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    MP through linear fit of randomly timed observations
    From: Jaap vd Heide
    Date: 2012 Jan 26, 09:53 -0800

    Hi all,

    I have been experimenting with fitting a straight line (linear function) in order to find MP / LAN or whatever you might call it, through points derived from randomly timed observations near transit.

    These experiments are based on the assumption the function for the change of altitude near transit in "Bowditch" is correct. Because if it is true that near transit the change of altitude C adhers to the function C=a/60*t^2, there apparently is an equation H(t)=H_transit - a/60*t^2 describing the altitude of the heavenly body near transit and thus a an equation H'(t)=-2a/60*t describing the rate of change of altitude. (Which is 0 when t=t_transit [M.P.])

    Fiddling around with these equations in the more generalized form y=at^2+b and y'=2at I found that for each pair t_I and t_II it is true that y'((t_I+t_II)/2)=(y(t_II)-y(t_I))/(t_II-t_I).
    Translated back to the "Bowditch function" and the altitude equation derived from it, each pair of observations made yields a point on the 'rate of change"-graph, with coordinates ((t_I+t_II)/2,(H(t_II)-H(t_I))/(t_II-t_I)).

    I have been doing some testing, first producing apparent altitudes with Stellarium 0.11.1 [www.stellarium.org], then determining the time of transit through the mechanism described above.

    Sets of 5 to 7 randomly timed "observations" yield a time of transit with an error of 10-15 seconds when the "observed" altitudes are just rounded to the nearest 0.2' (as if measured with a good quality "drum"-sextant) and an error up to 45 seconds when an extra random error of +/- 0.2' is introduced. (With the Excel function RAND())

    I have done the line-fitting based on the least-squares linear estimation function LINEST() in MS Excel, but it could offcourse also be done "eyeballing" with the aid of a ruler and pencil.

    I would appreciate comments.

    Regards,

    Jaap
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