We're going to calculate the blast
radius of an atomic bomb.
Fortunately without dropping any atomic
bombs, because that's one way to do it.
And that was exactly how the US
military were doing this;
1940s Manhattan project, Los Alamos, all
of that part of history;
interesting things happening in the
physics and maths world. The US military
are doing all these experiments and
were making measurements
of; as they changed the the size of the
bomb, or certain weather conditions,
as to how fast that, sort of, mushroom
cloud would expand.
Of course, quite useful information if
you're dropping a bomb, you know,
potentially near your own
soldiers or there are certain areas you
want to avoid - you want to have an idea
how much that's going to spread out.
Pretty important information.
And as - because atomic bombs were so new
back in the 1940s we didn't really know this,
so it did require this like trial and
error approach. Or
it required the clever mathematical
approach. And so this is a story of GI Taylor,
a British mathematician, who was asked by
the UK government to help out,
during the world wars, so he actually
helped out in World War 1 and World War 2.
In um World War 2 he was sent over as
part of the British delegation
to go and visit the test site, to visit
the Manhattan project. And he actually
saw the first
ever blast, the trinity test in 1945 he
actually witnessed.
One of - I think there were only 10 people
that were there and one of them was this
mathematician GI Taylor. What he did,
using some photographs taken from the
explosion,
was actually calculate the formula for
the blast radius
of the atomic bomb. Which, at the time,
was a US military secret.
So GI Taylor did these calculations and
just published this
in mathematical research journals, and it
was
quoted as pretty embarrassing for the
US military at the time, because this
was like their most closely guarded
military secret
about the inner workings of an atomic
bomb; and then this British mathematician
comes along and just kind of, you know,
does a pretty straightforward
calculation, as we'll see,
to just get the answers basically. So
it's a very very cool story
and also uses possibly my favorite tool
in all of applied maths, which is that of
scaling
analysis. It's very very applied maths,
should stress this, because you kind of go into
physics
and think about what is happening in a
problem,
to then write down the important
parameters in your problem,
and then combine them in such a way to
get the answer you want.
So you see, you do have to have that sort of
very applied physicist-like mind to be
able to use this, but it's such a good
tool because
you can calculate the formula for an
atomic bomb without setting off a single
atomic bomb.
And so the way Taylor did it; first of
all what
factors are going to influence the
spreading rate
of our bomb blast? First thing, at least
that came to my mind and to Taylor's
mind, was the energy.
You have a bigger bomb it's gonna blow
up more stuff.
It's reasonably straightforward to see.
So we don't know whether that
relationship is linear or anything yet,
but we know that the energy of the bomb
is going to affect that blast radius. So
we're going to write down
energy, we're going to label E. That's
definitely one of the parameters that's
going to be in our problem.
Now the other one that's reasonably
obvious is time.
If you've just set your bomb off it's not
gone very far, if you wait a few seconds
it's gone a lot further. We're trying to
work out the radius of the blast,
so as time increases the radius will
certainly increase. So we're going to
have time, which we label with
t. And now the sort of real fluid
dynamics and physics comes into it and
Taylor was a fluid dynamicist in
Cambridge, and this is where this really
helped him. Because he thought about the
problem as follows: you have
as the bomb detonates, you have a huge
amount of energy
being released from a tiny tiny volume.
And so you have a very, almost
instantaneous explosion,
from tiny tiny volume to massive amount
of energy. And what that's going to do
is create a massive massively high
pressure
within that explosion, that sort of cloud
of your bomb. So you can kind of think of
it as if this is
the sort of initial explosion; in here, if
you
take this to be pressure, this is really
high
inside this region. And as soon as you
get out of the bomb region
the atmospheric pressure is really low.
And you almost have like a,
what we would call a shock - this is where
the term shock wave
comes from, the shock wave of a blast.
Because here you've got super high
pressure,
from this giant explosion, and here
you've just got atmospheric pressure. And
these are like
orders of magnitude different. What this
tells you
is not what is important in the problem
but what is not
important. And this is the, sort of the
fluid dynamics, the physical intuition
of these problems; and Taylor was a fluid
dynamicist, this is why he knew this.
So because this is so high and this is
so small,
the pressure of the air is not going to
play a role
in the expansion of the bomb. So whether
it's high pressure or low pressure
weather system,
it's not actually going to matter. So it
tells you that pressure is crossed off.
