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Numerical Simulations of the Light Propagation in the Gravitational Field of Moving Bodies

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. . numerical simulations are desired. Reasons. Light propagation in the field of moving bodies is a complicated

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Numerical Simulations of the Light Propagation in the Gravitational Field of Moving Bodies

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  1. 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

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

  3. 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 • NUMERICAL: • Post-Minkowskian differential equations of motion • (specially derived for this investigation) 2. Post-Newtonian differential equations of motion

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

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

  6. 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

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

  8. The body‘s trajectories for analytical solutions

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

  10. Simulations: boundary problem • Vectors nfor the numerical and analytical solutions are compared • Distance is chosen so that the differences in n‘sare maximal

  11. Simulations: boundary problem For the most accurate light trajectory the impact parameteris the minimal one with Three series of the simulation for gravitating bodies on: • 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

  12. 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

  13. Results: parabolic motion

  14. Results: coplanar circular motion

  15. Results: realistic motion (DE405)

  16. Simulations: discussion (1) • The three series of the simulations are in reasonable agreement • Three solutions coincide within 0.002 as: • 1. Numerical post-Minkowskian • 2. Simplified analytical post-Minkowskian • 3. Analytical post-Newtonian • for uniformly moving bodies with tref=tca

  17. Results: realistic motion (DE405)

  18. Simulations: discussion (2) • Two post-Newtonian analytical models coincide within 0.001 as: • 1. Post-Newtonian for motionless bodies with tref=tca • 2. Post-Newtonian for motionless bodies with tref=tr • maximal difference: 0.00075 as for Jupiter • The error of these two analytical models: • 0.75 as for parabolic trajectories • 0.18as for reliastic motion

  19. Results: realistic motion (DE405)

  20. Simulations: discussion (3) • The simplest analytical post-Newtonian model for motionless bodies • with tref=tois 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 tref=to has errors between 0.1 and 1 as No reason to use these 3 models: better accuracy can be achieved for the same price...

  21. Results: realistic motion (DE405)

  22. 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 tref=tca • or at tref=tr

  23. 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 tref=tca

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