How to Convert Ecliptical coordinates to Equatorial And Vice Versa?

This article will derive equations for converting a planet’s ecliptic coordinates to equatorial coordinates. This conversion is necessary for calculating primary directions - an ancient technique of event prediction.

Notations

First, let’s define terms. We will denote

• Celestial longitude and latitude by $\lambda$ and $\delta$ letters
• Equatorial coordinates - right ascension and declination by $RA$ and $D$
• Ecliptic inclination by $\epsilon$

Coordinate rotation

Let’s denote the vector (a pointer) to the observed planet with the letter $\vec{v}$. Then the spherical coordinates of this vector in the ecliptic coordinate system will be

$$\vec{v} = (\lambda, \delta)$$

The Cartesian coordinates of the same vector, according to equation (2) will be equal to

$$\begin{aligned} \vec{v} &= \cos\delta\cos\lambda~\textbf X_\text{ecl} \\\ & + \cos\delta\sin\lambda~\textbf Y_\text{ecl} \\\ & + \sin\delta~\textbf Z_\text{ecl} \end{aligned}$$

Here we set $R$ (the radius of the celestial sphere) to be equal to 1 for simplicity. The $\textbf X_\text{ecl}$-axis directs to 0° Aries, the $\textbf Y_\text{elc}$-axis to 0° Cancer, and the $\textbf Z_\text{ecl}$-axis to the northern celestial hemisphere.

The equatorial plane is inclined at an angle $-\epsilon$ relative to the plane of the ecliptic in the $\textbf Y_\text{ecl}\textbf Z_\text{ecl}$ plane.

We can use the rotation matrix, which we introduced earlier, substituting ($-\sin\epsilon$) instead of $\sin(-\epsilon)$.

$$\mathbf{A}_{YZ} = \left[\begin{array} {rrr} 1 & 0 & 0 \\\ 0 & \cos\epsilon & -\sin\epsilon \\\ 0 & \sin\epsilon & \cos\epsilon \end{array}\right]$$

It gives us cartesian coordinates of the vector $\vec{v}$ in an equatorial coordinate system:

$$\begin{cases} x_{eq} = \cos\delta\cos\lambda \\\ y_{eq} = \cos\epsilon\cos\delta\sin\lambda - \sin\epsilon\sin\delta \\\ z_{eq} = \sin\epsilon\cos\delta\sin\lambda + \cos\epsilon\sin\delta \end{cases}$$

From the conversion equation (1) it follows that

$$\begin{cases} \tan(RA) = y_{eq} / x_{eq} \\\ \sin(D) = z_{eq} \end{cases}$$

It gives us the final equations for converting $(\lambda, \delta)\rightarrow(RA, D)$:

$$\begin{aligned} \tan(RA) = \frac{\cos\epsilon\sin\lambda - \sin\epsilon\tan\delta} {\cos\lambda} \\\ \sin(D) = \sin\epsilon\cos\delta\sin\lambda + \cos\epsilon\sin\delta \end{aligned}\tag{1}$$

Invert Conversion

For invert conversion we change $\epsilon$ to $(-\epsilon)$. It gives us the following:

$$\begin{aligned} \tan(\lambda) = \frac{\cos\epsilon\sin(RA) + \sin\epsilon\tan(D)} {\cos RA} \\\ \sin(\delta) = -\sin\epsilon\cos(D) \sin(RA) + \cos\epsilon\sin(D) \end{aligned}\tag{2}$$