Expert: Philip Stahl - 9/5/2008

Question
QUESTION: What is the exact definition of a day?

If a day is defined as a full spin of the earth on its axis, which seems to be a common conception, obviously the side of the earth facing the sun at any particular point in time will be the dark side exactly 182 full spins later. It will be the side facing the sun 182.5 full spins later, but that doesn't jibe with this conception of days at all.

On the other hand, if a day is the time from noon to noon in the sense that the sun is directly overhead, then a day isn't a full spin, it is more than a spin, accounting for the change in angle with respect to the sun as the Earth revolves, if I'm not mistaken. Consequently, doesn't a 24 hour clock have to measure more than one spin, ie 1/365.25th of a year instead of a day? If so, how long does it really take in hours and minutes for the earth to spin exactly one time, ie, so a vertical object in the ground is exactly parallel to its position on the previous spin?

If a day is defined as sunup to sunup or sundown to sundown, as it is in various cultures, days have different lengths, which does not seem to be all that scientific or logical.

Thanks.



ANSWER: Hello,

You have way way too many questions, and I only allow one to be answered at one setting, time. (Else I end up answering a half dozen and get credit for only answering one!) So I will boil everything down to one: Is there a day based on a system of time that is independent of the vagaries of motion peculiar to Earth's rotation?

Yes, there is. It is called "ephemeris time" (ET) and the day based upon it is called the "ephemeris day". This system removes the problems posed by the irregular motion of the real Sun in the sky and the efforts to render it uniform using "mean solar time" and the "mean solar day". (All of which you can google).

ET is strictly a uniform measure of time in the Newtonian sense, and is employed for purposes that require an exact (or more exact) measure such as in celestial mechanics (e.g. predicting the future position of a given planet).  More basically, it is the independent variable in the orbital theories of the Earth, Moon and other planets. It is also the argument for the fundamental tables in any astronomical ephemeris.

Key to arriving at an ephemeris day (or by extension year) is the value for the geometric longitude of the Sun at a given epoch. It is the constant term originally given by Newcomb, "Tables of the Motion of the Earth on its Axis and arond the Sun', 1895 cf.:

L(S) = 279 o 41' 48."04 + (terms in t_E_ )

The primary unit of ephemeris time is the length of the tropical century at the standard epoch of ET (e.g. where t_E = 0) termed the ephemeris century.

Key to smaller units of ET is the instantaneous rate of change of L(S) with respect to time, viz.:

dL(S)/ dt =  0."0410686389744 +  6."9017 x (10^-10)t_E

Now, during one tropical year, the Sun's geometric longitude increases by (2 pi) = 1,296,000 " (seconds of arc)

Then the length of a tropical year exprsssed as a function of t_E in ephemeris seconds is:

1,296,000"/  dL(s)/dt = 31,556, 925.s9747 (E) - 0.53032t_E (s)

where the 1st term is an absolute constant.

Thereby, the duration of the ephemeris second can be obtained:

E(s) =  1/ (31,556,925.9747)  T(Y)

where T(Y) denotes the tropical year at the epoch (e.g. at January 0.5 d ET, 1900).

The ephemeris day can thereby be obtained:

1d(E) =  86400 (E(s))

where E(s) is the ephemeris second (as defined above)

The Ephemeris hour is:

E(h) =  3600 E(s)


Thus, we have a means (albeit complex!) to avoid the problems you noted that arise from using solar time.

As you can see the answer is not 'simple' nor should anyone expect such. The matter of astronomical position and time keeping is an entire field (astrometry) and workers have spent many decades developing the mathematical basis and the measuring techniques for it.

---------- FOLLOW-UP ----------

QUESTION: Whoa! What a great answer!

I had no idea that there was an entire science relating to what seems like a simple question!

After Googling some of the terminology you used, I came to yet another concept -- the sidereal day. I'm pretty sure that was what I was conceptualizing; I didn't know about the variations in speed around the sun and in spin that makes even that more complex. The wiki entry I saw had a good graphic of what I was thinking about and noted that a sidereal day is 23 hours, 56 minutes and change, the rest being an averaged adjustment for the new position of the earth as the meridian lines up with the sun.

But I must confess that at various sites that I perused, I had to fight off complex terminology (like "diurnal" and "ascension" and "declination") commonly used in discussions without prior explanations as to what is meant. Unhappily, when things are "written in Greek" that is a turn off to noobs in a given subject; I wish it was otherwise.

Long and short, with your help, I did confirm that the "day" as non-scientists look at it is not really 24 hours; that is the approximation of a solar day, and I confirmed there is one more sidereal than solar days per trip around the sun.

So here is the next query; why do we use 24 hours and 60 minutes and so forth, rather than a metric-like system, ie, why aren't there 100 hours per day, 100 minutes per hour, 100 minutes per second, and like that? Americans are chastised world wide for not using the metric system, but the metering of time seems just as arbitrary.

Thanks.

ANSWER: Hello,

Okay - first I don't think "metering" (metric system) is in any way arbitrary. It's actually a much more rational system than foot-pounds etc. and makes more sense. It is also easier to spot inconsistencies when they emerge, say in the SI system. The multiples also make conversions from smaller to larger units (and vice versa) much easier, thus:

10 mm = 1 cm

100 cm = 1 m

1000 m = 1 km

Also: 1000 g = 1 kg

1 Newton (N) = 1 kg-m/s^2

1 Joule = 1 N-m

1 Watt = 1 Joule/s  =  1 N-m/s

Note how more complex units easily build on others based on the physical defintions. For example, Work (energy) = force x distance

Thus, W = F(N)  x  d (m)     (unit is the N-m or Joule)

This makes it vastly easier to check for consistency in unit for any given physical problem.

