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Okay, How do I fully answer this question??

Started by swsands, 03/12/2006 03:56AM
Posted 03/12/2006 03:56AM Opening Post
I received an email from a relative a few days ago. I explained seasons, the approximately 22.5 degree tilt of the earth, latitude etc., I really can't fully answer his question. Anyone care to help out?

"As you well know, on December 21st, the days (in Alton, at least) start getting longer. However, on the clock, the extension in time is greater at the “end” of the day, compared to the mornings. That is, the sunset times, e.g, extend by 2 minutes per day, but sunrise time are only 1 minute earlier each day. So, yes the days are getting longer, but not equally at both ends of the day. Why is this, and does this reverse when days begin to get shorter on June 21st or thereabouts?
Posted 03/12/2006 03:22PM #1
Steve,

A couple of factors that play into this are that the Earth's orbit is not circular, and the speed of motion around the Sun changes. The true day from a rotation of Earth standpoint is the Siderial day. The solar day is a bit longer as the Earth has moved in its orbit around the sun, and hence needs to rotate more than 1 rev on its axis to bring the sun back overhead. For calendar purposes, we've adopted the mean solar day, which is the mean value over the course of a year. The amount more that it needs to rotate varies, it is larger near perihelion in January. We see these variations as shifts in the sunrise and sunset times relative to the mean solar day.

Secondly, from anyone off of the equator, the Sun's movement in declination during each day causes a bit of the discrepancy as well, as it moves northward, it will set a bit later in the northern hemisphere. The fastest rate of declination chage is at the equinoxes.

Hope that helps.

Dave

Posted 03/12/2006 09:41PM | Edited 03/13/2006 11:08PM #2
If you truly want the question "fully" answered, then read what I have written below. It describes the factors that create differences between the lengths of the true solar day and the mean solar day leading to apparent anomalies in the times of sunrise and sunset. Sidereal time plays no role in my explanation.

There are two factors. The primary one is simply an effect of spherical geometry on the apparent celestial sphere surrounding the earth. The secondary factor is the eccentricity of the Earth’s orbit and the resultant changes in its angular velocity around the Sun. Even if the Earth’s orbit were not eccentric, the primary effect due to spherical geometry would still be quite noticeable regarding sunset and sunrise times and the time of the Sun’s meridian crossing (true noon). The earliest sunset would still occur a number of days before the winter solstice. The latest sunrise would still occur days after the winter solstice. The earliest sunrise would still occur before the summer solstice. The latest sunset would still occur after the summer solstice.

The combination of the two factors results in what is known as the Equation of Time. That is the difference between sundial time and that of our standardized clocks after a constant adjustment for longitude. In February sundials are 14 minutes slow. In May they are 4 minutes fast. In July they are 7 minutes slow. In November they are 16 minutes fast. Again, the main reason is spherical geometry. However, if the Earth’s orbit were not eccentric, the absolute values of those four numbers of minutes would be identical. The Equation of Time equally alters the times of sunrise and sunset for observers at all locations on Earth, including those on the equator.

A true solar day is the length of time between local meridian crossings by the Sun (true noon to true noon.) The Equation of Time is actually the sum of daily variations from 24 hours (a mean solar day) in the length of a true solar day: 23:59:39 hr to 24:00:30 hr as measured by mean time.

If we used sundial time, the apparent anomalies of sunrise, sunset and meridian crossing times would disappear, because we would be observing the true solar day. With standard time we utilize the mean solar day. Of course, in either case the graphs of sunset and sunrise times would loosely approximate sine waves. At the extremes of the graphed times, the daily changes of those times are minimal. Midway between the dates of extreme sunrises or sunsets, the daily changes would reach their maximums. But when reading a sundial, the daily change in both the sunrise time and the sunset time would be virtually identical in absolute value between any particular pairs of consecutive dates. Thus if your relative had been using a sundial, he would not have posed his question.

The geometric effect can be imagined this way. At the solstices the Sun reaches its maximum declinations on the celestial sphere. The meridians of right ascension are more closely packed than at the celestial equator. The Sun in its apparent motion through the equatorial coordinate system gains more each day in right ascension than it does around the equinoxes. Also, at the equinoxes the Sun would appear to be moving in a slant relative to the parallels of declination. Our standardized clocks are based on a fictitious mean Sun that advances in right ascension at a uniform rate. As a result of the spherical geometry, the true Sun’s right ascension is ahead of the mean Sun’s right ascension in February and July; it is behind in May and November.

The variable angular speed of the Earth in its eccentric orbit modifies the previously described geometric effect. The fact that the Earth is moving faster (and gaining more in right ascension) in January than July results in greater extremes in the Equation of Time in November and February than in May and July.

The basic sunrise or sunset times as measured by a sundial obviously go through one complete wave cycle each year. The orbital effect on the Equation of Time also goes through one wave cycle each year. The geometric effect goes through two wave cycles each year. When the three waves are combined the result is somewhat complex. If that is bothersome, then throw away your watch and use a sundial. wink

P.S. In the interest of relative brevity, I've left out any discussion of how the Equation of Time and the times of sunrise and sunset change over the centuries due to the precession of the equinoxes and changes in the Earth's orbital eccentricity and axial tilt. BTW, the current mean value of that tilt is 23.438° (not 22.5°)and will continue declining until it reaches a minimum of 22.6° in 10,000 years.

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