The measured position
Longitude is counted from the March equinox point round the ecliptic, with Capricorn beginning at two hundred and seventy degrees. On 30 December, measured at noon in every year from 1930 to 2035, the Sun sits between about seven point nine and nine point three degrees past that line.
So the answer to where the Sun was on your birthday is not a thirty degree band but a window of a little over a degree, and it holds for anyone born on this date in the last hundred years. That is a far more specific statement than a sign name, and it costs nothing more than a calculation to produce. It is also the form the tradition actually uses, since aspects, decans and house placements are all reckoned in degrees rather than in sign names.
Why a century of drift comes to so little
A tropical year is close to three hundred and sixty five and a quarter days. A calendar year of three hundred and sixty five days therefore ends slightly early, so each ordinary year the same date arrives about a quarter of a degree earlier in the Earth's orbit and the Sun appears a little further back.
Every fourth February adds a day, which pushes the date forward again by roughly three quarters of a degree in one move. Averaged over the cycle the two effects almost cancel, leaving a slow residue that the century and four hundred year rules mop up. This is why a fixed calendar date can be tied to a solar position at all, and why the tie is good for centuries rather than decades. The correction is not perfect, which is why the range for this date is a degree wide rather than a single figure, but it is close enough that the sign has never been in question.
Still the first decan, but near its edge
Ten degrees of Capricorn is where the first decan ends and the second begins. Eight and a bit degrees is inside the first, so the Chaldean ruler for this date is Jupiter rather than Mars, and that holds across the whole range of years measured. It is close to the line, though, and the next date along is where the answer stops being uniform.
The tradition treats a decan as a modifier on the sign rather than a replacement for it. Saturn rules Capricorn throughout, and Jupiter colours the opening third. What is read into that combination is interpretation; the band and the ruler are arithmetic. Sources vary in what they attach to a band, and none of that variation touches the boundary itself.
From the average to your own figure
The range given here is what noon produces across a century. Your own position depends on your birth year, on the hour, and on the longitude of your birthplace, because the local clock and the universal instant are not the same thing. The widget above takes all three and returns the exact figure, together with the distance to the nearest sign boundary.
That distance is the useful part to see rather than assume. On this date it will be large in both directions, so the sun sign is not in question. The placements that genuinely are in question, the Moon and the Ascendant, need a birth time before anything can be said about them at all.
Questions people ask about this
What sign is 30 December?
Capricorn. The Sun has been in the sign since the solstice on the twenty first or twenty second, so this date is more than a week inside it in every year.
How many degrees into Capricorn is the Sun?
About eight. Across the years from 1930 to 2035 the noon position ranges from roughly seven point nine to nine point three degrees.
Which decan does this date fall in?
The first decan of Capricorn, ruled by Jupiter in the Chaldean order. The position sits below the ten degree line in every year measured, though not by a wide margin.
Why does the position barely move over a century?
The leap year rule keeps the calendar and the orbit in step. The date slips back a quarter of a degree each ordinary year and the leap day pushes it forward again, so only a small residue accumulates.
Does my birthplace change the degree?
Slightly, because it changes the universal time corresponding to your local birth hour. The effect is a fraction of a degree, which matters for precision rather than for the sign.
Is a chart with a late decan Sun read differently?
The decan ruler changes the emphasis in a traditional reading, and here it remains Jupiter. Nothing about being near the edge of a band weakens it, since the bands have hard boundaries rather than gradients.