Requesting Math Help/Feedback for my Web Page

I would like to have some help and feed-back about the math I am developing for one of my web pages.On my site, I just uploaded a new version of my Earth's Rotation is Slowing page which addresses the common creationist "leap second" claim. That claim was created 1979 (the year of its earliest occurance) and was soundly refuted in 1982, but it's still with us nearly four decades later.Here's the original version by Walter Brown in 1979:

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Atomic clocks, which have for the last twenty-two years measured the earth's spin rate to the nearest billionth of a second, have consistently found that the earth is slowing down at a rate of almost one second a year. If the earth were billions of years old, its initial spin rate would have been fantastically rapidso rapid that major distortions in the shape of the earth would have occurred.

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The actual rate at which the earth's rotation is slowing down is 1.4 to 2 milliseconds per day per century, meaning that after 100 years a day would have become longer by 1.4 to 2 milliseconds. Another statement of the actual rate is given as 5 milliseconds per year per year, which would make Brown's rate of 1 second per year per year about 200 times too great.Brown's mistaken rate is because he did not know what leap seconds are nor what purpose they serve. Another problem with his claim is that the dire consequences he gives for his inflated rate are pure hand-waving with absolutely no indication that he had performed any actual calculations. I traced one line of descent from this claim in which the next link, R. L. Wysong (1980), employed hand-waving to describe the ancient earth as spinning so fast that it would be a "flat pancake", but he left out what the rate of slowing down was supposed to be. The rest of the versions of the claim descended from that one and leadin up to Kent Hovind were just restatements of Wysong's version devoid of any actual rate and consisting purely of hand-waving.However, Hovind also has a second version, a long version from his seminar videos, based on a different rate he got from his local newspaper: one millisecond per day per day. He also includes the calculated length of a day 6,000 years ago based on that rate, which I calculate to be over 18,000 times greater than the actual rate (which I determined to be about 55 nanoseconds per day per day).There's more information and links to sources on my page at http://cre-ev.dwise1.net/earth_rotation.html.I should also point out that CRR had messaged me with complaints about the previous version of that page, many of them spurious. However, he was correct in that I had made a mistake in my description of Walter Brown's claim, though I found that the statement in question was very much in line with Hovind's second claim, though I was off that by about a factor of two -- it's covered in the text and also in the caveat at the top of the page.Among other things, that raised a more general set of questions about the mathematics involved in these claims:

1. How do the different claimed rates really compare? They're all in different units, so I will need to convert the rates to common units in order to properly compare them. Therefore, I will need to ensure that I perform those unit conversions correctly.
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2. What are the consequences of those different rates? The treatments of this claim that I've found which present actual calculated values deal in the length of a day or the number of days in a year when we go this far into the past. I will need to ensure that I perform those calculations correctly. There might also be a question of whether there's a better way to evaluate a claimed rate.
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3. So, are the models that I will be working with valid and are there better models to use? For example, a creationist inflated rate can lead to a day of length zero a few millions of years ago, which seems to indicate that that approach may be flawed or at the very least rather limited.
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   For example, I could start with the earth's current rotational velocity (about 7.2921150×10^-5 radians per SI second) and translate the rate of slowing to the same units. Then I could calculate the earth's rotational velocity at any point in the past. This would also require a method for converting rotational velocity to the length of a day then.
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4. Also, my approach will be primarily algebraic. If the problem lends itself more to calculus or to another mathematical method, you could point that out to me.

At present, my "Doing the Math" section is only a statement of what I plan for that section and that I have a lot of work to do in preparing it.Just to intercept an obvious point: Yes, I do realize that the rate at which the earth's rotation is slowing is not a constant rate. First, the earth's rotation varies day-by-day due to many different factors, sometimes even speeding up. It is only over a long period of time that we have that rate of 1.4 to 2.0 milliseconds/day/century. But even then, geological evidence shows that even that rate has varied in the past (in part due to the distribution of the continental land masses, as I understand it). I feel justified in this discussion to assume a constant rate going back billions of years because that is the very same assumption that the creationists make in their claim.I look forward to working with any interested parties on this project. I intend for this message to be the topic proposal. I will follow it immediately (ie, before topic approval) with the first part of the problem: unit conversions of the rates.I have written my first message describing the unit conversion methods I've been using so far. I could post it either before or after this topic has been approved. For the moment, I will wait until after.

