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Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
Orbital Mechanics: 12. Lunar And Interplanetary Trajectories
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Orbital Mechanics: 12. Lunar And Interplanetary Trajectories

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  • 1. 12 Lunar and Interplanetary Trajectories12.1 Introduction The solar system consists of a single star (the sun) and nine principal planetsthat move around the sun in orbital paths that are nearly circular except for that ofPluto, which is highly eccentric. Early man considered the Earth to be the centerof the universe and the five planets, Venus, Mercury, Mars, Jupiter, and Saturn,to be divine. The most ancient observations of the planets date back 2000 yearsB.C. and appear to come from the Babylonian and Minoan civilizations. The termplanet means "wanderer" in Greek and refers to the celestial objects that moverelative to the stars. The Egyptians, Greeks, and Chinese once thought of Venus, for example, as twostars because it was visible first in the morning and then in the evening sky. TheBabylonians called Venus "Istar," the personification of woman and the motherof the gods. In Egypt, the evening star was known as Quaiti and the morningstar Tioumoutiri; to the Chinese, Venus was known as Tai-pe, or the BeautifulWhite One. The Greeks called the morning star Phosphorus and the evening starHesperos but, by 500 B.C., the Greek philosopher Pythagoras had come to realizethat the two were identical. As time passed, the Romans changed the name of theplanet to honor their own goddess of love, Venus. It was not until the Golden Age of Greece that astronomy as a science wasplaced on a firm foundation and the Earth and the planets were regarded as globesrather than flat surfaces. Had Greek quantitative analysis taken one more step anddethroned the world from its position as the center of the universe, the progressof human thought and logic would have been accelerated. The Greek philosopherand mathematician Aristarchus held a heliocentric view of the solar system, buthis ideas were opposed on religious grounds, and the later Greeks reverted to theidea of a central Earth. Ptolemy, who died in about A.D. 150, left a record of the state of the universeat the end of the classic Greek period. In his Ptolemaic system, the Earth lies inthe center of the universe, with the various heavenly bodies revolving around itin perfect circles. First comes the moon, the closest body in the sky; then comeMercury, Venus, and the sun, followed by the three other planets then known(Mars, Jupiter, and Saturn), and finally the stars. 1 An interesting law of planetary spacing was suggested by Bode in 1771, whostated that the normalized mean distances of the planets from the sun are given bythe terms of the series 0, 3, 6, 12, etc., when added to 4 and divided by 10. This lawimplied that there should also be a planet between Mars and Jupiter, which led to thediscovery of the asteroid Ceres and the asteroid belt. Bodes law holds for the plan-ets closest to the sun [e.g., for Earth (6 ÷ 4)/10 = 1 A.U. (astronomical unit, meanradius of the Earth orbit)] but fails for Neptune, where it predicts 38.3 vs 30.2 actualdistance. It also appears to be valid for the satellites of the planets as, for example,those of Uranus. The planets and their relative size are illustrated in Fig. 12.1, andthe planetary and satellite data are given in Tables 12.1 and 12.2, respectively. 265
  • 2. 266 ORBITAL MECHANICSThis picture represents the relative sizes of the planets in the Solar System, compared with the Suns disk.The distances from the Sun are given in millions of miles. As a visual reference, the Earth is about 110 Sundiameters away or about 11,700 times Earths own diameter away. On the scale of this picture, Earth wouldactually be about 20 meters (65 ft) away from the Sun. Jupiter is 5 times as far, and Saturn nearly 10 times asfar. Since Plutos orbit is highly eccentric, it now is closer to the Sun than Neptune, which would be 30 timesfarther away than the Earth. Fig. 12.1 The planets relative sizes and mean distance from the sun. Fundamental reasons for study of the solar system are exploration, questionsrelated to the origin of the solar system, and asteroid mining missions, for ex-ample. Other potential applications include nuclear waste disposal missions andeventual colonization of the solar system. In this chapter, after a short historicalbackground, the sphere of influence and the gravity-assist concepts are examined.Simple patched-conic approaches are then used to describe missions to the moon,and Mars.12.2 Historical Background During the 20-yr period from 1962 to 1981, the exploration of the solar systemwas initiated by means of unmanned spacecraft. Flybys of the inner planets Venusand Mars were accomplished (by Mariners 2 and 4-7 in the United States). Inthe 1970s, multiple flybys of the inner planets were performed, including orbitersabout Mars and Venus (Mariner 9, Viking, and Pioneer Venus in the United States).Delivery of soft-landing vehicles to Mars and Venus was accomplished and flybysof Jupiter performed. The 1980s began with the encounters of Pioneer 11 and two Voyager spacecraftwith Saturn. During this period, the Soviet Union also sent a number of flyby,orbiting, atmospheric entry, and landing spacecraft to Venus and Mars. Complexterrain across several thousand miles of the surface of Venus was, for example,photographed by the Soviet Venera 16 imaging radar spacecraft, which revealed a
  • 3. LUNAR AND INTERPLANETARY TRAJECTORIES 267 ~o <~ ~+~?~ -d ~°~ = .~=~ r.z3e4i X X X X X X X X X b b =2 -~ ,,.-,
  • 4. 268 ORBITAL MECHANICS {30 OO I == ¢t3 O g c4 ~ c4 II o ,-4,- Cq O. © == uh O~ ¢q ,30 C,I c~ © .o C, t3
  • 5. LUNAR AND INTERPLANETARY TRAJECTORIES 269 tt3 © tt3 ~3 tt3 ~ ~- £q .~- ~ trh ~ ~ £q ,t"q tt3 trh t¢3 r~ r.~ O- S © o r.z) © ~ ~.~o= o=. ~ r.~ 2:
  • 6. 270 ORBITAL MECHANICSnearly 11-km-high mountain topped with a 100-km-diam crater, believed to be amassive meteorite impact crater. 2 Preliminary explorations of comets and the continuing Voyager 2 mission toUranus and Neptune were performed, with intensive investigation of the outerplanets Saturn, Uranus, and Neptune, along with Saturns satellite Titan. 3 Voyager2s encounter with Uranus on 24 January 1986 showed, for example, that theplanets magnetic pole is inclined from the pole of rotation by an angle of about55 deg, which is the largest inclination in the solar system. Images of Miranda andUmbriel, the satellites of Uranus, revealed three terrain types of different age andgeology, such as hills, groved valleys, and craters. One of the major exploration projects in the United States was the Galileomission to Jupiter originally planned for 1986 but subsequently delayed to October1989 because of the grounding of the Space Shuttle fleet and the new safetyrequirements posed by loss of Shuttle Centaur upper stage. A possible backupmission was also planned for July 1991 if the 1989 mission was not possible. The trajectory to Jupiter employed a two-stage inertial upper stage (IUS) usingone gravity-assist maneuver at Venus and two maneuvers at Earth, requiring morethan 6 years of travel time/ The new trajectory, which seemed to send Galileo on a cruise through the solarsystem, soon earned the name "Solar Cruiser." GALILEO VEEGA TRAJECTORY TO JUPITER / ~ FLYBY (2) DEc90FLYBY(1)~, / DEC92 ~P"JB=~,,~L,. ~ LAUNCH - . OCT/NOV 89 ,OA,f" "--,,5. BELT VENUS PROBE LAUNCH E GASPRA E IDA RELEASE JUPITER "~~K~, ~ PROBE RELEASE ~ TIMETICKS: S/C 30 days EARTH = 30 days ARRIVAL VENUS = 30 days*LAUNCH PERIOD OPENING DEC 7, 1995 JUPITER = 100 days Fig. 12.2 Galileos route to Jupiter.
