PLANETARY ORBITS According to MATTER (Re-examined) Nainan K. Varghese, matterdoc@gmail.com http://www.matterdoc.info Abstract: It is an established fact that sn is a moving macro body in space. By simple mechanics, it is physically impossible for a free planetary body to orbit arond a moving central body, in any type of geometrically closed path. Both, a circle and an ellipse, are closed geometrical figres. Hence, elliptical planetary orbital paths (closed geometrical figres) arond sn are false or apparent. A planetary body moves in same direction and along with its central body. It is only when we imagine reversing the direction of planetary body s motion, on one side of central body s path; we can get a geometrically closed figre for planet s apparent orbital path. Use of a reference frame, related to a static central body, cases planetary orbital paths to appear as closed geometrical figres arond central body. As central body is a moving body, this does not reflect physical reality. Althogh they help to explain apparent phenomena, all properties attribted to elliptical/circlar planetary orbital path are nreal. Real physical actions are restricted to real entities and they have to be nderstood with reference to an absolte reference. Since elliptical shape of a planetary orbit is an imaginary aspect, it has its limitations to explain real actions in natre. Mechanism of orbit-formation and limitations of orbiting macro bodies, described in this article, are based on a radically different dynamics from an alternative concept presented in book, MATTER (Re-examined). Keywords: Planetary orbits, apparent orbit, real orbit. Apparent orbital path: Real orbital paths of all macro bodies, in a galaxy, is arond galactic centre. It is very large and contains many points of similar appearance in relation to central body of respective planetary system. Hence, it is convenient for s to se mch smaller strctre, the apparent orbit, with niqe reference points on it for all practical prposes. Apparent orbit is a small part of real orbital path, between two identical appearances of central body, looking from planetary body (e.g.: one solar year). It is an imaginary concept, where shape of path, speed of planetary body and directions of motions are maniplated to sit observations. As sch, it has no logical mechanical basis. It depicts appearance of a system, where it is assmed that central body by some imaginary mechanism (change of reference frame) is held stationary at centre of apparent orbit and planetary body moves at a (constant) linear speed by an eqally imaginary mechanism. Only case of actions, within a planetary system, is central force, de to gravitational attraction, which accelerates planetary body towards centre of apparent orbit the central body. Parameters of this action are mathematically maniplated to prodce the reqired orbital motion
arond central body that matches observations. [In case of earth, they are also sed to establish proofs of validity for Laws of motion and Laws of niversal gravitation ]. While doing this, greater motions of planetary body before it became a planet and motion or path of central body are ignored. Apparent orbit is convenient to predict cyclic featres that take place annally. However, taking apparent orbit as real orbit of a planetary body is highly illogical and incorrect. Mathematical treatments of apparent planetary orbit, arond a central macro body, assme direction of central force on planetary body is always (almost) perpendiclar to direction of its linear motion. This is an essential reqirement for all laws derived from apparent orbits. In real orbital motion, it is not so. Radial motion of planetary body is perpendiclar to orbital path only at datm points, sitated farthest and nearest to galactic centre. At all other points on orbital path, angle between radial motion and orbital path varies as sine of relative anglar position of planetary body with respect to central body and median path. In circlar (elliptical with no eccentricity) orbits, central force does not affect planetary body s linear motion at all. It can prodce only centripetal acceleration and displacement towards central body. Centrifgal displacement by an imaginary centrifgal force is essential to nllify centripetal displacement of planet towards central body. Unlike in apparent orbit, in case of real (wavy) orbital motion, direction of action of central force on planetary body changes throgh a fll circle, dring its passage throgh two sbseqent segments of (one wave of) orbital path. This behavior will not sstain mathematical proofs derived from calclations, sing parameters of apparent orbit. Central force not only prodces centripetal acceleration of planetary body towards central body bt also affects its linear speed. Magnitdes of these effects depend on relative positions of planetary body and central body. Since direction of centripetal acceleration changes throgh fll circles, depending on relative positions of macro bodies, it has its components assisting or opposing planetary body s linear motion. A non-circlar apparent orbit has two reference points on it, periapse and apoapse. They are sitated diametrically opposite on apparent orbit. Apoapse is the point on apparent orbital path, where planetary body is considered slowest and farthest from central body and periapse is the point on apparent orbital path, where planetary body is considered fastest and nearest to central body. In real orbital motion of planetary body, periapse is a point on its path, where it is nearest to central macro body bt it needs not be fastest at this point. And apoapse is a point on its path, where it is farthest from central body, bt it needs not be slowest at this point. In case of earth, these points are called perihelion and aphelion, respectively. Althogh perihelion(s) are points on orbital path, at which planetary and central bodies approach nearest, highest linear speed of planetary body occrs at points on orbital path, farthest from galactic centre. Similarly, althogh aphelion(s) are points on orbital path, at which distance between planetary and central bodies is farthest, lowest linear speed of planetary body occrs at points on orbital path, nearest to galactic centre. While considering apparent planetary orbits (arond static central body), linear speed of planetary body is redced to a small fraction of its real speed in space (eqal to its linear speed relative to central body), with corresponding redction in planet s kinetic energy. Action, by associated kinetic energy on a planet is likely to show p in miniatre form. Figre 1 compares real path of an orbiting macro body and its apparent orbit for dration of one apparent orbital period. Grey central line shows central body s path. Black, wavy line is path of planetary body. Larger black circle shows central body and circles in dotted line show its ftre positions. Small black circle shows orbiting macro body and grey circles show its ftre positions. Doble headed arrows show central force between macro bodies at varios positions as they move along their paths.
