Angles-Only Orbit Determination Copyright 2006 Michel Santos Page 1

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1 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 1 Abstract This ocument presents a re-erivation of the Gauss an Laplace Angles-Only Methos for Initial Orbit Determination. It keeps close track of the units (imensional length) of the various measurements, an thus provies a more logical form of the equations (compare to erivations in Vallao an Escobal). This form of the equations also makes it easier to perform the calculations without much consieration for rescaling measurements an constants arbitrarily. It is left up to the user to scale their own units, or use a tool which keeps track of units automatically.

2 Angles-Only Orbit Determination Copyright 6 Michel Santos Page Introuction The general approach for performing an angles-only initial orbit etermination, by either the Laplace or Gauss metho, is compose of the following steps: Step 1 - Perform angular observations Perform the observations at a certain time an place. Both the location of the observer an the angles must be mae, or calculate, relative to any inertial reference frame. The observe angles form LOS vectors There must be a minimum of three angular observations to fin a solution; more observations increase accuracy All observations can be mae from multiple locations (in fact, it woul be quite challenging to make all of the observations from one spot in inertial space) Step - Select Time of Interest (TOI) One of the times of observation is selecte as the Time of Interest. Often, with only three observations taken at three ifferent times, the TOI is the mile observation. Step - Determine magnitue of position vector at TOI The vector equation of the position vector is Satellite SlantRange LOS + Observer (Main-1)... an the magnitue of this equation, obtaine by performing the ot-prouct on itself, is... Satellite Satellite Satellite SlantRange LOS LOS + Observer Observer Satellite SlantRange LOS LOS SlantRangeLOS + Observer + Observer Observer (Main-) Formulate a moel of the motion of the satellite, an combine it with the equation above to etermine the magnitue of the position vector at the TOI. Laplace: Uses Lagrange-Interpolations an two-boy motion as a moel Gauss: Uses the series forms of the f an g Functions to estimate the motion of the satellite. Step 4 - Determine magnitue of slant-range vector(s) Calculate the magnitue of the istance from the observer to the satellite. Laplace: The slant-range will be calculate only for TOI Gauss: The slant-ranges will be calculate for all of the observation times Step 5 - Determine the full position vector(s) Calculate the the position vector(s) for the satellite relative to the reference frame selecte Laplace: Only the vector for the TOI is calculate Gauss: The vectors for all of the position vectors are calculate Step 6 - Determine the velocity vector at TOI Laplace: This metho extens the approach taken in etermining the position vector to calculate the velocity vector at the TOI Gauss: At this point, any of numerous methos can be use to calculate the velocity at the TOI using the positions an time etermine in the previous step. Three such methos are the Gibbs, Herrick-Gibbs, an Lambert-Universal Methos.

3 Angles-Only Orbit Determination Copyright 6 Michel Santos Page etermine in the previous step. Three such methos are the Gibbs, Herrick-Gibbs, an Lambert-Universal Methos. Step 7 - Improving solution Laplace: If more than three observations are available, improve the estimates for the Line-of-Sight vectors an its next two erivatives with respect to time (eg LOS, LOS Velocity, LOS Acceleration) by using higher-orer Lagrange Interpolations. Gauss: Improve the estimate for the motion of the satellite by using either higher-orer series forms of the f an g Functions, or now using analytical form of the f an g functions (Prussing an Conway (p. 18) Important assumptions of both methos The satellite moves within a plane in the inertial space Caution shoul be taken for special orientation cases. One such example is that of the observer being situate within the satellite's plane of motion uring all measurements. This causes the minimum of three LOS vectors to be linearly epenent (as two linearly-inepenent vectors are sufficient to escribe a plane.) This is iscusse in Escobal (p. 67) with further etails cite from Moulton.