  • 7. LUNAR AND INTERPLANETARY TRAJECTORIES 271: ..."= r, ~, E .-, ,.= .,=
  • 8. 272 ORBITAL MECHANICS Galileo was launched in October 1989 from Earth on a Space Shuttle and aninertial upper stage, a rocket whose energy is low compared to the Centaur. Insteadof heading toward Jupiter or the asteroid belt, Galileo took a flight path that carriedit to Venus. Galileo arrived there in February 1990. Venus gravity acceleratedGalileo and sent it on a flight path back toward Earth. When Galileo passed Earthin December 1990, the Earths gravitational field added energy to send Galileoout to the asteroid belt. A propulsive maneuver, performed in December 1991,brought Galileo past Earth again in December 1992 for a last gravity assist beforethe spacecraft began its final path to Jupiter. Arrival at Jupiter occurred late in 1995. A view of the Galileo trajectory and its spacecraft are shown in Figs. 12.2 and12.3, respectively. About 150 days before arrival of the Galileo spacecraft, the atmospheric probeseparated from the Orbiter. The Orbiter then flew within 1000 km of the satelliteIo, whose gravitational field helped to slow the spacecraft. The probe is designedto sample Jupiters equatorial zone, which consists primarily of ammonia. According to the plan, the orbiter entered a Jovian orbit ranging from 200,000km to more than 10 million km for a 20-month study of Jupiters environmentand its satellites. Current knowledge of this environment is illustrated in Figs.12.4-12.6. MAGNETOSPHERE BOUNDARY MAGNETOSPERE STREAMS OUT IN COMET-LINE TAIL AWAY FROM SUN O MAGNETIC AXIS SHEET OF N IONIZED PARTICLES MAGNETIC LINES OF FORCE CURRENT SHEET ENERGETIC CREATED BY PARTICLES MOVING PARTICLES ESCAPE OUTER RADIATION BELT O TO SUN INNER RADIATION BELT BOW SHOCK AXIS OF ROTATION P Fig. 12.4 Jupiters magnetosphere (from Ref. 8).
  • 9. LUNAR AND INTERPLANETARY TRAJECTORIES 273 IONOSPHERE /TOP OF / ATMOSPHERE ~ CLOUD TOPS ~ LIQUID HYDROGEN / ~ LIQUID E A LC Y R G N MT LIH DO E ~J(/A V "cL°u° J ~ ";; ~ TOPS ""AMMONIA LIQUID WATE ~-~ ICE DROPLETS r. CRYSTALS WATER ICE AMMONIA CRYSTALS HYDROSULFIDE CRYSTALSFig. 12.5 Jupiter atmosphere model (scale exaggerated); atmosphere depth to liquidzone is 1000 km (600 miles) (from Ref. 8). ATMOSPHERE: 82% HYDROGEN J 17% HELIUM ~ 1% OTHER ELEMENTS ~ ~CLOUD TOPS _ ",; ~..::....~,~j /240kn, 150,,IIUOWN ~ = i i i i i r:L:!:::.".; _ - / i i i i [ i i i i,~*,:;.;,=/. " " -~ - -- ¢-~.,¢_ti~ i i i i i i i i i i ~ :,: ol . iii1[i i i i i t,ic: :::. ~,. . - . ~ . . ~ .,I i ,= i i i i i i.i:.;{::.:~.~y/~ TRANSITIONFROM ",;.--- - - - - ~ . ~ ~ I , I ~-...T.~/~-~~ GAS TO L,OUID HYDROGEN " "~"- "~-L~ [=]::[~-i":?:~ . 1000km 600mi)DOWN -TRANSITION FROM LIQUID HYDROGEN -~=B==,,,~...~, ~ ~...~ ~ To ,,QU,D METALL,C HYDROGEN ~ ~ ~t."-:~ ~j~/~ ~ 25.000 km (15 000 mi) DOWN TEMPER.T;RE: 20,ooo°F [,yy777/~, ~ PRESSURE: 3 MILLION EARTH ~ ATMOSPHERES POLAR DISTANCE TO CENTER 66,750 km (41,480mi) NSMALL ROCKY CORE TEMPERATURE: 54.000°F Fig. 12.6 Interior of liquid Jupiter; planet is mainly hydrogen (from Ref. 8).
