REVOLUTION OF NEUTRAL BODY IN A CLOSED ELLIPSE UNDER GRAVITATIONAL FORCE OF ATTRACTION

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1 REVOLUTION OF NEUTRAL BODY IN A CLOSED ELLIPSE UNDER GRAVITATIONAL FORCE OF ATTRACTION Musa D. Abdullahi, U.M.Y. Univesity P.M.B. 18, Katsina, Katsina State, Nigeia musadab@msn.com, Tel: Abstact A body of mass m evolves in an obit though angle φ in time t, at a distance fom a much bigge mass M, with constant angula momentum, as: dϕ = m k whee k is a unit vecto pependicula to the plane of the obit. The equation of obit is: 1 = Acos ( ϕ + β ) + whee G is the gavitational constant and amplitude A and phase angle β ae detemined fom the initial conditions. The obit of motion is a closed ellipse with the cente of foce of attaction, the cente of gavity of the masses m and M, as the focus. Keywods: Angula momentum, eccenticity, foce, obit, peihelion, evolution, 1 Intoduction Revolution of a body, ound a cente of foce of attaction, is the most common motion in the univese. This comes with evolution of planets ound the Sun, binay stas ound thei cente of mass, a moon ound a planet o an electon ound the nucleus of an atom. The Geman astonome, ohannes Keple [1, ] ealy in the 17th centuy, fomulated thee laws, named afte him, concening the motions of planets. Keple based his laws on astonomical data painstakingly collected in 30 yeas of obsevations by the Danish astonome Tycho Bahe [3], to whom he was an assistant. Keple s poposals boke with a centuies-old belief based on the Ptolemaic system advanced by the Alexandian astonome Ptolemy [4], in the nd centuy AD, and the Copenican system put fowad by the Polish astonome, Nicolaus Copenicus [5], in the 16th centuy. The Ptolemaic cosmology postulated a geocentic univese in which the Eath was stationay and motionless at the cente of seveal concentic otating sphees, which boe (in ode of distance away fom the Eath) the Moon, the planets and the stas. The majo pemises of the Copenican system ae that the Eath otates daily on its axis and evolves yealy ound the Sun and that the planets also cicle the Sun. Copenicus's heliocentic theoies of planetay motion had the advantage of accounting fo the daily and yealy motions of the Sun and stas and it neatly explained the obseved motions of the planets. Howeve, the eigning dogma in the 16 th centuy, that of the Roman Catholic Chuch, was in favou of the Ptolemaic system and it abhoed the Copenican theoy. The Copenican theoy had some modifications and vaious degees of acceptance in the 16 th and 17 th centuies. The most famous Copenicans wee the Italian physicist Galileo Galilei [6] and his contempoay, the astonome ohannes Keple [1, ]. 1 Revolution of a Body unde Gavitational Foce of Attaction 011 Musa D. Abdullahi

2 By Decembe 1609 Galileo [7] had built a telescope of 0 times magnification, with which he discoveed fou of the moons cicling upite. This showed that at least some heavenly bodies move aound a cente othe than the Eath. By Decembe 1910 Galileo [7] had obseved the phases of Venus, which could be explained if Venus was sometimes neae the Eath and sometimes fathe away fom the Eath, following a motion ound the Sun. In 1616, Copenican books wee subjected to censoship by the Chuch [8]. Galileo was instucted to no longe hold o defend the opinion that the Eath moved. He failed to confom to the uling of the Chuch and afte the publication of his book titled Dialogue on the Two Chief Wold Systems [9], he was accused of heesy, compelled to ecant his beliefs and then confined to house aest. Galileo s Dialogue was odeed to be buned and his ideas banned. The ideas contained in the Dialogue could not be suppessed by the Roman Catholic Chuch. Galileo s eputation continued to gow in Italy and aboad, especially afte his final wok. Galileo s final and geatest wok is the book titled Discouses Concening Two New Sciences, published in It eviews and efines his ealie studies of motion and, in geneal, the pinciples of mechanics. The book opened a oad that was to lead Si Isaac Newton [10] to the law of univesal gavitation, which linked the planetay laws discoveed by astonome Keple with Galileo s mathematical physics. Keple [1, ] stamped the final seal of validity on the Copenican planetay system in thee laws, viz.: (i) (ii) (iii) The paths of the planets ae ellipses with the sun as one focus. The line dawn fom the sun to the planet sweeps ove equal aeas in equal time. The squae of the peiods of evolution (T) of the diffeent planets ae popotional to the cube of thei espective mean distances () fom the sun (T 3 ) The impot of Keple s fist law is that thee is no dissipation of enegy in the evolution of a planet, in a closed obit, ound the Sun. Any change of kinetic enegy is equal to the change of potential enegy. The second law means that a planet evolves ound the Sun with constant angula momentum, which is the case if thee is no foce pependicula to the adius vecto. Fom the thid law (T 3 ) and the elationship between the centipetal foce and speed (F v /), it can be deduced that the foce of attaction F on a planet is invesely popotional to the squae of its distance fom the Sun as discoveed by Newton in 1687 [10].. Keple s laws played an impotant pat in the wok of the English astonome, mathematician and physicist, Si Isaac Newton [10]. The laws ae significant fo the undestanding of the obital paths of the moon, the natual satellite of the Eath, and the paths of the atificial satellites launched fom space stations. The pupose of this pape is to deive equation of the obit of motion of bodies, such as the planets, like the Eath, evolving ound a cente of foce of attaction, the Sun. The stating points ae Newton s univesal law of gavitation and second law of motion. Newton s univesal law of gavitation. The foce of attaction F between two bodies of masses M and m, distance apat in space, is given by Newton s univesal law of gavitation: F = û (1) Revolution of a Body unde Gavitational Foce of Attaction 011 Musa D. Abdullahi

