Traditionally the start and stop times of Carrington rotations are derived assuming that the solar rotation period as viewed from the Earth is a constant. This assumption is perfectly adequate for most studies. We've had several inquiries asking how we derived these times, so for academic purposes we describe below both the original derivation and the presently-used technique that accounts for the variable length of Carrington rotations.
The following quote (emphasis added) is from from the " Carrington and Bartels Calendars" page at Stanford University:
The rotation of the Sun and the orbital motion of the Earth are in the same direction (both counterclockwise when viewed from north of Earth's orbital plane). As a consequence the apparent (synodic) solar rotation period is a bit longer than the true (sidereal) period. (Hint: it may help to think of the sideREAL period as the 'real' period.) If you think about it you can convince yourself that the sidereal period is equal to the synodic period divided by (1+frac), where frac = 27.2753/365.25 = the fraction of a year the Earth moves in its orbit during one synodic solar rotation.
Lord Carrington determined the solar rotation rate by watching low-latitude sunspots in the 1850s. He defined a fixed solar coordinate system that rotates in a sidereal frame exactly once every 25.38 days (Carrington, Observations of the Spots on the Sun, 1863, p 221, 244). The synodic rotation rate varies a little during the year because of the eccentricity of the Earth's orbit; the mean synodic value is about 27.2753 days. See the back of an Astronomical Almanac for details.
The Earth is farthest from the Sun in July and closest in January. Objects that are farther from the Sun have a smaller angular velocity (a longer 'year') than do closer objects. Hence the synodic period is a bit smaller in July than in January.
Previously we determined the start and stop times of Carrington
rotations by assuming a constant period of 27.2753 days.
The phase was taken from the Astronomical Almanac, which gives a
value for Carrington longitude of 349.03 degrees at 0000 UT on 1 January 1995.
One can then derive the Carrington longitude in degrees
(call it OLD) as a function of time:
OLD = 349.03 - (360.* X / 27.2753),
where X is the number of days since 1 January 1995. It is understood that OLD is to be taken modulo 360. Note that the Carrington longitude decreases as time increases. If one now compares the values of OLD with the values listed in the Almanac one finds reasonable agreement, with maximum discrepancies of about 4 hours.
To get a better estimate of the start and stop times, we find the difference
between OLD and the values listed in the Astronomical Almanac, and then
fit the difference (ALMANAC - OLD) with a sine-cosine series:
Fit#1 = f + X/g + a*SIN(2*π*X/e) + b*SIN(4*π*X/e) + h*SIN(6*π*X/e)
+ c*COS(2*π*X/e) + d*COS(4*π*X/e) + i*COS(6*π*X/e)
In Figure 1 the red data points are the values of (ALMANAC
- OLD), and the blue line reperesents Fit#1. The improved estimate is then
NEW = OLD + Fit#1
The values of (NEW - ALMANAC) are plotted in Figure 2. The maximum discrepancy is now about 2 minutes. Notice that the data points between Days ~ 350 and 700 appear disjoint from the rest of the data set. This appears to be caused by a small offset in the Almanac tables for 1996. Each volume of the Almanac presents data for the stated year, for the last day of the previous year and the first day of the next year. The last day of the previous year is referred to as "January 0", and the first day of the subsequent year is called "December 32". This repetition of data allows a comparison of data from the 1995, 1996, and 1997 almanacs, shown in Table 1.
It appears that the longitudes listed for 1996 is systematically very slightly
larger than for either 1995 or 1997. The fitting procedure described above can
be generalized to allow for a different offset in 1996. We define:
Fit#2 = Fit#1 + j for the year 1996
Fit#2 = Fit#1 for all other years
NEW2 = OLD + Fit#2
Values of (ALMANAC - NEW2) are shown in Figure 3. The residuals are less than 1 minute nearly all of the time. The function NEW2 is then used to generate the Carrington rotation start and stop times in our plots and listings.
Longitude data from the almanac were added for years past 2001. A plot of residuals using the 2001 fit shows that the fit is not as good for recent dates as it is for the original time period. So a new fit was done with the same fitting formula. The residuals are more stable with the new fit.