Next: Dates Up: Spherical Astronomy Previous: Azimuth of Ecliptic Ascension

Ecliptic Altitude and Orientation

Consider a point on the ecliptic circle of ecliptic longitude . We wish to determine the altitude of this point, as well as the angle subtended there between the ecliptic and the vertical, hours before or after it culminates at the meridian, as seen from an observation site on the earth's surface of latitude .

The situation is as shown in Fig. 13. Here, is the point in question, and an altitude circle (i.e., a great circle passing through the zenith) drawn through it. We wish to determine the altitude of point , as well as the angle . Note that is defined such that it lies between the ecliptic in the direction of increasing ecliptic longitude and the altitude circle in the direction of increasing altitude. Moreover, is acute when increasing altitude, , corresponds to increasing ecliptic latitude, , and obtuse when increasing corresponds to decreasing . See Figs. 13 and 14. Incidentally, this definition is adopted in order to simplify the calculation of lunar parallax--see Sect. 6.4. In the following, we shall refer to as the parallactic angle. However, it should be noted that, according to the modern definition, the parallactic angle is .

From Eqs. (15) and (16), the declination and right ascension of point are given by

 (44) (45)

respectively. We can also write , where is the right ascension of the point on the ecliptic which is culminating (i.e., point in the diagram), and is measured in time-degrees. Note that if is positive then it measures time before culmination, whereas if it is negative then its magnitude measures time after culmination. It follows from Eqs. (30) and (31) that the altitude and azimuth of point satisfy
 (46) (47)

respectively.

From Eq. (8), the unit vector

 (48)

is directed from the observation site to point . Furthermore, the unit vector
 (49)

is tangent to the ecliptic circle, at point , in the direction of increasing ecliptic longitude. From Eq. (27), the unit vector
 (50)

is directed from the observation site to point . Here, and are the altitude and azimuth, respectively, of this point in the sky. Moreover, the unit vector
 (51)

is a tangent to the altitude circle passing through point in the direction of increasing altitude. It follows from the definition of parallactic angle, and elementary vector algebra, that
 (52)

However, according to Eqs. (24), (25), and (48),
 (53) (54) (55) (56)

The previous five equations can be combined to give
 (57)

Now, it follows from Eq. (26) that

 (58)

This quantity is significant because if then increasing altitude corresponds to increasing ecliptic latitude, whereas if then increasing altitude corresponds to decreasing ecliptic latitude. Thus, in the former case, is the solution of (57) which lies in the range , whereas in the latter case it is the solution which lies in the range .

According to Eq. (46), the critical value of at which point reaches the horizon is given by

 (59)

Of course, the above equation is only soluble if . However, it is easily demonstrated that if then point never sets, whereas if then point never rises.

Note that the value of at represents the inclination of the ecliptic to the vertical as point culminates. Furthermore, the values of at (corresponding to ) represent the inclination of the ecliptic to the vertical as point rises and sets.

Tables 18-26 show the altitudes of twelve equally spaced points on the ecliptic, as well as the parallactic angle at these points, as functions of time, calculated for a series of observation sites in the earth's northern hemisphere with equally spaced terrestrial latitudes. The twelve points correspond to the start of the twelve zodiacal signs, and are named accordingly. Thus, Aries" corresponds to ecliptic longitude , Taurus" to ecliptic longitude , etc. For each point, four columns of data are provided. The first column corresponds to the time (in hours and minutes) either before or after the culmination of the point, the second column gives the altitude of the point (which is the same in both cases), the third column gives the parallactic angle, , for the case in which the first column indicates time prior to the culmination of the point, and the fourth column gives the parallactic angle for the opposite case. Data is only provided for cases in which the various points on the ecliptic lie on or above the horizon.

