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Astronomy/Rates of Rotational and Revolution of Planets

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QUESTION: Hi Professor Seligman!

I recently had a series of curious thoughts that eventually led me to your site. Succinctly, they were:
1) How many rotations of the earth must there be per year to experience one night/day cycle? You'd think it'd be one, but it's gotta be two to account for the one revolution of the earth per year.
2) I wonder what the rotation/revolution rates of other planets are.
3) In researching No. 2, I learned that the rotation of earth is not 24 hours but 24 hours minus roughly four minutes, which roughly makes up 24 hours per year, which makes up for Earth's revolution!
4) More research eventually brought me to your explanation of Mercury's strange orbit, which is FASCINATING!!! Thank you for such an engrossing explanation.

Which finally brings me to my question:
Is it complete coincidence that the deficit in minutes in Earth rotation period makes up one complete day in a year? Has there been discussion in the astronomy field about how to keep a day/night cycle consistent with Earth's revolution around the sun as Earth's rotation slows to a 24-hour sidereal day, assuming humans are still around?

Thanks for you time in answering!

Best,

Artem

ANSWER: I'm glad that you found the discussion of the Rotation of Mercury useful and interesting; but your remaining questions indicate that you didn't run across Rotation Period and Day Length (at http://cseligman.com/text/sky/rotationvsday.htm ), which explains why the rotation rate of a planet is not the same as the length of its day, why the Earth has a 3 minute and 56 second difference between the two, and how to calculate the difference for all the other planets (and our Moon). That will almost certainly take care of the first part of your question.

As far as the second part, involving the gradual slowing of the Earth's rotation, there is something called a Leap Second, which is used to keep the calendar days exactly in line with the actual days despite the gradual slowing, which is usually close to 2 milliseconds increase in the length of the day over a period of a century; so 5000 years or 50 centuries ago, the day (and rotation rate) would have been one-tenth of a second shorter than now. That seems like a very small amount, but some fraction of that error in the length of a day, over a period of 5000 times 365 days, adds up; and when we calculate where eclipses should have taken place in the distant past, historical records show that they actually occurred as much as a third of the way around the Earth because the Earth used to rotate faster than now.

For a detailed discussion of the various reasons that the Earth's rotation rate changes, and the fairly consistent gradual slowing which is primarily due to lunar and solar tides, take a look at Leap Seconds (at http://cseligman.com/text/sky/leapseconds.htm ). As noted at the end of that page, it was not as fully edited as the page about Rotation Period and Day Length; but I think it will probably answer your question better than any reasonably long answer posted here could do; and of course if either of the two pages referenced here do not completely clear things up, please feel free to contact me again.

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

QUESTION: Thanks for the quick reply! Those links definitely shed some light on my questions. But I'm specifically interested in when these small discrepancies (corrected by leap seconds) become larger discrepancies.

When the rotation period gets to 24h, for example - and I'm assuming orbital period doesn't change significantly or am I wrong there? - then having a 24h defined-day would see the sun overhead at noon in summer, but the dead of night at noon half a year later. Will leap seconds every once in a while turn into leap-12-hours every six months? But while that'll correct the lag, it won't curb it throughout the year.

When the discrepancy becomes noticeable to laypeople every day, how will timekeeping change?

(Also how long will it take to slow the sidereal day by 4 minutes?)

Artem

ANSWER: I don't know whether any consideration has been given to what to do when the rotation rate becomes so slow that there is a significant fraction of a second error in the length of the day; and if there has, I doubt that there has been any agreement about what to do at that time, since it would be hundreds of years in the future, and would involve considerable inconvenience to one group of people or another. However, there were some recent complaints about the current adding of Leap Seconds from the tech community, because it requires awkward, unpredictable changes in their timekeeping software in order to keep proper track of the total amount of time between events when the number of seconds in an interval is changed by the addition of leap seconds. So there may be some alteration in the method of dealing with the problem within our lifetimes; but odds are, any adjustments made in that short a time scale would not be big enough to worry most people.

