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.