Now, it can be seen, from the above analysis, that if , , then , , , , , and so , . It follows that Tables 18-26 can also be used to calculate altitudes and parallactic angles of points on the ecliptic, as functions of time, for observation sites in the earth's southern hemisphere. For example, suppose that we wish to determine the altitude and parallactic angle of the first point of Gemini (i.e., ), as seen from an observation site of terrestrial latitude , 3 hours before and after it culminates at the meridian. In order to do this, we must examine the Sagittarius (i.e., ) entry in the ecliptic altitude table: i.e., Table 19 (since when ). The fourth row of this entry tells us that 03:00 hrs. before culmination the altitude and parallactic angle of the first point of Gemini are and , respectively (since and as ). This row also tells us that 03:00 hrs. after culmination the altitude and parallactic angle of the first point of Gemini are and , respectively

Table 1: Ecliptic longitudes (relative to the mean equinox at the J2000 epoch), ecliptic latitudes, and visual magnitudes of selected bright stars lying within of the ecliptic plane.
 Aries Libra Mag. Name Mag. Name PEG VIR PSC VIR VIR VIR VIR Taurus Scorpio Mag. Name Mag. Name ARI LIB ARI LIB Gemini Sagittarius Mag. Name Mag. Name TAU SCO TAU SCO TAU SCO SCO SCO SCO OPH Cancer Capricorn Mag. Name Mag. Name GEM SGR GEM SGR GEM SGR GEM SGR GEM SGR SGR Leo Aquarius Mag. Name Mag. Name LEO AQR LEO CAP LEO Virgo Pisces Mag. Name Mag. Name LEO AQR LEO AQR LEO AQR VIR PSC

Table 2: Declinations and right ascensions of points on the ecliptic circle (a).
 Aries Taurus Gemini 00 +00 000 00 +11 027 00 +20 057 02 +00 001 02 +12 029 02 +20 059 04 +01 003 04 +12 031 04 +20 062 06 +02 005 06 +13 033 06 +21 064 08 +03 007 08 +14 035 08 +21 066 10 +03 009 10 +14 037 10 +21 068 12 +04 011 12 +15 039 12 +22 070 14 +05 012 14 +16 041 14 +22 072 16 +06 014 16 +16 043 16 +22 074 18 +07 016 18 +17 045 18 +22 076 20 +07 018 20 +17 047 20 +23 079 22 +08 020 22 +18 049 22 +23 081 24 +09 022 24 +18 051 24 +23 083 26 +10 024 26 +19 053 26 +23 085 28 +10 026 28 +19 055 28 +23 087 30 +11 027 30 +20 057 30 +23 090 Cancer Leo Virgo 00 +23 090 00 +20 122 00 +11 152 02 +23 092 02 +19 124 02 +10 153 04 +23 094 04 +19 126 04 +10 155 06 +23 096 06 +18 128 06 +09 157 08 +23 098 08 +18 130 08 +08 159 10 +23 100 10 +17 132 10 +07 161 12 +22 103 12 +17 134 12 +07 163 14 +22 105 14 +16 136 14 +06 165 16 +22 107 16 +16 138 16 +05 167 18 +22 109 18 +15 140 18 +04 168 20 +21 111 20 +14 142 20 +03 170 22 +21 113 22 +14 144 22 +03 172 24 +21 115 24 +13 146 24 +02 174 26 +20 117 26 +12 148 26 +01 176 28 +20 120 28 +12 150 28 +00 178 30 +20 122 30 +11 152 30 +00 180

Table 3: Declinations and right ascensions of points on the ecliptic circle (b).
 Libra Scorpio Sagittarius 00 -00 180 00 -11 207 00 -20 237 02 -00 181 02 -12 209 02 -20 239 04 -01 183 04 -12 211 04 -20 242 06 -02 185 06 -13 213 06 -21 244 08 -03 187 08 -14 215 08 -21 246 10 -03 189 10 -14 217 10 -21 248 12 -04 191 12 -15 219 12 -22 250 14 -05 192 14 -16 221 14 -22 252 16 -06 194 16 -16 223 16 -22 254 18 -07 196 18 -17 225 18 -22 256 20 -07 198 20 -17 227 20 -23 259 22 -08 200 22 -18 229 22 -23 261 24 -09 202 24 -18 231 24 -23 263 26 -10 204 26 -19 233 26 -23 265 28 -10 206 28 -19 235 28 -23 267 30 -11 207 30 -20 237 30 -23 270 Capricorn Aquarius Pisces 00 -23 270 00 -20 302 00 -11 332 02 -23 272 02 -19 304 02 -10 333 04 -23 274 04 -19 306 04 -10 335 06 -23 276 06 -18 308 06 -09 337 08 -23 278 08 -18 310 08 -08 339 10 -23 280 10 -17 312 10 -07 341 12 -22 283 12 -17 314 12 -07 343 14 -22 285 14 -16 316 14 -06 345 16 -22 287 16 -16 318 16 -05 347 18 -22 289 18 -15 320 18 -04 348 20 -21 291 20 -14 322 20 -03 350 22 -21 293 22 -14 324 22 -03 352 24 -21 295 24 -13 326 24 -02 354 26 -20 297 26 -12 328 26 -01 356 28 -20 300 28 -12 330 28 -00 358 30 -20 302 30 -11 332 30 -00 360

