Improving the inverse of Carrington longitude to corresponding central meridian crossing time: A nec

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Improving the inverse of Carrington longitude to
corresponding central meridian crossing time A
necessary step to estimate the effect of
differential rotation

• Xuepu Zhao

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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.

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• 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).

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2. Rogers 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
  Tc1-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)(srp2/2PI), (2)
• where srp denotes sidereal rotation period, 25.38
  days, Rse the
• Sun-Earth distance, Ve the Earths speed when the
  Earth is located at
• Rse.

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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.
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• 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).

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3. Improved Rogers 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.

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Figure 2. Comparison of different methods of
inverting Carrington longitude to central
meridian crossing time.
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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 xuepus
  code,
• and green is suninfo. The difference between
• black and green is slight, though the computation
  
• time using xuepus code is significantly smaller
  than
• the suninfo in IDL system.

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Figure 4. Comparison of results from different
method.
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Figure 2. Comparison of different methods of
inverting Carrington longitude to central
meridian crossing time.