Showing posts with label tropical year. Show all posts
Showing posts with label tropical year. Show all posts

Monday, February 29, 2016

Of Leap Days, Leap Years & Altered Calendars

Today, February 29th -  the 'extra' day this February - is designated a 'leap day' in this leap year of 2016. Few people, however, understand how leap days and leap years arose and just how they are tied to astronomical timekeeping and the calendars we use.

Time keeping by calendar.

One might say it all began with the development of the Julian calendar, named after Julius Caesar. He had introduced a calendar with 365 days and a leap year every 4 years with 365 days. This meant an average length of year of 365.25 days.  But that extra 0.25 days for the mean Julian year made it in fact a smidgeon longer than what we define as the tropical year of 365.242374 days (or 365 days 5 h 48 min 46 s).

This set the stage for future problems, especially as the  role of the astronomical point known as the vernal equinox (also associated with a date) came to prominence. (The vernal equinox marks the intersection of the ecliptic with the celestial equator. The ecliptic is the projected path of the Sun - as it appears from Earth - onto the celestial sphere, but in reality is the plane of the Earth's own orbital motion).

In the wake of the Edict of Milan, the Council of Nicaea in 325 A.D. defined the dates of Easter and certain other religious holidays by reference to the vernal equinox. . In particular, March 21 was re-specified as the date of the vernal equinox while Easter was defined to occur on the first Sunday after the 14th day of the Moon (e.g. days after the full Moon). It ought to be noted here that the Christians at Nicaea didn't willy- nilly just change the date of the vernal equinox from March 25 back to March 21. No, what happened is that between 45 B.C. and 325 A.D. that date had slipped back from March 25th to March 21st. The reason is based on simple math: Because the Julian year was defined with an average of 365 ¼ days and is 11 mins. 14 secs longer than the tropical year of 365 days 5 h 48 min 46 ec then the slight difference had accumulated to 3 days in those 4 centuries.

Anyway, the ecclesiastical choice of March 21 as the vernal equinox did not always agree with the precise astronomical definition, i.e. when the astronomical point described above transited the meridian. This year, for example, that occurs on March 20th. The proximity is close, but because it isn't exact divergence occurs.

The divergence was such that by 1582 Pope Gregory XIII was tasked with another adjustment. This is because the original tiny deviation between Julian and Tropical year, e.g. of 11 mins and 14 seconds,  had grown to another 10 days so that the first day of spring was now occurring on March 11 and not the prescribed March 21.  What to do? In order to re-align his now (Gregorian) calendar with the actual vernal equinox (or the Church's approximation to it) Pope Gregory removed ten days from the middle of October.  This was undertaken on October 4, 1582 when the next day was proclaimed as October 15, 1582. (Some idiots at the time actually complained that the pope had "taken 10 days out of our lives".)

The effect of the Gregorian change was to realign the seasons but the primary goal was to calculate the  date of Easter based on the Church's definition of the vernal equinox on March 21.  Alas, not everyone in the world was committed so that much confusion reigned with different calendars and dates being kept in different places. Britain, for example, kept the old Julian calendar until 1752.

In the wake of the above the rule for the leap year was changed so that the average length of the year would closely approximate the length of the tropical year. The rule then applied was that only century years divisible by 400 would be leap years. Thus, 1700, 1800 and 1900 - all leap years under the Julian calendar - were not under the Gregorian, while 1600 and 2000 were.

Meanwhile, the average length of the Gregorian calendar, at 365.2425 days, was correct to within 1 day every 3300 years.

Incredibly, a more rational World Calendar was proposed in the 1950s, according to USF visiting astronomy professor Anthony Aveni. The device will replace the Gregorian calendar with its numerous peculiarities. The benefit of this calendar would mainly be that it is identical from one year to the next - so fewer calendars would have to be printed.  The design gives each quarter of three months of 31, 30 and 30 days, respectively, for a total of 91 days or 91 days x 4 = 364 days for each year.   The 365th day, deemed, "Year end day" is simply denoted December W or "Worldsday".  The economy is thereby built into the design so you get just one calendar and can follow it for x years.

  Alas, the U.S.  - after objections  from religious groups (about the dates of Easter, Christmas, etc.), rejected it.  Some religious reactionaries have gone so far as to claim "adoption will mean the end of religious liberty for million" since special holydays will be absorbed into the design and no longer stand out. 

The takeaway? While we often view timekeeping as a precise enterprise, the interjection of politics and religious' dogmas often undermines that assumption. Today, people need to remember that as they do whatever on this special day.
 
Enjoy today as the leap day it is and be thankful you weren't living back in 1650 or 1700.

Sunday, July 1, 2012

Where Did That Extra Second Come From?

As most readers are probably aware, yesterday (Saturday) featured 86, 401 seconds rather than the usual 86, 400. One leap second was added to compensate for a slowing Earth rotation (actually - if factored in from 1972, 24 full seconds slower than at that time). A cutesy explanation (on ABC News) last night was that "just as humans slow down when they age so has the Earth." Well, yes...and no.

The actual physical reason has less to do with aging per se and more to do with what we call "tidal friction" resulting from the Moon's cumulative gravitational effects (part of which causes our spring and neap tides as well as regular tides on Earth). Over time those gravitational effects add up....and a leap second must be added to correct it.

