1. Improving the inverse of Carrington longitude to corresponding central meridian crossing time: A necessary step to estimate the effect of differential rotation Xuepu Zhao

2. 1. Introduction Synoptic charts of the photospheric magnetograms have been used as a proxy of the entire solar-surface distribution of the photospheric magnetic field. Because of the differential rotation of magnetic features, the synoptic charts actually cannot cover entire solar surface at any single time of solar rotations, though the large-scale photospheric magnetic field might be assumed to be time-independent in one solar rotation [Ulrich and Boyden, 2006; Zhao et al., 2009].

3. To correct the synoptic charts for the effect of the differential rotation and construct the "snapshot heliographic map" or the "synchronic map" that covers entire solar surface at a specific time, a concept of "Carrington time" has been introduced to approximately estimate the time difference corresponding to two Carrington longitudes so that the effect of differential rotation can be estimated on the basis of the differential rotation formula (Ulrich and Boyden, 2006).

4. 2. Roger’s method and the variable synodic rotation period of the Sun The Carrington time is measured in Carrington rotation units, and for a Carrington longitude, L, in Carrington rotation number N, the Carrington time may be simply calculated as Tc=1-L/360, and the time difference from L1 to L2 can be calculated as follows: dt = (Tc2 - Tc1)*27.2753 (1) Here 27.2753 days is the Carrington rotation period, or the average of the synodic rotation period. Because of the eccentricity of the Earth's orbit, the synodic rotation period of the Sun, snp, varies from one rotation to the next, can be estimated snp = srp + (Ve srp)/(Rse 2Pi/srp) = srp + (Ve/Rse)(srp^2/2PI), (2) where srp denotes sidereal rotation period, 25.38 days, Rse the Sun-Earth distance, Ve the Earth’s speed when the Earth is located at Rse.

5. Figure 1. Synodic rotation period of Sun in 2008. The maximum and minimum are 27.34 and 27.20 days, occurred near winter and summer solstices.

6. Figure 1, which is calculated on the basis of the date of commencement of each Carrington rotation from CR2065 to CR2079 (See Astronomical Almanac 2008), shows the variation of the synodic rotation period between 2007.12.29.05 and 2009.01.13.94. It shows that the maximum and minimum synodic period occured around winter and summer solstice, it is consistent with the fact that the Earth moves in winter solstice faster than in summer solstice (see Eqation (2).

7. 3. Improved Roger’s method By replacing the Carrington period with variable synodic periods, i.e., dt = (Tc2 - Tc1) snp (3) the calculated time difference is expected to be more accurate than that obtained using formula (1). The dashed blue line and solid blue curve in Figure 2 show the results estimated by Eqs (1) and (3). The maximum and minimum time difference between Carrington longitudes of 150.550 and 190.505 degrees do occur at the winter and summer solstice, respectively.

8. Figure 2. Comparison of different methods of inverting Carrington longitude to central meridian crossing time.

9. 4. Iteration method The black curve in Figure 2 is obtained using the basic formula for calculating Carrington longitude from time and the iteration algorithm. The blue curve matches the black curve better than the blue dashed line. Figure 3 shows results of iteration method using different code: black is xuepu’s code, and green is “suninfo”. The difference between black and green is slight, though the computation time using xuepu’s code is significantly smaller than the suninfo in IDL system.

10. Figure 4. Comparison of results from different method.

11. Figure 2. Comparison of different methods of inverting Carrington longitude to central meridian crossing time.