SLS-RFM_14-18 Orbital Considerations For A Lunar Comm Relay Fall Technical Meeting, London 11/10 – 11/14/2014 SLS-RFM_14-18 Orbital Considerations For A Lunar Comm Relay H. Garon, V. Sank - NASA/GSFC/ASRC RFM-PCOM 12 Nov 2014

Lunar Satellite Comm Relay Problem Statement Potential lunar base placement where Earth may be below lunar horizon majority portion of a month (slide 4) Due to Earth’s gravitational pull, high-altitude (>1200km) orbits regardless of eccentricity are inherently unstable while circular orbits at lower altitudes will not remain circular for very long. Time scales associated with substantive perturbation may be on the order of weeks. Crash into Moon Initially elliptical a semi major axis Angle from X axis to line of nodes Angle from line of nodes to periapsis M mean anomaly e = c/a (e) Initially circular Initially circular RFM-PCOM 12 Nov 2014 10/9/14

Ely*: “New class of stable high-altitude lunar orbits” Ely, T.A., Stable Constellations of Frozen Elliptical Inclined Lunar Orbits, Journal of the Astronautical Sciences, vol. 53, No. 3, July-Sept 2005, pp. 301-316 Ely, T.A. and Lieb, E., Constellations of Elliptical Inclined Lunar Orbits Providing Polar and Global Coverage, AAS 05-343, AAS/AIAA Astrodynamics Specialists Conference, August 7-11, 2005. [http://trsnew.jpl.nasa.gov/dspace/bitstream/2014/37462/1/05-2140.pdf] * Proposal – 3 satellites 120° apart Ellipse with e~0.6 and i~51° Periapsis (North lunar pole) ~ 700km (Note: figure directly from http://science.nasa.gov/science-news/science-at-nasa/2006/30nov_highorbit/ ) RFM-PCOM 12 Nov 2014 10/9/14

. . Orbit of Moon around Earth Ecliptic Earth Moon Moon 4 Moon North pole, spin axis, tipped 6.8º from ecliptic normal. Moon rotates on axis once per month Moon orbit period = 27. 317 days, (360º /27.3 = 13.2º / day) but during that time Earth rotates 27º around Sun so period appears to be ~ 29 days. Moon spin axis orientation fixed due to conservation of angular momentum 6.8º Moon . 6.8º Moon . Ecliptic Plane of Earth orbit around Sun Earth 5.14º Moon South pole in view of Earth. From south pole, Earth appears at elevation of 6.8º . Moon South pole NOT in view of Earth. From south pole, Earth appears at elevation of - 6.8º . Moon orbit plane inclined ~ 5.14º from ecliptic. Varies from 4.98º to 5.30º Shackleton crater location: 89.6 S, 110 E 0.4 from south pole Earth will be at least 1º above Horizon for about 12 days out of ~ 28 day cycle. From Moon South pole, Earth will appear to. rise to elevation of 6.8º and fall over a ~ 28 day period. From Shackleton crater, Earth will appear to. rise to elevation of 7.2º and fall to - 6.4º over a ~ 28 day period. Elevation to Earth center from Moon south pole (degrees) + 6.8º 14 28 42 56 12 days - 6.8º Time (days) 14 days RFM-PCOM 12 Nov 2014 4 12/10/05

Terminology from Kepler 1+ e cos x2 a2 y2 b2 + = 1 For point on ellipse Basic Orbit Dynamics . X Y  a b p Q V S t F E r Object in orbit perigee apogee c = ea x,y v = ( ) 1 1 r 2a Definition of terms (in plane only, 2D) a = semi major axis b = semi minor axis a2=b2+c2 c = semi distance between foci e = eccentricity = c/a p = semi latus rectum = a(1-e2) ra = “radius” to apoapsis rp = “radius” to periapsis m = mass of central body G = Newton's gravitational constant  = G m T = orbit period = 2 a3/2 / = True anomaly = angle centered at focus, from perigee to object E = Eccentric anomaly = angle at center of superimposed circle, from perigee to object line M = Mean anomaly = E - e sin E = 2 (time fraction of orbit) n = Mean Motion = 2  /T = “” Area of ellipse =  a b Central body x RFM-PCOM 12 Nov 2014 5 2/6/10