Apparent orbit of planetary body, when it is at position P with central body at S, is shown by the oval in figre 1. As planetary body moves, its apparent orbit moves along with central body. Planet s perihelion is at P and aphelion is at E. In real motion, highest and lowest linear speeds of orbiting macro body occr, when it is at 90 away from path of central body, at M and position B (represented by N on apparent orbit), respectively. All parameters of apparent orbit and orbital motion are related to perihelion and aphelion. From its position at C, ntil B, orbiting macro body is in front of central body and hence it is retarded in its linear motion. From B to A, orbiting macro R A Real orbit P M C Median path S Apparent Orbit B N E Figre 1 T body is behind central body and hence it is accelerated in its linear motion. Line RST is radial line connecting central body to centre of its crved path (galactic centre). Acceleration and deceleration of planetary body change over at points M and B. These points are fixed relative to path of central body. Point M, on oter side of central body s path, is oter datm point and point B (corresponding to point N on apparent orbit), on inner side of central body s path, is inner datm point. Datm points have only a vage relation to apparent orbit. A circlar apparent orbit arond static central body is datm orbit. Real orbital path: In figre, planetary body is at point P on its real orbital path, represented by crved arrow X 1 PP 1. XX 1 is tangent to orbital path at P. Line xx 1 is parallel to line XX 1. Arrow PA represents absolte instantaneos linear speed of planetary body, V, in magnitde and direction. De to inertia, planetary body tends to maintain present direction of its linear motion. Angle between V and path of planetary body varies as planetary body moves in its crved path. Total anglar displacement, W, prodces crvatre of planetary body s path and hence its orbital motion. Direction of planetary body s present instantaneos linear speed, V, is deflected otwards from crved path (from tangent XX 1 ). At any instant, V is deflected from XX 1 by angle α ( sign indicates otward deflection), which is otward drifting rate of planetary body. Arrow PB represents central force on planetary body towards central body. Magnitde of radial speed de to central force is PB =. Direction of is towards central body. Angle between and parallel line xx 1, angle XPB, is eqal to angle PBx 1 = θ. Arrow PP 1 is resltant instantaneos speed of planetary body, V R, and it makes angle W with V. De to action of central force (radial motion ), instantaneos linear motion of planetary body is deflected from PA to PP 1 at the rate of angle W per nit time. Angle between V and VR = APP1 = W, Direction of radial motion with respect to tangent XX 1, QPB = PBx1 = θ A C V X Q W α P X 1 V R P 1 θ x Y B x 1 Figre Y 3
APB = θ + α, P 1 Bx = APX = α, = PBP = ( 180 θ α) Line CP 1 is perpendiclar to line PA. In ACP, PAP1 1, AP 1 = PB =, le Δ 1 1 = PAP1 = PBP = 180 θ AP 1 =, CAP 1 ( α) CP1 = AP1 Sin CAP1 = Sin( 180 θ α) = Sin[ 180 ( θ + α) ] = Sin( θ + α) = AP1 Cos CAP = Cos ( 180 θ α) = Cos [ 180 ( θ + α) ] = Cos ( θ + α) = Cos ( θ + α) PC = PA CA = V ( Cos ( θ + α) ) = V + Cos ( θ + α) Tan W CP1 Sin( θ + α) = PC V + Cos ( θ + α) In parallelogram APBP 1, Diagonal = V = ( AP) + ( PB) PB AP Cos( α) CA 1 = (1) PP R 1 + ( AP) + ( PB) PB AP Cos( + α) = V + V Cos( α) VR = + VR V + V Sinα = V + + V ( ) Sinα = () Factors; deflection rate, W, and radial velocity,, are inward (towards centre of circlar path) and drifting rate, α, is otward (away from centre of circlar path). Hence, relative directions may be assigned to these factors. W and are assigned positive direction and α is assigned negative direction. Ths, eqation (1) becomes: Sin [ θ + ( α) ] Sin ( θ α) Tan W = = (3) V + Cos θ + α V + Cos θ α [ ( )] ( ) W is rate of inward anglar deflection between directions of present velocity, V, and resltant velocity, V R. It is deflection rate. Rate of otward anglar deflection, α, between direction of present motion, V, and tangent at P is drifting rate of planetary body, away from tangent to orbital path. In order to make orbital path crve towards median path of planetary (and central) body, resltant of inward deflection rate, W, and otward drifting rate, α, shold be in the same direction as that of radial motion of planetary body,, with respect to tangent XX at P. [As shown in figre, magnitde of W shold be more than magnitde of α]. Vertical component (to tangent XX) of present speed V is a real motion, sbstitting for effect of (presently) imaginary centrifgal force. This real motion prodces drifting rate, α. Circlar orbit: Circlar orbit is a possible path that can be traced only arond a central body that has no translational motion. Centers of stable galaxies are the only points (entities in niverse) withot translational motion. In A all other cases, circlar orbit is an apparent orbit related to a V central body that is assmed static. For motion in a circlar X α W P X path, instantaneos resltant linear speed, V R, shold be eqal to present instantaneos linear speed, V. Both linear V R speeds are with respect to absolte reference. That is, at any instant, resltant linear speed of a planetary body in its P 1 crved path is eqal to its present linear speed. In a circlar orbit, deflection rate, W, is constant. Planetary body has no Figre 3 B anglar acceleration in orbital path. It maintains constant distance from central body. Hence, drifting rate remains constant and eqal to α all arond orbital path. If vale of negative (clockwise) drifting rate, α, can be maintained constant at half the vale of positive (anti-clockwise) deflection rate, W, by external means or by natral process, direction of resltant linear motion of 4
