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The Orbital Debris Collision Hazard for Satellites in Geostationary Orbit

The Orbital Debris Collision Hazard for
Satellites in Geostationary Orbit

Duncan Steel
2015 April 29

Introduction

In my two previous posts (here, and here) regarding the orbital debris collision hazard I have been considering only test satellites in low-Earth orbit (LEO). In this post I turn my attention to satellites in geostationary orbit (GEO); that is, satellites in orbits that have periods of one sidereal day (i.e. altitudes near 35,800 km) and are in near-circular (i.e. low eccentricity) paths close to the equatorial plane (i.e. low inclination). Such orbits are used for many communications satellites, and are sometimes termed Clarke orbits.

 

A crude estimation of the typical collision probability in GEO

In principle it is straightforward to derive an estimate of the collision probability with space debris for a satellite in GEO. Consider the diagram below. The white circle shows the orbit of a geostationary communications satellite: there are many such satellites in GEO, but they remain in set positions due to their operators performing station-keeping burns/manoeuvres and so cannot collide with each other. Even a GEO satellite which is no longer under control only has a small drift speed (much smaller than the GEO orbital speed of  3.075 km/sec) and so the risk of inter-satellite collisions is very small, if we are considering only true geostationary satellites.

GEO hazard

This is not true, however, for orbiting objects that cross the GEO altitude. The red ellipse in the diagram is intended to represent a rocket body in a geostationary/geosynchronous transfer orbit, having performed its purpose in carrying a GEO satellite to the necessary altitude. That defunct rocket body will pose a collision hazard to all functioning satellites in the geostationary band, as its argument of perigee and longitude of the node precess under gravitational perturbations due to the Moon and the Sun, and also the Earth’s oblateness (dependent upon the altitude of its perigee).

Imagine a sphere with radius (r) equal to the distance of the geostationary band from the centre of the Earth (about 42,000 km). That sphere has an area of 4πr2. The rocket body, having an apogee above the geostationary band as shown, crosses that band twice per orbit, giving it two chances to hit that one particular communications satellite. If we estimate the mutual collision cross-section to be 100 square metres, then the probability of a collision is given by 2 × 100 / 4πr2 which is a shade less than one part in 1014 per orbit of the GTO object (the rocket body). That GTO will have a period lower than one sidereal day, perhaps about 15 hours, or one part in 584 of a year, so that our estimate of the mutual collision probability becomes about 5 or 6 × 10-12 per year.

To put that figure into some perspective, the universe is about 13.8 billion years old, and so a mutual collision would have a probability of occurrence of less than ten per cent over such a timescale. (Of course, the solar system is only about one-third the age of the universe; and over such phenomenal timescales all manner of other things happen.)

This, however, was a calculation for a mutual collision between only one pair of orbiting objects. The reality is that there are hundreds of satellites in geostationary orbit; and hundreds of rocket bodies and fuel tanks and other debris items crossing the GEO altitude, so that a more sophisticated assessment of the collision hazard is warranted.

 

Specific example orbits

The Inmarsat 5F2 communications satellite was launched on 2015 February 01 and inserted into geostationary orbit. The Briz-M rocket booster that was used to carry it to its final altitude remains in a GTO similar to that shown in the preceding diagram. Orbital parameters for the two objects in question are shown in the table below; note that the inclination and eccentricity of Inmarsat-5F2 in particular will vary from day-to-day and week-to-week under the influence of gravitational (and radiation-induced) perturbations, plus station-keeping burns by the operating company.

Name International Designator Satellite Catalogue Number (SCN) Perigee altitude (km) Apogee altitude (km) Eccentricity Inclination to the Equator (degrees)
Inmarsat-5F2 2015-005A 40384 35,551 36,029 0.000165 0.0338
Inmarsat-5F2 Rocket Body 2015-005B 40385 2,775 63,223 0.767648 27.6213
Inmarsat-5F2 Fuel Tank 2015-005C 40386 355 14,849 0.518453 50.7385

The final line in the table, for the fuel tank left in a lower orbit, plays no further part in the present calculations and is included simply for completeness. The orbits of the first two objects are shown in the graphic below, with three views being given so as to indicate the three-dimensional nature of the geometry. In addition a short movie portraying the movements of the two objects over five days is available for viewing and download from here (1.44MB).

