Numerical Simulations of the Light Propagation in the Gravitational Field of Moving Bodies Sergei A. Klioner & Michael Peip GAIA RRF WG, 3rd Meeting, Dresden, 12 June 2003

Reasons  numerical simulations are desired Light propagation in the field of moving bodies is a complicated theoretical problem Many possible „points of view“ and corresponding solutions Effects are much larger than 1  as Not easy to compare analytically

Possible solutions I. I.NUMERICAL: 1. 1.Post-Minkowskian differential equations of motion (specially derived for this investigation) 2. Post-Newtonian differential equations of motion both can be integrated numerically (initial value or two point boundary problem) the post-Newtonian equations are contained in the post-Minkowskian equations only for checks 

Possible solutions II.ANALYTICAL 1. 1.Post-Minkowskian analytical model (Kopeikin, Schäfer, 1999). some non-integrable parts are dropped

The Kopeikin-Schäfer solution in a nutshell body unperturbed light perturbed light „body-rest frame“ at rest uniform rectilinear post-Newtonian Schwarzschild solution uniform rectilinear uniform rectilinear the Kopeikin solution for uniformly moving bodies BCRS  Lorenz transformation Klioner, 2003: A&A, 404, 783

The Kopeikin-Schäfer solution in a nutshell For uniformly moving bodies: The solution can be derived and understood from almost trivial calculations The retarded moment is not essential for the solution The same technique can be applied for bodies with full multipole structure Klioner, 2003: A&A, 404, 783

Possible solutions II.ANALYTICAL (continued) 1. 1.Post-Minkowskian analytical model (Kopeikin, Schäfer, 1999). some non-integrable parts are dropped 2. 2.Post-Newtonian analytical model for uniformly moving bodies (Klioner, 1989):

The body‘s trajectories for analytical solutions

Possible solutions II.ANALYTICAL (continued) 2. 2.Post-Newtonian analytical model for uniformly moving bodies (Klioner, 1989): 6 choices of the constants

Simulations: boundary problem Vectors n for the numerical and analytical solutions are compared Distance is chosen so that the differences in n ‘s are maximal

Simulations: boundary problem For the most accurate light trajectory the impact parameter is the minimal one with Three series of the simulation for gravitating bodies on: 1. 1.parabolic trajectories with realistic velocities and accelerations 2.coplanar circular orbits with realistic semi-major axes 3.realistic orbits (DE405) All possible mutual configurations of the observer and the body are checked on a fine grid

Technical notes ANSI C program with „long double“ arithmetic: up to 18 decimal digits on INTEL-like and 34 digits on SUN SPARC Everhart integrator efficient even for 34-digit arithmetic: accuracy is checked by backward integration Highly optimized code (partially with CODEGEN): about 1 million light trajectories for each body

Results: parabolic motion

Results: coplanar circular motion

Results: realistic motion (DE405)

Simulations: discussion (1) The three series of the simulations are in reasonable agreement Three solutions coincide within  as: 1. Numerical post-Minkowskian 2. Simplified analytical post-Minkowskian 3. Analytical post-Newtonian for uniformly moving bodies with t ref =t ca

Results: realistic motion (DE405)

Simulations: discussion (2) Two post-Newtonian analytical models coincide within  as: 1. Post-Newtonian for motionless bodies with t ref =t ca 2. Post-Newtonian for motionless bodies with t ref =t r maximal difference:  as for Jupiter The error of these two analytical models: 0.75  as for parabolic trajectories 0.18  as for reliastic motion

Results: realistic motion (DE405)

Simulations: discussion (3) The simplest analytical post-Newtonian model for motionless bodies with t ref =t o is too inaccurate (up to 10 mas or even more) The simplified algorithm to compute the retarded moment increases the error to 0.3  as The analytical post-Newtonian model for uniformly moving bodies with t ref =t o has errors between 0.1 and 1  as No reason to use these 3 models: better accuracy can be achieved for the same price...

Results: realistic motion (DE405)

Conclusions (I) If an accuracy of 0.2  as is sufficient: 1. Simple post-Newtonian analytical model for motionless bodies. 2. The position of the body can be taken either at t ref =t ca or at t ref =t r

Conclusions (II) If an accuracy better than 0.2  as is required: 1. The analytical post-Minkowskian solution with the non-integrable parts dropped or 2. The post-Newtonian analytical solution for uniformly moving bodies with t ref =t ca