Thread copied here from the Requesting Math Help/Feedback for my Web Page thread in the Proposed New Topics forum.

Since I am not versed in LaTeX, my mathematical notation will be a mix of ASCII and HTML. I hope that does not cause any confusion.First, I'd like to address the problem of unit conversions. All the rates are given in different units -- eg, seconds per 12 months, seconds per 18 months, milliseconds per year, milliseconds per day, milliseconds per day per century, milliseconds per year per year -- so that you cannot compare the various rates to each other directly. I'm not taking apples and oranges here, but rather fruit cocktail!Here are the rates that I will be working with:

1. Actual rate as per USNO: 1.4 to 2.0 ms / day / century. I nominally go with the 2.0 value as a kind of upper-bound worst-case figure.
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2. Thwaites and Awbrey: 5 ms / year / year. They used this rate in their 1982 refutation of Walter Brown's claim (As the World Turns: Can Creationists Keep Time? by William M. Thwaites and Frank T. Awbrey, Creation/Evolution, Issue IX, Summer 1982, pp.18-22), but I'm not clear where they got it from. Using the unit conversion method I've worked out, it does agree with the actual rate, 1.4 ms / day / century.
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3. Walter Brown: 1 sec / year / year. "the earth is slowing down at a rate of almost one second a year."
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4. Kent Hovind: 1 sec / 1.5 year / 1.5 year. Given in his seminar video as "a leap second about every year and a half." Even though he does nothing with that figure, adding one leap second every 18 months is what should happen in theory, so this rate should be included.
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5. Kent Hovind: 1 ms / day / day. From his seminar video: "{The earth} is actually slowing down 1000th of a second everyday." His source was the local newspaper, Pensacola News Journal (1990 Dec 06).

Let's start by converting ms/day/century to ms/year/year.First, let's define 1 year as 365.25 days, which is what I understand is the value normally used in astronomy even though it's about 11 minutes too large for calendar purposes.Now, how to interpret that for unit conversion? I read it as (ms/day)/century, which would be (ms/day) × (1/century), which would in turn be ms/(day × century). I see some justification for that in that acceleration written as m/s/s becomes m/s².So then, we multiply the units successively by 1:(ms/(day × century)) × (century / 100 year) = (1/100) × (ms / (day × year))(1/100) × (ms / (day × year)) × (365.25 day / year) = (365.25/100) × (ms / year²)(365.25/100) × (ms / year²) = 3.6525 ms / year / yearUsing that as a conversion factor:1.4 ms / day / century => (1.4 × 3.6525)ms / year / year = 5.1135 ms / year / year.If we were to go with the upper bound:2.0 ms / day / century => (2.0 × 3.6525)ms / year / year = 7.305 ms / year / year.Since 5.1135 ms / year / year very nearly equals 5 ms / year / year, that would confirm Thwaites and Awbrey's rate as being equivalent to the actual rate from the US Naval Observatory.Does everybody concur?Converting Walter Brown's rate is simple:
(1 sec / year / year) × (1000 ms / sec) = 1000 ms / year / yearKent Hovind's rate is a bit more involved, but not by much:1 sec / 1.5 year / 1.5 year = 1 sec / (1.5 year)² = (1 / 1.5²) × sec / year / year(1 / 1.5²) × sec / year / year = 0.4444 sec / year / year = 444.4 ms / year / yearKent Hovind's other rate is even more involved:
1 ms / day / day = 1 ms / day²(1 ms / day²) × (365.25 day / year)² = 365.25² ms / year²365.25² ms / year² = 133.4 sec / year / yearDoes everybody concur? Did I combine too many steps? I should lay it out more explicitly step-by-step when I publish.Now, here are the converted rates we just determined and how they compare:

