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The moment of mean conjunction can easily be computed from an expression for the mean ecliptical longitude of the Moon minus the mean ecliptical longitude of the Sun (Delauney parameter D). Jean Meeus gave formulae to compute this in his Astronomical Formulae for Calculators based on the ephemerides of Brown and Newcomb (ca. 1900); and in his 1st edition of Astronomical Algorithms based on the ELP2000-85 (the 2nd edition uses ELP2000-82 with improved expressions from Chapront et al. in 1998). These are now outdated: Chapront et al. (2002) published improved parameters. Also Meeus’s formula uses a fractional variable to allow computation of the four main phases, and uses a second variable for the secular terms. For the convenience of the reader, the formula given above is based on Chapront’s latest parameters and expressed with a single integer variable, and the following additional terms have been added:

We live in a Cosmic "shooting gallery". Objects inhabiting our Solar System have been profusely and mercilessly blasted by showering asteroids and comets for billions and billions of years. However, planets and large moons have their way of smoothing away the scars--their strong gravity pulls them into a nice ball-like spherical shape. Furthermore, some of these larger spheres possess sufficient internal heat to cause flows of fiery lava and other volcanic features that can fill in the scars of impact craters. A few such large bodies are blasted by strong winds and pouring rains, which also erode away the pockmarks left on their surfaces by showering impactors. There was a time when Earth had no Moon. About 4.5 billion years ago, when our ancient Solar System was still forming, the dark night sky above our primordial planet was moonless. At this time, the Earth was about 60 percent formed, although it did have a differentiated crust, mantle, and core. This was a very chaotic and violent era in our Solar System's past, with planets first forming out of blobs of primordial dust, gas, and rock. During this era, frequently likened to a "cosmic shooting gallery", collisions between the still-forming planets were commonplace. Orbits were not as orderly as they are now. On March 27, 2012, Cassini made its closest flyby yet over Enceladus's "tiger stripes". In a string of enticingly close passes over the dazzling moon, the spacecraft saw more hints that watery jets may be shooting out into Space from an immense subsurface sea. The jets, tearing through cracks in the moon's icy crust, could lead back to a zone harboring living tidbits.