Table 4: Terrestrial climes in the earth's northern hemisphere. The superscripts , , and stand for hours, minutes, and days, respectively. The symbols S and N indicated that the upper transit of the sun occurs to the south and north of the zenith, respectively.
 Longest Day Summer Solstice Noon Equinoctial Noon Winter Solstice Noon Altitude of Sun Altitude of Sun Altitude of Sun N S S N S S N S S N S S N S S N S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S

Table 5: Right ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 000 00 027 00 057 00 090 00 122 00 152 02 001 02 029 02 059 02 092 02 124 02 153 04 003 04 031 04 062 04 094 04 126 04 155 06 005 06 033 06 064 06 096 06 128 06 157 08 007 08 035 08 066 08 098 08 130 08 159 10 009 10 037 10 068 10 100 10 132 10 161 12 011 12 039 12 070 12 103 12 134 12 163 14 012 14 041 14 072 14 105 14 136 14 165 16 014 16 043 16 074 16 107 16 138 16 167 18 016 18 045 18 076 18 109 18 140 18 168 20 018 20 047 20 079 20 111 20 142 20 170 22 020 22 049 22 081 22 113 22 144 22 172 24 022 24 051 24 083 24 115 24 146 24 174 26 024 26 053 26 085 26 117 26 148 26 176 28 026 28 055 28 087 28 120 28 150 28 178 30 027 30 057 30 090 30 122 30 152 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 207 00 237 00 270 00 302 00 332 02 181 02 209 02 239 02 272 02 304 02 333 04 183 04 211 04 242 04 274 04 306 04 335 06 185 06 213 06 244 06 276 06 308 06 337 08 187 08 215 08 246 08 278 08 310 08 339 10 189 10 217 10 248 10 280 10 312 10 341 12 191 12 219 12 250 12 283 12 314 12 343 14 192 14 221 14 252 14 285 14 316 14 345 16 194 16 223 16 254 16 287 16 318 16 347 18 196 18 225 18 256 18 289 18 320 18 348 20 198 20 227 20 259 20 291 20 322 20 350 22 200 22 229 22 261 22 293 22 324 22 352 24 202 24 231 24 263 24 295 24 326 24 354 26 204 26 233 26 265 26 297 26 328 26 356 28 206 28 235 28 267 28 300 28 330 28 358 30 207 30 237 30 270 30 302 30 332 30 360

Table 6: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 000 00 025 00 054 00 085 00 118 00 150 02 001 02 027 02 056 02 087 02 120 02 152 04 003 04 029 04 058 04 089 04 122 04 154 06 005 06 031 06 060 06 092 06 124 06 156 08 006 08 033 08 062 08 094 08 127 08 158 10 008 10 034 10 064 10 096 10 129 10 160 12 010 12 036 12 066 12 098 12 131 12 162 14 011 14 038 14 068 14 100 14 133 14 164 16 013 16 040 16 070 16 103 16 135 16 166 18 015 18 042 18 072 18 105 18 137 18 168 20 017 20 044 20 074 20 107 20 139 20 170 22 018 22 046 22 076 22 109 22 141 22 172 24 020 24 048 24 079 24 111 24 143 24 174 26 022 26 050 26 081 26 114 26 145 26 176 28 024 28 052 28 083 28 116 28 148 28 178 30 025 30 054 30 085 30 118 30 150 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 209 00 241 00 274 00 305 00 334 02 181 02 212 02 243 02 276 02 307 02 335 04 183 04 214 04 245 04 278 04 309 04 337 06 185 06 216 06 248 06 280 06 311 06 339 08 187 08 218 08 250 08 283 08 313 08 341 10 189 10 220 10 252 10 285 10 315 10 342 12 191 12 222 12 254 12 287 12 317 12 344 14 193 14 224 14 256 14 289 14 319 14 346 16 195 16 226 16 259 16 291 16 321 16 348 18 197 18 228 18 261 18 293 18 323 18 349 20 199 20 230 20 263 20 295 20 325 20 351 22 201 22 232 22 265 22 297 22 326 22 353 24 203 24 235 24 267 24 299 24 328 24 354 26 205 26 237 26 270 26 301 26 330 26 356 28 207 28 239 28 272 28 303 28 332 28 358 30 209 30 241 30 274 30 305 30 334 30 360

