1 Satellite observation systems and reference systems (ae4-e01) Orbit Mechanics 2 E. Schrama

2 Contents Perturbed Kepler orbits –Linear C20 perturbations and classification of orbits –Orbit determination, solve the equation of motions –Effects of other acceleration models Numerical implementation –Example 1: Bullet physics –Example 2: Kepler and higher order physics –Orbit determination Parameter estimation –Parameters in function model –Parameter estimation procedure –Variational equations –Organisation parameter estimation

3 Perturbed Kepler Orbits Please remember that the Kepler problem assumes a central force field with U=GM/r In reality the gravity potential U is more difficult than that and spherical harmonics are involved. Moreover there are other conservative and non-conservative forces that determine the motion of a spacecraft

4 Linear perturbations by C 20 Ref: Seeber p 84 C 20 not normalised, n: mean motion

5 Classification of orbits Sun synchronous orbits:  runs as fast as the Earth’s rotation around the Sun. This is possible by tuning the a, e and I. Golden inclination: Perigee is frozen in time Repeating: Ground tracks reoccupy the same geographic locations after a certain time (a cycle) Polar orbits: the orbit plane is fixed in inertial space despite the presence of gravitational flattening.

6 Orbit determination Kepler’s theory happens to be a very good approximation to describe the motion of small particles in a gravity field as a result of the presence of a large body like the Earth or the Sun. In reality there are higher order multipoles in the gravity field and other accelerations play a role. The more complete equations of motion are therefore:

7 This is Y200

8 Y300

9 Y210 and Y211

10 Y320

11 Y330

12 Solution equations of motion Analytic –Lagrange planetary equations –Gravity Potential in Kepler elements –Isolate first order solution –Approximate higher order perturbations Numeric –Conversion to system of first order ODE –Integration of system of equations

13 What other accelerations? Tidal forces cause by Sun and Moon Gravity effect of air, water in motion etc Radiation pressure as a result of sun light and light reflected from Earth (Albedo) Heat radiating away from the spacecraft Atmospheric drag Relativistic mechanics

14 Effect of perturbing accelerations The table below lists various acceleration terms that act on the orbit of a GPS satellite, gravitational flattening is by far the largest contributor. Ref: Seeber table 3.4

15 Hard to model perturbations The remaining perturbations always result in oscillating functions. There are cos/sin series from which the amplitudes and phases are defined Numerical integration is the way to go, all orbit determination s/w uses this method. Required is an initial state vector and an acceleration model for the satellite. To classify satellite orbits a first-order analytical solution can be used.

16 Numerical implementation Keplerian physics is easy to understand, essentially follows from a central force field with a point-mass potential The real world is more difficult, essentially because there are higher order terms in the potential and because there are other accelerations Orbit dynamics can be described in the form of ordinary differential equations. You should formulate the problem as a system of first-order ODEs There are efficient numerical tools to solve ODEs, in particular single-step and multi-step integrators

17 Demonstration numerical solution ordinary diff. eq. Example gun bullet physics function f = bullet( t,state ) % implements bullet dynamics xp = state(1); yp = state(2); xv = state(3); yv = state(4); g = 9.81; dia = 0.44*2.54; length = 1.5*2.54; dens = 8000; area = pi*(dia/2).^2; mass = dens*area*length; cd = 1; h = yp; f = exp(-h/6000*log(2)); rho = 1.2 *f ; v = sqrt(xv*xv+yv*yv); ad = 0.5*rho*(area/mass)*v*v*cd; nx = xv/v; ny = yv/v; xa = -nx*ad; ya = -ny*ad - g; f = [state(3) state(4) xa ya]'; In reality

18 Demonstration Numerical Implementation (2) function f = satdyn( t,state ) % implements Kepler dynamics xp = state(1); yp = state(2); xv = state(3); yv = state(4); mu = 4e14; r = sqrt(xp.^2+yp.^2); factor = mu/r/r/r; xa = -factor*xp; ya = -factor*yp; f = [state(3) state(4) xa ya]'; Example Kepler orbit physics: In reality

19 Orbit prediction (1) During orbit perdiction one needs to integrate the equations of motion. Suitable numerical techniques are used to treat differential equations of the following type: There are numerical procedures like the Runge-Kutta single step integrator and Adams-Moulton-Bashforth multi step integrator that allow the state vector y 0 to be propagated from y 0 till y n. In this case a state vector at index j coincides with the time index t 0 +(j-1)*h where h is the integrator step size.

