Within the uncertainty of solar system models and their computation, what and when was the closest occurrence (in terms of solid angle) of the inner eight planets forming a radial alignment outward from the Sun?
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I am thinking this to be more a statistical problem than one of astronomy in particular. For instance, if one required all these planets "align" in the same semicircle bisecting the orbits of the solar system, the probability of that occurring would be 2-7.
Would time constraints, such as orbital periods, be of secondary (if any at all) concern to the fundamental problem? If the time considered were on the order of 5 billion years, what would the minimum angular dispersion from alignment be?
It is a matter of time - you need orbital period to calculate alignment frequency. My geometry isn't good enough without putting some effort into it to calculate the frequency from the orbital periods, but for example, Mars has an orbital period of 1.9 years and an opposition frequency of 2.6 years. That means that once every 2.6 years, for 1.3 years, the Earth and Mars are in the same half of the solar system. And once every 2.6 years, for 2.6 days, they are within the same degree.