I3F2

Graphic above: The orbits of the Inmarsat-5F2 satellite (in a near-circular geostationary orbit) and also the rocket body that carried it there, that rocket body (labelled I5F2RB above) having been left in an orbit with eccentricity near 0.77 and a perigee altitude of a few thousand kilometres. 

Using the orbital parameters listed above I can now calculate the separate collision probabilities involving (a) Inmarsat-5F2 itself; and (b) the Briz-M rocket body used to carry Inmarsat-5F2 to its geostationary orbit, against all objects in the tracked orbiting object catalogue. In a previous post I described the list of 16,167 tracked items that I have been using in calculating collision probabilities. Again using that list I have derived the following results:

Object Number of tracked objects crossed by the I5F2 objects  Net collision probability per square metre per year
Inmarsat-5F2 1,543 3.06 × 10-8
Inmarsat-5F2 R/B 3,763 1.15 × 10-9

 

Discussion

The collision probabilities derived above are in units of per square metre per year, and are based on a model test satellite in each case having a spherical shape and a geometrical cross-section (and therefore collision cross-section) of one square metre. Of course, the actual objects in question do not have such a size and shape.

The two are shown in the graphic below (the source of which is here). The Briz-M booster might be modelled as being near-enough to spherical for present purposes, with a characteristic size (diameter) of almost five metres, and so a cross-sectional area of about 20 square metres.

I5F2_and_Breeze-M

Graphic showing the Inmarsat-5F2 communications satellite (upper right) and the Briz-M rocket booster (centre). Courtesy Khrunichev.

The Inmarsat-5F2 satellite is much larger. Once its solar cells were deployed it attained a greatest linear dimension of 33.8 metres, and the thermal radiators and antennas that fold out from the main bus render linear dimensions of 8.08 metres. Looking at the satellite from orthogonal views its greatest cross-sectional area is therefore about 270 square metres, and its smallest about 65 square metres.

During its orbit around the Earth the satellite rotates its solar cells so as to keep these pointing towards the Sun, and the effect of this is to ‘average out’ the cross-section from the perspective of potential colliders such that again we might approximate its collision cross-section in terms of a sphere with a cross-sectional area of about 150 square metres, as a rough estimate.

Using that as a collision cross-section one derives a collision probability for the Inmarsat-5F2 satellite of about 5 × 10-6 per year (i.e. 150 × 3.06 × 10-8) and so a characteristic event rate of one collision per 200,000 years.

This value, however, is misleading: as noted earlier, most objects in the geostationary belt are moving in step, making it infeasible for them to collide with each other. One approach I could take here would be to sort through the 1,543 objects crossing the orbit of Inmarsat-5F2 and weed out those that are also truly geostationary or geosynchronous, making collisions with each other impossible in the present epoch, but it is sufficient to say simply that the true collision probability against other tracked objects is likely an order of magnitude less than the value calculated here (and so around 5 × 10-7 per year, or a timescale of two million years).

Turning to the Briz-M rocket body, there is nothing to protect it from collisions with objects in the geostationary belt (or anywhere else). The collision probability for this object is therefore about 2.3 × 10-8 per year (i.e. 20 × 1.15 × 10-9) and the characteristic collision time is of order 40 million years.

In my first post here on orbital debris I discussed three different ways in which the ‘true’ collision probability could and would be higher than my calculations indicate, these ways being:
(i) The Satellite Situation Report and Two-Line Elements that are publicly-available do not include any CLASSIFIED orbits, these comprising almost 800 objects compared to the 16,167 in my listing used in these calculations;
(ii) Many of the tracked objects are larger than my model spherical satellite of cross-sectional area one square metre, so that the true collision cross-section for specific cases will be substantially larger than one square metre; and
(iii) The tracked objects are larger than about 10 cm in size, else they are not detectable with the ground-based optical and radar sensors employed by the U.S. military to keep tabs on anthropogenic orbiting objects, but fragmentation events such as remnant fuel explosions and hypervelocity collisions will produce many orbiting objects smaller than that 10 cm limit.

