Now, for 0 a random variable, such that r varies from a(l — e)<r<a(l+e). This bounding on r constitutes the Borel set of r, that is to say, the probabilitized set of values r may have [3], It is natural only to consider 6 from when working towards statistics on r, because in the interval the values of r replicate themselves and give no new data (or insight) on r itself. Continuing from before (equation 1.4): This is to say that for a given orbit, if 6 were to be considered a random variable, the average value of r would be a V 1~ e2, the value of the semiminor axis of the ellipse. While this is an interesting statistic, it may be of little practical use. From an applications point of view, the position of a satellite (as a function of time) is far more prone to be at the furthermost point of its orbit with respect to the orbit (or force) cente than at its closest point; this is a result of Kepler’s second law (equal areas are swept out in equal time). Rather than true anomaly being considered a random variable, it is of more value, and of greater physical importance, that we consider time (since perifocal passage) a random variable. Manipulating equation 1.2b (conservation of angular momentum) we have:
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