<--
.. title: Counting Critters
.. date: 2011-06-28 00:19:00
.. tags: circle of life, scott bakula, jimmy johns
.. category: old
.. slug: counting-critters
.. author: Corky
.. has_math: true
-->
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[![image](http://4.bp.blogspot.com/-H4DbCndvDRw/TgfuQZt7aJI/AAAAAAAAAM8/WLfVRd4USK8/s320/marx_horse.jpg)](http://4.bp.blogspot.com/-H4DbCndvDRw/TgfuQZt7aJI/AAAAAAAAAM8/WLfVRd4USK8/s1600/marx_horse.jpg)
This picture allows us to set a lower bound on the number of [creatures](http://www.youtube.com/watch?v=9IrCgCKrv8U) that ever lived of \~4.
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We recently had a big book sale [1] here in town where books were being
sold for about a quarter. Needless to say, I bought far more than I'll
probably ever need or read. One of the books I bought was called
*General Paleontology* by A. Brouwer [2]. Anyways, I didn't really make
it too far in the book. In fact, I only made it to the first sentence of
the second paragraph of the first chapter, when I encountered this line:
*"The number of individuals which has populated the Earth since life
began is beyond estimation."* ** Horse feathers, I say! Horse Feathers!
The number of things that ever lived may very well be *unknowable*, but
it's certainly not beyond estimation. So below, Alemi and I each provide
an estimate for the total number of creatures that have ever lived on
Earth. We flipped a coin and I lost, so I guess I'll go first. My
estimation will be a genuine guess-y kind of estimate that doesn't draw
too heavily on too many physical considerations. Instead, I will
formulate a series of assumptions and base my final answer on that. So
assuming my assumptions are valid, the answer should give a reasonable
estimate to the total number of creatures that have ever lived. My
assumptions are as follows: (1) The number of individuals that have ever
lived will be almost entirely dominated by the number of bacteria that
have ever lived on Earth. So to leading order, all the life that has
ever lived on Earth is bacteria (or something similar). (2) Life began
at some time, T, in the past and immediately spread to all places on
Earth. Hey, man, it's the power of geometric progression! (3) The
majority of life is found within h = 1 m from the surface of water. I
picked this number since it's roughly (order of magnitude) how far down
I can see in really clear water. Most of the life will be photosynthetic
and thus need a fair amount of sunlight. (4) The number density of
organisms in water is n \~ 10^5. I have no real justification for this.
(5) The average lifetime of an organism is t \~ 1 hr. Alright, so if
these assumptions are valid (a big if [3]), then the following
prediction should be fairly accurate. So the total volume in which these
creatures may live is just the shell of the Earth down to about a meter:
$$ V = 4 \pi R_{\oplus}^2 h $$ where R = 6 * 10^6 m is the radius
of the Earth. Alright, so the number of creatures at any given moment
will be the volume times the number density which I will take to be n \~
10^5. That will give us the total number of creatures at any given
moment. But we want it for *all* the moments. So I will take the total
number of "generations" to be the time life has been around divided by
the average lifetime of a given organism. Putting this all together
gives $$N = 4 \pi R_{\oplus}^2 h \times \frac{T}{t} $$ Plugging in
some numbers I get: $$ N \sim 10^{39}
\left(\frac{h}{1\~\mbox{m}}\right)\left(\frac{n}{10^5\~\mbox{cm}^{-3}}\right)\left(\frac{T}{3\times10^9\~\mbox{yrs}}
\right)\left(\frac{t}{1\~\mbox{hr}} \right)^{-1}$$ So for the
nominal values I've plugged in, I'll get that about 10^39 creatures
have ever lived on Earth [4]. I've left my equation in a dimensionless
form above, so if you think my individual estimates are garbage, you can
easily plug in your own estimates to see how things change. Except for
the completely arbitrary factor for the average number density of
organisms per cubic centimeter of water, I feel alright about this
estimate. And I'm fairly confident that the number density will not be
off more than about 3 orders of magnitude either high or low. So my
final estimate is: $$ N \sim 10^{39 \pm 3} $$ I promised that there
would be two estimates, so I present below in picture form, Alemi's
back-of-the-wrapper estimate.
