Ecliptic coordinate system

Remarks

The ecliptic is tilted against the celestial equator by the obliquity $\varepsilon \approx 23.44^\circ$. Both systems share the primary direction, so they differ by a rotation by $\varepsilon$ about it. In terms of equatorial coordinates $(\alpha, \delta)$: $$\sin\beta = \sin\delta\cos\varepsilon - \cos\delta\sin\varepsilon\sin\alpha,$$ $$\cos\beta\cos\lambda = \cos\delta\cos\alpha, \qquad \cos\beta\sin\lambda = \sin\delta\sin\varepsilon + \cos\delta\cos\varepsilon\sin\alpha.$$ Example: the north celestial pole, $\delta = 90^\circ$, gets $\sin\beta = \cos\varepsilon$, so $\beta = 90^\circ - \varepsilon$. Non-example: the system with the plane of the Milky Way as fundamental plane is the galactic system, not the ecliptic one. Heliocentric and geocentric ecliptic coordinates use the same angles with different centres.