Calculating Perihelion Passage
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When does Earth's center pass through the point on Earth's orbit that comes closest to the sun? That is, when does Earth pass through its orbit's perihelion?
Over many centuries astronomers have observed the sun's location on the background of the starry sky over the entire span of the year. From those data by way of Niklas Koppernigk's concept of the solar system they have inferred the relationship between Earth's rotation and Earth's revolution about the sun: they have gained the knowledge to determine to great precision the time and date when Earth occupies any point on its orbit. Thus we expect that we can simply read the date we want out of the appropriate ephemeris.
But suppose that we don't have an ephemeris at hand. How then shall we determine the time and date of a given year's perihelion passage? We shall have to resort to calculation based on our knowledge of Earth's dynamics.
Let's start by determining the time and date of Earth's perihelion passage in AD2000. We know that at AD2000 Jan 01 AM 12:00 (noon) Earth occupied ecliptic longitude 100.46435°. At that same time Earth's perihelion occupied the ecliptic longitude 102.94719°. The difference between those two longitudes, 2.48284°, is small enough that, to an excellent approximation to the true value, we can calculate the difference between Earth's times of passage through them by converting that angle to radians (0.043333732) and dividing it by the ratio of Earth's perihelion speed (30.272 kilometers per second) to the perihelion radius of Earth's orbit (147,099,302 kilometers). We calculate Δt = 210,570 seconds = 2.437 days (2 days, 10 hours, 29 minutes), which puts the occurrence of perihelion passage at AD2000 Jan 03 PM 10:29.
With that as a starting point we can calculate the time and date of Earth's perihelion passage for other years in the few centuries (as far as I trust my approximations) prior to or after AD2000. To set up such a calculation we need to know the difference between an apsidal year and a calendar year.
The Gregorian year (365.2425 mean solar days or 365 days, 5 hours, 49 minutes, 12 seconds), the basis for the calendar, reflects the minimal manipulation of whole days needed to keep March 21 on or as close as possible to the vernal equinox. That manipulation involves inserting an extra day at the end of February every fourth year (the year evenly divisible by four) except for those century years not evenly divisible by four hundred (e.g. AD1800, AD1900, AD2100). The westward shift of the equinoxes due to the precession of Earth's axis makes the Gregorian year, on average, 19 minutes and 57 seconds shorter than the sidereal year.
The anomalistic year (365.259635864 mean solar days or 365 days, 6 hours, 13 minutes, 52 seconds) gives us the time that Earth takes after passing through one of its apsides to return to that same apsidal point. Due to gravitational interaction with the other planets, with the oblateness of the sun, and due to General Relativistic considerations, Earth's orbit precesses in the prograde direction at a rate that makes the anomalistic year 4 minutes and 43 seconds longer than the sidereal year.
On average, then, the anomalistic year runs 24 minutes and 40 seconds longer than the Gregorian year. Thus the date of perihelion passage shifts through the Gregorian calendar by one full day every 58.378 years, which means that in AD2058 perihelion passage will occur at about 10:30 PM on Jan 04.
But when we want to calculate a time and a date on the Gregorian calendar, we don't get to use "on average". Astronomers devised the calendar to use whole days, using leap days to compensate the calendrical drift due to the accumulation of the fractional day in each year. To see how we can accommodate that fact let's calculate the time and the date of Earth's perihelion passage for a specific year, AD2159.
Between AD2000 Jan 01 and AD2159 Jan 01 the Gregorian calendar marks 39 leap years minus the non-leap year of AD2100, se we can count the elapse on the calendar of 365 days x 159 years plus 38 leap days. Between the perihelion passage of AD2000 and that of AD2159 the anomalistic year puts 365 days x 159 years plus 41 days, 6 hours, 46 minutes. Thus to the time and date of AD2000's perihelion passage we must add the difference, 3 days, 6 hours, 46 minutes. That puts the perihelion passage of AD2159 into Jan 07 AM 5:15.
However, at that date 2.8654 years will have elapsed since the last leap day was inserted into the Gregorian calendar. In that time the calendar will have drifted 16 hours and 41 minutes out of synchrony with the vernal equinox. That time represents the amount by which the calendar shifts west relative to the vernal equinox before the next leap day kicks it back east to start the four year cycle over again. Thus it also represents the time that the calendar shifts relative to the perihelion, the time that we must add to the raw calculation above. Thus we find that our desired perihelion passage occurs at AD2159 Jan 07 PM 09:56.
By this means we can
calculate to reasonable accuracy the time and date of Earth's perihelion passage
for any year within perhaps 500 years of AD2000.
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