The motions of the planets in the Solar system are simulated.
Each planet is simulated as a two body problem, planet - Sun.
The Newtonian gravitational acceleration is replaced by
the post-Newtonian approximation of the acceleration predicted by GR.

For each planet is the advance of the perihelion measured to
show what GR predicts the rate of perihelion advance would be
if the only existing bodies in the Universe were the planet
and the Sun.

https://paulba.no/pdf/GRPerihelionAdvance.pdf

https://paulba.no/PerihelionAdvance.html

--
Paul

https://paulba.no/

On Thursday, 14 October 2021 at 12:25:01 UTC+2, Paul B. Andersen wrote:
> The motions of the planets in the Solar system are simulated.

Sure; they are simulated in - a banned by your ideology euclidean
3 dimensional space with a non local, independent on observer
time. But you will impudently lie that the simulation is using
your GR shit.

Paul B. Andersen wrote:

> The motions of the planets in the Solar system are simulated.
> Each planet is simulated as a two body problem, planet - Sun.
> The Newtonian gravitational acceleration is replaced by the
> post-Newtonian approximation of the acceleration predicted by GR.
> For each planet is the advance of the perihelion measured to show what
> GR predicts the rate of perihelion advance would be if the only existing
> bodies in the Universe were the planet and the Sun.
> https://paulba.no/pdf/GRPerihelionAdvance.pdf
> https://paulba.no/PerihelionAdvance.html

great, now model the galaxy rotation without dark matter and dark energy.
Win that noble prize.

On Thursday, October 14, 2021 at 7:25:01 AM UTC-3, Paul B. Andersen wrote:

You can't use 2-Body calculations for each planet, to obtain a general result. So: what do you want to probe with your formulae?

The general equation for the N-Body problem is highly NON LINEAR, and imply to use about 8x7 formulae (Newton or GR) computed
along 100 years to obtain a true value of Mercury's perihelion shift. Not to mention the influence on other planets perihelion shift, with
the impact of the longest orbital period (Neptune).

Only by approximations, you add each 2-Body result to obtain 575.31 arcsec/Julian century for Mercury shift, as it was obtained in
this paper:

Precession of Mercury’s Perihelion from Ranging to the MESSENGER Spacecraft
Ryan S. Park, William M. Folkner, Alexander S. Konopliv, James G. Williams, David E. Smith, and Maria T. Zuber

The Astronomical Journal, 153:121 (7pp), © 2017 AAS
March https://doi.org/10.3847/1538-3881/aa5be2

--------------------------------- Excerpt from Page 6, Table 3 (simplified by me for this post) -------------------------------------------------------

Table 3
The Breakdown of Estimated Contributions and Uncertainties from the Planets, Asteroids, GE effect, LT effect, and
Solar Quadrupole Moment to the Precession Rate of Mercuryʼs Perihelion Computed in the coordinate Frame Defined
in Section 4 (i.e., along Mercuryʼs Mean Orbit Plane)

Mercury: 0.005 arcsec/century
Venus: 277.4176 arcsec/century
Earth/Moon: 90.8881 arcsec/century
Mars: 2.4814 arcsec/century
Jupiter: 153.9899 arcsec/century
Saturn: 7.3227 arcsec/century
Uranus: 0.1425 arcsec/century
Neptune: 0.0424 arcsec/century

Total for Newtonian gravity: 532.2896 arcsec/century

Gravitoelectric (Schwarzschild-like): 42.9799 arcsec/century

Total for GR gravity: 575.2695 arcsec/century

Total for other perturbations: 0.0405 arcsec/century

Total for Newtonian and non-Newtonian contributions: 575,31 arcsec/century

-------------------------------------------------------------------------- End of excerpt ------------------------------------------------------------------------------------------

In your case, the result of 2-Body results per orbit (rad/orbit) with your formula:

δφᵤ = 6π (GM)² /[G(M + mᵤ) aᵤ..(1 −eᵤ²) c²] = Σᵤ (6π.G.M²/c²)/[(M + mᵤ) aᵤ.(1 −eᵤ²)] , for u = 1, 2, 3, ....,8

dismiss the effects of the other bodies plus non-linear perturbations, which worsen as you go far away from the Sun and
closer to massive planets, which is a simplification in a HIGH DEGREE. In your Table 3: Perihelion advance of the planets

