PLANETARY ORBITS (According to Hypothesis on MATTER )

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1 PLANETARY ORBITS (Accoding to Hypothesis on MATTER ) Atho: Nainan. K. Vaghese, mattedoc@gmail.com Abstact: De to constant motions of fee bodies, it is pactically impossible fo a fee body to obit aond anothe. Howeve, they may obit abot each othe and follow a common median path in space. Sch motion, when viewed fom a planet appeas to be its obital motion aond the cental body. This is meely an illsion ceated by the elative motions of the bodies. Taking this appaent motion as a scientific fact and applying it to pove the validity of laws of gavity is illogical. Mechanism of obit fomation and the limitations of obiting bodies descibed in this aticle ae based on a adically diffeent dynamics fom an altenative concept pt fowad in Hypothesis on MATTER. Despite its constant adial acceleation, a planet maintains constant speed towads the cental body and yet neve eaches any neae. Planet s paametes ding initial enty into its datm obit detemine the size and eccenticity of its appaent obit. Only those bodies, which appoach the cental body fom the ea, on the ote side of its cved path, thogh a small window can fom stable obits. Hence, it is impeative that all bodies of a planetay system obit in the same sense and ae (almost) in the same plane. Peihelion / aphelion cold be anywhee in the obit, bt the point at which the obiting body has its highest / lowest linea speeds ae fixed in elation to the cental body. All natal planets, whose peihelion ae in font of thei point of enty; aive fom otside the planetay system. Keywods: Obits, Obital mechanism, Planetay obits, Sola system, Celestial mechanism, Astonomy, Cosmology, Hypothesis on MATTER. Intodction: Hypothesis on MATTER descibes an altenative concept. In it: the matte content of a body and the enegy abot a body ae distinctly sepaate. Matte content is the total sm of thee dimensional matte in a body. Enegy is the stain developed de to distotions in the natal aangements of basic matte paticles in and abot a body. Matte content and enegy content of

2 a body case and sppot each othe fo thei existence and stability. They ae not convetible into each othe. Entie space is filled with D enegy fields, two dimensional latticewok fomations by basic 1D matte paticles. Althogh, D enegy fields ae made of (appaently) solid matte paticles it has all popeties of an ideal liqid. Pats of D enegy fields, within the body-dimensions, contain sfficient distotions to sstain integity and stability of a maco body in its cent state. This pat of D enegy fields is the matte field of the body. Distotions in the matte field ae the wok existing in the body and it detemines the state of the body. Foce is the ate of wok being stoed in a body with espect to its displacement. Action of a foce is simple stctal eshaping of the matte field and the eslting motion of any matte paticles pesent in the egion. State of a body depends on the wok (enegy stoed) in it athe than on the foce applied on it. All appaent inteactions between matte paticles take place thogh the medim of D enegy fields. This avoids the assmption of actions at a distance. Thee ae no pll foces o igid bodies in this concept. All foces, classified into vaios types, ae diffeent manifestations of only one type of foce and it is of psh nate. Wok is tansmitted only in staight lines and sepaately in each plane. Foces in diffeent planes do not fom esltant. Foces in the same plane in diffeent diections intefee to edce/incease each othe s efficiency to podce the body s motion. Independent displacements of a body podced by extenal foces in diffeent diections o in diffeent planes may be egaded, togethe, to be esltant motion of the body in D space system. In this aticle, pesent conventions of pll foces and thei esltants ae sed fo claity. A fee body is that maco body, which is fee fom all intefeences othe than the foces/actions consideed. Tendency of a D enegy field to attain seene state does not allow static distotions in it. Tansfe of distotions in the matte field of a maco body caies the associated matte paticles and ths podces body s motion. This inetial action, abot a maco body, maintains its state. A change in the inetial actions abot a body podces its acceleation. If cetain wok is invested into a body, the body will attain a stable state only afte inetial delay, ding which the wok within the body stabilizes. This is te even afte the application of foce is teminated. Matte is inet; it has no ability to move o act on its own. Associated matte field-distotions podce all appaent actions, pesently assigned to the matte. Pesence of D matte paticles in a D enegy field beaks its continity. Discontinity cases imbalance in the D enegy field. Pesses applied by the D enegy field latticewok fom the sides, in an attempt to estoe its continity, compess a matte paticle. [Pimay D matte paticles ae of nifom size and they constitte all othe geate matte bodies]. If the extents of D enegy field on opposite sides of a matte paticle ae neqal, the matte paticle expeiences a esltant foce, which tends to move the paticle towads the side of lowe pesse/foce. Extent of D enegy fields between two matte paticles is less than the extent of D enegy fields on thei ote sides. As a eslt, matte paticles ae pshed towads each othe. Motions of constitent paticles move the whole body. This action gives ise to the appaent gavitational attaction between bodies. Appaent gavitational attaction between two bodies is, elatively, a mino podct of gavitational actions. It takes place between (spinning and disc shaped) pimay matte paticles of both the maco bodies, which ae in the same plane at the given instant. Appaent gavitational attaction, at any instant, is podced between extemely small nmbes of pimay matte paticles in two maco bodies. An aveage appaent attaction is deived fom spoadic actions between vaios matte paticles, which happen to be in the same plane. Contay to pesent belief, gavitational