4 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 4 Gauss Angles-Only Orbit Determination Derivation Gauss' angles-only metho of orbit etermination relies on a couple of characteristic assumptions: the satellite moves within an inertial plane the motion of the satellite within that plane can be moele by the f an g functions; the first iteration of this technique requires the low-orer series form of the functions; to improve the solution as suggeste in Prussing an Conway (p. 18), later iterations can make use of higher-orer forms of the series functions or the full analytical functions by using the results from the first iteration the angular observations (which form the LOS vectors) must be taken at ifferent times Some less istinctive assumptions are that: the origin of the inertial reference frame lies within the inertial plane the three LOS vectors o not lie within a single plane (this assumption is also true for the Laplace metho) Step 1 - Perform angular observations Assume complete Step - Select Time of Interest (TOI) Assume complete Step - Determine magnitue of position vector at TOI Since the satellite moves within an inertial plane, the three position vectors also lie within that plane. Therefore, there exists a linear combination of these position vectors such that their vector-sum is the zero-vector: c 1 r 1 + c r + c r (Gauss-1) The position vectors, r i, are of any length-imension esire The coefficients, c i, are imensionless We will insert into this equation the representation of the position vector, Eq. (Main-1), as the sum of the vector to the observer an the slant-range vector from the observer to the satellite. Satellite SlantRange LOS + Observer However, to make the later equations more succinct, we will make a change of variable names to shorten the equation. r ρ L + robs (Gauss-) Each of the vectors above are of a length-imension (eg meters, kilometers, astronomical units, etc.) The coefficient for the LOS vector, ρ, is imensionless. It will be scale appropriately by the length-imension of the LOS vector.

5 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 5 So, Eq. (Gauss-1) becomes... c 1 ρ 1 L 1 + r obs_1 + c ρ L + r obs_ + c ρ L + r obs_ Rearranging the equation results in... c 1 ρ 1 L 1 + c ρ L + c ρ L c 1 r obs_1 c r obs_ c r obs_ Rewriting the equation in matrix form results in... L 1 L L c 1 ρ 1 c ρ c ρ r obs_1 robs_ robs_ c 1 c c Let... L mat L 1 L L L 1x L 1y L 1z L x L y L z L x L y L z... an... r obs r obs_1 robs_ robs_ Leaing to... c 1 ρ 1 c ρ c ρ c 1 ρ 1 c ρ 1 L mat r obs c ρ c 1 c c... an finally to values for slant-ranges alone... ρ 1 ρ ρ c 1 c c 1 1 L mat r obs c 1 c c (Gauss-b) We wish to focus only on the equation for the mile slant range, ρ. As an intermeiate step, consier the matrix forme from the multiplications the inverse LOS matrix, L mat, with the site observation matrix, R obs. Consier its prouct as compose of 9 elements in a x matrix. 1 M L mat r obs M 11 M 1 M 1 M 1 M M M 1 M M

6 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 6 This allows a specific inspection of only the slant range for the mile slant range, ρ. ( ) 1 ρ M c 1 c 1 M c M c (Gauss-) Re-expressing Eq. (Main-) aroun the mile observation an with shorter variable names. r ρ L L + ρ L r obs_ + r obs_ robs_ Inserting Eq. (Gauss-) into the equation yiels... 1 r M c 1 c 1 M c M c What remains to be foun are the values of the coefficients, c i. The first step in etermining their values are to take the cross-prouct of the first an thir position vectors with Eq. (Gauss-1), assuming that c oes not equal, to get the two equations... r 1 r ( c 1 ) r r c Since there is some freeom in etermining the values for these coefficients, let c -1, an the other two coefficients become... r c 1 r 1 ( ) r r ( )... an an... 1 L L ( M c 1 c 1 M c M c ) L + r obs_ + r obs_ r 1 r ( c ) r 1 r c r 1 c r 1 r r ( ) robs_ (Gauss-4) We have now exhauste the utility of first primary assumption for the Gauss metho: that the satellite moves within an inertial plane. We now move on to using the secon primary assumption: that the f an g functions to approximate the motion of the satellite within the inertial plane. The f an g functions are functions of a reference position, velocity, an time. They are use to etermine the position an velocity of a satellite at some other time. In this case, the reference position to be use is that of the satellite's mile position, r, which we are still in the process of etermining. So, the position vectors for r 1 an r are... r 1 f1 r + g 1 v... an... r f r + g v Since the position vectors have imensions of Length an the velocity vectors have imension of Length over Time, observe that: the f function is imensionless the g function has imension of Time

