The Last 15 Minutes of Flight of MH370

2015/04/28 Duncan

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.

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:

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;
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.)

Astronomy & Space Science, Orbital Debris

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

2015/04/28 Duncan

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:

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.
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).
The highest collision probabilities are at altitudes of 700-900 km, consistent with my previous post.
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.

Astronomy & Space Science, Orbital Debris

Assessment of the Orbital Debris Collision Hazard for Low-Earth Orbit Satellites

2015/04/27 Duncan

Assessment of the Orbital Debris Collision Hazard for Low-Earth Orbit Satellites

Duncan Steel
2015 April 26

Introduction

The fact that anthropogenic orbital debris (or ‘space junk’) poses a collision hazard to functioning satellites in geocentric orbits is well-known, although the level of the risk is not always recognised.

For example, there is often talk of the chance of communications satellites in geostationary orbit being struck by debris, disrupting global links and TV broadcasts, and from that notion stems the belief that once such satellites reach the end of their useful lives they should be boosted into higher orbits simply so as to obviate the risk of collisions with functioning payloads. This belief is false, as I will show in another post: at geostationary altitudes (circa 35,800 km above Earth’s surface) the probability of being struck by man-made debris, or other satellites, is tiny; and the impact risk is dominated by the hazard posed by natural meteoroids and interplanetary dust, which does not depend on the altitude.

On the other hand, in low-Earth orbit (LEO: altitudes below about 2,000 km) the population of debris and the relative velocities are such that the impact hazard posed by material we have launched into orbit is substantial, and too high to be ignored. The intent of the present paper is simply to illustrate the sorts of collision probabilities that exist in the present epoch.

The space debris problem has attracted the attention of the world’s space agencies, each of which has produced its own reports and recommendations over the past two or three decades, and also various national and international organizations such as the United Nations. There are too many reports that have been published to reference any but a few of them here. As an introduction I would just recommend to readers two substantial reports published in the last few years by US organisations: the RAND Corporation (Baiocchi & Welser, 2010) and the National Research Council (2011).

For my own part, my own work on the orbital debris problem began in 1988 when I established companies called Spaceguard Proprietary Limited (P/L) and SODA Software P/L (SODA meaning Spaceguard Orbital Debris Analysts) in South Australia. Much of the research I did then, having commercial applications, was not openly published. However, a few of the papers I did publish in following years, based on the software which I have used in deriving the results that appear in this post, are listed in the references below as Steel (1992a,b) and Dittberner et al. (1991).

The intent of the present post is not to give a final word on this matter, but rather to provide a primer for interested readers; I will be preparing further posts on the overall subject of orbital debris and the hazard it poses to present and planned satellites.

Input orbital data

In this analysis I have used all the objects available in the Satellite Situation Report (SSR), which is freely available from Space-Track.org (specifically, here). There were various inconsistencies that I identified in the SSR by comparison with the Two-Line Elements (TLEs: available from here). The SSR and TLE datasets used were the most recent available as at 2015 April 09.

My dataset compiled as above contained 40,586 objects, about 60 per cent of which have departed geocentric orbit: some of those have been sent on Earth-escaping paths (e.g. interplanetary probes) but the majority of the objects that are no longer in orbit are those which have re-entered the atmosphere. A total of 16,167 objects were found to have usable orbits for present purposes (i.e. determining collision probabilities with test/strawman satellite orbits).

Apart from that 16,167 there are about 800 objects still in geocentric orbit but listed in the publicly-available SSR as having “NO ELEMENTS AVAILABLE”. In general these are CLASSIFIED orbits: that is, they are Defence- or intelligence agency-related objects of the United States and its allies (including here launches by the United Kingdom, Germany, Israel and Japan, for example). That is, these are payloads, rocket bodies and debris associated with CLASSIFIED launches. Based on 800 being about five per cent of 16,167, one might imagine that the overall collision probability derived from the 16,167 orbits that are available might be about five per cent lower than the value which would result if all orbits were available for analysis; however, personal knowledge indicates that the CLASSIFIED launches include a disproportionate number of surveillance satellites placed into LEO, and many of these are high-mass and capable of producing a large amount of debris on fragmentation. In view of this I would anticipate that the net collision probability derived using the available 16,167 orbits is likely to be about 25 to 50 per cent lower than that which would result if the approximately 800 tracked CLASSIFIED orbits were also available for use here.

