Planets orbital parameters

Our chemical experiences suggest that differential equations seem to be something stable, and by that we mean that, if there is a small change in one of the conditions, either initial concentrations or rate constants, we expect small changes in the outcomes as well. The classical example for a stable system is our solar system of planets orbiting the sun. Their trajectories are defined by their masses and initial location and velocity, all of which are the initial parameters of a relatively simple system of differential equations. As we all know, the system is very stable and we can predict the trajectories with an incredible precision, e.g. the eclipses and even the returns of comets. For a long time, humanity believed that the whole universe behaves in a similarly predictable way, of course much more complex but still essentially predictable. Descartes was the first to formally propose such a point of view. 

The basic physical and orbital parameters of the solar system s giant planets (Cox, 2000) are summarized in Table 1. Included as well in the table are data on the one giant planet beyond... 

Table IV. - Average radii of planet orbits (in astronomical units) as measured and calculated (according to Bode s law), together with the mass and number of known satellites (in parentheses) for each planet (in units of the Earth mass, 5.976 10 Gkg). Although many astronomers today consider this law as accidental, optimizing its two parameters yields a quasiperfect fit, with a correlation coefficient of 0.9997.

What we called the distance of closest approach is what was already known to the ancients as the perihellion of the orbit of a planet around the Sun. Modem chemists know this concept in terms of how close the electrons get to the nucleus of the atom. For collisions under a realistic potential, if the impact parameter is quite low the molecules will get all the way in to the range of the repulsive forces. For high-impact parameters the molecules will only sample the long-range attraction and not penetrate much beyond R = b. But for any impact parameter the colliding particles will feel a mutual force. The only exceptions are potentials whose influence extends over only a finite range, such as a hard-sphere potential. 

The epi -cycle for the motion of a particle or a planet means that its trajectory is composed of two superimposed circular motions. The center of the epi -cycle or epi -circle orbits around another annulus, the deferent, the center of which is fixed. This linear superposition of cycles can also be considered as a Fourier series. With the time t as a parameter, the two... 

Fig. 8.5.2 Variation of flux and linear polarization with orbital longitude due to a single standard hot spot of area 2000 km (T = 450 K) located at 180° W, 15°N on a model planet with n = 1.5 and a true geometric albedo of 0.8. In each frame of the figure, the solid line is the signature of this standard spot and the broken lines show the changes in this signature if one of the parameters is varied from the standard configuration, (a) The quasi-geometric albedo p (b) the degree of linear polarization V (dotted line, n = 1.8) (c) azimuth of linear polarization f (dashed line, latitude 45°N dotted line, latitude 15°S) (Goguen Sinton, 1985).

Table 3.1 Some important parameters of the planets in the solar system. D denotes the distance from the Sun, P the orbital period, R the radius and Pnot the rotation period...

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