Mercury: 42.982717 "/century
Venus: 8.624984 "/century
Earth: 3.838873 "/century
Mars: 1.350975 "/century
Jupiter: 0.062276 "/century
Saturn: 0.013674 "/century
Uranus: 0.002382 "/century
Neptune 0.00077 "/century

It is evident that your result is ALMOST THE SAME as the Gravitoelectric effect (Schwarzschild-like): 42.9799 arcsec/century
that the quoted paper display in Table 3, but it doesn't imply that the effect on Mercury can be generalized as a 2-Body problem
for the rest of the planets.

If true, it could be solved with an ENIAC computer by 1945.

So, GR can be used to explain Mercury's perihelion advance but, unless an N-Body solution is used, it can't be generalized.

As far as I know, gravitational effects on the Solar System (with advanced solutions within agencies for Space Dynamics) are
calculated using discrete approximations in supercomputers. And this computation is beyond the "simplicity" of GR, when
complex perturbations have to be accounted.

And, at any case, Newton is more reliable than GR for PERMANENT CALCULATIONS of trajectories of Deep Space Sondes or
for Sondes sent to Mars, for instance.

Here is other paper, from a Russian physicist, who details the problems related above. He and his staff have more than 20 years
of experience on in research on the evolution of orbital and rotary movement of the Earth and planets. Watch the equations that
are used for numerical integration of the equations of interaction of N bodies under Newton’s law of gravity. The computer program
is recursive, with each run adapting parameters as time evolves. It's used for a finite number of loops, due to the time involved.

Gravitation, Field, and Rotation of Mercury Perihelion
Joseph J. Smulsky
Institute of the Earth Cryosphere of the Russian Academy of Sciences
Siberian Branch, Tyumen, RUSSIA

https://www.researchgate.net/publication/228428991_Gravitation_field_and_rotation_of_mercury_perihelion

QUOTE: "It is established that under the influence of Newtonian gravity, the perihelion turns in a motionless reference system by
529.9" per century, whereas the data of observation shows that it turns on 582.3" per century. In early work, the effect rotation
of the Sun on the movement of planets was not taken into account."

> The motions of the planets in the Solar system are simulated. Each planet is simulated as a two body problem, planet - Sun.
> The Newtonian gravitational acceleration is replaced by the post-Newtonian approximation of the acceleration predicted by GR.
>
> For each planet is the advance of the perihelion measured to show what GR predicts the rate of perihelion advance would be
> if the only existing bodies in the Universe were the planet and the Sun.
>
> https://paulba.no/pdf/GRPerihelionAdvance.pdf" rel="nofollow" target="_blank">https://paulba.no/pdf/GRPerihelionAdvance.pdf
>
> https://paulba.no/PerihelionAdvance.html
>
> --
> Paul
>
> https://paulba.no/

On Thursday, October 14, 2021 at 3:02:01 PM UTC-3, Richard Hertz wrote:
Typo corrected. Σᵤ should be there:
δφᵤ = 6π (GM)² /[G(M + mᵤ) aᵤ.(1 −eᵤ²) c²] = Σᵤ (6π.G.M²/c²)/[(M + mᵤ) aᵤ.(1 −eᵤ²)] , for u = 1, 2, 3, ....,8
It is, instead:
δφᵤ = 6π (GM)² /[G(M + mᵤ) aᵤ.(1 −eᵤ²) c²] = (6π.G.M²/c²)/[(M + mᵤ) aᵤ.(1 −eᵤ²)] , for u = 1, 2, 3, ....,8 (8 Planets)

Richard Hertz wrote:

> On Thursday, October 14, 2021 at 3:02:01 PM UTC-3, Richard Hertz wrote:
>
> Typo corrected. Σᵤ should be there:
>
> δφᵤ = 6π (GM)² /[G(M + mᵤ) aᵤ.(1 −eᵤ²) c²] = Σᵤ (6π.G.M²/c²)/[(M + mᵤ)
> aᵤ.(1 −eᵤ²)] , for u = 1, 2, 3, ....,8
>
> It is, instead:
>
> δφᵤ = 6π (GM)² /[G(M + mᵤ) aᵤ.(1 −eᵤ²) c²] = (6π.G.M²/c²)/[(M + mᵤ)
> aᵤ.(1 −eᵤ²)] , for u = 1, 2, 3, ....,8 (8 Planets)

you cant, you have to sum them up.