Taking each of these considerations in turn:
(i) There are certainly various CLASSIFIED payloads in geostationary orbit, and rocket bodies used to take them there also have GEO-crossing paths, but the effect of including these is unlikely to boost the derived net collision probability for objects in geostationary orbit by more than 50 per cent;
(ii) The communications satellites in GEO tend to be the largest objects in such orbits (unlike in LEO, where the International Space Station, the Hubble Space Telescope and various classified surveillance platforms dwarf most other satellites) so that the appropriate collision cross-sections to use are indeed the sizes of the communications satellites themselves, with areas dominated by solar cell arrays and typically being 100-300 square metres; and
(iii) There is very little small debris in orbits crossing the geostationary band, and so the enhancement in the collision probability due to sub-decimetre projectiles is small.

Overall my analysis leads to an expectation that the individual lifetimes of satellites in the geostationary band against catastrophic collisions with artificial space debris are of order millions to tens of millions of years.

 

Conclusions

I will now repeat what I have just written: the individual lifetimes of satellites in the geostationary band against catastrophic collisions with artificial space debris are of order millions to tens of millions of years.

If this conclusion is correct then, from the perspective of satellite protection and safety, there is no real need to boost defunct satellites into higher orbits, as is current practice. Indeed one might argue that to do so may have the effect of increasing the collision hazard: in any orbital manoeuvre there is a finite likelihood of something going wrong, perhaps with disastrous consequences. For example, re-igniting a rocket booster attached to a communications satellite, when that booster had been dormant for a decade or more since the satellite was placed in its desired station, might  lead to an explosion and thus the spreading of myriad pieces of debris around the geostationary belt with high relative velocities. It would be better to leave the satellite as it was, simply closing it down until future generations with better technical capabilities can clear it up.

If the debris impact timescale for geostationary orbits is indeed of order millions of years, the collisional impact risk for satellites in that high-altitude band is dominated by the flux of natural meteoroids. I will say nothing more on that subject here, and simply refer the reader to a recent report on the subject.

 

The Last 15 Minutes of Flight of MH370

The Last 15 Minutes of Flight of MH370

Brian Anderson
Report prepared 2015 April 24

 

 Introduction

Determining what happened to MH370 during the last 15 minutes of flight is difficult, given the limited information available, yet it is important to arrive at some estimates of the aircraft altitude and path during this time. The path is needed in order to establish where the end point might be.

At the time of the Independent Group’s (IG) September 26 statement last year, the information available to help with the analysis of this period was limited to the satellite-derived BTO and BFO data at 00:11 and 00:19 UTC (on 2014 March 08), and the expectation that the engine failures with fuel exhaustion would not occur at precisely the same instant in time.

It was expected at that stage that knowledge of the engine Performance Degradation Allowance (PDA) figures would provide evidence as to which engine failed first, and also enable the remaining flight time before the second engine failed to be determined. Unfortunately the PDA information has not been made public by Malaysia Airlines.

The analysis of several simulation runs in a certified B777 level 4 simulator showed that knowing which engine fails first, and hence being able to determine the direction of the Thrust Asymmetry Compensation (TAC) deflection at the time of the second engine failure, is not a predictor of the flight characteristics of the aircraft following the second engine failure. Hence, knowing the PDAs is not immediately helpful in this regard (i.e. knowing which engine first ran out of fuel is not a necessity here). Rather, the flight characteristics, and in particular the propensity of the aircraft to bank into a turn immediately following the second engine failure, is a function of precisely how the aircraft was trimmed immediately before the first engine failure. The simulator trials showed that seconds after the second engine failure the TAC is reset to the cruise rudder trim position.

We know that each B777 aircraft is subtly different (as are most other aircraft) with respect to rudder trim in particular. One might say that they may be slightly ‘bent’ in terms of aerodynamics.  Some require a little left trim, some a little right, and some require neither in order to fly straight. In addition, individual pilots often adopt slightly different regimes in order to manage the trim (TAC) requirements. Without knowing how this particular aircraft handled historically – and even if we did know – it may not be possible to say with certainty if the aircraft would depart into a turn to the left or to the right following the second engine failure.

