Satellite Motion 475. p t. Low pressure Air faster (Streamlines are dense) Bottom

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1 Satellite Motion 475 X Satellite Motion Miljenko Solaić Univesity of Zageb - Faculty of Geodesy Coatia 1. Intoduction What is satellite? The wod satellite is coming fom the Latin language (Latin satelles escot, companion). Satellites ae objects that otate aound the planets unde the influence of the gavitational foce. Fo example, the Moon is natual satellite of the Eath. What is the atificial Eath satellite? The atificial Eath satellites ae atificial objects which ae launched into obit aound the Eath by a ocket vehicle. This kind of satellites was named, the Human-Made Eath Satellites. How do ockets function? Aeoplanes wok on the pinciple of buoyancy diffeence on thei wings. This is the eason why aeoplanes can fly only in the ai but not in the vacuum. Thus, an aiplane cannot be used fo launching satellites in thei obit aound the Eath (Fig. 1). ai Top p t Low pessue Ai faste (Steamlines ae dense) p b > p t Bottom Ai slowe (Steamlines ae aely) p b High pessue Fig. 1. When the aeoplanes have velocity in the ai on the bottom of his wings they have highe pessue then on the top of wings. This diffeence of pessue is giving the foce of buoyancy and the aeoplanes can fly. It is also not possible to launch a human-made satellite into the obit aound the Eath with a cannon o a gun because a cannon-ball has the velocity of about 0.5 km/s. It means that this velocity of cannon-ball is about 15 times smalle than the fist cosmic velocity (7.9 km/s). So fo it has been possible to launch a satellite into an obit aound the Eath whee thee is vacuum is possible only with using ockets. The wod ocket comes fom the Italian Rocchetta (i.e. little fuse), a name of a small fiecacke. It is commonly accepted that the fist ecoded use of a ocket in battle was by the Chinese

2 476 Satellite Communications in 13 against the Mongol hodes at Kai Feng Fu. The Mongols wee the fist to have applied ocket technology in Euope as they conqueed some pats of China and of Russia, Easten and Cental Euope. Konstantin Tsiolkovsky ( ) (Fig. ) fom the Impeial Russia and afte fom the Soviet Union published the fist seious scientific wok on space tavels titled The exploation of Cosmic Space by Means of Reaction Devices in He is consideed by many to be the fathe of theoetical astonautics. He also advocated the use of liquid hydogen and oxygen fo popellant, calculating thei maximumm exhaust velocity. His wok inspied futhe eseach, expeimentation and the fomation of Society fo Studies of Inteplanetay Tavel in 194. Also in 194, Tsiolkovsky wote about multi-stage ockets, in Cosmic Rocket Tains. Fig.. Konstantin Fig. 3. Robet Tsiolkovsky Goddad Fig. 4. Segei Koolev Fig. 5. Wenhe von Baun In the USA, Robet Goddad (Fig. 3) began a seiouss analysis of ockets in 191. It can thus be concluded that conventional solid-fuel ockets needed to be impoved in thee ways. One of them is that ockets could be aanged in stages. He also independently developed the mathematics of ocket flight. Fo his ideas, caeful eseach, and geat vision, Goddad was called the fathe of moden astonautics. Afte the Wold Wa II in the USA Wenhe von Baun (Fig. 5) and Segei Koolev (Fig. 4) in the Soviet Union wee the leades in the advancing ockets technology. The opeational pinciple a ocket can be explained by means of a balloon (Fig. 6). In a balloon the pessue of gas is pactically equal on all sides (Fig. 6 a). When this balloon has an apetue then a paticle of gas on this apetue will unde pessue of gas be thown out with velocity v g (Fig. 6 b). The pessue in the balloon in opposite diection of the apetue will poduce the pessuee on the balloon and will give it the velocity v B. The pessues in the othe diections will be mutually cancelled in opposite diection. The ocket opeates on this pinciple. A ocket tavelling in vacuum is acceleated by the high-velocity expulsion of a small pat of its mass (gas). The Fig. 7. epesents a ocket with the situationss befoe and afte the explosion. This is closed mateial system and fo this system linea momentum needs to be conseved. So we can say that the momentum in the beginning position (M v) is equal to momentum of this system afte the explosion when a paticle with the mass (dm) is be thown out with velocity (u el ) in the opposite diection of this ocket velocity.

