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by Dr. Irv Bromberg, University of Toronto, Canada
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Twilight is that period of dusk after sunset or dawn before sunrise during which the sky is partially lit by atmospheric scattering of sunlight. The duration of twilight after sunset or before sunrise depends on atmospheric conditions (clouds, dust, air pressure, temperature, humidity) and on the parallactic angle (the angle between the path of the setting or rising sun and the local horizon), both of which vary with the seasons (specifically the solar declination) and the terrestrial latitude.
Astronomical sunrise is the moment when the upper limb of Sun first appears at the local easterly horizon. Astronomical sunset is the moment when the upper limb of Sun just disappears below the local westerly horizon. From sunrise to sunset a horizontally mounted sundial in an open area will display the local apparent time of day, if the sky is clear.
Generally the brightness of the sky after sunset or before sunrise correlates with the degree of solar depression, that is the angle between the sea level horizon, the center of Earth, and the center of the solar disk (ignoring the elevation of the locale and the slightly raised apparent level of the local horizon caused by atmospheric refraction). Thus various stages of twilight are defined in terms of the solar depression angle, in degrees.
Clouds can either shorten the duration of twilight or darken its stages, if the clouds are dense and darken the sky, especially if they obstruct sunlight, or they may prolong the duration or brighten its stages, if there is clear sky to the west below the horizon allowing sunlight to reflect from the clouds.
(A planet or moon without an atmosphere has no twilight periods before sunrise or after sunset, and stars are visible at all times, even during the daytime.)
Civil twilight ends after sunset or begins before sunrise when the solar depression angle is 6° = solar zenith angle 96° = solar elevation angle -6°. There is enough light during civil twilight for most outdoor activities, and only the brightest stars are visible. When the solar depression angle is greater than 6° artificial lighting is required for reading and for most outdoor activities. The end of civil twilight after sunset or the beginning of civil twilight before sunrise are considered the beginning or end of night for aviation purposes, respectively.
Nautical twilight ends after sunset or begins before sunrise when the solar depression angle is 12° = solar zenith angle 102° = solar elevation angle -12°. At sea during nautical twilight the horizon is visible as a line separating the sky from the water. When the solar depression angle is greater than 12° the sky is too dark to distinguish from the water at the horizon.
Astronomical twilight ends after sunset or begins before sunrise when the solar depression angle is 18° = solar zenith angle 108° = solar elevation angle -18°. During astronomical twilight it is not dark enough to see dim stars, galaxies, or nebulae, even with a good telescope. When the solar depression angle is greater than 18° the sky is as dark as it ever can get (depending on the lunar location and phase) and visibility of dim stars, galaxies, and nebulae are visible by telescope, as the atmospheric scatter of sunlight is minimal.
Modern portable computer-controlled telescopes must be aligned with known bright stars before they are ready to automatically point at more challenging targets. It is convenient to set up such a telescope during the evening nautical twilight when the brightest stars are visible and easy to find, then go for a coffee break until the end of astronomical twilight, and then get down to the business of star gazing. Unfortunately, during the most pleasant summer months for star gazing in my home country of Canada (June and July) astronomical twilight continues for most or all of the night time, therefore chilly autumn nights are best for star gazing in Canada. The 49th parallel that separates western Canada from the USA corresponds very closely to the latitude north of which astronomical twilight continues all night near the summer solstice.
At high latitudes during the summer time astronomical twilight after sunset may blend into astronomical twilight before sunrise, if the solar depression angle is less than 18° at local apparent midnight. Likewise, at higher latitudes during the summer time the evening nautical twilight may continue into the morning nautical twilight if the solar depression angle is less than 12° at local apparent midnight. At even higher latitudes during the summer time the evening civil twilight may continue into the morning civil twilight if the solar depression angle is less than 6° at local apparent midnight. Of course, at even higher latitudes there may be no twilight at all on those dates when any part of the solar disk remains above the horizon at local apparent midnight.
Instead of solar depression angles, some prefer to use zenith angles, that is the angle between the point that is directly overhead and the center of the solar disk. The civil, nautical, and astronomical twilight zenith angles are 96°, 102°, and 108°, respectively. I prefer not to use zenith angles because most people are unable to accurately pinpoint the zenith without instrumental assistance. Alternatively, the civil, nautical, and astronomical twilight angles correspond to –6°, –12°, and –18° of solar altitude, respectively.