planetary body along its crved path will deflect at a constant rate. Magnitde of resltant linear speed of planetary body remains constant, eqal to its present (instantaneos) linear speed. In figre 3; arrow PA represents present (absolte linear) speed, V, of a planetary body P. Direction of its present linear motion is deflected from tangent XX to planetary body s crved path at P by an angle, α. Arrow PB represents planetary body s radial motion,, perpendiclar to tangent XX to its crved path at P. Instantaneos resltant speed of planetary body, V R, is represented by arrow PP 1. Its direction is deflected from present linear speed, V, by an angle + W. AP 1 = PB =. APB = + ( α), For circlar path, V = V R ; Sbstitting V for V R in eqation (); V = V + + V Sinα = V Sinα (4) Sinα = V 1 α = Sin radian (5) V When orbital path is circlar, central force acts in perpendiclar direction to tangent at any point on it. Hence, θ =, Sbstitting vale of from eqation (4) in eqation (3), ( V Sinα) Sin( α) V Sinα Sin( α) Sinα Cosα TanW = = = V + ( V Sinα) Cos ( α) V V Sinα Cos ( α) 1 Sinα ( Sinα) Sinα Sinα Sinα = = = 1 Sin α Cos α + Sin α Sin α Cos α Sin α Sinα = = Tanα, W = α Cos α For circlar orbit, where deflection rate W is positive, drifting rate α is in negative direction, with magnitde α = W 1 Sbstitting from eqation (5), W = α = Sin (6) V W Sin 1 W =, Sin = (7) V V For a circlar orbit, where direction of W is positive: Drifting rate, α = W in negative direction. (8) This is the condition, reqired for circlar orbital path of a planetary body arond a static central body or circlar parts of other orbital paths at points on their orbital path, where they exhibit properties of circlar orbit. Resltant linear speed of planetary body, along crved path remains a constant, whose magnitde is eqal to its present (instantaneos) linear speed. Anglar speed of planetary body abot centre of circlar path (deflection rate) is eqal to twice its drifting rate (in opposite anglar direction) and its magnitde is constant. Direction of linear motion of a planetary body that desires to enter circlar orbital path has to be otward from tangent at the point of entry. Drifting rate of a planetary body at the point of initial entry on datm orbit (clockwise from tangent at P, as shown in figre 3), reqired to achieve circlar orbital path (in this case) is precisely eqal to half of deflection rate prodced by central force at present distance between centre of orbital path and point of entry. Hence, planetary body is reqired to approach entry point P from within datm orbit. These conditions can be flfilled only in cases, where orbital path is formed arond a static central body. 5
All natral planetary bodies are mch smaller than their central bodies and they approach their orbital paths from otside datm orbits. A planet in real orbital motion, traces segments of crved paths on either sides of its median path (which is also the median path of central body). A circlar orbital path reqires semi-circlar paths on either sides of median path. De to constantly changing relative direction of central force, it is also impossible to maintain constant anglar speed by a planetary body abot a moving central body. Conseqently, natral planetary bodies cannot have circlar orbital paths. However, certain points on their orbital path may exhibit properties of circlar orbital path. Exceptions to the above are probable cases of static binary systems at the centers of stable galaxies or other planetary systems formed by explosion of a static parent macro body, also at the centre of a stable galaxy, where planetary bodies are thrown away from a static central body to enter their orbital paths from within. Since, no macro body, except stable galaxies, can remain in space withot translational motion; this is only a theoretical consideration. Circlar orbit is a critical condition. Parameters of planetary body (maintained in circlar orbit by external means, in addition to central force ) are very precise. Once in orbit, its drifting rate can be easily changed by external factors. Changes in matter-contents of planetary and central bodies or their linear speeds de to external inflences are bond to affect stability of circlar orbit, de to changes in drifting rate. Collision with debris in space or even neven distribtion of matter contents of planetary bodies can inflence state of a circlar orbit. It shold also be noted that no macro bodies smaller than a stable galaxy can remain static in space. To form a circlar apparent orbit, parameters of an orbiting planetary body shold apparently 1 satisfy eqation (6); W = Sin ( V), at every point on its orbital path. All factors in eqation shold remain constants. This is achieved by ignoring state of motion of central body and original state of motion of planetary body. Apparent circlar orbit of a planet is its smallest apparent orbit. It is the datm orbit of a planetary body for present parameters of central and planetary bodies. This eqation is also applicable to (apparently) circlar parts of non-circlar orbits. Elliptical Orbit: Like circlar orbit, elliptical orbit, arond a static central body, is also an apparent orbit. Variations in parameters of an orbiting planetary body change its datm orbit. Conseqently, even if a planetary body was in an apparent circlar orbit, its datm orbit changes on variation of any parameter. Sch changes or difference in magnitde of drifting rate change shape of its orbital path. Non-circlar apparent orbits are based on datm orbits of planetary bodies. A deformed datm orbit becomes non-circlar apparent orbit. Deformation of datm orbit is with respect to two points (mid-points) that are on diametrically opposite sides. Either forward or rearward part of non-circlar apparent orbit is placed within and other part is placed otside datm orbit, as shown in figre 4. Figre 4 shows two possible representative deformations of datm orbit of into elliptical apparent orbits. Circle in dashed line shows planetary body s datm orbit arond static central body. Oval shapes in red and green show elliptical apparent orbits of planetary body, arond static central body, formed corresponding to its parameters dring entry into orbital path. Since a planet apparently moves in a non-circlar path, tangent to a point on apparent orbital path is not perpendiclar to its instantaneos radis, along which central force is acting. Bt, there are two points that lie on apparent orbital path, at which conditions reqired for circlar apparent orbits are satisfied. At these points, direction of radial motion (action of central force ) is perpendiclar to tangents to apparent orbital path and variation in length of its radis reverses direction. 6
