Sets the time zone so your birth time converts correctly. City data from GeoNames, CC BY 4.0.

Black Moon Lilith is not a body. It is the apogee of the Moon's orbit - the far end of the ellipse the Moon travels, and the empty focus of that ellipse, where nothing is. Astrology attached the name Lilith to it from the figure in Jewish folklore who refused to submit and was written out of the story, and reads the placement accordingly: the sign it occupied at your birth describes the part of yourself you were taught to suppress, the appetite or anger or refusal that was made unacceptable, and what happens when it surfaces anyway.
Because it is a geometric point rather than an object, it falls straight out of the Moon's orbital elements. The mean apogee is the mean perigee turned through half a circle, and the mean perigee is the Moon's mean longitude less its mean anomaly - so computing it needs nothing the Moon's own position does not already require. It moves at about a degree every nine days, spending roughly nine months in each sign and completing a full circuit in just under nine years. A birth date is all the calculation needs.
The word "mean" is doing real work there, and it is the thing most calculators leave unsaid. This tool reports mean Black Moon Lilith, which is what almost all astrology means by Lilith and what almost all software reports by default. There is also a true or osculating apogee, which tracks the actual wobbling apogee of the moment rather than the smoothed average, and it can differ from the mean by more than thirty degrees over a single month - a completely different point, in a different sign, not a refinement of the same one. If a reading you have seen elsewhere disagrees with this one, that difference is the first thing to check. There are also two unrelated points sometimes called Lilith: the asteroid 1181 Lilith, and a hypothetical "Dark Moon" that has never been observed. Neither is this.
Black Moon Lilith spends about nine months in a sign, so the date settles it on all but the days it crosses a boundary.
It moves about a ninth of a degree a day, so a time matters only if you were born as it changed sign. The calculator says when that applies.
Optional. With a time and a city the tool also reports the whole-sign house, which astrologers read as the area of life the suppressed material shows up in.
See the sign with its exact degree, its element and modality, and a reading of what the placement traditionally describes.
Reports mean Black Moon Lilith and says so - the true or osculating apogee can be thirty degrees away, in a different sign entirely
Computed from the Moon's own mean orbital elements rather than a separate table, so it cannot drift out of step with the Moon
Reports the position to the arcminute, with the whole-sign house when a birth time and place are supplied
Says when your birth fell on the day Lilith changed sign, and shows both possibilities rather than picking one
Find your Chiron sign - the "wounded healer" - from your birth date, computed from a JPL Horizons ephemeris rather than an approximate date table
Find your North and South Node signs from your birth date - the lunar nodes, read as the direction of growth and what already comes easily
Find your Part of Fortune from your exact birth time and place, using the day or night formula your chart's sect actually calls for
Find your Venus sign from your birth date - how you love, what you find beautiful, and what you value in a partner - computed from Venus's real position
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