3 Satellite Motion 477 vb Reaction u el dm t = 0 M Mv t = M - dm v v + dv (M-dm)v dmuel a) b) Action vg Fig. 6. A balloon closed unde gas pessue and afte opening an apetue. Fig. 7. The pinciple of a ocket opeation. It means that the momentum fo a paticle is negative (-dm u el ) and we can wite the equation (Caton, 1965) and (Danby, 1989): M v ( M dm) ( v dv) dm uel. (1) v 0000 km/h (5555 m/s) Active II v Stage Vetical Passive peiod α O Active I Stage Passive Potected cape I Stage v 7500 km/h (083 m/s) Active III Stage vi = 8800 km/h (8000 m/s) 500 km III Stage Satellite II Stage III Stage II Stage I Stage 100 ton 8 ton 1 ton Stat Sepaation v=0 I stage 18 kg 960 kg Satellite 51. kg 76.8 kg Sepaation III stage Sepaation II stage Fig. 8. Imagey of launching a satellite by ocket with the thee stages. Fig. 9. Fo example duing the stat a thee stages ocket may have the mass 100 t but mass of a space vehicle will be only 51. kg. It follows fom this equation that the velocity of ocket inceased fo the elementay magnitude

4 478 Satellite Communications dm( v u dv el ). () M dm Fom this equation it is possible to see that the incease of the ocket velocity is lage if the velocity u el of the paticles of gas is maximally the geate when the mass of paticle dm has some magnitude. This is the eason why the constuctos of ockets like to make ockets with vey high (maximal) velocity of the paticles (gas) of the ocket. Usual ockets have vetical stat. Longe delaying of ockets in the Eath gavitation field causes the loss of velocity but also to big thust duing the stat is not suitable. So ockets ae usually made in same stages (Fig. 8 and 9). Duing the stats of ockets the consummation of fuel is vey lage so that a satellite o a space vehicle entes into the obit with a mass pactically next to nothing (see fo example Table 1). Table 1. Data on initial mass of a ocket in a stat, spend fuel, thown pats fo a launching space vehicle of the mass 51. kg Stage 1 3 Initial mass 100 t 8 t 640 kg Spend fuel 80 t 6.4 t 51 kg Thown pat - 1 t 960 kg Definitely mass 0 t 1.6 t 18 kg Velocity 3860 (m/s) 770 (m/s) (m/s). Planet Motion Until the 17 th centuy people wee thinking that the Sun and planets ae otated aound the Eath by cicles. Such opinions wee pactically usual until Johann Keple..1 Keple s Laws of Planetay Motion Johann Keple ( ) discoveed the laws of planetay motion empiically fom Tycho Bahe s ( ) astonomical obsevations of the planet Mas. The fist and the second laws he published in Astonomia Nova (New Astonomy) in 1609, and the thid law in Hamonices mundi libi V (Hamony of the Wold) in a) Keple s Fist Law of Planetay Motion This law can be expessed as follows: The path of each planet descibes an ellipse with the Sun located at one of its foci. (The Law of Ellipse) This fist Keple s Law (Fig. 10 and 11) is sometimes efeed to as the law of ellipse because planets ae obiting aound the Sun in a path descibed as an ellipse. An ellipse is a special

5 Satellite Motion 479 cuve in which the sum of the distances fom evey point on the cuve to two othe points (foci F 1 and F ) is a constant. Planet A Aphelion θ p A-P apsidal line F F 1 = SUN e lin =ε a a a b P Peihelion max min Fig. 10. The Obit of a planet is an ellipse and its elements. Befoe Keple the Geek astonome Ptolemy and many othes afte him wee thinking that the Sun and planets tavel in cicles aound the Eath. The ellipse can be mathematically expessed in the pola coodinate system by this equation: p 1 cos, (3) whee (, θ) ae heliocentic pola coodinates fo the obit of planet ( the distance between the Sun and a planet, θ angle fom the peihelion to the planet as seen fom the Sun, espectively known as the tue anomaly), p is the semi-latus ectum, and ε is the numeical eccenticity. At θ = 0 the minimum distance is equal to p min. (4) 1 At θ = 90 the distance is equal p. At θ = 180 the maximum distance is p max. (5) 1 The semi-majo axis is the aithmetic mean between min and max: : max min p a. (6) 1 The semi-mino axis is the geometic mean between min and max :

