Satellite Posi+oning. Lecture 5: Satellite Orbits. Jan Johansson Chalmers University of Technology, 2013

Transcription

1 Lecture 5: Satellite Orbits Jan Johansson Chalmers University of Technology, 2013

2 Geometry Satellite Plasma Posi+oning physics Antenna theory Geophysics Time and Frequency GNSS Orbital mechanics Physics of the Atmosphere Relativity Stochastic processes Kalman Filtering Wave propagation

3 GPS ERROR BUDGET AND DILUTION OF PRECISION (DOP) Total root sum square when SA off: 9.5

4 Satellite geometry at two loca+ons GLONASS & GPS coverage in Kiruna

5 Orienta+on of the Earth in Iner+al Space The oscilla+on of the earth s rota+onal vector with respect to iner+al space is called nuta+on. It is par++oned into: Secular precession Periodic Nuta+on. The orienta+on of the earth s crust with respect to its rota+onal vector is called polar mo+on Varia+on in length of day (LOD) cause varia+on in orienta+on

6 Astronomical " Latitude also shown" Satellite Posi+oning Geode+c La+tude

7 Internal Satellite Mo+ons Posi+oning of Earth Solid Earth +des Ocean, atmosphere +des and loading Ocean, atmosphere, and cryosphere mass mo+ons and loading Glacial isosta+c rebound Geophysical phenomena

8 Internal Mo+ons of Earth Solid Earth +des Ocean, atmosphere +des and loading Ocean, atmosphere mass mo+ons and loading Glacial isosta+c rebound Geophysical phenomena

9 Satellite Orbits Treat the basic descrip+on and dynamics of satellite orbits Major perturba+ons on GPS satellite orbits Sources of orbit informa+on: SP3 format from the Interna+onal GPS service Broadcast ephemeris message Accuracy of orbits and health of satellites

10 Important satellite orbit laws

11 Implica+ons Satellite Posi+oning of Kepler Laws

12 Dynamics of satellite orbits Basic dynamics is described by F=Ma where the force, F, is composed of gravita+onal forces, radia+on pressure (drag is negligible for GPS), and thruster firings (not directly modeled). Basic orbit behavior is given by

13 Simple dynamics GM e = µ = x10 8 m 3 s - 2 The analy+cal solu+on to the central force model is a Keplerian orbit. For GPS these are ellip+cal orbits. Mean mo+on, n, in terms of period P is given by For GPS semimajor axis a ~ 26400km

14 Solu+on for central force model We will deal only with closed ellip+cal orbits. The orbit plane stays fixed in space One of the foci of the ellipse is the center of mass of the body These orbits are described Keplerian elements

15 Kepler Satellite elements: Posi+oning Orbit plane Ω, ω, and ί: orienta+on of ellipse

16 Satellite Kepler elements Posi+oning in plane a and e : Size and shape of orbital ellipse

17 mo+on The mo+on of the satellite in its orbit is given by T o is +me of perigee

18 True anomaly Difference between true anomaly and Mean anomaly for e (rad) (seconds)

19 Satellite Eccentric Posi+oning anomaly Difference between eccentric anomaly and Mean anomaly for e (rad) (seconds)

20 Satellite Vector to Posi+oning satellite At a specific +me past perigee; compute Mean anomaly; solve Kepler s equa+on to get Eccentric anomaly and then compute true anomaly. Vector r in orbit frame is

21 Final conversion Satellite Posi+oning to Earth Fixed XYZ Vector r is in satellite orbit frame To bring to iner+al space coordinates or Earth fixed coordinates, use This is basically the method used to compute posi+ons from the broadcast ephemeris

22 Perturbed mo+ons The central force is the main force ac+ng on the GPS satellites, but there are other significant perturba+ons. Historically, there was a great deal of work on analy+c expressions for these perturba+ons e.g. Lagrange planetary equa+ons which gave expressions for rates of change of orbital elements as func+on of disturbing poten+al Today: Orbits are numerically integrated although some analy+c work on form of disturbing forces.

23 Perturba+on from Flaoening J 2 The J 2 perturba+on can be computed from the Lagrange planetary equa+ons

24 J 2 Perturba+ons No+ce that only Ω, ω, and n are effected and so this perturba+on results in a secular perturba+on The node of the orbit precesses, the argument of perigee rotates around the orbit plane, and the satellite moves with a slightly different mean mo+on For the Earth, J 2 = x10-3

25 Gravita+onal Satellite perturba+on Posi+oning styles Parameter Secular Long period Short period a No No Yes e No Yes Yes i No Yes Yes Ω Yes Yes Yes ω Yes Yes Yes M Yes Yes Yes

26 Term Other perturba+on on orbits and Satellite Posi+oning approximate size Acceleration (m/sec 2 ) Central 0.6 Orbit error J 2 5x10-5 (10000 m) Other gravity 3x10-7 (200 m) Third body 5x10-6 (3000 m) Earth tides 10-9 (0,3 m) Ocean tides (0,04 m) Drag ~0 Solar radiation 10-7 (200 m) Albedo radiation 10-9 (0,03 m)

