Perihelion, rotation

An example will at once make clear what is meant. Let us take the relativistic Kepler motion, or, in other words, motion in an ellipse with a perihelion rotation. In general, the path fills a circular ring and, therefore, a two-dimensional region, densely everywhere. The boundaries for the libration of the radius vector are here concentric circles. 

Figure 19. It is assumed here that the concept of the ellipsoids of observation apply to all fields moving with the velocity of light, such as electric or gravitational fields. Thus, precession of the perihelion of Mercury (the rotation of its elliptic orbit) can be explained by the asymmetry of the gravitational forces as the planet advances toward (resp. retreats from) the sun.

The rotational axis of the Earth gradually precesses like that of a spinning top, and the ellipitical orbit of the Earth also precesses. The periods of these two precessions are 19 and 23 kyr, respectively, yielding an average of c.21 kyr. Precession controls the distribution of insolation over the Earth s surface by varying the timing of the seasons relative to the perihelion, but, like obliquity, does not affect the total insolation. 

Since the motion consists of a libration of r, combined with a uniform rotation of the perihelion, the form of the orbit is that of a rosette (cf. fig. 12). 

The field of force of the core of an atom is, at a sufficiently great distance, a Coulomb- field of force. In the case of the neutral atom it corresponds to the effective nuclear charge Z=l, in the case of the 1-, 2-. . . fold ionised atom Z=2, 3. . . respectively. The orbits of the radiating electron at a large distance are therefore similar to those in the case of hydrogen. They differ from the Kepler ellipses only by the fact that the perihelion executes a slow rotation in the plane of the orbit. The semi-axes and parameter of the ellipses are, by (9), (10), and (11) of 22,... 

The equation of the path differs from that of an ellipse with the parameter l and eccentricity e by the factor y. While r goes through one libration, the true anomaly 0 increases by 2ir/y. The path approaches more nearly to an ellipse the smaller the coefficient cl of the additional term in the potential energy, and for c1=0 it becomes an ellipse. For small values of c1 we can regard the path as an ellipse, whose perihelion slowly rotates with the angular velocity... 

The smaller the principal quantum number the larger is the relativity correction (8), and it is therefore greatest for the l1-orbit. For the same value of n it is greater the greater the eccentricity of the orbit. The frequency of rotation of the perihelion will be... 

The stationary motions in a weak electric field are, however, essentially different from those in a spherically symmetrical field differing only slightly from a Coulomb field. In the latter (for which the separation variables are polar co-ordinates) the path is plane it is an ellipse with a slow rotation of the perihelion. In the former (separable in parabolic co-ordinates) it is likewise approximately an ellipse, but this ellipse performs a complicated motion in space. If then, in the limiting case of a pure Coulomb field, k or nf be introduced as second quantum number, altogether different motions would be obtained in the two cases. The degenerate action variable has therefore no significance for the quantisation. 

In 1930, Tombaugh discovered Pluto, the outermost known planet (Reaves, 1997 Marcialis, 1997). Several authors have derived the radius of Pluto with very small uncertainties unfortunately, the derived values do not overlap. Consequently, only a broad range can be quoted (1145 to 1200 km) within which the true radius of Pluto may fall (Tholen Buie, 1997). Pluto is by far the smallest planet of our Solar System it is even smaller than many planetary satellites. Pluto s orbit is highly eccentric and inclined by more than 17° to the ecliptic plane (Malhotra Williams, 1997). At perihelion (29.7 AU), Pluto is closer to the Sun than Neptune (30.1 AU), and at aphelion it reaches a heliocentric distance of almost 50 AU. Pluto s orbital period, 248.35 sidereal years, is locked in a 3 2 ratio with that of Neptune (Cohen Hubbard, 1965). The axis of rotation is nearly in the orbital plane therefore, this small planet undergoes rather complex seasonal changes (Spencer et al., 1997). Malhotra (1993, 1999) provides interesting discussions of the possible evolution of Pluto s orbit and that of other planets (see also Stem etai, 1997). 

In 1978, Christy Harrington (1978) discovered Pluto s rather large, but close-by satellite Charon. The radius of Charon is between 600 and 650 km, which is more than half of that of Pluto (Tholen Buie, 1997). In comparison, the lunar radius is 0.27 that of Earth. Charon orbits Pluto at a distance of 16.5 Pluto radii with an orbital period of 6.4 Earth days. It can safely be assumed that the bodies are tidally locked to each other, which means the rotation periods of Pluto and of Charon equal the orbital period of Charon (Dobrovolskis etal, 1997). Charon must be an impressive sight observed from Pluto hovering over the same equatorial area, it would appear nearly 7.5 times the diameter of the Moon as seen from Earth. Even more dramatic would be Pluto observed from the surface of Charon its apparent diameter would be nearly 14 times the lunar diameter. In contrast to this, the diameter of the Sun subtends only 38 and 48 arcsec as seen from the aphelion and perihelion positions of Pluto, respectively. The maximum angular diameter of Jupiter seen from Earth is about 46 arcsec. 

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