4                                                          Astrodynamics4.1   Introduction    Astrodynamics is the study o...
104                         SPACE VEHICLE DESIGNless than a kilometer. Planets such as Mars, Venus, and Jupiter, which hav...
ASTRODYNAMICS                                   105                 ~                     M                               ...
106                         SPACE VEHICLE DESIGN   2) The angular momentum of the system is constant; as a result, the mot...
ASTRODYNAMICS                                                                                     107                     ...
108                         SPACE VEHICLE DESIGNwhere                                    rp -- a(1 - e)                   ...
ASTRODYNAMICS                                    109   One of the more elegant features of the solution for central-force ...
110                          SPACE VEHICLE DESIGN4.2.2   Circular and Escape Velocity   From Eq. (4.17), and noting that f...
ASTRODYNAMICS                            111              Auxiliary               circle/                         Empty   ...
112                           SPACE VEHICLE DESIGN   I f E = 0 = 27r, then   (t - tp) = ~, the orbital period. Equation (4...
ASTRODYNAMICS                                     1134.2.4   Motion in Hyperbolic Orbits   The study of hyperbolic orbits ...
114                              SPACE VEHICLE DESIGNVo~ = hyperbolic approach and      departure velocity relative      t...
ASTRODYNAMICS                                    115   The magnitude of the hyperbolic velocity V~ is found from the vis-v...
116                          SPACE VEHICLE DESIGN                                   V•   Oo                               ...
ASTRODYNAMICS                                  117while the B-plane offset required to obtain a desired periapsis is      ...
118                         SPACE VEHICLE DESIGNeliminated, for parabolic orbits. Thus,                                   ...
ASTRODYNAMICS                                    119   In classical celestial mechanics, position and velocity information...
120                         SPACE VEHICLE DESIGN                                                                          ...
ASTRODYNAMICS                                     121influences or perturbations are always present and result in a depart...
122                               SPACE VEHICLE DESIGN   For orbits about or observations from Earth (or, when appropriate...
ASTRODYNAMICS                                     123plane about the sun, the ecliptic plane, as the reference plane for t...
124                          SPACE VEHICLE DESIGN                              Polaris                                Z   ...
ASTRODYNAMICS                                    125                                                    I                 ...
126                         SPACE VEHICLE DESIGNcalculation would be required following a midcourse maneuver in aninterpla...
ASTRODYNAMICS                                       127                                                    I             L...
128                           SPACE VEHICLE DESIGNSince i is defined as the angle between h and k,                        ...
ASTRODYNAMICS                                     129precisely over the equator, an equatorial orbit is impossible, becaus...
130                            SPACE VEHICLE DESIGNnot colinear, any other vector in the orbit plane can be written as a l...
ASTRODYNAMICS                                    131and Ui(X, a) is the universal function of ith order. The universal fun...
132                         SPACE VEHICLE DESIGNtransformations required to relate the observing site location to the iner...
ASTRODYNAMICS                                     133be made for the position of celestial objects at specific past or fut...
134                         SPACE VEHICLE DESIGN   Since 1 January, 1958, the basis for timekeeping has been international...
ASTRODYNAMICS                                    135   4.2.11.3 Ephemeris time. From earlier discussions, it is clear that...
136                         SPACE VEHICLE DESIGNwhere  I = year  J = month  K = day of month   The Julian day system has t...
ASTRODYNAMICS                                     137where We is the inertial rotation rate of Earth and was 7.292116 x 10...
138                        SPACE VEHICLE DESIGNsphere of influence about the small body has the approximate radius        ...
ASTRODYNAMICS                                   139of this method is consistent with the basic sphere-of-influence calcula...
140                         SPACE VEHICLE DESIGNSpacecraft can occupy these positions for extended periods, but only if a ...
ASTRODYNAMICS                            141potential is 25                                                               ...
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Space Vehicle Design: 4. Astrodynamics
Upcoming SlideShare
Loading in...5
×

Space Vehicle Design: 4. Astrodynamics

2,878

Published on

Published in: Education, Technology, Design
0 Comments
2 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
2,878
On Slideshare
0
From Embeds
0
Number of Embeds
4
Actions
Shares
0
Downloads
129
Comments
0
Likes
2
Embeds 0
No embeds

No notes for slide

Space Vehicle Design: 4. Astrodynamics

  1. 1. 4 Astrodynamics4.1 Introduction Astrodynamics is the study of the motion of man-made objects in spacesubject to both natural and artificially induced forces. It is the latter factor thatlends a design element to astrodynamics that is lacking in its parent science,celestial mechanics. The function of the astrodynamicist is to synthesizetrajectories that, within the limits imposed by physics and launch vehicleperformance, accomplish desired mission goals. Experience gained since thedawn of the space age in 1957, together with the tremendous growth in the speedand sophistication of computer analyses, have allowed the implementation ofmission designs not foreseen by early pioneers in astronautics. This trend wasdiscussed briefly in Chapter 2 and shows every sign of continuing. The use ofhalo orbits ~ for the International Sun-Earth Explorer (ISEE) and WilkinsonMicrowave Anisotropy Probe missions, the development of space colonyconcepts using the Earth-moon Lagrangian points, 2 together with the analysis byHeppenheimer 3 of "achromatic" trajectories to reach these points from the moon,and the extensive modem use of gravity-assist maneuvers 4 for interplanetarymissions are but a few examples. Astrodynamics, through its links to classical astronomy, has its roots in thevery origins of the scientific revolution. The mathematical elegance of the fieldexceeds that found in any other area of astronautics. Problems posed in celestialmechanics have been a spur to the development of both pure and appliedmathematics since Newton s development of the calculus (which he used, amongother things, to derive Keplers laws of planetary motion and to show that a spherically symmetric body acts gravitationally as if its mass were concentrated at a point at its center). Hoyle 5 comments on this point and notes that it has notbeen entirely beneficial; the precomputer emphasis on analytical solutions led to the development of many involved methods and "tricks" useful in the solution of celestial mechanics problems. Many of these methods persist to the present as an established part of mathematics education, despite having little relevance in an era of computational sophistication. Because of the basic simplicity of the phenomena involved, it is possible in astrodynamics to make measurements and predictions to a level of accuracy exceeded in few fields. For example, it is not unusual to measure the position of an interplanetary spacecraft (relative to its tracking stations) to an accuracy of 103
  2. 2. 104 SPACE VEHICLE DESIGNless than a kilometer. Planets such as Mars, Venus, and Jupiter, which have beenorbited by spacecraft, can now be located to within several tens of meters out ofhundreds of millions of kilometers. Such precision is attained only at the price ofextensive data processing and considerable care in modeling the solar systemenvironment. We shall not engage in detailed consideration of the methods by which thehighest possible degree of precision is attained. Although it is true that the mostaccurate methods of orbit prediction and determination are desirable in the actualexecution of a mission, such accuracy is rarely needed at the levels of missiondefinition appropriate to spacecraft design. What is required is familiarity withbasic orbit dynamics and an understanding of when and why more complexcalculations are in order. We take the view that the spacecraft system engineerrequires a level of competence in astrodynamics approximately defined by therange of methods suitable for solution via pocket calculator. Any analysisabsolutely requiring a computer for its completion is in general the province ofspecialists. Of course, this threshold is a moving target. "Pocket" calculators available asthis goes to press (2003) offer a level of capability substantially exceeding that ofdesktop personal computers of the mid-1980s, allowing many formerlyprohibitive computations to be completed with ease, even during preliminarydesign.4.2 Fundamentals of Orbital Mechanics4.2.1 Two-Body Motion The basis of astrodynamics is Newtons law of universal gravitation: GMm F = (4.1) r2which yields the force between two point masses M and m separated by a distancer and directed along the vector r (see Fig. 4.1) between them. G is the universalgravitation constant, a fundamental (and very difficult to determine) constant ofnature. It is a sophomore-level exercise in physics 6 to show that M and m may beextended spherically symmetric bodies without affecting the validity of Eq. (4.1). A necessary and sufficient condition that F be a conservative (nondissipative,path-independent) force is that it be derivable as the negative (by convention)gradient of a scalar potential. This is the case for the gravitational force law, asseen by differentiating the potential function (per unit mass of m) given by GM U - (4.2) r
  3. 3. ASTRODYNAMICS 105 ~ M f mass r m Fig. 4.1 Two-body motion in inertial space.U has dimensions of energy per unit mass and is thus the potential energy ofmass m due to its position relative to mass M. Note that the singularity at theorigin is excluded from consideration, because M and m cannot be coincident,and that the potential energy is taken as zero at infinity, an arbitrary choice,because Eq. (4.2) could include an additive constant with no change in the forcelaw. With the negative-gradient sign convention indicated earlier, the potentialenergy is always negative. Using Newtons second law, F = ma (4.3)in an inertial frame, and equating the mutual force of each body on the other leadsto the familiar inverse square law equation of motion dZr G ( M + m) } r = 0 (4.4) dr 2 r3where r is defined as shown in Fig. 4.1. Several key results may be obtained 7 for auniverse consisting of only the two masses M and m: 1) The center of mass of the two-body system is unaccelerated and thus mayserve as the origin of an inertial reference frame.