Apart from that consideration, of course the objects/orbits available are restricted to those which are large enough to be tracked using ground-based optical and radar sensors. For LEO this leads to a size limitation of about 10 cm (i.e. the size of a baseball, or a large orange) although the precise limit is dependent on several unknown factors separately for optical and radar detections. The fact is that several dozens of catastrophic on-orbit fragmentation events have occurred, and following such events only a minor fraction of the resultant debris items are detectable, those below 10 cm in size being expected to follow broadly similar orbits to the original object but not be trackable. It is believed that the number of debris items larger than 1 cm in size in LEO is above 100,000 and may well be substantially greater than that. Since a 1 cm object striking a functioning satellite at a speed of around 10 km/sec would be anticipated to cause calamitous damage, the net collision probability derived from only the 16,167 large tracked objects very much represents a lower bound for the overall debris impact risk. The real hazard is likely at least ten times higher.

For my own early assessment of the importance of smaller (untracked) fragments, please see Steel (1992a).

Method

The technique used to derive the collision probabilities between orbiting objects is that developed by Kessler (1981) and also described by Steel & Baggaley (1985). In the context of collisions between objects in heliocentric orbits (e.g. impacts of asteroids or comets on planets) the utility and possible limitations of Kessler’s method has been investigated by various researchers, for example Milani et al. (1990).

In the present instance I have used this method to derive collision probabilities between objects in geocentric orbits. Those orbits are described in terms of the semi-major axis (a), eccentricity (e) and inclination (i, referred to the equatorial plane) of each. Rather than a and e it is often simpler to provide input in terms of the perigee and apogee altitudes for all objects, and convert those into the geocentric perigee and apogee distances, and thence semi-major axes and eccentricities, for each object. In this report all altitudes are referred to the Earth equatorial radius of 6,378.137 km.

Kessler’s method involves a numerical integration of the collision probability values across all small volumes of space around the Earth which both orbiting objects can access. These small volumes or cells have dimensions defined in the software, and in these computations I have used radial steps of one kilometre and (geocentric) latitudinal steps of one degree. It is found that the results obtained for the collision probability asymptotically approach a constant value (for pairs of test orbits) as the fineness of these steps in radial distance and latitude are decreased, and the above adopted values render an acceptable trade-off between precision and computer execution time.

An essential assumption inherent in Kessler’s method is that the argument of perigee (AOP) and the longitude of the ascending node (LAN) are random, so that all AOP and LAN values have equal probabilities of occurrence. For objects in geocentric orbits that do not have controlled parameters (e.g. station-keeping) this will be the case, with the AOP and LAN values cycling quite rapidly. For the sake of interest and understanding I will make available some example movies here, demonstrating how the orbits of objects disperse upon fragmentation due to differential gravitational perturbations caused by the Earth’s non-sphericity.

The 3D view below introduces a set of modelled fragments of a satellite, originally in a circular orbit at altitude 900 km and with inclination 60 degrees, which disintegrates for some reason with the fragments having speeds relative to the original object (which was moving at close to 7.4 km/sec in orbit) of up to 0.8 km/sec. Enhanced orbital speeds render apogee altitudes up to 5,000 km; reduced orbital speeds drop the perigee, but any with perigee altitude below 200 km would soon re-enter due to atmospheric drag. I have assumed (for simplicity) that all fragments start off in the same orbital plane, but these soon spread due to differential precession rates. This graphic indicates the fragment positions about half-an-hour after the disintegration:

A movie is available here (27.8 MB) showing how these fragments disperse over the next eight hours. The 2D map that follows below indicates the spread of paths attained in just eight hours in terms of the sub-satellite/fragment tracks across the Earth’s surface, with a movie (4.45 MB) again being available for download from here.

The following 3D view shows how much the orbits have spread over the next 45 days, with a movie being available here (25.3 MB) which illustrates how quickly and widely this dispersal occurs.

It was mentioned above that Kessler’s method requires an assumption that both the AOP and the LAN of the two orbiting objects under consideration are random. Clearly this is not the case for, say, a satellite in a sun-synchronous polar orbit; however, the relative values of the AOP and LAN from one object (the satellite of interest, perhaps in such a controlled orbit) referred to the others (the debris in uncontrolled orbits) will be random, making the method a viable means for deriving collision probabilities.