6 480 Satellite Communications p b min max a 1. (7) 1 The semi-latus ectum p is equal to The aea A of an ellipse is b p. (8) a A πab. (9) In the special case when ε = 0 then an ellipse tuns into a cicle whee = p = min = max = a = b and A= π. Using ellipse-elated equations Keple s pocedue fo calculating heliocentic pola coodinates, θ, fo planetay position as a function of the time t fom Peihelion, and the obital peiod P, follows fou steps: 1. Compute the mean anomaly M a fom the equation M πt a P. (10). Compute the eccentic anomaly E by numeically solving Keple s equation: M E sin E. (11) a 3. 1 E Compute the tue anomaly θ by the equation tan tan. 1 (1) 4. p Compute the heliocentic distance fom the equation. 1 cos (13) Fo the cicle ε = 0 we have simple dependence θ = E = M a. CIRCLE ELLIPSE n mean angula velocity (mean motion) Ma mean anomaly E eccentic anomaly θ tue anomaly Ma = n (t-t0) positions A 3 positions A 3 A A 1 1 = A = A 3 SUN 1 position Planet Planet (t) Ma E SUN Fig. 11. Elements of paametes of a satellite obit. θ n t t0 = 0 Peihelion 1 da v Fig. 1. The adius vecto dawn fom the Sun to a planet coves equal aeas in equal times.

7 Satellite Motion 481 b) The Second Keple s Law of Planetay Motion This law can be expessed as follows: The adius vecto dawn fom the Sun to a planet coves equal aeas in equal times. (The Law of equal aeas) Mathematically this law can be expessed with the equation: d 1 0, (14) whee 1 is angula velocity of tue anomaly, is the aeal velocity that the adius vecto dawn fom the Sun to the planet sweeps in one second (Fig. 1). Fom this law it follows that the speed at which any planet moves though space is continuously by changing. A planet moves most quickly when it s close to the Sun and moe slowly when it is futhe fom the Sun. c) The Thid Keple s Law of Planetay Motion This law can be expessed as follows: The squaes of the peiodic times of the planets ae popotional to the cubes of the semi-majo axes of thei obits. (The hamonic law) This law is giving the elationship between the distance of planets fom the Sun and thei obital peiods. Mathematically and symbolically it s possible to expess as follows: P α a 3, whee P is the obital peiod of ciculate planet aound the Sun and a is the semi-majo axis of this obit. Because this popotionality is the same fo any planet which otates aound the Sun it s possible to wite the next equation: P planet1 P planet, namely 3 3 a planet1 a planet P planet1 P planet 3 a planet1. (15) 3 a planet 3. The Physical Laws of Motions Si Isaac Newton s fomulated thee fundamental laws of the classical mechanics and the law of gavitation in his geat wok Philosophieæ Natualis (Pincipia Mathematica) published on July 5, Befoe Isaac Newton the geat contibution to the advance of mechanic was given by Galileo, Keple and Huygens. 3.1 The Fist Law of Motion Law of Inetia This law can be expessed as follows:

8 48 Satellite Communications Eveybody pesists in its state of being at est o moving unifomly staight fowad, except insofa as it is compelled to change its state by foce impessed. Newton s fist law of motion is also called the law of inetia. It states that if the vecto sum of all foces acting on an object is zeo, then the acceleation of the object is zeo and its velocity is constant. Consequently: An object that is at est will stay at est until a balanced foce acts upon it. An object that is in motion will not change its velocity until a balanced foce acts upon it. If the esultant foce acting on a paticle is zeo, the paticle will emain at est (if oiginally at est) o will move with constant speed in a staight line (if oiginally in motion). 3. The Second Law of Motion Law of Foce This law can be expessed as follows: Foce equals mass times acceleation. If the esultant foce acting on a paticle is not zeo, the paticle will have acceleation popotional to the magnitude of the esultant and in the diection of this esultant foce. This law may be expessed by the equation: F = ma, (16) whee F is the vecto of foce, m the mass of paticle and a is the vecto of acceleation (Fig. 13. a). a m F Rope ω F O F ma whee the mass m is a facto of popotionality and a measue of inetial a) b) Tangent F v = const F Little ball Fig. 13. a) An acceleation a of a fee body on a hoizontal plane unde influence of a foce F, b) In the otation when the ope is boken, a little ball shall stat moving with constant velocity along the line of tangent in the hoizontal plane. Really, this is diffeential equation which epesents a basic equation of motion o basic equation of dynamic. Altenatively this law can be expessed by the equation: d F mv, (17)