  4. 4. 106 SPACE VEHICLE DESIGN 2) The angular momentum of the system is constant; as a result, the motion isin a plane normal to the angular momentum vector. 3) The masses M and m follow paths that are conic sections with their center ofmass as one focus; thus, the possible orbits are a circle, an ellipse, a parabola, or ahyperbola. We note that the two-body motion described by Eq. (4.4) is mathematicallyidentical to the motion of a particle of reduced mass: mm (4.5) m,--M+ msubject to a radially directed force field of magnitude G M m / r 2. The two-bodyand central-force formulations are thus equivalent, which leads to the practice ofwriting Eq. (4.4) as dZr /x dt-- + ~ r - 0 2 (4.6)where/x = G(M + m). In nearly all cases of interest in astrodynamics, m ~ M,which leads to mr ~-m, tx "~ GM, and a blurring of the physical distinctionbetween two-body and central-force motion. The system center of mass is then infact the center of mass of the primary body M. For example, a satellite in Earthorbit has no measurable effect on the motion of the Earth, which appears asthe generator of the central force. This approximation is still quite valid in thedescription of planetary motion, although careful measurements can detect themotion of the sun about the mutual center, or barycenter, of the solar system.Interestingly enough, the Earth-moon pair provides one of the few examples inthe solar system where the barycenter of the two masses is sufficiently displacedfrom the center of the "primary" to be readily observable. The formulation of Eq. (4.6) is especially convenient in that/x is determinableto high accuracy through observation of planetary or spacecraft trajectories,whereas G is itself extremely difficult to measure accurately. As an aside, recenttheoretical and experimental work 8 suggests that G may not be a constant, butdecreases gradually over cosmologically significant time scales. We now proceed to quantify the results cited earlier. Figure 4.2 depicts thepossible orbits, together with the parameters that define their geometricproperties. Note that the different conic sections are distinguished by a singleparameter, the eccentricity e, which is related to the parameters a and b or a and pas shown in Fig. 4.2. It is also clear from Fig. 4.2 that a polar coordinaterepresentation provides the most natural description of conic orbits, as a singleequation, P r - (4.7) 1 + e cos 0accounts for all possible orbits. The angle 0 is the true anomaly (often v in theclassical celestial mechanics literature) measured from periapsis, the point of
  5. 5. ASTRODYNAMICS 107 Hyperbola X2 e > l y2 1~ Parabola e=l y2 = 4 px a-~ ~ - b-~ = ij, k P,I / Ellipse 0<e<l y/ x2 y2 a-~ + ~-~ = / / Circle / e=0 x 2 + y2 = a ~ / / / / Hyperbola e = Eccentricity Circle Ellipse Parabola- - a = Semi-major or a=b>0 a>b>0 a=oo a<0 semi-transverse axis b2 b2 e=0 e2 = 1 -~-2 e= 1 e2 = 1 + a2 b = Semi-minor or semi-conjugate axisi p rp (l+e) a (1-e 2) p = Parameter or r=l +ecosO = 1 + e c o s O = 1 + e cos 0 ~ semi-latus rectum Fig. 4.2 Conic section parameters.closest approach of M and m, as shown in Fig. 4.2. The parameter, or semilatusrectum p, is given by p - a(l - e 2) (4.8)It may be useful to combine Eqs. (4.7) and (4.8) to yield r -- r p ( 1 + e) (4.9) 1 + e cos 0
  6. 6. 108 SPACE VEHICLE DESIGNwhere rp -- a(1 - e) (4.1 O)is the periapsis radius, obtained at 0 - 0 in Eq. (4.7). Elliptic orbits also have awell-defined maximum or apoapsis radius at 0 - 7r given by ra - - a(1 + e) (4.11) Combining Eq. (4.10) and (4.11) yields the following useful relationships forelliptic orbits only: ra + rp a - ~ (4.12) 2and e ~ ra - rp (4.13) ra + rp It remains only to specify the relationships between the geometric parametersa, e, and p, and the physical variables energy and angular momentum. Thesolution of Eq. (4.6) establishes the required connection; this solution is given ina variety of texts 9 and will not be repeated here. We summarize the results in thefollowing. The total energy is simply the sum of kinetic and potential energy for eachmass. Because the two-body center of mass is unaccelerated and we are assumingthat m < M, the energy of the larger body is negligible; thus, the orbital energy isdue to body m alone and is V2 /x Et -- T + U = (4.14) 2 rwhere T is the kinetic energy per unit mass. In polar coordinates, with velocitycomponents Vr and Vo given by r and r dO/dr, respectively, r 2 4- (r dO/dt) 2 tz Et = (4.15) 2 rwhich is constant due to the previously discussed conservative property of theforce law. The polar coordinate frame in which Vr and Vo are defined is referred to as theperifocal system. The Z axis of this system is perpendicular to the orbit plane withthe positive direction defined such that the body m orbits counterclockwise aboutZ when viewed from the + Z direction. The origin of coordinates lies at thebarycenter of the system, and the X axis is positive in the direction of periapsis.The Y axis is chosen to form a conventional right-handed set. As discussedearlier, this axis frame is inertially fixed; however, it should not be confused withother inertial frames to be discussed in Sec. 4.2.7 (Coordinate Frames).
  7. 7. ASTRODYNAMICS 109 One of the more elegant features of the solution for central-force motion is theresult that Et = ~ (4.16) 2ai.e., the specific energy of the orbit (energy per unit mass of the satellite) dependsonly on its semimajor (or semitransverse) axis. From Eqs. (4.14) and (4.16), weobtain V2 - / z ( ~ - ~) (4.17)which is known as the vis-viva or energy equation. The orbital angular momentum per unit mass of body m is dr h = r × -- = r × V (4.18) dtwith magnitude given in terms of polar velocity components by rZdO h = rVo = (4.19) dtand is constant for the orbit, as previously discussed. This is a consequence of theradially directed force law; a force normal to the radius vector is required for atorque to exist, and in the absence of such a torque, angular momentum must beconserved. From the solution of Eq. (4.6), it is found that h2 : / x p (4.20)Thus, the orbital angular momentum depends only on the parameter, or semilatusrectum, p. It is also readily shown that the angular rate of the radius vector fromthe focus to the body m is dO h h(1 + e c o s 0) 2 dt - - r 2 -- p2 (4.21) Equations (4.16) and (4.20) may be combined with the geometric result (4.8)to yield the eccentricity in terms of the orbital energy and angular momentum, e2--l+2Et(h) 2 (4.22) This completes the summary of results from two-body theory that areapplicable to all possible orbits. In subsequent sections, we consider specializedaspects of motion in particular orbits.
  8. 8. 110 SPACE VEHICLE DESIGN4.2.2 Circular and Escape Velocity From Eq. (4.17), and noting that for a circular orbit r = a, we find that therequired velocity at radius r, Vcir- ~ (4.23)where Vcir is circular velocity. If Et -- 0, we have the condition for a parabolicorbit, which is the minimum-energy escape orbit. From Eq. (4.14), Wes -- ~ / ~ - ~/2Vcir c (4.24)where Vesc is escape velocity. Of course, circular velocity can have no radialcomponent, whereas escape velocity may be in any direction not intersecting thecentral body. Circular and parabolic orbits are interesting limiting cases corresponding toparticular values of eccentricity. Such exact values cannot be expected inpractice; thus, in reality all orbits are either elliptic or hyperbolic, with Et < 0 orEt > O. Nonetheless, circular or parabolic orbits may be used as referencetrajectories for the actual motion, which is seen as a perturbation of the referenceorbit. We will consider this topic in more detail later; for the present, we examinethe features of motion in elliptic and hyperbolic orbits.4.2.3 Motion in Elliptic Orbits Figure 4.3 defines the parameters of interest in elliptic orbit motion. The conicsection results given earlier are sufficient to describe the size and shape of theorbit, but do not provide the position of body m as a function of time. Because it isawkward to attempt a direct solution of Eq. (4.21) to yield 0 (and hence r) as afunction of time, the auxiliary variable E, the eccentric anomaly, is introduced.The transformation between true and eccentric anomaly is 1+ 1/2 tan(0)--(1-;) tan( E) (4.25)It is found ~° that E obeys the transcendental equation f (E) = E - e sin E - n ( t - tp) = 0 (4.26)where n -- mean motion = v / t x / a 3 tp - time of periapsis passagewhich is known as Keplers equation. The mean motion n is the average orbitalrate, or the orbital rate for a circular orbit having the same semimajor axis as the
  9. 9. ASTRODYNAMICS 111 Auxiliary circle/ Empty I focusApoa a Fig. 4.3 Elliptic orbit parameters.given elliptic orbit. The mean anomaly M =- n ( t - tp) (4.27)is thus an average orbital angular position and has no physical significance unlessthe orbit is circular, in which case n - dO/dt exactly, and E - 0 = M at all times. W h e n E is obtained, it is often desired to know r directly, without theinconvenience of computing 0 and solving the orbit equation. In such a case, theresult r -- a(1 - e cos E) (4.28)is useful. The radial velocity in the orbit plane is rVr = na 2e sin E - e(/xa)l/2 sin E (4.29)The tangential velocity Vo is found from rdO h Vo - = - (4.30) dt r
  10. 10. 112 SPACE VEHICLE DESIGN I f E = 0 = 27r, then (t - tp) = ~, the orbital period. Equation (4.26) then gives "r-- 27ra ~ (4.31)which is Keplers third law. The question of extracting E as a function of time in an efficient manner is ofsome interest, especially prior to the modern era with its surfeit of computationalcapability. A numerical approach is required because no closed-form solution forE(M) exists. At the same time, Eq. (4.26) is not a particularly difficult specimen;existence and uniqueness of a solution are easy to s h o w . 9 Any common root-finding method such as Newtons method or the modified false position method 11will serve. All such methods are based on solving the equation in the "easy"direction (i.e., guessing E, computing M, and comparing with the known value)and employing a more or less sophisticated procedure to choose updatedestimates for E. The question of choosing starting values for E to speedconvergence to the solution has received considerable attention. 12 However, theavailability of programmable calculators, including some with built-in rootfinders, renders this question somewhat less important than in the past, at least forthe types of applications stressed in this book. When the orbit is nearly circular, e _~ 0, and approximate solutions ofadequate accuracy are available that yield 0 directly in terms of mean anomaly.By expanding in powers of e, Eqs. (4.25) and (4.26) can be reduced to the result 1° O~M+2esinM+(~-~-)sin2M+ ".. (4.32)In many cases of interest for orbital operations, the orbits will be nearly circular,and Eq. (4.32) can be used to advantage. For example, an Earth orbit of200 x 1000 km, quite lopsided by parking orbit standards, has an eccentricity of0.0573, which implies that, in using Eq. (4.32), terms of order 2 x 10 -4 are beingneglected. For many purposes, such an error is unimportant. When the orbit is nearly parabolic, numerical difficulties are encountered inthe use of Keplers equation and its associated auxiliary relations. This can beseen from consideration of Eq. (4.25) for e _~ 1, where there is considerable lossof numerical accuracy in relating eccentric to true anomaly. The difficulty is alsoseen in the use of Keplers equation near periapsis, where E and e sin E will bealmost equal for near-parabolic orbits. Battin 13 and others have developeduniversal formulas that avoid the difficulties in time-of-flight computations forKeplerian orbits. However, in spacecraft design the problems of numericalinaccuracy for nearly parabolic orbits are more theoretical than practical and willnot concern us here.