1. USNO: 5.1135 to 7.305 ms / year / year. This one is golden since it comes straight from the Time Lords at USNO, NIST, and BIPM.
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2. Thwaites & Awbrey: 5 ms / year / year. Compared to USNO, it is either 0.6845 or 0.9778 what it should be. A little low, but fairly close to equivalent to the actual rate.
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3. Walter Brown: 1000 ms / year / year. 136.89 to 195.56 times greater than the actual value.
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4. Kent Hovind: 444.4 ms / year / year. 60.835 to 86.9 times greater than the actual value.
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5. Kent Hovind: 133.4 sec / year / year. 18,261.5 to 26,087.8 times greater than the actual value.

Next step would be to use those rates to calculate how long a day would be at various times in the past. The only rates that have been used for that purpose in claims or articles are Thwaites & Awbrey's and Hovind's second rate (133.4 sec / year / year). Walter Brown never uses his rate to perform such a calculation, but rather just hand-waves his conclusions at us, and Hovind doesn't use his first lower rate at all.So then, comments? Criticisms? Suggestions? Edited by dwise1, : Corrected comparison of USNO rates to Thwaites and Awbrey's rate.

Keep talking nerdy....its sexy!

I don't comprehend what a "per day per century" would be.You do clarify things in the second part of the sentence - "meaning that after 100 years a day would have become longer by 1.4 to 2 milliseconds."Moose

Although here the units don't match (day vs century) that can be easily corrected by converting either to the other so they match.But dwise1 is only writing out the long way what can be considered an "acceleration."For example:
9.8 m/s^2 = 9.8 meters per second per secondExcept, if these are all 'slowing down' then, really, it should be a deceleration and therefore have a negative sign in front of it.Other than that, I didn't really get into the actual maths

I agree that the format makes it appear to be an acceleration, but I'm not so sure that it isn't more of a velocity. Part of this effort is to work out those definitions so that we can examine and explain just exactly what's happening. FWIW, I'm certain that the creationists using this claim have no clue themselves, so yet again we have to do their work for them.However, I did take advantage of the equivalent ways to express acceleration; eg, meters/sec/sec versus meters/sec².While the phenomenon in question is the long-term deceleration of the earth's rotation, what we are measuring it by is the increasing length of the day. For that reason, these rates are positive.That also means that what we're expressing is the rate at which days are lengthening, that rate being constant in the long term, at least over the past several millennia. Geological evidence shows that it has been different in the distant past.My main goal in this first step is to get everybody's rates into the same units so that we can compare them directly. Then we can work out valid methods for calculating the length of days in the ancient past. Yes, I know that it has changed, but the creationists are assuming the rate to have been constant for billions of years, so I'm operating on the same assumption in order to compare my results with theirs.Have to rush off to work. Will be able to post something during lunch but I won't be home until late tonight.

Well, to begin with, I had gotten turned around in comparing Thwaites and Awbrey's rate with the golden USNO rates. I have gone back and corrected Message 3 and that correction is reflected in the quote box above. Sorry about that.But the reason for this message is that, if our ultimate goal is to find the length of a day a given number of years ago, then my target units were wrong: instead of ms/year/year they should have been ms/day/year.Quickly converting the above by multiplying the top two terms by (1 year)/(365.25 day):

1. USNO: 0.014 to 0.02 ms / day / year. This one is golden since it comes straight from the Time Lords at USNO, NIST, and BIPM.
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2. Thwaites & Awbrey: 0.0137 ms / day / year. Compared to USNO, it is either 0.685 or 0.97857 what it should be. Still close enough to equivalent to the actual rate.
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3. Walter Brown: 2.738 ms / day / year. 136.9 to 195.57 times greater than the actual value.
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4. Kent Hovind: 1.2167 ms / day / year. 60.835 to 86.9 times greater than the actual value.
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5. Kent Hovind: 365.23 ms / day / year. 18261.5 to 26087.857 times greater than the actual value.