Collision cross-sections

Unlike in encounters between interplanetary objects and the planets, no gravitational focussing is applicable in the case of these low-mass objects in geocentric orbit, and so only the geometrical cross-sections are of import.

For present purposes I have assumed that the test satellite is of spherical shape, so that the collision cross-section is the same no matter which direction the debris approaches it. This is shown schematically in the graphic below.

Because I wish to derive a collision probability per square metre per year, I model this spherical satellite to have a cross-sectional area of one square metre, which implies a radius of 56.4 centimetres.

Using this geometrical cross-section of one square metre as being the collision cross-section against each of the 16,167 tracked objects enables an overall collision probability to be derived, but again the value arrived at will be substantially lower than the true value because in reality the collision cross-section will be enhanced by the finite sizes of the colliding objects. That is, in effect I am assuming the potentially-colliding objects to have infinitesimally-small sizes, whereas in fact many of them are very substantial in size and indeed much bigger than this small model test satellite; consider the sizes of the Hubble Space Telescope or the International Space Station, for example.

To give a numerical example, consider a cylindrical rocket body two metres in diameter (and perhaps ten metres long) that happens to pass the spherical test satellite with its long axis aligned with its relative motion vector. If that long axis passes anywhere within one metre of the surface of the spherical test satellite then there will be a collision. This renders a minimum ‘miss’ distance of 1.564 metres, and so an enhancement of the collision cross-section by a factor of (1.564/0.564)² = 7.69. Given that the rocket body is far longer than it is wide, as it tumbles through space whilst moving along its orbit about Earth we should anticipate that its collision cross-section against the model spherical test satellite is substantially higher than this factor of 7.69, and will in fact be perhaps 20 or 30 times higher than the collision cross-section I am using here (i.e. simply the geometrical cross-section of the sphere). On the other hand, many of the tracked objects in the SSR are indeed much smaller than this model spherical test satellite, consisting of shards from exploded rocket bodies, an astronaut’s glove, a mislaid wrench, circuit boards, and so on. Overall the effect of the finite sizes of tracked objects might be to increase the collision probability for a small object (the one-square-metre sphere considered here) by a factor of two or three. In later analyses I will be attempting to assess this effect more accurately than this guesstimate of two or three used in the present post.

Results

For the present post I have restricted my analysis to test satellite orbits in two inclinations only: 30 degrees and 98 degrees. The former is typical for many LEO satellites (launches due eastwards attain inclinations equal to the geocentric latitude of the launch site, and Cape Canaveral is at 28.5 degrees latitude), whilst the latter is the approximate inclination required for a sun-synchronous polar orbit. In a future post I will examine how the collision probabilities vary with inclination for LEO satellites.

In Figure 1 below I show the collision probabilities obtained for test (spherical) satellites in circular orbits at different altitudes. All 16,167 tracked objects (as described earlier) are included in the calculations, but only limited numbers will have orbits crossing each test satellite altitude, and thus contribute to the overall collision probability. For example, for the test satellite orbits at altitude 800 km (the peaks of the plots in Figure 1) there were 4,535 tracked objects in all that might collide with those test satellites; the other 11,632 do not cross that 800 km altitude.

Figure 1: Total collision probability with test satellites in
circular orbits and with inclination 30 degrees (solid line)
and 98 degrees (dashed line).

An immediate question that might come to mind from a perusal of Figure 1 is this: why is the overall collision probability at all altitudes higher at inclination 98 degrees than at 30 degrees? The simple answer is that to first order the individual collision probabilities vary linearly with the collision speed, and the collision speeds are higher for high-inclination objects. Obviously this pushes up the overall collision probability, and also the likelihood of severe damage (due to the higher impact speed). Retrograde satellites and debris are disproportionately dangerous!

In Figure 2 I have plotted the collision probabilities for twenty test satellite orbits all with perigee altitude 300 km but apogees at heights between 300 km (i.e. a circular orbit) and 1,200 km (i.e. the highest eccentricity test orbit). A comparison of Figure 1 and Figure 2 indicates that the collision probabilities are reduced by a factor of about 2 in the latter. Why? The basic reason is that with reduced perigee altitudes the test satellites spend substantial time in each orbit lower down, where the population of tracked objects (debris etc.) is lower and this reduces the overall collision probability values, despite the fact that test satellite orbits with larger eccentricities (larger range between perigee and apogee) will cross the orbits of greater numbers of tracked orbiting items.

