Calculating the Times of Sunrise and Sunset with a Scientific Calculator

The first thing we need to do is determine the mean ecliptic longitude of the sun for your chosen day.  Use this table to get a first approximation.

this area intentionally blank MAR APR MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB
01 339.226 009.781 039.351 069.906 099.475 130.030 160.585 190.155 220.710 250.279 280.834 311.390
02 340.212 010.767 040.337 070.892 100.461 131.016 161.571 191.141 221.696 251.265 281.820 312.375
03 341.198 011.753 041.322 071.877 101.447 132.002 162.557 192.126 222.681 252.251 282.806 313.361
04 342.183 012.738 042.308 072.863 102.432 132.987 163.542 193.112 223.667 253.236 283.791 314.346
05 343.169 013.724 043.293 073.849 103.418 133.973 164.528 194.098 224.653 254.222 284.777 315.332
06 344.155 014.710 044.279 074.834 104.404 134.959 165.514 195.083 225.638 255.208 285.763 316.318
07 345.140 015.695 045.265 075.820 105.389 135.944 166.499 196.069 226.624 256.193 286.748 317.303
08 346.126 016.681 046.250 076.805 106.375 136.930 167.485 197.054 227.610 257.179 287.734 318.289
09 347.112 017.667 047.236 077.791 107.361 137.916 168.471 198.040 228.595 258.165 288.720 319.275
10 348.097 018.652 048.222 078.777 108.346 138.901 169.456 199.026 229.581 259.150 289.705 320.260
11 349.083 019.638 049.207 079.762 109.332 139.887 170.442 200.011 230.566 260.136 290.691 321.246
12 350.068 020.624 050.193 080.748 110.317 140.873 171.428 200.997 231.552 261.122 291.677 322.232
13 351.054 021.609 051.179 081.734 111.303 141.858 172.413 201.983 232.538 262.107 292.662 323.217
14 352.040 022.595 052.164 082.719 112.289 142.844 173.399 202.968 233.523 263.093 293.648 324.203
15 353.025 023.581 053.150 083.705 113.274 143.829 174.385 203.954 234.509 264.078 294.634 325.189
16 354.011 024.566 054.136 084.691 114.260 144.815 175.370 204.940 235.495 265.064 295.619 326.174
17 354.997 025.552 055.121 085.676 115.246 145.801 176.356 205.925 236.480 266.050 296.605 327.160
18 355.982 026.537 056.107 086.662 116.231 146.786 177.341 206.911 237.466 267.035 297.590 328.146
19 356.968 027.523 057.093 087.648 117.217 147.772 178.327 207.897 238.452 268.021 298.576 329.131
20 357.954 028.509 058.078 088.633 118.203 148.758 179.313 208.882 239.437 269.007 299.562 330.117
21 358.939 029.494 059.064 089.619 119.188 149.743 180.298 209.868 240.423 269.992 300.547 331.102
22 359.925 030.480 060.049 090.605 120.174 150.729 181.284 210.854 241.409 270.978 301.533 332.088
23 000.911 031.466 061.035 091.590 121.160 151.715 182.270 211.839 242.394 271.964 302.519 333.074
24 001.896 032.451 062.021 092.576 122.145 152.700 183.255 212.825 243.380 272.949 303.504 334.059
25 002.882 033.437 063.006 093.561 123.131 153.686 184.241 213.810 244.366 273.935 304.490 335.045
26 003.868 034.423 063.992 094.547 124.117 154.672 185.227 214.796 245.351 274.921 305.476 336.031
27 004.853 035.408 064.978 095.533 125.102 155.657 186.212 215.782 246.337 275.906 306.461 337.016
28 005.839 036.394 065.963 096.518 126.088 156.643 187.198 216.767 247.322 276.892 307.447 338.002
29 006.824 037.380 066.949 097.504 127.073 157.629 188.184 217.753 248.308 277.878 308.433 338.988
30 007.810 038.365 067.935 098.490 128.059 158.614 189.169 218.739 249.294 278.863 309.418  
31 008.796   068.920   129.045 159.600   219.724   279.849 310.404  

Next we need to correct this approximation for your particular year.  For the purposes of this correction, January and February are included with the previous year.  (e.g. February of 1900 uses the correction for 1899.)  Given that your year has four digits, find the first two in these tables.  Add the correction at the top of the column and the left of the row.  Don't forget to designate Old Style or New Style.  (As a general rule, the red areas of the tables should seldom (if ever) be used.)