  • 10. 274 ORBITAL MECHANICS12.3 Important ConceptsSphere of Gravitation 5 Consider the motion of a mass point (P, m) under the influence of two largermasses ml and m2, as shown in Fig. 12.7. The larger masses are termed centers ofattraction A1 and A2. It is assumed that ml << m2 and m << ml. The values of the gravitational forces F1 and F2 acting on mass m toward Amand A2 are with reference to Fig. 12.7 given by Gmml Gmm2 /1 -- - - F2 -- - - (12.1) [A1PI2 IA2PI2where G = universal constant of gravitation. The locus of points where/1 > F2 defines the sphere of gravitational attractionof mass m l with respect to mass m2. The location and the radius of the sphere aredetermined from boundary condition FI=F2or AlP m ~ 2 = const <1 ~ 2 P -- (12.2)which indicates that the ratio of the distance from P to A1 and A2 is constant. Fromelementary geometry, the locus of points defined by this condition is a sphere, thediameter of which is defined by points C and D in Fig. 12.7. Thus, letting Rs, Ro be the radius and origin of the sphere of gravitation andusing Eq. 12.1 for the collinear points of attractions C and D, the ratio of thedistances A1C _ AID _ m~ m (12.3) AzC AzD Vm2or R, - Ro _ Rs + Ro __ mm/~l (12.4) A1A2 - (Rs - Ro) A1A2 + (Rs + Ro) A2PD°Tc A 2 ,m 2 Fig. 12.7 Sphere of gravitation.
  • 11. LUNAR AND INTERPLANETARY TRAJECTORIES 275which yields A1A2v/--~/m2 Rs- (12.5) 1 ml/m2 - A1A2(m]/m2) Ro-- (12.6) 1 - ml/rn2The notation A~A2 denotes the distance from A1 to A2. For example, the lunar sphere of gravitation is found as 384000~/]--/81 Rs = 1 - (1/81) 43000 kmand its location 384000(1/81) Ro-- 1 - (1/81) 4500 kmSphere of Influence 5 The sphere of influence is defined as the locus of points measured with respectto the gravitational centers of attraction A1 or A2, where the ratios of the pertur-bative to primary gravitational accelerations of the centers are equal. To find themagnitude of the sphere of influence, consider the notation in Fig. 12.8. The motion of point P (of mass m) with respect to the gravitational center ofattraction A1 can be expressed as d2p dt 2 = al + CI~I (12.7)where - G ( m l --1-m) al -- p3 /9 (12.8) = primary acceleration of A 1 r-p r) ~1 = Gm2 ~- r12 (12.9) = perturbative acceleration of A2 P~m ~ ~ ~ ~ ~ _ A2,m 2 A1,mI r12 Fig. 12.8 Sphere of influence.
  • 12. 276 ORBITAL MECHANICS The ratio c~l/al determines the magnitude of the deviation from the Keplerianorbit of point P. With respect to A2, the equation of motion becomes d2r d T = a2 q- ~2 (12.10)where -G(m2 q- m) r (12.11) a2 - r3 ¢~2=Gml(Pr3r ~3) (12.12) The sphere of influence with respect to A 1 is defined by the condition q~l q~2 - - < - - (12.13) al a2 The boundary of this region is defined by the equality sign in Eq. (12.13). Then, (ml + m)mlr2 r12 r32 p3 = (m2 q- m)m2p2 r1__2_2 p q32 _ (12.14)which, for the case of ml << m2, approximately yields - - ( l + 3 c o s 20)= (m, 4 - - (12.15) r12/ m2/where O is as defined in Fig. 12.8. Solving for p, P = r12 1 + 3cos2 0 (12.16) Equation (1 2.1 6) represents a slightly prolate spheroid that differs little from asphere when ml/m2 is small. Then, the radius of the sphere of influence becomes (ml~ 2/5 RSOI = r12 -- (12.17) m2/Thus, for the moon, [ 1 ,,2/5 Rsoi = 3 8 4 , 0 0 0 ~ - ) 66,280 km > Rs = 43,000 km
  • 13. LUNAR AND INTERPLANETARY TRAJECTORIES 277Velocity at "Infinity" Velocity at "infinity" V~ is the velocity of the vehicle at the sphere of influenceof the planet with respect to the coordinates of the planet. The geometry of the escape hyperbola from Earth is illustrated in Fig. 12.9,where the burnout velocity Vbo at the radius rbo and phase angle ~ required toachieve v~o is also known. The value of Vbo is found from the fact that the energye is constant along the geocentric escape hyperbola and, therefore, V2o /~ S-- 2 rbo 2 _ v~ ~ ~ v2/2 2 r~owhich yields 2 V2o = vo~ + 2#/rbo (12.18)where ro~ is the radius of the sphere of the influence of the Earth, and tt is thegravitationalconstant. The phase angle ~ is found from -i cos ~ -- (12.19) ewhere the eccentricity e of the departure hyperbola is given by e = V/1 + 2 e h 2 / # 2 (12.20) h = rbovbo = specific angular momentum (12.21)for injection at perigee. to ;un c3 = Iv[ 2 ¢* rn % Earth Earths ~ ~ orbital velocity ~ l 1 ~------n Vp departure asymptate / Fig. 12.9 Escape hyperbola (~om ReL 6).
  • 14. 278 ORBITAL MECHANICS Equation (12.18) may be solved for the velocity at infinity. Thus, / 2/~ voo = ,/V2o (12.22) V FboThe v~ velocity vector may then be added to the velocity of the planet in orderto obtain the velocity of the vehicle with respect to the sun as in Fig. 12.10, forexample. The characteristic energy C3 = v~. It is used to define the energy requirements 2for departure from the planets sphere of influence.Gravity Assist An important consequence of a spacecraft entering a sphere of influence of aplanet (secondary) is the possibility of gaining or losing energy with respect tothe sun (primary). The gain or loss of energy is caused by the turning (rotation) ofthe spacecraft velocity vector under the influence of the secondarys gravitationalfield. Thus, if the gravitational field of the secondary turns the relative velocityvector of the spacecraft toward that of the secondary, the resultant vector velocitywith respect to the primary is greater than that at the entry to the sphere of influenceof the secondary. This effect is illustrated in Fig. 12.11, where the notation v~denotes the spacecraft velocity at "infinity" at entry or exit from the sphere ofinfluence of the secondary. An example of the Mars flyby and gravity-assist vector geometry is shown inFig. 12.12, where V~o and V~ou are the incoming and outgoing velocity vectors twith respect to Mars at the radius of its sphere of influence. The velocity of thesatellite with respect to the sun is thus shown to be increased after gravity assistby the equivalent vector A V. Fig. 12.10 Heliocentric velocity of a vehicle.