3 whee G is the gavitational constant and û is a unit vecto in the adial diection. This foce is extemely feeble that it is only noticeable in espect of huge masses like the moons, the planets and the Sun. Two bodies unde mutual attaction will evolve ound thei cente of mass with the lage mass M evolving in an inne obit and the lighte mass m evolving in an oute obit. The two bodies will evolve with the same angula velocity unde equal and opposite foces of attaction. If M is vey much lage than m as in the case of the Sun (M = kg) and the Eath (m = kg), the Sun may be consideed as almost stationay at a point while the Eath evolves at a distance fom the cente of the Sun. The gavitational foce of attaction, on a moving body, is independent of its speed in a gavitational field. Newton s second law of motion, on a body of mass m moving at time t with speed v and acceleation dv/ in the adial diection, gives: dv F = û= m û () To obtain an expession fo the acceleation (dv/)û, let us conside the evolution of a body ound a cente of foce of attaction. 3 Velocity and acceleation unde a cental motion In Figue 1, the adius vecto OP makes an angle ϕ with the OX axis in space. The position vecto of the point P, in the diection of unit vecto û, (adial diection) and the velocity v at time t, ae espectively given by the vecto equations: = û (3) Y v d d dû v = = û + (4) û P m F M O ϕ X Figue 1. A body of mass m at a point P evolving anticlockwise, with angula displacementϕ, in an obit at velocity v unde the attaction of a stationay mass M at a cente of attaction O. Fo obital motion in the X-Y plane of the Catesian coodinates, dû/, the angula velocity, is given by the vecto (coss) poduct: dû dϕ = k û (5) The angle ϕ is the inclination of the adius vecto OP fom OX, k is a constant unit vecto, in the Z-diection, pependicula to the obital plane (out of the page in Figue 1) and (dϕ/)k is the angula velocity. The velocity v is: 3 Revolution of a Body unde Gavitational Foce of Attaction 011 Musa D. Abdullahi

4 The velocity in the adial û diection is: d d dϕ v = = û + k û (6) d vû = û (7) The acceleation (noting that k is a constant unit vecto) is obtained, a vecto in two othogonal diections, as: d v d dψ = d dψ d ψ û + + k û (8) The acceleation, in the diection of foce of attaction, is a û in the adial diection, so that: v ψ = a û = û (9) d d d 4 Angula momentum In equation (8), the foce pependicula to the adial diection is zeo and, theefoe, the acceleation of mass m is also zeo in this k û diection, so that the equation gives: d dϕ d ϕ m d dϕ m + 0 k û = k û = This equation can be expessed in tems of angula momentum, a vecto, as: whee m d dϕ 1 d = = 0 k û û dϕ = m k (10) Hee, is the constant angula momentum at any point in the closed obit, in accodance with Keple s second law. Equation (10) and equation (9) will be used to deive the equation of the obit of motion of the body at P in Figue 1. 5 Diffeential equation of motion The centipetal acceleation on a body evolving though angle ϕ, unde a cental foce, gives the acceleating foce (equation and 9) as: d dϕ F = û= m û ϕ = GM d d (11) 4 Revolution of a Body unde Gavitational Foce of Attaction 011 Musa D. Abdullahi

5 This is a diffeential equation of motion in and ϕ as functions of time t. We can educe it to an equation of as a function of ϕ only. The angula momentum of m, being a constant (Keple s second law), gives: Making the substitution = 1/u and with equation (1), gives: dϕ = m (1) d d dϕ du du = = (13) du dϕ m dϕ = = dϕ m dϕ = m dϕ d d d dϕ d du u d u (14) Substituting equations (14) and (13) in equation (11) gives: 3 u d u u GMu m dϕ m = d u d u ϕ + = (15) Equation (15) is a second ode diffeential equation with constant coefficients. Tying u = Aexp(zϕ) as a possible solution, we obtain the auxiliay equation:. z z + = 1 0 = 1 = ± j The geneal solution of equation (15) is: An appopiate solution is: 1 u = = Aexp( jϕ ) + (16) 1 = Acos ( ϕ + β ) + (17) whee the amplitude A and phase angle β ae detemined fom the initial conditions. If β = 0, equation (17) may be witten as: 1 A = 1 cos + ϕ (18) This is an ellipse with the majo axis along the X-axis. Equation (18) then becomes: 5 Revolution of a Body unde Gavitational Foce of Attaction 011 Musa D. Abdullahi