Now, let's try a back-of-the-envelope calculation of how much shorter a day should have been one million years ago using each rate.Assume a current day length (at least circa 1900) of 86,400 seconds (24×60×60). Let r be a rate from the list above. Therefore, the day-length from a given number of years ago should be given by the function: Using that function, the retrodicted day lengths a million years ago would be:

1. USNO: 0.02 ms / day / year ==> 86380 sec -> 23:59:40.
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2. Thwaites & Awbrey: 0.0137 ms / day / year ==> 86386.3 sec -> 23:59:46.3.
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3. Walter Brown: 2.738 ms / day / year ==> 83662 sec -> 23:14:22.
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4. Kent Hovind: 1.2167 ms / day / year ==> 85183.3 sec -> 23:39:43.3.
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5. Kent Hovind: 365.23 ms / day / year ==> -278830 sec. This rate already has us well past the point of spinning impossibly fast.

Now let's try a billion (10⁹, known to long-scale folk as a milliard) years ago:

1. USNO: 0.02 ms / day / year ==> 66400 sec -> 18:26:40.
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2. Thwaites & Awbrey: 0.0137 ms / day / year ==> 72700 sec -> 20:11:40.
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3. Walter Brown: 2.738 ms / day / year ==> -2651600 sec. This rate already has us well past the point of spinning impossibly fast.
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4. Kent Hovind: 1.2167 ms / day / year ==> -1130300. This rate already has us well past the point of spinning impossibly fast.

OK, we've now eliminated all the creationist rates, since they all bottom out long before a billion years ago. However, these kinds of results aren't very satisfying, because we've got to try to figure out how to interpret negative day lengths -- kind of like trying to figure out what negative mass is supposed to mean when you take relativity calculations above the speed of light. What they are telling me is that I need to find a different way to perform these calculations, like maybe looking directly at the rate at which the earth is spinning (degrees or radians per second) and finding and applying its deceleration (degrees or radians per second per year), then translating that to a day length.With the two rates that are left, let's go for broke and try four billion (4.0×10⁹) years ago:

1. USNO: 0.02 ms / day / year ==> 6400 sec -> 1:46:40.
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2. Thwaites & Awbrey: 0.0137 ms / day / year ==> 31600 sec -> 8:46:40.

I also checked out 4.5 billion (4.0×10⁹) which would bring the USNO upper figure that I've been using to -3600, hence less than a second, while the T&A rate would give us just under 7 hours -- please note that I've been using the higher figure for the USNO rate as a kind of worst-case and that the lower figure for the USNO rate is only marginally greater than the T&A rate.In their refutation of this claim (As the World Turns: Can Creationists Keep Time? by William M. Thwaites and Frank T. Awbrey, Creation/Evolution, Issue IX, Summer 1982, pp.18-22), they offer their formula based on their rate of 0.005 sec./year/year:

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Dprevious + (N x 0.005) = Dnew

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D: stands for the difference between the perfect clock and the earth (a clock that is gradually slowing down)
N: is the number of years that the perfect clock and the earth have been allowed to drift apart 0.005 is the measured slowing rate for the earth

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Their retrodicted rate 4.6 billion years ago is given here:

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Before we all join Brown as young-earthers, however, we should realize that Brown's deceleration value of one second per year per year is much greater than the accepted value of 0.005 second per year per year. Brown is off by 20,000 percent for two-hundred-fold! If one extrapolates back in time 4.6 billion years with the accepted estimate of 0.005 second per year per year, one gets a fourteen-hour day. This means that objects at the equator would have been traveling at rates considerably less than the escape velocity. The effect of such a spin rate can be seen with the planet Jupiter. It spins on its axis in ten hours and is only slightly oblate-hardly anything like the flattened earth to which Brown alludes. Hence, the earth's observed spin deceleration rate does not falsify the notion that the earth is 4.6 billion years old.

