The radioactive formula

 The math for calculating how old something is from a study of it's radioactive elements is often presented in a very obscure fashion when it is really quite simple.

A formula is a mathematical expression or 'sentence' that describes some phenomenon in nature (or not in nature).  There is a straight forward expression of any formula - the one that can be understood intuitively.  Then you can rearrange the formula (solve for) to make any of the other factors in the equation the subject.  These 'derived' formula are often less easy to understand intuitively. 

First, for any youngsters that haven't yet learned algebraic notation, let's divert for a moment.  Skip this if you already know algebra.

 

 Notation in Algebra

First why do we bother to make formulas with letters.  Why not just put in the numbers.  The reason is so that this gives us a general formula which can be used for the same phenomenon but for different numerical examples.  Also with a different letter representing each different factor, it is much easier to make any one of the factors the subject of the formula.

ab

First what does it mean when two letters are presented beside and touching each other.  For instance ab.  

This means you are to multiply the value of 'a' by the value of 'b'.  With letters, we don't separate them by a times sign such as axb because this can be confused for telling you to multiply a times x times b.  On the other hand if you are using numbers you can include the times sign so 7 x 9.  Alternatively, a dot is often used in algebra in the middle of the line like this 7^.9.  This notation says to multiply 7 by 9 while 7.9 with the dot at the bottom of the line means seven decimal (or point) 9 - nearly 8.

 

3a

How about 3a.  

This instructs you to multiply 'a' by 3 so it is the same as a+a+a.

 

a³

And then we have a³.  

This tells you to multiply 'a' by 'a' by 'a' which is also called 'a' to the third power 

a₃

We also have a_{3 also known as asub3}.  This isn't an operator.  It doesn't tell you to do anything.  Series are a powerful tool in math and this notation could mean the third 'a' in  a series.  You could also have a₀.  This could mean a at time 0.  We will use this just now.  Whatever the notation,  generally we described what each of the factors in a formula means at the beginning of the calculation unless it is such a well known formula that an explanation is not needed

The Physical Situation

Math is used to describe something in nature.  I fine it quite gob smacking that so many things in nature can be described by quite simple math.  In our case we are looking at the breakdown of a radioactive isotope.  Perhaps we should describe what an isotope is for the young ones.  There are about 92 separate natural occurring elements on the periodic table and a whole bunch more that are artificially created.  Their chemistry (how they react with other elements) and hence their identity is determined by the electrons whizzing about the nucleus and especially the outer shell of electrons.  In the nucleus are positive particles called protons and the number of electrons in an uncharged atom* exactly match the number of protons.  But there are also uncharged neutrons in the nucleus.  They have almost the same mass as a proton.  The sum of the protons and neutrons gives you the Atomic number of the element.  The Electrons are very light compared to protons and neutrons.  The number of  neutrons approximately equal the number of protons but can vary quite a bit.  The different forms of atoms of a given element, due to the different number of neutrons, are called isotopes of that element.  All the atoms of a specific element contain the same number of protons but the neutrons can vary.  Some of these isotopes are radioactive.  All this means is that they break down spontaneously into simpler, lighter elements. They aren't stable.   At break down, they give off alpha, beta or gamma radiation.  I'll tell you about them in an appendix.

* Perhaps you have heard of ions and might confuse an ion with an isotope.  An ion is an atom which is temporarily missing or has an excess of one or more electron.  Ions have a very strong tendency to gain or loose electrons to bring them back into equilibrium.  Actually ions have a part to play in the dating of radioactive isotopes as we shall see.

Here is where we get to the critical observation about these isotopes which allows us to date them.  Early researchers noted that if you observed a given radioactive element for a while, at some time in the future, half of it will be gone - changed into something else.  Nothing surprising here.  Then if you continued to observe this isotope, for the that same time period, half of what was left will be gone.  Starting with some initial amount, each time, that particular period of time (which is unique for each different isotope) elapses, you would have half, then a quarter, then an eighth, then a sixteenth and so forth until there is too little to observe.  This period has, not surprisingly, being called its half life.  Half lives vary from milliseconds to millions of years for different isotopes.

The Formula

So we can start to build a formula.  We will put each formula into words as well.

Suppose you know how much of a radioactive isotope you have today and want to know how much you will have after one 'half life' has gone by. 

A_1/2 = 1/2A₀  In words, the amount after one half life is equal to one half times the amount at time 0. This is what we observed in the physical world. So far pretty simple - no? Remember for this problem we define A_{1/2 to mean The Amount after one half life.  We could put 't' instead of 1/2 meaning the Amount after time t has gone by after the initial condition.}

Now suppose that two half lives have gone by.  We have to multiply again by 1/2 so we have A_t = (1/2)(1/2)A₀  Or we could write (1/2)²A₀.  ie. after two half lives one half squared times the Amount at time 0 which, of course, is one quarter as much as we started with.

We could go on like this or we could put n (number of half lives) in the formula so it becomes A_t = (1/2)ⁿA_{0.  In words, if you want to know how much of a radioactive element remains after a given time, raise 1/2 to the power of the number of half lives that have past and multiply this number by  the original amount.} 

Now we will have a short break and I'll ask you a question.  Suppose we have some radioactive isotope that has a half life of 5 (h = 5)years and 15(t = 15) years have gone by.  How many half lives have gone by and what fraction of the original isotope is left.