9 Satellite Motion 483 whee the poduct mv is the momentum: m - the mass of paticle and v - the velocity. So, we can say: The foce is equal to the time deivative of the body s momentum. 3.3 The Thid Law Law of action and eaction This law can be expessed in the following way: To evey action thee is an equal by magnitude and opposite eaction (Fig. 14. a). W F m Fig. 14. a) Unde the influence of the weight W of a body, a nomal eaction of its suppot occus, b) A beam unde loading by a foce F will be defomed as eaction to an active foce F. This law can be also expessed: N W = N a) b) The foces of action and eaction between bodies in contact have the same magnitude, same line of action and opposite sense. 3.4 The Law of Gavitation The Law of Univesal Gavitation Isaac Newton stated that two paticles at the distance fom each othe and, espectively, of mass M and m, attact each othe with equal and opposite foces F 1 and F diected along the line joining the paticles (Fig. 15). The common magnitude F of these two foces is: Mm F G, (18) whee G is the univesal constant of gavitation G m 3 kg -1 s - ),o appoximately ft 4 /lb-sec 4 in Bitish gavitational system of units (Bee & Johnston 196). The foce of attaction exeted by the Eath on body of the mass m located on o nea its suface is defined as the weight (W) of the body (Fig. 16) W =mg, (19) whee g is the acceleation of gavity, being also the acceleation of foce of weight.

10 484 Satellite Communications M F 1 = F F 1 F m m g M C M W mg G m R M R g G R Pole Ellipsoid C ωeath Foce of attaction g φ Pendulum Centifugal foce Fig. 15. Newton s Law of the univesal gavitation. Fig. 16. The weight of a body on the suface of the Eath and the influence of centifugal foces. Because this foce is eally the foce of univesal gavitation it s possible to say Fom this equation it follows that the acceleation of gavity is M W mg G m. (0) R GM g. (1) R The Eath is not tuly spheical so the distance R fom the cente of the Eath depends on the point selected on its suface. This will be the eason why the weight of the same body will also not be the same weight on diffeent geogaphical latitude and altitude of the consideed point. Fo moe accuate definition of the weight of a body it s necessay to include a component epesenting the centifugal foce due to the otation of the Eath. So, the values of g fo a body in est at the sea level vay fom m/s (3.09 ft/s ) at the equato to 9.83 m/s (3.6 ft/s ) at the poles. 3.5 D Alembet s Pinciple Jean le Rond d Alembet ( ) postulated the pinciple called by his name fom the basic equation of dynamic: This equation can be witten in this fom: F m a. () a 0 F m. (3) Fom this equation the magnitude m a is called inetial foce. So, this equation epesents an equation of fictive equilibium whee F epesents esultant of all active and eactive foces, and inetial foce which has the magnitude m a, but in opposite diection of the acceleation a. This equation is named equation of dynamic equilibium.

11 Satellite Motion 485 Fo example, when a body is otating in a cicle with constant velocity v estained by a ope length R then centifugal foce appeas (Fig. 17). C R a n v = const v an R Nomal component of acceleation C m F Rope Centifugal foce (inetial foce) L=(-ma n ) Centipetal foce (the foce in ope) Fig. 17. Centifugal foce at otation with constant magnitude of velocity v along a cicle, and then centifugal foce L appeas. 3.6 Potential due to a Spheical Shell A basic esult poved by Isaac Newton is that spheical shell which is homogeneous (with constant density) attacts an exteio point with mass m=1 as if all of the mass M of the spheical shell concentated at its cente C (Fig. 18). This is the same, as if we have homogenous concentic layes but with diffeent densities and whole masses M then an exteio mass point m attacts as if all of the mass M of the sphees was concentated at its cente (Fig. 19). M C σ = const GM U m=1 σ1= const1 σ= const σ3= const3 σ4= const4 σ5= const5 σ6= const6 M C GM U m=1 Fig. 18. The potential due to the solid spheical shell. Fig. 19. The potential due to the concentic solid homogeneous spheical shells. This fundamental esult allows us to conside that the attaction between the Eath and the Sun, fo example, to be equivalent to that between two mass points. So we can say: The solid sphee of constant density attacts an exteio unit mass though all of its masses wee concentated at the cente. The potential, theefoe, due to a spheical body homogeneous in concentic layes, fo a point outside the sphee is