3 foce is enomosly stong compaed to othe manifestations of foce. All conclsions expessed in this aticle ae taken fom the Hypothesis on MATTER [1]. Fo details, kindly efe to the same. Foces on a planetay body: In this aticle, all actions of a planet de to its inheent inetial motion ae cedited to linea motion / wok (attained by the planetay body befoe it enteed into its stable obital path) and all actions de to the cental foce ae cedited to adial motion / wok in it towads the cental body. A body is defined by the measements of space, occpied by its matte content and by its mass, epesenting the qantity of its matte content. A fee body tends to move in a staight line de to associated inetia. Appaent foce of attaction de to gavity between two bodies is the eslt of appaent attaction between thei matte paticles. Inetia of a body does not a apply foce on the body. While inetia maintains a moving body in staight line, it is the cental foce, by its action on the body, which changes the diection of planet s motion and podce its spin motion. Actions of each body ae between it and the sonding D enegy fields. Concent actions on two bodies, consideed togethe, may be intepeted as an appaent action between them. Althogh gavitational action on each body is sepaate, sch actions on the cental and planetay bodies, when consideed togethe, povide an appaent cental foce of attaction between them. A planet is appaently attacted towads the cental body. Gavitational actions between two bodies take place only in common planes occpied by them. Actions, simila to the obital motion of a planet analysed hee, takes place on cental body also. A moving body contains the wok eqied fo its linea and spin motions. A fee body, which is associated with sch wok, will contine its linea motion in a staight line and its spin motion at constant speeds. Wok contained in the body was invested into its matte field by extenal foces, inclding appaent attaction de to gavity towads the cental body befoe its enty into obit. Anothe extenal foce is eqied to change the state of constant motions of the obiting body. Magnitde of matte field distotions (enegy) in a body, moving in cicla path, does not change. Howeve, to keep changing the diection of motion at a constant ate, distotions in its matte field ae modified continosly. Changes in the matte field distotions podce body s acceleating stages. In motion in a cicla path, iespective of changes in the matte field, magnitde of total matte field distotions in the body is kept constant. Instantaneos velocity of the body depends on the magnitde of its matte field distotions. Acceleation depends on the vaiation in the magnitde o diection of matte field distotions. A planetay body, simltaneosly, maintains a constant acceleation towads the cental body, maintains motion at constant velocities in linea and adial diections and maintains (almost) constant acceleation of its spin motion. Foces in diffeent planes do not inteact. They act on the matte paticles independently. Matte paticles ae moved by each of the foces in its own diection, to podce thei esltant motion. A planet has two simltaneos linea motions, a linea motion nealy tangential to its obit and a adial motion towads the cental body. Linea motion of the planet is diected away fom obit. Angle between the diection of linea motion and the tangent at the obit, the difting ate, podces a pependicla component of the linea motion. This, a eal motion of the body, eplaces the assmed motion podced by the imaginay centifgal foce on the body. Majo pat of (linea) wok within a planet caies it thogh the obital path. Relative diection of adial motion to the tangential linea motion vaies at diffeent

4 points on the eal obital path. Most of the matte field distotions, podcing linea motion and adial motion of the body, ae in diffeent planes. Hence, they do not podce esltants. Howeve, independent displacements of the body, podced by the matte field distotions in diffeent planes, may be ndestood as the esltant motion of the body. In the following analysis, planetay obit abot a cental body (moving in a mch lage cicla path) is consideed. As a planet moves in its obit, its elative diection to cental body changes thogh half a cicle, altenately in eithe diection. This is in contast with pesent assmption of a planet moving aond the cental body in fll cicles. Changes in elative diection between the bodies case vaiations in foces and thei actions. Explanations given below ae fo elative position of the cental and planetay bodies, when the tangents to thei paths ae paallel and the bodies ae moving in the same diection. Action of the cental foce: Cental foce, between a planet and its cental body, is povided by appaent gavitational attaction between them. [Real gavitational foce, pshing the bodies towads each othe is consideed hee as appaent attaction between them]. Diection of this appaent attaction, at datm points in the obit (points on the obital path whee the planet is displaced by п/ adians fom the median path), is pependicla to body s linea motion. Action of the cental foce depends on the magnitde of (adial) matte field distotions it is able to invest into the body. Magnitde of matte field distotions, a body is able to stoe is govened by its absolte linea speed (with espect to the space/d enegy fields). Action of gavitational foce is instantaneos and continos. As long as the paticipating bodies occpy common planes, gavitational foce on the bodies, contines to invest distotions in thei matte fields. Fige shows a spheical (homogeneos) planetay body of adis and mass m with its cente at O and moving to the left (in elation to the cental body, the planetay body is ovetaking the cental body). Magnitde of matte field density depends on the dation of action of a foce. Density of (adial) matte field distotions inceases fom C to O. Planet moves in the diection fom O to C. Absolte linea speed of the body being V m/sec, whole body takes /V seconds to pass a point in space. De to diffeence in sizes of cental and planetay bodies, it takes moe than a second fo the planet to move acoss the cental body, in any tangential diection. Gavitational foce acts on the planet in adial diection, paallel to AB, fo whole of this time. Wok intodced by the cental foce in a coss section of the planet depends on the time, ding which it is nde gavitational inflence in a paticla diection. Take an elementay cicla section PQ (ct by planes paallel to AB at distances x and x+dx fom AB) of thickness dx, pependicla to the axis XX. Magnitde of matte field distotions in any pat of the body is popotional to its volme. PM = OP OM = x, CM = x Volme of PQ = π PM dx = π( x )dx P A X C M O X Font Q x B Fige Rea

