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Keplerian Orbit Elements to Cartesian State Vectors M.Eng. René Schwarz Master of Engineering in Computer Science and Communication Systems (M.Eng.) Bachelor of Engineering in Mechatronics, Industrial and Physics Technology (B.Eng.) Memorandum № 1 Keplerian Orbit Elements −! Cartesian State Vectors Inputs Outputs a traditional set of Keplerian Orbit Elements cartesian state vectors . Other – Semi-major axis a [m] – position vector r(t)[m] or [AU] m AU – Eccentricity e [1] – velocity vector r_(t)[ s ] or [ d ] – Argument of periapsis ! [rad] – Longitude of ascending node (LAN) Ω[rad] – Inclination i [rad] – Mean anomaly M0 = M(t0)[rad] at epoch t0 [JD] considered epoch t [JD], if different from t0 standard gravitational parameter µ = GM of the central body, if different from Sun (G…Newtonian constant of m3 gravitation [ kg·s2 ], M…central body mass [kg]) https://creativecommons.org/licenses/by-nc/4.0/ ), unless otherwise stated. 1 Algorithm 1. Calculate or set M(t): rene-schwarz.com ( a) If t = t0: M(t) = M0. 1 b) If t =6 t0: i. Determine the time difference ∆t in seconds with René Schwarz ∆t = 86 400(t − t0): (1) ©2017 M.Eng. This document is subjectLicense to (CC the conditions BY-NC 4.0), whichof the licenses can Creative may be Commons Attribution-NonCommercial apply examined for at 4.0 International particular contents or supplemental documents/files. ii. Calculate mean anomaly M(t) from r µ M(t) = M + ∆t (2) 0 a3 cbn · 20 · 10 m3 with µ = µÀ = 1:327 124 400 41 10 ( 1 10 ) s2 for the Sun as central body. Normalize M(t) to be in [0; 2π). 2. Solve Kepler’s Equation M(t) = E(t) − e sin E for the eccentric anomaly E(t) with an appropriate method numerically, e.g. the Newton–Raphson method2: f(E) = E − e sin E − M (3) f(E ) E − e sin E − M E = E − j = E − j j ;E = M (4) j+1 j d j − 0 f(Ej) 1 e cos Ej dEj 3. Obtain the true anomaly ν(t) from ( ) p E(t) p E(t) ν(t) = 2 · arctan2 1 + e sin ; 1 − e cos ; (5) 2 2 https://goo.gl/XB85LZ Errors, comments or ideas regarding! this paper? 1 Be aware that Orbit Elements change over time, so be sure to use one set of Orbit Elements given for a certain epoch t0 only for a small time interval (compared to the rate of changes of the Orbit Elements) around t0. 2 Argument (t) omitted for the sake of simplicity. Page 1 of 3 version: 2017/10/05 15:36 M.Eng. René Schwarz (rene-schwarz.com): Memorandum Series Keplerian Orbit Elements −! Cartesian State Vectors (Memorandum № 1) where arctan2 is the two-argument arctangent function 8 ( ) > y x > 0 >arctan ( x ) > y >arctan + π y ≥ 0; x < 0 <> ( x ) arctan y − π y < 0; x < 0 (y; x) = x : arctan2 > π (6) >+ 2 y > 0; x = 0 > >− π y < 0; x = 0 :> 2 undefined y = 0; x = 0 4. Use the eccentric anomaly E(t) to get the distance to the central body with rc(t) = a(1 − e cos E(t)): (7) 5. Obtain the position and velocity vector o(t) and o_ (t), respectively, in the orbital frame (z-axis perpendicular to orbital plane, x-axis pointing to periapsis of the orbit): 0 1 0 1 0 1 0 1 p − E ox(t) cos ν(t) o_x(t) µa p sin @ A @ A @ A @ 2 A o(t) = oy(t) = rc(t) sin ν(t) ; o_ (t) = o_y(t) = 1 − e cos E (8) rc(t) oz(t) 0 o_z(t) 0 6. Transform o(t) and o_ (t) to the inertial frame3 in bodycentric (in case of the Sun as central body: heliocentric) rectangular coordinates r(t) and r_(t) with the rotation matrices Rx(') and Rz(') using the transformation