  • 15. LUNAR AND INTERPLANETARY TRAJECTORIES 279 out in ~VsEcONDARY VSECONDAR Y out min Vm I mOUt --VsEcONDAR Y VA out VSECONDAR Y in V~in V B - Spacecraft Velocity Before Encounter V A - Spacecraft Velocity After EncounterFig. 12.11 Gravity-assist concept: a) energy increasing; b) energy decreasing V8 =spacecraft velocity before encounter; VA = spacecraft velocity after encounter.12.4 Lunar TrajectoriesThe Earth-Moon System The motion of the Earth-moon binary system is, in general, complex. The Earthand the moon revolve about their common center of mass, which is 4728 km fromthe center of the Earth. The mean distance between the Earth and the moon is384,400 kin, and the mass of the moon is 1/81.30 that of the Earth. The period ofrevolution about the mass center is 27.3 days. The orbital period of the moon is slowly getting longer because the distancebetween the Earth and the moon is increasing. This is the result of the tidalbulge in the Earths oceans, which are carried eastward by the Earths rotation. ~o ~ARS RBIT V VSAT WRT VMARS MARS N T WRT SUN SU OU VmOUT SUN IN QU IVAL ENT / EQUIVALENT -.°°7 .,., V~IN ~ ~ ~ / ~ ~ ~ TURNING V=IN -- / / ~ SUN IN Fig. 12.12 Mars flyby and gravity assist.
  • 16. 280 ORBITAL MECHANICSConsequently, a shift of the Earths center of mass relative to the Earth-moon linegives the moon a small acceleration in the direction of its orbital motion. Themoon speeds up and moves further away from the Earth. The loss of energy in thisprocess is known as the tidal friction effect. The mean eccentricity of the moons orbit is 0.0549. It is inclined to the ecliptic(the plane of Earths orbit) by about 5°8 . The line of nodes, or the intersection ofthe orbit planes, moves westward making one complete revolution in 18.6 yr. Theangle between the Earths equatorial plane and the moons orbital plane variesbetween 18°19 and 28 ° 35 within a period of 18.6 yr. The line of apsides rotatesin the direction of the moons orbital motion within a period of 8.9 yr.12.5 Analytical ApproximationsTwo-Body Motion Several analytical models may be employed for preliminary estimates of thevelocity requirements for lunar trajectories. The simplest model assumes the massof the moon to be negligibly small. In this model, the lunar trajectory is describedby an unperturbed two-body conic section relative to the Earth. The inclination ofthe moons orbital plane relative to the equator and nonequatorial launch sites canalso be included in this model. The velocity requirements for transfer between the Earth and the moon can beobtained approximately with the use of the vis-viva equation in the form 1) (12.23)where rbo is the radius at burnout and a is the semimajor axis of the lunar trajectory.The flight time can be computed as half of the period of the transfer ellipse or, moreaccurately, by the use of the Keplers equation. Typical injection velocities as afunction of injection altitude are shown in Fig. 12.13 for 50-, 66-, and 90-h transfertimes. The lower and upper values for the shaded regions represent perigee andapogee lunar radii, respectively. These velocities are impulsive inertial (absolute)velocities and may be partly composed of a component of velocity due to Earthsrotation. The range of permissible lifloff times (launch windows) on a given dateis determined by the launch azimuth, park orbit coast duration, and lifloff time.Typically, the launch windows may vary from 2 to 5 h, depending on the lunardeclination.Patched-Conic Approach The patched-conic approach makes use of the spheres of influence of the Earthand the moon. Very near the Earth, the effect of the moon is small and may beneglected regarding the satellite to be on an unperturbed orbit about the Earth.Near the moon, the opposite is assumed, and the trajectory is regarded as anunperturbed orbit about the moon. The "patching" of the trajectories or conics isat the sphere of influence of the moon. This is illustrated in Fig. 12.14, where thechange of the reference frames results in a selenocentric (lunar) hyperbola sincethe velocity of the satellite at the lunar sphere of influence is finite with respect tothe moon.
  • 17. LUNAR AND INTERPLANETARY TRAJECTORIES 281 VEHICLE PERFORMANCE ESTIMATION TECHNIQUES 11150 III00 11050 11000 v 1o95o o > 10900 ~ lo85o 10800 10750 10700 10650 10610 100 200 300 400 500 I N J E C T I O N A L T I T U D E (km) Fig. 12.13 Lunar mission injection velocity vs altitude (from Re~ 9).Patched-Conic Example Trajectory The basic patched-conic lunar transfer problem can be illustrated using theapproach of Ref. 6. Specifically, it may be desired to arrive at perilune (perigee) ata specified altitude. This can be achieved by selecting the translunar injection (TLI)velocity vector components r0, v0, ¢0 in Fig. 12.15 such that the desired periluneradius is obtained by iterating on the phase angle ~.1 at arrival to the moons sphereof influence. An algorithm required to solve this problem and determine the timeof flight may be constructed as follows: With reference to Fig. 12.15, select r0 (km), v0 (km/s), ¢0 (deg),)~l (deg). Then, Ee- V2 /~e (12.24) 2 r0
  • 18. 282 ORBITAL MECHANICS / I"- ~ ~ f Lunar Sphere of Influence / Selenocentric ..~--~. / Hyperbola / I/ [ ...._~ _..._. I r Orbit of I / ~ U//Si~ ~ "~ ~ ~_~ the V /" / ] , -,-a.. Moon ~ ~Elliptical Transfer Orbits / I Si -incoming I asymptote ~ ~-. S^- outgoing x ~~I ~k / v asymptote , I lil (not to s c a l e ) Fig. 12.14 Lunar trajectory patched-conic concept.where ~e = 3.986 X 10 5 km3/s 2 (12.25) h = rovo cos00 (12.26) rl = /0 2 + R 2 -- 2 D R s cos,k]where D = 384400 k m Rs = 66300 k m (12.27)
  • 19. LUNAR AND INTERPLANETARY TRAJECTORIES 283EARTHro I~ V0 MOONs SPHERE OF INFLUENCE AT t o I toeorth ~t - Vm to eorth / Fig. 12.15 Earth-moon patched-conic geometry (from Ref. 6).