On Thursday, October 14, 2021 at 1:02:01 PM UTC-5, Richard Hertz wrote:
> On Thursday, October 14, 2021 at 7:25:01 AM UTC-3, Paul B. Andersen wrote:
>
> You can't use 2-Body calculations for each planet, to obtain a general result. So: what do you want to probe with your formulae?

The simplest case, obviously. Anomalous precession does not require
additional planets to be manifest.

If you throw in all of the other planets, you get a lot of extra complexity
without really adding much in the way of extra insight:
https://drive.google.com/file/d/0B8XIf0XcrpOcSmdrY0VUVXh3dkE/view?usp=sharing&resourcekey=0-YaO_tG3fpDc5_KKg2zHmmQ

On Thursday, October 14, 2021 at 3:25:01 AM UTC-7, Paul B. Andersen wrote:
> The motions of the planets in the Solar system are simulated.
> Each planet is simulated as a two body problem, planet - Sun.

A genuine two-body solution takes into account the motions of both bodies, relative to their center of mass. The formula you are using in your simulation isn't a two-body formula, it assumes the Sun is stationary and just describes geodesic trajectories around a single gravitating body. That's the one-body solution. (Also, note that if you actually performed a separate two-body solution for each planet, the Sun's motion would be different in each of those individual solutions, so they wouldn't be coherently accounting for the Sun's motion.) Naturally, you're comparing your simulation results with the one-body analytic solution (to the first order) of general relativity. For purposes of assessing the relativistic contribution to the precession to within observation precision, the analytic solution of the one-body problem suffices.

On Thursday, October 14, 2021 at 8:47:47 PM UTC-3, Townes Olson wrote:

<snip>

> A genuine two-body solution takes into account the motions of both bodies, relative to their center of mass. The formula you are using in your simulation isn't a two-body formula, it assumes the Sun is stationary and just describes geodesic trajectories around a single gravitating body. That's the one-body solution. (Also, note that if you actually performed a separate two-body solution for each planet, the Sun's motion would be different in each of those individual solutions, so they wouldn't be coherently accounting for the Sun's motion.) Naturally, you're comparing your simulation results with the one-body analytic solution (to the first order) of general relativity. For purposes of assessing the relativistic contribution to the precession to within observation precision, the analytic solution of the one-body problem suffices.

δφᵤ = (6π.G.M²/c²)/[(M + mᵤ) aᵤ.(1 −eᵤ²)] , for u = 1, 2, 3, ....,8}
δφᵥ = (6π.G.M²/c²)/[(M + mᵥ) aᵥ.(1 −eᵥ²)] , for v = 1, 2, 3, ....,8}

δφᵤ/δφᵥ = [(1 + mᵥ/M) aᵥ.(1 −eᵥ²)]/[(1 + mᵤ/M) aᵤ.(1 −eᵤ²)] ≈ [(1 + mᵥ/M) aᵥ ]/[(1 + mᵤ/M) aᵤ ] , for u,v > 1

Mercury: 1
Venus: 2
Earth: 3
Mars: 4
Jupiter: 5
Saturn: 6
Uranus: 7
Neptune 8

It can be seen that the influence of contiguous planets is, on average, high. So, maybe they can't be neglected so easily.

For example, the quotient between these two planets gives a ratio:

δφ₂/δφ₃ = 2.247 for Venus and Earth

δφ₃/δφ₄ = 2.842 for Earth and Mars

δφ₅/δφ₆ = 4.554 for Jupiter and Saturn

I might be wrong with this interpretation, which is roughly of very first order (and as average within 100 years period).

