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Typically we can deliver astrometric positions of natural satellites witherrors in the 50-150 mas range. Apparent distances from mutual phenomena, havemuch smaller errors, less than 10 mas. However, this method can only be appliedduring the equinox of the planets. We developed a method that can provideaccurate astrometric data for natural satellites -- the mutual approximations.The method can be applied when any two satellites pass close by each other inthe apparent sky plane. The fundamental parameter is the central instant $t_0$of the passage when the distances reach a minimum.  We applied the method for the Galilean moons. All observations were made witha 0.6 m telescope with a narrow-band filter centred at 889 nm with width of 15nm which attenuated Jupiter's scattered light. We obtained central instants for14 mutual approximations observed in 2014-2015. We determined $t_0$ with anaverage precision of 3.42 mas (10.43 km). For comparison, we also applied themethod for 5 occultations in the 2009 mutual phenomena campaign and for 22occultations in the 2014-2015 campaign. The comparisons of $t_0$ determined byour method with the results from mutual phenomena show an agreement by lessthan 1-sigma error in $t_0$, typically less than 10 mas. This new method isparticularly suitable for observations by small telescopes.

Astrometry of mutual approximations between natural satellites. Application to the Galilean moons