) increases with time .However, its exact time dependence is not well determined form cosmology. One has to resort toother source of information. This is found to be the past earth rotation.It is known that the earth rotation is decreasing with time since the earth was formed. Scientistsattribute this to the tide rasing force by the moon on earth. Accordingly, the day is lengthening ata a rate of about 2 millisecond every century. Hence, the earth is losing angular momentum and themoon must increase its angular momentum, as due to the angular momentum conservation of theearth-moon system. This fact implies that the moon must be receding from the earth. We knowthat the motion of the earth around the sun conserves the angular momentum. One can satisfythis conservation by requiring the earth to accelerate in its orbit around the sun. According to thescale expanding cosmos (SEC) the present acceleration of the earth is about 2.8 arcsec per centurysquared (Kolesnik and Masreliez, 2004). Moreover, one can attribute the deceleration of the earthrotation as due to cosmic expansion. The variation of the length of day are normally thought as dueto the tidal dissipation raised by the moon on the earth. Others connect this deceleration with theinteractions of the Earth core. However, in the present scenario we only know the total contribution,which we trust to be a consequence of the acceleration of the universe. This accelerated expansionis counteracted by a growing gravitational force between celestial objects. This gradual increasein gravity force is the main consequence of the astronomical phenomena we now come to observe.Geologists observed that the length of the day has not been constant over the past million years.Besides, they observe a similar change in the number of days in a year, days in a month, distance2etween earth and moon. These variations can be calculated and their corresponding values canthen be confronted with observational data.We suggest that the cosmic expansion has an influence on the Earth-Sun-Moon system and similarsystems. For a bound system, like the Earth-Sun, to remain in a bound state, despite the cosmicexpansion (possibly accelerating), gravity strength has to increase to compensate for the cosmicexpansion consequences. This strengthening of gravity would manifest its self in some aspects, liketidal acceleration, or orbital acceleration. We anticipate the Earth-Sun distance to change withcosmic time too. This means in the remote past the planets were at different positions from the Sunwhen they were formed.The viability of this model will depend on the future astronomical or geological data that willemerge thereafter. The formulae we have obtained are not extrapolation, but rather emerge originallyfrom a Gravitational Theory of Relativity (GTR), and therefore are reliable. They represent empiricalrelations that account for the rotation and evolution of the Earth-Sun system and similar system.Present data can not be used to understand the full history of the Sun-Earth-Moon system by justextrapolating them over very distant past. Hence, the use of our data will be inventible. Our model isso far the only model that provides a temporal evolution of the Earth-Moon-Sun system parameters.The prediction of these formulae are overwhelming, however. Theoretical prejudice favors that theEarth primordial rotation is about six hours. Only our model can give this value.From the angular momentum ( L ) and the Kepler’s third law, one finds L ∝ G T , (1) L ∝ √ Gr , (2)and L ∝ Gv − , (3)where T is the number of days in a year, v is the orbital velocity of the earth (planet) rotation inits orbit, r is the earth (planet)-sun distance, and G is the gravitational constant. If the angularmomentum of the earth-Sun system is constant, then on find that T ∝ G − , (4) r ∝ G − , (5)and v ∝ G . (6)Eq.(2) implies that as long as G is constant then T, r , and v are constant too. However, there is apossibility that G might have been changing appreciably over cosmological time. In this case if oneknows the way how G varies the variation of the distance r can be calculated. Thus the variation3f G will mimic the tidal effects which people now attribute these changes to. If G changes withtime Newton’s gravitational law still holds. However, the equivalence principle of general theory ofrelativity is broken. The variation of G may not be real and it is due to an existence of dark mattercoexisting with normal matter. Its effect is to make the gravitational coupling (Newton’s constant)appear to be increasing. The effect of a little normal matter and increasing gravity in a universe isequivalent to that of more matter and normal gravity universe. We may dictate that Newton’s law ofgravitation (and Kepler’s law) to be applied to an evolving local system, like the planetary system,viz. Earth-Moon-Sun system.In our present study we rely on a general form for the variation of G with cosmic time (Arbab,2003). In this scenario a gravitating body interacts with an effective gravitational constant G eff . which differs from a bare Newton’s constant we used to know. We have, in particular, an increasing G = G eff . at the present epoch, viz. G eff . = G (cid:18) tt (cid:19) n (7)where n > G eff . rather than with bare G because of cosmic expansion. The effect of this constant is toreplace in all formulae the normal(bare) Newton’s constant with this effective constant. The Earthcouples with the rest of the universe with this value. This coupling follows from the idea of Machthat the inertia of an object is influenced by the rest of matter in the universe. In an evolvinguniverse this effective constant induces a cosmological effect on over planetary system while the bareconstant ( G ) stays invariant. This why we observe some cosmic effects exhibited in tidal effects, oreffects drawn from perturbation by other objects in the nearbye solar system. In this context onehas a calculable variation in the strength of gravity due to cosmic expansion. This variation can’tbe measured directly. We present here a new approach of detecting its variation with cosmic timein the way it has affected planetary system dynamics. An increasing gravitational constant maymimic an increasing mass of a gravitating body. Or alternatively, it mimics a dark matter nearbyethe gravitating body that makes the orbiting object to fall towards it. A universe with increasinggravitational constant may look indistinguishable from the one with dark matter. Hence, if