  11. 11. ASTRODYNAMICS 1134.2.4 Motion in Hyperbolic Orbits The study of hyperbolic orbits is accorded substantially more attention inastrodynamics than it traditionally receives in celestial mechanics. In celestialmechanics, only comets pursue escape orbits, and these are generally almostparabolic; hence, orbit prediction and determination methods tend to centeraround perturbations to parabolic trajectories. In contrast, all interplanetarymissions follow hyperbolic orbits, both for Earth departure and upon arrival atpossible target planets. Also, study of the gravity-assist maneuvers mentionedearlier requires detailed analysis of hyperbolic trajectories. Figure 4.4 shows the parameters of interest for hyperbolic orbits. Because ahyperbolic orbit of mass m and body M is a one-time event (possibly terminatedby a direct atmospheric entry or a propulsive or atmospheric braking maneuver toeffect orbital capture), the encounter is often referred to as hyperbolic passage.Although described by the same basic conic equation as for an elliptic orbit,hyperbolic passage presents some significant features not found with closedorbits. In this section, we consider only the encounter between m and M, i.e., the two-body problem; hence, motion of M is ignored. Thus, if m is a spacecraft and M aplanetary flyby target, then the separate motion of both m and M in solar orbit isneglected. This is equivalent to regarding the influence of M as dominating theencounter and ignoring that of the sun. This is, of course, the same approximationwe have used in the preceding discussions, and, when m and M are relativelyclose, it is not a major source of error in the analysis of interest here. For example,the gravitational influence of the sun on a spacecraft in low Earth orbit will notusually be of significance in preliminary mission design and analysis. Hyperbolic passage is fundamentally different. Although the actual encountercan indeed be modeled as a two-body phenomenon to the same fidelity as before,the complete passage must usually be examined in the context of the largerreference frame in which it takes place. This external frame provides the "infinityconditions" and orientation for the hyperbolic passage, and it is in this frame thatgravity-assisted velocity changes must be analyzed. For the present, however, weconsider only the actual two-body encounter. The results will be useful within aso-called sphere of influence (actually not a sphere and not sharply defined) aboutthe target body M. Determination of spheres of influence will be addressed in asubsequent section. When m is "infinitely" distant from M, the orbit equation may be solved withr = c~ to yield the true anomaly of the asymptotes. From Eq. (4.7), 0a -- COS-1 (-~-~) (4.33)and due to the even symmetry of the cosine function, asymptotes at ___Oa areobtained, as required for a full hyperbola. Since r > O, values of true anomaly for
  12. 12. 114 SPACE VEHICLE DESIGNVo~ = hyperbolic approach and departure velocity relative to target body M0 a = true anomaly of asymptotes M~/r/3 = B-plane miss distance; offset in plane ± to arrival asymptote = turning angle of passage a<o Arrival asymptote Fig. 4.4 Hyperbolic orbit geometry.finite r are restricted to the range [-Oa, Oa], and so the orbit is concave toward thefocus occupied by M. Knowledge of O~ serves to orient the orbit in the external frame discussedearlier. The hyperbolic arrival or departure velocity Voo is the vector difference,in the external frame, between the velocity of m and that of M. The condition thatit must lie along an asymptote determines the orientation of the hyperbolicpassage with respect to the external frame. This topic will be considered inadditional detail in a subsequent section.
  13. 13. ASTRODYNAMICS 115 The magnitude of the hyperbolic velocity V~ is found from the vis-vivaequation with r - oo to be V 2 = _ tx = 2 E t (4.34)where it is noted that the semimajor axis a is negative for hyperbolas. Since Voo isusually known for the passage from the infinity conditions, in practice onegenerally uses Eq. (4.34) to solve for a. Conservation of energy in the two-bodyframe requires V~o to be the same on both the arrival and departure asymptotes.However, the vector velocity Voo is altered by the encounter due to the change inits direction. This alteration of Voo is fundamental to hyperbolic passage and isthe basis of gravity-assist maneuvers. The change in direction of Voo is denoted by ~ , the tuming angle of thepassage. If the motion of m were unperturbed by M, the departure asymptotewould have a true anomaly o f - O a + "rr, whereas due to the influence of M thedeparture is in fact at a true anomaly: Oa -- --Oa + "rr + ~ (4.35)Hence, -~ = Oa - - -~ - - sin -1 (4.36)where the second equality follows from Eq. (4.33). The velocity change seen inthe external frame due to the tuming angle of passage is, as seen from Fig. 4.5, 2V~ V - 2Voo sin = -- (4.37) 2 e The eccentricity may be found from Eq. (4.10) with the semimajor axis agiven by Eq. (4.34), yielding e = 1 -} V 2 r p (4.38) /xThis result is useful in the calculation of a hyperbolic departure from an initialparking orbit, or when, as is often the case for interplanetary explorationmissions, the periapsis radius at a target plane is specified. However, Eq. (4.38) isinappropriate for use in pre-encounter trajectory correction maneuvers, which areconventionally referred to as the so-called B-plane, the plane normal to the arrivalasymptote. Pre-encounter trajectory corrections will generally be applied at aneffectively "infinite" distance from the target body and will thus, almost bydefinition, alter the placement and magnitude of Voo relative to M. The B-plane istherefore a convenient reference flame for such maneuvers.
