### **The Orbital Debris Collision Hazard for**

Proposed Satellite Constellations

Proposed Satellite Constellations

**Duncan Steel**

**2015 April 30**

**Introduction**

In my first post here on the calculation of orbital debris collision hazards I wrote the following in connection with reported plans to insert constellations of satellites into low-Earth orbit (LEO), in the following case at altitude 800 km:

*“If these enhancement factors are of the correct order, then the collision probability attains a value of around 5 × 10 ^{-3} /m^{2}/year, implying a lifetime of about 200 years against such collisions with orbital debris. This indicates that there is cause for concern: insert 200 satellites into such orbits and you should expect to lose about one per year initially, but then the loss rate would escalate because the debris from the satellites that have been smashed will then pose a much higher collision risk to the remaining satellites occupying the same orbits (in terms of a, e, i). This ‘self-collisional’ aspect of the debris collision hazard I highlighted in the early 1990s.”*

The enhancement factors in question are those that elevate the collision probability for any planned satellite above the value I had calculated on the basis of the orbits of the 16,167 objects listed in the *Satellite Situation Report* in early April. These factors may be summarised as follows: (a) The lack of available orbits for objects that are CLASSIFIED, this potentially increasing the collision probability by 25 to 50 per cent; (b) An increase by a factor of two or three in the collision probability due to the finite sizes of tracked objects compared to the one-square-metre spherical test satellite assumed in my calculations (with the potential impactors being taken to be infinitesimally small); and (c) The overall collision probabilities being higher perhaps by a factor of a hundred due to the large population of orbiting debris produced by fragmentation events that is smaller than the (approximately) 10 cm size limit for tracking from the ground by optical or radar means but nevertheless large enough to cause catastrophic damage to a functioning satellite in a hypervelocity impact.

Herein I consider in more detail two mooted satellite constellations, intended to deliver internet access to the entire globe, and assess the orbital debris collision hazard that they will face; and also the hazard that they would pose to themselves and other orbiting platforms should the plans go ahead.

**Proposed internet satellite constellations **

Several distinct satellite constellations have been proposed and are apparently undergoing development, although definitive information is sparse. As an initial illustrative example I will describe *WorldVu*, a proposal stemming from former employees of *Google* although subsequently spun off into a new company and rebranded as OneWeb. The original WorldVu plan apparently involved a constellation of 360 satellites in total, 180 each at altitudes of 800 and 950 km, and all having an inclination of 88.2 degrees. At each altitude the 180 satellites would be arranged in nine orbital planes each containing twenty satellites. Shown below is a visualization of just the 180 satellites at altitude 950 km, with the blue access cones beneath each satellite indicating the coverage of Earth’s surface subject to a limitation of the elevation angle from ground to satellite being at least 25 degrees (this being a conceivable limit for ground station access to each satellite).

*Graphic above: A Strawman satellite constellation of 180 platforms in polar orbits. A movie showing the movement of these satellites in orbit is available for download from **here** (14.4MB).*

My previous posts on the LEO debris collision hazard (here and here) have shown that the collision probability against tracked orbiting objects is particularly high for altitudes 800-1000 km, most specifically for any proposed satellite in a polar orbit (inclinations 80-100 degrees).

Perhaps in recognition of the crowded and congested nature of geocentric space at such altitudes, more recent (i.e. in 2015) proposed satellite constellations have involved slightly higher altitudes, at 1100 km and 1200 km. The two specific constellations that I consider here may be summarised as follows.

*OneWeb*: 648 (or possibly 700) satellites at altitude 1200 km; individual satellite masses of order 125-200 kg; strategy to evolve to a second-generation constellation of 2,400 satellites.*SpaceX*: 4,025 satellites at altitude 1100 km; individual satellite masses 200-300 kg.

Whether or not the above basic information is realistic or not, and whether or not either or both of these proposals go ahead, the two provide pertinent input data for a consideration of the orbital debris hazard that such constellations will face.

**Collision probability calculations**

I have derived collision probabilities between test satellites and my list of 16,167 tracked orbiting objects as described previously. Although the satellite altitudes (i.e. 1100 and 1200 km) are known, I do not know for certain the proposed inclinations. The previously-promoted WorldVu plan was for an inclination of 88.2 degrees. A graphic from OneWeb shown immediately below indicates polar orbits. Sun-synchronous orbits for altitudes of around 1110-1200 km require an inclination close to 100 degrees. In view of the above I have performed collision probability calculations for those two inclinations, 88.2 and 100 degrees. Note that this means that I have also assumed that the SpaceX constellation would be in polar orbits (and it might well be that lower-inclination orbits would be chosen so as to provide better coverage of low-latitude rather than polar ground locations).

