The above graph is the quadrupole gravity potential for a rotating sphere drawn along the rotation axis z. Newtonian potential is not included in the picture. The boundary of the star is where Z=1, -1. The square of the form factor is set at 0.02 and the rotational speed at the surface of the star is set at 0.05c, while M=G=c=R=1.
The length of the jets may be determined by the point where the peak height of the dipole potential meets the ordinary Newtonian gravity far out side of the Z axis when a horizontal line is drawn along the Z axis.
While the peak height of the potential is determined by the rotational frequency and the mass of the rotating center and its geometrical shape, there is a mathematical form factor etha that is an arbitrary parameter yet it determines the length of the jets critically. In case of the rotating sphere, however, Lenz-Thirring's result could be used to determine the accurate value of the form factor which turns to be about 0.3R.
In general case, since we do not know exactly how the rotation center may look like in terms of its precise geometrical shape, this arbitrary form factor may be inevitable for the time being.
In actuality, there is little chance that the rotating center will remain like a sphere even if it may have started as one in the beginning. Because of the strong dipole force near the equatorial center, when viewed from the side, it may as well look like two saucers attached together face to face in the horizontal direction.
A slight dimple in the middle of the saucer may define the jet's outlet and the main body spreads out around the equatorial plane which may be much longer than the radius of a typical saucer.
While predictions can be made regarding the length of the jets, other informations obtained from the orbital motion of the galactic system would be critical to determine the validity of its predictions.
Note: The above Z directional potential is a fixed one to make it consistent with Lens-Thirring force. The actual Z directional potential which is consistent with the dark matter problem should look like,
Friday, April 20, 2007
The Length of the Jets
Posted by Eue Jin Jeong at 1:19 PM