In performing the analysis presented in this paper I have made the following necessary assumptions:

  • There was no manual intervention to control the aircraft during the last 15 minutes of flight;
  • The aircraft was flying on autopilot during its passage south over the Indian Ocean until the second engine failure;
  • The flight path is similar to the path discussed in the IG statement dated September 26 (and also a number of other independently-derived flight paths ending at similar latitudes, so that the precise flight path taken over the Indian Ocean is not significant here);
  • The B777 level flight simulator runs previously studied by the IG (per Mike Exner) deliver a valid representation of how MH370 would be expected to have behaved near the end of its flight.

 

Fuel Data

With the release of the Factual Information statement by the Malaysian Safety Investigation Team on 08 March 2015 [reference 1], new information became available which enabled more analysis of the fuel consumption rates and better estimates of the engine failure timings, without requiring specific knowledge of the PDAs. This at least enables us to say with some certainty that it was the Right engine that failed (ran out of fuel) first.

When the first engine fails the TAC is immediately set to trim the aircraft to fly straight, and the auto-throttle increases thrust in order to maintain air speed and altitude, as far as possible.

The ATSB, in its 26 June 2014 report [2], described the Satellite Data Unit (SDU) power up process and concluded that the 0019:29 log-on would have occurred three minutes and 40 seconds (+/- 10 seconds) following the second engine failure, and that the autopilot would have been disengaged for this period due to the interruption in electrical power. The SDU boot-up and final log-on is assumed to be a result of an automatic Auxiliary Power Unit (APU) start due to the loss of both electrical generators driven by the jet engines. The time required for the entire process, from loss of the second engine until the log-on, is 3m40s. This time was confirmed during the B777 flight simulator trials mentioned previously.

Note that the APU is supplied from the left main (fuel) manifold, and will continue to run only for as long as there is sufficient fuel in the fuel lines to the APU (given that the fuel tanks are empty, resulting in the jet engine failure). It is not surprising therefore that the final communication received from the aircraft was at 00:19:38, a possible cause being complete loss of power again, as the APU shut down. (An alternative explanation for the incomplete log-on might be mis-direction of the satellite communications antennae due to a spiral dive;  however the receiver power indications provided by Inmarsat [3] remain stable throughout all signal exchanges, and so this possibility seems less likely.)

The possibility that the aircraft impacted the sea at or very shortly after this time (00:19:38) should not be ignored: an implication of this would be that at that time the aircraft was at a low altitude, affecting the calculated position of the 7th/final ping arc, and therefore the search region for wreckage on the sea floor. Taking the aircraft altitude to be near zero at this time rather than at 35,000 feet has the effect of shifting the 7th ping arc by about 10 km to the northwest; this is discussed in more detail below.

A separate analysis of the fuel data is being undertaken within the IG [4] and it is sufficient to recognize here that the rate of fuel consumption rate for both engines combined for the remainder of the flight, from 17:07 through to 0019:29, is an average of about 6,072 kg/hr, (13,386 lbs/hr), and hence significantly lower than that shown for the take-off and climb phases of the flight as reported in the Factual Information [1], Appendix 1.6B. It is expected that the difference in fuel consumption rates between the two engines will be lower too (information in the Factual Information report indicates differences of 1.9 and 1.3 per cent, for take-off and climb: reference 4), and for the purpose of the ongoing discussion I have assumed, based on a simplified analysis, that this difference is 0.8 per cent. Using this figure it is possible to deduce that the Right engine failed close to, and perhaps a few seconds before, 00:11 UTC on 2014 March 08.

 

Flight path after the first engine failure

The simulator trials mentioned earlier illustrated clearly that following the first engine failure the auto-throttle increased thrust in the remaining engine, altitude was maintained, and the (longitudinal) pitch increased as the speed reduced so as to provide adequate lift. The deceleration observed was noted, and the Indicated Air Speeds (IAS) were converted to True Air Speeds in Knots (KTAS) for illustration in Figure 1, below. The observed trend is linear, with a deceleration of approximately 19 knots per minute. (dy/dx = -0.315 ´ 60 = -18.9). A deduction from this result is that the aircraft would still have been well above the best single engine speed and would have continued to fly at the same altitude for some minutes after the engine failure.

