What the date does fix
The Sun is in Capricorn on 4 January in every year, without a boundary anywhere near. Its position at noon across the years 1930 to 2035 falls between about 13.0 and 14.4 degrees, a spread of well under a degree and a half. That stability is not a coincidence: the leap year rule exists to hold the calendar in step with the orbit, and it holds this date to a narrow band.
Thirteen degrees places the Sun in Capricorn's second decan, the band from ten to twenty. Its Chaldean ruler is Mars and its triplicity ruler is Venus by way of Taurus, and both attributions are traditional rather than measured.
Thirteen degrees is also far from any decan edge, which is not true of every birthday in this stretch of the month. Dates around the ninth and tenth sit close to the twenty degree line and genuinely need calculating. This one does not.
What the hour decides instead
The Moon travels roughly thirteen degrees a day, so it covers close to half a sign between midnight and midnight. A 4 January birth can have the Moon in either of two adjacent signs, and in some years three signs are technically in play across the twenty four hours. Only the birth time separates them.
The rising sign is more sensitive still. The whole zodiac passes the eastern horizon once a day, so a new sign begins rising roughly every two hours, and the intervals are uneven because of latitude. Two people born on the same 4 January four hours apart in the same city will have different rising signs and a completely different house layout.
Why the Sun is the least personal part
Everyone born within about a month of you shares your sun sign, and everyone born within about three days shares your solar degree to within a degree or two. That is a very large group. The Moon narrows it to a couple of days, the rising sign to a couple of hours, and the combination of the three narrows it a great deal further.
This is why a page that answers only the sun sign question can feel thin. The answer is correct and it is also the piece of information that distinguishes you least from anyone else.
The tradition itself has always treated the Sun as one factor among several rather than as a summary of a person. The idea that a birthday alone describes character is a twentieth century newspaper convention, not an older astrological claim, and it does not survive contact with the rest of the technique.
Closing the gaps
Two facts close most of them: the time on the birth record and the town. With those, the Moon becomes exact, the Ascendant becomes calculable, and every planet acquires a house.
The widget above starts with what the date alone can give, computing your solar degree from your year and birthplace and reporting how far your Sun sits from the nearest sign boundary. On 4 January that distance will be comfortable in either direction.
Where the record has no time on it, some things can still be narrowed. The Moon can often be pinned to one of two signs and sometimes to one outright, and a rising sign can occasionally be reasoned back from a known life event, though that is a working assumption rather than a calculation.
Questions people ask about this
What sign is 4 January?
Capricorn in every year. The date is roughly a fortnight from each end of the sign, so neither the year nor the hour of birth can move it.
Can my moon sign differ from my sun sign?
Almost always, and it usually does. The Moon changes sign every two and a half days on average, so on 4 January it may be in either of two signs depending on the hour of birth.
Do I need a birth time for this date?
Not for the sun sign, which is settled without it. You need one for the rising sign, for the houses, and to be certain of the Moon when the birth falls near a lunar sign change.
What degree is the Sun on 4 January?
About 13.0 to 14.4 degrees of Capricorn at noon across 1930 to 2035. The widget above computes the figure for your own year and location instead of the noon default.
What decan does this birthday fall in?
The second decan of Capricorn, ten to twenty degrees, with Mars as its Chaldean ruler and Venus as its triplicity ruler.
If the Sun says so little, why publish it?
Because it is asked for constantly and it is genuinely part of the chart. The point is not to discard it but to say plainly what it covers and what it does not.