THE SHORTEST AND THE LONGEST TWILIGHTS

The table lists when happen the shortest twilights, in the restricted problem and in the general problem

 

The restricted problem : the effect of atmosferic refraction and the size of the solar disk are neglected; sunrise and sunset are the instants when the geometric center of the Sun is on the horizon.

The general problem : it is considered the diameter of the Sun and the refraction.

CIVIL TWILIGHT : THE SUN IS AT -6░

Lat.  Dec.   Min.    Dec1.  Min1.
-90   3.00           3.42
-88   3.00           3.42
-86   3.00           3.41   322.0
-84   2.99   360.0   3.40   204.3
-82   2.97   194.7   3.39   151.2
-80   2.96   148.0   3.37   120.4
-78   2.94   120.7   3.35   100.2
-76   2.91   102.4   3.32    85.9
-74   2.89    89.1   3.29    75.3
-72   2.86    79.1   3.25    67.1
-70   2.82    71.2   3.21    60.6
-68   2.79    64.8   3.17    55.3
-66   2.74    59.6   3.12    50.9
-64   2.70    55.2   3.07    47.2
-62   2.65    51.5   3.02    44.1
-60   2.60    48.3   2.96    41.4
-58   2.55    45.5   2.90    39.0
-56   2.49    43.1   2.83    37.0
-54   2.43    41.0   2.77    35.2
-52   2.37    39.1   2.69    33.6
-50   2.30    37.4   2.62    32.2
-48   2.23    35.9   2.54    30.9
-46   2.16    34.6   2.46    29.8
-44   2.09    33.4   2.38    28.7
-42   2.01    32.3   2.29    27.8
-40   1.93    31.4   2.20    27.0
-38   1.85    30.5   2.10    26.2
-36   1.77    29.7   2.01    25.5
-34   1.68    29.0   1.91    24.9
-32   1.59    28.3   1.81    24.4
-30   1.50    27.7   1.71    23.9
-28   1.41    27.2   1.60    23.4
-26   1.32    26.7   1.50    23.0
-24   1.22    26.3   1.39    22.6
-22   1.12    25.9   1.28    22.3
-20   1.03    25.5   1.17    22.0
-18   0.93    25.2   1.06    21.7
-16   0.83    25.0   0.94    21.5
-14   0.73    24.7   0.83    21.3
-12   0.62    24.5   0.71    21.1
-10   0.52    24.4   0.59    21.0
 -8   0.42    24.2   0.48    20.9
 -6   0.31    24.1   0.36    20.8
 -4   0.21    24.1   0.24    20.7
 -2   0.10    24.0   0.12    20.7
  0   0.00    24.0   0.00    20.7
  2  -0.10    24.0  -0.12    20.7
  4  -0.21    24.1  -0.24    20.7
  6  -0.31    24.1  -0.36    20.8
  8  -0.42    24.2  -0.48    20.9
 10  -0.52    24.4  -0.59    21.0
 12  -0.62    24.5  -0.71    21.1
 14  -0.73    24.7  -0.83    21.3
 16  -0.83    25.0  -0.94    21.5
 18  -0.93    25.2  -1.06    21.7
 20  -1.03    25.5  -1.17    22.0
 22  -1.12    25.9  -1.28    22.3
 24  -1.22    26.3  -1.39    22.6
 26  -1.32    26.7  -1.50    23.0
 28  -1.41    27.2  -1.60    23.4
 30  -1.50    27.7  -1.71    23.9
 32  -1.59    28.3  -1.81    24.4
 34  -1.68    29.0  -1.91    24.9
 36  -1.77    29.7  -2.01    25.5
 38  -1.85    30.5  -2.10    26.2
 40  -1.93    31.4  -2.20    27.0
 42  -2.01    32.3  -2.29    27.8
 44  -2.09    33.4  -2.38    28.7
 46  -2.16    34.6  -2.46    29.8
 48  -2.23    35.9  -2.54    30.9
 50  -2.30    37.4  -2.62    32.2
 52  -2.37    39.1  -2.69    33.6
 54  -2.43    41.0  -2.77    35.2
 56  -2.49    43.1  -2.83    37.0
 58  -2.55    45.5  -2.90    39.0
 60  -2.60    48.3  -2.96    41.4
 62  -2.65    51.5  -3.02    44.1
 64  -2.70    55.2  -3.07    47.2
 66  -2.74    59.6  -3.12    50.9
 68  -2.79    64.8  -3.17    55.3
 70  -2.82    71.2  -3.21    60.6
 72  -2.86    79.1  -3.25    67.1
 74  -2.89    89.1  -3.29    75.3
 76  -2.91   102.4  -3.32    85.9
 78  -2.94   120.7  -3.35   100.2
 80  -2.96   148.0  -3.37   120.4
 82  -2.97   194.7  -3.39   151.2
 84  -2.99   360.0  -3.40   204.3
 86  -3.00          -3.41   322.0
 88  -3.00          -3.42
 90  -3.00          -3.42


