Astrodynamics 103 Part 1: Gauss/Laplace+ Algorithm

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1 Astrodynamics 103 Part 1: Gauss/Laplace+ Algorithm unknown orbit Observer 1b Observer 1a r 3 object at t 3 x = r v object at t r 1 Angles-only Problem Given: 1a. Ground Observer coordinates: ( latitude, longitude, altitude ) at t i or 1b. Space Observer position vectors: ( x, y, z ) with respect to Sun/Earth at t. Angles data of unknown object: ( Right ascension, Declination ) at t object i at t 1 i = 1,, 3 i Sphere/ Spheroid AMOS September 14, 01 Find: x is the range at t giving r, v, and ( TLE if desired ) of the unknown orbit Gim Der DerAstrodynamics July 1, 013

2 Analytic Astrodynamics Overview Astrodynamics 101: Kepler+ Algorithm Part1: Analytic Prediction Algorithms Part: Verifications Astrodynamics 10: Lambert+ Algorithm Part1: Analytic Multi-revolution Targeting Algorithms (Orbit Determination for Radar Data) Part: Verifications Astrodynamics 103: Gauss/Laplace+ Algorithm Part1: Analytic Angles-only Algorithms (Orbit Determination for Optical Sensor Data) Part: Verifications

3 Astrodynamics 103: Gauss/Laplace+ Algorithm Part 1. Analytic Angles-only Algorithms 1. What, Why, How. Physics 3. New Angles-only Algorithms 4. Details and Hints 5. Applications for SSA Part. Verifications 6. Numerical Examples

4 How Close Will It Get? Feb 15, 013 Asteroid Earth DA14 GEO Moon Planet Asteroid and Space Debris Collisions Happened and Will Happen Again

5 Range Guessing Current Angles-only Algorithms can process 10% or less of current Space Catalog ~,500 GEO and deep space objects GEO and Deep-Space Objects, if guessed correctly

6 Using optical sensors for 10% of cataloged space objects is not good enough! Range Solving 90% of the space objects is Near-Earth Gauss/Laplace+ can process optical sensor data for 100% of cataloged and unknown space objects in any Orbit Regime

7 Bigger Means Better? The European/World s Biggest Optical Telescope 39.3-metre dish, to be built in Chile by 0 Size Matters But Powerful Angles-only Algorithm Matters More

8 1. What, Why, How What? Why? How? Cataloging of 100,000+ space debris and satellites is a challenging Space Situation Awareness (SSA) problem Need new Angles-only algorithm to process and catalog efficiently millions optical sensor measurements of known and unknown objects Understand the Physics and Mathematics of Astrodynamics for orbit determination using angles-only data Building New Space Catalog of 100,000+ Objects Requires Analytic Gauss/Laplace+ Algorithm

9 Range Guessing (out of thin air) x Gauss or Laplace or Double-r -Body "known" orbit Observer 1b Observer 1a Improve with f and g r 3 Angles-only and Gauss/Laplace+ Algorithms x Angles-only Problem Range Solving Gauss/Laplace Given: 1a. Ground Observer coordinates: th 8 deg. Polynomial Equations ( latitude, longitude, altitude ) at t i or 1b. Space Observer position vectors: x ( x, y, z ) with respect to Sun/Earth at t Find: object at t 3 = r v. Angles data of unknown object: ( Right ascension, Declination ) at t x is the range at t, giving r, v and ( TLE if desired ) of the unknown orbit object at t r 1 object at t 1 unknown orbit Observer 1b Observer 1a i Gauss and Laplace and Double-r add perturbations analytically ( J, J, J, object at t J, J 31, Sun, Drag,.. ) i, i = 1,, 3 r 3 x = r v object at t r 1 object at t 1 Sphere Angles-only (-Body) solution (inaccurate but fast) Spheroid Gauss/Laplace+ solution for SSA (accurate and fast)

10 . Physics Physics and Primary Objective

accuracy th no use for the Gauss/Laplace 8 degree

11 Ranging with Old Optical Telescopes 18 th Century Optical Telescopes: Poor angles measurement accuracy th no use for the Gauss/Laplace 8 degree polynomial range equations FOV Resolution: Poor ~ 1.0 degree

12 1.0 degree Why Optical Sensors and Telescopes? Field-of-View Comparison (typical) Radar / Laser (very small FOV) 1.5 degree Optical Sensor (bigger FOV) Optical Sensor provides much larger FOV and angles data but no range. Until now Angles-only algs cannot consistently find range Optical Sensor is at least 10x cheaper to build and operate than comparable Radar or Laser systems

