1 / 23

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

gwyn
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related