Table 7: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 000 00 023 00 050 00 080 00 114 00 147 02 001 02 025 02 052 02 083 02 116 02 150 04 003 04 026 04 054 04 085 04 119 04 152 06 004 06 028 06 055 06 087 06 121 06 154 08 006 08 030 08 057 08 089 08 123 08 156 10 007 10 032 10 059 10 091 10 125 10 158 12 009 12 033 12 061 12 094 12 128 12 160 14 010 14 035 14 063 14 096 14 130 14 162 16 012 16 037 16 066 16 098 16 132 16 165 18 014 18 039 18 068 18 100 18 134 18 167 20 015 20 040 20 070 20 103 20 136 20 169 22 017 22 042 22 072 22 105 22 139 22 171 24 018 24 044 24 074 24 107 24 141 24 173 26 020 26 046 26 076 26 109 26 143 26 175 28 022 28 048 28 078 28 112 28 145 28 177 30 023 30 050 30 080 30 114 30 147 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 212 00 245 00 279 00 309 00 336 02 182 02 214 02 247 02 281 02 311 02 337 04 184 04 216 04 250 04 283 04 313 04 339 06 186 06 218 06 252 06 285 06 315 06 341 08 188 08 220 08 254 08 287 08 317 08 342 10 190 10 223 10 256 10 289 10 319 10 344 12 192 12 225 12 259 12 291 12 320 12 345 14 194 14 227 14 261 14 293 14 322 14 347 16 197 16 229 16 263 16 296 16 324 16 349 18 199 18 232 18 265 18 298 18 326 18 350 20 201 20 234 20 268 20 300 20 327 20 352 22 203 22 236 22 270 22 302 22 329 22 353 24 205 24 238 24 272 24 304 24 331 24 355 26 207 26 240 26 274 26 306 26 333 26 356 28 209 28 243 28 276 28 307 28 334 28 358 30 212 30 245 30 279 30 309 30 336 30 360

Table 8: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 000 00 021 00 045 00 075 00 109 00 145 02 001 02 022 02 047 02 077 02 112 02 147 04 002 04 024 04 049 04 079 04 114 04 150 06 004 06 025 06 051 06 082 06 117 06 152 08 005 08 027 08 053 08 084 08 119 08 154 10 006 10 028 10 054 10 086 10 121 10 156 12 008 12 030 12 056 12 088 12 124 12 159 14 009 14 031 14 058 14 091 14 126 14 161 16 011 16 033 16 060 16 093 16 128 16 163 18 012 18 035 18 062 18 095 18 131 18 166 20 013 20 036 20 064 20 098 20 133 20 168 22 015 22 038 22 066 22 100 22 135 22 170 24 016 24 040 24 069 24 102 24 138 24 173 26 018 26 042 26 071 26 105 26 140 26 175 28 019 28 043 28 073 28 107 28 143 28 177 30 021 30 045 30 075 30 109 30 145 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 214 00 250 00 284 00 314 00 338 02 182 02 216 02 252 02 286 02 316 02 340 04 184 04 219 04 254 04 288 04 317 04 341 06 186 06 221 06 257 06 290 06 319 06 343 08 189 08 224 08 259 08 293 08 321 08 344 10 191 10 226 10 261 10 295 10 323 10 346 12 193 12 228 12 264 12 297 12 324 12 347 14 196 14 231 14 266 14 299 14 326 14 348 16 198 16 233 16 268 16 301 16 328 16 350 18 200 18 235 18 271 18 303 18 329 18 351 20 203 20 238 20 273 20 305 20 331 20 353 22 205 22 240 22 275 22 307 22 332 22 354 24 207 24 242 24 277 24 308 24 334 24 355 26 209 26 245 26 280 26 310 26 335 26 357 28 212 28 247 28 282 28 312 28 337 28 358 30 214 30 250 30 284 30 314 30 338 30 360