I have built into my Kalendis Calendar Calculator, which is free software for Windows that you can download from <http://www.sym454.org/kalendis/>, the ability to display and export various time and astronomy values, including the moments of civil, nautical, and astronomical twilight for any specified locale. An example of such a report, in web page format and calculated for the locale of Greenwich, can be seen by clicking here.
For further introductory information about civil, nautical, and astronomical twilight please see the "Rise, Set, and Twilight Definitions" web page at the U.S. Naval Observatory at <http://aa.usno.navy.mil/faq/docs/RST_defs.php> and the Wikipedia "Twilight" web page at <http://en.wikipedia.org/wiki/Twilight>.
Various stages and durations of dusk and/or dawn twilight are of ritual significance to several religions, especially those that start calendar dates at sunset or sunrise.
Functions for the calculation of the moments of sunset and sunrise, the moment corresponding to a specified solar depression angle, the solar longitude and the moment when Sun is at a specified solar longitude, as well as calendar and time-of-day functions are explained byNachum Dershowitz and Edward M. Reingold in their excellent book entitled "Calendrical Calculations", 3rd edition published in 2008 by Cambridge University Press, and in their on-line errata at <http://www.calendarists.com/>.
Two excellent books written by Jean Meeus and published by Willmann-Bell, Richmond, Virginia, USA <http://www.willbell.com/> include important discussion of the mathematics of the duration of twilight.
The basis for and calculation of the shortest and longest twilight are explained by Jean Meeus on pages 368-379 (chapter 65: "The shortest and the longest twilight") in his book "More Mathematical Astronomy Morsels" (2002) <http://www.willbell.com/math/moremorsels.htm>.
The calculation of the parallactic angle, which is the primary predictable determinant of the duration of twilight, is explained by Jean Meeus on pages 97-100 (chapter 14: "The Parallactic Angle") in his book "Astronomical Algorithms" (second edition, 1998) <http://www.willbell.com/math/mc1.htm>. At sunrise or sunset only, the solar parallactic angle is given by:
solar parallactic angle at sunrise or sunset = arccosine[ sine( geographic latitude ) / cosine( solar declination ) ]
where all angles are in degrees.
Definitions:
Variations of the duration of sunrise and sunset are related to the duration of twilight and depend mainly on the angle made by the path of the rising or setting Sun with respect to the horizon, known as the solar parallactic angle. Although Sun rises at the true east direction and sets at the true west direction on the day of an equinox, on that day the solar parallactic angle would be 90° only at the equator.
The solar parallactic angle can only be 90° on days when the solar declination equals the observer's geographic latitude, which occurs twice per year for locales between the Tropic of Cancer and Tropic of Capricorn, but never occurs at latitudes further north or south of the tropics. Locales that are on the Tropic of Cancer see a 90° solar parallactic angle only when sunset or sunrise occurs near the moment of the north solstice, and locales that are on the Tropic of Capricorn see a 90° solar parallactic angle only when sunset or sunrise occurs near the moment of the south solstice.
Sunrise and sunset durations are maximal in both hemispheres at the time of both solstices, and minimal near the dates of both equinoxes. Sunrise and sunset durations increase with latitude further away from the equator, reaching extremes near the arctic circle around the date of the north solstice and near the antarctic circle around the date of the south solstice. On the day of the north solstice at about a degree south of the arctic circle the solar parallactic angle is almost parallel to the horizon so Sun takes a long time to rise, spends most of 24 hours above the horizon, then takes a long time to set, whereas on the same day about a degree north of the antarctic circle the solar parallactic angle there is also almost parallel to the horizon so Sun likewise takes a long time to rise but spends very little time above the horizon, then takes a long time to set. The converse occurs on the day of the south solstice.
On average, Sun appears to move 360° per mean solar day (by definition), and has a mean diameter of about 1/2° (more accurately, slightly more = 0° 31' 59.3"). The minimum sunrise or sunset duration occurs when the solar parallactic angle is 90°, so on such a day Sun might simply be predicted to take 86400 seconds × ( 0° 31' 59.3" / 360° ) = 2 minutes and 8 seconds to rise or set. The minimum sunrise or sunset duration in the tropics, as plotted in the charts above, is about 2 minutes and 30 seconds. This drops by 9 seconds at the equator (calculated separately but not shown). These figures are slightly longer than the simple prediction because of atmospheric refraction, which makes Sun appear above the horizon when it is actually just below it, and the amount of refraction decreases rapidly with increasing solar altitude, thus slightly stretching the duration of sunrise and sunset. By contrast, atmospheric refraction is not considered to play a role in the duration of twilight.