  • 20. 284 ORBITAL MECHANICS (12.28) gl = sin-1 (R~-lSsin)~l ) (12.29)For the velocity of the moon, Vm = 1.018 km/s 1)2 = /1)2 + V2 __ 21)lYre COS(q~I -- Yl) (12.30) Em- 1)2 ll, m (12.31) 2 Rswhere #m = 4.093 × 103 km3/s 2 hm = Rs 1)2 sin S 2 (12.32)where e2 sin -1 [ 1)~COS)~I -- -1)-1 1)2 cosO~l - ?/1 - qh)/ 1 (12.33) J em (12.34) #m Emh2 em = 1+2 /Z---~m (12.35) Fp m P~ (12.36) 1 +em ~ ( /Am) Vp = 2 Em +-~p (12.37)Here, h2 p~m (12.38) /Ze m#e a D (12.39) 2Ee e=~l Pa (12.40)For true anomalies, 0o, 0~, p -- r 0 cos 00 - - - ~- Oo (12.41) roe
  • 21. LUNAR AND INTERPLANETARY TRAJECTORIES 285 p -- F1 COS 01 - - - - ~ 01 (12.42) rle ( _e + cos 00 ) E0 = cos -1 1 + e cos 00 (12.43) El=COS -ll+ecos01 ( 5+-5°sfi ) (12.44)Flight time At is given by At = tl -- to = [(El -- e sin El) -- (Eo - e sin Eo)] (12.45)Phase angle at departure is found from }to = Ol - - Oo - - Y l - - c o m A t (12.46)where o)m = 2.649 x 10 -6 rad/s. Typical flight time as a function of injection velocity at 320 km altitude forthe phase angle at arrival Xl = 65 deg is shown in Fig. 12.16. The correspondingperigee radius and velocity at the moon (perilune) are shown in Figs. 12.17 and12.18, respectively. Flight Time versus Injection Ueloci%y a% 328 km A l % i t u d e 118 188 ,~ 98 I•- 88-- 78 68 58 18.81 18.815 18.82 18.825 18.83 18.835 18.84 18.845 18.85 18.855 Inlet%ion Oeloci%9-km/s Fig. 12.16 Flight time vs injection velocity at Ro = 320 kin.
  • 22. 286 ORBITAL MECHANICS x18 4 Perilune Radius for Injection at 328 k~ Rltitude 3 ! 2.5 1.5 B.5 8 18.81 18.815 11t,82 18.825 18.B3 11t,835 10.84 11].845 18.85 18.855 Injection Ue Iocity-kM/s Fig. 12.17 Perilune radius for injection at 320 k m altitude. Perilune Ueloci~y for InJection a~ 328 kn Rl~itude 18 9 8 7 Iu 6o 5,,t"r, 4 3 2 1 18.81 18.815 18.8Z 18.825 18.83 18.835 18.84 18.845 18.85 18.855 Injec£ion Ueloci£y-kM/s Fig. 12.18 Perilune velocity for injection at 320 k m altitude.
  • 23. LUNAR AND INTERPLANETARY TRAJECTORIES 28712.6 Three-Dimensional Trajectories The three-dimensional trajectories require the specification of several parame-ters in addition to those for the two-dimensional case discussed previously. Theadditional parameters are the initial latitude fi, the initial launch azimuth 7t, thelongitude difference between the initial point and the moon at impact A;~, theinstantaneous declination of the moon 3, and its maximum value 6m. The rela-tionships among these parameters determine the launch window at any launchsite and the conditions at arrival at the moon. Reference [10] shows, for example,that these families of Earth-moon trajectories departing in either co- or counter-direction with Earth rotation have a common vertex on the far side of the moon(see Fig. 12.19). The corotational direction takes advantage of Earth tangentialvelocity at the launch site. An example of a Ranger-type (lunar impact) trajectoryis illustrated in Fig. 12.20. The targeting parameter B is defined as a vector originating at the center of thetarget body (e.g., moon) and directed perpendicular to the incoming asymptoteof the target-centered approach hyperbola. The targeting parameter B is resolvedinto two components that lie in a plane normal to the incoming asymptote Si.The orientation of the reference axes in this plane is arbitrary but is usuallyselected to lie in a fixed plane. For interplanetary trajectories, the unit vector T isin the ecliptic plane, and R is normal to it, as shown in Fig. 12.21.12.7 interplanetary TrajectoriesTypes of Transfers Feasibility-type trajectories that are used for preliminary vehicle performancestudies involve a simple two-body problem. The classical Hohmann trajectories,for example, are ellipses that are tangent to both the launch and the arrival orbit.The energy for transfer from the launch orbit to the target orbit is, in most cases, aminimum for this trajectory, but the transfer time is usually quite long. The timecan be found from Keplers third law. There are three main groups of trajectories that can be evolved from theHohmann transfer: 1) staying tangential to the larger orbit but intersecting thesmaller one, 2) intersecting the larger orbit and staying tangential to the smallerone, and 3) intersecting both orbits. Parabolic or hyperbolic transfer trajectoriesintersect the larger orbit and may, but do not need to be, tangential to the smallerorbit. The flight times for parabolic or hyperbolic trajectories are generally shorterthan those for Hohmann transfers. Examples of several types of Earth-Mars tra-jectories are illustrated in Fig. 12.22.Mars Landing Mission Example Taking the approach of Ref. 6, we will examine the example of a spacecraftlander (probe) on Mars. For this case, the spacecraft is subject to the gravitationalactions of Earth, sun, and Mars, as shown in Fig. 12.23. The probe of mass mp is subject to the gravitational forces Fs, Fe, Fm of sun,earth, and Mars, respectively; thus, the equation of motion is mpip = Fs + Fe + Fm (12.47)