  14. 14. 116 SPACE VEHICLE DESIGN V• Oo Departure asymptote / / .. __] rrival asymptote Fig. 4.5 Velocity vector change during hyperbolic passage. The orbital angular momentum is easily evaluated in terms of the B-plane missdistance/3, yielding h =/3Voo (4.39)which follows readily from the basic vector definition of Eq. (4.18) applied atinfinity in rectangular coordinates. Using this result plus Eq. (4.34) in Eq. (4.22),we find e 2 = 1 q- ( @ ) 2 (4.40)From Eqs. (4.38) and (4.40), the periapsis radius is given directly in terms of theapproach parameters/3 and Voo as rp /x /x (4.41)
  15. 15. ASTRODYNAMICS 117while the B-plane offset required to obtain a desired periapsis is ~ -- rp 1 + rp V~ (4.42) Equations (4.33-4.42), together with the basic conic section results given inSec. 4.2.1 (Two-Body Motion), suffice to describe the spatial properties ofhyperbolic passage. It remains to discuss the evolution of the orbit in time. Aswith elliptic orbits, the motion is most easily described via a Kepler equation, f(F) = e s i n h F - F - n ( t - tp) = 0 (4.43)where n = mean motion = [tz/(-a)3] 1/2 tp = time of periapsis passageand again it is recalled that a < 0 for hyperbolic orbits. The hyperbolic anomalyF, as with the eccentric anomaly E, is an auxiliary variable defined in relation to areference geometric figure, in this case an equilateral hyperbola tangent to theactual orbit at periapsis. 14 The details are not of particular interest here, becausethe analysis has even less physical significance than was the case for the eccentricanomaly. The transformation between true and hyperbolic anomaly is given by tan(0)= [~ -t- 11]1/2tanh ( 2 ) (4.44)Analogously to Eqs. (4.28) and (4.29), it is found that r = a(1 - e cosh F) (4.45)and rVr =naZe sinh F = e(-atx)l/Zsinh F (4.46)with the tangential velocity again given by Eq. (4.30).4.2.5 Motion in Parabolic Orbits Parabolic orbits may be viewed as a limiting case of either elliptic orhyperbolic orbits as eccentricity approaches unity. This results in somemathematical awkwardness, as seen from Eq. (4.8), a - lim[ p ]-oo (4.47) e---~ 1 1 - - e2The semimajor axis is thus undefined for parabolic orbits. The result is ofsomewhat limited concern, however, and serves mainly to indicate thedesirability of using Eq. (4.9), from which the semimajor axis has been
  16. 16. 118 SPACE VEHICLE DESIGNeliminated, for parabolic orbits. Thus, 2rp (4.48) 1 + cos 0and by comparison with Eq. (4.8), it is seen that p = 2rp (4.49)It may be shown that the motion in time is given by D3 + D = M = n(t- tp) (4.50)The parabolic anomaly D is an auxiliary variable defined as 0 D- ~ tan- (4.51) 2and the mean motion is n- ~p3 (4.52)As with hyperbolic orbits, n has no particular physical significance. In terms ofparabolic anomaly, r - - r p ( 1 +~_~2) (4.53)while rVr - (tZrp)l/2D - nrZD (4.54)As always, the tangential velocity Vo is given by Eq. (4.30). It should be notedthat the exact definition of D varies considerably in the literature, as does the formof Kepler s equation [Eq. (4.50)]. Care should be taken in using analytical resultsfrom different sources for parabolic orbits.4.2.6 Keplerian Orbital Elements Orbital motion subject to Newtonian laws of motion and gravitational forceresults in a description of the trajectory in terms of second-order ordinarydifferential equations, as exemplified by Eq. (4.6). Six independent constants arethus required to determine a unique solution for an orbit; in conventionalanalysis, these could be the initial conditions consisting of the position andvelocity vectors r and V at some specified initial time to, often taken as zero forconvenience. In fact, however, any six independent constants will serve, with thephysical nature of the problem usually dictating the choice.
  17. 17. ASTRODYNAMICS 119 In classical celestial mechanics, position and velocity information are neverdirectly attainable. Only the angular coordinates (fight ascension and declination)of objects on the celestial sphere are directly observable. Classical orbitdetermination is essentially the process of specifying orbital position and velocitygiven a time history of angular coordinate measurements. Direct measurement ofr and V (or their relatively simple calculation from given data) is possible inastrodynamics, where ground tracking stations and/or onboard guidance systemsmay, for example, supply position and velocity vector estimates on a nearlycontinuous basis. However, even when r and V are obtained, information in this form is ofmathematical utility only, because it conveys no physical "feel" for or geometric"picture" of the orbit. It is thus customary to describe the orbit in terms of sixother quantities plus an epoch, a time to at which they apply. These quantities,chosen to provide a more direct representation of the motion, are the Keplerianorbital elements, defined graphically in Fig. 4.6 and listed in the following: a or p = semimajor axis or semilatus rectum e = eccentricity i = inclination of orbit plane relative to a defined reference plane - longitude or right ascension of ascending node, measured in the reference plane from a defined reference meridian c o - argument of periapsis, measured counterclockwise from the ascending node in the orbit plane 00, M0, or tp = true or mean anomaly at epoch, or time of relevant periapsis passage As seen, there is no completely standardized set of elements in use, acircumstance in part due to physical necessity. For example, the semimajor axis ais undefined for parabolic orbits and observationally meaningless for hyperbolicorbits, requiring use of a more fundamental quantity, the semilatus rectum p, oroccasionally the angular momentum h. Nonetheless, the geometric significanceand convenience of the semimajor axis for circular and elliptic orbits will not bedenied, and it is always employed when physically meaningful. Other problems occur as well. For nearly circular orbits, w is ill defined, as is1~ for orbits with near-zero inclination. In such cases, various convenientalternate procedures are used to establish a well-defined set of orbital elements.For example, when the orbit is nominally circular, co = 0 is often adopted byconvention. When i _~ 0, accurate specification of 1~ and hence co is difficult, andthe parameter H, the so-called longitude of periapsis, is often used. Here,H = 1~ + co, with 1~ measured in the X - Y plane and co measured in the orbitalplane. In such cases II may be accurately known even though ~ and co are eachpoorly specified.
  18. 18. 120 SPACE VEHICLE DESIGN /I hy I 7--- / f Periapsis hx Ascending node X Fig. 4.6 Orbital elements. It may be seen that a (or p) and e together specify the size and shape or,equivalently, the energy and angular momentum, of the orbit, and i, 1~, and o)provide the three independent quantities necessary to describe the orientation ofthe orbit with respect to some external inertial reference frame. The final requiredparameter serves to specify the position of body m in its orbit at a particular time.As indicated, the true or mean anomaly at epoch, or the time of an appropriateperiapsis passage, may also be used. For spacecraft orbits, conditions at injectioninto orbit or following a midcourse maneuver may also be employed, as may thetrue or mean anomaly at some particular time other than the epoch. In a simple two-body universe, the orbital elements are constant, and theirspecification determines the motion for all time. In the real world, additional
  19. 19. ASTRODYNAMICS 121influences or perturbations are always present and result in a departure frompurely Keplerian motion; hence, the orbital elements are not constant. Perturbinginfluences can be of both gravitational and nongravitational origin and mayinclude aerodynamic drag, solar radiation, and solar wind; the presence of a thirdbody; a nonspherically symmetric mass distribution in an attracting body; or, incertain very special cases, relativistic effects. One or more of these effects will beimportant in all detailed analyses, such as are necessary for the actual executionof a mission, and in many cases for preliminary analysis and mission design aswell. Indeed, it is common practice in mission design to make use of certainspecial perturbations to achieve desired orbital coverage, as discussed briefly inChapter 2 and in Sec. 4.3.3 (Aspherical Mass Distribution). It commonly happens that a spacecraft or planetary trajectory is predomi-nantly Keplerian, but that there exist perturbations that are significant at somelevel of mission design or analysis. When approximate analyses of such cases arecarried out, it may be found that the perturbing influences alter various elementsor combinations of elements in a periodic manner, or in a secular fashion with atime constant that is small compared to the orbit period. Such analyses may beused to provide relatively simple corrections to a given set of orbital elementsdescribing the average motion, or to a set of elements accurately defined at someepoch. The result of procedures such as those just described is a description of theorbit in terms of a set of osculating elements, which are time varying and describea Keplerian orbit that is instantaneously tangent to the true trajectory. In this wayincreased accuracy can be obtained while still retaining a description of themotion in terms of orbital elements, i.e., without resorting to a numericalsolution. These topics will be considered in more detail in a later section.4.2.7 Coordinate Frames Within the fixed orbit plane of two-body Keplerian motion, the coordinatesystem of choice is the polar coordinate system depicted in Fig. 4.2. The positionof the object is given by the coordinates (r, 0), with the true anomaly 0 measuredfrom periapsis. This system of perifocal or orbit plane coordinates is both naturaland sufficient as long as the orientation of the orbit in space need not beconsidered. However, we have seen in the preceding discussion that a particular orbit isdefined through its elements in relation to a known inertial reference frame, asshown in Fig. 4.6. There are two major inertial reference frames of interest. The X axis in all cases of interest in the solar system is defined in the directionof the vernal equinox, the position of the sun against the fixed stars on (presently)March 21, the first day of spring. More precisely, the X axis is the line from thecenter of the Earth to the center of the sun when the sun crosses the Earthsequatorial plane from the southern to northern hemisphere.