7 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 7 The values for the first an thir position vectors are inserte into the equations for coefficients, c 1 an c. r f r + g v c 1 f 1 r + g 1 v f r + g v f 1 r + g 1 v r c f 1 r + g 1 v f r + g v g f 1 g f g 1 g 1 f 1 g f g 1 As escribe in Vallao's 1997 (p. 9), if we knew the position an velocity vectors at the mile observation, the f an g functions coul be applie to fin the values for the two coefficients, c1 an c. However, lacking this information we must use the low-orer series form of the f an g functions f i 1 τ r... an... g i τ 1 τ 6 r... where τ in this case is... τ i t t This leas to an approximation for the coefficients c 1 an c. τ 1 τ τ τ 1 c 1 + a τ τ a 1u τ r τ 1 r τ 1 1 τ τ τ 1 c a τ τ a u τ r τ 1 r Vallao (1997 p.55) emphasizes that, "Of course, truncation limits the time interval over which this approach is useful." This limite shorter time interval is equivalent to a shorter traversal of the orbital path. We can now finally place these newly etermine coefficients into Eq. (Gauss-4). r M 1 c 1 M + M c r ( ) L M 1 a 1 + a 1u M + M a + a u r r... M 1 a 1 + a 1u M M a + a u + + L r r obs_ + r obs_ r ( ) τ ( ) τ 1 L + ( M 1 c 1 M + M c ) L r obs_ + r obs_ robs_ L L +... robs_

8 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 8 Multiplying both sies of the equation by r 6 8 r... M 6 1 a 1 r + a 1u r 6 6 M r + M a r + a u r 6 + L r obs_ + r robs_ r obs_ 8 r... + ( M 1 a 1u + M a u ) ( M 1 a 1 M + M a ) L L + M ( 1 a 1u + M a u ) L r obs_ r ( M 1 a 1u + M a u ) + L L Letting... M 1 a 1 r + a 1u M r + M a r + a u ( ) L M 1 a 1 M + M a L L ( M 1 a 1 M + M a ) L r obs_ + r obs_ r M 1 a 1 M + M a M 1 a 1u + M a u We finally get an eigth-orer equation for the magnitue of the mile-observation position vector. 8 r 1 L L + 1 L robs_ + r obs_ robs_ r 6 1 L L + L robs_ + r + L L 8 r 1 L L + 1 L robs_ + r obs_ robs_ r 6 1 L L + L robs_ r L L Then a value for the magnitue of the mile position vector, r, is etermine by some means (i.e. numerical, graphical, etc.). Caution must be taken when etermining the value of r that satisfies the equation (Pines). Step 4 - Determine magnitue of slant-range vectors Having calculate the magnitue of the position vector, return to the earlier calculations for the coefficients, ci. Use the low-orer series form of the f an g functions for this initial estimate. Recall, that the coefficient for c has alreay been fixe at -1. τ 1 τ τ τ 1 c 1 + a τ τ a 1u τ r τ 1 r ( ) τ ( ) τ 1 τ 1 1 τ τ τ 1 c a τ τ a u τ r τ 1 r ρ 1 ρ ρ c 1 c c 1 1 L mat r obs c 1 c c (Gauss-b)

9 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 9 Step 5 - Determine the full position vector(s) Calculate Eq. (Gauss-) for the mile observation. r. ρ L + r obs_ Step 6 - Determine the velocity vector at TOI Gibbs metho (Vallao 1997 p. 414) Calculate the mile observation's velocity Z 1 r 1 r Z r r Z 1 r r 1 α cop 9eg acos Z r 1 Z r 1 α 1 acos α acos r 1 r r 1 r r r r r N r 1 Z + r Z 1 + r Z 1 D Z 1 + Z + Z 1 ( ) r 1 S r r B D r ( ) r ( ) r + r r 1 + r 1 r L g ND v L g r B + L g S

10 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 1 Step 7 - Improving solution ELORB Algorithm (Vallao 1997 p.146) Calculate the semi-parameter, p h r v ζ ( v ) r e a ( v ) r ζ ( ) v r r v p a 1 e ( ) Use the full f an g functions to estimate the location of the position vectors at the first an thir observations (Vallao 1997 pp ) f 1 1 r 1 p ( ( )) 1 cos ν 1 f 1 r p ( ( )) 1 cos ν ( ) r 1 r sin ν 1 g 1 p ( ) r r sin ν g p g c 1 f 1 g f g 1 g 1 c f 1 g f g 1

11 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 11 ρ 1 ρ ρ c 1 c c 1 1 L mat r obs c 1 c c (Gauss-b)... continue looping through from Step 5 to her until the slant-ranges "converge".

