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 Relativistic precession - numerology? (Read 9362 times)
 wil Uploader I Love YaBB 2! Posts: 41 Relativistic precession - numerology? 11/04/11 at 22:24:04   I tested the precession of Mercury and I see it's lower, but probably because loss of precision, anything else.   Some numbers.   Only Jupiter. Official contribution to Mercury's precession 152'' / cy.   ===== J- Euler-Crommer t = 512 2011-6: 29.12698, 48.31617 2111-6: 29.15977, 48.27495 0.03279 -0.04122 = -0.00843 = -30'' !!! big precission loss - method is too weak   ===== J-RK4 -> t = 512 2011-6: 29.15462, 48.31617 2111-6: 29.23939, 48.27495 0.08477 -0.04122 = 0.04355 = 156.8   ->1024 2211-6: 29.325356, 48.233859 0.085966 -0.041091 = 0.044875 = 161.5 > 152 ->2048 2311-6: 29.408837, 48.192791 0.083481 -0.041068 = 0.042413 = 152.7 ->4096 2411-6: 29.49163, 48.15208 0.082793 -0.040711 = 0.042082 = 151.5 ->8192 2511-6: 29.574675, 48.110483 0.083045 -0.041597 = 0.041448 = 149.2 -> 16384 2611-6: 29.656582, 48.069174 0.081907 -0.041309 = 0.040598 =  146.2   Precession RK4 with step t=1024s is optimal, i.e. truncation error =~ roundoff error. Lower timestep needs more steps, then roundof error acumulates faster. And bigger timestep grows truncation error (fourth order).   We can check this: period of Mercury = 88 days =~ 7000000 s number of steps: 7000000/1024 = 7000   1/7000^4 = 4e-16 -> this is about precission of double (53 bits), i.e. maximal precission of calculations - more is impossible! Then we can't recreate of the full precession of Mercury using only double precission arithmetic! ---   And from direct calculation we get more precesson: 3/2 pi * 1/1047 * (0.387/5.2)^3 / sqrt(1-0.2^2) = 162'' / cy [415 orbits] it's only part - quadrupole... not all.   --- With full sol system it's simmilar, but now optimal step is larger (because other planets have longer periods).   t = 4096: 2211-9: 29.7065, 48.0647 0.2792 -0.1263 = 0.1529 = 550.44'' > 532'' - officjal !     I don't known what Le Verrier... trained - probably Vulcanic numerology! Back to top IP Logged
 wil Uploader I Love YaBB 2! Posts: 41 Re: Relativistic precession - numerology? Reply #2 - 11/05/11 at 10:22:21   http://www.coolissues.com/gravitation/Perihelia/peri.htm and here: http://www.mathpages.com/home/kmath280/kmath280.htm   It's evident that in modern astronomy all want about 40'' of additional precession and they get, wath want!   Why 40''? Because light speed: c = 3e8 m/s, and all theories based on retarded forces give about 40'' additional precession. Weber force, Ether theories, and GR - based on field theory analogous to Maxwell electromagnetism (based on Ether theories of coures - propagation of forces across some medium, or indirectly transfered by virtual particles).   Unfortunately Jupiter causes more than 162'' and remains now only ~30''. Additional 30'' we can explain also... it's not too difficult. Just a little look around and quickly find the actual cause ... which are not included in the calculation.     ----   Error curve has only one extremum - minimum. E-C method accumulates gigantic roundoff errors, and then all little forces dissapear completeley in those errors.   And errors always generate backward rotation of orbits. It is effect of natural resonance of errors in circular Kepler problem.   Can You make analogous curve for RK4, and second - only Jupiter? Maybe then we can see something more interesting. Back to top IP Logged
Tony

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Re: Relativistic precession - numerology?
Reply #3 - 11/06/11 at 22:37:17

Quote from wil on 11/05/11 at 10:22:21:
...Can You make analogous curve for RK4, and second - only Jupiter?
Maybe then we can see something more interesting.

I just made one for RK4, Jupiter only:
 Time Step arcsec/cy 64K 612 32K 180 16K 154 8K 152 4K 152 2K 152 1K 152

Leap frog (2nd order), Jupiter only
 Time Step arcsec/cy 1K 41 512 124 256 145 128 151

I'll see if I can fill in a few more slower times for leap frog, but it takes the computer a while to do the slower runs.  In any case, we can see where its asymptotically heading... ~152, and it will probably reach there at t ~64.