Once one gets used to using the metric system it is no biggie.

There is also no confusion, for example, between units - such as mass and weight, or having to resort to some oddball unit like "slugs".

Thus, mass in the metric is in kg

Weight (or any force) is in N.

As far as time, it wouldn't make sense to go metric since time units are not natural multiples of 100, but rather 60. As we know from geometry and circular measure, the circle is the basis and we have 360 degrees, which can be reduced to minutes and seconds of arc.

The Earth undergoes (essentially, and if one isn't picky) one complete revolution in 24 hours.

So 360 deg in 24 hours translates into:   360 deg/ 24 h  = 15 deg/h

In other words, a longitude difference of 15 degrees easily translates into one hour.

Thus, the basis for a 24 hour natural time scale is already there - again, so long as one isn't picky about fine details.

In the approach to early temporal measures, the difference between sidereal and solar days wasn't yet known and primitive clocks like sundials were used. Later timepieces simply adopted a 24 hour clock. To register 23h 56m would have immensely complicated analog clock-time devices for only small practical gain. Hence, the circular measure was already well established well before cesium clocks and digital devices entered the picture.

The world was already divided into these (natural) longitude zones too, which demarcate time zones.

Further, the natural terrestrial circular measure fit well with the celestial sphere in astrometry which already had its "longitude" demarcated into 24 hours of Right Ascension - the celestial coordinate corresponding to longitude on Earth. All positions of celestial objects are reported in Right Ascension and Declination (the analog to latitude on Earth)

Changing all that to a "metric" system would have been horrendous and probably counter-productive.

Thus, the bottom line is that the metric system is indeed highly rational for most uses of measurement, but not for time which will in all probability remain the way it is for the forseeable future. Even given small differences in actual day length!



---------- FOLLOW-UP ----------

QUESTION: Thanks for the answer, but I think you misunderstood my point; I freely admit the American measurement system of inches/feet/miles/yards is more arbitrary than the metric system, but also say that time based on 24/60/60 is just as arbitrary. Both because of the complexity of doing math in a non-base 10 system.

Moreover, it seems to me that your point that "complex units easily build on others based on the physical defintions" is a wash; if the equations were built around cubits and stones instead of meters and grams, they would work just as surely.

If anything, it seems to me that we would all be better off if the measurement system was based on a system of ten derived off of some universal constant, like the speed of light, something that could arbitrarily be forced into a number divisible by 10. If the speed of light was declared to be 1 million meters per second instead of 299,792,458, we could easily establish liters and grams around the volumes of 1 square meter based on that in lieu of what we have, and the system would work just as well, perhaps better because of the neat number brought into the relationship.

Though the American system does have this going for it: it is easier to understand and request a "cup" of milk and a teaspoon of sugar than .25 liters and .05 grams of it, and easier for a kid to recognize the meaning of being four feet tall than 1.3 meters; our system minimizes fractions by its very nature. Both obviously came from the time when careful measuring tools were hard to come by and measurements could be premised on the resources available, cups and feet; as it is there is nothing absolutely natural about the designated magnitude of a gram, the only natural thing to the metric system is the easily designated relationships between grams, liters and meters.

Answer
Hello again,

You wrote:

"Thanks for the answer, but I think you misunderstood my point; I freely admit the American measurement system of inches/feet/miles/yards is more arbitrary than the metric system, but also say that time based on 24/60/60 is just as arbitrary. Both because of the complexity of doing math in a non-base 10 system."

Sorry, I just don't agree. The 24/60/60 system is in fact *less arbitrary* because it is based on a natural system which roughly approximates a 360 degree full cycle (Earth rotation) occurring in 24 hours - and also which approximates the celestial sphere doing the same.

To undo the system in favor of some other would be to undo the celestial (coordinate-time)system as well, just ain't gonna happen.

The beauty of the 24-60-60 system is it works both on Earth and for the stars. The entire astrometric system, based on tranformation of coordinates - from Equatorial (RA, declination) to horizon (altitude, azimuth) etc. is also based on it. There is absolutely no way the workers in those disciplines will surrender the *rational simplicity* of their system for some newfangled one based on the speed of light etc.

The bottom line is, yes, the circular-angular system does have its limitations (and is indeed approximate when one gets into the details) but it more generally fulfills and meets all the demands made by all the sub-disciplines of astronomy. Unless all the stars and planets suddenly turn into "cubes" and the sky itself displays a cubic or other feature there will be no alterations, nor any admission the system is "arbitrary".

Lastly, just because people operate more naturally in the decimal system doesn't mean the binary or hexadecimal (or other) systems are "alien" or more contrived. Look at the applications in computers, etc. In truth, the decimal system is merely an extension from the biology of a numerate biped that happens to possess 10 digits - nothing more. If we had 6 digits, the hexdecimal system would be the "natural one".

Earlier you mentioned yourself perusing some of the literature brought up by google and saying you didn't get some of the lingo such as: "right ascension" and "declination". This may also explain why you feel as you do that the 24-60-60 is somehow "arbitrary" - not realizing our celestial coordinate system is predicated on the projection of poles, equator etc. into the sky to form those features there - providing a natural coordinate system by which to plot sky positions easily. (Not to mention the fact that in the sky, the most natural system is angular measure - NOT linear! One thus articulates star distances in *degrees* - not meters, which is useless given no knowledge of true distance)

I simply cannot remotely think of an alternative system that would yield the same simple results!

Anyway, I suspect we shall probably have to agree to disagree on this.