If radis of apparent orbit was increasing before planet reaches the point, it will gradally decrease after crossing this point, till planetary body reaches a similar point on diametrically opposite side of the apparent orbit. At this point, direction of change in distance between central and planetary bodies reverses and radis of apparent orbit gradally increases till planetary body reaches original point on apparent orbit. Periodic changes in length of radis of apparent orbit, abot a mean vale, sstain planetary body in its non-circlar stable apparent orbital path. Points, where reversals of changes in radis of apparent orbit occr, are periapse (perihelion) and apoapse (aphelion) at which planetary body is nearest or farthest from central body. Other reference points on real orbital path are oter datm point and inner datm point, where planetary bodies attain highest and lowest linear speeds. In case of real orbital motion, planetary orbital path is not arond central body bt it oscillates abot a common median path, shared by central and planetary bodies. Anglar speed of planetary body does not correspond to motion in circlar or elliptical path. Deflection rate of orbital path from median path is limited, alternating on both sides; whereas, total deflection of radians in same direction is reqired for every apparent orbit. Planetary body accelerates pto oter datm point, where it moves to front of central body or it decelerates pto inner datm point, where it moves behind central body. Highest and lowest linear speeds of planetary body occr at datm points. Datm points need not coincide with either perihelion or aphelion of apparent orbit, assmed above. Datm points of orbital path are sitated on radial lines of galactic radis (line perpendiclar to median path of central and planetary bodies), passing throgh centers of both, central and planetary bodies. Periapse and apoapse indicate points, on orbital path of a planetary body, which are nearest and farthest from central body. They have no other relations to motions of orbiting planetary body. Periapse and apoapse may be displaced along orbital path withot affecting other parameters (excepting planetary spin speed) of orbital motion. Bt points, at which planetary body s linear acceleration and linear deceleration changes-over, are fixed in space, with respect to median path of central body and depend on relative position of planetary body. Figre 5 shows a planetary body P at periapse of apparent orbital path. Arrow PA represents present instantaneos linear speed, V, of planetary body. Arrow PB represents its radial speed,. Arrow PP 1 represents its resltant (ftre) instantaneos linear speed, V R. Otward drifting rate is eqal to α. W is inward deflection rate of resltant motion. Changes de to anglar acceleration and deceleration add to drifting and deflection rates. Motion of planetary body, at perihelion of its X Mid-point X Orbit Periapse P Oter V datm point A α W B V R P 1 Figre 5 Elliptical orbit Datm orbit Elliptical orbit Figre 4 7 Mid-point
non-circlar apparent orbit, exhibits properties of motion in circlar apparent orbit. Angle between PB and tangent XX at periapse is 90. Figre 6 shows a planetary body P at apoapse of apparent orbital path. Arrow PA represents present instantaneos linear speed, V, of planetary body. Arrow PB represents its radial speed,. Arrow PP 1 represents its resltant (ftre) instantaneos linear speed, V R. Otward drifting rate is eqal to α. W is inward deflection rate of resltant motion. Changes de to anglar acceleration and deceleration add to drifting and deflection rates. Motion of planetary body, at aphelion of its non-circlar apparent orbit, exhibits properties of motion in circlar apparent orbit. Angle between PB and tangent XX at apoapse is 90. Conditions for circlar orbital paths exist at aphelion and perihelion. Angle between tangent to orbital path and central force = θ = / Sin W Ptting Tan W = and θ = / in eqation (3) Cos W SinW Sin( α) Cos α = = (9) Cos W V + Cos ( α) V + Sinα Dring circlar orbital conditions, drifting rate ( α) is eqal to half of deflection rate W. Ptting drifting rate, α = W/ in eqation (9); W W SinW Cos ( ) Cos W W = =, V SinW SinW Sin W W CosW Cos = 0 Cos W V + Sin( ) V Sin W W V sinw ( CosW Cos + SinW Sin ) = 0, V SinW Cos( W W ) = 0 W W W W V sin( ) Cos( ) = 0, V sin( ) Cos( W ) Cos( ) = 0 W V Sin Cos W W W Cos = 0, V Sin = 0, W Sin = V (10) 1 W = Sin (11) V Eqation (10) is same as eqation (7) for circlar apparent orbit. Therefore, a non-circlar apparent orbital path has two points on it, where it behaves like a circlar path. Since crvatre of orbital path varies continosly, behavior of non-circlar apparent orbit as circlar apparent orbit lasts only for an instant. As soon as point of periapse or point of apoapse is passed, planetary body will prse its non-circlar apparent orbital path. Drifting rate at periapse, α peri, is half of deflection rate, W peri, and at apoapse; drifting rate, α aphe, is half of deflection rate, W aphe, in magnitdes. As shown in figres 5 and 6, directions of drifting rates, with respect to radial motion, are always same. Inclded angles between and direction of approach of planetary body are of same sense. In a non-circlar orbit; eqation (11) gives deflection rate. Eqation (7) for circlar apparent orbit is applicable to a non-circlar apparent orbit at its periapse and apoapse, where circlar apparent orbital conditions exist. Deflection rates, W, for periapse and for apoapse are in opposite directions. If planetary body entered its datm orbital path from within (with negative drifting rate), it will move towards its apoapse, where conditions of circlar apparent orbit take place. Similar conditions (reqired for circlar apparent orbit) shold also repeat at a point 180 away from periapse / apoapse, as the case P 1 X V R B α W A V inner datm point P X Orbit Figre 6 8