27 GPS Orbits Orbit characteris+cs are Semimajor axis km (12 sidereal hour period) Inclina+on 55.5 degrees Eccentricity near 0 (largest 0.02) 6 orbital planes with 4-5 satellites per plan Design life+me is 6 years, average life+me 10 years Genera+ons: Block II/IIA

28 GLONASS Constellation and Comparison to GPS

29 Galileo Constellation Satellite Posi+oning and Comparison to GPS Galileo GPS No of satellites (+?) No of orbital planes 3 6 Orbital inclination 56º 56º Orbit al+tude 23222km 20200km Period of revolution ~14h05m 11h58m

30 Broadcast Ephemeris Satellites transmit as part of their data message the elements of the orbit These are Keplerian elements with periodic terms added to account for solar radia+on and gravity perturba+ons Periodic terms are added for argument of perigee, geocentric distance and inclina+on The message and its use are described in the ICD- GPS- 200

31 Broadcast Satellite ephemerides Posi+oning 1 (2) Parameter t e a e M 0 ω 0 i 0 l 0 Δn i Explanation Ephemerides ref. epoch Square root of semi-major axis Eccentricity Mean anomaly Argument of Perigee Inclination Longitude of node at weekly epoch Mean motion difference Rate of inclination angle

32 Broadcast Satellite ephemerides Posi+oning 2 (2) Parameter ΏΏ C uc C us C rc C rs C ic C is t c a 0 a 1 a 2 Explanation Rate of node s right ascension Correction coefficients (arg. perigee) Correction coefficients (geoc. dist.) Correction coefficients (inclination) Satellite clock reference epoch Satellite clock offset Satellite clock drift Satellite clock frequency drift Time given in GPS Week & Seconds

33 Rela+vis+c effects General rela+vity affects GPS in three ways Equa+ons of mo+ons of satellite Rates at which clock run Signal propaga+on In our GPS analysis we account for the second two items Orbits only integrated for 1-3 days and equa+on of mo+on term is considered small

34 Clock effects GPS is controlled by MHz oscillators On the Earth s surface these oscillators are set to 10.23x( x10-10 ) MHz (39,000 ns/day rate difference) This offset accounts for the change in poten+al and average velocity once the satellite is launched. The first GPS satellites had a switch to turn this effect on.

35 Propaga+on and clock effects Our theore+cal delay calcula+ons are made in an Earth centered, non- rota+ng frame using a light- +me itera+on i.e., the satellite posi+on at transmit +me is differenced from ground sta+on posi+on at receive +me. Two correc+ons are then applied to this calcula+on

36 Satellite Correc+ons Posi+oning terms Propaga+on path curvature due to Earth s poten+al (a few cen+meters) Clock effects due to changing poten+al For e=0.02 effect is 47 ns (14 m)

37 Rela+vis+c Effects

38 Distribu+on Satellite of Posi+oning Ephemerides The broadcast ephemeris is decoded by all GPS receivers and for geode+c receivers the souware that converts the receiver binary to an exchange format outputs an ASCII version The exchange format: Receiver Independent Exchange format (RINEX) has a standard for the broadcast ephemeris. Form [4- char][day of year][session].[yy]n e.g. brdc n up://igscb.jpl.nasa.gov/ igscb/data/format/rinex2.txt

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40 Primary ITRF2005 sites and co- located techniques Satellite Posi+oning (48) (25) (6) VLBI sites SLR/LLR laser GPS station DORIS

41 Current geode+c Satellite networks Posi+oning contribu+ng to ITRF

42 Interna+onal Terrestrial Reference Satellite Posi+oning Frame (ITRF) Cartesian system (XYZ) Realized by space geode+c sta+ons with known coordinates and velocity vectors

43 Example of precise ephemerides from IGS igs12980.sp3 analysis center GPS week day of GPS week format Descrip+on of the sp3 and sp3c format: up://igscb.jpl.nasa.gov/igscb/data/format/ (X,Y,Z, dt) every 15 minutes terrestrial based reference frame!

44 Satellite Orbit Dissemina+on Source of Information Quality Availability Dissemination Predicted (broadcast) 3 m Real time Signal-in-Space IGS predicted (48 hours) 0,2 m Real time Internet/FTP IGS final orbit 0,05 m 1 week delay IGS Data centers Orbit Error Baseline length Baseline Error 3 m 10 km 1,2 mm 3 m 100 km 12 mm 0,5 m 10 km 0,2 mm 0,5 m 100 km 2 mm 0,05 m 10 km 0,02 mm 0,05 m 100 km 0,2 mm Satellite clocks also important => 10 ns corresponds to 3 meter

45 Satellite Summary Posi+oning The important things from this lecture are: Orbit characteris+cs for GPS, GLONASS and Galileo Perturbed mo+on Broadcast ephemerides Precise ephemerides (from IGS)