12 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 1 Laplace Angles-Only Orbit Determination Derivation Laplace' angles-only metho of orbit etermination relies on a couple of characteristic assumptions: the satellite moves within an inertial plane the motion of the satellite within that plane can be moele by a two-boy gravitational moel there are a minimum of three LOS vectors (forme by three angular measurements) which must be taken at ifferent times the LOS vector at any time can be etermine by using the Lagrange Interpolation Formula to interpolate an extrapolate vectors base on the observations mae the velocity an acceleration of the the LOS vectors are etermine by ifferentiating, with respect to time, the Lagrange Interpolate values for the LOS vectors A less istinctive assumptions is that the minimal, three LOS vectors o not lie within the inertial plane iscusse in the as the first assumption above. This is iscusse further below. Given at least three observations vectors, this metho is use to estimate the mile position an velocity vectors. Step 1 - Perform angular observations Assume complete Step - Select Time of Interest (TOI) Assume complete Step - Determine magnitue of position vector at TOI The first equation to be iscusse is the representation of the position vector, Eq. (Main-1), as the sum of the vector to the observer an the slant-range vector from the observer to the satellite. Satellite SlantRange LOS + Observer However, to make the later equations more succinct, we will make a change of variable names to shorten the equation. r ρ L + robs (Laplace-1) Each of the vectors above are of a length-imension (eg meters, kilometers, astronomical units, etc.) The coefficient for the LOS vector, ρ, is imensionless. It will be scale appropriately by the length-imension of the LOS vector. Taking the ot-prouct of Eq. (Laplace-1) with itself aroun the mile observation yiels... r ρ L L + ρ L r obs_ + r obs_ robs_ (Laplace-)

13 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 1 We are trying to solve for the magnitue of the mile position vector, r. We alreay know the LOS vector an the observer's location, r site_, uring the mile observation, t. What remains to be foun is the magnitue of the mile slant-range vector, ρ. The first step in attaining this is to ifferentiate Eq. (Laplace-1) with respect to time, twice, to obtain velocity an acceleration vectors for the mile observation vectors. t r t ρ L ρ + L t t r + obs (Laplace-a) r t ρ L t t ρ t L + + ρ L + t r obs t (Laplace-) The primary assumption of a two-boy gravitational moel means that the acceleration of the satellite is... r r t r r r r r ρ L + robs This is equation for the acceleration is inserte into Eq. (Laplace-)... r ρ L + robs ρ L ρ + L + ρ L + t t t t r obs t Rearranging yiels... ρ L + ρ L + ρ t t t Formulate this equation in matrix form, as escribe by Vallao (1997 p. 9). (The following set of equations are borrowe from Vallao's erivations. L L t The values of the slant-range vectors can be calculate by assuming that the magnitue of the position vector, r, is known. Subsequently, this permits the calculation of the slant-range values to be etermine by use of eterminants. Consier the eterminant of the left-han sie matrix D L L t L + t r L L + t r L L + t r L ρ t t ρ ρ r obs t r r obs r obs t r r + obs (Laplace-4)

14 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 14 "Do some column reuction by subtracting /r times the first column from the thir. This will reuce the complexity slightly. Factoring out the from the secon column... yiels... D L t L L t (Laplace-5) Now, apply Cramer's Rule to Eq. (Laplace-4) by also using Eq. (Laplace-5) to solve for ρ Dρ L L r t obs t r r + obs L t L Split the eterminant of the right-han sie of the equations. r obs t r r + obs Dρ L t L r obs t r L t L r obs (Laplace-6) Let... D 1 L t L r obs t... an... D L t L r obs (Laplace-7) Finally, etermine the value of the slant-range... ρ D 1 D r D D (Laplace-8) Escobal (p. 67) iscusses the possibility that the eterminant, D, may be zero. When use with the Lagrange Interpolation Formula, the eterminant, D, is actually a multiple of "the volume of a tetraheron forme by the [attracting boy] an the three positions of the satellite with respect to the [observer]. The volume of this tetraheron will therefore vanish only if the three positions of the satellite as seen from the [observer] lie on the arc of a great circle." However, this is not a eal-breaker for the Laplace metho as can be viewe by noting Eq. (Laplace-6). Its left-han sie woul be, leaing to... r L L t L t L r obs r obs t However, assuming that the eterminant, D, was not zero, insert Eq. (Laplace-8) into Eq. (Laplace-) to get... r Multiply this equation by r 6 8 r D 1 D r D D D 1 D r D D D 1 L L D D r D L + r obs_ + r obs_ D 1 L L D r 6 D 6 + r D L r obs_ + r obs_ robs_ r robs_