  20. 20. 122 SPACE VEHICLE DESIGN For orbits about or observations from Earth (or, when appropriate, any otherplanet), the natural reference plane is the planetary equator. The orbitalinclination i is measured with respect to the equatorial plane, and the longitude ofthe ascending node 1~ is measured in this plane. The positive Z direction is takennormal to the reference plane in the northerly direction (i.e., approximatelytoward the North Star, Polaris, for Earth). The Y axis is taken to form a right-handed set and thus lies in the direction of the winter solstice, the position of thesun as seen from Earth on the first day of winter. The coordinate frame thus defined is referred to as the geocentric inertial(GCI) system. Though fixed in the Earth, it does not rotate with the planet. It isseen that, in labeling the frame as "inertial," the angular velocity of the Earthabout the sun is ignored. Because the frame is defined with respect to the"infinitely" distant stars, the translational offsets of the frame throughout the yearare also irrelevant, and any axis set parallel to the defined set is equally valid. It will often be necessary to transform vectors (r, V) in orbit plane coordinatesto their equivalents in inertial space. From Fig. 4.6, it is clear that this can beaccomplished in three steps, starting with the assumption that the orbit plane iscoincident with the inertial X - Y plane, and that the abscissas are co-aligned:1) Rotate the orbit plane by angle f~ about the inertial Z axis (colinear with the angular momentum vector h).2) Rotate the orbit plane about the new line of nodes by inclination i.3) Perform a final rotation about the new angular momentum vector (also the new Z axis) by angle w, the argument of periapsis. In the terminology of rotational transformations (see also Chapter 7), this is a3-1-3 Euler angle rotation sequence composed of elementary rotation matrices: Tp--+I - - So) Co) 0 0 Ci -Si $1~ C1~ 0 (4.55) 0 0 1 0 Si Ci 0 0 1This yields the rotation matrix that transforms a vector in perifocal coordinatesinto a vector in the inertial frame. In combined form, with S0 and C0 representingsin0 and cos0, we have [ C1~Co9 - SI~SoJCi -Cl~Soo- SI"~CoJCi S~Si 1 Tp~I -- S ~ - ~ C w + Cl~SooCi -SI~Sw + Cf~CwCi -CI~Si ~ (4.56) S ooSi C ogSi Ci Transformation matrices are orthonormal, and so the inverse transformation(in this case, from inertial to orbit plane coordinates) is found by transposing Eq.(4.56): TI___> P T~__,I ~ -1 t Ttp__+l (4.57) For heliocentric calculations, planetary equators are not suitable referenceplanes, and another choice is required. It is customary to define the Earths orbital
  21. 21. ASTRODYNAMICS 123plane about the sun, the ecliptic plane, as the reference plane for the solar system.The Earths orbit thus has zero inclination, by definition, whereas all other solarorbiting objects have some nonzero inclination. The Z axis of this heliocentricinertial (HCI) system is again normal to the reference plane in the (roughly)northern direction, and the Y axis again is taken to form a right-handed set. TheEarth s polar axis is inclined at approximately 23.5 ° relative to the ecliptic, and sothe transformation from GCI to HCI is accomplished via a coordinate rotation of23.5 ° about the X axis. The relationship between these two frames is shown inFig. 4.7. In either system, a variety of coordinate representations are possible inaddition to the basic Cartesian (X, Y, Z) frame. The choice will depend in part onthe type of equipment and observations employed. For example, a radar or otherradiometric tracking system will produce information in the form of range to thespacecraft, azimuth angle measured from due North, and elevation angle abovethe horizon. Given knowledge of the tracking station location, such informationis readily converted to standard spherical coordinates (r, 0, ~b) and therefore to(X, Y, Z) or other coordinates. When optical observations are made, as in classical orbit determination orwhen seeking to determine the position of an object against the background offixed stars, range is not a suitable parameter. All objects appear to be located atthe same distance and are said to be projected onto the celestial sphere. In thiscase, only angular information is available, and a celestial longitude-latitudesystem similar to that used for navigation on Earth is adopted. Longitude andlatitude are replaced by fight ascension and declination (c~, 6). Right ascension ismeasured in the conventional trigonometric sense in the equatorial plane (aboutthe Z axis), with 0 ° at the X axis. Declination is positive above and negative belowthe equatorial (X-Y) plane, with a range of + 90 °. Figure 4.8 shows therelationships between Cartesian, celestial, and spherical coordinates. Usefultransformations are X - r sin 0 cos ~b - r cos 6 cos c~ (4.58a) Y - r sin 0 sin 05 - r cos 6 sin a (4.58b) Z- rcos 0 - rsin 6 (4.58c) r 2 -- X 2 _~_ y2 _+_Z 2 (4.58d) 0- E z COS -1 (X 2 -+- y2 _+_Z2)1/2 ] (4.58e) ~b-a~-tan-l(Y) (4.58f) 6 - - s i n - 1 (X 2 + Y 2 + Z 2 ) l / 2 (4.58g)
  22. 22. 124 SPACE VEHICLE DESIGN Polaris Z GCl Z HCl Y HCl 23.5 ° Y GCl X Fig. 4.7 Relation of GCI and HCI coordinates.where line-of-sight vectors only are obtained, as with optical sightings, and r isassumed to be of unit length in Eqs. (4.58). The Earths spin axis is not fixed in space but precesses in a circle with aperiod of about 26,000 yr. This effect is due to the fact that the Earth is notspherically symmetric but has (to the first order) an equatorial bulge upon whichsolar and lunar gravitation act to produce a perturbing torque. The vernal equinox
  23. 23. ASTRODYNAMICS 125 I I ~J Fig. 4.8 Relationship between Cartesian, spherical, and celestial coordinates.of course processes at the same rate, with the result that very precise or verylong-term observations or calculations must account for the change in the"inertial" frames that are referred to the equinox. There is also a small deviationor about 9 arc-seconds over a 19-yr period due to lunar orbit precession that mayin some cases need to be included. Specification of celestial coordinates forprecise work thus includes a date or epoch (2000 is in common current use) thatallows the exact orientation of the reference frame with respect to the "fixed"stars (which themselves have measurable proper motion) to be computed. In many spacecraft applications, corrections over the mission lifetime aresmall with respect to those relative to the year 2000. For this reason, spacemissions are commonly defined with reference to true of date (TOD) coordinates,which have an epoch defined in a manner convenient to a particular mission.4.2.8 Orbital Elements from Position and Velocity As mentioned, knowledge of position and velocity at any single point and timein the orbit is sufficient to allow computation of all Keplerian elements. As animportant example, it is possible given the position and velocity of the ascentvehicle at burnout to determine the various orbit injection parameters. A similar
  24. 24. 126 SPACE VEHICLE DESIGNcalculation would be required following a midcourse maneuver in aninterplanetary mission. As a matter of engineering practice, the single-measurement errors in vehicleposition and velocity are of such magnitude as to result in a rather crude estimateof the orbit from one observation, and so rather elaborate filtering and estimationalgorithms are employed in actual mission operations to obtain accurate results.However, for mission design and analysis such issues are unimportant, and it is ofinterest to know the orbital elements in terms of nominal position and velocityvectors. This information is also of use in sensitivity studies, in which theorbit dispersions that result from specified launch vehicle injection errors (seeChapter 5) are examined. It is assumed that r and V are known in a coordinate system of interest, such asGCI for Earth orbital missions or HCI for planetary missions. In practice, one ormore coordinate transformations must be performed to obtain data in the requiredform, because direct measurements will be made in coordinates appropriate to aground-based tracking station or network. It is assumed here that i,j, and k are theunit vectors in the (X, Y, Z) directions for the appropriate coordinate system. Given r and V in a desired coordinate system, the angular momentum is, fromEq. (4.18), h=r× V (4.59)with magnitude h=rvsin(2- y)-rVcosy (4.60)where y, the flight-path angle relative to the local horizon, is defined in Fig. 4.9.Thus, rV sin 3 = ~ (4.61) rVEccentricity may be found from Eq. (4.22), with Et given by Eq. (4.14).Alternatively, it may be shown 9 that, in terms of flight-path angle y, e (r 2 = - 1 cos 2 y + sin 2 y (4.62)or - 11 c°s2 y + 1 (4.63)and (rV 2//.t)sin ycos 3 tan 0 = (4.64) (rV2/t.z)cos2y - 1
  25. 25. ASTRODYNAMICS 127 I Local horizon Fig. 4.9 Motion in orbit plane showing flight path angle.Equation (4.62) avoids the ambiguity in true anomaly inherent in the use of theinverse cosine when Eq. (4.7) is used. Note that y - - 0 implies 0 = 0 ifrV2/I~ > 1, and 0 = vr, if rV2/i~ < 1. Thus, the spacecraft moves horizontallyonly at perigee or apogee if the orbit is elliptic. If rV 2/~ -- 1, the orbit is circular,and Eq. (4.61) will yield y = 0, which implies that 0 is undefined in Eq. (4.64).As discussed in Section 4.2.6, this difficulty is due to the fact that w, and hence0, are undefined for circular orbits. Defining ~o = 0 in this case will resolve theproblem. If defined, the semimajor axis is found from Eq. (4.8): a -- p (4.65) 1 - e2or, if the orbit is nearly parabolic, we may use Eq. (4.20): h2 p = (4.66) U
  26. 26. 128 SPACE VEHICLE DESIGNSince i is defined as the angle between h and k, h.k-- h c o s / = hz (4.67)and hz cos/= -- (4.68) hwhere it is noted that 0 ° < i < 180 °. m The node vector 1 lies along the line of nodes between the equatorial and orbitplanes and is positive in the direction of the ascending node. Thus, 1= k × h (4.69)Since 12, the right ascension of the ascending node, is defined as the anglebetween i and l, I. i = 1cos f~ = lx (4.70)and cos f~ = _lx (4.71) 1where ly < 0 implies fl > 180 °. It may also be noted that tan 1"), - hx (4.72) -h vwhich avoids the cosine ambiguity in Eq. (4.71). Finally, it is noted that to is the angle from the node vector 1 to perigee, in theorbit plane, whereas 0 is measured from perigee to r, also in the orbit plane. Thus, 1. r = lr cos(to + 0) (4.73)Hence, the argument of perigee is to - c°s-1 (--~-r)l" _ 0 r (4.74)In Eq. (4.74), k . r > 0 implies 0 ° < t o + 0 < 180 °, and k . r < 0 implies180 ° < t o + 0 < 360 ° . If desired, the periapsis time tp may be found from the Kepler time-of-flightrelations (4.26), (4.43), or (4.50), plus the auxiliary equations relating E, F, or Dto true anomaly 0 The location of the launch site (or, more accurately, the location and timing ofactual booster thrust termination) is a determining factor in the specification ofsome orbital elements. Of these, the possible range of orbital inclinations is themost important. Qualitatively, it is clear that not all inclinations are accessiblefrom a given launch site. For example, if injection into orbit does not occur
