Results are sensitive to planetary body angular diameter and the relative position of the North Planetary Pole as reported in an ephemeris.

I was unable to replicate North's result for a 1978 sunspot based on Carrington's method. Table 2 indicates that the accuracy of this method is in the range of 2° to 8°.

As with most astronomical applications that depend on the sine and cosine functions, their accuracy deteriorates the closer a value approaches 0°s; or 90°s;. Areas of probable inaccuracy are shown in Figure 3. Sidgwick (1971) recommends limiting measurements to within 60° of the apparent disk's center.

Figure 3 - Schematic of sin and cos generated inaccuracy zones

A feature on a planet or a sunspot on the Sun is presented in the amateur astronomer's telescope as a two dimensional disk of a three dimensional object. Such eyepiece views give only a few reference points:

• the direction of the North Celestial Pole. This direction is found by temporarily turning the tracking drive off and with an eyepiece reticule, orienting along an east-west celestial sphere line.
• the center of the planetary and solar disk.
• the position angle of the feature on the disk, measured from the North Celestial Pole by using a reticule.
• the linear from the center of the disk to the feature, again measured by using a reticule, a sketch or an astrophotograph.

Figure 4 diagrams a typical situation and the data to be collected at the eyepiece.

Figure 4 - Data collected at the
eyepiece in order to use the calculator

If you add data from an ephemeris -

• the position angle of the planet's or Sun's North Planetary Pole, or longitude 0°, latitude 90° on the body;
• the distance that the north planetary pole is along the NPP line (in arcsecs); and,
• the body's latitude and longitude of the center of disk as seen from Earth, also called the sub-Earth or sub-Observer latitude and longitude,

- then you have enough data to complete several spherical right triangles. Figure 5 diagrams the data to be collected from ephemeris sources.

Figure 5 - Data collected
from ephemeris sources in
order to use the calculator

From this data, several spherical right triangles can be constructed. The spherical right triangles ultimately allow the body-centric latitude and longitude of the feature to be calculated. Figure 6.

The traditional reduction procedure is Carrington's method that was developed in the early 1900s. North (1997) and Sidgwick (1971) detail Carrington's algorithm in the following equations:

sin(R-rho) = r/r_theta {Eq. 1}

rho = arcsin(r / r_theta) - R {Eq. 2}

sin(B) = sin(B_theta)*cos(rho) + cos(B_theta)*sin(rho)*cos(P-theta) {Eq. 3}

sin(L) = sin(rho)*sin(P-theta)/Math.cos(B) {Eq. 4}

I was unable to implement Carrington's method in Javascript ECMA code, principally because of the number of quadrant checks that needed to be done to assure that the Javascript asin and acos functions would function in all cases. Carrington's equation seemed to work only in the first cartesian quadrant of the apparent view of the Sun.

Figure 6 - Spherical right
triangles constructed from
observational and ephemeris data

An alternative method was implemented. The apparent two dimensional presentation in the eyepiece of the three-dimensional planetary or solar body includes a fundamental 2d right triangle. Figure 7. This 2d right-triangle is based on the origin of the North Celestial Pole and Celestial East-West lines, the position angle to the feature, and a perpendicular line extended from the feature to the Celestial East-West line seen in the eyepiece.

Once the included angle is found from the position angle of the feature, the right triangle can be solved for distances x and y.

Projecting these two edges into three-dimensional space that defines a coordinate system of the planet, edges x and y are the sides of two right triangles that define the azimuth and the altitude of the feature. Figure 8. For lack of a better term, we will call the altitude azimuth coordinate system seen in the eyepiece the "eyepiece coordinate system."

Figure 7 - Fundamental apparent 2D triangle in eyepiece view

Once Triangle I and Triangle II in Figure 8 are solved, we know the azimuth and altitude of the feature in this 3D "eyepiece coordinate system." A similar set of triangles is solved for the planetary or solar north body pole - again yielding an altitude and azimuth in the eyepiece coordinate system. Figure 9.

From the coordinates of the feature and planetary north pole, the angular distance between them can be computed using basic spherical trigonometry after Duffett-Smith (1988). By definition, the geographical latitude of an object is based on the angular distance of a location from the planet's north celestial pole. (For northern hemisphere objects, the latitude is 90° minus the distance from the north pole; for southern hemisphere objects, the distance from the north pole minus 180°. By this method, the geographical planetary latitude of the feature is determined.

Geographical longitude is determined from ephemeris data and our computation of the azimuth of a feature in the eyepiece coordinate system. Ephemeris data gives the planetary or solar geographical latitude and longitude of the center of the apparent disk seen in the eyepiece. The distance between the planetary north pole and the zenith in eyepiece (the north celestial pole) can be found by spherical trigonometry. This angular distance also defines the obliquity of the planetary latitude line that runs through the apparent center of the disk. Figures 10 and 11.

The computed azimuth gives an hour angle between the center of the disk and the feature. The obliquity of the meridian line gives the included angle shown in Figure 10. From this information the hour angle between the apparent center of the disk and feature can be found in the planet's geographical coordinate system with:

H'= H / cos(obliquity of the meridian) {Eq. 5}

H' is the hour angle along the planetary meridian;

H is the hour angle equal to the feature's azimuth.

The feature size computation is based on the foreshortening equation after Sidgwick (1971):

f_scale = sec(angular distance from center) * object apparent diameter {Eq. 6}

The secant is the reciprocal of the cosine, therefore:

North (1997) also provides an iteration method for adjusting the apparent latitude of a feature for the oblateness of the body's spheroid. This technique applies to the Sun and major gas giants and is based on three iterations of the following equation based on eyepiece measurements shown in Figure 2:

sin(B) = [ 2 ( D/2 - X ) ] / [ D + (D + (oblateness * D * cos(B_est)] {Eq. 7}, where -

B_est is an initial estimate of the feature's latitude, usually B, computed in the feature position calculator, above; and,

D and X are drawing unit measurements made at the eyepiece or from an astrophotograph as shown in Figure 2.

A supplemental calculator performs the tedious reduction computations.

The final accuracy achieved by the position calculator was not what was hoped (see Table 2), but is sufficient for the amateur purposes of generally describing the latitudes of sunspots or resolving the approximate locations of uncertain features on the Moon or Mars.

Figure 8 - Fundamental apparent 2D triangle seen in eyepiece projected into 3D space

Figure 9 - Solving the alt-az coordinate system for feature's geographic latitude

Figure 10 - Solving the alt-az coordinate system for feature's geographic longitude