Atmospheric refraction increases with lower air temperature, higher relative humidity or high rate of moisture evaporation from the land or water in the direction of the horizon, higher air pressure, and lower altitude (elevation). Atmospheric refraction also raises the apparent level of the horizon in all directions, causing the illusion that even a perfectly flat plain or the open sea is within a shallow bowl, apparently curved in the opposite direction to the actual curvature of Earth's surface, perhaps explaining the prolonged historical reluctance to accept that Earth is spherical. Light from celestial objects is subjected to greater refraction because that light passes obliquely through the full thickness of Earth's atmosphere, whereas light from the horizon passes only from the horizon to the observer's eyes [see "Variability in the Astronomical Refraction of the Rising and Setting Sun" in Publications of the Astronomical Society of the Pacific (PASP) 2003 October, 115:1256–61].
The maximum sunrise or sunset duration at the equator (calculated separately but not shown) is 2 minutes and 34 seconds, at both solstices.
Each of the following sets of charts are plotted from the day before March 1, 2007 to the day after March 31, 2008. The charts show the durations of dusk twilight but the durations of dawn twilight are essentially symmetrically equal for any given degree of solar depression angle.
Each PDF is about 140KB and contains 6 pages (Civil, Nautical, and Astronomical twilight durations for the northern and southern hemispheres).
| # | Charts | Description and Comments |
|---|---|---|
| 1. |  |
If the duration of twilight is expressed in terms of mean solar minutes (normal clock time), then seasonal variations are evident for all locales, with the shortest twilight durations a few days before the spring equinox and a few days after the fall equinox, and the longest twilight durations near the summer solstice, with a lesser maximum of durations near the winter solstice. The seasonal variation and duration of twilight is minimal at the equator, where the solar parallactic angle is close to 90° all year round (±23.5°), and the variations increase in amplitude and duration for locales that are progressively further away from the equator. |
| 2. | |
If the duration of twilight is expressed in terms of daytime temporal minutes (the time span from sunrise to sunset divided into 720 equal portions), then each stage is almost constant and minimal in duration throughout the spring and summer, except near the equator and high latitudes, the shortest twilight durations are 2-3 weeks after the spring equinox and 2-3 weeks before the fall equinox, and the longest twilight durations are near the winter solstice. |
| 3. | |
If the duration of twilight is expressed in terms of night time temporal minutes (the time span from sunset to sunrise divided into 720 equal portions), then each stage is almost constant and minimal in duration throughout the fall and winter, except near the equator and high latitudes, and the longest twilight durations are near the summer solstice. |
| 4. | |
If the duration of twilight is expressed in terms of daytime or night time temporal minutes (whichever is longer), then each stage is almost constant in duration all year round, except near the equator and high latitudes. The switchover date for "whichever is longer" occurs a few days before the spring equinox and a few days after the fall equinox, when the actual lengths of daytime and night time are truly equal. |
This freeware Excel spreadsheet with VBA macro (Visual Basic for Applications) uses information about a user-selected or user-provided locale to compute the moments of sunrise/set, mid day/night, and start/end of civil/nautical/astronomical twilight (as well as zmanim) for an entire year. It produces two single-page graphic charts plus a "List" worksheet. The listing could be printed, although it is rather dense and was intended for reference purposes. The multiple plotted series on the charts point at the various columns of the listing worksheet.
The user needs to enable running of VBA macros to see it run. Nevertheless, you can open the file without enabling VBA macros just to view the various worksheets and charts.
Click here to download the Twilight Charter spreadsheet for Microsoft Excel 
about 1MB
The "Setup" worksheet is the main control sheet. The "Locales" worksheet lists some built-in locales, any of which can be selected for plotting. The "Twilight" chart shows the plot of twilight moments. The "Zmanim" chart shows the plot of zmanim moments. The "Events" sheet just contains computed equinox and solstice moments, used to plot those moments as vertical lines on the charts, for reference purposes.
I have saved it with the capital of Iceland as the selected locale, which makes for some interesting curves during the course of a full year, and consequently some cells of the listing worksheet were left blank because they never happen on those dates.
The user can choose to have the plots calculated in Zone Clock Time, Local Mean Time, or Local Apparent Time.
This page updated Apr 4, 2011 (Symmetry454) = Apr 4, 2011 (Symmetry010) = Apr 7, 2011 (Gregorian)
