New SFL implementation in GILDAS

This appendix is based on the formalism of the CG02 paper, applied to the particular case of the SFL projection. See Table 1 for the coordinates involved here. According to the CG02 section 5.3, the pseudocylindricals are constructed with the native coordinate system origin at the reference point. Accordingly we set (their eq. 89):

$$(\phi_0,\theta_0) = (0,0)$$ (13)

The native longitude of the celestial pole $\phi_p$ is defined as (their section 2.2):

$$\phi_p = \begin{cases} 0^\circ & \text{for $\delta_0 \geq 0$} \\ 180^\circ & \text{for $\delta_0 < 0$} \end{cases}$$ (14)

This implies $\sin\phi_p = 0$. Their eq. 8 is simplified as:

$$\delta_{p,1} = (0,\cos\phi_p) + \cos^{-1}(\sin\delta_0)$$ (15)

$$\delta_{p,2} = (0,\cos\phi_p) - \cos^{-1}(\sin\delta_0)$$ (16)

where atan2$(y,x)$ is the inverse tangent function that returns the correct quadrant. $\delta_{p,1}$ and $\delta_{p,2}$ are two solutions for $\delta_p$. The choice between the two is ruled by several conditions detailed in the CG02 paper section 2.4. The rule 4 states that the northernly solution is the default if the native latitude of the celestial pole ($\theta_p$, LATPOLEa in FITS headers) is not defined, which we implement as:

$$\delta_p = \begin{cases} \delta_{p,1} & \text{if } \delta_{p,... ... \leq 90^\circ \\ \delta_{p,2} & \text{otherwise.} \end{cases}$$ (17)

If $\delta_p = \pm 90^\circ$, their rule 2 applies to determine $\alpha_p$, otherwise their generic equations 9 and 10 are used:

$$\alpha_p = \begin{cases} \alpha_0 + \phi_p - 180^\circ & \tex... ...a_0}{\cos\delta_p\cos\delta_0}) & \text{otherwise.} \end{cases}$$ (18)