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Orbital Mechanics

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Orbital Mechanics Danish Space Research Institute Danish Small Satellite Programme DTU Satellite Systems and Design Course Orbital Mechanics Flemming Hansen MScEE, PhD Technology Manager Danish Small Satellite Programme Danish Space Research Institute Phone: 3532 5721 E-mail: [email protected] Slide # 1 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Planetary and Satellite Orbits Johannes Kepler (1571 - 1630) 1st Law • Discovered by the precision mesurements of Tycho Brahe that the Moon and the Periapsis - Apoapsis - planets moves around in elliptical orbits Perihelion Aphelion • Harmonia Mundi 1609, Kepler’s 1st og 2nd law of planetary motion: st • 1 Law: The orbit of a planet ia an ellipse nd with the sun in one focal point. 2 Law • 2nd Law: A line connecting the sun and a planet sweeps equal areas in equal time intervals. 3rd Law • 1619 came Keplers 3rd law: • 3rd Law: The square of the planet’s orbit period is proportional to the mean distance to the sun to the third power. Slide # 2 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Newton’s Laws Isaac Newton (1642 - 1727) • Philosophiae Naturalis Principia Mathematica 1687 • 1st Law: The law of inertia • 2nd Law: Force = mass x acceleration • 3rd Law: Action og reaction • The law of gravity: = GMm F Gravitational force between two bodies F 2 G The universal gravitational constant: G = 6.670 • 10-11 Nm2kg-2 r M Mass af one body, e.g. the Earth or the Sun m Mass af the other body, e.g. the satellite r Separation between the bodies G is difficult to determine precisely enough for precision orbit calculations. ⋅⋅⋅ 14 3 2 GMEarth can be determined to great precision: GMEarth = µ =3.986004418 10 m /s Slide # 3 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Coordinate Systems Geocentric Inertial System Heliocentric Inertial System The reference changes with time ..... 50.2786” westerly drift of the Vernal Equinox per year Slide # 4 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Kepler Elements, Orbital Elements (Earth Orbits) Five orbital elements are neded to determine the orbit geometry: a Semi-Major Axis Determines the size of the orbit and the period of revolution e Eccentricity – Determines how elongated the ellipse is i Inclination Angle between the equator and orbital planes 0° ≤ i ≤ 90° : The satellite has an easterly velocity component – prograde orbit 90° ≤ i ≤ 180° : The satellite has a westerly velocity component – retrograde orbit Ω Right Ascension of Ascending Node - RAAN Angle from Vernal Equinox to the point where the orbit intersects the equatorial plane passing from South to North (the Ascending Node). ω Argument of Periapsis / Perigee Angle Between the line of nodes (the line between the ascending and descending nodes) and periapsis / perigee Slide # 5 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Two more elements are needed to calculate the position of the satellite at any time: T0 Epoch Time – Reference time for orbital elements and start of orbit propagation ν 0 True anomaly at Epoch – Angle from periapsis / perigee to the satellite position at T0 Normally the parameter “Mean Motion” is used instead of the semi-major axis in orbital calculations: 3 1 µ . a Mean Motion:n . Orbit period: Tp 2 π . µ 2 π a3 Normally the parameter Mean Anomaly instead true anomaly in an orbital parameter set: M - M0 = n (t - T0) = E - e sin E (Kepler’s equation) where E is the Eccentric Anomaly Slide # 6 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Kepler’s problem Given: r0, v0 at time T0 Find: r, v at time t Slide # 7 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Perturbations The fact that the Earth is not a sphere but an ellipsoid causes the orbit of a satellite to be perturbed Geopotential function: Φ µ Σ n (r,L) = ( /r) •[1 - Jn(RE/r) Pn(sinL)] where: L is latitude RE is the Earth radius at Equator Pn is the Legendre polynomials Jn are dimensionless coefficients: J2 = 0.00108263 J3 = -0.00000254 J4 = -0.00000161 ∆Ω ... is the drift of the ascending node ∆ω is the drift of periapsis / perigee Slide # 8 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Drift of Ascending Node − 7 2 − 2 14 a 2 deg NodalDriftRate() a, i , e := −2.06474⋅10 ⋅ ⋅cos() i ⋅()1e− ⋅ 1km⋅ day The Sun moves easterly by 0.9856° per day. An orbit exhibiting the same drift of the ascending node is said to be sun-synchronous (Helio-synchronous). This requires i > 90° i.e. a retrograde orbit. The satellite crosses the Equator at the same local solar time every orbit. Slide # 9 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Drift of Periapsis / Perigee Note that the drift is zero at i = 