  27. 27. ASTRODYNAMICS 129precisely over the equator, an equatorial orbit is impossible, because theorbit plane must include the injection point. This is shown quantitatively byBate et al. TM cos i -- sin ~bcos A (4.75)where 4 ) - injection azimuth (North = 0 °) h - injection latitude.If the boost phase is completed quickly so that injection is relatively close to thelaunch site, conditions for tb and A as determined for the initial launch azimuthand latitude approximate those at vehicle burnout. Equation (4.75) implies that direct orbits (i < 90 °) require 0°_< 4~ < 180 °, andfurthermore that the orbital inclination is restricted to the range iil >_ [AI. The longitude of the ascending node 1) also depends on the injectionconditions, in this case the injection time. This is because launch is from arotating planet, whereas 1) is defined relative to the fixed vernal equinox. Whenthe choice of 120 is important, as for a sun-synchronous orbit (see Sec. 4.3.3,Aspherical Mass Distribution), the allowable launch window can become quitesmall.4.2.9 State Vector Propagation from Initial Conditions In practical work it is often desired to compute the orbital state vector (r, V) attime t given initial conditions (r0, V0) at time to. The material presented thus farallows us to do so as follows: 1) From (r0, V0), compute the orbital elements using Eqs. (4.59-4.74). 2) Use the appropriate form of Keplers equation, depending on eccentricity,to find true anomaly O(t) given 00 - Oo(to). 3) Determine r from Eq. (4.7), the orbit equation. 4) Use (for example) Eqs. (4.29-4.30) to find Vr, VO, hence (r, V) in orbit-plane coordinates. 5) Apply the appropriate coordinate transformation Eq. (4.56) to convert (r, V)from orbit plane coordinates to GCI or HCI. This process, while conceptually clear, is undeniably awkward. Someimprovement may be realized after O(t) is obtained in step 2 through the use of theLagrangian coefficients, 1°13~5 where the dependence on initial conditions isexpressed as r - fro + gVo (4.76a) V = fro + gVo (4.76b) This relationship is mandated by the fact that, because (r0, V0) are coplanar but
  28. 28. 130 SPACE VEHICLE DESIGNnot colinear, any other vector in the orbit plane can be written as a linearcombination of (r0, V0). Various forms o f f and g exist depending on whether theindependent variable is taken to be true anomaly 0 or eccentric anomaly D, E, or F.For example, with A 0 = (0 - 00), /,. f -- 1 - (1 - cos A 0 ) - (4.77a) P rro sin A 0 g -- (4.77b) /xp j~ /-~[1-cosAO 1 1] (_~) - tan (4.77c) p r r0 -- 1 - (1 - cos A0) r° (4.77d) PEquations (4.77) are independent of eccentricity, but of course it is necessary touse the appropriate form of Keplers equation, depending on eccentricity, toobtain O(t) given 00 and ( t - t o ) . This is inconvenient in many analyticalapplications, an issue to which we have alluded previously. As mentioned earlier, Battin 13 pioneered the development and application of"universal" formulas to eliminate this inconvenience. Numerous formulationsexist; we cite here a particularly elegant approach (without its rather tediousderivation), easily implemented on a programmable calculator. Beginning withthe end result, it is found that Uz(x, c~) f- 1 - ~ (4.78a) /"0 U1 (/~/, c~) + Or0 U 2 (,¥, t~) g - (4.78b) j~ = -- ~/r-~Ul(/~ ~) (4.78c) rro - 1 - Uz(x, c~) ~ (4.78d)where 1 2 V2 (4.79) a r0 /x ro. Vo or0 - (4.80)
  29. 29. ASTRODYNAMICS 131and Ui(X, a) is the universal function of ith order. The universal functions, whichare analogous to the trigonometric functions, satisfy aX2 (aX2)2 (4.81) Uo(X, a) = 1 - -~. + 4------7[-- . . . . . [ aX2 (aX2)2 J (4.82) Wl (X, Og) -- X 1 -- 7 . + 5-----~- . . . .and satisfy the recursion relation x" U.(x, ~) Un+2(X, oL) - ~ - an! ol (4.83)The universal Kepler equation is ~ / ~ ( t - to) = roU1 (X, a) + ooU2(x, a) + U3(X, a) (4.84)The variable X is the universal form of the eccentric anomaly and can be shown tobe X = (o"- o0) + a v / - ~ ( t - to) (4.85)although usually this is unknown and is to be found via iteration. The application of these results, as opposed to their derivation, isstraightforward. Assuming that the functions Uo(X, a) and UI(X, a) can beprogrammed as executable subroutines, in the manner of standard trigonometricfunctions, the procedure is as follows: 1) Given initial conditions (r0, V0), compute a and Oo. 2) Given ( t - to), solve the Kepler equation for X, i.e., guess X, computethe Ui(X, a), and apply conventional root-finding techniques to iterate toconvergence. 3) Given X, compute U0, U1, U2, and U3. 4) With Uo(X, a), UI(X, a), U2(X, a), and U3(X, a) now known, compute theLagrangian coefficients. 5) Use f, g, )k, and g to find the new state vector (r, V) at time t. Note that the orbital eccentricity is not used. This approach, of course, fails toconvey the visual picture of the orbit offered by use of the classical elements.However, it is unrivaled in its computational efficiency and utility.4.2.10 OrbitDetermination In earlier sections we have given procedures for obtaining orbital elementswhen r and V are known in an inertial frame such as GCI or HCI. The problem oforbit determination then essentially consists of obtaining accurate values of r andV at a known time t. Doppler radar systems and certain other types of equipmentallow this to be done directly (ignoring for the moment any coordinate
  30. 30. 132 SPACE VEHICLE DESIGNtransformations required to relate the observing site location to the inertialframe), whereas other types of radar or passive optical observations do not.Indeed, classical orbit determination may be thought of as the process ofdetermining r and V given angular position (a, 6) observations in an inertialframe at known times. Classical orbit determination remains important today. Radar observations arenot possible for all targets and are seldom as accurate on any given measurementas are optical sightings. Given adequate observation time and sophisticatedfiltering algorithms (two days of Doppler tracking followed by several hours ofcomputer processing are required to provide the ephemeris data, accurate towithin 10 m at epoch, used in the Navy Transit navigation system), radiometricmeasurement techniques are the method of choice in modem astrodynamics.However, for accurate preliminary orbit determination, optical tracking remainsunsurpassed. This is evident from the continued use and expansion of suchsystems, as for example in the Ground-Based Electro-Optical Deep SpaceSurveillance (GEODSS) program. 16 As discussed, six independent pieces of time-tagged information are requiredto obtain the six orbital elements. Several types of observations have been used tosupply these data: 1) Three position vectors r(t) may be obtained at different known times. Thiscase is applicable to radar systems that do not permit Doppler tracking to obtainvelocity. 2) Three line-of-sight vectors (angular measurements) may be known atsuccessive times. A solution due to Laplace gives position and velocity at theintermediate time. 3) Two position vectors r(t) may be known and the flight time between themused to determine position and velocity. This is known as the Gauss problem (orthe Lambert problem, when viewed from the trajectory designers perspective),and remains useful today in part for its application to ballistic missile trajectoryanalysis and other intercept problems. We note in passing that in few cases would an orbit be computed from only theminimum number of sightings. In practice, all data would be used, with themultiple solutions for the elements filtered or smoothed appropriately to allow abest solution to be obtained. However, practical details are beyond the scope ofthis text. Many references are available, e.g., Bate et al., 15 Gelb, 17 Nahi, 18 andWertz. 19 An excellent survey of the development of Kalman filtering foraerospace applications is given by Schmidt. 2° We will not consider the details of orbit determination further in this text.Adequate references (e.g., Danby l° and Bate et al. 15) exist if required; however, such work is not generally a part of mission and spacecraft design.4.2.11 Timekeeping Systems Accurate measurement of time is crucial in astrodynamics. The analyticresults derived from the laws of motion and presented here allow predictions to
  31. 31. ASTRODYNAMICS 133be made for the position of celestial objects at specific past or future times.Discrepancies between predictions and observations may, in the absence ofa priori information, legitimately be attributed either to insufficient fidelity in thedynamic model or to inaccuracy in the measurement of time. Obviously, it isdesirable to reduce uncertainties imposed by timekeeping errors, so thatdifferences between measured and predicted motion can be taken as evidence for,and a guide to, needed improvements in the theoretical model. The concept of time as used here is that of absolute time in the Newtonian ornonrelativistic sense. In this model, time flows forward at a uniform rate for allobservers and provides, along with absolute or inertial space, the framework inwhich physical events occur. In astrodynamics and celestial mechanics, thisabsolute time is referred to as "ephemeris time," to be discussed later in more detail. Special relativity theory shows this concept of time (and space) to befundamentally in error. Perception of time is found to be dependent on the motionof the observer, and nonlocal observers are shown to be incapable of agreeing onthe exact timing of events. General relativity shows further that the measurementof time is altered by the gravity field in which such measurements occur. Theseresults notwithstanding, relativistic effects are important in astrodynamics only invery special cases, and certainly not for the topics of interest in this book. For ourpurposes, Newtonian concepts of absolute space and time are preferable toEinsteinian theories of space-time. 4.2.11.1 Measurement of time. Measurement of absolute time isessentially a counting process in which the fundamental counting unit isderived from some observable, periodic phenomenon. Many such phenomenahave been used throughout history as timekeeping standards, always with aprogression toward the phenomena that demonstrate greater precision in theirperiodicity. In this sense, precision is obtained when the fundamental period isrelatively insensitive to changes in the physical environment. Thus, pendulumclocks allowed a vast improvement in timekeeping standards compared tosundials, water clocks, etc. However, the period of a pendulum is a function of thelocal gravitational acceleration, which has a significant variation over the Earth ssurface. Thus, a timekeeping standard based on the pendulum clock is truly validonly at one point on Earth. Moreover, the clock must be oriented vertically, andhence is useless, even on an approximate basis, for shipboard applications whereaccurate timekeeping is essential to navigation. Until 1 January, 1958, the Earths motion as measured relative to the fixedstars formed the basis for timekeeping in physics. By the end of the nineteenthcentury, otherwise unexplainable differences between the observed and predictedposition of solar system bodies led to the suspicion that the Earths day and yearwere not of constant length. Although this was not conclusively demonstrateduntil the mid-20th century, the standard or ephemeris second was nonethelesstaken as 1/31,556,925.9747 of the tropical year (time between successive vernalequinoxes) 1900.