5 π m Matte density of PQ = m = (1) π m m Matte content of section PQ π( ( x ) dx = x )dx = π Using invese sqae law fo appaent attaction de to gavity; m( x ) dx MG MGm( x ) Cental foce on section PQ = = Whee, M is the mass of the cental body, G is the gavitational constant in D space system and D is the centes of section PQ and the cental body. Cental foce is the ate of investment of (pependicla) matte field distotions into section PQ of the planetay body with espect to the distance moved, towads the cental body. Cental foce acts on section PQ fo the time, ding which it exists nde the foce. Since, we ae consideing the motion of the planet acoss its obital path; we ae inteested only in those common planes pependicla to planet s obital path containing both the cental and planetay bodies. As soon as the font edge of the planet eaches the pependicla line passing thogh the ea edge of the cental body, both bodies stat to have common planes. The nmbe of common planes inceases as the planet moves fowad to ovetake the cental body. Dation of action of the cental foce on the planet, in pependicla diection to obital path, befoe the section PQ in in the common planes with the cental body is fom the time planet s fowad edge entes the common plane with the cental body to the time, when the section PQ ente the common plane. Distance between the font edge of the planet and section, PQ = ( x). Time dation = displacement / speed = ( x) / V Let the constancy of popotion between foce and magnitde of distotions intodced by it is eqal to k. This constant of popotion fo each body is diffeent. It depends on the size of the body in the diection of foce, consistency of the body and the body s matte density. MGm( x ) dx ( x) Magnitde of (adial) matte field distotions invested in PQ = MGmk = D V D ( x )( x)dx Magnitde of total (adial) matte field distotions in the hemisphee ACBO A of the dynamic planet, when the whole planet is within the common planes with the cental body, W ( x )( x)dx x MGmk 1 = = x= 0 D V x = ( x x + x ) MGmk MGmk = dx = D V D V x= 0 MGmk = D V x= x= 0 D D MGmk = ( x )( x) dx D V x x MGmk 5 + = D V 1 = x 16D 5MGmk V dx x + 0 V k 5

6 Since the vale of the gavitational constant G is detemined expeimentally, we can take that the opeation by the constant of popotion, k, is also atomatically acconted fo in the vale of G. Hence, we may neglect the facto k in the above eqation. 5MGm Ths, W1 = 16D V () Similaly, taking othe (ea) hemisphee AO BEA, of the planetay body, as shown in fige, we may detemine the wok the wok invested in it, as follows; Take an elementay cicla section PQ (ct by planes paallel to AB at distances x and x+dx fom AB) of thickness dx, pependicla to the axis XX. Magnitde of matte field distotions in any pat of the body is popotional to its volme and its distance fom C. ( PM) ( O P) ( O M) = x =, CM = + x, Volme of section PQ = π( x )dx Matte density of section PQ π = m m = π kg/m m m Matte content of section PQ = ( ) ( x ) π x dx = π m( x ) dx MG MGm( x ) Cental foce on section PQ = = Cental foce is the ate of investment of matte field distotions in the section PQ, in the diection of cental body. Time dation in which PQ is nde cental foce = ( + x) V (Taking the constant of popotion eqal to k), Magnitde of (adial) matte field distotions MGmk( x ) dx ( + x) invested in PQ = D V MGmk = D V D ( x )( x)dx + Total magnitde of (adial) matte field distotions in the hemisphee AOBEA of the dynamic planet, x x= MGmk MGmk W = ( x )( + x) = ( x )( + x) = x= 0 dx D V x = ( + x x x ) MGmk = dx D V x= 0 MGmk = D V + x= 0 D dx dx D Font V MGmk x = x + D V = MGmk 11 = D V 1 x 11MGmk 16D V A x dx 0 P Rea X C O M E X X B Fige Q 6

7 Since the vale of the gavitational constant G is detemined expeimentally, we can take that the opeation by the constant of popotion, k, is also atomatically acconted fo in the vale of G. Hence, we may neglect the facto k in the above eqation. 11MGm Ths, W = 16D V () Sm total (adial) wok held in the body, fom eqations (.8/) and (.8/5), W 5MGm 11MGm MGm = W1 + W = + = () 16D V 16D V D V Uneqal momenta of foces abot the cente of gavity, of the fee planet, case its simltaneos adial and spin motions. Eqal momenta on eithe side of cente of gavity, togethe, case the adial motion of the planetay body. They act as single set of wok (foce) thogh the cente of gavity. Remaining one-sided momentm podces a cople abot cente of gavity and cases spin motion of the body. Wok, in the left-hand hemisphee and eqal pat of wok in the ight-hand hemisphee, togethe, podce planet s motion in pependicla diection to its obital path. Total (adial) wok acting thogh the cente of gavity, 5MGm 5MGm Wg = = (5) 16D V 8D V This wok, 5MGm 8D V, acts to podce planet s motion towads the cental body. No body can stay motionless in space. Hence, the facto V is always of positive vale. Remaining (adial) wok, acting abot the cente of gavity of the body and podcing spin motion of the planet, W MGm 5MGm MGm s = = (6) D V 8D V 8D V {[] See the aticle Planetay Spin } Magnitde of adial velocity: Cental body of a planetay system is vey lage, compaed to a planet. Theefoe, it takes some time fo the planet to move acoss the cental body, in any tangential diection. They maintain common planes, paallel to the adial diection, ding this time. As long as the common planes ae pesent, they ae nde appaent gavitational attaction in that adial diection. Appaent attaction de to gavity, in any adial diection, begins as soon as the fowad pat of planet comes in line with the cental body and contines to be pesent as the planet advances in its obital path, moving acoss the cental body. Cental foce in adial diection ceases when the planet has flly cossed the cental body in that tangential diection. At the end of this time, all the wok invested into the planet, fo the podction of its adial velocity in this diection, has been tilized (to change the diection of linea motion and to spin the planetay body) and the planetay body will end its adial motion in this diection. Actions of the cental foce on the planetay body ovelaps fo nea-by points on the obital path. Radial displacement of the planetay body towads the cental body, at evey instant, is along diffeent diections and it (in any adial diection) stops as soon as wok intodced into the body fo motion in that paticla diection is lost fom the body. Conseqently, despite the continos displacement towads the cental body, a planet neve eaches any neae to the cental body (disegading vaiations eqied fo eccenticity of the obital path). 7