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So part of my task is to also verify their calculations.I will also plan on a better set of "years ago" to include ones for which we have geological evidence of the length of the day; eg, 400 days in the year in the Devonian, about 400 million years ago.

You're measuring the rate of change of the length of the day with respect to time; a velocity is change in position with respect to time. A change in the length of the day over time means a change in the speed of the earth's rotation over time, and of course the rate at which speed changes is acceleration.

I have about five examples of the claim from creationists, only two of which offer an actual rate (Brown and Hovind, with Hovind providing two different rates) and only one of which shows any evidence of having actually attempted to calculate the effects of his rate (Hovind).I would appreciate hearing about other claims that provide rates and especially such claims that state how long a day would have been x years ago (or how many days would have been in the year, from which we should be able to derived the day length).Of course, bibliographic references for such claims would help immensely.

"You're measuring the rate of change of the length of the day with respect to time; a velocity is change in position with respect to time. A change in the length of the day over time means a change in the speed of the earth's rotation over time, and of course the rate at which speed changes is acceleration."Not really. What is being discussed is the rate at which the length of a day is increasing. It does not make sense to call such a thing an acceleration.We would instead talk about the rate at which the earth's rate of spin is decreasing. The spin is an angular velocity. It would make sense to call the rate at which an angular velocity is increasing acceleration.

Curiously I have an entry on just this in Message 7 of my The Age of the Earth (version 3 no 1 part 3) thread:

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Coral Growth and Geochronometry⁽³⁾

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The two chief approaches to geochronometry are based on radioactive isotopes and on astronomical data. The most recent estimates 2 of geological time based on the rates of radioactive decay give the Cenozoic a length of 65 million years, the Mesozoic 165 million years, and Palaeozoic 370 million years, and so on. The beginning of the Cambrian is placed about 600 million years ago. To accept these figures is an act of faith that few would have the temerity to refuse to make. The other approach, radically different, involves the astronomical record. Astronomers seem to be generally agreed that while the period of the Earth's revolution around the Sun has been constant, its period of rotation on its polar axis, at present 24 h, has not been constant throughout Earth's history, and that there has been a deceleration attributable to the dissipation of rotational energy by tidal forces on the surface and in the interior, a slow-down of about 2 sec per 100,000 years according to the most recent estimates. It thus appears that the length of the day has been increasing throughout geological time and that the number of days in the year has been decreasing. At, the beginning of the Cambrian the length of the day would have been 21 h (ref. 1). After the first, a second act of faith is easy, and we can develop a simple relation between the geological time-scale and the number of days per year (Fig. 1)

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Accepting this relation, in the absence of evidence to the contrary, it now follows that if we can find some means of determining the number of days per year for the different geological periods, we have a ligation between the results of geophysical and astronomical deductions. If we could determine, for example, that the number of days in the Cambrian year was of the order of 420, it would seem to confirm an indicated age of 600 million years based on tidal friction and in turn that since isotopic methods give the same result they too are in harmony. Is it possible to obtain estimates of the number of days per year from data preserved in recent and fossil organisms ? I would like to think so.

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... Physiological studies on calcium carbonate secretion in recent corals, first made some years ago by Kawaguti and Sakumotos 6, and recently and more elegantly by Goreau 7, who used the isotope calcium 45, show that in reef corals the rate of calcium carbonate uptake in coral tissues falls at night or in darkness and rises during the day, but the reflexion of this in the skeleton ton has not yet been investigated. A much less-sophisticated approach would be to compare the number of ridges with the annual growth-rate; but this has not been directly tried on living corals in the field. I have tested it indirectly on one or two recent corals the annual linear growth-rate of which is fairly well known, and to my gratification found that the number of ridges on the epitheca, of the living West Indian scleractinian Manicina areolata (Fig. 5) hovers around 360 in the space of a year's growth. This strongly suggests, subject to experimental confirmation, that the growth-lines are diurnal or circadian in nature. It may be noted in passing that they may provide a much more sensitive caliper for measurement of annual growth-rates than the larger yearly simulations.