63.435°. This type of orbit with e ≈ 0.75, ≈12 hour period and apogee on the Northern hemisphese is denoted Molniya orbit and is used by Russian communication satellites − 7 2 − 2 14 a 2 2 deg ArgPerigeeDriftRate() a, i , e := 1.03237⋅ 10 ⋅ ⋅()4− 5⋅ sin() i ⋅()1e− ⋅ 1km⋅ day Slide # 10 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Molniya Orbit Drift of periapsis / perigee is zero at i = 63.435°. This type of orbit with e ≈ 0.75, ≈12 hour period and apogee on the Northern hemisphese is denoted Molniya orbit and is used by Russian communi- cation satellites The advantage is that the apogee stays at the Northern hemisphere and is high in the sky for around 8 hours A geostationary satellite far North (or South) is near the horizon and is easily blocked by mountains or buildings. Slide # 11 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Other Orbits Geostationary Orbit (GEO or GSO). Circular orbit with inclination = 0º, semi-major axis = 42164.172 km, h = 35768 km Period = 23 h 56 m 4.0954 s Geostationary Transfer Orbit (GTO) Elliptical Orbit with inclination ≈ 7º (from Kourou), ≈ 28.5º (from Cape Canaveral) Perigee ≈ 500 – 600 km, Apogee = 35768 km) Low Earth Orbit (LEO) Orbits in the altitude range 250 – 1500 km Medium Earth Orbit (MEO) Orbits in the altitude range 10000 – 25000 km Slide # 12 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Software for Orbit Calculations NOVA for Windows 32 Shareware at USD 59.95 Download from http://www.nlsa.com Free demo which may be registered for legal use. Slide # 13 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme NOVA for Windows Orbital Elements Slide # 14 FH 2001-09-16 Orbital_Mechanics.ppt Danish Space Research Institute Danish Small Satellite Programme Orbital Elements in NASA TLE (Two-Line Elements) format ØRSTED = Name of spacecraft (Max. 11 characters) Col. No. ....:....1....:....2....:....3....:....4....:....5....:....6....:....7 1234567890123456789012345678901234567890123456789012345678901234567890 Legend L AAAAAU YYNNNPPP XXDDD.FFFFFFFF CCCCCCCC UUUUUUU VVVVVVV E SSSSZ TLE 1 1 25635U 99008B 99132.50239417 +.00000000 +00000+0 +00000+0 1 12344 Col. No. ....:....1....:....2....:....3....:....4....:....5....:....6....:....7 1234567890123456789012345678901234567890123456789012345678901234567890 Legend L AAAAA III.IIII RRR.RRRR EEEEEEE PPP.PPPP NNN.NNNN MM.MMMMMMMMKKKKKZ TLE 2 2 25635 96.4836 68.6614 0153506 8.1032 183.1749 14.40955463 11246 TLE Data Line 1 TLE Data Line 2 L: Line Number L: Line Number A: Catalog Number A: Catalog Number U: Unclassified (C: Classified) I: Inclination Y: Last Two Digits of Daunch Year R: Right Ascension of Ascending Node N: Launch Number of the Year E: Eccentricity, assuming implicitly P: Piece of Launch a decimal point at left X: Last two digits of Epoch Year P: Argument of Perigee D: Day number in Epoch Year N: Mean Anomaly F: Fraction of day in Epoch Time M: Mean Motion C: Decay Rate (First Time Derivative of the Mean Motion divided by 2) K: Orbit Number U: Second Time Derivative of Mean Motion divided by 6. (Blank if N/A) Z: Checksum V: BSTAR drag term if GP4 general perturbation theory was used. Otherwise, radiation pressure coefficient. E: Ephemeris Type (=1 normally) S: Element Set Number Z: Checksum Slide # 15 FH 2001-09-16 Orbital_Mechanics.ppt for MissionPlanning Use ofOrbitCalculations Satellite Programme Danish Small Danish SpaceResearch Institute Elevation at ground station versus time Elevation atground station versus versus time Separation– satellite station ground orbit Molniya in mission Rømer Eksempel: FH 2001-09-16 Orbital_Mechanics.ppt Range [km] 10000 15000 20000 25000 30000 35000 40000 45000 Elevation [deg.] 5000 10.0 20.0 30.0 40.0 50.0 0.0 0 1 1 89 101 Ballerina MONS Orbit. and Spacecraft Copenhagen to Molniya in Ground Station Range 177 201 265 301 353 Ballerina and MONS in Molniya Orbit. Elevation Angle from Copenhagen 401 441 501 529 617 601 705 701 793 801 881 901 969 1001 1057 1101 1145 1201 1233 Time [min.] Time Time [m in.] 1321 1301 1409 1401 1497 1501 1585 1601 1673 1701 1761 1801 1849 1901 1937 2001 2025 2113 2101 2201 2201 2289 2301 2377 Slide #16 2401 2465 2501 2553 2601 2641 2701 2729 2817 2801 Danish Space Research Institute Danish Small Satellite Programme Ballerina and MONS in Molniya Orbit. Attained Channel Data Rate 64000 56000 48000 40000 32000 Downlink bit-rate versus time 24000 Bit Rate [bit/s] Rate Bit 16000 8000 0 1 90 179 268 357 446 535 624 713 802 891 980 1069 1158 1247 1336 1425 1514 1603 1692 1781 1870 1959 2048 2137 2226 2315 2404 2493 2582 2671 2760 2849 Time [m in] Ballerina & MONS Eclipse Durations 01:00:00 00:55:00 00:50:00 00:45:00 00:40:00 Satellitten in eclipse (in Earth’s shadow).
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