8 Eqation (5) gives (adial) total matte field distotions, held (o wok) in the planet and podcing its constant adial motion,, towads the cental body. Unlike in the nomal cases, whee an extenal foce intodces matte field distotions in a body ding its action, the case of planetay system is diffeent. This is becase of the constant change in the diection of motion of the planetay body. In any adial diection, magnitde of matte field distotion is constant, that is, thee is no natal acceleating stage fo the body. The body moves at a constant adial velocity along any adis in consideation (this consideation lasts only fo an instant). Acceleating stage to develop this constant velocity took place befoe the body came in the line of diection consideed. Kinetic enegy of a body, moving at constant speed, = m /. Mass of the planetay body m is constant and its kinetic enegy depends on its velocity. m 5MGm Compaing these two; = 8D V 5MG Radial velocity of the planetay body towads the cental body, = m/sec (7) D V Althogh this (adial) velocity appeas to be of constant magnitde (disegading changes in D and V), it is being enewed at evey instant. Wok is sed p and new wok of eqal magnitde is invested thoghot the planetay body. Continos loss of wok keeps the velocity of the body constant despite continos investment of wok. Investment of wok podces a body s acceleation. Yet, in this case, final velocity is constant iespective of body s acceleation. This is becase of the limitation on body s ability to stoe moe wok (in adial diection) than a constant maximm magnitde, de to its linea motion. Planetay body stats to acceleate at a ate a in its planes towads the cental body, when it stats to coss a common plane with the cental body. Acceleation in this diection ceases when the whole body has cossed the common planes with the cental body in that adial diection. Theeafte it is nable to stoe moe wok of this nate. Long befoe this time, it wold have stated simila actions in neaby planes also. Obital motion: Actions of the cental foce and a planet s obital motion ae independent of all othe bodies, inclding the cental body. Role of the cental body o any othe body in the vicinity is to limit the extent of D enegy fields acting on one side of the planet. Rest of all actions ae pefomed by the actions of D enegy fields on the planet. Althogh a planetay body appeas to move in obital path aond a cental body, in eality, it has independent motion of its own. Appaent attaction by the cental body cases it to move along with the cental body in its motions. De to the gavitational actions, obiting bodies appea to inflence the diection of each othe s motion and ceate petbations in thei paths. Since a planet is vey small compaed to the cental body, deviations in its path ae moe pominent. When these deviations ae obseved abot a cental body that is assmed to be static, path of the planet appeas to be an obit aond the cental body. This is the appaent obit of the planet, which we obseve in eveyday life. Obital motion is appaent only with espect to the paticipating bodies. With espect to absolte efeence, a planet does not obit aond the cental body. Motion of the planet is wave-like along the cental body s path, peiodically moving to the font and to the ea of the cental body. In fige, path of the cental body is shown by the aow in thick dotted line. This cved path, also, is wavy to a smalle extent, cving in the same diections as the 8

9 path of planet. Aow in black wavy-line shows planet s path. Path of a satellite of the planet will be a wavy-line abot planet s path. Cental body and the planet ae shown by black cicles and thei fte positions ae shown by gey cicles. In this sense, it can be seen that a Path of planet 9 Path of cental body Fige planet (o a satellite) obits aond the cente of the cental body s cved path and the wave patten in its path is cased by the pesence of the cental body. Sch changes in the path of a fee body may be attibted to petbations cased by pesence of neaby bodies. These petbations look like obital motion aond a cental body, only when they ae efeed to a elatively small system of bodies. This agment can be caied fthe to show that with espect to absolte efeence thee is no natal obital motion (aond cental bodies) at all, except obital motions of bodies aond the (static) galactic centes. Both, the planet and the cental body move in the same diection abot the same median path in space. Since the cicla obital motion is an appaent phenomenon, eithe of the bodies can be consideed as the cental and the othe as its planet. Planetay laws ae eqally valid in eithe case. Althogh it is geneally stated that the eath obits aond the sn in eastwad diection, it is eqally valid to state that the sn obits aond the eath in westwad diection. Howeve, when moe than two bodies ae consideed as a single system, it is moe convenient to take the common and most pominent body as the cental body and to take othe bodies as planetay o satellite bodies. Lage obital path of a planet is aond the galactic cente. It is vey lage and contains many points of simila appeaance in elation to the cental body of a planetay system. Hence, a mch smalle stcte the appaent obit with niqe points on it, fo efeence, is sed. Appaent obit is a small pat of the lage obit between two identical appeaances of the cental body fom the planet (eg: one sola yea). It is an imaginay concept. As sch, it has no logical basis. It depicts the appeaance of a system, whee it is assmed that the cental body, by some imaginay mechanism, is held stationay at the cente of the obit. Only case of actions within the system is the cental foce, de to mtal gavitational attaction and it acceleates a planet towads the cente of the obit. Paametes of this action ae mathematically maniplated to podce the eqied obital motion aond a cental body that matches the obsevations. [In case of eath, sch mathematical teatments ae also sed to establish poofs of validity fo Laws of motion and Laws of nivesal gavity ]. While doing this, mch geate motions of the planetay body befoe it became a planet and the motion o path of the cental body ae ignoed. An appaent obit is convenient to pedict cyclic feates that take place annally. Howeve, taking an appaent obit as the eal motion of a planet is highly illogical and incoect. A non-cicla appaent obit has two efeence points on it, peiapse and apoapse. They ae diametically opposite. Apoapse is the point whee the planet is consideed to be slowest and fathest fom the cental body and peiapse is the point, whee the planet is consideed to be fastest and neaest to the cental body. In eal motion of the planet, peiapse is a point on its path, whee it is neaest to the cental body bt the body need not be fastest at this point. And apoapse is the point on its path, whee it is fathest fom the cental body, bt the body