I suppose that when the number of seconds in a day becomes significantly different from now (say a few hundredths of a second per day different), the most practical way of handling things would be for those who need to keep accurate track of time differences to continue to use current seconds, and specify particular dates and times as so many umpteen seconds of that sort (and some fraction of a second); while the seconds used on clocks could be adjusted so that there were still 24 hours in a day, and 86400 seconds in 24 hours, just the same as now; but those seconds would be a little longer than the ones used for long-term timekeeping, which means that all existing timepieces would have to be replaced (need a new watch? get a brand-new model (insert century year here)!). So though 'practical', such a change would make a difference in everyone's life; and I suspect that means it would be put off as long as possible. (If and when this happens, there would undoubtedly be something similar to Old Style and New Style dates, as used in the late 1700's; so you might have a particular moment specified as one Old Time value, and a different New Time value.)

Slowing the sidereal day by 4 minutes, which is 240 seconds or 240,000 milliseconds, would take about 120,000 centuries if the rate of slowing is a constant 2 milliseconds per day per century. The actual rate appears to be closer to 1.7 milliseconds per day per century, so the actual time period required would be about 14 million years. Of course, even at that time the day and rotation period will still be different, by nearly the same 4 minutes as now; but at that time if we kept seconds, minutes and hours with Old Time the rotation period would be 24 hours, and the day 24 hours and 4 minutes (approximately), whereas if we kept seconds, minutes and hours with New Time the day would still be 24 hours, and the rotation period about 4 minutes less.

(Wrapping my mind around that was a bit confusing, so I went over this several times to make sure it is correctly stated but cannot absolutely guarantee that it is. So if it doesn't make sense, send me another query.)



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

QUESTION: Awesome! I hadn't considered that we might just change the definition of a second. That would, of course keep noon with the sun overhead, and - as you mention - throw so many industries and economies for a loop!

One thing I can't quite figure the math on: if the rate of slowing is so slow itself, how is it that in 2500 years the rotation has reduced by a fifth of a rotation, as per your website, and how do you get the 1/20 second discrepancy you mentioned at 5000 years ago? Is my calc just failing me right now?

Thanks!

Answer
The 1/20th of a second in 5000 years is a mistake (sorry). The actual amount is about 2 * 50 = 100 milliseconds, or about 1/10th of a second (or more accurately, since it is about 1.7 instead of 2 msec/day/century, about 85% of that).

The fifth of a way round the world (and I think a third of a rotation for the oldest more or less reliable eclipse report) is due to the fact that although the difference in any given day is small there are a lot of days involved, so the cumulative error is surprisingly large. For instance, in the last century the rotation has only slowed by about 2 milliseconds per day; but the cumulative error in a clock continually running at the rate the Earth rotated in 1900 is well over a minute(!) Over longer periods the total time error increases in the same way as compound interest, so the error can reach surprisingly large amounts in surprisingly "short" periods of time. It reminds me of the story of the fellow who was offered a king's ransom to do something, and asked instead to be paid only a penny for the first square on a chessboard, 2 pennies for the next square, 4 pennies for the following one, and so on. By the time you get to the end of the board the amount is staggering (unfortunately, once the sultan realized how he'd been tricked, he had the fellow beheaded, instead of rewarded).

Sorry about the mistake in the calculation in the first paragraph; I was looking for grammatical errors that would make it hard to understand, and missed the simple math error. I hope I didn't make a similar error in THIS reply, but don't have time to check it, as I picked this up just before leaving for an appointment. So if you find another mistake, please let me know.

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Courtney Seligman

Expertise

I can answer almost any question about astronomy and related sciences, such as physics and geology. I will not answer questions about astrology and similar pseudo-scientific rubbish.

Experience

I have been a professor of astronomy for over 40 years, and am working on an online text/encyclopedia of astronomy, and an online catalog of NGC/IC objects.

Publications
Astronomical Journal, Publications of the Astronomical Society of the Pacific (too long ago to be really relevant, but you could search for Courtney Seligman on Google Scholar)

Education/Credentials
I received a BA in astronomy and physics and a MA in astronomy, both from UCLA. I was working on my doctoral dissertation when I started teaching, and discovered that I preferred teaching to research.

Awards and Honors
(too long ago to be relevant, but Phi Beta Kappa and Sigma Xi still keep trying to get me to become a paying member)

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