may be. A stable apparent orbit can be formed only if these conditions are met. At periapse, direction of change in distance between central and planetary bodies reverses. Planetary body moves towards apoapse, where condition for circlar apparent orbit is flfilled once again. At mid-points between periapse and apoapse, inward deflection rate, W, becomes eqal to otward drifting rate, α. Resltant anglar speed of planetary body becomes zero. For an instant, planetary body moves in straight line. Direction of linear motion of planetary body is tangential to apparent orbital path at these points bt directions of radial motion,, are not perpendiclar to tangents at those points. Therefore, conditions for circlar apparent orbit are not flfilled at midpoints in non-circlar apparent orbits. Tangents at mid-points are not parallel to major axis of noncirclar apparent orbit. In a theoretical elliptical path, tangents at the ends its minor axis are parallel to major axis. In cases, explained above, they are not so. Hence, apparent orbital path of a planet is oval (with its narrower end towards apoapse) in shape rather than elliptical. However, de to very small eccentricity of apparent orbits, we come across in natre; they are sally considered as elliptical or circlar. Present laws on planetary motion are formed for elliptical apparent orbits. In real orbital path, points at which orbital path of planetary body crosses median path of central body may be considered as mid-points. After mid-point in real orbital path, between periapse and apoapse, anglar difference between radial and linear motions of planetary body diminishes. When planetary body has moved from mid-point by anglar displacement eqal to deflection of periapse or apoapse from datm points, linear and radial motions of planetary body become co-linear. At this point, deflection rates, W, of planetary body de to central force become zero. However, planetary body contines to move in its crved orbital path nder inflence of drifting rate, α, which contines to decrease in magnitde. Once, this point on real orbital path is passed, direction of anglar difference between linear and radial motions of planetary body reverses. Deflection rate, W, and drifting rate, α, are in same direction for a short while ntil drifting rate, α, changes its sense. Planetary body anglarly accelerates till it reaches another point, where conditions for circlar orbital motion are flfilled, where deflection rate, W, and drifting rate, α, are in opposite directions and magnitde of W is twice that of α. This point is apoapse of real orbital path. Thereafter, similar processes contine to sstain stable orbital motion of planetary body in its real orbit abot its central body. Resltant orbital anglar speed at perihelion = W peri αperi = ωperi Resltant orbital anglar speed at mid-point = Wmid αmid = 0 Resltant orbital anglar speed at aphelion = W aphe αaphe = ωaphe Time to move from periapse to apoapse = T. Where, T is orbital time period. 9 Resltant orbital anglar speed of a planetary body, in its real orbital path, decreases from ω peri at periapse to zero at mid-point, increases in opposite direction from zero at mid-point to ω aphe at apoapse, decreases from ω aphe at apoapse to zero at mid-point and increases in opposite direction from zero at mid-point to ω peri at periapse. Total anglar deflection of a planetary body dring every cycle of real orbital motion is zero. [Total anglar deflection of a planetary body dring every cycle of apparent orbital motion is fll circle radians]. Therefore, real orbital path of a planetary body is wavy abot median path of central body. Location of periapse or apoapse of real orbital path depends on location of point of entry of planetary body on datm orbit and drifting rate at the time of entry. For appropriate drifting rate, point of entry can be at periapse or apoapse of orbital path. Location of periapse and apoapse can
shift later de to external inflences, whereas, locations of datm points remain at their relative positions with respect to central body. Limits of anglar speeds at the point of entry: Real orbit of a planetary body is a path in space, whose strictres are related parameters of central and planetary bodies. It is improbable for planetary bodies to originate in their orbital paths. They have to come to their orbital paths from space, away from orbital paths. For smooth transition from a planetary body s motion otside into motion in an orbital path, all parameters of central and planetary bodies, at the time and at the point of entry shold be same, as if the planetary body was moving in a stable orbital path, at that point. Every potential planetary body has a datm orbit abot its central body. Parameters of datm orbit depend on matter-contents of central and planetary bodies, present linear speeds of centre and planetary bodies and relative direction of approach of planetary body. Planetary body has to approach datm orbit in near-tangential direction, rather than in collision corse towards central body. If planetary body s parameters do not sit appropriate datm orbit, it will be nable to move in a stable orbit abot the central body. Central force on a planetary body (towards central body) is active even when it is very far from its ftre orbital path. Hence, parameters of a planetary body s motion are modified continosly, even before it enters orbital path abot the central body. A planetary body enters its orbital path in near-tangential direction, sbject to following limits. Macro bodies, approaching point of entry into their datm orbits (otside certain limits of anglar speeds with respect to central body), are nable to form stable planetary bodies. To approach a moving central body, potential planetary body from rear has to move at a greater linear speed in a direction almost parallel to direction of central body s motion. Magnitde of (otward) drifting rate of a planetary body depends on angle of approach and linear speed of planetary body. Magnitde of (inward) deflection rate depends on magnitde of central force. As