10 need not be slowest at this point. In case of eath, these points ae called peihelion and aphelion, espectively. In an appaent obit, diection of adial motion of the planet towads the cental body is always (almost) pependicla to its obital path. In eal obital motion, it is not so. Radial planetay motion is pependicla to the obital path only at datm points sitated fathest and neaest to the cental body. At all othe points on the obital path, angle between adial motion and obital path vaies as the sine of elative angla position of the planet with espect to the cental body and the median path. A Median path B Fige 5 Peihelion P M S Obit Fige 5 compaes the eal path of an obiting body and its appaent obit fo the dation of one appaent obital peiod. Black cental line is Cental body s path. Gey, wavy line is the path of the planet. Lage black cicle shows the cental body and the cicles in dotted line show its fte positions. Small black cicle shows the obiting body and the gey cicles show its fte positions. Doble headed aows show the appaent cental foce between the bodies at vaios positions as they move along thei paths. As a planet moves, its appaent obit moves along with the cental body. Appaent obit of the planetay body, when it is at position P with the cental body at S, is shown by the oval in fige 5. Planet s peihelion is at P and aphelion is at E. In eal motion, highest and lowest linea speeds of the obiting body occ, when it is at 90 away fom the path of the cental body, at M and B, espectively. All paametes of an appaent obit and the obiting motion ae elated to peihelion and aphelion. Fom its position at C, ntil B, the obiting body is in font of the cental body and hence it is etaded in its linea motion. Fom B to A, the obiting body is behind the cental body and hence it is acceleated in its linea motion. Line RST is the adial line connecting the cental body to the cente of its cved path (galactic cente). Acceleation and deceleation of the planet change ove at points M and B. These points ae fixed elative to the path of the cental body. Point M, on the ote side of cental body s path may be called ote datm point and point B (coesponding to point N on the appaent obit), on the inne side of cental body s path may be called inne datm point. Datm points have only a vage elation to appaent obit. A cicla appaent obit aond a cental body epesents an ideal obit abot the cental body, which may be called datm obit. In fige 6, P is the planet at a point on its obital path. XX is the tangent to the obit at P. PA is the absolte linea speed of the body, V, in magnitde and diection. De to inetia, the planet tends to maintain the diection of its linea motion. Angle between V and the path of the body vaies as the body moves in its cved path. Total angla displacement podces body s obital motion. At any instant, V is deflected fom XX by difting ate, α (clockwise deflection). Radial motion de to the cental foce is PB =. Diection of is towads the cental body. Angle between and the tangent XX, at P is + θ, eqal to the N T R E Aphelion C 10

11 angla displacement of the planet in its obit fom a efeence point, X 1 on the median path. PP 1 is the esltant motion of the body, V R, and it makes an angle W with V. De to the A V X Q W α P X Y 11 Obit P 1 V 1 θ action of the cental foce (adial motion ), linea motion of the body is deflected fom PA to PP 1. Angle between V and VR = APP1 = W, QPB = PBX1 = θ Angle between V and = APB = θ + [ θ + ( α) ] [ θ + ( α) ] ( α) Sin tanw = (8) V + Cos W is the ate of angla deflection between pesent speed, V, and esltant speed, V R. It may be called the deflection ate. α, the difting ate is the ate of angla deflection between the pesent speed, V, and the tangent. In ode to make the path cve towads the median path, esltant of W and α shold be in the same diection as that of. Vetical component (to the tangent XX) of pesent speed V is a eal motion, sbstitting fo the effect of the (pesently) imaginay centifgal foce. This eal motion podces the difting ate, α. Cicla obit: X 1 Y B Fige 6 Fo motion in a cicla path, V R = V. That is, at any instant, the esltant linea speed of the body in its cved path is eqal to its pesent speed. In a cicla obit, W is constant. Thee is no angla acceleation. Hence, the difting ate emains A constant and eqal to α all aond the path. If a α V negative difting ate ( α) can be maintained constant X W P X by extenal means o by natal pocess, the esltant linea motion of the body along its cved path deflects V R at a constant ate and its magnitde emains a constant eqal to its pesent (instantaneos) linea speed. P 1 In fige 7; V is the pesent (absolte linea) speed, deflected fom the tangent XX at P by an angle Fige 7 B ( α), is the adial motion pependicla to the tangent. PP1 = VR, is the esltant motion of the body, deflected fom V by an angle W. π APB = + α ( ) π = V + + VCos[ + ( α) ], V = V + VSin( α) VR Fo motion in a cicla path; V R = V, Ptting V in place of V R, R

12 = V + VSin( α), = VSin( α), VSin( α) V = (9) 1 Sin ( α) = o ( α) = Sin ad V V (10) Ptting the vale of fom eqation (11) in eqation (10), VSin( α) Cos( α) VSinα Cosα Sinα Sinα tanw = = = = V VSin( α) Sin( α) V VSin α 1 Sin α Cosα 1 Sinα 1 W = tan = tan ( tanα) = α Cosα (11) 1 W Ptting vale of α fom eqation (10), W = α = Sin, Sin V V = (1) Fo a cicla obit, whee diection of W is positive: α = W in negative diection. (1) This is the condition, eqied fo a cicla obital path aond a cental body o cicla pats of othe obits. Resltant linea speed of a planet along the cved path emains a constant, eqal to its pesent (instantaneos) linea speed. Angla speed of the planet (deflection ate) is eqal to twice the difting ate (in opposite angla diection) and it is a constant. Difting ate of a planet, eqied to achieve a cicla obit (in this case) is less than the angle of contingence at the point of initial enty on the datm obit (clockwise fom the tangent at P) and it is pecisely eqal to half the ate of deflection ate podced by the cental foce at that distance. Hence, the body is eqied to initially appoach the enty point P fom within the datm obit. These conditions can be met only in cases, whee the obit is fomed aond a static cental body. All natal planetay bodies ae mch smalle than thei cental bodies and they appoach thei obits fom otside thei datm obits. In eal obital motion, a planet taces segments of cved paths on eithe sides of its median path. A cicla obital path eqies semi-cicla paths on eithe sides of median path. De to constantly changing elative diection of cental foce, it is also impossible to maintain constant angla speed by a planet abot a moving cental body. Conseqently, natal planetay bodies cannot have cicla obits. Exceptions to the above ae pobable cases of static binay systems o othe planetay systems fomed by explosion of a static paent body, whee the planets ae thown away fom a static cental body to ente thei obits fom within. Cicla obit is a citical condition. Paametes of a body (maintained in a cicla obit by extenal means) ae vey pecise. Once in the obit, the difting ate can be easily changed by extenal factos. Changes in the masses of the planetay and cental bodies o thei speeds de to extenal inflence ae bond to affect the stability of a cicla obit de to changes in the difting ate. Collision with debis in space o even neven distibtion of mass of the bodies can inflence the state of a cicla obit. It shold also be noted that no bodies smalle than a galaxy cannot emain static in space. To fom a cicla appaent obit, paametes of a planet shold satisfy the eqation 1 W = Sin ( V) at evey point on its obital path. All factos in the eqation emain constants. Appaent cicla obit is the smallest appaent obit of a planetay body. It is the datm obit of the body fo its pesent paametes. This eqation is also applicable to cicla pats of non-cicla obits. Evey stable obital path has two points on it, which coesponds to cicla paths. 1