If you have understood the concept, you calculated 3(n = 3) half lives  and there is one eighth left of the original amount.   Putting this into letters, n = t/h or the number of half lives that have passed equals the time elapsed divided by the half life.  Since n= t/h we can put t/h where n appeared in our formula.

We now have A_t = (1/2)^(t/h)A₀  and that is the whole radioactive formula.  One of the main uses of this formula is to find t, the time that has elapsed since, say, Carbon 14 was incorporated into a plant or animal or perhaps the time since a rock melted and solidified and reset the clock.  Below a quick description of what is meant and then we will learn how to rearrange the formula to make t or h or A₀ the subject of the formula instead of At

Note that your little hand held calculator isn't bothered at all if 't' is not an even number of half lives.  Not an easy thing to do by hand without another branch of math called logarithms but the calculator takes it in its stride.

The physical situation

There are two main fields where radioactivity is used for dating.  First Carbon fourteen.

Carbon 14

The usual, (common) non-radioactive form of Carbon is Carbon twelve.  This means that the sum of the number of protons plus the number of neutrons in the nucleus is 12. In this case the number of neutrons and protons are equal.  That is to say, 6 of each.   This sort of Carbon is stable, it is not radioactive.  However, when high energy particles know as cosmic rays hit the atmosphere from outer space and high energy particles from the sun do likewise, some of the Nitrogen 14 is changed into carbon 14.  and this diffuses into the atmosphere.  It becomes part of the biosphere and any plant or animal takes up some of this radioactive Carbon along with the non-radioactive type.  The quantities are very small but the methods of detecting the relative proportions of radioactive and non radioactive carbon in a plant or animal are very very accurate.  I will describe them later.  

 

What this results in, is that any living animal or plant is more or less in equilibrium with its environment and to a close approximation all have the same proportion of radioactive carbon to non radioactive carbon in their bodies. 

 

However, when an organism dies, it is no longer taking up either form of carbon and the proportion of  Carbon 14 begins to decrease as it changes back into N14.  The half life of C14 is 5730 years and at a pinch, the amount can be measured with modern techniques to about 10 half lives.  Of course the accuracy decreases, the older the sample but this takes us back to about 50,000 years.  To put this into perspective, we can date materials back to half way into the most recent glacial period but not to the most recent interglacial, the Eemian which occurred about 125,000 years ago. 

 

Rocks 

Rocks can also be dated using radioactive isotopes.  Here you can do a wee experiment in your kitchen to illustrate the process.  Get some Copper sulfate which is a blue crystal.  Dissolve as much as you can in some water.  Now add sugar and dissolve as much of this as you can in the same water.  Pour off the clear liquid (colored blue) into a clean glass and suspend a piece of string that you have dipped in powdered crystals of both solutes.  Watch what happens.  The Copper Sulfate and the sugar will crystalize separately as the water evaporates so you will have sugar and copper sulfate crystals back again.  The sugar crystalizes with the sugar and the Copper sulfate with the Copper sulfate.  The same thing happens with rocks.  If you melt them and let them  cool and solidify, the separate mineral crystalize  separately.  Think of a granite rock.  melting re-sets the clock.  If there is a Uranium containing mineral in the melt, it will crystalize out separately.  Then it will continue to break down and the final result of a series of radioactive decays is Pb (Lead).  By measuring the relative proportions of U and Pb, you can date when the rock cooled from a molten state.  However we need to re-arrange the formula to make 't' the subject of the formula.  Let's do it.

 

Solving for 't'

We start with the basic formula  A_t = (1/2)^(t/h)A₀

      and we want to get 't' by itself and all the rest on the other side of the equation.  This is called 'solving for t'.  So let's divide both sides by A0.  Remember, we can do anything we want to an equation as long as we do exactly the same to both sides.  Of course, the trick is to choose the right thing to do.

               A_t/A0 = (1/2)^(t/h)

^{This cancels out Ao on the right side and leaves it in the denominator (bottom) of the left side.}  

^{Now we need a wee log identity.  I will give you a hint at the bottom of this blog of how logs work but for the moment, take my word for it that:}

^{log_abc = clog_ab}         Or in words, Log to the base 'a' of 'b' raised to the 'c'th power equals c times log to the base a of b.  ie You can move the power to the front.  I'll explain more about this in an appendix.  Just remember that we can do anything to one side of an equation if we do the same to the other side.  So the formula becomes.

log At/A0 = log (1/2)(t/h)   and this becomes

log At/A0 = (t/h)log(1/2)

 

Oh, I nearly forgot to mention something.  When you use the log function without any explanation, 'log' it always means to the base 10.  If you want it to a different base, you must note it and ln is the natural log to base 2.718. Both Log and ln are on your computer.