12 486 Satellite Communications GM U, (4) whee is the distance fom the point with mass m to the cente C of the mass of the homogeneous sphee o to the cente C of the concentic homogeneous sphees. 4. Detemination of Obits Jacques Philippe Maie Binet ( ) deived the diffeential equation in the pola coodinate system of the motion fee mateial paticle unde action of the cental foce when aeal velocity by the second Keple s law is constant. This Binet s diffeential equation can be put down in witing d u mc u u F ad d, (5) 1 whee is u, C - double aeal velocity ( C ), F ad gavitation foce of the cental body with mass M on the fee paticle with mass m ( Fad GMmu ), whee minus sign indicates that this is an attacting foce, and plus sign stands fo the epulsive foce. This diffeential equation is equation of fee paticle motion in a plane displayed in the pola coodinate system. Thus, inhomogeneous diffeential equation is obtained d u GM u. (6) d C Homogeneous Pat Inhomogeneous Pat The solution fo the homogeneous pat of this equation is u 1 Bcos( 0 ), (7) whee B and θ 0 ae the constants of integation. Choosing the pola axis so that θ 0 = 0 we can wite and fo the inhomogeneous pat of the equation The solution of this inhomogeneous diffeential equation (6) is u1 Bcos (8) GM 1 u. (9) C p

13 Satellite Motion C u u u1 1 B cos p GM. (30) The equation fo the ellipse and fo the othe conic section in the pola coodinate system can be witten in the fom cos. (31) p Afte compaing the equations (30) and (31) it is possible to see that the equation (30) tuly epesents the equation of a conic section. The poduct of the constants B and C /(GM) defines the eccenticity ε of the conic section. So it can be expessed by the equation (Bee & Johnston, 196) B BC. (3) GM GM C CIRCLE ELLIPSE PARABOLA HYPERBOLA α. α Fig. 0. The conic sections: cicle, ellipse, paabola and hypebola. Fou cases may be distinguished fo diffeent eccenticities (Fig. 0). 1) The conic section is a cicle when is ε = 0. ) The conic section is an ellipse when 0 < ε < 1. 3) The conic section is a paabola when ε = 1. 4) The conic section is a hypebola when ε > 1. Of cause fo the planets and fo the satellites obits can be only ciculas o ellipses.

14 488 Satellite Communications 5. The Two-Body Poblem It is possible to investigate the motion of two bodies that ae only unde thei mutual attaction. It can also be assumed that the bodies ae symmetical and homogeneous and that they can be consideed to be point masses. So we can do analysis of the motion of planets and the Sun. FP S z O S (Sun) (M) C (cente of mass of the Sun and a planet) S C P FS P 0 0 = / P (planet) (m ) y x Fig. 1. Motion of the Sun and a planet in two-body poblem. The diffeential equation of the Sun motion (Fig. 1) is d M S G Mm. (33) The sign + is because the foce F P S has the same oientation as the vecto 0. The diffeential equation of the planet motion is P G m d Mm. (34) The sign is because the foce F S P has the same oientation as the vecto 0. Afte summing up the equations (33) and (34) we can wite S d P m d d M 0 o MS m P 0. (35) Fom the static it is known that the sum of the moment foces is equal to the moment of esultant. So we can say that the sum of the moment masses is equal to the moment of esultant mass. Now it is possible to wite S P C Afte the fist and the second deivation we have M m ( M m). (36)

