Basics of Orbital Mechanics I (cont)

Transcription

1 Basics of Orbital Mechanics I (cont) p. 1/10 Basics of Orbital Mechanics I (cont) Modeling the Space Environment Manuel Ruiz Delgado European Masters in Aeronautics and Space E.T.S.I. Aeronáuticos Universidad Politécnica de Madrid April 2008

2 Basics of Orbital Mechanics I - Practicum Basics of Orbital Mechanics I (cont) p. 2/10 Reference Systems Satellite Reference Systems and Time Element Set State Vector State Vector Element Set

3 Reference Systems Basics of Orbital Mechanics I (cont) p. 3/10 Origin Fundamental plane Principal direction Orientation (rh)

4 Reference Systems Origin Fundamental plane Principal direction Orientation (rh) Basics of Orbital Mechanics I (cont) p. 3/10

5 Reference Systems Origin Fundamental plane Principal direction Orientation (rh) GST Basics of Orbital Mechanics I (cont) p. 3/10

6 Reference Systems Origin Fundamental plane Principal direction Orientation (rh) GST Basics of Orbital Mechanics I (cont) p. 3/10

7 Reference Systems Basics of Orbital Mechanics I (cont) p. 3/10 Origin Fundamental plane Principal direction Orientation (rh) Mean Ecliptic of J2000 Mean Equinox of J2000 GST Mean Equator of J2000

8 Reference Systems Basics of Orbital Mechanics I (cont) p. 3/10 Origin Fundamental plane Principal direction Orientation (rh) Far radio sources: Quasars (VLBI) Mean Ecliptic of J2000 Mean Equinox of J2000 GST Mean Equator of J2000

9 Reference Systems Basics of Orbital Mechanics I (cont) p. 4/10 System Origin F. Plane P.Direction Definition Use Interplanetary Systems ICRS Barycenter Star-Fixed Star-Fixed IAU00/Hipp Cat Ephem HCS Sun MEcl J2000 MEqx J2000 IAU76/FK5 Interpl HCS Sun MEcl B1950 MEqx B1950 FK4 Obsolete Earth-Centered Systems GCRS Earth COM Star-Fixed Star-Fixed IAU00/Hipp Cat Perturbations GECS Earth COM MEqu J2000 MEqx J2000 IAU76/FK5 Orbit prop ITRF-xx Earth COM Crust-fixed Greenwich IAU00/Hipp Cat Astron obs ECEF Earth COM Earth-fixed Grw/Local IAU76/FK5 WGS84,MSIS International Celestial Reference System (/Frame) Heliocentric Coordinate System Geocentric Equatorial C. S. International Terrestrial RF, Earth-Centered, Earth-Fixed Inertial Inertial Inertial Rotating Rate of Change

10 Change of Reference Systems: SOFA routines Basics of Orbital Mechanics I (cont) p. 5/10 MJD TT UT1 GST Time conversions Modified Julian date (MJD 0 +days) Terrestrial Time Universal time Greenwich Sidereal Time Frame convesion matrices GCRF Mean Equ Eqx J2000 Frame Bias Constant Mean Equ Eqx of date Precession Long period True Equ Eqx of date Nutation Short period(s) True Equ of date, Greenw. Earth Rotation Daily ITRF Polar Motion Yearly

11 Change of Reference systems Basics of Orbital Mechanics I (cont) p. 6/10 Ε (") Data computed with SOFA routines Nutation Precession Jan Jan Ψ (") Precession Chandler Pole Motion (IERS data) Nutation m 2 Data from IERS I I 97 90ºE North Pole Greenwich I 98 I 99 I 00 I I 03 I ºW m 0º Frame bias GST GW Daily Earth Rotation: Greenwich Sidereal Time

12 Satellite Reference Systems and Time Basics of Orbital Mechanics I (cont) p. 7/10 h i Sat. θ e Ω ω i Ω u N y 1 x 1 Line of nodes z 1 Peric. Equatorial i 1 () j 1 k 1 Nodal u N ( ) h u N h Perifocal e (Per) h e h Orbital u r u θ h Julian Date (JD): Days from Jan 01, 4713BC, 12:00 noon Modified Julian Date (MJD): JD-2,400,000.5 J2000=JD 2,451,545.0 Epoch 1 Jan :00 TT J2000=MJD 51,544.5 Epoch 1 Jan :00 TT Solar (24h) and Sidereal (23h56m4.091±0.005s) day

13 Element Set State Vector Basics of Orbital Mechanics I (cont) p. 8/10 h i z 1 i, Ω,ω,a,e,τ, t r,v, t e ω Ωu N y 1 x 1 t M ( Kepler Eq. ) Æ θ p 1+ecos θ p = a(1 e 2 ) r = cos θ r = r sin θ 0 v = µ p per sin θ e + cos θ 0 Q = R(k 1, Ω) R(u N, i) R(h, ω) = cωcω sωsωci cωsω sωcωci sωsi = sωcω + cωsωci sωsω + cωcωci cωsi sωsi cωsi ci per r Eq = Q r pef v Eq = Q v pef

14 State Vector Element Set Basics of Orbital Mechanics I (cont) p. 9/10 h i z 1 r,v, t i, Ω,ω,a,e,τ, t,θ Ω u N x 1 y 1 e ω h = r v ω = ATAN2 [ (u N e) hh,e u ] N θ = ATAN2 [ (e r) hh,e r] Æ Æ (Kepler Eq.) M u N = k 1 h e = r r h v µ Circ? e E m = v2 2 µ r Ell/Par/Hyp? a i = ACOS(h z /h) τ Ω = ATAN2 (u Ny,u Nx )

15 Canonical Units Basics of Orbital Mechanics I (cont) p. 10/10 Grazing orbit: Circular orbit with r g = R E, v g = µ R E Units L T µ SI (km) km s km3 s 2 C. U. R E TU 1 R3 E TU 2 µ = km3 s 2 = 1 R3 E TU 2 TU = s C. U. : r g = 1 R E v g = 1 R E TU µ = 1 R3 E TU 2