# Calculating Carrington longitude¶

Carrington coordinates are a heliographic coordinate system, where the longitude is determined assuming that the Sun has rotated at a sidereal period of ~25.38 days starting at zero longitude at a specific time on 1853 November 9.
For a given observer at a given observation time, the center of the disk of the Sun as seen by that observer (also known as the apparent 1 sub-observer point) has a Carrington latitude and longitude.
When the observer is at Earth, the apparent sub-Earth 2 Carrington latitude and longitude are also known as B_{0} and L_{0}, respectively.

## Quick summary¶

SunPy calculates the sub-observer Carrington longitude in a manner that enables co-alignment of images of the Sun’s surface from different observatories at different locations/velocities in the solar system, but that results in a ~20-arcsecond discrepancy with *The Astronomical Almanac*.

## Observer effects¶

An observer will perceive the locations and orientations of solar-system objects different from how they truly are. There are two primary effects for why the apparent sub-observer point is not the same as the “true” sub-observer point.

### Light travel time¶

It takes time for light to travel from the Sun to any observer (e.g., ~500 seconds for an observer at Earth). Thus, the farther the observer is from the Sun, the farther back in time the Sun will appear to be. This effect is sometimes called “planetary aberration”.

As an additional detail, the observer is closer to the sub-observer point on the Sun’s surface than to the center of the Sun (i.e., by the radius of the Sun), which corresponds to ~2.3 lightseconds.

### Observer motion¶

The motion of an observer will shift the apparent location of objects, and this effect is called “stellar aberration”. A more subtle question is whether observer motion also shifts the apparent sub-observer Carrington longitude.

Stellar aberration causes the apparent ecliptic longitude of an observer to be different from its true ecliptic longitude. This shift in ecliptic longitude also means a shift in the heliographic longitude of the observer. For example, for an Earth observer, the apparent heliographic longitude of Earth is ~0.006 degrees (~20 arcseconds) less than its true heliographic longitude. For some purposes, this shift in the heliographic longitude of the observer should also shift the apparent sub-observer Carrington longitude.

However, shifting the sub-observer Carrington longitude due to observer motion is not always appropriate.
For certain types of observations (e.g., imagery of features on the surface of the Sun), stellar aberration does not shift the positions of solar features **relative** to the center of the disk of the Sun; everything shifts in tandem.
Thus, for the purposes of co-aligning images from different observatories, it is critical to **exclude** any correction for observer motion.

## Background information¶

### The IAU definition¶

Since the original definition of Carrington coordinates 3, there have been a number of refinements of the definition, but there have been inconsistencies with how those definitions are interpreted. Possible points of contention include the reference epoch, the rotation period, and the handling of observer effects.

The most recent definition was essentially laid down in 2007 by the IAU 4:

The reference epoch is J2000.0 TDB (2000 January 1 12:00 TDB). The difference between Barycentric Dynamical Time (TDB) and Terrestrial Time (TT) is very small, but both are appreciably different from Coordinated Universal Time (UTC).

The longitude of the prime meridian (W

_{0}) at the reference epoch is 84.176 degrees. This longitude is the “true” longitude, without accounting for any effects that would modify the apparent prime meridian for an observer (e.g., light travel time).The sidereal rotation rate is exactly 14.1844 degrees per day. This is close to, but not exactly the same as, a sidereal rotation period of 25.38 days or a mean synodic period of 27.2753 days.

The ICRS coordinates of the Sun’s north pole of rotation is given as a right ascension of 286.13 degrees and a declination of 63.87 degrees.

### The tuning of W_{0}¶

Importantly, the IAU parameters (specifically, W_{0}) were tuned following an investigation by *The Astronomical Almanac* 5.
The prescription of *The Astronomical Almanac* included both the correction for light travel time and the correction for stellar aberration due to Earth’s motion.
When using this prescription, *The Astronomical Almanac* found that a W_{0} value of 84.176 degrees minimized the differences between the modern approach and earlier approaches.
However, for those purposes where one should use sub-observer Carrington longitudes without the stellar-aberration correction, there will be a discrepancy compared to *The Astronomical Almanac* and to calculations made using earlier approaches.

