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Old Mar 10th 2015, 12:14 PM   #1
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Tracking the Precession of Mercury's Orbit

I didn't exactly know where to put this one, since it involves mechanics, circular motion, and General Relativity. To start off, I have to create a series of differential equations of motion to describe Mercury's orbit, then come up with another series to describe the effects another planet has on its orbit and track the orbit of the other planet, also accounting for effects of Mercury's gravitational pull on that planet as well. This is the part I am stuck on, as I am not too familiar with generating differential equations to track the motion of an object. Later, I'll also have to account for the effects of General Relativity, and then use a computer to get numerical data.
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Old Mar 10th 2015, 05:14 PM   #2
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Originally Posted by SiegDerMaus View Post
I didn't exactly know where to put this one, since it involves mechanics, circular motion, and General Relativity. To start off, I have to create a series of differential equations of motion to describe Mercury's orbit, then come up with another series to describe the effects another planet has on its orbit and track the orbit of the other planet, also accounting for effects of Mercury's gravitational pull on that planet as well. This is the part I am stuck on, as I am not too familiar with generating differential equations to track the motion of an object. Later, I'll also have to account for the effects of General Relativity, and then use a computer to get numerical data.
Wow. This one could turn into one time consuming topic.

Starting out the concept is simple enough. We are only dealing with Newtonian gravity so the form of the equations should be simple enough. The problem is deriving it. For simplicity I would assume that the Sun doesn't move and that "Venus" is more along the mass of Jupiter so we can effectively assume that Mercury's orbit isn't going to have an effect on Venus'. Then it's a problem of adding two gravitational force vectors on Mercury: one from the Sun and one from "Venus."

The problem is that Venus isn't in a stationary position...it's got an orbit of its own. If you are going to be stuck using a computer model anyway the approach I would take is "point by point." Assume Venus and Mercury are in some specific position. Calculate the forces on Mercury (via Netwon's Law) and let the two forces predict where Mercury is going to be headed for in the next time increment. Then let Venus move in its orbit for a time increment as well, then rinse and repeat.

Unfortunately I can't give you a method to write out a differential equation for the motion. I'd love to say that we could use a Lagrange Multipliers method (the constraint being Venus' motion in its orbit) but I can't figure out how to apply it.

-Dan
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Old Mar 10th 2015, 05:38 PM   #3
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Thank you for your reply! I'm afraid I am required by the problem to use the differential equations of motion method, starting out by deriving them from Newton's Law of Universal Gravitation. Also! I could use some help casting said equations in dimensionless form, since I am unfamiliar with that as well.

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Old Mar 10th 2015, 07:08 PM   #4
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Actually the differential equations for simple planetary motion aren't too difficult to set up. Using polar coordinates, from basic mechanics you have in the radial direction:

m[(d^2r/dt^2) - r(d theta/dt)^2] = -F

Where F is the force of gravity from the sun, or GMm/r^2. And in the circumferential direction:

r(d^2 theta/dt^2) + 2(dr/dt)(d theta)/dt =0

Solving these is a mess, but as suggested by Dan using numerical techniques through computer simulation should be straight forward, as long as you start with a valid set of data for the planet's initial coditions of r, theta, and their first and second derivatives.

Adding in the effects of the 2nd planet would simply add an additional force vector to the above equations, which will vary with distance and direction between the planets. You will need a model of the orbit of the 2nd planet to be able to figure that out.

Good luck - sounds like an interesting project!
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Old Mar 10th 2015, 07:25 PM   #5
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Yes! Thank you! Would it be alright if I kept this thread open in the event that I run into another roadblock?
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Old Mar 10th 2015, 08:15 PM   #6
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We'd be more than happy to hear how you are doing.

-Dan
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Old Mar 10th 2015, 10:49 PM   #7
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Also, I know this isn't so much physics as it is computer programming, but do you know of anyone on this site experienced in the use of IDL for data manipulation and plotting? I'll need to use that to keep track of the orbits once I enter the computational/numerical phase of things.

Edit: I hit another roadblock. Given what I already have, I need to derive the relation:
M(d^2R/dt^2)=-GSM[R/(||R||^3)]-GmM[(R-r)/(||R-r||^3)]
Where M is the mass of Mercury, m is the mass of Venus, S is the mass of the sun, R is the radius vector for Mercury and r is the radius vector for Venus. I have no idea where to start this derivation.

Last edited by SiegDerMaus; Mar 10th 2015 at 11:58 PM.
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Old Mar 11th 2015, 05:52 AM   #8
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One of the issues with these multibody orbit solutions is that the analytic solution rapidly becomes impossible (or at least impractical)
This is a reason numerical methods are employed, then each mutual interaction can be treated as quasi-independent.
Rather like a series expansion, you deal with the principle interactions first, then secondary, then... continue until the next level of interaction is negligible.

The problem with this is time; over eons the "negligible" adds up...
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Old Mar 11th 2015, 09:32 AM   #9
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Originally Posted by SiegDerMaus View Post
Also, I know this isn't so much physics as it is computer programming, but do you know of anyone on this site experienced in the use of IDL for data manipulation and plotting? I'll need to use that to keep track of the orbits once I enter the computational/numerical phase of things.
Sorry - can't help with that.

Originally Posted by SiegDerMaus View Post
Edit: I hit another roadblock. Given what I already have, I need to derive the relation:
M(d^2R/dt^2)=-GSM[R/(||R||^3)]-GmM[(R-r)/(||R-r||^3)]
Where M is the mass of Mercury, m is the mass of Venus, S is the mass of the sun, R is the radius vector for Mercury and r is the radius vector for Venus. I have no idea where to start this derivation.
It comes directly from Newton's law of gravity, expressed in vector form. Start with the equation for the magnitude of gravitational force:

F = GMm/x^2

where x is the distance between the two objects. This gives the magnitude of F, but not its direction. To make F a vector (which I'll indicate in bold text), you need to indicate the direction of x. We can do that using the notation "x_hat," where the "hat" indicates a unit vector in the x direction:

F = -(GMm/ |x^2|) x_hat

Note the minus sign - this is because we define x as pointing from the sun toward Mercury, or from Venus toward Mercury, but the force of gravity on Mercury is in the opposite direction. Since x = |x| x_hat , you have x_hat = x/|x|:

F = -(GMm/|x^2|) (x/|x|)= - GMm x/(|x|^3)

Simply substitute the appropriate values for x and the appropriate masses. For the case of gravitational attraction of the sun on Mercury, which I'll call F_s:

F_s = -GSM R/(|R|^3)

Again, the R vector points from the sun to Mercury, hence the minus sign.

For the case of gravitational attraction of Venus on Mercury the x vector pointing from Venus towards Mercury is R-r, and the gravitational force F_v is:

F_v = -GMm (R-r)/(|R-r|^3)

The total force acting on Mercury is then the sum of F_s + F_v.
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Old Mar 11th 2015, 01:28 PM   #10
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Thank you! Now I think all I need is an expression for Theta for Kepler's orbit equation so I can start plotting it. Though from the equations I have so far, it seems that any expression for theta would be r dependant, making it infinitely recursive when plugged into Kepler's orbital equation. Is there something I'm missing?

Edit: And I don't have initial values for theta and its derivatives. I was only given mass, eccentricity, and semi-major radius of orbit.

Edit: I desperately need help on plotting this orbit and determining the effects of other planets. My deadline is fast approaching and the computational section of this problem is no closer to being finished than it was at the beginning.

Last edited by SiegDerMaus; Mar 11th 2015 at 03:35 PM.
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general relativity, gravitation, mercury, orbit, orbital momentum, precession, tracking



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