If the air pressure isn't important, then
you think well what other properties of
the air
are going to affect the spread of the
bomb? You would expect there to be a
difference in the spreading rate
depending on the fluid or the air that
you're moving through. The other
possibilities are density
and maybe temperature. We have an
equation relating these two, it's called
the Ideal Gas Law,
and you often will model air as such and
it says that these two,
if the pressure is constant or
negligible, these two just depend on each
other. So he just picked density.
So this, sort of, physical intuition
told you pressure didn't matter and that
meant it had to be
density. That was the key insight that
came from Taylor being a fluid
dynamicist. - (Brady: the pressure of the-)
(this atmosphere doesn't matter, but)
(doesn't this pressure matter?) - This
pressure will absolutely matter, but this
will basically be controlled by the
energy.
This is the, the thing with scaling
analysis, and it's a very good question
because
there are so many different things you
can pick, and sometimes it is almost
like black magic that you pick the right
ones and put them together in such a way.
So I'm sure there are many many examples
of scaling analysis not working,
but this just happens to be like
possibly my favorite example of it
working and discovering a military
secret. So
we've got energy, we've got time, and then
we've got density. Density of the outside
air
which we label with rho. The scaling
analysis is what comes in here. Because
what you do
is you look at the units of your three
parameters.
We denote this with a square bracket in
maths, so let's start with time because
that's easy.
The units of time are time - seconds. I'm
going to just call it a time which is a
t.
Now energy, the way I remember the units
of energy is to think of - let's take my
favourite energy, kinetic energy, so
kinetic energy is
a half times mass times velocity squared,
so the units
are a mass times a velocity squared.
Velocity is distance divided by time so
that's
distance over time squared.
So I now have a mass, a length squared
over a time squared for the units for
energy. I've got just a t for time.
And density is a mass per
length cubed, because we're working in 3D.
Kilograms per meter cubed for example. So
we've now got
the units of energy, the units of time
and the units of density. And now the
scaling analysis
argument is to say: well I have
determined through
knowledge of physics and fluid dynamics,
as Taylor did, that these three
are the most important parameters that I
think control the blast radius.
So if I can arrange these in such a way
that I get rid of the mass
and get rid of time from the units, and
all I'm left with is a length,
that must be the radius - because what
other possible length could it be?
That's the kind of argument and then of
course you have to validate this
and test this with some amount of
experiments. So you don't need to do
hundreds like the US military,
but Taylor happened to have one or two
photographs from that one test he went
to and he checked it against that and
showed that it was correct. So what we
just have to do now is combine
the three sets of units in such a way
that we just get length. So we've got a mass,
a length and a time. So I think if we
start by removing mass,
so if we do energy E divided by density
rho, then that is equal to - square
brackets for units;
M L squared over t squared, divided by
the density,
puts an M on the bottom and an L cubed
on top.
So the M's cancel and so this is equal
to
L to the 5 over t squared. So now we've
got rid of mass, which is very good, but
we still have time.
So we get rid of time by introducing t.
So we just are going to have to multiply
on the top
by time squared. And that gives us now,
with the time squared on the top,
we just have L to the five, because it
will cancel, and finally
this is a length to the power five.
We just want a radius which is just
L, so we just take the whole thing to the
power of one-fifth.
So if we want our time-dependent radius
of an atomic bomb, the radius of an
atomic bomb,
is proportional to the energy to the
one-fifth,
the time after explosion to the
two-fifths,
and the air density to the minus
one-fifth.
And that right there was the most
closely guarded secret
in US military in the 1940s.
(Brady: Had the US military figured it out with)
(mathematics? Or had they just used a)
(number of tests and sort of)
(you know, guessed roughly?) - So, so if you
were doing this experimentally, which is
what they did, you would do lots of
different explosions
and you would take measurements at
different times, as the thing is
exploding and expanding,
you'd make- take do it on different days,
with different air densities, and you'd
use different sized bombs.
And you'd have all these data points and
then you do
sort of try and find a scaling that
makes the data collapse
into a straight line. That's generally
how you do applied maths research today,
that's how i did
my experiments doing my research in fact.