*Above: The proposed OneWeb satellite constellation’s orbits and ground coverage (source: **OneWeb website**). *

In Figure 1 are shown the collision probabilities for test satellites in circular orbits in 50 km steps between altitudes 650 and 2000 km, for the two inclinations discussed above. Purely on the basis of minimising the debris impact risk it is clear that it would be wise to avoid altitudes below 1000 km or even 1050 km.

* ***Figure 1**: Collision probabilities against all tracked objects in the publicly-available *Satellite Situation Report* as a function of altitude for circular test orbits and for two different near-polar inclinations; a spherical test satellite of cross-sectional area one square metre is used, and all possible impactors (the tracked objects) are taken to be infinitesimally small (i.e. the mutual collision cross-section is assumed to be one square metre in every case).

The form of Figure 1 might, in isolation, be thought to imply that there are comparatively few objects whose orbits take them above 1,000 km altitude, and indeed it has been reported in the media that this is one reason for the SpaceX choice of an altitude of 1100 km. However, this is not the case: there are many tracked objects in orbits well above 1000 km. In Figure 2 I plot the numbers of tracked objects with orbits that cross each of the discrete altitudes in Figure 1. Whilst that plot peaks at 850 km with just over 5,000 tracked objects, there are still about 3,000 tracked objects crossing 1100-1200 km, and over 2,000 objects right the way up to altitude 2000 km.

In view of the information in Figure 2 it might seem surprising that the collision probabilities as shown in Figure 1 drop rather quickly at altitudes above 1000 km. The lesson to be learned from this is that mere numbers of objects do not provide a reliable indication of the collision risk: their orbits (and especially their inclinations) greatly affect the formal collision probabilities. It happens that polar orbiters have disproportionately high collision probabilities (with extreme likely impact speeds), and most polar orbiters (e.g. in sun-synchronous orbits) circuit our planet at altitudes below 1000 km.

**Figure 2**: Numbers of tracked objects with orbital paths crossing discrete altitudes between 650 and 2,000 km.

In order to move from a collision probability per square metre per year (as in Figure 1) to a collision probability for specific satellites we need to have some information regarding their size. Although ballpark masses have been stated above for individual satellites in the proposed OneWeb and SpaceX constellations, specific linear dimensions are not known (at least by me!). The graphic below shows an artist’s representation of a OneWeb satellite, and that I have used to estimate that a size of about 5 × 1 × 1 metres is of the correct order.

*Above: Representation of a OneWeb satellite in orbit
(source: *

*OneWeb website*

*).*

During each orbit the satellites will rotate their solar panels so as to maintain their direction towards the Sun, and this will have the effect of ‘averaging out’ the cross section with regard to debris approach directions. In view of that I will adopt a characteristic cross-section of three square metres for the OneWeb satellites.

Media reports of the masses and planned capabilities of the SpaceX constellation satellites indicate that these may well be larger. For these I will adopt a characteristic cross-section of five square metres.

Regardless of the choice of inclination (88.2 or 100 degrees) the collision probabilities in Figure 1 at altitudes of 1100 and 1200 km are approximately 3 × 10^{-6} and 2 × 10^{-6} respectively, per square metre per year. These lead to the following estimates for the collision probabilities, when the cross-sectional areas of the satellites are included:

OneWeb satellites: 6 × 10^{-6 }per year

SpaceX satellites: 1.5 × 10^{-5 }per year

It is emphasized that these values should be considered to be lower limits on the orbital collision probabilities, due to factors that have been previously mentioned: (i) There is no allowance for CLASSIFIED orbiting objects; (ii) The finite (often large) sizes of other objects will enhance the collision cross-sections by an average of perhaps two or three; (iii) No allowance has been made for small (less than 10 cm) untracked debris items, and these can increase the collision risk by a large amount; note also that the capability to detect and track small debris items falls off with altitude, given that the US DoD sensors are ground-based.

I would have difficulty in making a case for overall enhancements in the collision probabilities due to the preceding considerations being less than by a factor of ten, and quite likely the true enhancement is by 20 to 50. It may only be through statistics of satellite losses that we will be able to obtain a true assessment of the hazard, or else satellite-borne sensors counting the frequency of near-misses by small orbiting debris (plus natural meteoroids).

