Sunrise and Sunset worksheet

1. Nautical dawn / nautical dusk: 9:25 AM / 7:44 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}\mathrm{-23.2121}\text{\hspace{0.17em}°}\text{, EoT =}5.452\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-100}\right)+\left(\mathrm{-240}\right)-5.452$$=874.55\text{=}\text{2:35 PM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 102° from overhead

   $\frac{\mathrm{-0.207912}}{0.573576\times 0.919052}-1.428148\times \left(\mathrm{-0.428851}\right)$$=0.218053$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 77.4053$$=309.62\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for nautical dawn and add them for nautical dusk. Each time is then fine-tuned with the Sun's position at that moment

   $\text{9:25 AM / 7:44 PM}$

6. Subtract one time from the other to see how much time lies between them

   $\mathrm{1,184.19}-564.74$$=619.45\text{\hspace{0.17em}min}\text{(10 h 19 min)}$

2. Sunrise / sunset: 4:42 AM / 5:39 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}9.6646\text{\hspace{0.17em}°}\text{, EoT =}\mathrm{-0.164}\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-62.5}\right)+\left(\mathrm{-300}\right)-\left(\mathrm{-0.164}\right)$$=670.16\text{=}\text{11:10 AM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 90.833° from overhead

   $\frac{\mathrm{-0.014538}}{0.843391\times 0.985807}-0.63707\times 0.170298$$=\mathrm{-0.125978}$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 97.2372$$=388.95\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for sunrise and add them for sunset. Each time is then fine-tuned with the Sun's position at that moment

   $\text{4:42 AM / 5:39 PM}$

6. Subtract one time from the other to see how much time lies between them

   $\mathrm{1,059.3}-281.54$$=777.77\text{\hspace{0.17em}min}\text{(12 h 58 min)}$

3. Nautical dawn / nautical dusk: 9:06 AM / 10:01 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}\mathrm{-10.3373}\text{\hspace{0.17em}°}\text{, EoT =}\mathrm{-13.558}\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-110}\right)+\left(\mathrm{-240}\right)-\left(\mathrm{-13.558}\right)$$=933.56\text{=}\text{3:34 PM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 102° from overhead

   $\frac{\mathrm{-0.207912}}{0.737277\times 0.983768}-0.916331\times \left(\mathrm{-0.182404}\right)$$=\mathrm{-0.11951}$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 96.8638$$=387.46\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for nautical dawn and add them for nautical dusk. Each time is then fine-tuned with the Sun's position at that moment

   $\text{9:06 AM / 10:01 PM}$

6. Subtract one time from the other to see how much time lies between them

   $\mathrm{1,321.33}-546.49$$=774.85\text{\hspace{0.17em}min}\text{(12 h 55 min)}$

4. Sunrise / sunset: 6:45 AM / 8:47 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}21.4269\text{\hspace{0.17em}°}\text{, EoT =}\mathrm{-6.044}\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-100}\right)+\left(\mathrm{-300}\right)-\left(\mathrm{-6.044}\right)$$=826.04\text{=}\text{1:46 PM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 90.833° from overhead

   $\frac{\mathrm{-0.014538}}{0.848048\times 0.930884}-0.624869\times 0.392438$$=\mathrm{-0.263639}$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 105.2861$$=421.14\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for sunrise and add them for sunset. Each time is then fine-tuned with the Sun's position at that moment

   $\text{6:45 AM / 8:47 PM}$

6. Subtract one time from the other to see how much time lies between them

   $\mathrm{1,247.07}-404.73$$=842.34\text{\hspace{0.17em}min}\text{(14 h 2 min)}$

5. Astronomical dawn / astronomical dusk: 4:25 AM / 4:11 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}\mathrm{-23.4327}\text{\hspace{0.17em}°}\text{, EoT =}2.309\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-80}\right)+\left(\mathrm{-420}\right)-2.309$$=617.69\text{=}\text{10:18 AM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 108° from overhead

   $\frac{\mathrm{-0.309017}}{0.573576\times 0.917528}-1.428148\times \left(\mathrm{-0.433416}\right)$$=0.031802$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 88.1776$$=352.71\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for astronomical dawn and add them for astronomical dusk. Each time is then fine-tuned with the Sun's position at that moment

   $\text{4:25 AM / 4:11 PM}$

6. Subtract one time from the other to see how much time lies between them

   $970.51-264.85$$=705.67\text{\hspace{0.17em}min}\text{(11 h 46 min)}$

6. Sunrise / sunset: 4:02 AM / 7:55 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}22.6004\text{\hspace{0.17em}°}\text{, EoT =}1.429\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-75}\right)+\left(\mathrm{-300}\right)-1.429$$=718.57\text{=}\text{11:59 AM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 90.833° from overhead