Den 15.10.2021 01:47, skrev Townes Olson:
> On Thursday, October 14, 2021 at 3:25:01 AM UTC-7, Paul B. Andersen wrote:
>> The motions of the planets in the Solar system are simulated.
>> Each planet is simulated as a two body problem, planet - Sun.
>>
>> https://paulba.no/pdf/GRPerihelionAdvance.pdf
>>
>> https://paulba.no/PerihelionAdvance.html
>
> A genuine two-body solution takes into account the motions of both bodies, relative to their center of mass. The formula you are using in your simulation isn't a two-body formula, it assumes the Sun is stationary and just describes geodesic trajectories around a single gravitating body. That's the one-body solution. (Also, note that if you actually performed a separate two-body solution for each planet, the Sun's motion would be different in each of those individual solutions, so they wouldn't be coherently accounting for the Sun's motion.) Naturally, you're comparing your simulation results with the one-body analytic solution (to the first order) of general relativity. For purposes of assessing the relativistic contribution to the precession to within observation precision, the analytic solution of the one-body problem suffices.
>

It is a two body _simulation_. It means that the motion
of each body is caused by the gravitational acceleration
from the other body.
So it is _obviously_ done in the centre of gravity frame.
Both bodies are moving in elliptic orbits around the barycentre
- of course.

https://paulba.no/pdf/GRPerihelionAdvance.pdf

If the acceleration of the planet according to equation (3) is a⃗,
then the acceleration of the Sun is -(m/M)⋅a⃗

Note however that the advance of perihelion is measured
relative to the Sun.

The point with the simulation is really to evaluate
the acceleration in equation (3). Since this
acceleration gives a perihelion advance so close to
the theoretical value, the conclusion is that
it is a very good approximation of the gravitational
acceleration predicted by GR.

I will use this in a simulation of the 9-body simulation
of the Solar system. (A much more complex task, obviously.)

A Newtonian simulation of the solar system can be seen here:

https://paulba.no/SolarSystem/SolarSystem.jar
https://paulba.no/SolarSystem/Results.pdf
https://paulba.no/SolarSystem/Screenshots.pdf
https://paulba.no/SolarSystem/SolarSystem.pdf

--
Paul

https://paulba.no/

On Friday, 15 October 2021 at 09:49:42 UTC+2, Paul B. Andersen wrote:

> It is a two body _simulation_. It means that the motion
> of each body is caused by the gravitational acceleration
> from the other body.

What a heresy is that, Paul, poor idiot? Your insane guru
has discovered and announced that gravitational
acceleration doesn't exist and free-falling objects
are inertial.

On Friday, October 15, 2021 at 12:49:42 AM UTC-7, Paul B. Andersen wrote:
> It is a two body _simulation_. It means that the motion
> of each body is caused by the gravitational acceleration
> from the other body.

I guess one could call *anything* a two-body "simulation", including just treating it as a one-body problem. But, to be clear, the equation you're using is a one-body equation, in the sense that (as explained in your source) it is simply derived as an approximation to the paths of geodesics in Schwarzschild's solution around the Sun, which is purely a one-body solution. That's how it's used in your source, for comets and asteroids around the Sun. Now, although you didn't mention it in your write-up, you could alternately apply this equation (in terms of a naively defined but incoherent system of coordinates) first to an incremental motion of the planet in the Sun's field, and then to an incremental motion of the Sun in the planet's field (strictly two different coordinate systems), and you could call this a two-body simulation, but the whole reason there is no simple analytic two-body solution in general relativity is that it's non-linear, so solutions don't just "add up".

> So it is _obviously_ done in the centre of gravity frame.

You could add that to your write-up, and state that you are doing the alternate incremental procedure described above, and then provide some rationale (if you can think of one) for why this should be regarded as a valid "two-body simulation" with better accuracy than just using the Schwarzschild one-body solution (on which your equation is based).

> The point with the simulation is really to evaluate
> the acceleration in equation (3). Since this
> acceleration gives a perihelion advance so close to
> the theoretical value, the conclusion is that
> it is a very good approximation of the gravitational
> acceleration predicted by GR.

