Why is Jan. 1, the start of a new year, several days after the Winter Solstice, instead of coinciding with a solstice or equinox or other natural annual event?
Note: The question does not ask why New Year's is on January 1 instead of March 1, June 1, etc.
Because the various "calendar modifications" in Western Europe during the ages were driven by political and religious "interests" and not by scientific ones.
European calendar is basically due to Ancient Rome; see Roman calendar :
The Roman calendar changed its form several times between the founding of Rome and the fall of the Roman Empire.
The original Roman calendar is believed to have been a lunar calendar, which may have been based on one of the Greek lunar calendars. [...] Roman writers attributed the original Roman calendar to Romulus, the mythical founder of Rome around 753 BC. Romulus' calendar had ten months with the spring equinox in the first month : Martius (31 days). [...] The regular calendar year consisted of 304 days, with the winter days after the end of December and before the beginning of the following March not being assigned to any month.
Numa Pompilius, the second of the seven traditional kings of Rome, reformed the calendar of Romulus around 713 BC. The Romans considered even numbers to be unlucky, so Numa took one day from each of the six months with 30 days, reducing the number of days in the 10 previously defined months by a total of six days. There were 51 previously unallocated winter days, to which were added the six days from the reductions in the days in the months, making a total of 57 days. These he made into two months, January and February, which he prefixed to the previous 10 months [emphasis added].
Julius Caesar, as Pontifex Maximus, reformed the calendar in 46 BC. The new calendar became known as the Julian calendar. Quintilis was renamed as Iulius (July) in honour of Julius Caesar in 44 BC by Mark Antony. The calendar reforms were completed during the reign of his successor Augustus, when the Senate renamed Sextilis as Augustus (August) in 8 BC.
The calendar year originally began on 1 March, as is shown by the names of the six months following June (Quintilis = fifth month, Sextilis = sixth month, September = seventh month, etc.). It is not known when the start of the calendar year was changed to 1 January. Ancient authors attributed it to Numa Pompilius. [...] A surviving calendar from the late Republic proves the calendar year started in January before the Julian reform.
How years were identified during the Roman monarchy is not known. During the Roman Republic, years were named after the consuls, who were elected annually. Thus, the name of the year identified a consular term of office, not a calendar year. For example, 205 BC was "The year of the consulship of Publius Cornelius Scipio Africanus and Publius Licinius Crassus", who took office on 15 March of that year, and their consular year ran until 14 March 204 BC. [...] The first day of the consular term changed several times during Roman history. The Senate changed it to 1 January in 153 BC in order to allow consul Quintus Fulvius Nobilior to attack the city of Segeda (in Aragon, Spain) during the Celtiberian Wars. Before then, it was 15 March. Earlier changes are a little less certain. There is good reason to believe it was 1 May for most of the third century BC, until 222 BC.
For more details, see :
page 43 :
In fifth-century Athens [...] The 12 lunar months in Athens were: 1. Hekatombaion, 2. Metageitnion, [...]. The year started on Hekatombaion 1, which occurred on the evening of the first sighting of the new moon's crescent following the summer solstice. Another way of expressing this practice is that the last month of the year included the summer solstice. [...] To maintain alignment with the seasons, a lunar calendar eventually requires the intercalation of a 13th month. In Athens it is usually thought that this was achieved by repeating the sixth month, Poseideon, but the evidence is not so emphatic on this point [...].
A fundamental component of the Athenian democracy was a Council of 500 (the Boule), comprising 50 citizens drawn by lot annually from each of the city-state's ten tribes. The representatives of each tribe acted as a Standing Committee (prytany) of the Council for a tenth of the year. The year was thus divided into ten 'months' (prytanies). In the political calendar of Athens, dates were counted according to these ten prytanies of each year's Council. [...] How long the two calendars were independent of each other we do not know.
page 98 :
The Roman calendar still forms the core of our own western calendar, in terms of the month-names, the lengths of those months and - give or take a tweak or two since antiquity - the length of the year. But the route to the present-day version of the Roman calendar is, as ever, not straightforward.
The sequence of the Roman months was the same as it still is in the western calendar: January, February, March, April, May, June, July, August, September, October, November, December. All but July and August derive directly from the calendar of the Roman Republic - those from September to December are in fact the Latin forms of the names, while the rest are anglicised versions of the originals. July and August, on the other hand, stem from two changes in the late Republic/early Empire, when the original names of Quintilis and Sextilis were altered to reflect the names of the two leading political figures of the time, Julius Caesar and Augustus (the former Octavian). In 44 BC, before Julius Caesar's assassination, the Roman Senate decreed that the month Quintilis should be called Iulius after him, because he was born in that month. The month Sextilis was named Augustus, in the lifetime of the emperor of that name, because it was the month in which he gained his first consulship and most important victories.
[...] The Romans ascribed the creation of their own calendar to the shadowy figures of Romulus and Numa, respectively the founder and the second king of Rome (ostensibly in the eighth-seventh centuries BC). According to Macrobius, Numa made the calendar lunar, by increasing the Roman year first to 354 days and then to 355, and divided the year into 12 months. The length of the year was increased to 354 days to accord with the time 'in which 12 circuits of the moon are completed', but then Numa afterwards added an extra day 'in honour of the odd number'. To the pre-existing ten months were added January, which was made the first of the year, and February to follow it ahead of March.
page 102 :
The sole surviving Republican calendar predating the reforms of Julius Caesar is that from Antium, south of Rome, the Fasti Antiates Maiores. This dates to 84-55 BC, and its preserved fragments cover just over half the year. [...] The whole calendar starts on 1 January [...].
page 112 :
In 46 BC Julius Caesar ordered a wholesale revision of the calendar to the point of dismissing the quasi-lunar calendar of the Republic and adopting a purely solar one. The new calendar had an average year of $365 \frac 1 4$ days, with the quarter-day being absorbed into an extra single whole day added every fourth year. [...] Significantly, in his own calendar reform Caesar had the services of Egyptian astronomers, whose expertise he called on while he was in Alexandria helping to put Cleopatra back on the throne.
There are calendars where the New Year does correspond with a “natural” (or more specifically: astronomical) event. In luni-solar calendars (Babylonian, Ancient Greek, Chinese, Jewish, Indian etc.) the year begins in principle with the new moon following the vernal (or autumnal) equinox. Such calendars are however difficult to work with, as you need to intercalate a thirteenth month at irregular intervals and also because the length of each month varies depending on the time of the new moon. By contrast, the Roman calendar operates with a fixed year of 365 days, adding an extra day every four years. This means that it is relatively easy to manipulate, but it also means that the beginning of the year (and of the months) does not and indeed cannot correspond to any astronomical event.
PS. In theory, it should be possible to begin the year at the winter solstice. But to determine the date of the solstice you need to be able to measure the length of the day and the night, which was not easy before the invention of the clock. It is possible that the Roman New Year on 1 January is based on a faulty calculation of the solstice.
The Roman year started when consuls entered office or vice versa. When the Julian reform was done this was Jan. 1st. The position of the sun on the celestial sphere changes by 1 degree per day; around the 1st c BCE it was known that the vernal equinox is at 7-8 degrees from the beginning of the zodiacal constellation Aries, that is, there was a displacement of one week between the 12 sideral/zodiacal divisions and the beginnings of months. The Gregorian reform ensures that equinoxes and solstices fall within a two day fixed interval, e.g. Dec 21-22 for winter solstice; achieving absolute precision would be impractical.