Table 9: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 000 00 018 00 039 00 068 00 104 00 142 02 001 02 019 02 041 02 070 02 106 02 144 04 002 04 020 04 043 04 073 04 109 04 147 06 003 06 022 06 045 06 075 06 111 06 149 08 004 08 023 08 046 08 077 08 114 08 152 10 005 10 024 10 048 10 079 10 116 10 154 12 007 12 026 12 050 12 082 12 119 12 157 14 008 14 027 14 052 14 084 14 121 14 159 16 009 16 029 16 054 16 087 16 124 16 162 18 010 18 030 18 056 18 089 18 127 18 164 20 011 20 031 20 058 20 091 20 129 20 167 22 013 22 033 22 060 22 094 22 132 22 169 24 014 24 035 24 062 24 096 24 134 24 172 26 015 26 036 26 064 26 099 26 137 26 175 28 016 28 038 28 066 28 101 28 139 28 177 30 018 30 039 30 068 30 104 30 142 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 217 00 255 00 291 00 320 00 341 02 182 02 220 02 258 02 293 02 321 02 343 04 185 04 222 04 260 04 295 04 323 04 344 06 187 06 225 06 263 06 297 06 324 06 345 08 190 08 227 08 265 08 299 08 326 08 346 10 192 10 230 10 268 10 301 10 328 10 348 12 195 12 232 12 270 12 303 12 329 12 349 14 197 14 235 14 272 14 305 14 330 14 350 16 200 16 238 16 275 16 307 16 332 16 351 18 202 18 240 18 277 18 309 18 333 18 352 20 205 20 243 20 280 20 311 20 335 20 354 22 207 22 245 22 282 22 313 22 336 22 355 24 210 24 248 24 284 24 314 24 337 24 356 26 212 26 250 26 286 26 316 26 339 26 357 28 215 28 253 28 289 28 318 28 340 28 358 30 217 30 255 30 291 30 320 30 341 30 360

Table 10: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 000 00 013 00 031 00 058 00 096 00 138 02 000 02 014 02 033 02 061 02 098 02 140 04 001 04 015 04 034 04 063 04 101 04 143 06 002 06 017 06 036 06 065 06 104 06 146 08 003 08 018 08 038 08 068 08 107 08 149 10 004 10 019 10 039 10 070 10 110 10 152 12 005 12 020 12 041 12 072 12 112 12 154 14 006 14 021 14 043 14 075 14 115 14 157 16 007 16 022 16 044 16 077 16 118 16 160 18 008 18 023 18 046 18 080 18 121 18 163 20 009 20 025 20 048 20 082 20 124 20 166 22 010 22 026 22 050 22 085 22 126 22 168 24 010 24 027 24 052 24 088 24 129 24 171 26 011 26 029 26 054 26 090 26 132 26 174 28 012 28 030 28 056 28 093 28 135 28 177 30 013 30 031 30 058 30 096 30 138 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 221 00 263 00 301 00 328 00 346 02 182 02 224 02 266 02 303 02 329 02 347 04 185 04 227 04 269 04 305 04 330 04 348 06 188 06 230 06 271 06 307 06 332 06 349 08 191 08 233 08 274 08 309 08 333 08 350 10 193 10 235 10 277 10 311 10 334 10 350 12 196 12 238 12 279 12 313 12 336 12 351 14 199 14 241 14 282 14 315 14 337 14 352 16 202 16 244 16 284 16 316 16 338 16 353 18 205 18 247 18 287 18 318 18 339 18 354 20 207 20 249 20 289 20 320 20 340 20 355 22 210 22 252 22 292 22 321 22 341 22 356 24 213 24 255 24 294 24 323 24 342 24 357 26 216 26 258 26 296 26 325 26 344 26 358 28 219 28 261 28 298 28 326 28 345 28 359 30 221 30 263 30 301 30 328 30 346 30 360