Thus did the concept of Ephemeris time (ET) enter the picture and ET thereby became part of the lexicon of astronomers, and integrated within the specialized sub-discipline of astrometry. The job of astrometry is to keep careful quantitative track of any changes that would lead to time corrections, specially related to ET. This isn't so stirring an idea now, but a couple of centuries ago it was almost a given that the Earth's period of rotation about its axis was constant. (Apart from a slow secular increase due to tidal friction). 

Note that "secular changes" are essentially non-reversible, and hence proportional to time passed.

Meanwhile, the value of ET at a given instant is reckoned by extremely accurate observations of sudden variations in the longtiudes of Sun, Moon and planets which of course emanate from variations in Earth's rate of rotation. Hence, if I detect a rather sudden "jump" in the Sun's (heliographic) longitude - say during lengthy observations of a sunspot group, I can tie that in to variation in Earth's rotation. (Generally, however, such changes wouldn't occur that fast.)

To get into the nitty gritty, Simon Newcomb (1895) originally provided us with the detailed expression for the longitude of the Sun (cf. 'Spherical and Practical Astronomy Applied to Geodesy', p. 168):

L= 279 deg 41'48."04 + 129, 602, 768."13 t_e  + 1."089 t_e^2

where t_e is the length of the tropical century from the standard epoch. (Recall the 'tropical year' is the time taken by the fictitious Sun to make one passage between the mean Vernal equinox. Hence, a tropical century refers to 100 such passages.)

Now, differentiating the preceding with respect to time t_e,  the instantaneous rate of change of L per second of ET is obtained, viz.

dL/ dt_e  = 0."0410686389744 + 6."9017 x 10^-10t_e

The last factor featuring the 10 raised to the negative 10 power is an important marker. It allows us to estimate the frequency at which time corrections, say to Earth's rotation, need to be made. Thus, the equation shows us that to define Ephemeris time correctly to 1 part in 10^10 (or one part in ten billion) observations of the Moon are required over at least 5 years.

Another quantity that makes its way into such computations is the Right Ascension of the fictitious mean Sun;

RA = 18 h 38 m 45.s 836  = 8, 640, 184.s 542 t_e  =0.s 0929 t_e^2

What exactly is the fictitious mean Sun? What, for that matter, is the fictitious Sun (referenced earlier in connection with the tropical year, and century)? Basically, the former is that fictitious entity which is presumed to travel always at a uniform rate every hour and every day. This fictional creation enables us to fabricate "time zones" based on the fiction that the Earth turns uniformly through 360 degrees every 24 hours, hence through 15 degrees every hour. Hence, we set out longitude markers to reference the mean time for all locations, say within 15 degrees of longitude.

Hence, in regular standard time (not 'daylight saving') one has the time zone for the Greenwich meridian registering say 12 noon, then at a longitude of 15 deg W. the time is one hour earlier, and at 30 W two hours earlier, and so forth. All locations within the time zone agree (with few exceptions) to keep the same time.

By contrast, the fictititous Sun is not so regularized so its motion is erratic. This is also referenced to what we call "apparent solar time" or "sundial time".  Based on comparisons of the two, one can then compute what is called the "equation of time", see e.g.

http://brane-space.blogspot.com/2011/07/tackling-simple-astronmoyh-problems.html

Now, if we reckon Ephemeris time into the context of the "fictitious Suns" we will be using two key hour angles: h (the hour angle of the fictitious mean Sun) and h' (the ephemeris hour angle of the fictitious mean Sun. Based on this, Ephemeris time (ET) would be expressed:

ET = h + 12h

While the equation of time (EQN T) is:

EQN T =   h"   -  h'

where h" is the hour angle of the fictititous Sun.(Those who'd like to review hour angles and how to work them out can consult two earlier blogs:

http://brane-space.blogspot.com/2011/03/more-spherical-astronomy.html

and

http://brane-space.blogspot.com/2011/03/solutions-to-spherical-astronomy.html

I will also give here, without derivation (though I welcome energized readers attempting it using material from prior blogs) the relation between the corrected time delta T, Ephemeris time and Universal Time (UT):

delta T = ET - UT

Generally, delta T is computed - or at least used to be before the era of atomic clocks - through inverse interpolation in the ephemeris of the Sun, Moon & planets...and deducting the recorded UT time of observation. (Note: An 'Ephemeris' is a manual giving all the changing positions correlated with times, dates for celestial objects).

In the modern atomic clock -time epoch, things are (fortunately!) much more streamlined and we have powerful Cray and other supercomputers to boot. Basically, one associates the atomic time (AT) and Ephemeris time (ET) via the generic expression:

AT - ET = a + bt  + ct^2

where t denotes the time from the epoch when AT = ET + a

Hence, the coefficient 'a' determines the epoch of atomic time in relation to Ephemeris time. The coefficient 'b' is the division ratio adopted for the atomic resonator (as we know, different atomic clocks, e.g. cesium , quartz etc. have different frequencies, hence differing resonator rates). Finally, coefficient c is a cosmic constant which value may be 0 or more likely on the order of 0.s001 per year.

As we can see from the above considerations, for the most recent leap second correction:

ET - UT = 1 s  =  AT  - ET

Just be glad you only have to savor that extra second of time, not figure out when exactly you have to add it!