magnitde of drifting rate, α, increases to a limit in negative (otward) direction, deflection rate, W, becomes insfficient to overcome drifting rate and direction of macro body s resltant motion, V R, becomes parallel to (or deflected otward from) tangent to datm orbit at the point of entry. Macro body will fly away in tangential direction to (or away from) datm orbit. Sch a macro body is not able to form a stable orbital path abot central body. Figre 7 shows anglar speeds of planetary body at point of entries near oter X P 1 V R X datm point and inner datm points on W α P datm orbit. Upper figre shows planetary V Inside median path body entering datm orbit otside median A Figre 7 path and lower figre shows planetary body entering datm orbit inside median path. Crved dashed lines represent parts of datm orbit. P is point of entry. PA is present instantaneos linear speed, V and PP 1 is ftre instantaneos linear speed of planetary body, V R. PB = = AP1 is motion de to central force. PB = AP 1, in figre. Ftre instantaneos linear speed, V R is resltant of present instantaneos linear speed V and motion de to central force. In both cases, deflection rate W is eqal to drifting rate α and direction of resltant linear A V otside median path W α P X X P 1 V R B 10 datm orbit
speed PP 1 is along tangents XX at point of entry. Hence, present drifting rate α is the highest anglar speed for any prospective planetary body. Only those macro bodies, approaching with anglar speed less than α, can form a stable orbital path abot the central body. When resltant linear speed of planetary body, V R is along tangent to datm orbit at point of entry, deflection rate W is jst sfficient to overcome otward drifting rate α. W = α. AP PA 1 From figre 7, SinW = = = Sin( α) Highest permitted drifting rate, α = Sin radian (1) Eqation (1) gives higher limit of (otward) drifting rate at point of entry for a macro body, which approaches datm orbit from within and may enter into sccessfl orbital path abot a central body. Macro bodies, approaching datm orbit from within, which have greater otward drifting rate than this vale will fly away from central body. Eqation (5), α = Sin 1 V V 1 V radian, gives condition reqired for an orbiting planetary body to have periapse and apoapse in its orbital path. This eqation shold be satisfied two times in every completed cycle (apparent orbit) of orbital path. If planetary body is entering its datm orbit from otside, by time it reaches its periapse, drifting rate of planetary body shold attain a vale of 1 Sin V. As long as this vale is not reached, planetary body will contine to move towards periapse in its orbital path. That is, distance between central and planetary bodies contines to redce, ntil planetary body reaches periapse in its orbital path. Shold drifting rate exceeds vale of 1 Sin V 11, planetary body will move presmably towards periapse in its orbital path. However, drifting rate, being greater than the vale reqired for stable orbital path, planetary body will move towards central body at a higher rate and spiral down into it, withot ever attaining the condition reqired for periapse. Even if planetary body were to enter datm orbit at the point of periapse, its drifting rate shold not exceed this limit. Ths, the oter limit of drifting rate dring entry (for a planetary body entering datm orbit from otside) to form a stable orbital path abot a central body. V V 1 Figre 8 shows anglar D limits of a planetary body s approach to its datm orbit. S F Grey dashed crve shows part of datm orbit, near point of entry of planetary body. Only one T Figre 8 possible point of entry on oter side of median path is discssed. Point of entry on inner side of median path is also similar. 1 Sin X 1 V 3 R P B A X Point P is point of entry of planetary body in datm orbit. Central line X 1 X is tangent to datm orbit at point of entry. Directions of planetary body s approach along arrows AB, CD and EF are shown in figre 8. Present instantaneos linear speeds of macro bodies are shown by arrows PV 1, PV and PV 3 in magnitde and direction. Arrows V 1 R, V S and V 3 T show radial linear speeds,, of macro bodies de to central force. Considering parameters of approaching macro bodies are E C V is
identical and parameters of central body do not change, magnitdes and directions of are identical. Resltant linear speeds of macro bodies are shown by arrows PR, PS and PT. Macro body, approaching in the direction of arrow EF has otward drifting rate eqal to angle V 1 PX 1. Part of central force acts in opposition to macro body s linear speed and resltant linear speed is redced to PR. Macro bodies, which have greater otward drifting rate than angle V 1 PX 1 will fly away from central body. For greater drifting rate, redction in resltant linear speeds will be larger. Unless, approaching linear speed is very high, macro body will be retarded sfficiently to retrn it towards central body and fall into it. Macro body, approaching in the direction of arrow CD has otward drifting rate eqal to angle V PX 1. This is the lowest limit of otward drifting rate for a macro body approaching datm orbit from within and highest limit of otward drifting rate for a macro body approaching datm orbit from otside. Magnitdes of drifting rate, lower than this vale are for macro bodies approaching datm orbit from otside. Therefore, direction of arrow CD is the border between approach angles for macro bodies from within and withot datm orbit. All macro bodies approaching in directions between arrows EF and CD can form stable orbital paths abot central body, by proceeding towards their apoapse. For macro bodies approaching in the direction of arrow CD, present linear speed and resltant linear speed are eqal in magnitde. Radial motion of macro body is sed solely to change its direction of motion. Macro body will trace semicirclar orbital paths, alternately on either side of median path (circlar apparent orbit). Macro body, approaching in the direction of arrow AB, same as tangent X 1 X at point of entry, has otward drifting rate eqal to zero. This is the lowest limit of otward