Now all we have to do is to divide both sides by log(1/2) to move log(1/2) to the other side and  multiply both sides by h to move h to the left side.  't' remains in glorious isolation on the right side.  It is conventional to put the subject of the formula on the left side so we can reverse them.  After all if a = b then b = a

Our formula for t then becomes

t = (logA_t/A₀₎

       (log1/2)

The Analysis System

  I suppose one could extract the lead from a rock and the Uranium and weigh how much of each there was.  These figures could be inserted into the formula to age a rock.  But that wouldn't work with Carbon.  You are comparing the amounts of Carbon fourteen with the amount of Carbon twelve and both have the same chemistry.  There is a better and very sensitive method that works much like the first television sets.  

In the early TV sets, the screen was one side of a large tube which tapered to the back behind the screen.  At the back of the tube was a filament which, when heated, gave off electrons.  These were accelerated in an electrical field and then passed through two sets of magnets oriented at right angles to each other.  A charged particle moving through a magnetic field is bent.  One set of magnets bent the beam of electrons back and forth horizontally and the other set up and down.  The magnets were varied in such a way that the beam of electrons sped back and forth over the inside of the screen, causing the layer of phosphorescent material on the inside of the screen to light up.  The electrical field was varied to give a stronger field which would provide a brighter point or less strong field which would give a darker point.  The 'refresh' rate was so fast that you saw moving picture.  

A machine to measure, for instance, the relative amounts of Carbon 14 and Carbon 12 works pretty well the same way.  The sample of carbon to be measured is heated so hot that charged atoms of carbon are produced.  This can be done with a laser.  The charged atoms are accelerated through an electrical field and then between electric magnets, one North and the other South.  This bends the beam of Carbon atoms.  The neat part is that the heavier atoms are bent less than the lighter atoms.  Where the beams hit the side of the container are detectors and the electronics connected to them can measure individual atoms.  The strength of the magnets can be varied to bend the beam more or less to ensure each beam hits a detector.  Clearly, this is a very accurate method of measuring very tiny amounts of Carbon 14 and Carbon 12.  The number of hits on each detector can be inserted into the formula.

Logarithms

I promised I would try to explain logarithms.  First a bit of notation.  If you see a notation log1000 = 3, it is understood that this is the log to the base 10.  If you use another base, you have to state it.  There is another notation ln.  It means the log to the base 'e'.  e is the base of the natural logarithms, whatever that means.  I don't understand it.  e is 2.71828.  It is known as Euler's number.  If anyone out there has a good explanation for the natural logarithms, please put it in the comments.  Anyway the two following notations are equivalent.

10⁵ = 100000

log 100000 = 5

You can see that the log of a number is the exponent that you have to raise 10 to in order to get the number.  Note again that it is understood that if you use the term log, it is understood that it is to the base 10.  You could also write it this way.

log₁₀100000 = 5

Carbon dating and the Math

One would have to be a hermit not to have heard about carbon dating.  This is the dating, for instance, of a piece of wood in an old building or a piece of charcoal in an archaeological dig.

At a first approximation, the physics is pretty straight forward.  An atom consists of a nucleus with electrons whizzing around the nucleus.  Which element the atom is depends on the number of electrons and the number of electrons, in turn, depends on the number of protons in the nucleus.  In a normal, unionized atom, the number of electrons and protons are equal and the atom is neutrally charged.

The glue that holds these positively charged protons together in the nucleus (remember like charges repel each other) are the neutrons.  Don't ask me how they do this.  The explanation is way above my pay grade.  Very roughly speaking, there are the same number of neutrons as protons but this can vary.  Carbon, for instance, can exist in a state with 6protons and 6 neutrons for an atomic mass number of 12. It can also exist in a form with 6 protons and 8 neutrons for a mass number of 14.

These two types are called isotopes of Carbon.  There is a third one but it is not needed for this explanation.

Some isotopes are stable, some are not (why is also above my pay grade).  In the case of Carbon, 12 is stable, 14 is not. 

Carbon 14 disintegrates into Nitrogen 14 with the ejection of an electron from one of it's neutrons.  The neutron becomes a proton so the atom is now a new element with 7 protons and 7 neutrons, hence ¹⁴N.

No one knows when any individual Carbon 14 atom is going to disintegrate.  There is a very small probability at any one moment but when you have a lot of ¹⁴C, you can predict how many atoms will change to ¹⁴N in any given time period.  This results in something interesting which has been observed experimentally.  If you know how much of the radioactive element you have, you will observe that half of it will break down in a given time, referred to as it's half life.  The half life of various radioactive isotopes varies from tiny fractions of a second to many millions of years.

In the case of ¹⁴C, it's half life is 5730 years give or take 40 years.

In 5730 years you will have half left, in another 5730 years, a quarter of the original amount, in one more half life, one eighth of the original amount and so forth.


So now we need the math for this.  We will work out what I call the straight forward formula and then we can change it around (solve for other parts) so that each component of the formula becomes the subject.

First a note on mathematical notation.

What is meant when you see a symbol.

xA means multiply the A by x.  If A is 2 and x is 3 then xA is 6

A^x means multiply A by itself x times.  If A is 2 and x is 3 then A^x is 8.  In words, A is raised to the xth power.