NAUTICAL TWILIGHT : THE SUN IS AT -12░

Lat.  Dec.   Min.    Dec1.  Min1.
-90   6.03           6.45
-88   6.03           6.44
-86   6.02           6.43
-84   6.00           6.41   548.5
-82   5.97           6.38   354.8
-80   5.94           6.35   272.6
-78   5.90   360.0   6.31   223.2
-76   5.85   237.0   6.26   189.7
-74   5.80   195.9   6.20   165.4
-72   5.74   169.1   6.13   146.8
-70   5.67   149.7   6.06   132.2
-68   5.59   134.8   5.98   120.4
-66   5.51   123.0   5.89   110.7
-64   5.42   113.3   5.79   102.6
-62   5.32   105.1   5.69    95.7
-60   5.22    98.3   5.58    89.8
-58   5.11    92.4   5.46    84.6
-56   5.00    87.3   5.34    80.2
-54   4.88    82.9   5.21    76.2
-52   4.75    78.9   5.08    72.7
-50   4.62    75.5   4.93    69.6
-48   4.48    72.4   4.79    66.9
-46   4.34    69.7   4.63    64.4
-44   4.19    67.2   4.47    62.2
-42   4.03    65.0   4.31    60.2
-40   3.87    63.0   4.14    58.4
-38   3.71    61.2   3.96    56.7
-36   3.54    59.6   3.78    55.3
-34   3.37    58.1   3.60    53.9
-32   3.19    56.8   3.41    52.7
-30   3.01    55.6   3.22    51.6
-28   2.83    54.5   3.02    50.6
-26   2.64    53.5   2.82    49.7
-24   2.45    52.6   2.62    48.9
-22   2.26    51.8   2.41    48.2
-20   2.06    51.1   2.20    47.5
-18   1.86    50.5   1.99    47.0
-16   1.66    50.0   1.77    46.5
-14   1.46    49.5   1.56    46.0
-12   1.25    49.1   1.34    45.7
-10   1.05    48.8   1.12    45.4
 -8   0.84    48.5   0.90    45.1
 -6   0.63    48.3   0.67    44.9
 -4   0.42    48.1   0.45    44.8
 -2   0.21    48.0   0.22    44.7
  0   0.00    48.0   0.00    44.7
  2  -0.21    48.0  -0.22    44.7
  4  -0.42    48.1  -0.45    44.8
  6  -0.63    48.3  -0.67    44.9
  8  -0.84    48.5  -0.90    45.1
 10  -1.05    48.8  -1.12    45.4
 12  -1.25    49.1  -1.34    45.7
 14  -1.46    49.5  -1.56    46.0
 16  -1.66    50.0  -1.77    46.5
 18  -1.86    50.5  -1.99    47.0
 20  -2.06    51.1  -2.20    47.5
 22  -2.26    51.8  -2.41    48.2
 24  -2.45    52.6  -2.62    48.9
 26  -2.64    53.5  -2.82    49.7
 28  -2.83    54.5  -3.02    50.6
 30  -3.01    55.6  -3.22    51.6
 32  -3.19    56.8  -3.41    52.7
 34  -3.37    58.1  -3.60    53.9
 36  -3.54    59.6  -3.78    55.3
 38  -3.71    61.2  -3.96    56.7
 40  -3.87    63.0  -4.14    58.4
 42  -4.03    65.0  -4.31    60.2
 44  -4.19    67.2  -4.47    62.2
 46  -4.34    69.7  -4.63    64.4
 48  -4.48    72.4  -4.79    66.9
 50  -4.62    75.5  -4.93    69.6
 52  -4.75    78.9  -5.08    72.7
 54  -4.88    82.9  -5.21    76.2
 56  -5.00    87.3  -5.34    80.2
 58  -5.11    92.4  -5.46    84.6
 60  -5.22    98.3  -5.58    89.8
 62  -5.32   105.1  -5.69    95.7
 64  -5.42   113.3  -5.79   102.6
 66  -5.51   123.0  -5.89   110.7
 68  -5.59   134.8  -5.98   120.4
 70  -5.67   149.7  -6.06   132.2
 72  -5.74   169.1  -6.13   146.8
 74  -5.80   195.9  -6.20   165.4
 76  -5.85   237.0  -6.26   189.7
 78  -5.90   360.0  -6.31   223.2
 80  -5.94          -6.35   272.6
 82  -5.97          -6.38   354.8
 84  -6.00          -6.41   548.5
 86  -6.02          -6.43
 88  -6.03          -6.44
 90  -6.03          -6.45