20 Example in MATLAB span = [ ]; state = [ 1e e3]; option = odeset('RelTol',1e-10); [t,y] = ODE45('satdyn',span,state,option); plot(y(:,1),y(:,2))

21 Orbit prediction (2) The orbit prediction problem is entirely driven by the choice of the initial state vector y 0, the definition of F(y,t) and g(t). The basic question is of course, where does this information come from? F(y,t) and g(t) fully depend on the realism of your mathematical model and its ability to describe reality However, knowledge of the initial state vector should follow from 1) earlier computations or 2) launch insertion parameters The conclusion is that it is desirable to estimate initial state parameters from observations to the satellite.

22 Parameter estimation Terminology: –Here, a problem refers to an interesting case to study. Problems in satellite geodesy: –Type of problem does it contain orbit parameters? does it contain gravity field parameters? does it contain any other geophysical parameters? –How do you organize parameter estimation? it is a batch or a sequential least squares problem? can you solve it from one observation set or are more sets involved? Is preprocessing of observations involved or is it in the problem?

23 Function model (1) The function model aims to relate observations and parameters to another The unknowns are gathered in vector The observations are in vector Usually we begin to approximate reality by a priori estimates and

24 Function model (2) XeXe YeYe ZeZe B S RiRi RjRj  R ij

25 Function model (3): Examples The over-determined GPS navigation solution for one receiver VLBI observations of phase delay Two GPS receivers: double difference processing SLR network: station, orbit parameters, earth rotation parameters DORIS with orbit and gravity field improvement Spaceborne GPS receiver on a LEO

26 Implementation From our function model we conclude that: –it is by definition a non linear problem –it depends on a priori information –it almost always depends on orbit dynamics –orbit predictions are used to correct the raw observations and to set-up the design matrix –the orbit prediction model is not necessarily accurate the first time you apply it

27 Least squares

28 Minimize cost function The way the A matrix is computed completely depends on the type of observations and parameters in your problem. We will distinguish between problems that contain orbit parameters and problems that do not. Our first task will always be to model an orbit in the best possible way given the existing situation This task is called orbit prediction

29 Example” Initial state vector estimation in POD Task: determine the size, orientation and position of the arrow, it determines whether you hit the bull’s eye

30 Variational equations Example  : initial state vector component, terms in force model etc

31 Set-up parameter estimation program In reality orbit parameters are estimated from observations like range, Doppler or camera to the satellite or inbetween satellites Orbit prediction method –Numerically stable schemes are used –Choice initial state vector –Definition satellite acceleration model Variational method –Define parameters that need to be adjusted using least squares –Iterative improvement of these parameters –Use is made of the variational equations

32 Parameters in POD Station coordinates Station related parameters (clock, biases) Initial state vector elements of satellite orbits Parameters in acceleration models satellite Other satellite related parameters (clock, biases, etc) Signal delay related parameters Earth rotation related parameters Gravity field related parameters

33 Organization parameter estimation For large scale batch problems: –separation of arc -- and common parameters –combination of normal matrices and right hand sides –choice of optimal weight factors for combination –example: development of earth models like EGM96 Sequential problems –apart from the adjustment procedure there is a state vector transition mechanism –During transition state vector and variance matrix are advanced to the next time step (normally with a Kalman filter) –Example: JPL’s GPS data processing method