The number nobody publishes
Search this birthday and the answers converge on a single word and a handful of adjectives. That is not wrong, it is just the least of what can be worked out. The Sun has a position measured in degrees of longitude, and on 2 January at noon that position falls between about 10.9 and 12.3 degrees of Capricorn across births from 1930 to 2035.
A span of a degree and a half over more than a century is remarkably steady. It holds because the leap year rule was built to hold it. The Sun slips about six hours later against the calendar each year, then the extra February day claws that back, so the position oscillates within a narrow band instead of drifting away from it.
The degree also matters for anything computed from angles. Aspects are measured between exact positions rather than between sign names, so a Sun at eleven degrees of Capricorn is in a very different relationship to the rest of a chart than one at twenty six, even though both people answer the same question the same way at a party.
Eleven degrees means the second decan
Each sign splits into three bands of ten degrees, called decans or faces. Eleven degrees of Capricorn is early in the second band, which runs from ten to twenty.
The Chaldean order assigns rulers to the thirty six decans in a repeating sequence of Mars, Sun, Venus, Mercury, Moon, Saturn and Jupiter. Counting from the start of Aries, the second decan of Capricorn takes Mars. The triplicity scheme instead gives each decan to a sign of the same element in turn, so this band goes to Taurus and its ruler Venus.
Nothing about a 2 January birthday is close enough to a decan boundary for the answer to be in doubt, which is unusual and worth noting. Someone at nineteen degrees would have to check carefully. At eleven you are clear of the edges on both sides.
What the tradition attaches to this band
Capricorn is cardinal earth with Saturn as its ruler, and the tradition reads the sign as structural: long horizons, tolerance for delay, work as a method rather than a burden. The decan modifies that rather than replacing it.
Under the Chaldean reading, Mars in a Saturn sign is described as force applied to something slow, which older texts treat as productive and abrasive in equal measure. Under the triplicity reading, the Taurus colouring softens the band towards patience and physical comfort. These are interpretations from a written tradition, not findings, and they are given here as such.
Capricorn itself is cardinal earth, which the tradition describes as initiating and practical at once. Cardinal signs begin the seasons, and Capricorn begins winter at the solstice, which is where a good deal of the sign's imagery about endurance and the long dark comes from.
Where the date runs out
The Sun is one body among many and the date pins only that one. The Moon covers roughly thirteen degrees in a day, so a 2 January birth may have the Moon in either of two signs and sometimes a third, decided purely by the hour. The Ascendant cycles through the entire zodiac every twenty four hours, so without a birth time it is unknown outright.
The widget above works from your own year and birthplace rather than a noon default. It returns your exact solar degree and the distance from your Sun to the nearest sign boundary, which for this date will be a comfortable one.
None of that is a criticism of the sun sign. It is simply the reason a page can be accurate and still leave a reader with almost nothing they did not already know, unless it goes on to the degree and says plainly where the information runs out.
Questions people ask about this
Is 2 January always Capricorn?
Yes. The date falls well inside Capricorn season in every year of the modern calendar, so there is no year in which it becomes Sagittarius or Aquarius.
How many degrees of Capricorn is 2 January?
About 10.9 to 12.3 degrees at noon across the years 1930 to 2035. Your own figure depends on your year, your birth hour and your time zone, all of which the widget above accounts for.
What decan is 2 January in?
The second decan of Capricorn. The birthday sits near the start of that band and is not close enough to either edge for the year to change the answer.
Why does the degree matter?
It decides the decan, and it decides which aspects a planet makes to your Sun. Two Capricorns nine degrees apart have different charts in every respect except the sign name.
Can I be a Capricorn Sagittarius cusp on this date?
No. The Sagittarius boundary was crossed a fortnight earlier and the Sun sits well past it. Cusp language describes a gap in someone's information, and on this date there is no gap.
What else do I need for a full reading?
A birth time and a birthplace. Those two things fix the Moon precisely, produce a rising sign, and place every planet in a house, none of which the calendar date can supply.