Linearized Acceleration Fields

The linearized relative acceleration used to check the least squares fit of the 2012 DA14 observations was evaluated for a neighborhood centered on the Earth's position. One can derive the formula as follows,

GM is the gravitational constant for the Sun, X_E the Earth's position relative to the Sun and X is the position relative to the Earth where one wants to evaluate the acceleration. Beyond 500 R_E the error in the linear approximation is quite noticable. It might be better if one used the methods above to derive the linearized acceleration for a neighborhood about the position of the object at X_o.

The constraint that the two accelerations be the same is equivalent to requiring that the object follow a path determined by Newton's law, F=ma. The eigenvectors were used to search the neighborhood of the object for the point of best correspondence. All the eigenvectors can be used but the ones with smaller eignevalues tend to produce larger corrections. In the example given the change in both the position and velocity were along the line of sight.