drifting rate for a macro body approaching datm orbit from otside. Part of radial motion is sed to increase resltant linear speed, PT. Macro bodies approaching datm orbit between directions of arrows CD and AB (from otside datm orbit) may form stable orbital path abot central body by proceeding towards their periapse. A macro body, approaching datm orbit with negative drifting rate (inward deflection from tangent) cannot form stable orbital path abot a central body. It will spiral down into central body. Considering above limits together; to form stable orbital motion abot a central body, planetary body has to enter its datm orbit with vales of drifting rates between Sin 1 ( V) and zero. Angles, between Sin 1 ( V) and Sin 1 ( V) are for macro bodies, which are approaching datm orbit from within. Angles, between Sin 1 ( V) and zero are for macro bodies, which are approaching datm orbit from otside. Planetary bodies with (otward) drifting rate between Sin 1 ( V) and Sin 1 ( V) have their aphelion in front of entry point. When (otward) drifting rate is eqal to critical vale Sin 1 ( V) (when magnitde of resltant linear speed is eqal to magnitde of present linear speed), planetary body traces semi-circlar orbital paths or either sides of median path. Planetary bodies with (otward) drifting rate between Sin 1 ( V) periapse in front of their point of entry. Limits between Sin 1 ( V) 1 and zero have their and zero are for those macro bodies approaching from otside datm orbit. From point of entry they can move only towards periapse of their orbital path.
These stringent restrictions, in conjnction with restrictions on direction of approach (as explained in the next sb-section on Orbits abot moving central body ), considerably lowers nmber of macro bodies, which are able to form stable orbital paths and prevents profsion of planetary bodies abot a central body. Crvatre of orbital path and tangential speed of a planet, at any time, depend on its location on orbital path. Center of crvatre at any point on orbital path is its focs. In real orbital motion, centre of crvatres for alternate half cycle of orbital motion lie on opposite sides of median path. Magnitde of orbital path s crvatre is zero at mid-points and increases as planetary body moves towards periapse or apoapse. However, while considering apparent orbit, imaginary crved orbital path is assmed to close-in on itself to provide circlar or elliptical natre. A planetary body, approaching its datm orbit from otside, approaches on convex side of orbital path. Hence, it is not possible for this planetary body to enter its datm orbit within angles between Sin 1 ( V) and zero to tangent to datm orbit dring its first entry. Practically, planetary body has to cross datm orbit to reach the point of entry Orbits abot a moving central body: It is nlikely that macro bodies of considerable sizes move away from a central body to enter into orbital path abot it. All larger macro bodies, like planets, have to come from otside planetary system. Planets cold enter into orbital paths in any direction arond a static central body. However, if central body is moving (as is the case with all natral macro bodies), directions of approach of their planets are restricted. Following description is abot planetary bodies approaching a central body from otside their datm orbits. It is qite practical that many planetary bodies approaching a central body from otside their datm orbits may have to cross part of datm orbit before their entry into orbital path. That is, instead of approaching point of entry, directly, they may have to enter datm orbit, first and then on their way ot they reach the point of entry, proper. Somewhat similar conditions are applicable to planetary bodies that may approach datm orbit from within. To make explanation simpler, apparent orbit is sed as a reference and real orbital path is compared with it, later. All large macro bodies, in space, move at very high linear speed. (It is estimated that sn moves in a circlar path arond galactic center at a relative speed of abot 50000 m/sec, mch greater than relative speed of earth with respect to the sn, which is abot 30000 m/sec, in its orbital path abot sn). Relative speed between central body and a planetary body, trying to enter into orbital path abot central body, depends on relative directions of their linear motions, with limitation that linear components of motion of both macro bodies are always in identical direction. Planetary bodies, approaching a central body in opposite direction to its linear motion will find action of central force is to enhance their present linear speeds. Relative speed of macro bodies becomes too large for them to form a planetary system. Conseqently, no macro body that is approaching a moving central body in opposite direction to central body s linear motion can enter into a sccessfl orbital path abot the central body. Similarly, macro bodies approaching from sides (all arond) of a moving central body will be left far behind. Sch macro bodies have very little or no linear motion in the direction of linear motion of central body. Hence, their relative speeds are too large to be controlled by relatively small central force. They cannot form stable orbital paths abot a moving central body. In order to enter into a sccessfl orbital path abot a moving central body, a planetary body 13