Old Style –001.157 –000.386 +000.386 +001.157
–001.070 00 01 02 03
+002.015 04 05 06 07
+005.101 08 09 10 11
+008.186 12 13 14 15
+011.271 16 17 18 19
+014.356 20 21 22 23
+017.441 24 25 26 27
+020.526 28 29 30 31
+023.611 32 33 34 35
+026.697 36 37 38 39
 
New Style +000.322 +000.107 –000.107 –000.322
–000.577 00 01 02 03
–000.449 04 05 06 07
–000.320 08 09 10 11
–000.192 12 13 14 15
–000.064 16 17 18 19
+000.064 20 21 22 23
+000.192 24 25 26 27
+000.320 28 29 30 31
+000.449 32 33 34 35
+000.577 36 37 38 39

Now find the last two digits of your year in this table and add the correction at the top of the column and the left of the row (like before).

  +000.358 +000.119 –000.119 –000.358
–000.370 00 01 02 03
–000.339 04 05 06 07
–000.309 08 09 10 11
–000.278 12 13 14 15
–000.247 16 17 18 19
–000.216 20 21 22 23
–000.185 24 25 26 27
–000.154 28 29 30 31
–000.123 32 33 34 35
–000.093 36 37 38 39
–000.062 40 41 42 43
–000.031 44 45 46 47
+000.000 48 49 50 51
+000.031 52 53 54 55
+000.062 56 57 58 59
+000.093 60 61 62 63
+000.123 64 65 66 67
+000.154 68 69 70 71
+000.185 72 73 74 75
+000.216 76 77 78 79
+000.247 80 81 82 83
+000.278 84 85 86 87
+000.309 88 89 90 91
+000.339 92 93 94 95
+000.370 96 97 98 99

You now have the mean ecliptic longitude of the sun for midday (UT) of your chosen day.  Sit this aside.  We will come back to it later.

Now we also need to determine the ecliptic longitude of perigee of the sun on this same day.  This changes very slowly, thus we need know only the year to compute it with sufficient accuracy.  Find the first two (of four) digits in your year in this table.  Add the number at the column head and row head as before.

  –002.58 –000.86 +000.86 +002.58
251.97 00 01 02 03
258.85 04 05 06 07
265.73 08 09 10 11
272.61 12 13 14 15
279.49 16 17 18 19
286.37 20 21 22 23
293.25 24 25 26 27
300.14 28 29 30 31
307.02 32 33 34 35
313.90 36 37 38 39

Now find the last two digits in this table and add the corrections at the column and row heads as before.

  –000.03 –000.01 +000.01 +000.03
–000.83 00 01 02 03
–000.76 04 05 06 07
–000.69 08 09 10 11
–000.62 12 13 14 15
–000.55 16 17 18 19
–000.48 20 21 22 23
–000.41 24 25 26 27
–000.34 28 29 30 31
–000.28 32 33 34 35
–000.21 36 37 38 39
–000.14 40 41 42 43
–000.07 44 45 46 47
+000.00 48 49 50 51
+000.07 52 53 54 55
+000.14 56 57 58 59
+000.21 60 61 62 63
+000.28 64 65 66 67
+000.34 68 69 70 71
+000.41 72 73 74 75
+000.48 76 77 78 79
+000.55 80 81 82 83
+000.62 84 85 86 87
+000.69 88 89 90 91
+000.76 92 93 94 95
+000.83 96 97 98 99

You now have the ecliptic longitude of perigee of the sun for the mean summer solstice of your chosen year (and effectively for the entire year).

Now determine the mean anomaly of the sun using the following formula.

mean
ecliptic
longitude
of the sun
ecliptic
longitude
of perigee
of the sun
= mean
anomaly
of the sun

Now, to make sure that your scientific calculator is in "degree mode", punch in "90 cos".  Your result should be "0.000000".  If you get something else, then your calculator is not in degree mode.  (Refer to your calculators owner's manual to get in "degree mode".)