The fact that you reproduce the analytic *one-body precession* result to several significant digits doesn't lend support to the idea that this approach includes valid two-body effects. It's really just a plausibility argument confirming what we already knew, i.e., that the magnitude of the two-body effects for planets around the Sun is fairly negligible for purposes of evaluating the non-Newtonian precession.

On Friday, October 15, 2021 at 10:53:12 AM UTC-3, Townes Olson wrote:

<snip>

> The fact that you reproduce the analytic *one-body precession* result to several significant digits doesn't lend support to the idea that this approach includes valid two-body effects. It's really just a plausibility argument confirming what we already knew, i.e., that the magnitude of the two-body effects for planets around the Sun is fairly negligible for purposes of evaluating the non-Newtonian precession.

I think that the projection of the calculated precession for 1 orbit, as the original 1898 formula provides (rederived in 1915) to a 100 years
time lapse, with the multiplication of 1 orbit result by about 415 (orbits in 100 years) is also questionable since it has been proven recently
that such precession per orbit has random variations along time. So, a more serious study including other gravitational perturbations has
to be performed along a large enough number of Mercury's orbits, in order to obtain an approximation to the main value and its standard
dispersion. The final result can differ from the 43° written in stone since GR come out in Nov. 1915.

On Friday, October 15, 2021 at 7:26:56 PM UTC-7, Richard Hertz wrote:
> The final result can differ from the 43° written in stone since GR come out in Nov. 1915.

Utter Imbecile

43°is MEASURED.

On Friday, October 15, 2021 at 11:30:01 PM UTC-3, Dono. wrote:

<snip>

>
> 43°is MEASURED.

As usual, you appear as an opinionated ignorant, even with a BsC in Mathematics and a PhD in EinsteinLand.

The most accurate measurements about Mercury perihelion were done by the MESSENGER (MErcury Surface, Space
ENvironment, GEochemistry, and Ranging) probe in orbit, between 2011 and 2015, about Mercury to estimate the
precession of Mercury’s perihelion, along with many other parameters related to the planet.

Results about Mercury perihelion SHIFT were projected for a century and are ABSOLUTE: 575.3100 ± 0.0015″/century.

MESSENGER wasn't used to PROVE GR. The result from 4 years measurements (16 orbits around the Sun) didn't provide
a differential value for non-newtonian calculations (Schwarzschild-like effects).

Given the ABSOLUTE VALUE of 575.3100 ± 0.0015″/century, in 2017 a team of scientists generated the following paper,
at which they TRY to breakdown the 575.31 value into its THEORETICALLY computed components. I posted the values
in a recent post but you, as a cretin, don't acknowledge that post.

Get the paper and read it:

Precession of Mercury’s Perihelion from Ranging to the MESSENGER Spacecraft

Ryan S. Park, William M. Folkner, Alexander S. Konopliv, James G. Williams, David E. Smith, and Maria T. Zuber
(JPL, Caltech, NASA, MIT and Department of Earth, Atmospheric and Planetary Sciences). published 2017 February 21

I just post a summary of 3 subtotals (do your own work):

THEORETICAL COMPUTATION for Newtonian gravity (8 planets): 532.2896 ± 0.0008″/century
THEORETICAL COMPUTATION for Schwarzschild effect (Mercury): 42.9799 ± 0.0009″/century
THEORETICAL COMPUTATION for other perturbations (Newton): 0.0405 ± 0.0104″/century
----------------------------------------------------------------------------------------------------------------------------------------------------------
TOTAL PROJECTED value from MESSENGER mission 2011-2015: 575.3100 ± 0.0015″/century,

Do you UNDERSTAND? The most accuracy value measured on 4 years and projected to 100 years is the one above.

Only theoretical decomposition in EXACTLY 19 parameters was obtained by the teamwork. THEORETICAL!

And the contribution of GR (1 in 19) is based on the 1915 GR formula, presented by Einstein on Nov. 18, 1915 (his 43").

