Finding the value which minimises all residuals

Question

I have a series of observations, measurements made at various times $t$. I now need to determine the most likely value of $R$ (distance) using the model below. The guide says I should find the value of $R$ which minimizes the values of $\Delta RA$ across all times, $t$.

The model is: $$ \tan(\Delta RA)={ X\cos(Dec_0 + \mu t)\sin(RA_0 + \nu t)-\sin(wt) \over X\cos(Dec_0 + \mu t)\cos(RA_0 + \nu t)-\cos(wt) } $$

where; $$ X={R\over R_E\cos\lambda} $$

• $t$ is a variable of time
• $\Delta RA$ varies with time (i.e. there are different values for each row in the table)
• Every other variable, except for $R$, are already determined constants.

With the known values substituted, we have: $$ \tan(\Delta RA)={ \left({R \over 2115}\right) \cos(14.174550 - 0.003488 t) \sin(0.814907 - 0.000468 t) - \sin(15.04 t) \over \left({R \over 2115}\right) \cos(14.174550 - 0.003488 t) \cos(0.814907 - 0.000468 t) - \cos(15.04 t) } $$

How would I even begin to work this out?

Answer 1

You have a table of times and measurements of something (RA and declination?). For a given value of $R$, you can calculate $\Delta RA$ at each time, using the observations. Then you can take the absolute value of the $R$'s, or the squares, and add them up. This gives you a function $error(R)$ of one variable. Now adjust $R$ to minimize the function. Excel will let you Goal Seek to minimize the function, or you can use a routine from any numerical analysis text.