15 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 15 Collecting terms of r 8 r D 1 D 1 4 D L L 4 L D r obs_ + r obs_ robs_ 6 D 1 D D r 8 D L L 4 L D r + obs_ r 4 D + D L L 8 4D 1 r D L 4D 1 r obs_ D L L r obs_ robs_ 6 4D + r D L 8D 1 D r obs_ D L L 4 D + r D L L (Laplace-9) Still unknown are the rates-of-change of the LOS vectors which are embee in the eterminants D 1, D, an D. We will now make use of the primary assumption that the LOS vectors can be use with the Lagrange Interpolation formula to approximate the LOS vectors at any other time, an therefore to approximate the rates-of-change of this vectors. The general formulation of the Lagrange Interpolation Formula is, accoring to Vallao (1997 p. 89)... r( t) n i 1 r i k_not_i t t k t i t k Applying this formula to the LOS vectors... ( ) ( t t ) ( ) ( t 1 t ) t t L( t) t 1 t L 1 + ( t t 1 ) ( t t ) ( t t 1 ) ( t t ) L + ( t t 1 ) ( t t ) ( t t 1 ) ( t t ) L t L ( t) t t t ( ) ( t 1 t ) t 1 t L 1 + t t 1 t ( t t 1 ) t t ( ) L + t t 1 t ( t t 1 ) t t ( ) L L( t) t t 1 t ( ) ( t 1 t ) L 1 + ( t t 1 ) t t ( ) L + ( t t 1 ) t t ( ) L Vallao iscusses (1997 p. 89) that "if more observations are available, we can estimate the first an secon erivatives [of the LOS vectors] much more accurately by using Lagrange's interpolation formula with these aitional observations." In fact, Escobal points this out as being essential (p. 6, p. 65) "This actually must be one when... higher orer erivatives are not of negligible orer... for near-earth satellites, the metho of Laplace must be moifie consierably by the inclusion of higher-orer erivatives in L." These interpolate values for the LOS vectors, their velocities, an their accelerations are use in the calculation of Eq. (Laplace-6), Eqs. (Laplace-7), an those results are plugge into Eq. (Laplace-9). Then a value for the magnitue of the mile position vector, r, is etermine by some means (i.e. numerical, graphical, etc.). Caution must be taken when etermining the value of r that satisfies the equation (Pines).

16 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 16 Step 4 - Determine magnitue of slant-range vector Calculate the slant range magnitue by using Eq. (Laplace-) D 1 ρ D D D r Step 5 - Determine the full position vector(s) Calculate Eq. (Laplace-1) for the mile observation. r. ρ L + r obs_ Step 6 - Determine the velocity vector at TOI The final step involves etermining the velocity. Applying Cramer's Rule to Eq. (Laplace-) yiels... D ρ t ( L) Splitting the eterminant: D ρ t ( L) r obs t r r + site r obs t L t L t r ( ) L r obs L t Let... D ( L) r obs t L t... an... D 4 ( L) r obs L t t ρ D D 4 D r D Insert the above equation into Eq. (Laplace-a) to calculate the velocity vector t r t ρ L ρ + L t t r + obs