Anderson Fig1

Figure 1: Deceleration after first engine failure

The flight dynamics are such that although drag and the available thrust are lower at higher altitude, the minimum drag speed also increases. Hence, coincident with one engine failing, we would expect deceleration to commence immediately and a reduction in altitude to begin as the minimum drag speed is approached. This was not observed in the simulator trials commencing at FL350 (Flight Level 350, nominally 35,000 feet but in actuality the geometric altitude depends on the atmospheric temperature and pressure). In fact the speed continued to decrease (below the minimum drag speed) without a descent being automatically commenced, but the ensuing part of the simulator trial was perforce cut short by the failure of the second engine. The speed (and time) at which MH370 would have begun to descend is therefore unknown from the simulator trial.

Turning back to what we know about the actual flight, the BFO at 00:11 suggests that at that instant the aircraft was descending at about 250 feet per minute. Together then, the BFO descent rate, the estimated time of the Right engine failure, and the simulator trials, may all be reconciled if the altitude was greater than FL350, in which case it is possible that a shallow descent commenced just prior to 00:11.

The appendices in the ATSB report [2] provide an indication of possible vertical descent rates resulting from loss-of-control events at high altitude. Descent rates of greater than 20,000 feet/min have been observed.

At the rate of speed (and altitude) reduction observed in the simulator trials, and because the aircraft is observed to be within a normally-expected flight envelope, it seems clear that the aircraft would still be close to FL350 (or higher) at the time the second/Left engine failed, at 00:15:49 (+/- 10 seconds). If, as a result of total loss of control at that time (autopilot failure due to interruption of electrical power) it happened that the aircraft impacted the sea at 00:19:38, the average descent rate would have been about 9,500 feet/min, which is well within the observations referenced in the preceding paragraph.

An immediate conclusion should be drawn at this point. Taking a conservative view (conservative, that is, in terms of the plausible distance travelled from the 6th ping arc at 00:11), the 7th arc position calculation should assume that the aircraft was at or near sea level (the surface of the reference ellipsoid) at time 00:19:38, and hence only a little above sea level at 00:19:29, and descending rapidly. At the latitudes of interest this will position the 7th arc at a distance of about 56 nm from the 00;11 arc along the 186 deg (from True North) azimuth in the vicinity of a latitude of 38 degrees South.

Assuming that the deceleration is a linear function, as observed in the simulator trials, the distance travelled from time 00:11 until the Left engine flames out can be calculated. For a range of speeds at 00:11, and assuming that at that time the aircraft was exactly on the 00:11/6th ping arc, the distances are illustrated in Figure 2, below.

Anderson Fig2

Figure 2: Distance travelled from 00:11 before second engine failure

Assuming a ground speed of 480 knots at 00:11 (equivalent to a wind-corrected KTAS of about 500 knots), and noting that this is likely to be an optimistic value since various IG path models suggest ground speeds between 429 and 455 knots at this point, and with a linear deceleration of 19 knots/minute, the ground speed at 00:15:49 would have been 389 knots, and the distance travelled from the 6th arc at 00:11 is 35nm (nautical miles). On an azimuth of 186 degT this will put the aircraft 21nm short of the 7th arc calculated at sea level for when the Left engine fails. Continuing with that rate of deceleration in a straight line (allowing for the effect of wind following the second engine failure), the total distance covered before 00:19:29 is 56 nm, which is just on the 7th arc calculated at sea level, but 6nm/10km short of the 7th arc at FL350.

 Continuing deceleration after the Left engine fails is not a certainty. It is possible that the speed may increase, since it is governed primarily by the relationship between the drag and the component of the aircraft weight acting down the flight path. With the assumption that there is no further deceleration after the Left engine fails, then in 3m 40s the distance covered is 24 nm and hence the aircraft is capable of reaching the 7th arc calculated at sea level, but still not capable of reaching the (FL350) 7th arc at 00:19:29, on the 186 degT azimuth.

As a comparison it is useful to test the outcome assuming a lower ground speed, say 429 knots (rather than 480 knots) at 00:11, and assuming the same rate of deceleration. In this case the ground speed at 00:15:49 would be down to 338 knots, and the distance travelled from the 6th arc at 00:11 would be only 31nm, which is 25nm short of the 7th arc calculated at sea level. In order to reach the 7th arc in a straight line on the same 186degT azimuth, a linear acceleration of approximately 50 knots per minute would be required, reaching a ground speed of 485 knots at 00:19:29. This seems unlikely, and indicates a likelihood that the speed at 00:11 was greater than 429 knots. Perhaps even more likely is the possibility that the aircraft turned towards the 7th arc at the time of the second engine failure.

