**Planetary bodies and the Moon: ** The United States Naval Observatory Astronomical Applications Department has an online application that renders the current geocentric view of the Moon and other planets labeled with the sub-Earth latitude and longitude and a grid. Choose the option for "Data Services" and then "What The Moon Looks Like Today" or USNO "Apparent Disk of Solar System Object." The synthetic images can be printed to a transparency and then held over a printed copy of your image.

**Sun: ** The Stanford Solar Center Lat-Long Solar Screens provides online solar screen images that latitude and longitude grids that can be printed as transparencies and then held over your image.

Mars feature maps can be reported in planetographic (increasing longitude to the west) or planetocentric (increasing longitudes to the east). This calculator does not compensate for planetographic coordinate systems.

For the Sun, Jupiter and Saturn, oblateness of the spheroid may cause minor adjustments to latitude.

Online ephemeris generators (NASA/JPL Horizons Ephemeris Generator) or desktop programs (USNO Multi-year Interactive Computer Almanac v2.0 "MICA") provide the additional physical planetary or solar data that supplements the feature information gathered at the eyepiece or from an astrophotograph. The field names in two ephemeris sources, that contain the data for lookup, are listed as follows:

Table 1 - Obtaining ephemeris and other data needed to use the calculator

Variable letter used here | Item/Source__________________________________ | NASA/JPL Ephemeris_Generator fields | MICA_v2.0_fields |
---|---|---|---|

S | Angular radius of the disk (arcsecs) | 13. Target angular diam. [Ang-diam] | Semi-diameter (1/2 D in arcmins) |

P | Body's polar position angle | 17. N. Pole Pos. Ang & Dis [NP.ang] | Position Axis |

P_dist | Distance to the north body pole (arcsecs) | 17. N. Pole Pos. Ang & Dis [NP.dist] | Not available |

L_theta | Longitude of the center of the disk on the body's surface coordinate system (degrees) | 14. Obs sub-lng & obs sub-lat [Ob-lon] | Sub-Earth Pt. Long. |

B_theta | Latitude of the center of the disk on the body's surface coordinate system (degrees) | 14. Obs sub-lng & obs sub-lat [Ob-lat] | Sub-Earth Pt. Lat. |

Distance_au | Distance to the body in astronomical units (a.u.) | 20. Obsrv range & rng rate | delta |

**Notes:**

In the NASA/JPL Ephemeris_Generator use Table Settings: 13,14,17 and 20

The calculator default data is based on the Moon on August 19, 2006 at 7:00UTC from a geocentric position looking at Mare Crisium. One working example is by Gerald North (1997) at pp. 260-264, concerning a sunspot on June 1, 1978 at 9:35UTC from a geocentric position. Worked test examples in each cartesian quadrant of the apparent eyepiece view were tested based on the USNO "Apparent Disk of Solar System Object" online generator. Test ephemeris data files are from the NASA JPL Ephemeris Generator data for that date and time.. Test results are shown as follows:

Table 2 - Worked examples to test accuracy

Variable letter used here | Item/Source__________________________________ | G. North - Sun on June 1, 1978 at 9:35 UTC_______ | M. Crisium Aug. 19, 2006 7:00 UTC______ | Aristarchus Plateau Aug. 19, 2006 7:00 UTC_____ | C. Gassendi Aug. 19, 2006 7:00 UTC______ | C. Fracastorius Aug. 19, 2006 7:00 UTC_______ | GSR Jupiter Aug. 23, 2006 0:00 UTC_______ |
---|---|---|---|---|---|---|---|

n/a | Cartesian quadrant the feature is located in | 1 | 1 | 2 | 3 | 4 | 4 |

n/a | Data file link | Link | Link | Link | Link | Link | Link |

n/a | Construction image link | none | Link | Link | Link | Link | Link |

theta | Position angle of the feature measured counter-clockwise through west (CCW) (degrees) | 295.5° | 297.8° | 59.6° | 109.9° | 228.4° | 255.3° |

r | Distance from the center of the disk to the feature (drawing, pixels or reticule units) | 53.5mm | 154px | 162.4px | 128.7px | 106.9px | 92.4px |

r_theta | Radius of the disk (drawing, pixels or reticule units) | 76mm | 186px | 186px | 186px | 186px | 135px |

S | Angular radius of the disk (arcsecs) | 1895.6" | 1828.2" | 1828.2" | 1828.2" | 1828.2" | 35.1" |

P | Body's polar position angle | -15.59° | 2.14° | 2.14° | 2.14° | 2.14° | 18.7° |

P_dist | Distance to NPP from center (arcsecs) | -946.26° | -907.94° | -907.94° | -907.94° | -907.94° | -16.38° |

L_theta | Longitude of the center of the disk on the body's surface coordinate system (degrees) | 22.8° | 6.5° | 6.5° | 6.5° | 6.5° | 300.1° |

B_theta | Latitude of the center of the disk on the body's surface coordinate system (degrees) | -0.62° | -6.76° | -6.76° | -6.76° | -6.76° | -3.36° |

| Accuracy test | Results | |||||

B | Longitude of the feature in the body's surface coordinate system (degrees) | E63.8°E58.8°5.0° | E59.1E57.8°1.3° | W51.0°W54.2°-3.2° | W40.1W34.3°5.8° | E33.2°E32.2°1.0° | n/a °347.4 °n/a ° |

L | Latitude of the feature in the body's surface coordinate system (degrees) | N25.9°N8.1°17.8° | N17N16.60.4° | N26.0N21.7°4.3° | S17.6S18.7°-1.1° | S21.5S28.8°-7.3° | -22°-34.0°12.0° |

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 |

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 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 |

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 Projecting these two edges into three-dimensional space that defines a coordinate system of the planet, edges |
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 ( |
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 |

Cox, A. 2000 (4ed). Allen's Astrophysical Quantities. AIP Press. At Table 12.3, Sec. 14.1 and Sec. 12.7 (physical constants of planets and Sun)

Duffet-Smith, P. 1988 (3d). Practical Astronomy with Your Calculator. Cambridge Univ. Press. At p. 51, Sec. 32 (coordinate conversion, Carrington's method)

Gulliberg, J. 1997. Mathematics: From the Birth of Numbers. W.W. Norton Co. At 584-585. (Polar coordinates)

North, Gerald. 1997 (2d). Advanced Amateur Astronomy. Cambridge Univ. Press. At pp. 260-264 (Carrington's method) and 210-212 (adjustment of oblateness of spheroid body)

Sidgwick, G. 1971. Observational Astronomy for the Amateur. Dover.. At 34-36 (sunspot area), pp. 31-34 (Carrington's method), pp. 140-141 (latitude of features on Jupiter).

This javascript calculator implements a low accuracy formula for estimating the latitude and longitude of a feature on a solar system body. Because it is low precession and has not been extensively tested, use this calculator subject to this cautionary warning regarding accuracy.

This note is amateur astronomer product. Corrections to any errors are welcomed and appreciated.

No copyright is asserted as to any material in this web document including embedded astronomy javascript functions.

Prepared K. Fisher fisherka@csolutions.net org. 8/2006