If one adopts an enhancement by only a factor of ten, perhaps dominated by 1-10 cm untracked debris from disintegrations such as the Chinese ASAT demonstration in 2007 and that of DMSP-F13 just less than three months ago (see my notes in this post), the catastrophic collision probabilities against orbiting debris become:

OneWeb satellites: 6 × 10^{-5 }per year

SpaceX satellites: 1.5 × 10^{-4 }per year

(Note that although the two satellite disintegrations mentioned above occurred at around 850 km, tracked debris fragments from them have attained apogees crossing the altitudes of the above two proposed satellite constellations; and smaller, untracked debris items generally have larger relative speeds, making them more likely to attain higher apogees. I also note that at this stage the cause of the observed disintegration of DMSP-F13 is unknown, and it might well have been due to an impact by a fragment of the Chinese ASAT target, *Fengyun-1C.*)

Next I multiply the above probabilities by the number of satellites in each proposed constellation (648 and 4,025 respectively), to obtain an estimate of the loss rates:

OneWeb constellation: 0.04 per year

(one catastrophic collision per 25 years)

SpaceX constellation: 0.6 per year

(one catastrophic collision per 20 months)

A note here on collision speeds: the most likely impact speeds for mutual collisions between polar orbiters at these altitudes (1100-1200 km) are above 14 km/sec.

**Implications **

The above estimated satellite loss rates due to debris collisions might be considered to be tolerable, but there are other implications. The most important matter, which stems directly from the sorts of calculations performed here, is that any satellite (or other associated object such as a rocket body) in a constellation that suffers a catastrophic fragmentation event immediately elevates the collision hazard for all other satellites that remain in similar orbits. Essentially, the collision probability between two orbiting objects goes up markedly when they have similar orbits, in terms of their perigee/apogee altitudes and inclinations (or *a*, *e*, *i*).

As an example, consider one satellite in the proposed SpaceX constellation which undergoes a disintegration due to being hit by some random piece of orbiting debris. The figures above indicate that such an event is to be expected within the first two years of the deployment of the proposed constellation (and by the time that the constellation is deployed the orbital debris hazard will certainly be worse, not better). Let us assume that the satellite dry mass is 250 kg. Observations of the mass distributions of objects fragmenting in orbit, and indeed deliberate (chemical) explosions of spare satellites in laboratories, indicate that more than 10,000 fragments with masses above 1 gram are to be expected, and each will be moving on an orbit distinct from the parent satellite but nevertheless on quite a similar path. Each of those fragments now poses an impact hazard to all other satellites in the constellation: they are no longer subject to station-keeping, and they are no longer moving in step with the functioning satellites.

(There is an entirely different way of looking at this, that astrophysicists will likely understand. To first order one may say that if there are *n* satellites in a constellation then the collision hazard they pose to themselves goes up as *n²* rather than *n*.)

Assuming an original satellite orbit to be circular at 1100 km, and the debris initially to have perigee and apogee heights spread a few kilometres above and below that, the collision probability for each fragment with each of the other satellites is found to be about 5 × 10^{-7 }per year. For 10,000 fragments capable of colliding with 4,000 functioning satellites the collision rate is then:

5 × 10^{-7 } × 10,000 × 4,000 = 20 per year

What happens, of course, is not that twenty satellites per year are lost, but rather a rapid cascade occurs with a proliferation of debris in similar orbits obliterating all functioning satellites and leaving that region of LEO unusable for a very long time. This is the *Kessler syndrome* writ large.

**Final comments**

Although in this series of posts on orbital debris I have been making use broadly of test satellites assumed to be spherical, that is a non-necessary simplifying assumption. In fact it is possible to derive face-dependent impact probabilities and indeed impactor speed distributions for satellites that maintain their orientations during an orbit, and I not only have software to do this, but have exercised it in the past. Most of my results from the use of that software has not been published, either here or elsewhere; I am intending to prepare other posts for publication here, illustrating the sorts of understandings that can be derived (and it is only through understanding a problem that we can equip ourselves with the wisdom to act and react in certain ways).

There are at least two good reasons to make use of such software when planning future satellites. The first is so that a proper assessment of the orbital debris hazard can be made, making use of the real size and shape of the satellite (including its solar panels) rather than an assumed spherical form, as here. The second reason is that, at least for very small debris items (and indeed meteoroids and interplanetary dust), knowledge of the most likely arrival directions and speeds would enable appropriate shielding to be installed, and also the arrangement of the more sensitive parts of the satellite to be planned so as to minimise the overall mission risk. One cannot entirely obviate the orbital debris collision risk, but one can make judicious decisions on how to minimise it once one is well-informed.