  • 24. 288 ORBITAL MECHANICS w w J O O O •~ °~ d~ O
  • 25. LUNAR AND INTERPLANETARY TRAJECTORIES 289 ~Z rl- L, J O~ ,,~ C- u ~Z wJ N- a._z u.i m ~ 0 i - Z ~a2~
  • 26. 290 ORBITAL MECHANICS E n t r y to Lunar Sphere of I n f l u e n c e I z IL.. t nar Orbi, / - Plane - / l To Earth Fig. 12.21 Lunar targeting parameters.or mplp -- --k6s IZe(rp -- re) #m(rp -- rm) r3 rp (12.48) ] r p - - r e 13 Irp -- r m ]3Here s, e, and m refer to the sun, Earth, and Mars, respectively. This equation is not integrable in closed form and must be evaluated numerically.Approximate solutions for the trajectory can be obtained using a patched-conicapproach. In this approach, the trajectory is divided into three different phases:1) heliocentric, 2) Earth departure, and 3) Mars arrival. Each phase is a two-bodyKeplerian orbit, the conic section of which is "patched" with the following phase.Approximate velocity requirements for the mission can in this way be estimatedto determine the feasibility of the mission. Heliocentric phase. Assuming a Hohmann-transfer heliocentric trajectoryfrom Earth to Mars, as illustrated in Fig. 12.24, the required perigee velocity withrespect to the sun is given by the equation Ppt = ( ~s -- ) (12.49) = 32.74 km/swhere re --- 1 A.U. = 1.49597893 x 108 km re + rm 2.523 at -- - - -- 2 2 = 1.262 A.U. = 1 . 8 8 7 x 108km /~s = 1.32712499 x 1011 knl3/s 2
  • 27. LUNAR AND INTERPLANETARY TRAJECTORIES 291 MI II @ Orbit MI11 Earth at Launch Mars Mars at Launch Arrival Flight time, Earth-Mars Trajectory Vbo, km/s V®, km/s days distance I, 106 kmI Hohmann 11.6 2.94 259 236II Elliptic 11.9 3.75 165 133III Parabolic 16.7 12.33 70 79v ~ = burnout velocity at 500 km (vc = 7.62 kin/s) V® = velocity at infinity WRT Earth Fig. 12.22 Earth-Mars trajectory velocity requirements. The transfer-orbit apogee velocity at Mars is Vat = ~s at (12.50) = 21.45 km/s
  • 28. 292 ORBITAL MECHANICS ars at t = tf (rp-rm) Fig. 12.23 Mars mission trajectory. The v e l o c i t y at exit f r o m the Earth sphere o f influence is U~ ~ Upt - - Ue = 32.74 - 29.78 (12.51) = 2.96 k m / sTherefore, C3 = v~2 = 8.76 km2/s 2, and the energy Ee = C3/2 = 4.38 km2/s 2. v = Vpt - v ~ ~ ~ Hobmann TransferFig. 12.24 Heliocentric M a r s trajectory: r s o I ~ [ (ml/mo)Z/5; g = a for planet;m: = mass of planet; m o mass of sun.
  • 29. LUNAR AND INTERPLANETARY TRAJECTORIES 293 Earth departure phase. Assuming an injection burn at an altitude of 300 km,the burnout velocity required is Vbo = 2 ~ + Ee (12.52) = 11.32 km/swhere Re = 6378 km = equatorial radius of Earth h = 300 km = altitude at injection #e = 3.9860064 x 105 km3/s 2 = gravitational parameter of earth Ee = 4.38 km2/s 2 = energy of escape hyperbola Mars arrival phase. For a soft landing at Mars, the energy of the hyperbolicorbit at Mars is 2 veo Em -- 2 (Vm -- Vat) 2 2 (12.53) (24.13 - 21.45) 2 2 = 3 . 5 9 klTl2/s 2where vm = 24.13 km/s = velocity of Mars. Retro velocity at Mars surface is ,/2(l~,Rm#m.~ Vre~o = V Em) = 5.71 km/swhere # m = 43058 krn3/s 2 = gravitational parameter of Mars Rm = 3379 km = equatorial radius of Mars E m = 3.59 knl2/s 2
  • 30. 294 ORBITAL MECHANICS At Mars SOl Vo~ ~ 24.13 km/s To SunTotal velocity requirements for the mission are Burnout velocity at Earth 1)bo = 11.32 km/s Retro velocity at Mars Vretro= 5.71 km/s Total 17.03 km/s An equal amount of velocity impulse would be required for the return trip toEarth. Small thrust application may also be necessary for midcourse correctionof the heliocentric trajectory, and the probe would retrofire first to deboost to aparking orbit before landing. The velocities that were computed are feasibility-type numbers. Precision velocities would be somewhat different as they must beobtained by integration of the equations of motion.12.8 Galileo Mission During its six-year journey to Jupiter, the Galileo Orbiter examined the planetsVenus and Earth as well as the Moon and made several unique observations. Fol-lowing each Earth flyby, Galileo made excursions to the asteroid belt and studiedtwo asteroids, Gaspra and Ida, close-up.* The Galileo scientists also observedComet Shoemaker-Levy 9 fragment impacts on Jupiter.l IObjectives and Goals The primary objectives of the Galileo Orbiter were to 1) investigate the circulation and dynamics of the Jovian atmosphere; the upperJovian atmosphere and ionosphere; and the composition and distribution of surfaceminerals on the Galilean satellites; *Data available online at http://nssdc.gsfc.nasa.gov [cited 2 March 2002].