Table 11: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 000 00 011 00 026 00 051 00 090 00 135 02 000 02 011 02 027 02 053 02 093 02 138 04 001 04 012 04 028 04 056 04 096 04 141 06 002 06 013 06 030 06 058 06 099 06 144 08 002 08 014 08 031 08 060 08 102 08 147 10 003 10 015 10 033 10 063 10 105 10 150 12 004 12 016 12 034 12 065 12 108 12 153 14 004 14 017 14 036 14 068 14 111 14 156 16 005 16 018 16 038 16 071 16 114 16 159 18 006 18 019 18 039 18 073 18 117 18 162 20 007 20 020 20 041 20 076 20 120 20 165 22 007 22 021 22 043 22 079 22 123 22 168 24 008 24 022 24 045 24 082 24 126 24 171 26 009 26 023 26 047 26 084 26 129 26 174 28 010 28 024 28 049 28 087 28 132 28 177 30 011 30 026 30 051 30 090 30 135 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 224 00 269 00 308 00 333 00 348 02 182 02 227 02 272 02 310 02 335 02 349 04 185 04 230 04 275 04 312 04 336 04 350 06 188 06 233 06 277 06 314 06 337 06 351 08 191 08 236 08 280 08 316 08 338 08 352 10 194 10 239 10 283 10 318 10 339 10 352 12 197 12 242 12 286 12 320 12 340 12 353 14 200 14 245 14 288 14 321 14 341 14 354 16 203 16 248 16 291 16 323 16 342 16 355 18 206 18 251 18 294 18 325 18 343 18 355 20 209 20 254 20 296 20 326 20 344 20 356 22 212 22 257 22 299 22 328 22 345 22 357 24 215 24 260 24 301 24 329 24 346 24 357 26 218 26 263 26 303 26 331 26 347 26 358 28 221 28 266 28 306 28 332 28 348 28 359 30 224 30 269 30 308 30 333 30 348 30 360

Table 12: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 000 00 007 00 018 00 041 00 082 00 131 02 000 02 007 02 019 02 043 02 085 02 134 04 000 04 008 04 020 04 045 04 089 04 138 06 001 06 009 06 021 06 048 06 092 06 141 08 001 08 009 08 022 08 050 08 095 08 144 10 002 10 010 10 024 10 053 10 098 10 147 12 002 12 011 12 025 12 056 12 102 12 151 14 003 14 011 14 026 14 058 14 105 14 154 16 003 16 012 16 028 16 061 16 108 16 157 18 004 18 013 18 029 18 064 18 111 18 160 20 004 20 013 20 031 20 067 20 115 20 163 22 005 22 014 22 033 22 070 22 118 22 167 24 005 24 015 24 035 24 073 24 121 24 170 26 006 26 016 26 037 26 076 26 124 26 173 28 006 28 017 28 039 28 079 28 128 28 176 30 007 30 018 30 041 30 082 30 131 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 228 00 277 00 318 00 341 00 352 02 183 02 231 02 280 02 320 02 342 02 353 04 186 04 235 04 283 04 322 04 343 04 353 06 189 06 238 06 286 06 324 06 344 06 354 08 192 08 241 08 289 08 326 08 345 08 354 10 196 10 244 10 292 10 328 10 346 10 355 12 199 12 248 12 295 12 330 12 346 12 355 14 202 14 251 14 298 14 331 14 347 14 356 16 205 16 254 16 301 16 333 16 348 16 356 18 208 18 257 18 303 18 334 18 349 18 357 20 212 20 261 20 306 20 335 20 349 20 357 22 215 22 264 22 309 22 337 22 350 22 358 24 218 24 267 24 311 24 338 24 350 24 358 26 221 26 270 26 314 26 339 26 351 26 359 28 225 28 274 28 316 28 340 28 352 28 359 30 228 30 277 30 318 30 341 30 352 30 360

