Wednesday, January 16, 2013

The Advantages of a Stable Reference Point

  All distance observations are relative measurements. We have an object and an observer. We can't discuss the motion of an object by itself so we can never remove the observer from the observation completely. When we have a group of observations there is an advantage to having a common point of reference to which individual observations can be referred to. For a system of bound particles all the particles move relative to a center of mass and with it. The center of mass has only has the motion of the group and no individual motion so it is the most stable point for the group. The rotational motion of the Earth is relative to its center of mass and that of the Earth-Moon system is relative to its barycenter. This is why the two points are the natural choices for common reference points. They are points of relative stability. Observations from these points tend to be more consistent than those of individual observers. They simplify the relative motion of an object. Neither point is without some acceleration of its own so we can't completely remove the motion of the reference point from the object's motion but we have minimized it. Over short periods of time the acceleration of an object may have a negligible effect and we can make a better linear least squares fit of the position and velocity. When we notice some curvature in the path of the object it doesn't matter whether it is acceleration relative to a Galilean reference frame or acceleration relative to the reference point. A higher order polynomial may be needed to fit the data but we should be able to do this since the polynomial is linear in terms of its coefficients. A linear fit would be the minimum necessary to determine an orbit since the position and velocity are all that is needed to calculate the orbit's specific energy and specific angular momentum and consequentially the Keplerian orbital elements. Perturbing bodies like the Earth and the other objects in the solar system can cause an object's motion through the solar system to deviate from an ideal Keplerian orbit but we still have enough information to calculate an approximate path. Observational errors make the path somewhat uncertain but small deviations to the position and velocity can be used to compute alternative paths.

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