  • 31. LUNAR AND INTERPLANETARY TRAJECTORIES 295 2) characterize the morphology, the geology, and the physical state of theGalilean satellites; 3) determine the gravitational and magnetic fields and dynamic properties ofthe Galilean satellites; and 4) study the atmospheres, ionospheres, and extended gas clouds of the Galileansatellites and the interaction of the Jovian magnetosphere with the Galileansatellites.*The science goals of the Galileo Probe were to " . . . determine the chemical com-position of the Jovian atmosphere; characterize the structure of the atmosphere toa depth of at least 10 bars; investigate the nature of cloud particles and the locationand structure of cloud layers; (and) examine the Jovian heat balance."* The Jovianlightning activity and the flux of its energetic charged particles down to the top ofthe atmosphere were to be determined also.Results Several discoveries have been made during the trajectory of the Galileo Orbiterincluding a satellite, Dactyl, of an asteroid, Ida; an intense interplanetary duststorm; and a new radiation belt, approximately 50,000 km above Jupiters cloudtops. The existence of a huge ancient impact basin in the southern part of theMoons far side was confirmed and evidence of more extensive lunar volcanismwas found. Jovian winds in excess of 600 km/hr were detected but far less waterwas found in the Jovian atmosphere than was estimated from earlier Voyagerobservations. It was also determined that the individual lightning events on Jupiterare about 10 times stronger than on Earth but that the lightning activity is onlyabout 10% of that found in an equal area on Earth. The helium abundance in Jupiteris nearly the same as its abundance in the sun, 24% compared to 25%. Extensiveresurfacing of Ios surface due to continuing volcanic activity has been notedand evidence of intrinsic magnetic fields on Io and Ganymede have been seen.Evidence for liquid water ocean under Europas surface has also been seen. Thesame evidence of water under Ganymedes ice-reach surface has been obtainedby the Orbiter flybys.*Additional Discoveries Additional Jovian moons were also discovered. 12 With the use of the Universityof Hawaiis 2.2 m telescope on Manna Kea, scientists discovered 11 new satellitesof Jupiter. The new satellites are members of Jupiters outer "irregular" satellitesystem with large eccentric orbits inclined to its equator. Orbital radii are about300 times the radius of Jupiter and the inclinations are near 150 to 160 degrees.Nine of these objects and five previous known retrograde satellites of Jupiterconstitute a total of 14 such objects. The origin of the retrograde irregular satellitesappears to be a possible capture of objects from heliocentric orbit when Jupiter wasyoung. Friction with Jupiters atmosphere may have provided the energy requiredfor capture of such satellites3 *Data available online at http://nssdc.gsfc.nasa.gov [cited 2 March 2002]. *Data available online at http://www.ifa.hawaii.edu [cited 2 March 2002].
  • 32. 296 ORBITAL MECHANICSFig. 12.25 High-resolution picture of Saturn taken 2.5 million miles away. (Courtesyof Jet Propulsion Laboratory, California Institute of Technology/NASA).12.9 Cassini-Huygens Mission to Saturn and Titan Saturn, the sixth planet from the sun, is the second largest planet in the SolarSystem; Jupiter is the largest. With a volume about 760 times that of Earth, Saturnhas no solid surface but consists of a gas that compresses into fluid at great depthsbeneath the clouds. Its atmosphere is composed mostly of hydrogen and heliumgas and its rings and icy satellites appear to be made primarily of water ice. A layerof ammonia ice clouds forms at a pressure level that is comparable to sea-levelatmospheric pressure on Earth.* Figure 12.25 is an example of one of the high-resolution pictures of Saturnsrings that Voyager 2 obtained. Numerous offshoots from the rings or "spoke"features can be seen in the middle of this image. Such spoke features "... persistat times for two or three rotations of the ring about the planet" and appear to" . . . revolve around the planet at the same rate as the rotation of the magnetic field *Dataavailableonline at http://www.jpl.nasa.gov/cassini[cited2 March 2002].
  • 33. LUNAR AND INTERPLANETARY TRAJECTORIES 297 , he The Orbiter ProbeFig. 12.26 The Cassini Spacecraft. (Courtesy of Jet Propulsion Laboratory,California Institute of Technology/NASA).and the interior of Saturn, independent of their distance from the center of Saturn." *It is possible that the small dust particles that form the spokes are electricallycharged, while older spokes, which may have lost their electrical charge, revolvewith the underlying ring particles. Although it is not known exactly what causesthese spokes electromagnetic forces could be responsible.* Since 1997, the Cassini spacecraft consisting of the orbiter and the HuygensProbe (Fig. 12.26) has been traveling to the planet Saturn and its moon Titan.The mission is being managed by NASAs Jet Propulsion Laboratory, the de-signer and assembler of the Cassini Orbiter. The European Space Technology andResearch Center in cooperation with 14 European nations managed the develop-ment of the Huygens Titan Probe. The vehicle used to launch the spacecraft was" . . . made up of a two-stage Titan IV booster rocket, two strap-on solid rocketmotors, the Centaur upper stage and a payload enclosure..."* At launch, thespacecraft weighed about 5600 kg (12,346 lbs). Using 26 kg of Plutonium 238as fuel, radioisotope thermoelectric generators supplied the spacecrafts electricalpower. Two 445 N rocket engines provided maneuver capability when needed,and sixteen smaller thrusters were used to control the spacecrafts orientation.*The spacecraft must be built to withstand a thermal environment both inside theorbit of Venus (130°C) and at Saturn (-210°C). NASAs Deep Space Networkwas used for communicating with Cassini. Cassinis planned interplanetary flight path, illustrated in Fig. 12.27, beganwith the launch from Earth on 15 October 1997. Next there were two gravity assistflybys of Venus on 26 April 1998 and 24 June 1999; one of Earth on 18 August 1999; and one of Jupiter on 30 December 2000. Saturn arrival is scheduled for 1 July 2004 when a four-year orbital tour of the Saturn system will begin. "Withthe use of the VVEJGA (Venus-Venus-Earth-Jupiter Gravity Assist) trajectory, ittakes 6.7 years for the Cassini spacecraft to arrive at Saturn."* At arrival, the cone-shaped Huygens Titan Probe will detach from the main spacecraft and conduct its own investigations. The Probe holds six of Cassinis *Dataavailableonline at http://www.jpl.nasa.gov/cassini[cited2 March 2002]. tData availableonline at http://saturn.jpl.nasa.gov[cited 2 March 2002]. *Dataavailableonline at http://www.jpl.nasa.gov[cited2 March 2002].