Now determine the ecliptic longitude of the sun using the following formula.

mean
ecliptic
longitude
of the sun
+1.915( mean
anomaly
of the sun
sin)+0.020(( mean
anomaly
of the sun
2)sin)= ecliptic
longitude
of the sun

Now you can get the declination of the sun with the next formula.  (On some calculators "asin" will be labeled "sin-1".)

(( ecliptic
longitude
of the sun
sin)0.39777)asin= declination
of the sun

Also you will want the right ascension of the sun.  Use this for that.

ecliptic
longitude
of the sun
–(((( ecliptic
longitude
of the sun
2)sin)(23.2377+(( ecliptic
longitude
of the sun
2)cos)))atan)= right
ascension
of the sun

Another thing we will need is the equation of time.  Here is the formula for that.

right
ascension
of the sun
mean
ecliptic
longitude
of the sun
= equation
of time

Now we can compute the times of sunrise and sunset.  (Ignore the red spots for now.  If you get an error when you push "asin" it means that that event (sunrise or sunset) does not occur.)  (Longitudes are positive eastward and negative westward.  Latitudes are positive northward and negative southward.)

90– terrestrial
longitude of
observation
= mean time
of sunrise
 
mean time
of sunrise
–((( terrestrial
latitude of
observation
tan)( declination
of the sun
tan)·)asin)+ equation
of time
= time of
sunrise
 
270– terrestrial
longitude of
observation
= mean time
of sunset
 
mean time
of sunset
+((( terrestrial
latitude of
observation
tan)( declination
of the sun
tan)·)asin)+ equation
of time
= time of
sunset

These definitions for sunrise and sunset divide time equally into day and night.  Another common definition of sunrise and sunset is when the upper limb of the sun apparently crosses the horizon.  To get these times, one must correct for the semidiameter of the sun and for atmospheric refraction.  (These effects total about 50 minutes of arc.)  To accomplish this, insert the following where you see the red spots.

+0.01454( terrestrial
latitude of
observation
cos)( declination
of the sun
cos)

To calculate the times of other events, you can replace the red number as follows.

civil twilight this area intentionally blank 0.10453
nautical twilight   0.20791
astronomical twilight   0.30902

To convert your answers from degrees to regular clock time, divide by 360 and multiply by 24.  The whole part of this number is the hour.  Subtract off the whole hours and multiply the remaining fraction by 60.  The whole part of this number is the minutes.  Again, subtract off the whole minutes and multiply the remaining fraction by 60.  These are the seconds.  (You now have Universal Time in the hour/minute/second format.  Converting to your local time is up to you.  My local time is United States Eastern Standard Time, so I subtract 5 hours.)

Now you may notice that your times differ significantly from the times in an almanac.  This is because you used the mean ecliptic longitude of the sun for midday (UT) of your chosen day and not the actual time of the event (sunrise or sunset).  If you want your calculations to be perfect (i.e. to agree with the almanac), you will need to interpolate the mean ecliptic longitude of the sun for the actual time of the event.  Start by doing the whole calculation without interpolation.  This will give you very nearly the actual time of the event (likely within one minute).  Use this time to create the interpolating factor.  Now with your interpolated mean ecliptic longitude of the sun, go back and do the whole calculation again.  Your new answer will likely differ by less than a minute.  If you want it even more accurate, use this answer to interpolate and do the whole calculation yet again.  After doing this several times, you should notice that your final answer is no longer changing.  This is the actual time of the event.

The general accuracies involved in this algorithm are as follows.

mean ecliptic longitude of the sun this area intentionally blank 0.002
ecliptic longitude of perigee of the sun   0.02
mean anomaly of the sun   0.02
ecliptic longitude of the sun   0.01
declination of the sun   0.01
right ascension of the sun   0.01
equation of time   0.01
mean time of sunrise/sunset   0
time of sunrise/sunset (in the tropics)   0.01 ( 3 seconds)

For an example calculation, please follow this link.

March 20, 2000 –  I prepared this page in a way that it should not contain errors.  I have tested it once, the results of which agreed exactly with a standard almanac.  –author

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copyright 2000 Sean Barton, all rights reserved

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