Graphing the Orbit

It will be very interesting to graph the orbit for initial conditions X(0), Y(0), X dot (0) and Y dot (0). I would suggest using kappa sub X and kappa sub Y to define the initial conditions through:

kappa X sub X(0) + kappa sub Y Y(0) = -1.

Changing kappa sub X and kappa sub Y will change X sub (0) and Y sub (0), and the orbit will be changed. This is what I have in mind. For example, if kappa sub X = kappa sub Y inverse metres, then the initial condition is constrained by

X sub (0) + Y sub (0) = – 1

If kappa sub X = 100 inverse metres, and kappa sub Y = 1 inverse metre than the initial condition is constrained by

100 X sub (0) + Y sub (0) = – 1.

and the orbit will be completely different. Agreed about the factor 2 in Eqs. (21) and (22), but this is only a special case chosen for illustration only. In general there is no divergence problem. In general the orbit must be constrained by the gravitational field equations. I suggest trying out the code with the above two initial conditions. The idea is to try to get a variety of precessions, forward and retrograde.

Sent: 08/05/2017 13:04:41 GMT Daylight Time
Subj: Re: 377(2): ECE2 Orbital Theory in Terms of the Spin Connection

Note was checked. With the corrcted factor 2, eqs.(21,22) read

kappa_X = -1/(2X)
kappa_Y = -1/(2Y).

The factors have been corrected in note 277(3).
Since (21, 22) are obtained from

div g = kappa*g
div g = 4 pi rho_m

This seems to be the most general case, so (13) is most general. Since the orbit is completely defined by the equations on motion (17,18), it is not impacted by the spin connections kappa_X,Y. One of them can freely be chosen, the other follows from (13). It may be a bit misleading to write this in form of (19,20) because these equations are not independent form each other.
Another point is that (21, 22) diverge for X –> 0, Y –> 0.


Am 08.05.2017 um 10:34 schrieb EMyrone:

This is given by solving Eqs. (1) and (2) as shown. The resulting set of equations in Cartesian coordinates in a plane is Eq. (17) to (20). The X and Y components of the kappa vector are used as input parameters, and this might produce positive and negative precessions. The experimental precession is found by adjusting kappa sub X and kappa sub Y. The relativistic Newtonian force equation has been used in Eq. (1). The solution can be repeated with the Minkowski force equation in order to graph the effects of using the Minkowski force instead of the relativistic Newtonian force. The latter is the rigorously self consistent force equation in the frame of the field equations, because it gives the Einstein energy equation and relativistic hamiltonian (Marion and Thornton, third edition, chapter 14). Only one out of the four ECE2 gravitational field equations has been used, the gravitational Coulomb law (2). There are three more field equations of ECE2 gravitation in general, so all kinds of precessions in a plane can be computed.

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