  • 34. 298 ORBITAL MECHANICS VENUS SWINGBY E 26 APR 1998 VENUS SWINGBY - - 24JUN1999 ",. SATURNARRIVAL 1 JUL 2004 /", . . . .~ ORBITOF ORBIT OF / , ,/ ~ - , , .~..//EARTH JUPITER / / ~ , "~1~ DEEPSPACE I , .__.C~._. ~~ ! -1"- MANEUVER , "" ~ / ~ O j,_},,, R B ~ I j 3DEC1998 T OF -VENUS l i i / Jo O BIT SAT?RN 1 I n / ~ LAUNCHEARTHSWINGBY ~ 18 AUG 1999 15~CT 1997 / i / JUPITERSWlNGBY " , 30 DEC 2000 /t /Fig. 12.27 Cassini Interplanetary Trajectory. (Courtesy of Jet Propulsion Labora-tory, California Institute of Technology/NASA).eighteen science instruments including " . . . a gas chemical analyzer designed toidentify various atmospheric elements, a device to collect aerosols for chemicalcomposition analysis, a camera that can take images and make wide-range spectralmeasurements, and an instrument whose sensors will measure the physical andelectrical properties of Titans atmosphere." *12.10 Mars Odyssey Mission Mars Odyssey orbiter lifted off from Kennedy Space Center aboard a Delta IIrocket on 6 April 2001. The spacecraft carried scientific instruments to map thechemical and mineralogical composition of Mars and to assess potential radiationhazards to future human explorers. The orbiter reached Mars on 24 October 2001.For the next two and a half years, the orbiter will circle Mars looking for tracesof water and evidence of life-sustaining environments.* The mission timeline isshown in Fig. 12.28. The numbers on the figure are explained in further detail as follows: 1) Launch and Liftoff. Solar array deployed about 36 minutes after launch on7 April 2001. 2) Cruise phase. Payload health and status check, calibration and data collection.Flight path correction. Duration about 200 days. 3) Mars orbit insertion via aerobraking and main engine firing. Duration aboutone week after Mars arrival on 24 October 2001. *Data available online at http://www.jpl.nasa.gov/cassini [cited 2 March 2002]. tData available online at http://www.mars.jpl.nasa.gov/[cited 2 March 2002].
  • 35. LUNAR AND INTERPLANETARY TRAJECTORIES 299 1Fig. 12.28 Mars Odyssey Mission. (Courtesy of Jet Propulsion Laboratory,California Institute of Technology/NASA). 4) Mapping orbit. Thermal emission imaging of the Martian surface to determineits chemical and mineralogical composition. Communication support for landersand rovers in the future. Primary science mission duration is 917 days. References 1Mariner-Venus 1962 Final Project Report, NASA SP-59, 1965. 2Aviation Week and Space Technology, Feb. 1985. 3Wood, L. J., and Jordan, J. E, "Interplanetary Navigation Through the Year 2005: TheInner Solar System," Journal of the Astronomical Sciences, Vol. 32, Oct.-Dec. 1984. 4Aviation Week and Space Technology, Dec. 1987. 5Balk, M. B., Elements of Dynamics of Cosmic Flight, Naaka, 1965 (in Russian). 6Bate, R. R., Mueller, D. D., and White, J. E., Fundamentals of Astrodynamics, Dover,New York, 1971. 7Clarke, T. C., and Fanale, F. E, "Galileo: The Earth Encounters," Planetary Report, JetPropulsion Laboratory, CIT, Sept./Oct. 1989. 8"Pioneer to Jupiter," Bendix Field Engineering Corp., Palo Alto, CA, Nov. 1974. 9White, J. F., Flight Performance Handbook for Powered Flight Operations, Wiley,New York, 1963, pp. 2-54. l°Hoolker, R. E, and Brand, N. J., "Mapping the Course for the Moon Trip," Astronautics& Aeronautics, Feb. 1964. l1Johnson T. V., "The Galilean Mission to Jupiter and its Moons," Scientific American,Feb. 2000. 12Sheppard, S. S., Jewitt, D. C., Fernandez, Y. R., and Magnier, G., University of HawaiiReport in the International Astronomical Union Circular (IAUC 7555), 15 Feb. 2001. Problems12.1. Consider a nuclear waste disposal mission from Earth (1 A.U.) to a helio-centric circular orbit of radius 0.86 A.U. a) What is the charcteristic energy C3 required for a Hohmann transfer fromEarth to the circular orbit or radius 0.86 A.U.? b) What is the flight time via the Hohmann-transfer orbit from Earth to the 0.86A.U. orbit about the sun?
  • 36. 300 ORBITAL MECHANICS12.2. To accomplish certain measurements of phenomena associated with sunspotactivity, it is necessary to establish a heliocentric orbit with a perihelion of 0.85A.U. The departure from the Earths orbit will be at apohelion. What must theburnout velocity be at an altitude of 1300 km to accomplish this mission?12.3. A Venus probe departs from a 6378-km altitude circular parking orbit witha bumout speed of 8.69 km/s. Find the hyperbolic excess speed at infinity.12.4. Calculate the sphere of influence for the nine planets in the solar system.12.5. Compute the distance of L1, the Lagrangian liberation point, from thecenter of the moon along the Earth-moon line. To a first approximation, L1 is anequilibrium point between the gravitational and centrifugal accelerations of theattracting bodies.12.6. Compute the stationkeeping requirements (AV) to remain within 10 kmof L1 in Problem 12.5 for a year.12.7. Compute the velocity impulse (AV) for transfer from a 100-km circularorbit at Mars to a hyperbolic orbit Earth return trajectory with an eccentricitye = 1.5.12.8. Write the equation of motion for a solar sail in the solar system. Discussthe type of trajectories possible.12.9. If the Earth were stopped in its orbit, what would be the elapsed time, indays, until collision with the sun? Assume point masses, and assume the Earthsorbit to be circular, with r = 1.0 A.U. Selected Solutions12.1. 1.28 (knds) 2, 163.7 days12.2. 10.26 km/s12.3. 3.6 km/s

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