A boundary that actually falls here
Across noon positions from 1930 to 2035, the Sun on 10 January falls between about 19.1 and 20.5 degrees of Capricorn. The decan boundary sits at twenty, inside that range.
So the answer genuinely depends on the year and, within the year, on the hour. Later years in the leap cycle and later hours of the day push the Sun above twenty degrees and into the third decan. Earlier ones leave it below, in the second. This is not something a date range can resolve, which is precisely what a calculated position is for.
It is also a useful illustration of a general point. Boundaries in astrology are exact longitudes, not date ranges, and every question that looks blurry from the outside becomes a plain yes or no once a position is computed for a specific moment.
What changes across the line
The sign does not change. Capricorn covers the whole span from zero to thirty degrees and both decans are inside it, so the Sun is in Capricorn either way, under Saturn either way, cardinal earth either way.
What changes is the decan ruler. Under the Chaldean order the second decan belongs to Mars and the third to the Sun. Under the triplicity scheme the second belongs to Taurus and Venus, and the third to Virgo and Mercury. The traditional readings shift accordingly: the earlier band is described as effort pressed against structure, the later one as structure turned towards precision and craft.
All of that is inherited interpretation. What can be checked is the degree, and on this date checking it is worth the trouble.
Why the sun sign is still certain
None of this puts the sun sign in doubt. Aquarius begins at thirty degrees of Capricorn, around 19 or 20 January, and on this date the Sun is roughly nine or ten degrees short. That is nine or ten days of travel, far more than any leap year adjustment or time zone can account for.
The reason the degree stays in such a narrow band across a century is the leap year cycle. The calendar falls about six hours behind the orbit each year and the extra February day returns it, so the position oscillates within roughly a degree and a half instead of drifting.
The tropical zodiac is measured from the March equinox, so the boundaries used here do not move with the constellations. Claims that everyone's sign has shifted refer to the constellations behind the signs and have no bearing on a calculation of this kind.
Settling it, and what remains open afterwards
The widget above computes your solar degree from your own year, hour and birthplace rather than the noon default, and names the decan it falls in. On 10 January that is a real answer rather than a confirmation of one you already had.
Beyond the Sun the date settles nothing. The Moon moves about thirteen degrees a day and may be in either of two signs. The rising sign runs through all twelve in twenty four hours and needs a time and a place before it can be worked out at all.
Questions people ask about this
Which decan is 10 January?
Either the second or the third decan of Capricorn, depending on the year and the hour. The Sun sits within half a degree of the twenty degree boundary, so this is one of the few birthdays where the decan genuinely needs calculating.
What rules the third decan of Capricorn?
The Sun in the Chaldean order, and Mercury through Virgo in the triplicity scheme. The second decan, on the other side of the line, is given to Mars and to Venus respectively.
Is 10 January Capricorn or Aquarius?
Capricorn, in every year. Aquarius begins about ten days later, so no adjustment of hour or time zone brings the boundary into range.
Does a decan boundary work like a cusp?
It has the same structure and the same answer. The Sun occupies one longitude and that longitude is on one side of the line. There is no blended state, only a figure that has not been calculated yet.
What degree is the Sun on 10 January?
About 19.1 to 20.5 degrees of Capricorn at noon across 1930 to 2035. A morning birth in an early leap cycle year sits at the low end, an evening birth in a late one at the high end.
Does the decan affect anything practical?
It refines the description of the placement and nothing more. Aspects, houses and the rest of the chart are unaffected by which side of a decan line a planet falls on.