However, in the symbols A_x,  x is not an operator.  ie, it doesn't say to do anything.  It is a label.  It means the xth A.  For instance you could have A₁, A₂, A₃ etc.  This is the first, second and third A.  Or Ao and At which for our purposes will mean A at time zero and A at a specified future time.

There is a special one in Chemistry.  I'll use Carbon since this is what we are talking about.  For instance ¹⁴C.  This means the carbon atom with 14 nucleotides.   ie, The sum of neutrons and protons adds up to `14.  There also exist ¹²C and ¹³C.  Of course both have 6 protons or it wouldn't be Carbon.  The number of neutrons varies.

And one more in Math.  If the subscript is after the word log such as log₅ then it means log to the base 5.  If only log is used, it is understood it is to the base 10.   That is to say, log = log₁₀ and if ln is used it is to the base 'e'.  Don't worry about it, we don't need 'e'.  I only mention it because it is on your little hand held computer and you might wonder.

Lets go back to the basics.  Every half life period, (h) the amount is halved. In the case of Carbon, the half life is 5730 years but half lives for other isotopes varies hugely.   Lets call the amount we start with as A_o (A at time zero) and the amount we are left with as A_t (A at some time t in the future).  The amount we will have left after one half life is:

1.   A₁ = A_o(1/2)¹

After two half lives
2.    A₂ = Ao(1/2)²

After three half lives
3.    A₃ = Ao(1/2)³
Remember 1/2 times 1/2 is 1/4.   Multiply once more by 1/2 and you have 1/8.  When you see a times sign between fractions, replace it in your mind with "of".  then 1/2 x 1/2 becomes one half of one half.

The 1,2 and 3 are the number of half lives that have gone by.

4.  So A_n - A_o(1/2)ⁿ  or in words, to find the amount of a substance after n half lives have gone by, multiply A_o, the initial amount, times 1/2 raised to the nth power.

Note that in the notation A_x,  x means the amount at time x expressed in half lives.

Also note that even if the n is not a whole number and therefore would take a wee bit of higher math (knowing logarithms), to solve, your computer does this with no problem.  Your high school computer can solve, for instance, 6^3.22 without raising a sweat.

Suppose we start with one gram of a radioactive substance and one half life has gone by.  We simply multiply 1gram times 1/2

Suppose 4 half lives have gone back.  We multiply the one gram times (1/2)⁴.  that is to say by 1/2 times 1/2 times 1/2 times 1/2 which equals 1/16th times the original amount.

Now suppose we know what the half life (h) of a particular isotope is.  Say it is 10 years, for simplicity.  Say 30 years have gone by.  Obviously 3 half lives have past.  In other words n, the number of half lives equals the time elapsed (t) divided by the Half life (h).  In this case n = 30/10 = 3.

5.   n=t/h.

And, as I said, it doesn't have to be a whole number.  If the half life is 10 years and 75 years have gone by then n = 75/10 = 7.5.  With simple math we would have a problem raising a number to a fractional exponent but your computer has no such problem so don't sweat it.

You can see where this is leading.  Since n=t/h, we can substitute t/h into the formula where we see n.

The radioactive decay formula then becomes

6.  A_t = A_o(1/2)^t/h
or in words, to find the amount of radioactive material remaining after time t, multiply A_o, the initial amount, times one half raised to the power of t/h.


Good heavens!  I forgot to tell you where the radioactive Carbon comes from.  If it's half life is only 5730 years, in about 50,000 years there will be so little of it that carbon dating is out of the question and the world has been here for over 4b years.  Clearly, ¹⁴C must be being created somewhere.  the 'Where',, is in the upper atmosphere.  As cosmic rays hit the upper atmosphere, they are so energetic that they cause some nuclear reactions and one of these is to change some¹⁴N into ¹⁴C.  It is a very small amount but enough to be detected in living material with modern methods so we have a clock we can use.  When an organism dies it stops taking up carbon and the clock starts to tick.  If we  analyze it sometime in the future, we can know when it died (up to about 50,000 years).

Now we can do what a mathematician calls solving for A_o or for t or for h.  In other words we re-arrange the formula so that each of these terms in turn become the subject of the formula (ie. is by itself on the left and everything else is  on the right). I'll tell you what each variation of the formula is good for as we rearrange them.

The basic principle of solving for a factor (one of the letters) in a formula is that we can do anything we want to one side as long as we do the same to the other side.  After all if I have a formula that 7 = 3+4, if I multiply both sides by, say, 5, the formula is still correct.  Of course we don't just do random things to both sides of the formula. The trick is to do something that gets us closer to the solution we are looking for.

One other thing.  At one point in the procedure I am going to have to take a log of both sides.  Even if you don't understand logarithms, this should pose no emotional problem since I am doing the same to both sides.  Then, however, you are going to have to take my word for a 'log identity'.  If you are into logarithms, you will understand why the identity holds but if not, don't sweat it.  It is true.  This identity is:

log_ab^c = clog_ab.  Incidentally, the inverse of the left side of this formula is a^c =b.  That may give you a clue why the identity works.

In words:   log to the base 'a' of 'b' raised to the 'c'th power equals c times the log to the base a of b.