Celestial Equator Object Declination d d Vernal equinox a Right

13 Angles Data: Right ascension and Declination North Celestial Pole Input angles ( 3 sets ): a, d i, i = 1,, t i i Celestial Equator Object Declination d d Vernal equinox a Right ascension a The problem: Angles-only, no range! Optical Detection (D obs data)

14 The Classical Angles-only Problems Finding range of any space object by optical sensor A Non-Moving Object A Moving Object (a star) (a satellite or asteroid) B p x A star unknown orbit Observer r 3 object at t 3 v x = object at t r object at t 1 Earth (Jan) A 1 AU Sun B Earth (July) Earth r 1 Parallax: x = 1 AU p 00-year Angles-only riddle: Pick the correct range, x

15 Ranging with Modern Optical Sensors GEODSS Space Surveillance Telescope SST Accurate FOV Resolution: angles arc-seconds measurements sub- arc-second Very accurate angles measurements Gauss/Laplace+ Algorithm (Range Solving IOD) Cataloging to Near-Real-Time SSA

16 Primary Objective of Angles-only Algorithms Range, Range, Range 1. Given optical detection (D obs angles data). Innovative Range Solving Gauss/Laplace 8 th degree polynomial equations for any known or unknown object in any orbit regime 3. New analytic Angles-only Algorithm for fast and accurate IOD, TLE, Cataloging,.., SSA

17 3. New Angles-only Algorithms Angles-only Algorithms for Initial Orbit Determination

18 Incomplete Textbook Solutions Moulton, 1904 since Gauss 1801,... and in spite of hundreds of papers on orbit determination, very little is new... Prussing and Conway, 1993 The most significant of which is probably the solution of the eight-degree polynomial in the unknown range ( x ) f(x) = x a x b x c = 0 Vallado, multiple roots may exist... Selecting the correct root can be very difficult and many others Textbook Solutions: Multiple Roots Correct Range not found

19 Classical Gauss/Laplace Angles-only Problem unknown orbit Observer r 3 object at t 3 v x = object at t r r 1 object at t 1 Gauss / Laplace Angles-only Problem Input: 1. Observer coordinates: ( latitude, longitude, altitude ) at t. Angles data of unknown object: ( Right ascension, declination ) at t i = 1,, 3 Accurate angles are required to give accurate coefficients: a, b, c i i Sphere/ Spheroid th Gauss/Laplace 8 degree polynomial equations: f(x) = x a x b x c = 0 Output: x is the range at t, which gives ( r, v ) and TLE set of the unknown orbit

20 Guidelines for Angle/Time Spacing Normal object time spacing within an arc of the orbit Unknown Orbit object at t 3 object at t r 3 x = r object at t 1 Unknown Orbit 1 revolution object at t 3 Improper Obs data spacing: 1. Closely spaced data (Time span too close) Observer r 3 r r 1 object too close at t and t 1 Observer r 1 Given: * 3 sets of angle t a i i d i i = 1,, 3,.. * 1, and/or 3 sets of Observer coordinates Unknown Orbit revolutions r Observer object in first rev at t and t 1 r 1 Proper Obs data spacing: (1) Geocentric: minutes apart (max: 100 minutes) () Heliocentric: days or months apart (max: to be determined) object in second rev at t 3 r 3. Multi-revolutions data (Time span too far apart)

21 Guidelines for Total Time Span unknown orbit Observer r 3 object at t 3 x = r v t r 1 total time span object at t 1 t 3 t (Reasonable total angle spacing) 1 Sphere/ Spheroid Total Time Spacing for any Geocentric object in any orbit regime: 0 to 40 minutes

22 Step 1 of New Algorithm: Range Solving unknown orbit Observer r 3 object at t 3 x = r v t r 1 object at t 1. Proper spacing total time span t 3 t 1 1. Advanced optical sensors accurate angles Accurate coefficients: a, b, c Gauss/Laplace 8 th degree polynomial eqns: Sphere/ Spheroid Picking the correct root (range), x Unknown -body Keplerian orbit in terms of ( r, ) v

