The one boundary where the ruler does not change
Saturn rules Capricorn and, in the traditional scheme, rules Aquarius as well. This is the only sign boundary on the wheel where the Sun passes from one sign to the next without changing hands, which makes the popular idea of a dramatic hybrid here harder to sustain than anywhere else.
What does change is element and mode: cardinal earth becomes fixed air at three hundred degrees of ecliptic longitude, on 19 or 20 January. Modern practice assigns Uranus to Aquarius as well, and that is a nineteenth century addition rather than part of the older system.
Why the date alternates
A tropical year is longer than a calendar year by a little under a quarter of a day. The crossing therefore arrives later by about six hours each year, until the leap day drags it back some eighteen hours, and the pattern repeats. Over one cycle the instant covers close to thirty six hours of calendar ground, which is exactly the width of the disagreement between published tables.
Nobody occupies both signs
The Sun holds one ecliptic longitude at any moment and that longitude is inside one of the twelve arcs. There is no defined interval of joint membership, and no traditional or modern technique that produces one.
Cusp, used properly, is a house term. House boundaries come from the horizon and the meridian at the place of birth, so they depend on latitude and on the minute; sign boundaries are the same for everyone born at the same instant anywhere in the world.
Late Capricorn, early Aquarius
Mars is exalted at twenty eight degrees of Capricorn, so the closing days of the sign carry the one exaltation Capricorn has, and the Chaldean scheme gives that final decan to the Sun. Capricorn's last stretch is therefore its brightest and most assertive, not its most faded.
Aquarius opens under Venus in the same order, which is why the earliest Aquarius degrees are read as warmer and more sociable than the reputation of the sign. Both facts are decan facts, and neither is a trace of the neighbouring sign.
Working out which side a birth falls on
The year, the clock time and the birthplace are the only inputs required. Run against an ephemeris they give the solar longitude directly, and it is short of three hundred degrees or beyond it.
A range printed under a magazine column cannot do this work within about two days of the mark. Beyond that window, the ranges and the calculation agree and there is nothing to check.
Questions people ask about this
Am I a Capricorn Aquarius cusp?
No, because that placement does not exist. The Sun was either short of three hundred degrees at your birth or past it, which makes you Capricorn or Aquarius.
What date does the Sun move from Capricorn to Aquarius?
19 or 20 January, depending on the year and on the time zone the birth is dated in. The instant is the same worldwide; only the local calendar date differs.
Do I have traits of both signs?
Popular astrology says so and sells the blend as a type. The chart does not show it: one Sun, one longitude, one sign. Other placements can of course sit in either sign.
What is a cusp in real astrology?
A house boundary. Houses are divided using the horizon and the meridian of the birthplace, and their cusps are genuine dividing lines within a chart.
How close to the boundary do I need to be to bother checking?
About forty eight hours either side. Inside that window the year and the birth time decide the answer; outside it, any date range gives the same result.