 

Flight path after the second engine failure

Observations from the simulator trials suggests that following the failure of both engines the aircraft will bank into a turn almost immediately. The direction of the turn, and perhaps the rate of the initial turn, is a function of precisely how the aircraft was trimmed immediately before the first engine failure, and this is of course unknown. However, the observation that the aircraft may not be capable of reaching the 7th arc if that is assumed to be at 35,000 feet (as discussed above) may help in reaching a conclusion here.

For example, if the aircraft banked and turned to the right after the Left/second engine failed, then with the speed profile examined above, commencing at 480 knots at 00:11, it would not be possible to intercept the 7th arc until after 00:19:29, and even then the turn radius would have to be greater than about 48nm to intercept the arc at all.

Alternatively, if the aircraft banked and turned to the left after the Left/second engine failed then an intercept with the 7th arc at 00:19:29 is possible, but only if the turn radius is between 8 and 10 nm. At the assumed speed at which the turn commenced the required bank angle is approximately 15 degrees. Figure 3, below, illustrates these possibilities.

Anderson Fig3

Figure 3: Illustrating possible tracks after second/Left engine failure

The simulator trials illustrate that after entering a banked turn, even one with a bank angle as shallow as 15 degrees, the aircraft does not recover to a wings-level attitude. Rather, over a period of about three minutes, and interspersed with possible phugoids, the bank angle will continue to increase until the aircraft enters a spiral dive. At that point the bank angle may have increased to 90 degrees, the aircraft may have rotated through three complete turns, the aircraft speed with respect to the air would exceed the normally-allowed maximum operating speed (VMO), but not necessarily have increased beyond maximum Mach operating speed (MMO) at that altitude so that it would likely not have suffered severe structural damage, and the descent rate would be up to 15,000 feet/minute. A high-speed uncontrolled impact with the sea would be inevitable.

Note that a descent rate of 15,000 feet/minute at the airspeeds of interest requires an aircraft track dipping only about 20 degrees below the horizon. More extreme descent rates are certainly possible.

 

Conclusions

Based on this analysis one may conclude that:

  1. It is only with ground speeds greater than about 440 knots at 00:11 that it is possible subsequently to reach the 7th arc at sea level at all;
  2. Following the Left/second engine failure, the aircraft very likely entered a turn to the left, a turn which developed into a spiral dive over the course of a few minutes resulting in a high speed impact inside the 7th arc as calculated for sea level.

It is clear from this analysis that establishing precisely where the 7th arc is located is very important from the point of view determining the path of the final few minutes of flight, and the underwater search strategy to be used. Based on the corrected BTO (18,400 microseconds) at 00:19:29 provided by the ATSB (2), the advisable position would be to establish the 7th arc at the surface of the reference ellipsoid (i.e., sea level) and not at high altitude (say 35,000 feet) the distinction between these shifting the 7th arc by about 6nm/10km.

 

Acknowledgements

 I thank members of the Independent Group (IG) for the active discussion, and contributions, which have greatly helped me to present this analysis and to arrive at the above conclusions. It is hoped that these will assist the official search teams in their identification of where to concentrate their efforts to achieve the highest likelihood of timely success.

 

References

[1]       Factual Information: Safety Investigation for MH370, published 08 March 2015, Malaysian ICAO Annex 13 Safety Investigation Team for MH370, Ministry of Transport, Malaysia.

[2]       MH370 – Definition of Underwater Search Areas, 26 June 2014 (updated 18 August), ATSB Transport Safety Report.

[3]       The Search for MH370, Journal of Navigation, September 2014, authors Chris Ashton, Alan Shuster Bruce, Gary Colledge and Mark Dickinson (Inmarsat).

[4]       Fuel Burn Analysis, spreadsheet by Mike Exner, April 2015. (Note that this link to Exner’s spreadsheet was added on 2015 June 20.)

