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Additional info for Astrophysik IV: Sternsysteme / Astrophysics IV: Stellar Systems
It may be shown in a similar way, that at an effective limit Rs a. 2 ---+ °for R ---+ Rs. 12) 7. Ellipsoidal frequency function, based on certain integrals of the motion, as approximation in a limited region. Our experience of the velocity dispersion in our neighbourhood suggests that at least in a limited region about the Sun the function t may be a function of second order in the velocities. The most general function of 11 and 12 of the ellipsoidal type will be F(I1 - 2k1I2 + k2I~). 1) This gives, however, a velocity ellipsoid where the axis in the radial direction is equal to that in the z-direction.
Well known. Especially the part played by dark bodies of considerable size is an open question. For any physically de tined group of particles we may define a density function and a function describing the distribution of velocities, which may vary from one point to another. In the absence of exterior disturbances there will be a certain plane passing through the centre of gravity, "the invariable plane", for which the sum of the area velocities about the centre of gravity is a maximum. This plane may be taken as fundamental plane of reference.
1). A specification of all the six integrals would mean a detailed description of the actual motions in the system. The importance of the theorem as to the general state of motion in a stellar system depends on the fact that, if the system settles down to a state in which qJ is stationary (in a rotating or non-rotating system), certain simple integrals can be found. Moreover, the general complication of the individual motions along the surfaces defined by these integrals in phase space will mean a general process of mixing of matter, so that we can disregard a certain number of the six integrals in the frequency iunction t to be expected as a result of the mixing process.