13 Elliptical Obit: Vaiations in the paametes of an obiting body change its datm obit. Conseqently, even if a body was in an appaent cicla obit, its datm obit will change on vaiation of any paamete. Sch a change o a diffeence in the difting ate changes the shape of the obit. Noncicla appaent obits ae based on the datm obit of the body. A defomed datm obit becomes non-cicla appaent obit of the body. Defomation of the datm obit is with espect to two points (mid-points) that ae on diametically opposite sides on the appaent obit. Eithe fowad o eawad pat of non-cicla appaent obit is placed within and the othe pat is placed otside the datm obit. Since a planet moves in a non-cicla path, tangent to a point on the obit is not pependicla to the adis of the obital path, along which the cental foce is acting. Bt, thee ae two points that lie on the obit, at which the conditions eqied fo cicla obits ae satisfied. At these points, diection of adial motion is pependicla to the tangent and diection of change in the length of appaent obit s adis eveses. If the adis was inceasing befoe, afte cossing this point, it will gadally decease till the body eaches a simila point on the diametically opposite side of the appaent obit. At this point, diection of change in the distance between the bodies eveses and the adis gadally inceases till the body eaches the oiginal point on the appaent obit. Peiodic changes in the length of adis of the appaent obit abot a mean vale sstain the stable obital path. Points, whee these evesals occ ae peihelion and aphelion at which the bodies ae neaest o fathest. Othe efeence points ae ote and inne datm points, whee the planet attains highest and lowest linea speeds. In case of eal motion, an obital path is not aond the cental body bt it oscillates abot a common median path, shaed by the cental body. Angla speed of an obiting body does not coespond to cicla o elliptical motion. Deflection ate of obital path fom the median path is limited, altenating on both sides; whee as, a total deflection of π adians in the same diection is eqied fo evey appaent obit. Linea motion of the planet is acceleated p to the ote datm point, when the planet moves to the font of the cental body o it is deceleated p to the inne datm point, whee the planet falls behind the cental body. Highest and lowest speeds of the planet occ at these datm points. They need not coincide with eithe peihelion o aphelion of the appaent obit, assmed above. Datm points of obital path ae sitated on adial lines of the galactic adis (line pependicla to the median path) and passing thogh both the cental and planetay bodies. Points of peihelion o aphelion indicate the points, on the path of the planet, which ae neaest and fathest fom the cental body. They have no othe elations to the motions of the obiting body. Peihelion and aphelion of an obit may be displaced along the path withot affecting othe paametes (excepting the planetay spin speed) of obital motion. Bt the points, at which the acceleation / deceleation change-ove takes place, ae fixed with espect to the median path of the cental body and depend on the elative position of the planet. Pat A in fige 8 shows the planetay motion at the peihelion of an obit and pat B in fige 8 shows the planetay motion in the same obit at its aphelion. V is the pesent speed, is the adial speed and V R is the esltant speed of the body. Difting ates ae eqal to α. W is the deflection ates of the esltant motion. Changes de to the angla acceleation / deceleation add to the difting and deflection ates. Motions of the planet at the peihelion and at the aphelion of its non-cicla obit exhibit popeties of cicla obit. Angles between and tangents at peihelion and aphelion ae 90 each. 1

14 1 Peihelion P X Obit Ote datm point V W A α B A X P 1 P 1 V R V R X B α A W Inne datm point B P Aphelion Fige 8 Obit Ptting difting ate, α = W/ and θ = π/ in eqation (10); W Cos tan W = (1) W V Sin V W W SinW SinW Sin CosW Cos = 0, V SinW Cos( W W ) = 0, V W Sin Cos W W Cos = 0, V Sin W = W 1 0, Sin V =, W = Sin (15) V This is same as eqation (1) fo a cicla obit. Since the cvate of the path vaies continosly, this cicla behavio lasts only fo an instant. As soon as the points of peihelion o aphelion ae passed, planet will pse its non-cicla path. Difting ate at peihelion, α pei, is half of deflection ate, W pei, and at aphelion; difting ate, α aphe, is half of deflection ate, W aphe, in magnitdes. As shown in fige 8 A & B, diections of difting ates with espect to adial motion ae always the same. Inclded angle between and diection of appoach ae of the same sense. In a non-cicla obit; eqation (8) gives the deflection ate. Eqation (1) fo cicla obit is applicable to a non-cicla obit at its peihelion and aphelion, whee cicla obital conditions exist. Ptting vale of fom eqation (7) in eqation (15); 5MG 1 1 At peihelion o aphelion, D V 5MG W = Sin = Sin V 16D V (16) W fo peihelion and fo aphelion is of opposite diections. If the body has enteed its datm obit fom within (with negative difting ate), it will move towads its aphelion,