23 Step of New Algorithm: Adding Perturbations Kepler + solution (Vinti and other perturbations) (3) Vinti spheroidal solution only Numerical accurate solution (desired) V (t ) () Kepler solution with (other perturbations) a d General Equations of motion: d R d t = R 3 R a d, a d = 0 R (t) (1) Kepler solution only General method of solution => Kepler+ solution (1) Kepler solution only () Kepler + a d perturbed solution (3) Vinti spheroidal solution only Central Body R (t ) 0 V (t ) 0 object at t 0

24 New Angles-only IOD Algorithm unknown orbit Observer r 3 object at t 3 v x = r Sphere/ Spheroid object at t r 1 Gauss/Laplace f(x) = x a x b x c = 0 object at t 1 th Range solving ( The 00-year Riddle to find one correct root ) deg poly eqn: Range guessing ( Multiple or no solution ) 1. The Correct Root (-body or Keplerian) New angles-only algorithms find the correct root consistently without guesswork in a few micro-seconds on a computer. Fast and Accurate ( Kepler + ) Add analytically perturbations ( J, J, J, Sun/Moon, Drag,.. ) 3 4 resulting in accurate solution for any object in any orbit regime

25 Perturbations Compliant Gauss/Laplace+ Classical Kepler (-Body) Astrodynamics 101 Vinti (J, J3, J4 included) Kepler+ (J, J3, J4 and other perturbations)) Classical Lambert (-Body) Astrodynamics 10 Targeting by Kepler+ Lambert+ (J, J3, J4 and other perturbations)) Classical Gauss/Laplace (-Body) Astrodynamics Targeting by Kepler+ and Lambert+ Gauss/Laplace+ (J, J3, J4 and other perturbations))

26 4. Details and Hints Insight To New Angles-only Algorithm

27 Myths of Angles-only Algorithms 1. The straight edge of the Ocean is difficult to see (illusion vs knowledge) 1. Myth. The range of a very faraway Star is difficult to observe (perception vs knowledge) 3. The correct roots of Gauss and Laplace equations are difficult to select (guessing vs knowledge) 4. Analytic Kepler+ solutions with perturbations are difficult to believe (ignorance vs knowledge). Myth 3. Myth 4. Myth

28 The Correct Root of the 8 th Degree Poly. Eqn. Gauss/Laplace New Angles-only: It is all about picking the range, x f(x) = x a x b x c = 0, x is the range at t Simple facts: (1) For real trajectory, x must be a positive real root of f(x) () Only three terms can change signs, Descartes' Rule of Signs gives at most three real roots [1, ] (3) c is always negative, only values of a and b can change Unknown Orbit Observer r 3 object at t 3 x = r t a i i d i i = 1,, 3,.. r 1 object at t object at t 1 Reducing 8 roots to at most 3 positive real roots References: 1. Charlier, C. V. L., On Multiple Solutions in the Determ... From three obs., RAS, 1900&1911. Plummer, H. C., An Introductory Treatise on Dynamical Astronomy, Dover, 1918

29 Descartes Rule of Signs Down to either 1 or 3 positive real root(s) The second theorem of Descartes Rule of Signs implies that positive real roots cannot exist If a root solver is used, then you will not know how to pick the correct root when there are 3 roots

30 Determining the positive real root(s) Angles-only 8th degree polynomial (range) equation of Gauss/Laplace f(x) = x a x b x c = 0, x is the range at t Unknown Orbit object at t 3 f(x) f(x) not in closed-form, iterative method required x = i r [ 1 + ( i 1 ) /10 ] Earth i = 1,,... object at t object at t 1 x 1 one solution: f(x) = 0 x Observer three solutions: f(x) = 0 Need to pick! The 00-year angles-only IOD problem is solved: The correct root of f(x) is found without guesswork

31 Gauss Geometric Method Transform: f(x) = x a x b x c = 0 r = R + object at t 4 to: f( ) = sin M sin ( m = 0 M and m are computed from a, b and c Gauss: Determine the correct root r > R > r Earth and < (180 ) < 90 o o Observer R L Earth r Unknown Orbit r sin r = ECI position vector to object R = ECI position vector to observer = range vector from observer L = Line of sight vector from observer = = sin ( + R sin Hint: Laplace agreed? The correct positive real root can be identified

32 Double-r Algorithms (Der vs Gooding ) Der range solving th Gauss 8 - degree polynomial equation Unknown Orbit Observer r 3 object at t 3 x = r r 1 object at t object at t 1 Gooding range guessing 1. r and r estimates from range solving instead of range guessing 1 3. r, r and r are optimized instead of assumed with no errors r by Gibbs gives direct or retrograde instead of direct and retrograde 4. Analytic Kepler + perturbed solution instead of -Body solutions Double _ r (Der) Lambert + and Kepler + _ Double r (Gooding) Lambert and -Body 5. One (near SP) solution versus Multiple (-body) solutions