So, STOP being a jerk and assume it: nothing changed in 100 years. The 43" value is the one that Einstein presented that day.
In Einstein's words, on a translation ending his Nov. 18, 1915 paper about Mercury:

"This calculation leads to the planet Mercury to move its perihelion forward by 43 ′′ per century, while
the astronomers give 45 ′′±5′′, an exceptional difference between observation and Newtonian theory.
This has great significance as full agreement."

Now, try to DISPROVE both values (NASA 2017 and Einstein 1915), imbecile.

The famous and centenary 43" are A CALCULATION done by Einstein 105 years ago. And for JPL, NASA, MIT, Caltech, etc., nothing
has changed since then. And such a thing is extremely curious as it shows an effort to prove Einstein right one more time.

Who wrote the paper are suckers. All of them.

On 10/14/21 5:08 PM, Prokaryotic Capase Homolog wrote:
> If you throw in all of the other planets, you get a lot of extra
> complexity without really adding much in the way of extra insight:

Extra insight, perhaps not much. But if you want to actually TEST THE
THEORY, you must include all the planets, and even some large asteroids
-- many of them affect the precessions of perihelia more than the GR effect.

Tom Roberts

On Friday, October 15, 2021 at 9:19:22 PM UTC-7, Richard Hertz wrote:
> On Friday, October 15, 2021 at 11:30:01 PM UTC-3, Dono. wrote:
>
> <snip>
>
> >
> > 43°is MEASURED.
>

> The famous and centenary 43" are A CALCULATION done by Einstein 105 years ago.

Stubborn imbecile,

43° was a measurement KNOWN since the times of LeVerrier (and even before him). Keep eating shit.

On Friday, October 15, 2021 at 10:33:26 PM UTC-7, Dono. wrote:
> On Friday, October 15, 2021 at 9:19:22 PM UTC-7, Richard Hertz wrote:
> > On Friday, October 15, 2021 at 11:30:01 PM UTC-3, Dono. wrote:
> >
> > <snip>
> >
> > >
> > > 43°is MEASURED.
> >
> > The famous and centenary 43" are A CALCULATION done by Einstein 105 years ago.
> Stubborn imbecile,
>
> 43° was a measurement KNOWN since the times of LeVerrier (and even before him). Keep eating shit.
https://www.math.toronto.edu/~colliand/426_03/Papers03/C_Pollock.pdf

On Saturday, October 16, 2021 at 2:33:26 AM UTC-3, Dono. wrote:
> On Friday, October 15, 2021 at 9:19:22 PM UTC-7, Richard Hertz wrote:
> > On Friday, October 15, 2021 at 11:30:01 PM UTC-3, Dono. wrote:

<snip>

> > > 43°is MEASURED.
> >
> > The famous and centenary 43" are A CALCULATION done by Einstein 105 years ago.

> Stubborn imbecile,
>
> 43° was a measurement KNOWN since the times of LeVerrier (and even before him). Keep eating shit.

DONO: You are a disgrace as a lifeform and a pathetic and lame Einstein's pretorian guard.
He would spit in your face if he was alive.
You are a disgrace for everyone standing up to defend Einstein, due to your continuous tirade of lies.

Fucking retarded. I don't know how do you manage to live with yourself, but it might be your due punishment.

Now read this original, in custody by the University of Jerusalem, Princeton, Harvard and almost any site licensed to
post this document, presented by Einstein by Nov. 18, 1915, to the

Preussische Akademie der Wissenschaften,
Sitzungsberichte, 1915 (part 2), 831–839

Volume 6: The Berlin Years: Writings, 1914-1917 Page 242

https://einsteinpapers.press.princeton.edu/vol6-doc/270
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie.
Von A. Einstein.
...........................
"Die Rechnung liefert fur den Planeten Merkur ein Vorschreiten des Perihels um 43" in hundert Jahren, während die
Astronomen 45" ±5" als unerklärten Rest zwischen Beobachtungen und Newtonscher Theorie [17] angeben.
Dies bedeutet volle Übereinstimmung."
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Translation of this excerpt, the last part of such paper:
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Explanation of the perihelion movement of Mercury from the general theory of relativity.
By A. Einstein.
.........................
"For the planet Mercury, the calculation yields an advance of the perihelion by 43" in a hundred years, while the
Astronomers state 45 "± 5" as the unexplained remainder between observations and Newton's theory [17].
This means full agreement. "
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