So let's start.  I want to end up with a formula for each of the terms, in turn, on the left side of the equation.

The original equation is

A_t = A_o(1/2)^t/h

Let's divide each side by (1/2)^t/h.  Note that this cancels out the (1/2)^t/h on the right side and leaves it on the left in the denominator*.  It is more conventional to have the subject of the formula on the left so we will exchange them.  After all if 7 = 3+4 then 3+4 = 7.  Our formula then becomes

* The bottom part of a fraction.

A_o = A_t divided by (1/2)^t/h. Don't know how to get my computer to write this so I will leave you to write it down on a piece of paper.

Use
So what is this formula good for.  It was noted early on in the use of carbon dating that there were some discrepancies.  With artifacts for which the exact date was known, the Carbon date did not agree.  The hypothesis was that the rate of ¹⁴C production in the upper atmosphere might not have been constant over the years.  So cores were drilled into very old trees, the rings were separated and carbon dated.  The above formula was used to work out the concentration  of carbon 14 which had been present for each year  that a ring was laid down.  And indeed it was found that the true curve diverged by a small but significant amount over time from the theoretical curve.  When the true curve was used, the dates all fell into place.
 

Now let's work on t and h.  The first thing I will do is to divide both sides by A_o.  This cancels A_o on the right side and leaves us with

A_t/A_o = (1/2)^t/h

Now I'll take the log of both sides

log (A_t/A_o) = log[(1/2)^t/h]

Remember our identity.  I can take t/h to the front of the right side so

log(A_t/A_o) = t/h(log1/2)

Now it is simple.  I simply divide both sides by log1/2 and we have t/h by themselves on the right side.  You take it from here.  Isolate t and h.  If you do it right you will find that

t = [hlog(A_t/A_o]/[log(1/2)]

and

h = [tlog1/2}/[log(A_t/A_o}

Use
How about the formula for t.  This is pretty obvious.  Now that we have the needed correction of the production of 14C over the past , we can date any object that was once alive up to about 50,000 years.  This is carbon dating.

Use
How about h.  We can't actually wait around for 5730 years to see when we have half of a quantity of radioactive carbon left.  We can, thought, observe the rate of disintegration on a shorter time span.  Using the h formula we can work out the half life of each radioactive isotope and some of them are multi millions of years.

It is never that easy

There are always complications.  Charcoal, for instance, if it is in ordinary soils or even in a cave can be colonized by micro-organisms.  If in active soil, the micro-organisms will have a modern carbon signature.  One has to first clean the charcoal of the modern material in order to get the correct date for the charcoal

Add to that, that we have been spewing carbon into the atmosphere from fossil fuel.  This is old carbon and hence contains no Carbon 14.  On the other side we have had nuclear tests in the air.  They have added Carbon 14 to the air.  For future anthropologists, they will have to take this into account.

Other types of radioactive dating have their own special requirements.  For instance when a rock melt cools, crystals form and just as a solution of salt and sugar, as it crystallizes, will  produce crystals of pure salt and pure sugar, the  crystals in a melt are of one type of molecule.  If one of these is a radioactive species and it's end product is known you can measure the concentraton of both and calculate when the rock  was melted. 

Continental Glacier Meltdown

Over the past 2.5m year ice age numerous glaciated periods (glacials) and warm periods (interglacials) have come and gone.  The most recent continental glaciers began to melt around 20,000 years ago and really got underway away about 11,000 years ago leaving ice sheets only in Antarctica and Greenland.  The end of glacials appears to be synchronous with one of the Milankovitch cycles; namely the variation in the tilt (obliquity) of the earth's axis.  At the beginning of the present  ice age, which we are in the middle of, the cycle was 41,000 years.  However over the past million years of the present 2.5 million year Glacial Age, only every third or so obliquity nudge has resulted in an interglacial.  Glacials over the past million years or so have been lasting on the order of 100,000 years.  Coincidentally with the melting of continental glaciers there is a sharp rise in Carbon dioxide  

<sub>As suggested in a previous blog it seems unlikely that some sudden source of Carbon dioxide would occur exactly in sinc with the Milankovitch cycle.  As odd as it seems it is more likely that the rise in CO2 is somehow the result of the melting.  Of course, once sufficient CO2 is released, a run-away melting will occur.  One likely positive feed back (warming causing more warming) is the ability of the oceans to hold less Carbon dioxide when they are warm than when they are cold.  As warming starts, presumably as a result of obliquity, the oceans could give out Carbon dioxide or at the very least, not absorb as much.

In a previous blog, I suggested that methane clathrates and carbon dioxide clathrates could accumulate under an ice sheet once it had thickened to a few hundred meters.  Sources of methane and Carbon dioxide include coal measures, liquid and gaseous hydrocarbon deposits and shale beds as well as the decomposition of organic material buried by the accumulating ice.  If the ice covered over an area of permafrost**, the methane stored in the permafrost could also find its way to the bottom of the ice sheet.   Ice sheets cause the depression of the land by about a third of a km for every km of ice added^# and this might well act as a natural 'fracting', increasing the escape of such gases from all the mentioned sources.  All this carbon would be sitting there  at the bottom of the ice sheet ready to be released if the continental ice sheet started to melt.  If sufficient was released in a burst, the green house effect could lead to a feedback, melting more ice causing the release of more gas and causing more melting etc.  This run-away greenhouse effects would only end when the ice sheet was all melted.  Following the melt, the slow ever present sequestering of carbon in the various carbon sinks would continue until it was possible for the accumulation of snow to start again.

