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Relativistic precession - numerology? (Read 6856 times)
wil
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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!
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Tony
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Re: Relativistic precession - numerology?
Reply #1 - 11/04/11 at 23:13:57
 
Nice work Wil,
 
So I guess this means you got the new Alpha version working?
 
Where did you get the figure of 152"/cy for Jupiter's contribution?  I would have guessed it is more than that.  Perhaps Venus and Earth are larger contributors than I thought.
 
I've tried this before in Euler-Crommer.   The results are here:
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1158230793/5#5
 
As the time step approaches 0, the precession approaches the Newtonian-expected value of 531"/cy.  But it's way off until you get to below t=128s.
 
In making the above linked graph, I ran the sim for 10000 years.  As I looked closely at the data, I noticed that Mercury didn't always precess 531" every century.  There were some centuries where it actually precessed backwards.  But over a long period of time, the time-averaged rate is 531".  So you might want to try running it for more than a century.  You can use the "Output Data" feature in the File menu to create a comma-deliminated text file of your simulation.  Then you can use Excel to examine the data, adding columns precession.  Then you can see that it is not a smooth journey into the future.
 
Based upon your data, it looks like RK4's optimal time step is 2048, as 152.7 is closer to 152 than 161.5 is.
 
Tip:  you can set the time step to any value you want (rather than being limited to powers of 2) if you use the Autopilot.
 
The preferences menu has a setting for steps for radian.  RK4 variable time step uses this.  If any object is moving fast enough to complete a radian of its orbit in fewer than the stated value, then it slows the time step.  This way you can run sims with close encounters at higher speeds.  I set the default to 150 because that's the lowest I can make the number and still get good results for things like predicting solar eclipses, transits of Mercury or Venus across the Sun, centuries into the future or past.
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wil
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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.  Shocked
 
----
 
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.
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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 Steparcsec/cy
64K612
32K180
16K154
8K152
4K152
2K152
1K152

 
Leap frog (2nd order), Jupiter only
Time Steparcsec/cy
1K41
512124
256145
128151

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.
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« Last Edit: 11/07/11 at 16:25:49 by Tony »  
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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.
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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?
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wil
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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.
 
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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.
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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
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Re: Relativistic precession - numerology?
Reply #9 - 11/10/11 at 17:35:35
 
Quote from wil on 11/07/11 at 17:38:51:


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.

The problem was not resolved (numerically).
Need to use better methods (higher order, eg: 8) and higher precision of calculations.

 
Leapfrog should have higher amplitudes if you also used a time step of 1024.  As is demonstrated earlier, Leapfrog needs a much lower time step to compete with RK4.
 
If these were errors and not actual precession, I would expect that the same simulation run at different time steps would yield different answers.  I just tried RK4 @ t=2048 and RK4 @ t=1024.  The two were never more than 0.08 arcseconds apart from each other.  This difference probably would have been lower if I had chosen to compare t=1024 to t=512.  So you are seeing actual precession and insignificant numerical error.
 
The amplitudes you see are related to your other observation that "It should be measured in the same configuration".  I come up with 11.8625 years.  Try sampling your data every 11.8625 years, rather than every century, and you'll probably get a much flatter graph.  The method I was using was to just go 10K years into the future.  No need to average anything.  Just take final longitude of perihelion minus initial longitude of perihelion equals delta longitude of perihelion, and divide by 100 centuries, taking advantage of the fact that at t approaches infinity, the graph will asymptotically approach the correct answer.
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Re: Relativistic precession - numerology?
Reply #10 - 11/12/11 at 14:47:54
 
Quote from Tony on 11/10/11 at 17:35:35:

Leapfrog should have higher amplitudes if you also used a time step of 1024.  As is demonstrated earlier, Leapfrog needs a much lower time step to compete with RK4.

 
LF with 1024s shows random numers sequence, like this:
-45, 180, 6, 54, -100, 5, -37, ... Cheesy
and average is well below 150.
 
Quote:
If these were errors and not actual precession, I would expect that the same simulation run at different time steps would yield different answers.  I just tried RK4 @ t=2048 and RK4 @ t=1024.  The two were never more than 0.08 arcseconds apart from each other.  This difference probably would have been lower if I had chosen to compare t=1024 to t=512.  So you are seeing actual precession and insignificant numerical error. The amplitudes you see are related to your other observation that "It should be measured in the same configuration".  I come up with 11.8625 years.  Try sampling your data every 11.8625 years, rather than every century, and you'll probably get a much flatter graph.  The method I was using was to just go 10K years into the future.  No need to average anything.  Just take final longitude of perihelion minus initial longitude of perihelion equals delta longitude of perihelion, and divide by 100 centuries, taking advantage of the fact that at t approaches infinity, the graph will asymptotically approach the correct answer.

 
So, show me a period of time - a few thousand years - in which the precession increases (on average).
-------
 
Maybe calculation of orbital's elements is incorrect?
Mercury moves around center of mass of the Sun + Mercury, rather than the Sun directly, so here should  be some discrepancy.
 
Or measure directly of minimal distance, and get phase... but I'm not certain, if it was realy measured.
------
 
Leap Frog is exactly so slow, as RK4 (on my Pentium 4), and errors are bigger, of course. I don't see any sense to using it.
 
You can try improve simulation greatly, using only simple extrapolation. 'Leap Frog' (second order) is ideal for this purpose, alone is useles.
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Re: Relativistic precession - numerology?
Reply #11 - 11/14/11 at 09:51:19
 
Quote from wil on 11/12/11 at 14:47:54:

Maybe calculation of orbital's elements is incorrect?
Mercury moves around center of mass of the Sun + Mercury, rather than the Sun directly, so here should  be some discrepancy.

 
I think that it's realy a cause of deficit about 40'' / cy.
 
Can you correct calculation of orbital elements?
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Re: Relativistic precession - numerology?
Reply #12 - 11/14/11 at 10:15:25
 
Quote from wil on 11/14/11 at 09:51:19:
Quote from wil on 11/12/11 at 14:47:54:

Maybe calculation of orbital's elements is incorrect?
Mercury moves around center of mass of the Sun + Mercury, rather than the Sun directly, so here should  be some discrepancy.


I think that it's realy a cause of deficit about 40'' / cy.

Can you correct calculation of orbital elements?

 
I'm sure GravSim calculates correctly the orbital parameters , as per defintion the center of the Sun is the reference , to calculate every orbital parameter ( given an inertial frame)  . So there's is no need to choose for another formula . If the center of gravity of the solar system would be used as a reference things may become very very different . I can imagine in this case some precessions would be reduced to zero and other would evolve retrograde .
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Re: Relativistic precession - numerology?
Reply #13 - 11/14/11 at 13:11:44
 
Quote from frankuitaalst on 11/14/11 at 10:15:25:

I'm sure GravSim calculates correctly the orbital parameters , as per defintion the center of the Sun is the reference , to calculate every orbital parameter ( given an inertial frame)  . So there's is no need to choose for another formula . If the center of gravity of the solar system would be used as a reference things may become very very different . I can imagine in this case some precessions would be reduced to zero and other would evolve retrograde .

 
Yes. In astronomy the Sun is the reference object and, for this reason, orbital elements are approximate only.
But it is sufficient for most purposes, especially for meteorites, asteroids, comets, and probes - spacecrafts.
 
The exact orbit (instantaneous) is an ellipse with a focus in the barycenter of a given pair of bodies, and not in the center of the second body.
 
Differences are very small - practically unnoticeable.
For example Merrcury perihelion's precession discrepancy: about 40 arc seconds per hundred of years only!  Grin
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