Table 13: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 000 00 002 00 005 00 021 00 070 00 126 02 000 02 002 02 006 02 023 02 074 02 129 04 000 04 002 04 006 04 026 04 077 04 133 06 000 06 002 06 007 06 029 06 081 06 137 08 000 08 002 08 007 08 031 08 085 08 140 10 000 10 003 10 008 10 034 10 089 10 144 12 000 12 003 12 009 12 038 12 092 12 148 14 000 14 003 14 010 14 041 14 096 14 151 16 001 16 003 16 011 16 044 16 100 16 155 18 001 18 003 18 012 18 048 18 104 18 158 20 001 20 004 20 013 20 051 20 107 20 162 22 001 22 004 22 014 22 055 22 111 22 165 24 001 24 004 24 016 24 059 24 115 24 169 26 001 26 005 26 017 26 062 26 118 26 172 28 001 28 005 28 019 28 066 28 122 28 176 30 002 30 005 30 021 30 070 30 126 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 233 00 289 00 338 00 354 00 357 02 183 02 237 02 293 02 340 02 354 02 358 04 187 04 241 04 297 04 342 04 354 04 358 06 190 06 244 06 300 06 343 06 355 06 358 08 194 08 248 08 304 08 345 08 355 08 358 10 197 10 252 10 308 10 346 10 355 10 358 12 201 12 255 12 311 12 347 12 356 12 358 14 204 14 259 14 315 14 348 14 356 14 358 16 208 16 263 16 318 16 349 16 356 16 359 18 212 18 267 18 321 18 350 18 356 18 359 20 215 20 270 20 325 20 351 20 356 20 359 22 219 22 274 22 328 22 352 22 357 22 359 24 222 24 278 24 330 24 352 24 357 24 359 26 226 26 282 26 333 26 353 26 357 26 359 28 230 28 285 28 336 28 353 28 357 28 359 30 233 30 289 30 338 30 354 30 357 30 360

Table 14: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 360 00 354 00 -- 00 -- 00 032 00 118 02 359 02 353 02 -- 02 -- 02 044 02 122 04 359 04 352 04 -- 04 -- 04 052 04 126 06 358 06 352 06 -- 06 -- 06 059 06 131 08 358 08 351 08 -- 08 -- 08 065 08 135 10 358 10 351 10 -- 10 -- 10 070 10 139 12 357 12 350 12 -- 12 -- 12 076 12 143 14 357 14 349 14 -- 14 -- 14 081 14 147 16 357 16 348 16 -- 16 -- 16 086 16 151 18 356 18 347 18 -- 18 -- 18 091 18 155 20 356 20 346 20 -- 20 -- 20 095 20 159 22 355 22 344 22 -- 22 -- 22 100 22 163 24 355 24 342 24 -- 24 -- 24 104 24 167 26 355 26 340 26 -- 26 -- 26 109 26 171 28 354 28 335 28 -- 28 -- 28 113 28 175 30 354 29 327 30 -- 30 -- 30 118 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 241 00 -- 00 -- 00 032 00 005 02 184 02 246 02 -- 02 -- 02 024 02 005 04 188 04 250 04 -- 04 -- 04 019 04 005 06 192 06 255 06 -- 06 -- 06 017 06 004 08 196 08 259 08 -- 08 -- 08 015 08 004 10 200 10 264 10 -- 10 -- 10 013 10 003 12 204 12 268 12 -- 12 -- 12 012 12 003 14 208 14 273 14 -- 14 -- 14 011 14 002 16 212 16 278 16 -- 16 -- 16 010 16 002 18 216 18 283 18 -- 18 -- 18 009 18 002 20 220 20 289 20 -- 20 -- 20 009 20 001 22 224 22 294 22 -- 22 -- 22 008 22 001 24 228 24 300 24 -- 24 -- 24 007 24 001 26 233 26 307 26 -- 26 -- 26 007 26 000 28 237 28 315 28 -- 28 -- 28 006 28 000 30 241 29 327 30 -- 30 -- 30 005 30 000