# The basaltic rock on which the continents float has a specific Gravity (SG) of just over 3. Hence a km of ice with a specific gravity of  a tad under 1 would push down the continent about a third of a km.

**It is interesting to note that if snow is accumulating on an area of permafrost, the snow will insulate the underlying ground.  The 0 degree contour at the bottom of the permafrost will move upward as geothermal heat melts it.  Eventually, as the ice sheet deepens, all the permafrost, often rich in organic soil and methane clathrate, will be melted.  The permafrost undergoing anaerobic break down and any already stored up methane* would be available to combine with the bottom layer of ice and form methane clathrate.

 
In this blog, I would like to suggest  mechanisms which would explain why every nudge from the Milankovitch cycle does not end an ice age.  Lets do a mind exercise.  

Consider for simplicity a large continent like Australia.  It is shaped like a hockey puck, flat on top with very little slope in any direction.  Enough carbon has left the atmosphere and become sequestered in sinks for some snow to  last through the summer.    The process of going into a glacial is gradual.   The accumulation of snow can only proceed at the rate of precipitation in the area of accumulation minus sublimation and melting.  The process is somewhat accelerated by the albedo effect.  When a significant area is covered with white snow, incident light is mostly reflected back into space increasing the  cooling.

Snow occupies about 10 times as much volume as an equivalent weight of water but as the snow deepens, the weight of overlying snow on the bottom layers increases and air is squeezed out.  By the time there is a hundred or so meters of snow, the bottom layers have been squeezed into ice with some inclusions of air.  The ice at the bottom occupies about 10% more volume than an equivalent weight of water.


When the ice is only a few hundred meters thick it just sits there getting deeper and deeper.  The land is flat so it doesn't move down hill and there isn't enough pressure yet to squeeze ice outward.  At about 300m depth, there is enough pressure at the bottom of the ice layer for clathrates to form.  Any methane or Carbon dioxide which is coming from the underlying land combines with the ice and is trapped.

When the ice has reached a km or so in depth, the pressure is great enough that ice begins to be squeezed toward the edges.  Right in the middle of our hockey puck continent, there is no motion with respect to the underlying land.  As you go toward the edges, the motion is faster and faster.  Fast is all relative.  Even in glaciated continents such as Greenland with 3 or so km of ice at the center, the motion at the edges is only a few to a few tens of meters per year.  There are some individual glaciers which carry ice from the interior which are moving as much as 12km per year but this is down specific valleys and not along the entire perimeter of the glacier.   Averaged over the years, ice can only fall off the edges at the rate that it accumulates on top.   Since continental glaciers reach depths of at least 3km,  clearly, less ice was expelled than has was accumulated over the formation of the 3km deep ice sheet.   At some point, as the ice thickens, the rate of loss of ice will equal the rate of accumulation.  It is likely that toward the middle of the continent, the ice doesn't move at the bottom relative to the land but is rather squeezed out of the middle layers of the ice.  Toward the edges, ice would be moving over the ground.


Each Milankovitch nudge will probably result in some melting.  If it is correct that clathrates have been collecting at the bottom of the ice sheet, this will cause an increase in the output of green house gases  and the thicker the ice the faster this might occur due to faster rate of spread caused by the thicker ice. Also, the faster the ice is moving, the further into a melting climate the ice will be pushed.   This may be the explanation for the start of an interglacial only every few Milankovitch nudges.  Presumably a certain amount of green house gas is necessary to initiate a run away melting.  The older an ice sheet, the more clathrate could accumulate at the bottom and the thicker the ice sheet, the faster its borders are moving outward. Therefore, the older and thicker an ice sheet and the more clathrate it has accumulated, the greater the chance of a run away melting when an obliquity nudge occurs.

As a further factor, with a thick ice sheet, the glacier at the edge will be moving across the ground and expelling clathrate.  It would be expected that the concentration of clathrate would increase as you go toward the centre of an ice sheet.  As the ice sheet edge melts back, more and more clathrate breaks down into the atmosphere.  The thicker the ice, the more clathrate is likely to be stored under the ice and the faster the ice is moving at the edges.  Thick ice should be much less stable than thin ice.

Another factor which might be relevant is the heat coming out of the earth.  Although it varies widely from location to location, the temperature increases as you go down into the earth at about 25 degrees C per km.  Put a layer of ice on the ground and this heat has to work it's way up to the surface of the ice.  I haven't been able to find the factor for heat transmission in rock and in ice  in  order to compare them but for the sake of the argument let's say it is the same.  Let's also assume that the average temperature at the top of the ice sheet is -50C.  If the ice is 1km thick, the temperature at the bottom of the ice would then be -25degrees.  If the ice is 2km thick it would be 0 degrees.  If three km thick, +25 degrees.  Of course in this latter case this wouldn't be so.  The heat comes in contact with ice which melts at 0 degrees and absorbs a lot of heat doing so (latent heat of fusion).  Have a look at this link (maps half way down in the PDF file) which show calculations for the basal temperature of the Antarctic Ice sheet.

