Satellite observation systems and reference systems (ae4-e01)

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

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

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

Linear perturbations by C20 C20 not normalised, n: mean motion Ref: Seeber p 84

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.

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:

This is Y200

Y300

Y210 and Y211

Y320

Y330

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

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

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

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.

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

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]'; Demonstration numerical solution ordinary diff. eq. Example gun bullet physics In reality

Demonstration Numerical Implementation (2) Example Kepler orbit physics: 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]'; In reality

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 y0 to be propagated from y0 till yn. In this case a state vector at index j coincides with the time index t0+(j-1)*h where h is the integrator step size.

Example in MATLAB span = [0 14500]; state = [ 1e7 0 0 7e3]; option = odeset('RelTol',1e-10); [t,y] = ODE45('satdyn',span,state,option); plot(y(:,1),y(:,2))

Orbit prediction (2) • The orbit prediction problem is entirely driven by the choice of the initial state vector y0, 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.

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?

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

Function model (2) Ze S Rij B Rj Ri Ye Xe

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

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

Least squares

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

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

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

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

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

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 processingmethod

Satellite observation systems and reference systems (ae4-e01)