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[![image](http://2.bp.blogspot.com/-Eq37GVo3d8A/TggHx19J1pI/AAAAAAAAANA/Y0H3DTVb2so/s400/jj_phys.jpg)](http://2.bp.blogspot.com/-Eq37GVo3d8A/TggHx19J1pI/AAAAAAAAANA/Y0H3DTVb2so/s1600/jj_phys.jpg)
Click for the full Jimmy Johns experience
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To explain these scribbles, I now cede the floor (and the mic) to Alemi.
[ SEAMLESS TRANSITION ] So, when Corky posed this question to me while
we ate some tasty sandwiches, another approach came to mind. Namely, I
wanted to try to estimate the number of critters that have ever lived by
putting some kind of energy bound on the number. Ultimately, all
critters come from the sun. That is, all life on Earth is only able to
exist in so much as it consumes energy, and for almost all life
[ignoring the under ocean heat vent guys], the energy they consume, one
way or another comes from the sun. So, let's estimate the number of
critters in three parts (1) We need the energy the Earth recieves from
the sun. (2) We need to estimate the energy density of life (3) These
two things, combined with a characteristic length or volume scale for a
critter would enable us to figure out the rate at which the Earth could
produce critters. (4) Assuming this rate and a time scale for how long
life has been around on the Earth would give us a total number of
critters. Let's begin (1) Energy from the sun. Corky and I happen to
know that the solar flux on the sun is roughly 1000 W/m^2. Multiplying
this by half the surface area of the earth gives us a rough total solar
flux $$ (1000 \text{ W/m}^2) ( 2\pi R_{\oplus}^2) \sim 2 \times
10^{17} \text{ W} $$ (2) Energy density of critters For this we used
the bag of potato chips we had on hand, assuming that all life matter
has roughly the same energy content. The bag of chips was 150 calories
in a serving size of 28 grams. This and assuming that life forms are the
density of water gives us a life energy density $$ \left( \frac{ 150
\text{ kcal} }{ 28 \text{ g}} \right)\left( \frac{ 1 \text{ g}}{
\text{ cm}^3} \right) \sim 2 \times 10^4 \text{ J/cm}^3 $$ (3)
Length scale of critters We assumed that bacteria are the most abundant
life form, so we chose a length scale of 100 microns. Putting these
pieces together gives $$ \frac{ 2 \times 10^{17} \text{ W} }{
\left( 2 \times 10^4 \text{ J/cm}^3 \right) \left( 100\
\mu\text{m} \right)^3 } \sim 10^{19} \text{ 1/s} $$ Which is our
estimated critter creation rate (4) Time scale for life generation
Finally, we estimate that life creation has been chugging along on earth
for about 3 billion years, this gives us our final estimate for the
number of critters that have every lived $$ \left( 10^{19} \text{
1/s} \right) \left( 3 \times 10^9 \text{ years} \right) \sim
10^{36} $$ So, there we have it. If the Earth was 100% efficient at
converting solar energy into life, and that life is characteristically
the energy density of a potato chip and the size of a bacteria, we
should have had 1 billion billion billion billion critters ever. To make
us feel a little better we would like to tack on a 10% efficiency, since
we don't actually expect the Earth to be 100% efficient, and because 10%
seems to be the rule of thumb efficiency estimation used when it comes
to food chains and the like, so our final estimate, motivated purely by
physics is $$ \boxed{ \text{ Total number of critters ever } =
10^{35\pm 3} } $$ This number seems pretty good, and is in general
agreement with Corky's earlier method. Notice that the only parameter we
are a little worried about the is the length scale, especially because
our final answer depends on the inverse cube of this number, so, our
error is probably something around 3 orders of magnitude, as before,
since an order of magnitude goof in the size would cause 3 orders of
magnitude error in the final estimate. So there you have it. Two
not-egregiously-horrible estimates for the total number of critters that
have ever lived. All in all, I think that book was a quarter well spent!
Unnecessary footnotes: [1] This is a bit misleading. They actually sold
books of all sizes. [2] I like to read about paleontology and such just
in case I'm ever sent back in time. This way, I'll know what dinosaurs
are safe to eat. [3] Here's a bigger if: if [4] FUN FACT: The total
number of atoms in all the people on Earth is roughly 10^39. A proof of
this is left as an exercise for the reader.