The result is that with over 2km of ice depth and given some time to reach equilibrium, there should be water at the bottom of the ice sheet.  Here we run into another wee codicil.  If, as was suggested in a previous blog,  methane clathrate has accumulated at the bottom of the ice sheet, it can stay frozen up to 18 degrees centigrade with sufficient pressure.  Whatever the actual case, the general principle is that with a sufficiently deep ice sheet, the bottom layer should be melting.  This may be another part of the explanation as to why only every three or so nudges by the Milankovitch cycle sets off an interglacial period.  A sufficient depth of ice has to first collect to cause heat from the earth to liquefy its bottom and increase its horizontal movement on this lubricating layer .  So what sort of evidence would support this hypothesis.


Test

     A) at the bottom of present ice sheets, the temperature should be around 0 degrees.
     B)  There should be lakes below deep ice sheets where the topography allows.
     C)  It should be possible in some locations at least, to detect methane and possibly Carbon dioxide being evolved from the edges of  ice sheets where they are melting.
     D) Where ice sheets exit into the ocean and are at least 30m above sea level (and hence their base is 300m below sea level) there may be clathrates on the bottom layers in some locations.
     E) Since the carbon released from the bottom of a 100,000 year ice sheet would be "old carbon" (in other words, carbon depleted in C14) There should be a dating anomaly from the end of the last ice period, 11,000 years ago*.  This sudden influx of old carbon into the atmosphere should make wood, growing after the melting, look older.  For wood from about 11,000 years ago, one might see successive growth rings from a tree looking older and older despite the fact that they each successive growth ring is younger than the previous one.  A place to find suitable wood might be in tropical swamps where a log had sunk into the anaerobic mud or high mountains where trees such as the Bristle Cone Pine exist.
    F) if the bottom layer of an ice sheet is composed of clathrates, you might find that even though a core found solid ice right to the bottom of the core, the temperature could be above 0 degrees.  Clathrates can exist up to 18 degrees centigrade with sufficient pressure.  In fact, you might find a liquid layer at the bottom of the ice with a clathrate layer below the liquid layer. 

Of course, if the bottom of the glacier is moving horizontally at locations where there is permafrost, it will be scraping off the permafrost layer and carrying it toward the edge of the ice sheet with it's entrained load of clathrates.  If the ice sheet is frozen to the base and is only moving laterally due to the middle being squeezed out, the clathrates will only be released when that part of the glacier melts. 

A last contributor to sudden melt down and release of Carbon dioxide is Moulins. Moulins are  vertical shafts which are caused by melt water pouring down fissures in continental glaciers.  At present this phenomenon is best observed on the Greenland ice sheet where there is increased melting each summer.  Pools of water form on the surface of the ice and if they find a fissure, they pour down to the bottom of the ice sheet.  This water has to come out somewhere and presumably it will find its way out at the edges of the ice sheet.  It would be expected that it would carry with it the material from the bottom of the ice sheet.  Part of this would be the clathrates that have accumulated there.  At each Milankovitch nudge there would be expected to be surface melting and a wash out of some of the bottom material.  If great enough, this would result in a run away green house effect. It should be possible to detect methane where streams come out under continental ice sheets.


All the above scenarios depend on the supposition that clathrates will accumulate under continental glaciers ready to be released when the glacier melts.  The longer the glacier exists, the greater the accumulation should be.   If this is indeed happening, it should be observable under our two remaining continental glaciers on Antartica and Greenland and even possibly Iceland.

*At a pinch, carbon dating can go back 50,000 years.  Hence it would be perfectly useful for dating objects from the end of the last glacial but wouldn't extend back to the previous interglacial which was 125,000 years ago.

In summary
The older and thicker an ice sheet, the more unstable it should be.  This may  be due to:
1) Higher temperatures at the bottom of thicker ice sheets than more shallow ice sheets due to geothermal heat being insulated from escape due to the insulating properties of the ice.  At a sufficient thickness a layer of water at the bottom of the ice sheet would accelerate its flow outward.
2)  A greater accumulation of carbon in the form of clathrates the longer a glacial lasts and hence the larger available green house effect if the ice sheet starts to melt.
3) A greater speed of spread at the edges of an ice sheet, the deeper the ice is, pushing ice into geographical areas where it will melt.  If this ice has got a bottom layer of clathrate, this will be entering the environment.  Above some critical amount of carbon added to the atmosphere, a run away green house effect would occur. A nudge by the Milankovitch cycle would release more methane from a thick ice sheet than a shallower one.
4) Outwash of bottom material by surface melt and Moulins at each Milankovitch nudge.  The longer the ice sheet exists, the more carbon there should be available to be washed out.

