©2017 M.Eng. René Schwarz (rene-schwarz.com), unless otherwise stated. This document is subject to the conditions of the Creative Commons Attribution-NonCommercial 4.0 International Errors, comments or ideas regarding this paper? License (CC BY-NC 4.0), which can be examined at https://creativecommons.org/licenses/by-nc/4.0/. Other → https://goo.gl/XB85LZ cbn licenses may apply for particular contents or supplemental documents/files. 1 Inputs (B.Eng.) Technology Physics and Industrial (M.Eng.) Mechatronics, Systems in Communication Engineering and of Science Bachelor Computer in Engineering of Master M.Eng.   6. 5. 4. 3. 2. 1. Algorithm gravitation Sun ( from different if body, tnadgaiainlparameter gravitational standard vectors state cartesian – – bantelniueo h sedn node ascending the of longitude the Obtain eccentricity orbit the Determine inclination orbit the Calculate Preparations: ial,tesm-ao axis semi-major the Finally, tricity anomaly mean the Compute anomaly tric eoiyvector velocity vector position atsa tt Vectors State Cartesian eéSchwarz René b) a) c) eemn h vector the Determine body. central parameter gravitational standard with vector eccentricity the Obtain vector momentum orbital Calculate e : [ kg m = Ω · 3 s 2 E ] , n [1] M r ˙ r { (0 = ( ( : t t cnrlbd mass body …central arccos 2 [ ) [ ) π m m − , s 0 ] ] , arccos G 1) ∥ n n … n e x T ∥ a Newton [ = M × sfudfo h expression the from found is i m µ s ∥ yuigteobtlmmnu vector momentum orbital the using by ∥ h n 2 e n ihhl of help with e x = ] ∥ ( = [1] ∥ onigtwrsteacnignd n h reanomaly true the and node ascending the towards pointing GM e hc ssml h antd fteecnrct vector eccentricity the of magnitude the simply is which , for for − [1] a osatof constant ian [ h eoadm№2 № Memorandum kg from y n n ftecentral the of h h , y y ] ) [ Ω ≥ < x m µ s , Kepler n h rueto periapsis of argument the and 2 M 0 0 −→ 0) a ] = i : T = = e = µ = arccos À ∥ E sEuto rmteecnrcanomaly eccentric the from Equation ’s h 2 r r ∥ ˙ Kepler 1 = − = × − µ 1 e r h ∥ ∥ sin . × h r ˙ 2 2 0 18 400 124 327 Outputs µ ν h ω ∥ −  z 2 ∥ r = ˙ E = E . ∥ rdtoa e of set traditional a r r { – – – – – – { ∥ 2 = 2 arccos a ri Elements Orbit ian 2 arccos enanomaly Mean Inclination (LAN) node ascending of Longitude periapsis of Argument Eccentricity axis Semi-major π π arctan − − h where , arccos arccos ∥ ∥ ⟨ e ⟨ n · e ∥∥ n ∥∥ , 10 √ ω r , tan e r ⟩ e ∥ ⟩ : i ∥ 20 ∥ 1 1+ ∥ e h [ ⟨ e − ⟨ n rad e ∥∥ ν 2 z n ( [1] ∥∥ , e e r  , Kepler M stetidcmoetof component third the is e r ⟩ e ∥ ⟩ a ] 8 ∥ [ [ · m rad otherwise for 10 for for ] a ri Elements Orbit ian ] 9 ω ⟨ ) e e r [ m z z rad , s eso:21/00 14:55 2017/10/05 version: E 2 e r 3 ˙ ν ≥ < ≥ ⟩ n h eccen- the and , n h eccen- the and o h u as Sun the for ] [ rad 0 0 . 0 ae1of 1 Page ] with [ Ω rad (3) (2) (8) (6) (5) (4) (1) (7) h 2 ] : M.Eng. René Schwarz (rene-schwarz.com): Memorandum Series Cartesian State Vectors −→ Keplerian Orbit Elements (Memorandum № 2)

2 Constants and Conversion Factors

Universal Constants Symbol Description Value Source 1 · −11 m3 G Newtonian constant of gravitation G = 6.67428(67) 10 kg·s2 [1, 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 [3] ◦ ◦ · π ≈ Degrees → Radians 1 = 1 180◦ rad 0,017453293 rad

3 References

Equations 1 and 8:[2, p. 28]; Eq. 2:[8]; Eq. 3:[9, 12]; Eq. 5:[7, 10]; Eq. 4:[11]; Eq. 6:[6, 9]; Eq. 7:[5, p. 26]; Value for µÀ:[3]. [1] Mohr, Peter J.; Taylor, Barry N.; Newell, David B.: CODATA recommended values of the fundamental physical constants: 2006. In: Review of Modern Physics 80 (2): 633–730. American Physical Society, 2008. ISSN: 1539-0756. DOI: 10.1103/RevModPhys.80.633.[→ cited on page 2] [2] Montenbruck, Oliver; Gill, Eberhard: Satellite Orbits: Models, Methods, Applications. Corrected 3rd printing. Springer, Heidelberg, 2005. ISBN: 9783540672807. [→ cited on page 2] [3] NASA/JPL: Astrodynamic Constants. Online available at http://ssd.jpl.nasa.gov/?constants (retrieved 2011/04/26). [→ cited on page 2] [4] Pitjeva, E.; Standish, E.: Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit. In: Celestial Mechanics and Dynamical Astronomy 103 (4): 365–372. Springer Netherlands, 2009. ISSN: 0923-2958. DOI: 10.1007/s10569-009-9203-8.[→ cited on page 2] [5] Standish, E. Myles; Williams, James G.: Orbital Ephemerides of the Sun, Moon and Planets. Online available at ftp://ssd.jpl.nasa.gov/pub/ eph/planets/ioms/ExplSupplChap8.pdf (retrieved 2011/05/15). [→ cited on page 2] [6] Wikipedia (ed.): Argument of periapsis. Online available at http://en.wikipedia.org/w/index.php?title=Argument%5C_of%5C_periapsis& oldid=450370870 (retrieved 2011/10/29). [→ cited on page 2] [7] Wikipedia (ed.): Eccentric anomaly. Online available at http://en.wikipedia.org/w/index.php?title=Eccentric%5C_anomaly&oldid=416453169 (retrieved 2011/10/29). [→ cited on page 2] [8] Wikipedia (ed.): Eccentricity vector. Online available at http://en.wikipedia.org/w/index.php?title=Eccentricity%5C_vector&oldid=443226923 (retrieved 2011/10/29). [→ cited on page 2] [9] Wikipedia (ed.): Longitude of the ascending node. Online available at http://en.wikipedia.org/w/index.php?title=Longitude%5C_of%5C_the% 5C_ascending%5C_node&oldid=453399307 (retrieved 2011/10/29). [→ cited on page 2] [10] Wikipedia (ed.): Orbital eccentricity. Online available at http://en.wikipedia.org/w/index.php?title=Orbital%5C_eccentricity&oldid= 453223480 (retrieved 2011/10/29). [→ cited on page 2] [11] Wikipedia (ed.): Orbital inclination. Online available at http://en.wikipedia.org/w/index.php?title=Orbital%5C_inclination&oldid=452407742 (retrieved 2011/10/29). [→ cited on page 2] [12] Wikipedia (ed.): True anomaly. Online available at http://en.wikipedia.org/w/index.php?title=True%5C_anomaly&oldid=427251318 (retrieved 2011/05/15). [→ cited on page 2]

1 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.

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