15 whee conditions of cicla obit takes place. Simila conditions (as eqied fo cicla obit) shold also epeat at a point 180 (in appaent obit) away fom the peihelion / aphelion, as the case may be. A stable obit can be fomed only if these conditions ae met. At peihelion, diection of change in the distance between the bodies eveses. The planet moves towads its aphelion, whee condition fo cicla obit is flfilled once again. At midpoints between peihelion and aphelion, W becomes eqal to α. Resltant angla speed of the body becomes zeo. Fo an instant, planet moves in a staight line. Diection of linea motion of the body is tangential to the obital path at these points bt diection of adial motion is not pependicla to the tangent. Theefoe, conditions fo cicla obit ae not flfilled. Tangents at these points ae not paallel to the majo axis of the appaent obit. In an elliptical path, tangents at the ends its mino axis ae paallel to majo axis. In this case, they ae not so. Hence, appaent obital path of a planet is oval (with its naowe end towads the aphelion) athe than an ellipse. Howeve, de to vey small eccenticity of the appaent obits, we come acoss in nate; they ae sally consideed as elliptical o cicla. Pesent planetay laws ae fomed fo (the imaginay) elliptical appaent obits. Afte the mid-point, angla diffeence between the adial and linea motions diminishes. When the body has moved fom this point by an angla displacement eqal to the deflection of peihelion / aphelion fom the datm points, linea and adial motions of the body become co-linea. At this point, thee ae no deflection ates of the body de to cental foce. Howeve, the body contines to move in its cved obital path nde the inflence of difting ate, which contines to decease in magnitde. Once this point is passed, diection of angla diffeence between linea and adial motion eveses. W and α, both ae in the same diection fo a shot while ntil α changes its sense. The planet will anglaly acceleate till it eaches anothe point, whee conditions fo cicla obital motion ae flfilled, whee W and α ae in opposite diections and magnitde of W is twice that of α. This is the aphelion of the obit. Theeafte, simila pocesses contine to sstain stable obital motion. Resltant obital angla speed at peihelion = = W α = ω Resltant obital angla speed at mid-point = = Wmid αmid = 0 Resltant obital angla speed at aphelion = = W α = ω Time to move fom peihelion to aphelion = T, Whee, T is obital time peiod. Resltant obital angla speed deceases fom ω pei at peihelion to zeo at mid-point, inceases fom zeo at mid-point to ω aphe at aphelion, deceases fom ω aphe at aphelion to zeo at mid-point and inceases fom zeo at mid-point to ω pei at peihelion. Taking vaiation in the angla speed to be nifom; Total diffeence between angla speeds at peihelion and at aphelion = = ω + ω T Obital angla acceleation / deceleation = ( ω pei + ω aphe ) pei aphe pei aphe pei aphe pei aphe 15 = (17) Location of peihelion o aphelion of an obital path depends on the location of point of enty on the datm obit and difting ate at the time of enty. Fo the appopiate difting ate, the point of enty can be the peihelion o aphelion of the obit. Location of peihelion / aphelion can shift late de to extenal inflences, wheeas, datm points emain at thei elative positions with espect to cental body.

16 Limits of angla speeds at the point of enty: Obit of a planet is a path in space, whose paametes ae elated to the cental body. It is impobable fo planets to be bone in thei obits. They have to come to thei obits fom space away fom obital path. Fo a smooth tansition fom thei motion otside the obit into the obital path, all thei paametes of motion at the point of enty shold be same, as if they wee moving in the obit at that point. Cental foce is active on planetay bodies even when they ae vey fa fom obital path. Hence, paametes of planetay bodies motions ae modified continosly, even befoe they ente thei obits. A planetay body entes its obit in neatangential diection sbject to the following limits. Bodies, appoaching the point of enty into datm obit, otside cetain limits of thei angla speeds, ae nable to fom stable obits. As the magnitde of difting ate, α, appoaches a limit in negative (clockwise) diection, deflection ate, W, becomes insfficient to ovecome the difting ate and the diection of the esltant motion, V R, becomes paallel to the tangent. Sch a body is not able to fom an obit. 1 When V R is along the tangent, = Sinα, α = Sin V V This is the lowe limit of difting ate at the point of enty (fom within the datm obit) fo bodies, which may fom sccessfl obits abot a cental body. Bodies appoaching the datm obit fom within, with highe (negative) difting ate than this vale will fly away fom the cental body. 1 Eqation (10), α = Sin gives the condition eqied fo an obiting body to have V peihelion and aphelion in its obit. This eqation shold be satisfied two times in evey completed obit. If the body is enteing its datm obit fom otside, by the time it eaches its 1 peihelion, difting ate of the body attains a vale of α = Sin. As long as this vale is not V eached, the body will contine to move towads the peihelion. That is, distance between the 1 bodies contines to edce. If the difting ate exceeds α = Sin, the obiting body will V move towads the cental body at a highe ate and spial down into it, withot eve attaining the condition eqied fo peihelion. Even if the obiting body is to ente the datm obit at 1 the point of peihelion, its difting ate shold not exceed this limit. Ths, α = Sin is the V ote limit of difting ate ding enty (fo a body enteing the datm obit fom otside) fo a sccessfl obit. Consideing the above limits togethe, to fom a stable obital motion, a planet has to 1 ente its datm obit with difting ate between Sin and Sin 1 at the point of enty. V V Limit between Sin 1 and zeo is fo those bodies appoaching fom inside the datm V obit. Bodies with difting ate between Sin 1 and Sin 1 will have thei aphelion in V V font of the enty point. When the difting ate is eqal to the citical vale of Sin 1, the V body will tace a cicla obit. Bodies with difting ates between Sin 1 and Sin 1 V V 1 have thei peihelion in font of thei point of enty. Limits between zeo and Sin ae fo those bodies appoaching fom otside the datm obit. Fom the point of enty they can move only towads thei peihelion. These stingent estictions, in conjnction with estictions on the magnitde of angle of enty (diection of appoach as explained in next section), consideably lowes the nmbe of bodies, those ae able to fom stable obits and pevents pofsion of planetay bodies abot a cental body. V 16