gravityincreases for some reason the idea of dark matter need not be attractive. Milgrom modified Newton’slaw to account for the flattening of the rotation curve. In our present case the modification does notchange the form of Newton’s law.Here n determines the properties of the cosmological model proposed. If one assumes that thelength of the year remains constant, then the length of the day ( D ) should scale as D ∝ G . , (8)Hence, one has T D = T D , (9)4here the subscript ‘0’ on the quantity denotes its present value. Our model shows that the day wassix hours when the earth was formed. The angular velocity of the earth about the sun is (Ω = πT )Ω ∝ G . (10)This implies that the Earth is accelerating at a rate of˙ΩΩ = 2 ˙ G eff . G eff . (11)and at the same time the earth-sun distance decreases at a rate of˙ rr = − ˙ G eff . G eff . (12)If we know how G varies, one can calculate this variation. In our cosmological model, we know thegeneral variation of G depends on a parameter n that determines the whole cosmology. If n is knowthen the whole cosmological parameters are known. Our model [1] could not determine n exactly. Itplaces a weaker limit on the value of n . However, Wells [2] had found from a palaeontological studythe number of days in the year. To reproduce his result we require the age of the universe to be t ∼
11 billion years and n = 1 . G eff . = G (cid:18) tt (cid:19) . (13)Hence, eqs.(4)-(7) become, respectively T = T (cid:18) t t (cid:19) . , (14) r = r (cid:18) t t (cid:19) . , (15) v = v (cid:18) tt (cid:19) . , (16)and D = D (cid:18) tt (cid:19) . . (17)We remark here that there is an astrophysical system (Binary Pulsars) in which the decay of orbitis very prominent and attributed to the emission of gravitational waves. Can we assume here thatthere is a similar effect? Alternatively, may one suggest that the decay of orbit in the former system5s due to cosmic expansion in the manner we have identified above? This is quite plausible if theapparent acceleration of planetary system is the direct cause.It is worth to mention that Wells could not go far beyond the Precambrian (600 million yearsback). Our model gives a formula that determines the number of days in a year and the lengthof day at any time in the past. These data obtained from this formula are in full agreement withthose obtained by different methods (see Arbab 2004 and references therein). The correctness of theformula entitle us to say that our initial proposition that the expansion of the universe affects oursolar system is correct. If that is true, one can calculate the present acceleration of the earth (andother planets) and the rate at which their orbit decreases. For the earth orbital motion one findsthat the present acceleration amounts to˙Ω = (cid:18) . t (cid:19) Ω , ˙Ω = 3 .
05 arcsec / cy (18)At the same time the earth-sun distance decreases at a a rate of˙ r = − (cid:18) . t (cid:19) r , ˙ r = − . / year . (19)One can also write the above equations in terms of Hubble constant, as˙ΩΩ = 2 . H , ˙ rr = − . H , ˙ vv = 1 . H , ˙ DD = 2 . H , (20)since ˙ G eff . G eff . = 1 . H . (21)We see that the gravitational force increases as F = (cid:18) G eff . G (cid:19) F , (22)and upon using eq.(7), it becomes F = (cid:18) tt (cid:19) F , (23)We notice that the Newton’s and Kepler’s laws of gravitation do still work well, even in an expandinguniverse, with only a minor generalization that takes care of time evolution. The increase of thegravitational forces is such that to counteract the present universal expansion (acceleration) so thatthe universe remains in equilibrium (flat). The gravitational force between our Earth and the Sun4.5 billion years ago has been 12% less than now.6n increasing gravity would mean that in the past the gravity was weak. This might probablyprovide a comfortable life of gigantic animal, like dinosaurs, to roam freely on earth’s surface. Asgravity increases their wights would become heavier and finally may not support its growing weight.Thus it might not have been appropriate for them to survive and later they vanish when they areoverweight. This scenario may provide a rather convenient mechanism on how dinosaurs extinct.We thus see that all earth parameters vary as due to universe expansion. We have found thepresent Hubble constant to be H = 10 − y − so that (cid:16) ˙ G eff . G eff . (cid:17) = 1 . × − y − . This analysisimposes a new limit on G eff . and H which can be tested with observational data. There is only fewmodels that deal with increasing G . However, models in which G decreases with time lead to seriousdifficulties, when confronted with observations. Our model predicts a universal acceleration of allgravitating bodies. For instance, we found that Mercury accelerates at a rate of 12 . / cy , Venusat a rate of 4 .
95 arcsec / cy , Earth at a rate of 3 .
05 arcsec / cy and Mars at a rate of 1 . / cy . Weremark that the formulae pertaining to the planets motion are in good agreement with observation.We should also await the emergence of new data to test their applicability to these systems. Wesee from eq.(19) that the day ( D ) lengthens by 1.95 msec/cy. According to scale expanding cosmos(SEC) the planets spins down. If all of these data are found to be in accordance with observation,then our hypothesis that the cause of the present acceleration is due to gravity increase would beinventible! We have shown in this paper that the present cosmic acceleration induces its effect on our Earth-Moon-Sun system. This is apparent in the magnitude of the variation of the length of day, year,distance, angular velocity, etc which are all proportional to the Hubble parameter. The cosmologicaleffects show up in different forms some of them are understood as due to tidal effects. We anticipatethat the future observations will bring many puzzles and surprises with it.7able 1: Data obtained from fossil corals and radiometric timeTime a
65 136 180 230 280 345 405 500 600days/year 371.0 377.0 381.0 385.0 390.0 396.0 402.0 412. 0 424.0Table 2: Data obtained from our empirical formula in eqs.(14) and (17)Time
65 136 180 230 280 345 405 500 600days/year 370.9 377.2 381.2 385.9 390.6 396.8 402.6 412.2 422.6day (hr) 23.6 23.2 23.0 22.7 22.4 22.1 21.7 21.3 20.7Time a
715 850 900 1200 2000 2500 3000 3560 4500days/year 435.0 450.2 456 493.2 615.4 714.0 835.9 1009.5 1434day (hr) 20.1 19.5 19.2 17.7 14.2 12.3 10.5 8.7 6.1 a Time in million years before present
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