  32. 32. 134 SPACE VEHICLE DESIGN Since 1 January, 1958, the basis for timekeeping has been international atomictime (TAI), defined in the international system (SI) of units 21 as 9,292,631,770periods of the hyperfine transition time for the outer electron of the Ce 133 atom inthe ground state. Atomic clocks measure the frequency of the microwave energyabsorbed or emitted during such transitions and can yield accuracy of better thanone part in 1014. (For comparison, good quartz crystal oscillators may be stable toa few parts in 1013, and good pendulum clocks and commonly availablewristwatches may be accurate to one part in 106). The SI second was chosen toagree with the previously defined ephemeris second to the precision available inthe latter as of the transition date. 4.2.11.2 Calendar time. Calendar time, measured in years, months, days,hours, minutes, and seconds, is computationally inconvenient but remains firmlyin place as the basis for civil timekeeping in conventional human activities.Because spacecraft are launched and controlled according to calendar time, it isnecessary to relate ephemeris time, based on atomic processes, to calendar time,which is forced by human conventions to be synchronous with the Earthsrotation. The basic unit of convenience in human time measurement is the day, theperiod of time between successive appearances of the sun over a given meridian.The mean solar day (as opposed to the apparent solar day) is the average length ofthe day as measured over a year and is employed to remove variations in the daydue to the eccentricity of the Earths orbit and its inclination relative to the sunsequator. Because "noon" is a local definition, a reference meridian is necessary inthe specification of a planet-wide timekeeping (and navigation) system. Thereference meridian, 0 ° longitude, runs through a particular mark at the former siteof the Royal Observatory in Greenwich, England. Differences in apparent solartime between Greenwich and other locations on Earth basically reflect longitudedifferences between the two points, the principle that is the basis for navigationon Earths surface. Twenty-four local time zones, each nominally 15 ° longitudein width, are defined relative to the prime meridian through Greenwich and areemployed so that local noon corresponds roughly to the time when the sun is atzenith. The mean solar time at the prime meridian is defined as universal time (UT),also called Greenwich mean time (GMT) or Zulu time (Z). A variety of poorlyunderstood and essentially unpredictable effects (e.g., variations in theaccumulation of polar ice from year to year) alter the Earths rotation period;thus, UT does not exactly match any given rotation period, or day. Coordinateduniversal time (UTC) includes these corrections and is the time customarilybroadcast over the ratio. "Leap seconds" are inserted or deleted from UTC asneeded to keep it in synchrony with Earths rotation as measured relative to thefixed stars. Past corrections, as well as extrapolations of such corrections into thefuture, are available in standard astronomical tables and almanacs.
  33. 33. ASTRODYNAMICS 135 4.2.11.3 Ephemeris time. From earlier discussions, it is clear thatephemeris time is the smoothly flowing, monotonically increasing time measurerelevant in the analysis of dynamic systems. In contrast, universal time is basedon average solar position as seen against the stars and thus includes variations dueto a number of dynamic effects between the sun, Earth, and moon. The resultingvariation in the length of the day must be accounted for in computing ephemeristime from universal time. The required correction is ET = UT + AT ~ UT + AT(A) (4.86)where AT is a measured (for the past) or extrapolated (for the future) correction,and AT(A) is an approximation to AT given by AT(A) -- TAI - UT + 32.184 s (4.87)Hence, ET _~ TAI + 32.184 s (4.88)where AT = 32.184 s was the correction to UT on 1 January, 1958, the epoch forintemational atomic time. For comparison, AT(A) = 50.54 s was the correctionfor 1 January, 1980. The 18.36-s difference that accumulated between TAI(which is merely a running total of SI seconds) and UT from 1958 to 1980 isindicative of the corrections required. 4.2.11.4 Julian dates. Because addition and subtraction of calendar timeunits are inconvenient, the use of ephemeris time is universal in astronomy andastrodynamics. The origin for ephemeris time is noon on 1 January, 4713 B.C.,with time measured in days since that epoch. This is the so-called Julian day (JD).For example, the Julian day for 31 December, 1984 (also, by definition, 0January, 1985) at 0 hrs is 2,446,065.5, and at noon on that day it is 2,446,066.Noon on 1 January, 1985, is then JD 2,446,067, etc. Tables of Julian date aregiven in standard astronomical tables and almanacs, as well as in navigationhandbooks. Fliegel and Van Flandem 22 published a clever and widelyimplemented equation for the determination of any Julian date that is suitedfor use in any computer language (e.g., FORTRAN or PL/1) with integer-truncation arithmetic: J D - K - 32,075 + 1461, [I + 4800 + ( J - 14)/12] 4 + 367, { J - 2 - [ ( J - 14)/12],12} 12 - 3, {[I + 4900 + (J - 14)/12]/100] (4.89) 4
  34. 34. 136 SPACE VEHICLE DESIGNwhere I = year J = month K = day of month The Julian day system has the advantage that practically all times ofastronomical interest are given in positive units. However, in astrodynamics andspacecraft work in general, times prior to 1957 are of little interest, and the largepositive numbers associated with the basic JD system are somewhatcumbersome. Accordingly, the Julian day for space (JDS) system is definedwith an epoch of 0 hrs UT on 17 September, 1957. Thus, JDS = JD - 2,436,099.5 (4.90)Similarly, the modified Julian day (JDM) system has an epoch defined at 0 hrsUT, 1 January, 1950. These systems have the additional advantage for practicalspacecraft work of starting at 0 hrs rather than at noon, which is convenient inastronomy for avoiding a change in dates during nighttime observations. 4.2.11.5 Sidereal time. Aside from ephemeris time, the systemsdiscussed thus far are all based on the solar day, the interval betweensuccessive noons or solar zenith appearances. It is this period that is the basic24-h day. However, because of the motion of the Earth in its orbit, this "day" isnot the true rotational period of the Earth as measured against the stars, aperiod known as the sidereal day, 23 hrs 56 min 4 s. There is exactly one extrasidereal day per year. Sidereal time is of no interest for civil timekeeping but is important inastronautics for both attitude determination and control and astrodynamics,where the orientation of a spacecraft against the stars is considered. For example,it is the sidereal day that must be used to compute the period for ageosynchronous satellite orbit. Sidereal time is also needed to establish theinstantaneous relationship of a ground-based observation station or launch site tothe GCI or HCI frame. The local sidereal time is given by 0 = 0c + lie (4.91)where 0 = local sidereal time (angular measure) 0a = Greenwich sidereal time ~)e = east longitude of observing site The Greenwich sidereal time is given in terms of its value at a defined epoch,to, by OG = OGo -1 O ) e ( t - t- to) (4.92)
  35. 35. ASTRODYNAMICS 137where We is the inertial rotation rate of Earth and was 7.292116 x 10 -5 rad/s for1980. Again, tables of 0c0 for convenient choices of epoch are provided instandard astronomical tables and almanacs.4.3 Non-Keplerian Motion We have, to this point, reviewed the essential aspects of two-body orbitalmechanics, alluding only briefly to the existence of perturbing influences that caninvalidate Keplerian results. Such influences are always present and can often beignored in preliminary design. However, this is not always the case, nor is itindeed always desirable; mission design often involves deliberate use of non-Keplerian effects. Examples include the use of atmospheric braking to effectorbital maneuvers and the use of the Earths oblateness to specify desired (oftensun-synchronous) rates of orbital precession. These and other aspects of non-Keplerian orbital dynamics are discussed in the following sections.4.3.1 Sphere of Influence Of the various possible perturbations to basic two-body motion, the mostobvious are those due to the presence of additional bodies. Such bodies arealways present and cannot be easily included in an analysis, particularly atelementary levels. It is then necessary to determine criteria for the validity ofKeplerian approximations to real orbits when more than two bodies are present. If we consider a spacecraft in transit between two planets, it is clear that whenclose to the departure planet, its orbit is primarily subject to the influence of thatplanet. Far away from any planet, the trajectory is essentially a heliocentric orbit,whereas near the arrival planet, the new body will dominate the motion. Therewill clearly be transition regions where two bodies will both have significantinfluence on the spacecraft motion. The location of these transition regions isdetermined by the so-called sphere of influence of each body relative to the other,a concept originated by Laplace. For any two bodies, such as the sun and a planet, the sphere of influence isdefined by the locus of points at which the suns and the planets gravitationalfields have equal influence on the spacecraft. The term "sphere of influence" issomewhat misleading; every bodys gravitational field extends to infinity, and inany case, the appropriate equal-influence boundary is not exactly spherical.Nonetheless, the concept is a useful one in preliminary design. Although the relative regions of primary influence can be readily calculatedfor any two bodies, the concept is most useful when, as mentioned earlier, one ofthe masses is much greater than the other. In such a case, the so-called classical