ASTRONOMICAL TWILIGHT : THE SUN IS AT -18░

Lat.  Dec.   Min.    Dec1.  Min1.
-90   9.11           9.52
-88   9.11           9.52
-86   9.09           9.50
-84   9.06           9.47
-82   9.02           9.43
-80   8.97           9.38   474.1
-78   8.91           9.31   367.0
-76   8.84           9.24   304.7
-74   8.76           9.15   262.3
-72   8.66   360.0   9.05   231.0
-70   8.56   258.5   8.95   207.0
-68   8.44   222.3   8.83   187.8
-66   8.32   197.8   8.69   172.2
-64   8.18   179.3   8.55   159.2
-62   8.04   164.7   8.40   148.3
-60   7.88   152.7   8.24   138.9
-58   7.72   142.7   8.07   130.9
-56   7.55   134.2   7.88   123.8
-54   7.36   126.9   7.69   117.7
-52   7.17   120.5   7.49   112.2
-50   6.97   114.9   7.28   107.4
-48   6.76   110.0   7.06   103.1
-46   6.54   105.7   6.84    99.3
-44   6.32   101.8   6.60    95.8
-42   6.08    98.3   6.36    92.7
-40   5.84    95.2   6.11    89.9
-38   5.60    92.4   5.85    87.3
-36   5.34    89.8   5.58    85.0
-34   5.08    87.5   5.31    83.0
-32   4.81    85.5   5.03    81.1
-30   4.54    83.6   4.75    79.4
-28   4.26    81.9   4.46    77.9
-26   3.98    80.4   4.16    76.5
-24   3.69    79.1   3.86    75.2
-22   3.40    77.9   3.55    74.1
-20   3.11    76.8   3.24    73.1
-18   2.81    75.8   2.93    72.2
-16   2.50    75.0   2.61    71.5
-14   2.20    74.3   2.29    70.8
-12   1.89    73.7   1.97    70.2
-10   1.58    73.1   1.65    69.7
 -8   1.26    72.7   1.32    69.3
 -6   0.95    72.4   0.99    69.0
 -4   0.63    72.2   0.66    68.8
 -2   0.32    72.0   0.33    68.7
  0   0.00    72.0   0.00    68.7
  2  -0.32    72.0  -0.33    68.7
  4  -0.63    72.2  -0.66    68.8
  6  -0.95    72.4  -0.99    69.0
  8  -1.26    72.7  -1.32    69.3
 10  -1.58    73.1  -1.65    69.7
 12  -1.89    73.7  -1.97    70.2
 14  -2.20    74.3  -2.29    70.8
 16  -2.50    75.0  -2.61    71.5
 18  -2.81    75.8  -2.93    72.2
 20  -3.11    76.8  -3.24    73.1
 22  -3.40    77.9  -3.55    74.1
 24  -3.69    79.1  -3.86    75.2
 26  -3.98    80.4  -4.16    76.5
 28  -4.26    81.9  -4.46    77.9
 30  -4.54    83.6  -4.75    79.4
 32  -4.81    85.5  -5.03    81.1
 34  -5.08    87.5  -5.31    83.0
 36  -5.34    89.8  -5.58    85.0
 38  -5.60    92.4  -5.85    87.3
 40  -5.84    95.2  -6.11    89.9
 42  -6.08    98.3  -6.36    92.7
 44  -6.32   101.8  -6.60    95.8
 46  -6.54   105.7  -6.84    99.3
 48  -6.76   110.0  -7.06   103.1
 50  -6.97   114.9  -7.28   107.4
 52  -7.17   120.5  -7.49   112.2
 54  -7.36   126.9  -7.69   117.7
 56  -7.55   134.2  -7.88   123.8
 58  -7.72   142.7  -8.07   130.9
 60  -7.88   152.7  -8.24   138.9
 62  -8.04   164.7  -8.40   148.3
 64  -8.18   179.3  -8.55   159.2
 66  -8.32   197.8  -8.69   172.2
 68  -8.44   222.3  -8.83   187.8
 70  -8.56   258.5  -8.95   207.0
 72  -8.66   360.0  -9.05   231.0
 74  -8.76          -9.15   262.3
 76  -8.84          -9.24   304.7
 78  -8.91          -9.31   367.0
 80  -8.97          -9.38   474.1
 82  -9.02          -9.43
 84  -9.06          -9.47
 86  -9.09          -9.50
 88  -9.11          -9.52
 90  -9.11          -9.52

Lat = geographical latitude of the observer
Dec = declination of the Sun when happen the shortest twilight. From the declination we can resume the day of the year
Min = duration of the twilight
Dec1 e Min 1 = same values but in the general problem


MAXIMA DURATION OF THE TWILIGHTS

Civil twilight :
for latitudes between 0░ and 60░34' longest civil twilight occurs on the day of summer solstice
for latitudes between 60░34' and 87░ the day of the longest civil twilight is when the solar declination is 84░-latitude
for latitudes between 87░ and 90░ there is no proper civil twilight.

Astronomical twilight :
for latitudes between 0░ and 48░34' longest astronomical twilight occurs on the day of summer solstice
for latitudes between 48░34' and 81░ the day of the longest astronomical twilight is when the solar declination is 72░-latitude
for latitudes between 81░ and 90░ there is no proper astronomical twilight.