Note that methane is often quoted to be 20 to 30 times as effective a greenhouse gas as carbon dioxide.  This only holds on a 100 year basis.  Methane oxidizes in the atmosphere to Carbon dioxide with a half life of about 8 years.  If we look at the effect of, say, a cubic meter of methane over 100 years and calculate how much warming it will cause, it will cause 20 to30 times as much warming as a similar amount of CO2.  However, the actual strength of methane as a greenhouse gas is more like a hundred times as much as Carbon dioxide.  This is only important if methane is being introduced into the atmosphere in very large quantities (as seems to be the case now).


PS (Dec24, 2012)
A recent paper by German authors has shown that volcanic activity increases following strong ice melt.  This would also go some way to explaining the increase in carbon dioxide in the atmosphere following, rather than before ice melt.</sub>

 

Intelligent design - let it be taught

Intelligent design (the theory that the animals and plants we see around us were designed by a creator and are not the result of Darwinian evolution) should definitely be taught in school and taught as part of the science class. Scientists and science teachers are just as prone as anyone to rest on their laurels; to just assume everything they teach is true and to more or less teach their subject as revealed truth - in other words as a religion. Every scientific theory needs its sceptics and sceptics are every bit as important to the advancement of science as are the advocates of a particular theory.

A good dose of scepticism for Darwinian evolution from the intelligent design advocates would make each science teacher stretch a little further; to make that little bit more effort and to think of new ways of putting across his subject. It would make him work out ways of showing the evidence to his students rather than just jaw-boning the subject.

A warning to the Intelligent Design advocates though. If they come into the science class, the kid gloves are off. Often when a science type encounters someone with strong religious conviction, he will avoid confrontation. If the religious person expresses strong conviction, the science person will back off and keep quiet. I don't know why this is. There are probably a mix of reasons. Many of us have undergone some religious education as children and have been taught a degree of respect for religion. We all are all a little superstitious and blaspheming the lord feels a bit off for the most dyed in the wool atheist. Fair play, which we learn on the sports field may even play a part. Shooting down creationism is about as challenging as shooting fish in a barrel.

Whatever the reason, if the intelligent design people want time in the science classroom, they must not expect the same consideration they usually get. If they teach that the world was created some 5000 odd years ago, the children will be taken to a museum to see an exhibit of dinosaur skeletons. They will be taken to a nearby cliff face to see petrified varves from ancient glaciers. They will be taken to a Carbon 14 laboratory and the scientist in charge will explain how we can carbon-ate back almost 50,000 years, 10 times the religion-professed age of the earth and he will show a chronology of different artifacts which show these dates.  The student might be taken to a lab that studies the bristle cone pines and be shown that even they can go back some 5000 years.

If the Intelligent Design advocate tells how Noah saved two of every living thing on the earth, the science teacher will work out the estimated total weight of two of every known animal alive today with the food to keep them alive for the required time. A calculation will show what would be the minimum sized ship necessary to hold all these and the amount of dung Noah and his sons and daughters would have to clear up each day. Did they even have wheel barrows in Noa's time or would they have to transfer the dung to the side of the ship in baskets.

He will show his students the skeletons of a whole fauna of mega-mammals which evolved since the demise of the dinosaurs and which are no longer with us. He will show his students the skeletons of the wide variety of dinosaurs and ask how these two groups fits into these theory. And if the Intelligent Design advocate relates how man was created in god's image, all the information about an ever increasingly lineage of hominids which have been been discovered, which date over quite a few million years, including Lucy, Neanderthal man, Heidelbergensis, and so forth will be brought to the attention of the students.

The students will also be informed of some of the design boo-boos in the human frame from a poorly designed eye, through vestigial organs and an as yet to be perfected back bone. The science teacher may also talk to the students about a handful of other eye designs which are superior to ours. Some comments from the intelligent design advocate on where these beings fit into the picture would be enlightening.

A good start would be to allow one class each week by an advocate of intelligent design. Science teacher present of course. A main argument against this will be the huge amount of material that must be taught in the modern science class. This is true as far as it goes but it is far more important to teach the scientific method and the willingness to always listen to a contrary argument and to examine it on its merits. This is true science. It is just unfortunate that the intelligent design arguments are so easily shot down. A more robust opponent would make for  more robust debate.

At the end of the whole procedure, when little Johny asks the science teacher if he believes that god exists, the only possible 'scientific' answer is "I don't know". The lack of evidence for the existence of something isn't evidence of the lack of that thing. Imagine trying to explain radio waves to someone in Shakespeare's time. Radio waves are hard enough to explain now when we see the results in our portable transistor radio. The science teacher might feel moved to continue his explanation by saying that God could exist but if so in what form. You have thousands of choices from the wide variety of Christian faiths, from the other monotheistic religions, from animistic religions all the way back to the earliest ways man had of explaining a puzzling and often dangerous environment.

It would be just a touch egotistical to assume that the particular vision of god that the Intelligent Design teacher holds, is the correct one. One could even make the case that religion was an early form of science - a way of explaining our universe which, with limitations, gave a survival advantage to the group that advocated one creed over a different less useful creed. Religions have undergone Darwinian evolution and the varieties which conferred an advantage survived over those that didn't.

Creationism should definitely be taught in school.