33 Angles-only Algorithms (Der vs Others ) 1. Start: Range solving (No initial guess needed and no Newton iterative) 1. Start: Range guessing (Laplace/other methods to initiate and Newton iterative). Separated root resolution o 0 < < 90 (Ground, up) o o 90 < < 180 (Space, down) 3. Input data accuracy & req. (5+ significant digits for error-free, sparse data only) 4. Output Real orbit & motion rd (include: J, J, J, 3 body,.. ) Tests and comparison (00+ objects, all orbit regimes, test cases, simulated & real data, Accuracy: ASTAT-, 100%, ASTAT-1, 90% + ) 6. One Accurate solution. Confused roots o 0 < < 180 (up or down?) 3. Input data (No accuracy definition, sparse and/or dense data) 4. Output -body orbit & motion (Perturbations: none) 5. Tests and comparison ( lucky to find one, what accuracy? ) 6. Multiple guessed solutions

Unsuitable for real-time automatic processing Orbit Determination and Prediction/Propagation Osculating Orbital Elements at t o [ r (t ), v (t )] o Radars / Optical sensors o Future look angles (

34 Unsuitable for real-time automatic processing Orbit Determination and Prediction/Propagation Osculating Orbital Elements at t o [ r (t ), v (t )] o Radars / Optical sensors o Future look angles ( pointing prediction ) Others Object at t r Ephemerides Raw Observation data v Orbit Determination (Estimated future/past) position and velocity vectors Rise/Set Site visibility Astrodynamics 10: Lambert / 103: Gauss/Laplace Initial Orbit Determination Processed Observation data Differential Correction Batch / KF TLE conversion difficulties Singularity difficulties OscMean SGP4 Close approach (miss distance) Analytic algorithms in OD and P/P need to process 100,000+ Objects in less than 1 hours with accuracy of 10 km to centimeters for SSA (Estimated initial) positin and velocity vectors Orbital element set r and v Prediction / Propagation 1 SP 3 Numerical Integration Osculating Orbital Elements at t [ r, v ] Astro 101: Kepler Kepler+ 1 3 SGP4 needs TLE conversion (not efficient for SSA) SP is accurate, but slow for SSA Kepler+ is accurate and fast for SSA

35 SSA Applications 5. Applications

Radar Data Multi-sensor Multi-object UCT Cataloging SF SF I Correlating 90+ % of objects to catalog J K Fence or Radar UCT processing Solve by New angles-only

36 Radar Data Multi-sensor Multi-object UCT Cataloging SF SF I Correlating 90+ % of objects to catalog J K Fence or Radar UCT processing Solve by New angles-only algorithm: Example: I = J = K = 100 Correlation combination = I J K = 1,000,000 Takes a few seconds for a million combinations 3 computed? ranges = 3 detected ranges

37 Optical Sensor Data Multi-sensor Multi-object UCT Cataloging SF SF Correlating 90+ % of objects to catalog UCT processing Example: I = J = K = 100 Correlation combination = I J K = 1,000,000 Takes a few seconds for a million combinations A computed orbit = Solve by New angles-only algorithm:? Orbit type,,,...

Gauss/Laplace+ for SSA New Angles-only IOD Algorithms provide Future

unknown object in any orbit regime New Analytic and Semi-analytic

for All Objects in the Space Catalog 10% 90% or more are Near-Earth

38 Gauss/Laplace+ for SSA New Angles-only IOD Algorithms provide Future Sensors new capabilities to Track, Search and UCT cataloging for any unknown object in any orbit regime New Analytic and Semi-analytic Astrodynamics Algorithms are Key Tools for SSA with Collision Prediction for All Objects in the Space Catalog 10% 90% or more are Near-Earth Optical Sensor Track and Search UCT Cataloging: 50,000+ objects Near Real-Time SSA 100,000+ objects

39 Next Astrodynamics 103 Part : Verifications for SSA (Also please download iorbit: and run gau for Astrodynamics 103 Verifications)

