The Clarke Exobelt : an exercise in silly numbers

This is a fun little paper considering SETI by looking for sufficiently dense belts of satellites around other planets.

This paper puts forward a possible new indicator for the presence of moderately advanced civilizations on transiting exoplanets. The idea is to examine the region of space around a planet where potential geostationary or geosynchronous satellites would orbit (herafter, the Clarke exobelt). Civilizations with a high density of devices and/or space junk in that region, but otherwise similar to ours in terms of space technology, may leave a noticeable imprint on the light curve of the parent star. In some cases, a Clarke exobelt with a fractional face-on opacity of ~1E-4 would be easily observable with existing instrumentation.

But, there's a catch, which I wish they'd properly quantify :

A particularly useful database is the compilation of data from public sources made by the Union of Concerned Scientists. Currently, the list contains parameters for 1738 satellites, of which approximately one third are in geostationary or geosynchronous orbits. Assuming a typical radius of 1 m, we have that [maximum] opacity [the fraction of light in our line of sight that is blocked by a surface element] = 3E-13.

In order to become visible from nearby stars with our current observational capabilities, the Clarke belt of our planet would need to be about 1E-4. However, our belt is becoming increasingly populated. Figure 3 shows that the Earth belt opacity opacity has been growing exponentially over the last 15 years. If this trend is extrapolated into the future, we would reach the “observable” threshold around the year 2200.

Assuming that opacity is a linear function of the number of satellites, keeping them all the same size, then the required opacity requires an increase in the number of satellites by a factor of 300 million, or a total of about half a trillion satellites. Maybe I'm wrong. I'm feeling very lazy. It works out to an average of 2.5 billion satellite launches per year, but of course because it's exponential the peak rate will be much higher... I'm not sure it's fair to label a civilization with a launch capacity exceeding 2.5 billion per year (80 per second) as "moderately" advanced.

Obviously, this extrapolation should not be viewed as a prediction. There is no reason to assume that the current exponential growth will be sustained for another 200 years. In summary, the 2200 date is not even a rough guess of when humanity will reach detectability threshold but rather an indication that this outcome is a reasonable expectation for the near future, given current trends.

But is it though ? What would be the average density of a belt of 500 billion satellites ? What would their average separation be ? Would it even be feasible to able to add any more at that point without them colliding in a cascading catastrophe ? Anyone else wanna try the maths ?

Later, they note that the observations would be able to determine the radius of the belt accurately to tell if the objects were in "geo"synchronous orbit or not. This is nice, because : "Geostationary orbits are very interesting for a society but are not preferred by any known natural process."

The simulations presented here show that CEBs may in some situations be detectable with existing instrumentation. The best candidates are planets around red dwarfs in tidal locking, in line with the optimal conditions for habitable exoplanet search. An initial difficulty would be how to distinguish between a CEB and a ring system. However, once a candidate has been identified, detailed follow-up observations may resolve this ambiguity from the shape of the light curve. In any case, the detection of a dense belt of objects at the distance of geostationary orbit would be a very strong evidence for the presence of ETI, especially considering that rings around habitable rocky planets are probably rather uncommon.

Oh, and then we get something fun :

The total mass of the entire belt for all the cases considered here is between 1E12 and 1E14 kg, assuming average object radius and mass of 1 m and 100 kg. This range is between the mass of a comet and that of a mountain. It is not an unreasonable requirement for a moderately advanced civilization.

Which gives a number of satellites from 10 billion to 1 trillion, in agreement with my earlier calculation. Yay ! I'd still like to know the density of the belt or mean free path though.

What I think would maybe be more interesting to know is what it would take to be able to detect satellite belts of density much more comparable to that of our own - say, no more than 1,000 times greater. Does the sensitivity requirement scale linearly ? Will we be able to make significant progress with, say, the ELT or TMT ?

Ah screw it, let's have a stab at estimating the average distance between satellites. They say the maximum inclination of geo_synchronous_ satellites from the geo_stationary_ orbit is 15 degrees. Geostationary's at a radius of 36,000 km, so 36,000*tan(15) = 10,000 km (rounding terribly, because this ain't gonna need to be exact). They also say the allowed spread in altitude is 150 m. So we have a hollow cylinder, inner edge 36,000 km, outer edge 36,000 km + 150 m, height 10,000 km. Total volume = 3.4E11 cubic metres.

Total volume of satellites = 5E11, since their average diameter is given as 1m and assuming them to be cubes because why not. So the fraction of space occupied by the satellites is... greater than one.

Ahh, crap. Someone else can have a go.
https://arxiv.org/abs/1802.07723