sequence r(t) = R (−Ω)R (−i)R (−!)o(t) z 0 x z 1 ox(t)(cos ! cos Ω − sin ! cos i sin Ω) − oy(t)(sin ! cos Ω + cos ! cos i sin Ω) oz (t)=0 @ A (9) =========== ox(t)(cos ! sin Ω + sin ! cos i cos Ω) + oy(t)(cos ! cos i cos Ω − sin ! sin Ω) ox(t)(sin ! sin i) + oy(t)(cos ! sin i) r_(t) = R (−Ω)R (−i)R (−!)o_ (t) z 0 x z 1 o_x(t)(cos ! cos Ω − sin ! cos i sin Ω) − o_y(t)(sin ! cos Ω + cos ! cos i sin Ω) o_z (t)=0 @ A (10) =========== o_x(t)(cos ! sin Ω + sin ! cos i cos Ω) +o _y(t)(cos ! cos i cos Ω − sin ! sin Ω) : o_x(t)(sin ! sin i) +o _y(t)(cos ! sin i) 7. In order to obtain the position and velocity vector r(t) and r_(t), respectively, in the units AU and AU/d, calculate r(t) r_(t) r(t) = ; r_(t) = : (11) [AU] 1:495 978 706 91 · 1011 [AU/d] 86 400 · 1:495 978 706 91 · 1011 2 Constants and Conversion Factors Universal Constants Symbol Description Value Source 4 · −11 m3 G Newtonian constant of gravitation G = 6:67428(67) 10 kg·s2 [2, pp. 686–689] Conversion Factors Conversion Source Astronomical Units → Meters 1 AU = 1:495 978 707 00 · 1011 (3) m [4, p. 370 f.] Julian Days → Seconds 1 d = 86 400 s [5, p. 696] ◦ ◦ · π ≈ Degrees → Radians 1 = 1 180◦ rad 0;017453293 rad 3 W.r.t. the central body and the meaning of i, ! and Ω to its reference frame. 4 The numbers in parentheses in 6:67428(67)·10−11 are a common way to state the uncertainty; short notation for (6:674280:0000067)·10−11. Page 2 of 3 version: 2017/10/05 15:36 M.Eng. René Schwarz (rene-schwarz.com): Memorandum Series Keplerian Orbit Elements −! Cartesian State Vectors (Memorandum № 1) 3 References Equations 2–4, 7 and 8:[3, pp. 22–27]; Equations 9 and 10:[6, p. 26]; Equation 5:[7]; Value for µÀ:[1]. [1] IAU Division I Working Group on Numerical Standards for Unit. In: Celestial Mechanics and Dynamical Astronomy 103 (4): Fundamental Astronomy: Astronomical Constants: Current 365–372. Springer Netherlands, 2009. ISSN: 0923-2958. DOI: 10. Best Estimates (CBEs). Online available at http://maia.usno.navy. 1007/s10569-009-9203-8.[! cited on page 2] mil/NSFA/NSFA_cbe.html (retrieved 2017/10/05). [! cited on [5] Seidelmann, P. Kenneth (ed.): Explanatory Supplement to the page 3] Astronomical Almanac. First paperback impression. University [2] Mohr, Peter J.; Taylor, Barry N.; Newell, David B.: CODATA Science Books, Sausalito, California, USA, 2006. ISBN: 978-1- recommended values of the fundamental physical constants: 2006. 891389-45-0. [! cited on page 2] In: Review of Modern Physics 80 (2): 633–730. American Physical [6] Standish, E. Myles; Williams, James G.: Orbital Ephemerides Society, 2008. ISSN: 1539-0756. DOI: 10.1103/RevModPhys.80. of the Sun, Moon and Planets. Online available at ftp://ssd.jpl. 633.[! cited on page 2] nasa.gov/pub/eph/planets/ioms/ExplSupplChap8.pdf (retrieved [3] Montenbruck, Oliver; Gill, Eberhard: Satellite Orbits: Models, 2011/05/15). [! cited on page 3] Methods, Applications. Corrected 3rd printing. Springer, Heidel- [7] Wikipedia (ed.): True anomaly. Online available at http : / / en . berg, 2005. ISBN: 9783540672807. [! cited on page 3] wikipedia.org/w/index.php?title=True%5C_anomaly&oldid= [4] Pitjeva, E.; Standish, E.: Proposals for the masses of the three 427251318 (retrieved 2011/05/15). [! cited on page 3] largest asteroids, the Moon-Earth mass ratio and the Astronomical Page 3 of 3 version: 2017/10/05 15:36.
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