6 1 A = B 1+ cosϕ = B ( 1+ η cosϕ ) B (19) whee B = /. Equation (19) gives an ellipse, in the pola coodinates, with eccenticity η = A/B. The ellipse is shown as XYZW in Figue. In Figue, the lighte mass m evolves, in angle ϕ, at a point P, in a closed ellipse, at a distance fom a much heavie mass M at one focus F 1. The othe focus F is at a distance s fom P. A popety of an ellipse is that the distances s + = a, the length of the majo axis ZX. Othe popeties ae obtained as the angle ϕ takes values 0, π/ and π adians. The line ZX is the majo axis, WY is the mino axis and the chods CD and EG ae the latus ectums, with half length l = a(1 η ) = 1/B. The eccenticity of the ellipse is atio of the distance between the foci (F 1 F ) to the length of the majo axis (ZX). Fo a cicle, the two foci coincide and the eccenticity η is equal to zeo. In planetay motion, X, the point of closest appoach to the Sun, is called the peihelion and Z, the point of fathest sepaation, is the aphelion. A planet moves faste as it appoaches the Sun; it is fastest at the peihelion and slowest at the aphelion. At any time, a planet moves in such a way that the diffeence in kinetic enegy between two points is equal to the diffeence in potential enegy. v Figue. A body of lighte mass m evolving in an elliptic obit, of eccenticity η, ound a much heavie body of mass M at the cente of mass, the focus F 1, 6 Elliptic motion of an electically chaged body It should be noted that in equation (15), the adiation component containing du/dϕ, in the case of a chaged body evolving unde a cental foce, is missing. Whee the masses M and m ae electically chaged, thee is a possibility of adiation. A chaged body dissipates enegy wheneve it moves in the diection an electic field. Whee thee is adiation of enegy, the ellipse is not closed but the peihelion (point X in Figue ) otates about an axis 6 Revolution of a Body unde Gavitational Foce of Attaction 011 Musa D. Abdullahi

7 (point O in Figue ). This otation is called pecession of the peihelion. Thee is no adiation whee the evolution of a chaged body is in a cicle of constant adius. 7 Conclusion The obit of evolution of a neutal body, ound a gavitational foce of attaction, is a closed ellipse in one plane, without adiation of enegy. Such a peiodic motion may continue ad infinitum. In the event of the bodies caying opposite chages thee may be adiation of enegy and pecession of the peihelion, as obseved in the evolution of Mecuy, the planet neaest the Sun. The Sun may be positively chaged and Mecuy negatively chaged and the high eccenticity of Mecuy s obit makes pecession of its peihelion obsevable. If the Sun is positively chaged, the planets (including the Eath) may as well be negatively chaged but pecessions of thei peihelion ae masked because of thei lage distances fom the Sun o because of thei obits being nealy cicula.. 8 Refeences 1 planetay_motion Hyman, A., A Simple Catesian Teatment of Planetay Motion, Euopean ounal of Physics, Vol. 14, pp Chistianson,.R., On Tycho's Island: Tycho Bahe and His Assistants, , Cambidge Univesity Pess, 4 Pedesen, O. and Pihl, M., Ealy Physics and Astonomy: A Histoical Intoduction, nd ed. Cambidge Univesity Pess, London. 5 Thomas Kuhn, The Copenican Revolution: Planetay Astonomy in the Development of Westen Thought. Cambidge, MA: Havad Univesity Pess. 6 Raymond.S, Galileo Galilei, His Life and His Woks. Pegamon Pess, Oxfod. 7 Dake, S., Discoveies and Opinions of Galileo, New Yok: Doubleday & Company. 8 McMullin, A., 005. The Chuch and Galileo, pp Note Dame, IN: Univesity of Note Dame Pess. 9 Dake, S. (tanslato), Dialogue Concening the Two Chief Wold Systems. Univesity of Califonia Pess, Bekeley. 10 Newton, I., Mathematical Pinciples of Natual Philosophy (Tanslated by F. Cajoi, 1934), Univesity of Califonia Pess, Bekely. 7 Revolution of a Body unde Gavitational Foce of Attaction 011 Musa D. Abdullahi

History of Astronomy - Part II. Tycho Brahe - An Observer. Johannes Kepler - A Theorist

History of Astronomy - Part II. Tycho Brahe - An Observer. Johannes Kepler - A Theorist Histoy of Astonomy - Pat II Afte the Copenican Revolution, astonomes stived fo moe obsevations to help bette explain the univese aound them Duing this time (600-750) many majo advances in science and astonomy