Thread by @ProfHughLewis

Elon Musk told the Financial Times that "Tens of billions" of satellites can be accommodated in orbits close to Earth. Here's a thread looking at whether this is correct...
bbc.co.uk/news/business-…

1/ To investigate, I used the stability model developed by Don Kessler & Phillip Anz-Meador, which Phillip presented at the 3rd European Conference on Space Debris in 2001

2/ I will skip over the derivation of the model to go straight to the key result, the
critical number of intact objects above a specified altitude producing a
runaway environment:

3/ a "runaway environment" refers to unconstrained, exponential growth of the orbital (and trackable) population.

4/ In the paper, the authors present values for the model parameters based on the catalogued population from 1999. For my analysis, I will keep things simple and just use orders of magnitude.

5/ If we assume 4*pi*CD ~ 10, a ~ 10^7 metres, Vo ~ 10^4 metres/sec, rho_a ~ 10^-12 kg/cubic metre, then the numerator equates to ~ 10^14

6/ Taking a similar approach, we can assume N0 ~ 10^2, W ~ 1, (m/A)_a ~ 10^2 kg/square metre, V ~ 10^4 metres/sec & sigma_f ~ 10 square metres. These approximations allow us to easily calculate the critical number of intact objects. For these values, the answer is ~ 10^5

7/ This is somewhat lower than the "tens of billions" suggested by Elon Musk. However, the stability model assumes that the intact objects are derelicts & cannot perform collision avoidance manoeuvres. We need to look a little deeper.

8/ Kessler & Anz-Meador also provide an equation to calculate the collision rate:

9/ Unfortunately, I believe there is a mistake in this equation. A check of the units will reveal that the first term on the right hand side is faulty.

10/ If we just look at the collision rate for intacts-on-intacts, the actual collision rate is proportional to the square of the number of intact objects. In addition, if we focus on the altitude range 400 km to 1020 km, the volume of the orbital shell is ~ 10^21 cubic metres

11/ Hence, if we assume 10^5 intact objects in this orbital shell we arrive at an estimate for the collision rate of 10 per year. Note we still have not taken into account the presence of active spacecraft with collision avoidance capabilities.

12/ To gain an understanding of the manoeuvre burden, assuming all 10^5 objects have manoeuvring capability, we can estimate the number of close passes they will experience in one year. This value is proportional to the square of the number of intact objects

13/ We arrive at an estimate of 10^17 close passes per year (which seems like an extraordinarily high number, but ask @LeoLabs_Space how many CDMs are produced now...). A relatively small number of these would require a manoeuvre, but a small % of a large number is still large

14/ SpaceX has already said that Starlink satellites made more than 2000 collision avoidance manoeuvres in the 6 months from December 2020.

15/ Of course, I have assumed that all 10^5 objects can encounter each other, which is not going to be the case in reality. Nonetheless, even if the LEO environment could sustain 10^5 active objects, there would still be a huge traffic management problem to solve.

16/ If we come back to the "tens of billions" of spacecraft (~ 10^10 objects) & consider an optimistic failure rate of just 1% (which is better than we have seen with the current generation of Starlink satellites), we might expect ~ 10^8 derelict intact objects in LEO

17/ So not only is that far in excess of the 10^5 objects that the Kessler & Anz-Meador stability model predicts would lead to a runaway environment, the traffic management burden for the remaining active spacecraft population would be immense (practically impossible to achieve)

18/ Hence, Elon Musk's assertion that the environment can accommodate tens of billions of spacecraft is not correct, in my opinion.

To all suggesting that the deployment of Starlink will lead to the "Kessler Syndrome", you can also see from the 2001 paper that the runaway threshold was exceeded at least ~20 years before the first Starlink satellite was launched. Sorry to be the bearer of bad news.

To those suggesting I assume a lower failure rate, different cross-sections, or include active removal: of course these things can be done and they will change the critical number. But I can also include explosions, ASAT tests, a declining thermospheric density trend, etc.

What I'm saying is that we probably all want the critical number to be higher & we can make assumptions that enable this, but we also need to acknowledge the bias involved in the inclusion of only those assumptions.