Table 15: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 360 00 338 00 -- 00 -- 00 -- 00 102 02 358 02 336 02 -- 02 -- 02 -- 02 108 04 357 04 333 04 -- 04 -- 04 -- 04 114 06 356 06 329 06 -- 06 -- 06 -- 06 120 08 355 08 325 08 -- 08 -- 08 -- 08 125 10 354 10 316 10 -- 10 -- 10 -- 10 130 12 353 11 308 12 -- 12 -- 12 -- 12 135 14 351 14 -- 14 -- 14 -- 14 -- 14 140 16 350 16 -- 16 -- 16 -- 16 -- 16 145 18 349 18 -- 18 -- 18 -- 19 051 18 150 20 347 20 -- 20 -- 20 -- 20 061 20 155 22 346 22 -- 22 -- 22 -- 22 073 22 160 24 344 24 -- 24 -- 24 -- 24 082 24 165 26 342 26 -- 26 -- 26 -- 26 089 26 170 28 340 28 -- 28 -- 28 -- 28 096 28 175 30 338 30 -- 30 -- 30 -- 30 102 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 257 00 -- 00 -- 00 -- 00 021 02 184 02 263 02 -- 02 -- 02 -- 02 019 04 189 04 270 04 -- 04 -- 04 -- 04 017 06 194 06 277 06 -- 06 -- 06 -- 06 015 08 199 08 286 08 -- 08 -- 08 -- 08 013 10 204 10 298 10 -- 10 -- 10 -- 10 012 12 209 11 308 12 -- 12 -- 12 -- 12 010 14 214 14 -- 14 -- 14 -- 14 -- 14 009 16 219 16 -- 16 -- 16 -- 16 -- 16 008 18 224 18 -- 18 -- 18 -- 19 051 18 007 20 229 20 -- 20 -- 20 -- 20 043 20 005 22 234 22 -- 22 -- 22 -- 22 034 22 004 24 239 24 -- 24 -- 24 -- 24 030 24 003 26 245 26 -- 26 -- 26 -- 26 026 26 002 28 251 28 -- 28 -- 28 -- 28 023 28 001 30 257 30 -- 30 -- 30 -- 30 021 30 000

Table 16: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer Leo Virgo 00 360 00 -- 00 -- 00 -- 00 -- 00 -- 02 357 02 -- 02 -- 02 -- 02 -- 02 -- 04 354 04 -- 04 -- 04 -- 04 -- 04 066 06 351 06 -- 06 -- 06 -- 06 -- 06 089 08 349 08 -- 08 -- 08 -- 08 -- 08 100 10 346 10 -- 10 -- 10 -- 10 -- 10 110 12 342 12 -- 12 -- 12 -- 12 -- 12 118 14 339 14 -- 14 -- 14 -- 14 -- 14 126 16 336 16 -- 16 -- 16 -- 16 -- 16 133 18 331 18 -- 18 -- 18 -- 18 -- 18 140 20 327 20 -- 20 -- 20 -- 20 -- 20 147 22 321 22 -- 22 -- 22 -- 22 -- 22 154 24 313 24 -- 24 -- 24 -- 24 -- 24 160 25 294 26 -- 26 -- 26 -- 26 -- 26 167 28 -- 28 -- 28 -- 28 -- 28 -- 28 173 30 -- 30 -- 30 -- 30 -- 30 -- 30 180 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 180 00 -- 00 -- 00 -- 00 -- 00 -- 02 186 02 -- 02 -- 02 -- 02 -- 02 -- 04 192 04 -- 04 -- 04 -- 04 -- 04 066 06 199 06 -- 06 -- 06 -- 06 -- 06 046 08 205 08 -- 08 -- 08 -- 08 -- 08 038 10 212 10 -- 10 -- 10 -- 10 -- 10 032 12 219 12 -- 12 -- 12 -- 12 -- 12 028 14 226 14 -- 14 -- 14 -- 14 -- 14 023 16 233 16 -- 16 -- 16 -- 16 -- 16 020 18 241 18 -- 18 -- 18 -- 18 -- 18 017 20 249 20 -- 20 -- 20 -- 20 -- 20 013 22 259 22 -- 22 -- 22 -- 22 -- 22 010 24 270 24 -- 24 -- 24 -- 24 -- 24 008 25 294 26 -- 26 -- 26 -- 26 -- 26 005 28 -- 28 -- 28 -- 28 -- 28 -- 28 002 30 -- 30 -- 30 -- 30 -- 30 -- 30 000

Table 17: Oblique ascensions of the ecliptic for latitude .
 Aries Taurus Gemini Cancer