 

Dependence of the Orbital Debris Collision Hazard upon Inclination for Low-Earth Orbit Satellites

Dependence of the Orbital Debris Collision Hazard upon Inclination for Low-Earth Orbit Satellites

Duncan Steel
2015 April 28

 

Introduction

In my previous post concerning the orbital debris collision hazard for test satellites in low-Earth orbit (LEO) I used only two values of the inclination to the equator (30 and 98 degrees) as examples, that post being directed mainly towards elucidating how the collision probabilities vary with altitude. I wrote there that I would prepare another post that focusses upon the inclination-dependence: this is it.

The method and input orbital data for tracked objects used here are precisely the same as in that first post, and so I will not repeat what I presented there in that regard.

In Figure 1 below I show how the collision probability varies with the inclination of various test satellites (assumed spherical, cross-section one square metre) in circular orbits at differing altitudes in the LEO zone. The inclinations used go in ten degree steps from 0 to 180 degrees, except that three additional inclinations were inserted: 28.5 degrees (the geocentric latitude of Cape Canaveral), 45 degrees (near the latitude of the Baikonur launch site) and 98 degrees (the approximate inclination of satellites in sun-synchronous orbits in LEO).

Plot5

Figure 1: Variation of the collision probability against the publicly-available orbits of tracked objects in geocentric orbit as a function of inclination for circular test satellite orbits at altitudes of
500, 700, 800, 900, 1000 and 1500 km.


Discussion

The following salient points can be understood from Figure 1:

  1. Whilst all lines for any particular satellite altitude show peaks between 60 and 120 degrees, at higher inclinations (i.e. highly retrograde test orbits) the collision probabilities reduce. Given that the collision probability varies linearly with the relative velocity of potentially-colliding objects, and the fact that most of the tracked objects are in prograde orbits, this might not have been expected. The explanation for this feature (i.e. decreases in collision probability for highly retrograde orbits) is that if the orbital plane of the test satellite is close to the equator – whether prograde (i < 60 deg) or retrograde (i > 120 deg) – then that satellite spends much of each orbit at lower latitudes than the preponderance of tracked objects in near-polar orbits. This results in the overall collision probability being reduced.
  2. The lines for test satellite altitudes 700, 800 and 900 km show broad maxima centred on inclination 80 degrees, and may be understood as being a result of the collision probability varying with the relative velocity of potentially-colliding objects: a satellite at such an inclination (say 80-100 degrees) has an elevated probability of colliding head-on with a piece of tracked debris (or indeed a functioning payload) that has a similar inclination but is travelling in the opposite direction. In addition, the collision between Cosmos 2251 (inclination 74 degrees) and Iridium 33 (inclination 86 degrees) at an altitude of about 789 km resulted in almost 2,000 items of tracked debris (and many smaller).
  3. The highest collision probabilities are at altitudes of 700-900 km, consistent with my previous post.
  4. Subsidiary maxima are seen at inclinations near 98 degrees at altitudes 800, 900, 1000 and 1500 km. These may be understood as being due to the larger populations of tracked items at or near this inclination, the possibility of being temporarily coplanar during cycles of precession of the nodes leading to increased collision probabilities. The reason for the larger populations near inclination 98 degrees include: (a) More satellites being inserted into orbits at altitudes above 800 km with such an inclination (i.e. sun-synchronous orbits); (b) The Chinese anti-satellite demonstration in 2007 January, when the weather satellite Fengyun-1C was destroyed in an orbit at altitude near 865 km and inclination 98.9 degrees, resulting in almost 3,000 items of tracked debris (and many more too small to be detected from the ground); (c) On 2015 February 03 the U.S. Defense Meteorological Satellite DMSP-F13, which had been launched in 1995, exploded in orbit at an altitude near 850 km and inclination 98.6 degrees, leaving hundreds of trackable fragments in nearby orbits.

 

Concluding remarks

There is one core point that I intended to make in this post: the probability of losing a functioning satellite due to a collision with orbital debris is not just a function of the satellite altitude: there is also a strong dependence on the orbital inclination (to the equator) that is chosen for the satellite. The information plotted in Figure 1 indicates that the collision probability at any altitude can vary by a factor of two or three across the full inclination range, with near-polar orbits being the most dangerous to occupy.

This post is just the second in a series in which I intend to describe and discuss various specific points regarding the orbital debris impact hazard for functioning satellites. Further posts will follow.