Thread by @ProfHughLewis

I think the #KesslerSyndrome is too often presented as a tipping point or a threshold we have yet to cross, so I wanted to use some aspects of my paper at the 8th European Conference on #SpaceDebris to explain why I think that is wrong [1/n]

The starting point of my thinking was to look at how natural populations grow. The simple exponential model is a standard model that describes the growth of a single population [2/n]

If we know the initial number of individuals in the population N(0) then this model allows us to estimate the number of individuals at any future time t. Here, r is the intrinsic rate of natural increase, which depends on the birth rate, b, and death rate, d [3/n]

If the birth rate is greater than the death rate then r>0 and the population is growing. Conversely, if the birth rate is less than the death rate then r<0 and the population is declining [4/n]

A population that increases with a rate proportional to its current size will grow exponentially. This means that as the population increases so does the rate at which it grows [5/n]

We can write the exponential equation as a differential equation, where the rate of change of N is proportional to r [6/n]

However, observations of natural populations have shown that as the population density increases, the intrinsic growth rate tends to decrease (e.g. in a population of bacterium) possibly due to an impact on the available resources [7/n]

To capture this effect, we can modify our differential equation [8/n]

If we assume f(N) is linear (which is what is observed) then we can write f(N) = a - cN and the differential equation becomes: [9/n]

If we let a = r and c = r/K then via simple algebra we arrive at the logistic growth equation, which is another standard model describing the growth of a single population [10/n]

Here K is the carrying capacity (a term that is becoming familiar in the context of #SpaceDebris & #SpaceSustainability). Initially, when there are relatively few individuals, the population grows exponentially. As N continues to increase, the rate of change decreases [11/n]

N = K is a dynamic equilibrium. There are still births and deaths but the number of individuals in the population remains constant [12/n]

When we consider the #SpaceDebris population we can again start with the simple exponential model, replacing births with "breakups" and deaths with "decays" (we neglect launches for now). So, as before our initial model is [13/n]

Similar to our natural population, changing #SpaceDebris population density will also affect the intrinsic growth rate [14/n]

However, in this case an increase in density causes an increase in the growth rate, not a decline, and assuming this relationship is linear we can write f(N) = a + cN. So our differential equation becomes [15/n]

Although this might seem to be just a minor change compared with the corresponding logistic growth model, the effects are substantial. Instead of the characteristic S-shaped growth curve we have unconstrained growth for r>0 (dashed line shows classical exponential growth) [16/n]

It is still possible for the population to decline if the decay rate is sufficiently higher than the breakup rate (dashed line shows the population decay described by the simple exponential model for an equivalent value of r) [17/n]

But in any regime where the decay rate is small (or zero) the model indicates the population will grow in an unconstrained manner due to breakups (i.e. collisions) and the rate of growth will continue to increase non-linearly. [18/n]

What does this mean for a #KesslerSyndrome "tipping point" or "threshold"? Well, in terms of the number of individuals in the #SpaceDebris population, there is no meaningful threshold value for N in regimes where the decay rate is negligible or less than the breakup rate [19/n]

Adding just two objects to the environment where their orbits can intersect and where the effects of decay are negligible can lead to population increase (in reality, this cannot be unlimited because the individuals produced by breakups will get ever-smaller in size) [20/n]

So, IMHO, the #KesslerSyndrome doesn't describe some far-off tipping point. It describes the fundamental nature of the growth of the #SpaceDebris population in regions without any effective sinks (e.g. atmospheric drag) or where debris sources dominate [21/n]

Given our ongoing space activity, this description is true for pretty much all of near-Earth space except, perhaps, the region below 600 km altitude where the effects of atmospheric drag are important [22/n]

But as @mattkennybrown showed at the #SpaceDebris conference this week, anthropogenic CO2 emissions are having increasingly detrimental effects on this vital debris sink. Atmospheric density in this region has already declined by 20-25% since the year 2000 [23/n]

If our CO2 emissions continue then by the end of this century atmospheric density at 400 km could be 20% of what it was in the year 2000. The effects on #SpaceDebris will be significant [24/n]

This has been a long thread (thanks for reading to here!) & I hope it has shown the value of simple models to understand and explain concepts like the #KesslerSyndrome that are often misrepresented or misunderstood. [25/25 & the end]