On Saturday, October 16, 2021 at 2:53:27 AM UTC-3, Dono. wrote:

<snip>

> > 43° was a measurement KNOWN since the times of LeVerrier (and even before him). Keep eating shit.
> https://www.math.toronto.edu/~colliand/426_03/Papers03/C_Pollock.pdf

You and Pollock are two retarded of the same kind: a disgrace for humanity.

Even the formula (42) from Pollock is WRONG, imbecile both of you!

He wasn't even capable to copy the 1915 formula correctly. The formula (42) in Pollock's paper has 1/c missing
on the expression of precession.

Shame on you and shame on him.

IDIOTS!!!!

Formula from paper on Nov. 18, 1915

𝜖 = 24π³.a²/[c² . T². (1- e²)]

On Saturday, 16 October 2021 at 07:59:01 UTC+2, Richard Hertz wrote:
> On Saturday, October 16, 2021 at 2:33:26 AM UTC-3, Dono. wrote:
> > On Friday, October 15, 2021 at 9:19:22 PM UTC-7, Richard Hertz wrote:
> > > On Friday, October 15, 2021 at 11:30:01 PM UTC-3, Dono. wrote:
>
> <snip>
>
> > > > 43°is MEASURED.
> > >
> > > The famous and centenary 43" are A CALCULATION done by Einstein 105 years ago.
>
> > Stubborn imbecile,
> >
> > 43° was a measurement KNOWN since the times of LeVerrier (and even before him). Keep eating shit.
> DONO: You are a disgrace as a lifeform and a pathetic and lame Einstein's pretorian guard.
> He would spit in your face if he was alive.

He was actually quite similiar to Dono. There is an
interview, anyone opposing him was a fucken
antisemitist.

Den 14.10.2021 20:01, skrev Richard Hertz:
> On Thursday, October 14, 2021 at 7:25:01 AM UTC-3, Paul B. Andersen wrote:
>
> You can't use 2-Body calculations for each planet, to obtain a general result. So: what do you want to probe with your formulae?

https://paulba.no/pdf/GRPerihelionAdvance.pdf

If the only objects in the universe were a planet with
mass m and a star with mass M, and the planet and the star
were orbiting each other with semi-major axis a and
eccentricity e, then GR predicts that the perihelion
advance per orbit δϕ would be as shown in equation (8).

If we have a planet with mass m, and a star with mass M,
and the distance vector star->planet is r⃗ and the velocity of
the planet relative to the star is v⃗, then GR predicts that
the gravitational acceleration of the planet caused by
the star is a⃗ as shown in equation (3).

To test this equation, we can pretend that the only
objects in the universe are a planet with mass m
and a star with mass M, and make a two body simulation
where the acceleration of the planet is a⃗ , and the acceleration
of the star is -(m/M)⋅a⃗ . We can measure the perihelion
advance per orbit, and compare it with GR's prediction δϕ in eq.(8).

This is what I have done for all the planets, and the result
is that the accordance between the simulated values and eq.8
is very good for all the planets.

So what I have achieved is to show that equation (3)
is a very good approximation of the acceleration of
a planet caused by the Sun.

>
> The general equation for the N-Body problem is highly NON LINEAR, and imply to use about 8x7 formulae (Newton or GR) computed
> along 100 years to obtain a true value of Mercury's perihelion shift. Not to mention the influence on other planets perihelion shift, with
> the impact of the longest orbital period (Neptune).

Of course.
https://paulba.no/SolarSystem/SolarSystem.pdf
Equation (11) shows the Newtonian acceleration of object
number 'i' caused by all the other 8 objects in the solar system.