Using the 4K trial, I plotted the precession century-by-century.  Notice that not every century shows 152".  That was simply the final answer after 10K years.

Here's a similar graph, but with all the planets included.

I was wrong earlier when I said that some centuries it actually precesses backwards.  I should have said that some years it actually precesses backwards.  100 years is enough averaging to at least get it going in the right direction.

« Last Edit: 11/07/11 at 16:25:49 by Tony »
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 wil Uploader I Love YaBB 2! Posts: 41 Re: Relativistic precession - numerology? Reply #4 - 11/07/11 at 17:38:51   RK4 Jupiter     step 1024s:  160.7  154.3  156.3  148.7 ... x 10 ----- 155.0 average   Indeed, it decreases and then increases again, etc.   But these are errors, not the actual precession.   For RK4 amplitude is much smaller than the Leap frog. Besides, here the average has been steadily decreasing, giving a total function of type:   err = t * (A + B * cos (wt)) + C;   constant C is not seen here - is unknown.   But with the full system, I always have about 550'', never 530''!   1K: 552'' average - 400 years only. 32K: 556'' - over 1000 y.   The problem was not resolved (numerically). Need to use better methods (higher order, eg: 8) and higher precision of calculations. Back to top IP Logged
 Tony YaBB Administrator Posts: 1058 Gender: Re: Relativistic precession - numerology? Reply #5 - 11/08/11 at 09:28:36   Quote from wil on 11/07/11 at 17:38:51:RK4 Jupiter step 1024s: 160.7 154.3 156.3 148.7 ... x 10 ----- 155.0 average   Where did these numbers come from, and why are we multiplying by 10 to get an average? Back to top IP Logged
 wil Uploader I Love YaBB 2! Posts: 41 Re: Relativistic precession - numerology? Reply #6 - 11/08/11 at 15:41:12   Data is from Your program.   x10 means 10 items; Average of 10 measurements every 100 years = 1000 years. Back to top IP Logged
 Tony YaBB Administrator Posts: 1058 Gender: Re: Relativistic precession - numerology? Reply #7 - 11/08/11 at 17:37:59   Did you generate the numbers, or did you get them from the chart I posted?  When I run RK4 at t=1024, those aren't the numbers I get. Back to top IP Logged
 wil Uploader I Love YaBB 2! Posts: 41 Re: Relativistic precession - numerology? Reply #8 - 11/09/11 at 07:13:51   Jupiter RK4 1024s                                2005 01 01   02 09 16      29.108      48.325      77.432              2105 01 01   08 16 12      29.193      48.283      77.477      160.69        2205 01 01   14 23 08      29.277      48.242      77.520      154.35        2305 01 01   20 30 04      29.362      48.202      77.563      156.32        2405 01 01   02 37 00      29.444      48.160      77.604      148.74        2505 01 01   08 43 56      29.526      48.119      77.645      146.42        2605 01 01   14 50 52      29.609      48.077      77.687      149.63        2705 01 01   20 57 48      29.693      48.036      77.729      151.84        2805 01 01   03 04 44      29.779      47.994      77.773      159.47        2905 01 01   09 11 40      29.863      47.953      77.816      154.70        3005 01 01   15 18 36      29.949      47.912      77.861      160.78      154.29 average last 10  3105 01 01   21 25 32      30.031      47.871      77.902      149.03        3205 01 01   03 32 28      30.114      47.830      77.943      147.85        3305 01 01   09 39 24      30.197      47.788      77.985      149.58        3405 01 01   15 46 20      30.279      47.746      78.026      147.14        3505 01 01   21 53 16      30.365      47.704      78.069      157.49        3605 01 01   04 00 12      30.449      47.663      78.112      154.52        3705 01 01   10 07 08      30.536      47.622      78.157      162.85        3805 01 01   16 14 04      30.619      47.580      78.199      150.62        3905 01 01   22 21 00      30.702      47.539      78.241      150.47        4005 01 01   04 27 56      30.785      47.497      78.283      150.31      151.99  4105 01 01   10 34 52      30.867      47.456      78.323      143.39        4205 01 01   16 41 48      30.952      47.414      78.366      155.14         average 152.79 it decreases steadily... I check more.   It should rather be measured in the same configuration, ie multiple of Jupiter-Mercury synodic, about 12 years. Tmj = 0.24584;   49 * Tmj = 12.046 y 49 * 65 * Tmj = 783.0004 y Back to top IP Logged