When was the Jupiter-Saturn Conjunction?

 

  There's been a lot of talk about The Great Conjunction lately but little information on when it occurs. Given ephemeris data from JPL's Horizons on the positions of planets Jupiter and Saturn we can do a simple spreadsheet calculation to find the time. Horizons will output the data for the planet selected in text format and this can be copied and pasted into a spreadsheet. It's easiest to work with the both RA and Decl given in decimal degrees. When pasted into the spreadsheet the data appears in a number of lines. The value of each field, date, hour, RA and Decl, can be extracted from a line using the MID function. For small angles we can approximate the positions with x,y coordinates and determine the approximate angular displacements, Δr. To do a three point parabolic fit we need to determine the coefficients A, B and C, for the quadratic equation from which we can find the time of the conjunction.

The solution for the three coefficients, if the points are equally spaced in time, is fairly simple.

We just need to solve a linear matrix equation and use the coefficients to calculate Δr_min and t_min.

We can now assemble a simple table containing the required data and do the necessary calculations.

Supplemental (Dec 22): Since the coefficients of the matrix in the matrix equation are +/- 1, the solution is fairly straightforward.