Evolution of Glaxies

Yes all agree that they are incisive and very helpful to the reader, even the general reader with no mathematical training. I fully agree with this project, I have just started an analysis of a three dimensional hyperbolic orbit in note 270(4) and that is simple to work with. The usual dogma as in Marion and Thornton asserted that a constant angular momentum L = r x p means that both r and p are coplanar, so plane polar coordinates can be used. However, this is obviously wrong from the fundamental analysis of angular momentum in spherical polar coordinates given in UFT269. The usual dogma was simply an assertion that L is in the Z axis. This is only a special case.

To: EMyrone@aol.com
Sent: 01/09/2014 11:02:32 GMT Daylight Time
Subj: Re: Animation of elliptic 3D orbit

I am glad that my contributions are received so well.
Concerning the problem that spiral galaxies are mostly planar: A gaseous cloud is 3D, and if the angular momentum is conserved during building of galaxies and stars, so the galaxy structure should also be 3D. There could be a mechanism changing the angular momentum over time. What about assuming a spacetime or ether current that interacts with stars and gases over billions of years? This could be modeled by a simple approach with an inelastic interaction between current and star motion so that the angular component perpendicular to the current is annihilated. Perhaps this is worth some thoughts.

Horst

EMyrone@aol.com hat am 31. August 2014 um 17:00 geschrieben:

This is an excellent animation and shows immediately that the 3 -D orbit is a lot more intricate than the 2 – D orbit. The inversion problem is indeed a difficult one, and goes right back to Kepler himself. The computer might be able to find the answer straightforwardly. As Horst writes this should be regarded as a first attempt, but it is already full of originality. Obviously we have opened up a completely new subject area in cosmology upon which all can agree because it is based on the spherical polar coordinates.

To: EMyrone@aol.com
Sent: 31/08/2014 14:44:45 GMT Daylight Time
Subj: Animation of elliptic 3D orbit

This is an animation for the beta ellipse in 3D, parametrized by the theta angle. Since the theta grid has been chosen equidistant, the motion speed is not physical. I am not sure if the orbit is compatible with the first 3D surface r(theta,phi) shown in paper 269,3 (Figs. 3/4). Normally this should be a trace on the surface. This has further to be analysed. It would also not be straightforward to compute the right timing behaviour because the time function has to be inverted which has to be done numerically.

Horst

Four Notes To Date

Yes these are the four notes to date, I have just started work on three dimensional galactic dynamics.

In a message dated 01/09/2014 10:52:41 GMT Daylight Time, mail@horst-eckardt.de writes:

Is there also a note 270(2)?

My internet has been checked today morning, obviously someone had temporarily interrupted the line in the house, seems to work again now.

Horst

EMyrone@aol.com hat am 1. September 2014 um 10:59 geschrieben:

The condition for planar orbits is Eq. (17) or (27), the equation which results from the assumption that the constants of integration of Eq. (15) or (16) are zero. So this is the reason why planar orbits are planar. If one time dependent constant of integration is chosen as in Eq. (23), then the orbit is three dimensional and all previous work based on Eq. (24) remains valid. So Eq. (24) is the one to use in general. The three dimensional Binet equation is Eq. (26). The three dimensional orbit depends in general on the function (27) in which beta is a function of phi and theta. Computer algebra can be used to re express Eq. (27) in terms of phi and theta. I can also do this by hand using the chain rule of differential calculus. There are many interesting types of three dimensional orbits in galaxies, and they are all described by Eq. (27). All of this analysis is based on the spherical polar coordinates and lagrangians, so no one can object to the results, especially as they are checked many times by hand and computer algebra.

a270thpapernotes1.pdf

a270thpapernotes2.pdf

a270thpapernotes3.pdf

a270thpapernotes4.pdf

270(4): Three Dimensional Galactic Dynamics, Part 1

This note defines the three dimensional hyperbolic spiral in Eq. (3), ans that can be graphed in a spherical polar plot. The 3 – D Binet equation shows that it is produced by an inverse cube law of attraction in three dimensions. Lagrangian dynamics gives its trajectory in Eq. (16), and this can be animated straightforwardly. In this case the inversion is not a problem. An infinite variety of three dimensional galaxies can be produced with this type of analysis, but it is known experimentally that the trajectory in a 2 – D whirlpool galaxy is the two dimensional hyperbolic spiral, so it is meaningful to extend that analysis to three dimensions.

a270thpapernotes4.pdf

270(3) : The Condition for Planar Orbits

The condition for planar orbits is Eq. (17) or (27), the equation which results from the assumption that the constants of integration of Eq. (15) or (16) are zero. So this is the reason why planar orbits are planar. If one time dependent constant of integration is chosen as in Eq. (23), then the orbit is three dimensional and all previous work based on Eq. (24) remains valid. So Eq. (24) is the one to use in general. The three dimensional Binet equation is Eq. (26). The three dimensional orbit depends in general on the function (27) in which beta is a function of phi and theta. Computer algebra can be used to re express Eq. (27) in terms of phi and theta. I can also do this by hand using the chain rule of differential calculus. There are many interesting types of three dimensional orbits in galaxies, and they are all described by Eq. (27). All of this analysis is based on the spherical polar coordinates and lagrangians, so no one can object to the results, especially as they are checked many times by hand and computer algebra.

a270thpapernotes3.pdf

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