17 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 17 Comments The Line-of-Sight vector has a length imension Consier the following vector representing the the position of a satellite. It is etermine by aing the vector of the observer to the slant-range vector. Satellite SlantRange LOS + Observer It is imperative that the LOS vector have imensions of length. Otherwise, the above equation is illogical. Derivations of the Laplace an Gauss methos leave this out uner the unerstate assumption that all measurements an constants, which involve a length imension, have been rescale. Specifically, they have been rescale such that the magnitue of the LOS vector is one length unit long; it oes not matter what the length-unit is. Consier the following vector from the center of the Earth to the center of the moon. Moon km Uner the typical erivation, if the LOS vector is to have a imension of one Earth Raii, then the Moon Vector must be re-expresse as... Moon ER Similarly, the effect of this erivation also affects other variables containing imensions of length. One such example is Earth's gravitational parameter... Earth km s However, uner the typical erivation, if the LOS vector is to have a imension of one meter, it woul have to be rescale as.. Earth m s The effect of the LOS vector having a magnitue of one unit-length is evient when the LOS vector is ot-proucte with itself. LOS LOS 1 Length_Dimension This term occurs frequently in the Gauss an Laplace erivations. If all other variables have been scale appropriately, then this term can be roppe. On an acaemic level, by oing so, the equations lose the correct imensions. On a practical level, if one oes not want to rescale all of their measurements an constants, then this term must not be left out. As a slight counterpoint to this argument, it is often practical to perform calculations in one set of length, time, an mass units while using the Angle-Only methos. In other wors, a user will often rescale their measurements an constants anyway. However, the erivations in this ocument at least provie a user the freeom to select the units that he or she esires. On on another practical level, if one uses a tool such as Mathca, then the tool keeps track of the units an makes the life of user

18 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 18 On on another practical level, if one uses a tool such as Mathca, then the tool keeps track of the units an makes the life of user much easier. The user simply has to use the correct, full-imensional formulas an allow the tool to keep track of the scaling. Caution to be taken when etermine the roots of the eigth-orer equations for position vector magnitue There will be eight possible solutions. Six of the eight will be three sets of complex conjugate roots. If one root is positive, an the other is negative, then the positive root is answer. If both roots are negative, then there is no physical solution. If both roots are positive then: use the value for the position vector's magnitue to etermine the slant-range magnitue. Plug the slant-range magnitue into Eq. (Main-1) an confirm that the magnitue of this newly compute position vector matches up with the previously obtaine magnitue. If only one matches, then that one is the answer. If both roots match the above result, which is "rare," then take these values an stick them into the equations for the velocity vector to hopefully eliminate one of the two.

19 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 19 Summary A re-erivation of both the Gauss an Laplace methos for Angles-Only Initial Orbit Determination was presente. Dimensional analysis of the equations was iscusse. Critical assumptions use in the erivation of both methos were presente to point out potential pitfalls an limitations. Whereas the Gauss technique estimates a trajectory which matches equally well with all of the observations, the Laplace technique focuses on forming a goo solution for the central observation. The Gauss metho can appropriately be use for angular ata less than 6 egrees of an orbital trajectory, but best suite for angular separation less than 1 egrees (Long et a 9-9l). In contrast, the Laplace technique oes not work as well for near-earth satellites (Healy Lecture 5) however it has been use extensively to to etermine the orbits for comets an asterois. This angular separation can signify very ifferent elapse times ue to the time with which it may take a satellite to traverse that angular arc. The outer planets, for example will take much longer to traverse 1 egrees of their orbital trajectory compare to the inner planets.

20 Angles-Only Orbit Determination Copyright 6 Michel Santos Page References R. Bate an Donal Mueller an Jerry White, Funamentals of Astroynamics, Dover Publications, Inc., Toronto, Canaa, 1971 D. Pines, Class notes from Interplanetary Navigation an Guiance, University of Marylan - College Park, October 1, P.R. Escobal, Methos of Orbit Determination, Robert E. Krieger Publishing Company, Malabar, Floria, USA, 1965 L. Healy, Class notes from Space Navigation an Guiance, University of Marylan - College Park, Fall, Long, Anne C. et al., Goar Trajectory Determination System (GTDS) Mathematical Theory (Revision 1). FDD/55-89/1 an CSC/TR-89/61, Goar Space Flight Center: National Aeronautics an Space Aministration, 1989 O. Montebruck an Oberhar Gill, Satellite Orbits, Springer Verlag, F.R. Moulton, An Introuction to Celestial Mechanics, The Macmillan Company, New York, 1914 Prussing an Conway, Orbital Mechanics, Oxfor Press, New York, 199 D. Vallao an Wayne D. McClain, Funamentals of Astroynamics an Applications (1st Eition), McGraw-Hill, New York, 1997 D. Vallao an Wayne D. McClain, Funamentals of Astroynamics an Applications (n Eition), (Microcosm Press, El Seguno, California) an (Kluwer Acaemic Publishers, Dorrecht), 1

21 Angles-Only Orbit Determination Copyright 6 Michel Santos Page 1 Miscellaneous items ER km 844 Moon km Earth km s