  36. 36. 138 SPACE VEHICLE DESIGNsphere of influence about the small body has the approximate radius r ~ R(M) 2/5 (4.93)where r = sphere of influence radius R -- distance between primary bodies m and M m = mass of small body M -- mass of large bodyA spacecraft at a distance less than r from body m will be dominated by that body,but at greater distances, it will be dominated by body M. Table 4.1 gives sphere-of-influence radii from Eq. (4.81) for the planets relative to the sun and the moonrelative to the Earth. Other sphere-of-influence definitions are possible. A popular alternate to Eq.(4.93) is the Bayliss sphere of influence, which replaces the 2,/5 power with 1/3.This defines the boundary where the direct acceleration due to the small bodyequals the perturbing acceleration due to the gradient of the larger bodysgravitational field. Away from the sphere-of-influence boundary, Keplerian orbits are reasonablyvalid. Near the boundary, a two-body analysis is invalid, and alternate methodsare required. In some cases, it may be necessary to know only the generalcharacteristics of the motion in this region. The results of restricted three-bodyanalysis, discussed later, may then be useful. Generally, however, more detailedinformation is required. If so, the mission analyst has two choices. Accuratetrajectory results may be obtained by a variety of methods, all requiting acomputer for their practical implementation. Less accurate, preliminary resultsmay be obtained by the method of patched conics, discussed in Sec. 4.5. The use Table 4.1 Planetary spheres of influencePlanet Mass ratio (sun planet) Sphere of influence, kmMercury 6.0236 x 106 1.12 x 105Venus 4.0852 x 105 6.16 x 105Earth 3.3295 x 105 9.25 x 105Mars 3.0987 x 106 5.77 x 105Jupiter 1.0474 x 103 4.88 × 107Saturn 3.4985 × 103 5.46 × 107Uranus 2.2869 × 104 5.18 x 107Neptune 1.9314 x 104 8.68 × 107Pluto 3 × 106 1.51 × 107Moona 81.30 66,200aRelative to Earth.
  37. 37. ASTRODYNAMICS 139of this method is consistent with the basic sphere-of-influence calculation of Eq.(4.93) and thus does not properly account for motion in the transition regionbetween two primary bodies.4.3.2 Restricted Three-Body Problem The general case of the motion of three massive bodies under their mutualgravitational attraction has never been solved in closed form. Sundman 23developed a general power series solution; however, useful results are obtainedonly for special cases or by direct numerical integration of the basic equations. Aspecial case of particular interest involves the motion of a body of negligiblemass (i.e., a spacecraft) in the presence of two more massive bodies. This is therestricted three-body problem, first analyzed by Lagrange, and is applicable tomany situations of interest in astrodynamics. Earlier it was found 1° that, in contrast to the simple Keplerian potentialfunction of Eq. (4.2), the restricted three-body problem yields a complexpotential function with multiple peaks and valleys. This is shown in Fig. 4.10,where contours of equal potential energy are plotted. Along these contours,particles may move with zero relative velocity. The five classical Lagrangian points are shown in Fig. 4.10. L1, L2, and L3 aresaddle points, i.e., positions of unstable equilibrium. Objects occupying thesepositions will remain stationary only if they are completely unperturbed. Anydisturbances will result in greater displacement from the initial position. Contours of zero relative velocity L 4 (Stable) , ~ ~ ~ / / L 1 (Unstable) L 2 (Unstable) L 3" (Unstable) L 5 (Stable) Fig. 4.10 Lagrange points for restricted three-body problem.
  38. 38. 140 SPACE VEHICLE DESIGNSpacecraft can occupy these positions for extended periods, but only if a supplyof stationkeeping fuel is provided to overcome perturbing forces. L4 and L5 are positions of stable equilibrium; objects displaced slightly fromthese positions will experience a restoring force toward them. Small bodies canoccupy stable orbits about L4 or L5, a fact that is observationally confirmed by thepresence of the Trojan asteroids that occupy the stable Lagrange points 60 ° aheadof and behind Jupiter in its orbit about the sun. The properties of L4 and L5 haveled to considerable analysis of their use as sites, in the Earth-moon system, forpermanent space colonies and manufacturing sites that would be supplied withraw materials form lunar mining sites. 34.3.3 Aspherical Mass Distribution As indicated in Sec. 4.2.1 (Two-Body Motion), an extended body such as theEarth acts gravitationally as a point mass provided its mass distribution isspherically symmetric. That is, the density function p(r, O, 05) in sphericalcoordinates must reduce to a function p = p(r); there can be no dependence onlatitude 0 or longitude ~b. Actually, the Earth is not spherically symmetric, butmore closely resembles an oblate spheroid with a polar radius of 6357 km and anequatorial radius of 6378 km. This deformation is due to the angular accelerationproduced by its spin rate (and is quite severe for the large, mostly gaseous outerplanets such as Jupiter and Saturn), but numerous higher order variations existand are significant for most Earth orbital missions. The Earth is approximately spherically symmetric; thus, it is customary 24 todescribe its gravitational potential in spherical coordinates as a perturbation to thebasic form of Eq. (4.2), e.g., U(r, O, 6 ) - _Ix + B(r, O, 05) (4.94) Fwhere B(r, O, dp) is a spherical harmonic expansion in Legendre polynomialsPrim(COS 0). The complete solution 25 includes expansion coefficients dependenton both latitude and longitude. Both are indeed necessary to represent theobservable variations in the Earths field. However, low-orbiting spacecrafthaving short orbital periods are not sensitive to the longitudinal, or tesseral,variations because, being periodic, they tend to average to zero. Spacecraft inhigh orbits with periods too slow to justify such an averaging assumption are toohigh to be influenced significantly by what are, after all, very small effects. Thus,for the analysis of the most significant orbital perturbations, only the latitude-dependent, or zonal, coefficients are important. (An exception is a satellite in ageostationary orbit; because it hovers over a particular region on Earth, its orbit issubjected to a continual perturbing force in the same direction, which will beimportant.) If the longitude-dependent variations are ignored, the gravitational
  39. 39. ASTRODYNAMICS 141potential is 25 (4.95) n=2where Re : radius of Earth r = radius vector to spacecraft .In = nth zonal harmonic coefficient Pn(x) = Pno(x) = nth Legendre polynomial Table 4.2 shows that, for Earth, J2 dominates the higher order J~, whichthemselves are of comparable size, by several orders of magnitude. This is to beexpected, because J2 accounts for the basic oblateness effect, which is thesingle most significant aspherical deformation in the Earths figure. Furthermore,since (Re~r) < 1, it is to be expected that the second-order term in Eq. (4.95) willproduce by far the most significant effects on the spacecraft orbit. This is in fact the case. The major effect of Earth s aspherical mass distributionis a secular variation in the argument of perigee o) and the longitude of theascending node f~. Both of these depend to the first order only on the Earthsoblateness, quantitatively specified by J 2 . The results are (~) ( - ~ ) e 4 - 5sine i do) .,~ nJ2 _ )e (4.96) dt - (1 e 2 (~) (~)2 cos/ d O _~ _ n J2 e2)2 (4.97) dt (1 -where n- mean m o t i o n - v @ / a 3 a - semimajor axis i- inclination e- eccentricity Variations in a, e, i, and n also occur but are periodic with zero mean and small amplitude. In Eqs. (4.96) and (4.97) we have approximated these parameters by their values for a Keplerian orbit, whereas in fact they depend on J2. Errors implicit in this approximation may be as much as 0.1%. 19 Table 4.2 Zonal harmonics for Earth J2 1.082 x 10-3 J3 -2.54 x 10 - 6 J4 -1.61 x 10 - 6

×