15 Satellite Motion 489 d d d S P C and d d M m M m C S P C M m. (37) M m M m Fom the equations (35) and (37) it follows d C M m 0 d o C 0. (38) Theefoe the cente of this mateial system has no acceleation, namely this mateial system is in the inetial motion with the possibility to move with constant velocity v C, o emain at est. The equation (33) may be multiplied by m, and equation (34) by M and afte subtacting the equation (33) fom (34) we can wite d P d S mm Mm G M m, (39) d mm Mm P S G M m (40) C (cente of mass of the Sun and a planet) Sun 1 4 Fig.. The Sun is moving also aound the cente C of mass of the Sun and a planet by a small ellipse and a planet is moving about the same cente C by the bigge ellipse, not aound the geometical cente of Sun. Afte dividing the equation (40) by M and taking fom Fig. 1 that = P - S we can wite d m( M m) m G. (41)

16 490 Satellite Communications This is diffeential equation (41) of the planet motion when taken into account and the planet acting on the Sun. It is easy to pove that this planet is eally otating aound the cente of mass (C) of the Sun and the planet. Also the Sun s geometical cente is otating by the small ellipse aound the cente of mass (C) (Fig. ). Hence: The planet is otating aound the cente of masses (C) of the Sun and the planet by the bigge ellipse. The geometical cente of the Sun also otates aound the cente of masses (C) by a small ellipse. 6. Satellite Motion The poblem, of two bodies is solved exactly in the celestial mechanics, but only in the special case if both bodies ae having small dimensions, i.e. if the Sun and a planet can be thought of as paticles. In this special case the motions of paticles aound the body with finite dimensions is also included, if this body with finite dimensions has the cental spheical field of foces. (Fo example, as a homogeneous ball (Fig. 18) o concentic solid homogeneous spheical shells with diffeent densities (Fig. 19)). Just because ou Eath is not a ball and with homogeneous masses some discepancies appea at satellite motions aound the Eath fom the exact solutions of two bodies when we imagine whole mass of the Eath as concentated in it the cente of mass. Fo the solution of the poblem of the motion of bodies (two paticles) exactly valuable ae thee Keple s laws fom which fallow that the satellite would be moving constantly in the same plane by the ellipse with constant aeal velocity. Plane of equato (Descending node) N Plane of satellite O v S (Satellite) P (Peigee) u (Agument of longitude) (Apogee) A (Venal equinox) Line of node ight ascension of ascending node Fig. 3. Kepleian obital paametes. i (inclination) N (Ascending node) - Agument of peigee v Tue anomaly The positions of satellites ae detemined with six Kepleian obital paametes: Ω, i, ω, a, e and v o t (Fig. 3): The oientation of obits in space is detemined by: Ω - the ight ascension of ascending node (the angle measued in the equato plane between the diections to the venal equinox and ascending node N whee the satellite cosses

17 Satellite Motion 491 equatoial plane fom the south to the noth celestial sphee), i the inclination of obit, (the angle between the equatoial plane and obital plane) and ω the agument of peigee (the angle between the ascending node and the diection to peigee (as the neaest point of satellite)). The dimensions of obit ae detemined by: a - the semi-majo axis and ε the numeical eccenticity of an ellipse. The position of satellite on its obits is detemined by: v - the tue anomaly (as the angle between the diections to peigee and instantaneous position of satellite) o by t - the diffeence of time in instantaneous position and the time in peigee. All Keple s laws and Newton s laws fo a planet motion ae valued also fo the Eath s satellites motion but at satellites thee ae some moe petubations. 6.1 Requied Velocity fo a Satellite A body will be a satellite in a cicula obit aound the Eath if it has velocity in the hoizontal line so that centifugal foce is equal to centipetal foce which is poduced by the Eath s gavitation attaction (Fig. 4). So it can be expessed with the equation: v m I G R Hs mm R H s, (4) whee: m - mass of a satellite, M the Eath s mass, R - adius of the Eath, H s - altitude of a satellite above the suface of the Eath, G constant of univesal gavitation and v I velocity of a body which will become the satellite. v Centifugal foce m I vi (inetial foce) m R H s mm G Gavity R H s attaction of the Eath R (M) C Eath Hs Fig. 4. On the satellite in obit act the Eath s gavity attaction and the centifugal foce. Fom this equation (4) next the equation follows v I GM R H s. (43)

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