ParametricPlot celestial coordinates

Dear All,

I am trying to reproduce the top left figure 5 with q = 0 from the manuscript
https://arxiv.org/pdf/1707.09521 . In this case, for \theta = \pi/2 in equation 52, \alpha is equal to equation 45 in negative. Furthermore, I replace equation 44 to obtain \beta in equation 52. However, I am not getting the figure. I am attaching my worksheet.

Perhaps the goal would be to plot equation 55, but I can’t figure out how to plot the contour of an equation of the type \alpha(x)^2 + \beta(x)^2 = f(x) with \alpha and \beta on the horizontal and vertical axes.

Any help would be appreciated.

Best regards.

falla 51.nb (28.1 KB)