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The next step, of course, is an attempt to determine, the number of growth-lines per annum in fossil corals. Here, as is usually the case, hypothesis is easier than practice. Few fossil corals are sufficiently well preserved to show clearly the supposed diurnal growth-lines, and it is not easy to determine the annual rate. In epithecate recent corals the growth-lines are commonly abraded or corroded even before death of the polyp. The best of the limited fossil material I have examined so far is from the Middle Devonian. of New York and Ontario, especially specimens of Heliophyllum, Eridophllum .(Fig. 6) and Favosites. Diurnal and annual growth-rates vary in the same individual, adding to the complexity, but in every instance there are more than 365 growth -lines per annum. usually about 400, ranging between extremes of 385 and 410. It is probably too much, considering the crudity of these data, to expect a narrower range of values for the number of days in a year in the Middle Devonian; many more measurements will be necessary to refine them.

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A few more data may be mentioned: Lophophllidium from the Pennsylvanian (Conemaugh) of western Pennsylvania gave 390 lines per annum, and Caninia from the Pennsylvanian of Texas, 385. These results imply that the number of days a year has decreased with the passage of time since the Devonian, as postulated by astronomers. and hence that values of the isotopic dates of the geophysicists agree well with the astronomical estimates of the age of the Earth. ...

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So these samples with diurnal growth rings showing faster spin and more shorter days per year, right in line with astronomical predictions.

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Coral-algae metabolism and diurnal changes in the CO2-carbonate system of bulk sea water⁽⁴⁾

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Abstract

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Precise measurements were conducted in continuous flow seawater mesocosms located in full sunlight that compared metabolic response of coral, coral-macroalgae and macroalgae systems over a diurnal cycle. Irradiance controlled net photosynthesis (P_net), which in turn drove net calcification (G_net), and altered pH. P_net exerted the dominant control on [CO₃²⁻] and aragonite saturation state (Ω_arag) over the diel cycle. Dark calcification rate decreased after sunset, reaching zero near midnight followed by an increasing rate that peaked at 03:00 h. Changes in Ω_arag and pH lagged behind G_net throughout the daily cycle by two or more hours. The flux rate P_net was the primary driver of calcification. Daytime coral metabolism rapidly removes dissolved inorganic carbon (DIC) from the bulk seawater and photosynthesis provides the energy that drives G_net while increasing the bulk water pH. These relationships result in a correlation between G_net and Ω_arag, with Ω_arag as the dependent variable. ...

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This confirms how and why there are diurnal growth rings in corals.

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Earth's rotation: Change in rotational velocity⁽⁵⁾

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Over millions of years, the Earth's rotation slowed significantly by tidal acceleration through gravitational interactions with the Moon. In this process, angular momentum is slowly transferred to the Moon at a rate proportional to r^-6 , where r is the orbital radius of the Moon. This process gradually increased the length of day to its current value and resulted in the moon's being tidally locked with the Earth.

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This gradual rotational deceleration is empirically documented with estimates of day lengths obtained from observations of tidal rhythmites and stromatolites; a compilation of these measurements[41] found the length of day to increase steadily from about 21 hours at 600Myr ago[42] to the current 24 hour value. By counting the microscopic lamina that form at higher tides, tidal frequencies (and thus day lengths) can be estimated, much like counting tree rings, though these estimates can be increasingly unreliable at older ages.[43]

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And 21 hour days would mean 24x365.24/21 = 417 days per year.

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So it looks like your calcs are in the right ballpark.Enjoy