   $\frac{\mathrm{-0.014538}}{0.669131\times 0.923207}-1.110613\times 0.416269$$=\mathrm{-0.485847}$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 119.068$$=476.27\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for sunrise and add them for sunset. Each time is then fine-tuned with the Sun's position at that moment

   $\text{4:02 AM / 7:55 PM}$

6. Subtract one time from the other to see how much time lies between them

   $\mathrm{1,195.11}-242.45$$=952.66\text{\hspace{0.17em}min}\text{(15 h 53 min)}$

7. Astronomical dawn / astronomical dusk: 7:08 AM / 8:59 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}\mathrm{-12.271}\text{\hspace{0.17em}°}\text{, EoT =}15.994\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-80}\right)+\left(\mathrm{-180}\right)-15.994$$=844.01\text{=}\text{2:04 PM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 108° from overhead

   $\frac{\mathrm{-0.309017}}{0.866025\times 0.977153}-0.57735\times \left(\mathrm{-0.217505}\right)$$=\mathrm{-0.239588}$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 103.8622$$=415.45\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for astronomical dawn and add them for astronomical dusk. Each time is then fine-tuned with the Sun's position at that moment

   $\text{7:08 AM / 8:59 PM}$

6. Subtract one time from the other to see how much time lies between them

   $\mathrm{1,259.21}-428.38$$=830.83\text{\hspace{0.17em}min}\text{(13 h 51 min)}$

8. Civil dawn / civil dusk: 4:14 AM / 7:43 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}22.4954\text{\hspace{0.17em}°}\text{, EoT =}1.599\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-90}\right)+\left(\mathrm{-360}\right)-1.599$$=718.4\text{=}\text{11:58 AM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 96° from overhead

   $\frac{\mathrm{-0.104528}}{0.809017\times 0.92391}-0.726543\times 0.41412$$=\mathrm{-0.440721}$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 116.1499$$=464.6\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for civil dawn and add them for civil dusk. Each time is then fine-tuned with the Sun's position at that moment

   $\text{4:14 AM / 7:43 PM}$

6. Subtract one time from the other to see how much time lies between them

   $\mathrm{1,183.2}-253.89$$=929.31\text{\hspace{0.17em}min}\text{(15 h 29 min)}$

9. Sunrise / sunset: 11:36 AM / 9:33 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}\mathrm{-15.5021}\text{\hspace{0.17em}°}\text{, EoT =}\mathrm{-14.073}\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-110}\right)+\left(\mathrm{-180}\right)-\left(\mathrm{-14.073}\right)$$=994.07\text{=}\text{4:34 PM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 90.833° from overhead

   $\frac{\mathrm{-0.014538}}{0.694658\times 0.963621}-1.03553\times \left(\mathrm{-0.277364}\right)$$=0.2655$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 74.6033$$=298.41\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for sunrise and add them for sunset. Each time is then fine-tuned with the Sun's position at that moment

   $\text{11:36 AM / 9:33 PM}$

6. Subtract one time from the other to see how much time lies between them

   $\mathrm{1,292.8}-695.94$$=596.85\text{\hspace{0.17em}min}\text{(9 h 57 min)}$

10. Nautical dawn / nautical dusk: 5:14 AM / 11:37 PM

1. Find how far north or south the Sun is (its declination, δ) and the clock correction (equation of time, EoT) at solar noon

   $\text{δ =}18.5508\text{\hspace{0.17em}°}\text{, EoT =}\mathrm{-6.488}\text{\hspace{0.17em}min}$

2. Find solar noon in minutes after midnight. Start at 720 (12:00), then adjust for your longitude, the EoT and your time zone

   $720-4\times \left(\mathrm{-95}\right)+\left(\mathrm{-240}\right)-\left(\mathrm{-6.488}\right)$$=866.49\text{=}\text{2:26 PM}$

3. Use your latitude (φ) and the declination to find cos H, where H is how far the Earth turns from noon until the Sun is 102° from overhead

   $\frac{\mathrm{-0.207912}}{0.642788\times 0.948042}-1.191754\times 0.335581$$=\mathrm{-0.74111}$

4. Turn cos H back into the angle H (arccos). The Earth turns one degree every 4 minutes, so times 4 for minutes

   $4\times 137.8261$$=551.3\text{\hspace{0.17em}min}$

5. Subtract those minutes from solar noon for nautical dawn and add them for nautical dusk. Each time is then fine-tuned with the Sun's position at that moment

   $\text{5:14 AM / 11:37 PM}$

6. Subtract one time from the other to see how much time lies between them

   $\mathrm{1,416.99}-314.41$$=\mathrm{1,102.58}\text{\hspace{0.17em}min}\text{(18 h 23 min)}$