has to approach central body from the rear and nearly in orbital plane, at a linear speed greater or lesser (within very small margin) than linear speed of central body. Dring approach, relative speed of planetary body varies only by a small fraction with respect to central body s absolte speed. All macro bodies, in a galaxy, move in crved paths arond galactic centre. Directions of their linear motions may be modified slightly by gravitational interactions with other macro bodies in the galaxy. A central body sally has a crved path as shown by line NOM in figre 9, where direction of approach of a planetary body is shown with respect to its apparent orbit, prevailing at the time of planetary body s entry into orbital path. A planet approaching its datm orbit from inside of crved path, NOM, will find that it has an additional relative motion away from galactic center. This is prodced by crvatre of central body s path. Planetary body s drifting rate is enhanced by crvatre of central body s median path. Additional relative motion enhances radial motion prodced by central force. These factors prevent a macro body, approaching from concave side of central body s path, to enter into a sccessfl orbital path (abot central body that itself is moving in a crved path). Above mentioned factors leave only two small windows, shown by APEC and aqec in figre 9 (widths of windows shown are highly exaggerated), throgh which a planetary body may enter into sccessfl orbital path abot a central body. As central body is assmed static, similar windows cold be anywhere arond apparent orbit. Hence we shall limit discssion only to one window APEC on oter side of central body s path NOM. This window is on oter (convex) side of crved path, NOM, and to the rear of central body. Window, APEC, in 3D space, is somewhat conical in shape, with its apex towards oter datm point in planetary body s orbital path. Girth of conical window restricts entry to macro bodies, whose orbital plane can be gradally stabilized into central body s orbital plane. Similar window, aqec, available on opposite side of apparent orbit is described later with respect to real orbital path. All macro bodies, entering into sccessfl orbital paths abot a central body, enter throgh this window. Entry is frther restricted by limits of P drifting rate of planetary body s anglar speed D A dring entry. Therefore, there are no planets orbiting in opposite to direction of central body s own orbital motion arond galactic E B C centre or having its orbital plane too far from central body s orbital plane. Direction of F G O H K displacement of apparent orbit of a planetary body is same as the direction of central body s M c N orbital motion abot galactic center. Ths, all b e orbiting planetary bodies in a planetary Q system move in same apparent anglar a directions and in planes not mch different Figre 9 S R from central body s orbital plane. As shown in figre 9, O is the center of datm orbit arond (static) central body. Circle in green line, PFQHP, is datm orbit corresponding to central and planetary bodies parameters, at the instant, when planetary body enters datm orbit. Datm orbit is smallest possible circlar apparent orbit sitable to parameters of central and planetary body, when magnitde of otward drifting rate is half the magnitde of inward deflection rate. Black circle at O is central body and black circle at D is the planet at its periapse. Elliptical figre, DGRKD, in dashed line is planetary body s apparent orbit, corresponding to position of central body at O. 14
P and S are datm points, where highest and least linear speeds of planetary body occr. D is periapse and R is apoapse of orbital path, shown on apparent orbit. Extension of line, POS, joins central body with galactic centre. At any instant, datm points, central body and galactic centre are sitated on line POS. Crved arrow, BE, shows direction of linear motion of planetary body, while it is entering datm orbit corresponding to apparent orbit DGRKD. Planetary body may enter into datm orbit throgh conical window shown by region between A and C (shown in plane of paper by space between APEC). Relative position of periapse on apparent orbital path, D, depends on location of point of initial entry, E, and magnitde of drifting rate. Greater the drifting rate, farther from point of entry is periapse. Location of apoapse is always diametrically opposite on apparent orbit. Distance between central and planetary body is least, when planetary body is at periapse and it is highest, when planetary body is at apoapse. Linear speed of planetary body is fastest, when it is near oter datm point, P and it is slowest at point S (shown on apparent orbit), when it is near inner datm point, Q. Only those macro bodies, from otside datm orbit, whose otward drifting rate at the time of 1 entry into datm orbit is within limits; 0 < α < Sin ( V) and whose direction of entry is throgh permitted windows can prodce stable orbital paths abot central body that is itself moving in a crved path. Changes in rest mass or linear speed of an orbiting planetary body may change size of its orbital path bt not its eccentricity and anglar position. In order to change eccentricity or relative anglar position of apparent orbit, separate external effort has to act on orbiting planetary body to deflect it from its corse and change deflection rate of instantaneos present linear speed, V, while moving in stable orbital path. Figre 10 shows entry zones for planetary bodies on their real orbital paths. Both windows of entry are depicted in figre. Thick dotted arrow NOM represents path of central body. Wavy arrow KDQK represents one fll cycle of real orbital path of planetary body abot central body, moving along median path NOM. Part KD of orbital path, shown in dotted line, represents extension of orbital path before point of entry to show one fll cycle. 15 Orbital path D P K E B A C O c M e b a N Q K Figre 10 Entry zones are represented (in plane of paper) by regions between APEC and aqec. Arrows BE and be represents directions of approach of a macro body that may become sccessfl planetary body abot central body that is moving along median path NOM. This figre may be compared with figre 9. Description of window aqec is also similar to window APEC, given above. When figre 10 is sed to depict real orbital path of a planetary body, instead of apparent orbital path, as is sed presently, it is clearly evident that a planetary body approaching a central body from any qarters other than throgh the entry windows APEC and aqec cannot enter into sccessfl orbit abot the central body. Entry window APEC is otside apparent orbit of planetary body. Entry window aqec cold be represented on (opposite side) otside apparent orbit, bt in opposite direction. Therefore, it is not described, as shown in figre 9.