>
> Only by approximations, you add each 2-Body result to obtain 575.31 arcsec/Julian century for Mercury shift, as it was obtained in
> this paper:

I have done the Newtonian simulation.
The simulation:
https://paulba.no/SolarSystem/SolarSystem.jar
Some screenshots:
https://paulba.no/SolarSystem/Screenshots.pdf
The results:
https://paulba.no/SolarSystem/Results.pdf

If you look at the latter, you will see that according to
my simulation, the perihelion of advance of Mercury caused
by the Newtonian pull of the other planets is ≈ 532.3 "/century.
If you add 42.98 "/century, you get 475.28 "/century which
is very close to the value in the paper below, which includes
the relativistic effect.

>
> Precession of Mercury’s Perihelion from Ranging to the MESSENGER Spacecraft
> Ryan S. Park, William M. Folkner, Alexander S. Konopliv, James G. Williams, David E. Smith, and Maria T. Zuber
>
> The Astronomical Journal, 153:121 (7pp), © 2017 AAS
> March https://doi.org/10.3847/1538-3881/aa5be2

Since adding the relativistic value to the Newtonian
value seems to give such a good value, I will make a new
simulation of the solar system where I use equation (3) in:
https://paulba.no/pdf/GRPerihelionAdvance.pdf
for the acceleration of the planet caused by the sun,
and equation (11) in:
https://paulba.no/SolarSystem/SolarSystem.pdf
for the acceleration caused by the other planets.

--
Paul

https://paulba.no/

On Saturday, 16 October 2021 at 09:35:59 UTC+2, Paul B. Andersen wrote:
> Den 14.10.2021 20:01, skrev Richard Hertz:
> > On Thursday, October 14, 2021 at 7:25:01 AM UTC-3, Paul B. Andersen wrote:
> >
> > You can't use 2-Body calculations for each planet, to obtain a general result. So: what do you want to probe with your formulae?
> https://paulba.no/pdf/GRPerihelionAdvance.pdf
>
> If the only objects in the universe were a planet with
> mass m and a star with mass M, and the planet and the star
> were orbiting each other with semi-major axis a and
> eccentricity e, then GR predicts that the perihelion
> advance per orbit δϕ would be as shown in equation (8).

Stop lying, trash. The model you've used for simulation
and prediction was euclidean with galilean time. Even
you are not stupid enough to try applying your idiocies
for real, you just pretend you do, for some ideological
reasons.

On Saturday, October 16, 2021 at 12:04:02 AM UTC-5, tjrob137 wrote:
> On 10/14/21 5:08 PM, Prokaryotic Capase Homolog wrote:
> > If you throw in all of the other planets, you get a lot of extra
> > complexity without really adding much in the way of extra insight:
> Extra insight, perhaps not much. But if you want to actually TEST THE
> THEORY, you must include all the planets, and even some large asteroids
> -- many of them affect the precessions of perihelia more than the GR effect.

That last is an interesting point. I hadn't realized...

Den 16.10.2021 07:03, skrev Tom Roberts:
> On 10/14/21 5:08 PM, Prokaryotic Capase Homolog wrote:
>> If you throw in all of the other planets, you get a lot of extra
>> complexity without really adding much in the way of extra insight:
>
> Extra insight, perhaps not much. But if you want to actually TEST THE
> THEORY, you must include all the planets, and even some large asteroids
> -- many of them affect the precessions of perihelia more than the GR
> effect.
>
> Tom Roberts

I don't think the asteroids' contribution to the precession of
the perihelion of Mercury is very significant.

See Park & al: Precession of Mercury’s Perihelion from
Ranging to the MESSENGER Spacecraft

https://iopscience.iop.org/article/10.3847/1538-3881/aa5be2/pdf

Table 3: estimated contribution from asteroids: 0.0012 "/century
The estimated total precession of Mercury's perihelion is:
575.3100±0.0015 "/century, so the contribution from the asteroids
is less than the uncertainty in the total precession.

The contribution may be a bit bigger for the planets closer
to the asteroid belt, but I think I will keep ignoring
the asteroids in my simulation of the Solar system.

It would however be a simple matter to include the biggest
asteroids Ceres, Vesta, Pallas and Hygiea (as long as I don't
measure their perihelion advance), so I will consider it.

--
Paul

https://paulba.no/