## The approach in SunPy¶

SunPy determines apparent sub-observer Carrington longitude by including the correction for the difference in light travel time, but **excluding** any correction for stellar aberration due to observer motion.
The exclusion of a stellar-aberration correction is appropriate for purposes such as image co-alignment, where Carrington longitudes must be consistently associated with features on the Sun’s surface.

## Comparisons to other sources¶

### Compared to *The Astronomical Almanac* and older methods of calculation¶

*The Astronomical Almanac* publishes the apparent sub-Earth Carrington longitude (L_{0}), and these values include the stellar-aberration correction.
Consequently, SunPy values will be consistently ~0.006 degrees (~20 arcseconds) greater than *The Astronomical Almanac* values, although these discrepancies are not always apparent at the printed precision (0.01 degrees).

Since *The Astronomical Almanac* specifically tuned the IAU parameters to minimize the discrepancies with older methods of calulation under their prescription that includes the stellar-aberration correction, SunPy values will also be ~20 arcseconds greater than values calculated using older methods.
Be aware that older methods of calculation may not have accounted for the variable light travel time between the Sun and the Earth, which can cause additional discrepancies of up to ~5 arcseconds.

*The Astronomical Almanac* does not appear to account for the difference in light travel time between the sub-Earth point on the Sun’s surface and the center of the Sun (~2.3 lightseconds), which results in a fixed discrepancy of ~1.4 arcseconds in L_{0}.
However, this additional discrepancy is much smaller than the difference in treatment of stellar aberration.

### Compared to SPICE¶

In SPICE, the apparent sub-observer Carrington longitude can be calculated using the function subpnt.
subpnt allows for two types of aberration correction through the input argument `abcorr`

:

“LT”, which is just the correction for light travel time

“LT+S”, which is the correction for light travel time (“LT”) plus the correction for observer motion (“S”)

SunPy calculates the apparent sub-observer Carrington longitude in a manner equivalent to specifying “LT” (as opposed to “LT+S”). The discrepancy between SunPy values and SPICE values is no greater than 0.01 arcseconds.

### Compared to JPL Horizons¶

In JPL Horizons, one can request the “Obs sub-long & sub-lat”. JPL Horizons appears to start from the IAU parameters and to include the correction for light travel time but not the correction for observer motion (i.e., equivalent to specifying “LT” to subpnt in SPICE). The discrepancy between SunPy values and JPL Horizons values is no greater than 0.01 arcseconds.

### Compared to SunSPICE¶

In SunSPICE, one can convert to and from the “Carrington” coordinate system using the function convert_sunspice_coord. However, these Carrington longitudes are “true” rather than “apparent” because the observer is not specified, so there are no corrections for light travel time or for observer motion. For example, for an Earth observer, SunPy values will be consistently greater than SunSPICE’s coordinate transformation by ~0.82 degrees.

## Footnotes¶

- 1
The term “apparent” (as opposed to “true”) is used to indicate that the observer perceives the locations and orientations of objects different from how they truly are due to effects that include light travel time.

- 2
“Sub-Earth” is a special case of “sub-observer”, where the observer is at Earth (specifically, Earth center). That is, the sub-Earth point is the point on the Sun’s surface “below” the center of the Earth along the Sun-Earth line.

- 3
Carrington (1863),

*Observations of the Spots on the Sun*, p. 16 (https://archive.org/details/observationsofsp00carr/page/16/mode/1up) and p.27 (https://archive.org/details/observationsofsp00carr/page/27/mode/1up)- 4
Seidelmann et al. (2007), “Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2006”, http://dx.doi.org/10.1007/s10569-007-9072-y

- 5
Urban & Kaplan (2007), “Investigation of Change in the Computational Technique of the Sun’s Physical Ephemeris in The Astronomical Almanac”, http://asa.hmnao.com/static/files/sun_rotation_change.pdf