Figure 2: As Figure 1 but for non-circular test satellite orbits.
For each the perigee altitude is 300 km, with apogee altitudes as plotted along the x-axis.

In Figure 3 (below) the test satellites’ perigee altitude has been pushed up to 500 km, and this increases the net collision probability calculations by a small factor (compare Figures 2 and 3). Again one asks: why? The answer is that now the test orbits entail the satellites spending less time far from the densest region of tracked objects at altitudes between 700 and 900 km (cf. Figure 1).

Figure 3: As Figure 2 but perigee altitude 500 km.

In Figure 4 (below) the test orbit perigee has been increased to altitude 800 km. Now the net collision probability values are increased again from Figure 3, reaching comparable values to those presented in Figure 1. Of course the test orbits with perigee equal to apogee and equal to 800 km in Figure 4 are identical to those in Figure 1.

Figure 4: As Figure 2 but perigee altitude 800 km.

Discussion

The maximum values for net collision probabilities derived here are close to 1.4 x 10^-5 /m²/year, for orbits around 800 km altitude and high inclination. Previously it has been argued that the ‘real’ orbital debris collision probability would be substantially higher than the value calculated in this way for the tracked catalogue of objects in orbit, the enhancement appropriate being of the following order:

Higher by 25-50 per cent due to CLASSIFIED objects properly needing to be included in the assessment;
Higher by a factor of two or three due to the finite sizes of tracked objects compared to the one-square-metre spherical test satellite considered here; and
Higher by a factor of perhaps 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.

If these enhancement factors are of the correct order, then the collision probability attains a value of around 5 x 10^-3 /m²/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 (Steel, 1992a), and I will be re-examining it in future posts.

Acknowledgements

I would like to thank Dr T.S. Kelso at the Analytical Graphics, Inc. (AGI) Center for Space Standards and Innovation in Colorado Springs for his assistance and advice on the contents and veracity of the Satellite Situation Report and Two-Line Elements files. I also thank AGI for the free use of the wonderful STK application, with which I produced the various graphics and movies that form part of this post.

References

Baiocchi, D. & Welser, W., ‘Confronting Space Debris’, RAND Corporation, Santa Monica, California, Report MG-1042-DARPA (2010).

Dittberner, G., J. Elder & D. Steel, ‘Orbital Debris Damage Estimates Using Coupled Pre- and Post-Impact Calculations’, 29th Aerospace Sciences Meeting of the American Institute of Aeronautics and Astronautics, Reno, Nevada; paper AIAA 91-0302 (1991).

Kessler, D. J. ‘Derivation of the collision probability between orbiting objects: The lifetimes of Jupiter’s outer moons,’ Icarus, volume 48, pp.39-48 (1981).

Milani, A., M. Carpino & F. Marzari, ‘Statistics of Close Approaches between Asteroids and Planets: Project SPACEGUARD,’ Icarus, volume 88, pp.292-335 (1990).

National Research Council, ‘Limiting Future Collision Risk to Spacecraft: An Assessment of NASA’s Meteoroid and Orbital Debris Programs’, National Academies Press, Washington, D.C. (2011).

Steel, D. ‘The Space Station Will Eat Itself’, pp.271-276 in Hypervelocity Impacts in Space (edited by J.A.M. McDonnell), University of Kent, Canterbury, U.K. (1992a).

Steel, D. ‘Space Debris: Getting the Hazard into Perspective’, International Space Year in the Pacific Basin (edited by P.M.Bainum, G.L.May, M.Nagatomo, Y.Ohkami & Yang Jiachi), Advances in the Astronautical Sciences, volume 77, pp.243-254 (1992b).

Steel, D. I., & W. J. Baggaley 1985. ‘Collisions in the solar system-I. Impacts of Apollo-Amor-Aten asteroids upon the terrestrial planets. Monthly Notices of the Royal Astronomical Society, volume 212, pp. 817-836 (1985).

Duncan Steel

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