40 6. Numerical Examples w and w/o Input Errors Geocentric Objects More examples in iorbit application software Der, G. J., New Angles-only Algorithms for IOD, AMOS, 01 Readers should solve the example problems without errors first More examples can be provided upon request Examples of multiple sensors can be provided upon request

41 Notes on the Examples ANSWER in Numerical examples = Reference Data Simulated data from scenario generators GEODSS observation data (Geocentric objects) JPL DE405 and IAU data (Heliocentric objects) Input data Angles data > 5 significant digits => accurate Total time span between t1 and t3 reasonable Range Solving Objective Compute position ( r ) and velocity ( v ) at t Compare ( r, v ) of Answer against ( r, v ) of Gauss, Laplace, Double-r and Gauss/Laplace+

42 Example 1: LEO Object Input: (no errors in Right asc & Declination) One site, # 1: (lat, lon, alt) = ( deg, deg, 0.0 km) a Time (s) year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Output: DerAstro Algs r_eci (t) (km) v_eci (t) (km/s) Gauss (-Body) Laplace (-Body) Double-r (-Body) Gauss/Laplace ANSWER LEO object ( x = eci range(t) = km ) Coefs of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semi-major = 6704 km, ecc = 0.001, incl = deg, approx. alt = 30 (km)

43 Example : Sun Synchronous Debris Input: (no errors in Right asc & Declination) One site, # 1: (lat, lon, alt) = ( deg, deg, 0.0 km) a Time (s) year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Output: DerAstro Algs r_eci (t) (km) v_eci (t) (km/s) Gauss (-Body) Laplace (-Body) Double-r (-Body) Gauss/Laplace ANSWER MEO Sun-synchronous object (x = eci range(t) = 747. km) Coefs of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semi-major = 746 km, ecc = , incl = deg, approx. alt = 900 (km)

44 Example 3: MEO Iridium Satellite Input: (no errors in Right asc & Declination) One site, # 1: (lat, lon, alt) = ( deg, deg, km) a Time (s) year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Output: DerAstro Algs r_eci (t) (km) v_eci (t) (km/s) Gauss (-Body) Laplace (-Body) Double-r (-Body) Gauss/Laplace ANSWER MEO (Iridium SID 479) object (x = eci range(t) = km) Coefs of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semi-major = 715 km, ecc = , incl = deg, approx. alt = 900 (km)

45 Example 4: MEO Global Star Satellite Input: (no errors in Right asc & Declination) One site, # 1: (lat, lon, alt) = ( deg, deg, km) a Time (s) year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Output: DerAstro Algs r_eci (t) (km) v_eci (t) (km/s) Gauss (-Body) Laplace (-Body) Double-r (-Body) Gauss/Laplace ANSWER MEO (GlobalStar SID 516) object (x = eci range(t) = km ) Coefs of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semi-major = 7883 km, ecc = , incl = 5.01 deg, approx. alt = 1,400 (km)

46 Example 5: GPS Satellite Input: (Data with unknown errors in Right asc & Declination) One site, # 1: (lat, lon, alt) = ( deg, deg, km) a Time (s) year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Output: DerAstro Algs r_eci (t) (km) v_eci (t) (km/s) Gauss (-Body) Laplace (-Body) Double-r (-Body) Gauss/Laplace ANSWER GPS (SID 3575) object (x = eci range(t) = km) Coefs of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semi-major = 669. km, ecc = , incl = deg, approx. alt = 1,000 (km)

47 Example 6: Molniya Satellite Input: (Data with unknown errors in Right asc & Declination) One site, # 1: (lat, lon, alt) = ( deg, deg, km) a Time (s) year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Output: DerAstro Algs r_eci (t) (km) v_eci (t) (km/s) Gauss (-Body) Laplace (-Body) Double-r (-Body) Gauss/Laplace ANSWER Molniya (SID 09880) object (x = eci range(t) = km) Coefs of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semi-major = 7139 km, ecc = , incl = 63.7 deg, approx. alt = 8,000 (km)

48 Example 7: GEO Satellite Input: (Data with unknown errors in Right asc & Declination) One site, # 1: (lat, lon, alt) = ( deg, deg, km) a Time (s) year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Output: DerAstro Algs r_eci (t) (km) v_eci (t) (km/s) Gauss (-Body) Laplace (-Body) Double-r (-Body) Gauss/Laplace ANSWER GEO (SID 3018) object (x = eci range(t) = km) Coefs of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semi-major = 483 km, ecc = , incl = deg, approx. alt = 35,000 (km)