17 Cvate of obit and the tangential speed of a planet depend on its location on the obital path. Cente of cvate at any point on the obit is the focs of the obital path. In eal-motion, cente of cvates fo obital motion on eithe side of median path, lie on the opposite sides of the median path. Cvate is zeo at mid-points and inceases as the body moves towads the aphelion o peihelion. Howeve, while consideing the appaent obit, the cved path is assmed to close-in on itself to povide cicla / elliptical nate. Appaent angla motion of a planet: While a planet is pefoming its obital motion abot the cental body, it is also obiting aond the cente of cental body s obital cente (galactic cente). Conside a planetay system as a single nit (of evolving bodies abot a cental body) that is obiting the galactic cente. Galaxies ae spinning bodies bt they have no obital motion abot any othe body. They ae static in space othe than fo small movements ding thei stabilization. By the time the cental body completes an obit aond the galactic cente, evey planet in the nit appaently loses one obit each, abot the cental body. This povides an appaent loss of obital motion to the planet. A planet appaently loses pat of its obital motion at a constant ate. Obits abot a moving cental body: It is nlikely that bodies of consideable sizes move away fom a cental body to ente into obital path abot it. All lage bodies, planets, have to come fom otside the planetay system. Planets may ente into thei obits in any diection aond a static cental body. Howeve, if the cental body is moving, diections of appoach of the planets ae esticted. Following desciption is abot planetay bodies appoaching the cental body fom otside thei datm obits. To make the explanation simple, an appaent obit is sed. All lage bodies move at vey high speed. (It is estimated that the sn moves in a cicla path aond the galactic cente at a elative speed of abot m/sec, mch geate than the elative speed of eath with espect to the sn, which is abot 0000 m/sec, in its obit). Relative speed between the cental body and a planet tying to ente into an obit depends on the elative diections of thei motions. Planetay bodies, appoaching in opposite diection to the motion of the cental body will find the inteaction de to cental foce enhancing thei pesent speed. Relative speed of the bodies will become too lage fo them to fom a planetay system. Conseqently, no planet that is appoaching the cental body in opposite diection to the diection to its own motion can ente into a sccessfl obit. Similaly, bodies appoaching fom the sides (all aond) will be left fa behind the cental body. Sch bodies have vey little o no motion in the diection of motion of the cental body. Hence, thei elative speed is too lage. They cannot have stable obits. In ode to ente into a sccessfl obit, a body has to appoach the cental body fom the ea and nealy in its obital plane. Ding sch an appoach, elative speeds of the planetay bodies with espect to the cental body s absolte speed will be only a small faction. Relative speed of the appoaching body is with espect to the absolte speed of the cental body. A cental body sally has a cved path as shown by line NOM in fige 9, whee the diection of appoach is shown with espect to an appaent obit. A planet appoaching its datm obit fom the inne side of this cved path will find that it has an additional elative motion away fom the galactic cente. This is podced by the cvate of cental body s path. Planet s difting ate is enhanced by the cvate of cental body s path. Additional 17

18 elative motion will also enhance the adial motion podced by the cental foce. These factos pevent a body, appoaching fom the concave side of the cental body s path to ente into an obit (abot the cental body that itself is moving in a cved path). P D E B A 18 F G O H K C M N Q S R Fige 9 The above factos leave only a small window, shown by APEC in fige 9 (width of the window shown in the fige is highly exaggeated), thogh which a planet may ente into sccessfl obit abot a cental body. This window is on the ote (convex) side of cved path and to the ea of the cental body. It is somewhat conical in shape, with its apex towads the ote datm point. Gith of the cone esticts enty to the bodies, whose obital plane can be gadally stabilized into cental body s obital plane. All planets, enteing into sccessfl obit, ente thogh this window, which is fthe esticted by the limits of difting ate. Theefoe, thee ae no planets obiting in opposite diection to cental body s own obital diection o having its obital plane too fa fom cental body s obital plane. Diection of appaent obital motion of a planet is the same as the diection of cental body s obital motion abot the galactic cente. Ths, all obiting bodies in a planetay system move in the same angla diection and in planes not mch diffeent fom cental body s obital plane. As shown in fige 9, O is the cente of appaent obit. Black cicle at O is the cental body and the black cicle at D is the planet at its peihelion. PFQH is the datm obit coesponding to body s paametes. P and S ae the datm points, whee highest and least linea speeds of the obiting body occ. Obiting body may ente into the datm obit anywhee thogh the conical window shown by shaded egion between A and C. Position of peihelion of the obit, D, depends on the point of initial enty, E, and the difting ate. Geate the difting ate, fathe fom the point of enty is the peihelion. Only those planetay bodies, whose difting ate, at the time of thei enty, ae within the 1 1 limits Sin ( V) < α < Sin ( V) and whose diection of enty is thogh the pemitted window can ente into a stable obit abot a cental body that is itself moving in a cved path. Changes in the mass o speed of an obiting body may change the size of its obit, not its eccenticity and angla position. In ode to change the eccenticity o angla position of an obit, an extenal foce has to be applied on the obiting body to deflect it fom its cose and change the deflection ate of V by an effective change to the difting ate, while in the obit.

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