Yield of an Explosion Based on Its Radius (Demonstration)

***A) FROM WHERE WE START***

In order to calculate the yield (*Y*) of an explosion based on its radius (*r*), we need to start from a simple assumption:

# Y
k
⋅
V

{\displaystyle Y = k\cdot V}

Where:

*Y*: Yield of the explosion on its epicenter, J.

*V*: Volume of the blast, m³.

*k*: An unknown proportionallity constant, J/m³.

Then:

# Y
k
⋅

3
4

⋅
π
⋅

r
3

{\displaystyle Y = k\cdot \frac{3}{4}\cdot \pi\cdot r^3}

,

3
4

⋅
# π
c
o
n
s
t
a
n
t

{\displaystyle \frac{3}{4}\cdot \pi = constant}

# Y
K
⋅

r
3

{\displaystyle Y = K\cdot r^3}

Where:

*K*: Another constant (which we will calculate), J/m³

*r*: Radius of the blast, m.

Whose Error is:

σ

Y

=

r

4

(

9

k

2

σ

r

2

r

2

σ

k

2

)

{\displaystyle \sigma\_{Y}^{} = \sqrt{r^{4} \left(9 k^{2} \sigma\_{r}^{2} + r^{2} \sigma\_{k}^{2}\right)}}

Note: Sigma is the error of a certain variable.

***B) DEVELOPMENT***

Now, in order to get that unknown constant and complete the formula, we will use the best nuclear explosions simulator that is publicly available. First, within the numerous data it provides, we only have to use the radius in which there is near-total number of fatalities, since this will represent the radius of the blast we want to calculate, and we set the explosion to take place on the surface so the radius would be maximum. So we introduce an example there and get:

Now we just have to use those values on our formula and we will know the value of *K ...*

The formula in order to get "K" is:

# K

Y

r
3

{\displaystyle K=\frac{Y}{r^3}}

# Y
(
4.184
±
0.001
)
⋅

10

15

J

{\displaystyle Y = (4.184 \pm 0.001)\cdot10^{15} J}

# r
(
2180
±
10
)
⋅

10

0

m

{\displaystyle r = (2180 \pm 10)\cdot10^{0} m}

Y = Yield of the explosion

r = Radius of the blast

Whose Error is:

σ

K

=

9

Y

2

σ

r

2

r

2

σ

Y

2

r

8

{\displaystyle \sigma\_{K} = \sqrt{\frac{9 Y^{2} \sigma\_{r}^{2} + r^{2} \sigma\_{Y}^{2}}{r^{8}}}}

Note: Sigma is the error of a certain variable.

Final Value:

# K
(
4.04
±
0.06
)
⋅

10

5

J

/

m
3

{\displaystyle K = (4.04 \pm 0.06)\cdot10^{5} J/m^3}

***C) RESULTS***

We now know the value for the constat *K*, so the formula for the yield of a nuclear explosion based on its radius would be:

# Y
K
⋅

r
3

{\displaystyle Y = K\cdot r^3}

But, what if we the explosion we wanted to calculate is not nuclear? It is simple: only a 40-50% of a nuclear explosion actually goes for the blast, while for a normal one it is almost 100%, then the *K* constant for a non-nuclear explosion is half of which we calculated early.

# Y
0.5
⋅
K
⋅

r
3

{\displaystyle Y = 0.5\cdot K\cdot r^3}

***D) LIMITATIONS***

Based on the method we used to obtain this formula, it's important to note that it applies for the standard density of air and any other density could make the results wrong (even though it can be always supposed that the density of air in any feat is the same as the standard). Another condition in which it was calculated was where it took place, it was a ground explosion, so for any other altitude of the epicenter, the result must be divided by two, as this article explains.

***E) REFERENCES***

* Atomic Archive. (n.d.). Energy from a Nuclear Weapon. https://www.atomicarchive.com/science/effects/energy.html
* Myk 25. (2024, January 19). Types of Explosions and How to Calculate Them. *Physics Fiction Wiki*. Types of Explosions and How to Calculate Them
* Wellerstein, A. (2012, February 3). NUKEMAP. *The Nuclear Secrecy Blog*. https://nuclearsecrecy.com/nukemap/