49 Examples with and without Angles Errors Heliocentric Objects More examples in DerAstrodynamics iorbit application software iorbit will allow you to build your own examples verifiable with JPL or IAU data Der, G. J., New Angles-only Algorithms for IOD, AMOS, 01 Visit JPL and IAU website, planets and minor planets angles data can be easily obtained for verification with the computed values of these examples. Need help? See the document, IAU Minor planet ephemeris data extraction procedures, which is part of the iorbit package. km to AU conversion, use DE405 value: 1 AU = 149,597, km

50 Example 8: Saturn Input: (no angular errors in Right asc & Declination) a Time (day) Year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Time (day) r_earth_sci (t), (km) source JPL DE405 t1 = t = t3 = Output: sci= Sun Centered Inertial DerAstro Algs. r_sci (t) (km) v_sci (t) (km/s) Gauss (-Body) Laplace (-Body) Gauss/Laplace ANSWER Helio object, Saturn ( x = sci range(t) = 1,450,559,81. km ) Coef of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semimajor = km, ecc = , incl = deg wrt Ecliptic

Example 9: Jupiter Input: (no angular errors in Right asc & Declination) a Time (day) Year mo day hr min sec Right asc, deg d Declination, deg t1 = 0 1990 1 13 0 0 0.0 94.

51 Example 9: Jupiter Input: (no angular errors in Right asc & Declination) a Time (day) Year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Time (day) r_earth_sci (t), (km) source JPL DE405 t1 = t = t3 = Output: sci= Sun Centered Inertial DerAstro Algs. r_sci (t) (km) v_sci (t) (km/s) Gauss (-Body) Laplace (-Body) Gauss/Laplace ANSWER Helio object, Jupiter (x = sci range(t) = 780,053,495. km) Coefs of Gauss 8 th deg. Polynomial Eqn: a =.869, b = , c = Estimated: semi-major = km, ecc = , incl = deg wrt Ecliptic

52 Input: (no angular errors in Right asc & Declination) a Time (day) Year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Time (day) r_earth_sci (t), (km) source JPL DE405 t1 = t = t3 = Output: Example 10: Ceres, Asteroid sci= Sun Centered Inertial DerAstro Algs. r_sci (t) (km) v_sci (t) (km/s) Gauss (-Body) Laplace (-Body) Gauss/Laplace ANSWER Helio object, Ceres (x = sci range(t) = 40,571,834. km) Coefs of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semi-major = km, ecc = , incl = deg wrt Ecliptic

53 a Time (day) Year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Time (day) r_earth_sci (t), (km) source JPL DE405 t1 = t = t3 = Output: Example 11: Jupiter (with large errors) Input: (Data with approximately > 0.% errors in Right asc and Declination) sci= Sun Centered Inertial DerAstro Algs. r_sci (t) (km) v_sci (t) (km/s) Gauss (-Body) Laplace (-Body) Gauss/Laplace ANSWER Helio object, Jupiter (x = sci range(t) = 675,968,435. km) Coefs of Gauss 8 th deg. Polynomial Eqn: a = , b = , c = Estimated: semi-major = km, ecc = , incl = deg wrt Ecliptic

54 a Time (day) Year mo day hr min sec Right asc, deg d Declination, deg t1 = t = t3 = Time (day) r_earth_sci (t), (km) source JPL DE405 t1 = t = t3 = Output: Example 1: Ceres, Asteroid (with small errors) Input: (Data with approximately 0.1% errors in Right asc and Declination) sci= Sun Centered Inertial DerAstro Algs. r_sci (t) (km) v_sci (t) (km/s) Gauss (-Body) Laplace (-Body) Gauss/Laplace ANSWER Helio object, Ceres (x = sci range(t) = 40,410,759. km) Coefs of Gauss 8 th deg. Polynomial Eqn: a = 8.370, b = , c = Estimated: semi-major = km, ecc = , incl = deg wrt Ecliptic

ON SELECTING THE CORRECT ROOT OF ANGLES-ONLY INITIAL ORBIT DETERMINATION EQUATIONS OF LAGRANGE, LAPLACE, AND GAUSS

AAS 16-344 ON SELECTING THE CORRECT ROOT OF ANGLES-ONLY INITIAL ORBIT DETERMINATION EQUATIONS OF LAGRANGE, LAPLACE, AND GAUSS Bong Wie and Jaemyung Ahn INTRODUCTION This paper is concerned with a classical