Appendix F

Re-Entry Risk Formulas

Fernand Alby

The risk from falling debris during an atmospheric re-entry can be expressed in different forms:

• Probability of one casualty P(1)

• Probability of n casualties P(n)

• Probability of at least one casualty P(≥1)

P(≥1) = P(1)+P(2)+…+P(N)

• Expected value for the number of casualties E

E = 1P(1)+2P(2)+…+NP(N)

where N is the number of inhabitants in the area concerned.

The purpose of this appendix is to provide the formulas for calculating those various values and then to compare their orders of magnitude in a few typical scenarios.

Take a cell on the ground with an area equal to S that contains N inhabitants. Consider the fall of a single piece of debris inside this cell. The casualty area Ac for this debris corresponds to the danger zone on the ground, i.e. the area covered by the debris plus the cross-section area of a human being.

In the cell the inhabitants are represented by dots, with their area having been taken into account in the casualty area depicted by a circle.

image

Consider 1 inhabitant:

• the probability of being hit by falling debris corresponds to the probability of being in the danger zone = Ac/S.

• the probability of not being hit corresponds to the probability of being outside the danger zone = (S-Ac)/S.

If we now consider the N inhabitants:

1 Probability of no casualties P(0)

The probability of no casualties P(0) corresponds to the scenario in which all the inhabitants are outside the danger zone:

inhabitant 1 is outside Ac

AND

inhabitant 2 is outside Ac

AND

…

AND

inhabitant N is outside Ac

hence image

2 Probability of one casualty P(1)

The probability of just one casualty corresponds to the following scenarios:

inhabitant 1 is in the danger zone AND the N-1 others are outside

OR

inhabitant 2 is in the danger zone AND the N-1 others are outside

OR

…

OR

inhabitant N is in the danger zone AND the N-1 others are outside:

image

hence  image

Considering that the casualty area is much smaller than the area of the population cell (Ac ent S), we again get the classical risk formula = image

3 Probability of two casualties P(2)

In the total population there are image different groups of two people. There will be two casualties when any one of these groups of two people are in the danger zone AND when the N-2 other inhabitants are outside the danger zone

image

4 Probability of n casualties

Using the same reasoning, and considering groups of n people, leads to the following general formula:

image

5 Verification

• n = 0

image


hence image we again get the formula seen previously

• n = 1

image

and image and we again get the formula seen previously

• calculation of expected value:

By definition image

image

image

hence image

therefore image

or image

We can write image

to obtain image

but knowing that image

we obtain image and so we again get the familiar result image

6 Comparison between P(1), P(n), P(≥1) and E

For such purpose we evaluate the decrease in the terms of the series P(n) by calculating the ratio image

image

giving image

Considering that n ent N and that Ac ent S we obtain image

As image represents the expected value for the number of casualties (a low value in principle), we can conclude that the terms of the series P(n) should decrease rapidly in general.

7 Numerical examples

Consider the case of falling debris in two 1 km2 cells:

• A relatively sparsely populated cell, representing the average population density on Earth

S = 1 km2, N = 11 inhabitants

• A densely populated cell, corresponding to the density of 20,000 inhabitants/km2 in Paris

S = 1 km2, N = 20,000 inhabitants

Consider three sizes of debris: small (Ac = 1 m2), medium (Ac = 10 m2) and large (Ac = 100 m2)

We obtain the following results:

Small debris scenario (Ac = 1 m2)
Sparsely populated cell Densely populated cell
P(0) 0.99998900 0.98019866
P(1) 1.1E-05 0.01960399
P(2) 5.5E-11 0.00019603
P(3) 1.65E-16 1.3067E-06
P(4) 3.3E-22 6.5327E-09
P(5) 4.62E-28 2.6126E-11
P(≥1) 1.1E-05 0.01980134
E 0.00001100 0.02000000
Medium debris scenario (Ac = 10 m2)
Sparsely populated cell Densely populated cell
P(0) 0.99989001 0.81872993
P(1) 0.00010999 0.16374762
P(2) 5.4995E-09 0.01637411
P(3) 1.6499E-13 0.00109151
P(4) 3.2998E-18 5.4568E-05
P(5) 4.6197E-23 2.1823E-06
P(≥1) 0.00010999 0.18127007
E 0.00011000 0.20000000
Large debris scenario (Ac = 100 m2)
Sparsely populated cell Densely populated cell
P(0) 0.99890055 0.13532175
P(1) 0.0010989 0.27067057
P(2) 5.4951E-07 0.2706841
P(3) 1.6487E-10 0.18045607
P(4) 3.2977E-14 0.09022352
P(5) 4.6172E-18 0.0360858
P(≥1) 0.00109945 0.86466996
E 0.00110000 1.99990718
9 Conclusions

In most cases of falling debris generated during the atmospheric re-entry of a spacecraft, the debris danger zone (casualty area, Ac) does not exceed a few m2; in such cases the probability of one casualty, the probability of at least one casualty and the expected value for the number of casualties are approximately equal. However, in the event of impact from a complete launch vehicle stage containing propellant, the danger zone would be much larger; in that case, if the debris were to fall in a densely populated area, the values for the probability of one casualty, the probability of at least one casualty and the expected value for the number of casualties would be significantly different.

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