Satellite Orbits - Principles of the Global Positioning System

Satellite Orbits • Treat the basic description and dynamics of satellite orbits • Major perturbations on GPS satellite orbits • Sources of orbit information: – SP3 format from the International GPS service – Broadcast ephemeris message • Accuracy of orbits and health of satellites • Logistics: Who can attend lecture on Fridays at 11-12:30? 2 Dynamics of satellite orbits • Basic dynamics is described by F=Ma where the force, F, is composed of gravitational forces, radiation pressure (drag is negligible for GPS), and thruster firings (not directly modeled). • Basic orbit behavior is given by GM e rÝ Ý= − 3 r r 3 Simple dynamics • GMe = µ = 3986006x108 m3s-2 • The analytical solution to the central force model is a Keplerian orbit. For GPS these are elliptical orbits. • Mean motion, n, in terms of period P is given by n= 2π µ = 3 P a • For GPS semimajor axis a ~ 26400km 4 Solution for central force model • This class of force model generates orbits that are conic sections. We will deal only with closed elliptical 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 5 Keplerain elements: Orbit plane Z Satellite perigee ν Vernal equinox θ0 ω Ω i Node equator Greenwich i Inclination Ω Right Ascension of ascending node ω Argument of perigee ν True anomaly 6 Keplerain elements in plane Satellite b Apogee a ae r E ν Perigee Focus Center of Mass a semimajor axis b semiminor axis e eccentricity ν True anomaly E Eccentric anomaly M Mean anomaly 7 Satellite motion • The motion of the satellite in its orbit is given by M (t) = n(t − T0 ) E(t) = M (t) + esin E(t) 2 ⎡ 1− e sin E(t) /(1− ecos E(t))⎤ −1 ν (t) = tan ⎢ ⎥ ⎣ (cos E(t) − e) /(1− ecos E(t)) ⎦ • To is time of perigee 8 True anomaly 0.25 Difference between true anomaly and Mean anomaly for e 0.001-0.100 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 4 x 10 9 Eccentric anomaly 0.25 Difference between eccentric anomaly and Mean anomaly for e 0.001-0.100 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 4 10 Vector to satellite • At a specific time past perigee; compute Mean anomaly; solve Kepler’s equation to get Eccentric anomaly and then compute true anomaly. See Matlab/truea.m • Vector r in orbit frame is ⎡ cos E − e ⎤ ⎡cos ν ⎤ r = a⎢ ⎥ = r⎢ ⎥ 2 sin E sin ν 1− e ⎦ ⎣ ⎣ ⎦ a(1− e2 ) r = a(1− ecos E) = 1+ ecos ν 11 Final conversion to Earth Fixed XYZ • Vector r is in satellite orbit frame • To bring to inertial space coordinates or Earth fixed coordinates, use ri = R3 (−Ω)R1 (−i)R3 (−ω )r re = R3 (−Ω + θ )R1 (−i)R3 (−ω )r • This basically the method used to compute positions from the broadcast ephemeris 12 Perturbed motions • The central force is the main force acting on the GPS satellites, but there are other significant perturbations. • Historically, there was a great deal of work on analytic expressions for these perturbations e.g. Lagrange planetary equations which gave expressions for rates of change of orbital elements as function of disturbing potential • Today: Orbits are numerically integrated although some analytic work on form of disturbing forces. 13 Perturbation from Flattening J2 • The J2 perturbation can be computed from the Lagrange planetary equations 3 2 cosi Ý Ω = − nae 2 J 2 2 2 a (1− e ) 2 3 2 5 cos2 i −1 ωÝ= nae 2 J 2 2 2 a (1− e ) 4 2 3cos i −1 3 J MÝ = n + nae2 2 2 3 2 4 a (1− e ) 14 J2 Perturbations • Notice that only Ω ω and n are effected and so this perturbation results in a secular perturbation • The node of the orbit precesses, the argument of perigee rotates around the orbit plane, and the satellite moves with a slightly different mean motion • For the Earth, J2 = 1.08284x10-3 15 Gravitational perturbation 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 16 Other perturbation on orbits and approximate size Term Acceleration (m/sec2) Central J2 Other gravity Third body Earth tides Ocean tides Drag Solar radiation Albedo radiation 0.6 5x10-5 3x10-7 5x10-6 10-9 10-10 ~0 10-7 10-9 17 GPS Orbits • Orbit characteristics are – Semimajor axis 26400 km (12 sidereal hour period) – Inclination 55.5 degrees – Eccentricity near 0 (largest 0.02) – 6 orbital planes with 4-5 satellites per plan • Design lifetime is 6 years, average lifetime 10 years • Generations: Block II/IIA 9729 kg, Block IIR 11000 kg 18 Basic Constellation Orbits shown in inertial space and size relative to Earth is correct 19 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 radiation and gravity perturbations • Periodic terms are added for argument of perigee, geocentric distance and inclination • The message and its use are described in the ICDGPS-200 icd200c123.pdf (page 105 in PDF) 20 Distribution of Ephemerides • The broadcast ephemeris is decoded by all GPS receivers and for geodetic receivers the software 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. brdc0120.02n 21 RINEX standard • Homework number 1 also contains description of navigation file message (other types of RINEX files will be discussed later) 22