So you just get lots of data points and
try and
plug it all together and then sort of,
it's kind of trial and error and then
suddenly it
it perfectly fits this pattern.
(Brady: And Taylor was able to check he got it right)
(because)
(he had a few pictures from the Trinity Test?)
Yes, yes he did. He had these two pictures. He
actually published them in a magazine
and when he released this, this paper he
published these two images.
And so he using, um, he could check
this
against, against these actual images to
check the radius. And of course what we
are missing here
is, there's going to be a constant here.
So this is proportional to.
So the units match up but you can still
have a number
in this gap. And now that's the kind of
thing you need to do experiments to get.
And Taylor only had the one to compare
to, and so i think he was trying to
estimate
the size of the bomb dropped in this
Trinity Test in 1945,
and he believed it was 20 kilotons. Sorry,
he believed it was 22
kilotons; and the actual size was 20.
Which, given the - like it's pretty
good, let's be honest.
Given that, you know, he didn't have
access to
highly classified US military secrets, he
was able to get within a 10
accuracy the size of the bomb that was
dropped in that first ever
nuclear test. - (Brady: This isn't going to help)
(you build an atomic bomb,)
(this is just going to give you an idea)
(how devastating they could be?) - Absolutely.
So this, this would be one of the first
things, given some new,
some new weapon or some new even sort of
like a new source of energy;
one of the first things you would want
to do would be to try it at different
values of energy, different sizes,
different densities; and see what happens.
Sort of, before you were able to actually
use it safely;
well I guess a bomb's not really safe- - (Brady: In a controlled way?)
In a controlled way,
thank you. Without unintended
circumstances. You would need to know
this kind of information.
So this was, as I said this was literally
done based on the first
ever detonation of an atomic bomb.
(Brady: Is this constant, this unknown number here,)
(dependent on the type of bomb or? Would a)
(TNT be different to an atomic bomb or?)
I believe so. That's the sort of, the
limitation of the scaling analysis.
Whilst you can get which parameters are
important,
and the powers of those parameters, you
can never get that constant. - (Brady: But once)
(you've got the constant it works for all)
(atomic bombs?)
Yeah, there's going to be some error.
Absolutely going to be some error. But
yes once you have that constant
it hopefully would hold for, for a
particular - at least a particular type of
atomic bomb.
You had the same bomb and you dropped it
and then you doubled your energy,
this would predict how much more the
radius would be, and you would absolutely
expect that to follow.
(Brady: What does it tell us?) - Well, time is
certainly increasing the most,
in the sense of it has the highest power.
If time doubles, versus energy double or
density doubling,
you're going to see the biggest
change being in time. But what it does
tells us is as you increase the energy
the, the radius will increase; but it's
also telling you here if you were to go
from E to E squared
you're only going to get quite a
small increase in your radius.
So this one fifth is sort of saying that
you really have to
ramp up the energy to get that large
radius where- you know, compared to this
being a larger power because it's quite
a small power.
And because the density is negative
that's on the denominator, so as the
density gets
higher the radius decreases. And that's
always a good
sanity check for these things making
sense. If the energy goes up
you expect a bigger bomb. If the you wait
longer you expect a bigger
radius. And if you increase the density
of the air, so you imagine there's more
water in the air it's like
thicker, it's harder for the explosion to
move through thicker air,
so you would expect it to shrink. So
there's always that nice; as I said the
sanity check that it does make physical sense
and and mathematically
the unit's balance, the scaling works, and
then you would then need to do some
exp- at least one experiment
to get a grasp on that number. But, for
the US, the way you would think about it
is the, the density of the air is reasonably
constant. Say you're dropping
the bomb in New Mexico. The air density
in the summer will be
approximately the same across the summer
period, so you know that.
The time - that just will constantly
increase.
So the question would then be, say you
wanted a bomb
such that within two seconds of it being
dropped
it had destroyed a one kilometer radius
area.
You then plug in one kilometer as your
radius,
you say i want within two seconds it to
be gone, I plug in my air density,
it tells you the size of the bomb you
have to drop to do that damage.
So, so that's this really is quite
valuable information if you're trying to
do like a controlled sort of targeted bomb.
[Preview] We're releasing energy
and it produces two neutrons, each of
which
go on to react or to split
other atoms of plutonium. So you get a
chain
reaction and the explosion just builds
up and up and up.