Could you help to calculate precession analytically?

[Alvydas]

Hello. I have a formula for gravitational acceleration
http://www.part.lt/img/eb4c24f748f6e9a2797ed354bfbe6051663.png

And I have tested it by numerical integration.
It looks it gives perfect results. For example 0.104 arc seconds per revolution for the Mercury precession.
But I have no an idea how to calculate it analytically.

Maybe this could be interesting to note:
As you see dm can be positive (when mass increase) and negative (when mass decrease).
Therefore additional force v*dm/dt acts against the changing of the velocity v.
This is very similar to magnetic induction.
Maybe even magnetic induction itself can be explained in the same way?

 

[Alvydas]

Here is corrected version of the mentioned acceleration.
http://www.part.lt/img/db0d42e1474f160a32a4c0bcd95deb4a335.png

It gives the same precession for the Mercury ( 0.103 arc seconds per revolution )
but now it has much more clear meaning.
Also it is simpler. Maybe now somebody will be able to help with analytical derivation of the precession?
Or maybe someone can offer some other interesting cases to test this formula?

 

[edguy99]

Not sure what you mean by "calculate it analytically"?

 

[Alvydas]

Not sure what you mean by "calculate it analytically"?

I mean to derive some formula for the precession by using this acceleration formula. I am not good in this area.
But for numerical integration I just use my own very simple program with Verlet algorithm,
time step 0.005 seconds and 10100100100 loops to imitate a few revolutions of the Mercury.
I am detecting the points where I have the biggest distance from the Sun.
Than it is easy to find the angle between these points.
But I do not know how to go to obtain solution analytically.
Would be nice to compare it with the precession formula derived from General Relativity.

 

[edguy99]

...time step 0.005 seconds and 10100100100 loops to imitate a few revolutions of the Mercury. I am detecting the points where I have the biggest distance from the Sun. Than it is easy to find the angle between these points...

How close are you getting to the actual precession? You may find this helpful to test on other planets. From http://www.mathpages.com/rr/s6-02/6-02.htm

 

[Alvydas]

Thank you for the info.
Currently my program fails to calculate almost circular orbits.
As I mentioned my program is based on the detection of the maximum of the distance from the Sun.
When orbit is close to circular motion
the distance r at time step n becomes equal to the distance at time step n+1
(maybe there is no enough digits ). So accuracy is lost.
I will think how to overcome this problem.
But likely better computer and maybe better compiler is needed.
For the Mercury as I noted I have got about 0.103 arc seconds per revolution.
Maybe 0.1032 +/-0.0002

 

[edguy99]

That is pretty amazing accuracy you have achieved. Another one talked about in that same article that is not a circular orbit and may work to test your software is the asteroid Icarus (from 28 million km out to almost 300 million km).

 

[Alvydas]

That is pretty amazing accuracy you have achieved. Another one talked about in that same article that is not a circular orbit and may work to test your software is the asteroid Icarus (from 28 million km out to almost 300 million km).

Yes, I will try to calculate. Maybe you can help with two numbers for it:
the biggest distance from the Sun and velocity at this biggest distance ?
(alternatively the smallest distance and velocity at this smallest distance)

Wikipedia today do not works and the data from other sources like this
http://ssd.jpl.nasa.gov/sbdb.cgi?orb=1;sstr=1566
are a bit puzzled to me.

 

[edguy99]

Not sure I can help you much on that. I see the link you posted has Q= 1.969273957422264 +/- 8.145e-10 AU for the aphelion distance. One way to try this is to "guess" the velocity with ever more accuracy until you get the correct period, then check if the precession is correct. Good luck!

 

[Alvydas]

Not sure I can help you much on that. I see the link you posted has Q= 1.969273957422264 +/- 8.145e-10 AU for the aphelion distance. One way to try this is to "guess" the velocity with ever more accuracy until you get the correct period, then check if the precession is correct. Good luck!

According to
http://en.wikipedia.org/wiki/1566_Icarus and http://en.wikipedia.org/wiki/Apsis
aphelion = 2.94581631e11 meters and v = 8832.414 meters per second

Mentioned acceleration formula
http://img864.imageshack.us/img864/7905/mercury.png
gives precession 0.1129 arc seconds per revolution
(10.088 arc seconds per century).

 

[Alvydas]

I do not want to violate the rules of this forum, but I really need some help.

The initial question about analytical derivation likely it is too difficult.
I was asked the same question on math forum but moderator suggested to ask on physics forum.
His motivation was: the question is too physical. But here maybe too mathematical.

Anyway maybe with next question I can get some help.
As I mentioned previously two formulas well (very well) account orbital precession.
I have made some corrections so now it can account Pioneer anomaly also and still the same well predict the precession. I have calculated for Mercury and Icarus.
http://img810.imageshack.us/img810/5618/mercurypioneer.png
(I can post very simple code in C so everyone who just started to learn programming may test it.)

But Pioneer anomaly is not my main interest. It just happened by accident.
My main interest is precession in ProbeB.
I want to make some simplest mathematical model of some simplest gyroscope ant to test how mentioned formula effects such gyroscope.

All possible information how General Relativity accounts simplest gyroscope
lets say with two point masses spinning around the axis would be much useful.
Also maybe some ideas how to simulate the gyroscope itself?

 

[Alvydas]

Could you say what values I could use at this formula
http://apod.nasa.gov/apod/ap110510.html
to obtain given results for both precessions?

Could you put them to http://www.wolframalpha.com for testing?

 

[Alvydas]

It looks I understand by myself.
Mentioned Ω
http://apod.nasa.gov/apod/ap110510.html
we need to multiply by the orbit time in seconds to obtain precession in radians.
But the next maybe can be interesting also.
The same result for geodetic precession can be obtained without relativity. See
http://www.wolframalpha.com/input/?i=lim+r-%3E0+%28%28sqrt%28+1+-+%28+G*M%2F%28+%28R%2Br%29*c^2%29+%2B+2*G*M%2F%28+%28R%2Br%29*c^2+%29+%29+%29+-+sqrt%28+1+-+%28G*M+%2F+%28R*c^2%29+%2B+2*G*M%2F%28R*c^2+%29+%29+%29++%29+*+v%2F+r+%29

I have used small body with radius r and circular orbit with radius R around big mass M.
I also have used momentum conservation law.
If mass of a free body decrease k times then velocity increase k times.
p=mv = (m/k) * (v*k)
I quess this is also the reason why Universe expands with acceleration.
Just total mass becomes smaller and v becomes bigger.

But the help with more data about frame dragging still would be very useful.