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1. Summary Analysis of the USSTRATCOM TLEs for the period 2011 Nov 10-20 UTC reveals evidence of at least 10 orbit changes. They were the result of coplanar, posigrade manoeuvres that cumulatively rotated the line of apsides +14.1 deg and increased the perigee and apogee 10.9 km and 1.4 km, respectively. Their total delta-V was approximately 18.3 m/s. The manoeuvres almost certainly are attributable to the four 11D457F bipropellant 53.9 N thrusters that faced -X (aft) in the vehicle's coordinate system. Their main purpose was attitude control, but also propellant settling (ullage burns) prior to main engine firings. Consumption of fuel plus oxidizer was about 98 kg. Total burn duration was about 1289 s. In the planned mission, the main engine would have performed two lengthy burns 127 min apart to exit LEO and head for Mars. Both burns would have been preceded by 58 s ullage burns, that would each have consumed 4.4 kg of fuel plus oxidizer and imparted delta-V of 0.82 m/s. Ullage burns best fit the magnitude of the manoeuvres, their precisely in-plane, posigrade thrust vector, and their frequency, so I am 95 percent confident of that explanation. Since the total manoeuvre delta-V is equivalent to approximately 22 single ullage burns - roughly double the 10 orbit changes evident in the TLEs - it is reasonable to speculate whether the ullage burns occurred in pairs at the mission-nominal 127 min spacing. My analysis reveals that this was a possibility, but I cannot prove it. Sections 2 and 3 describe how I identified and quantified the manoeuvres. Section 4 relates them to the propulsion system, and quantifies the fuel consumption and burn duration. Section 5 discusses the ullage burn hypothesis, and speculates about the potential value of a more sophisticated version of this analysis to the Russian failure investigation. 2. Statistical Estimate of Effect of Manoeuvres The manoeuvres began late on 2011 Nov 10 UTC and ended late on Nov 20 UTC. I followed the evolution of the orbit closely during that time, and it appeared that they occurred roughly every day, but at irregular intervals. When it became clear that they had ended, I began a new analysis, intended to estimate their contribution to the evolution of the orbit. The analyzed TLEs span epoch 11314.33481518 - 11325.37814289 (2011 Nov 10 08:02 - Nov 21 09:04:32 UTC). Separating the effects of the manoeuvres from drag perturbations required estimating the normal evolution of the orbit in the absence of manoeuvres. The drag perturbations depended on the spacecraft's area to mass ratio (A/m), which was determined through analysis of TLEs after the manoeuvres ended and before the final decay from orbit. The analysis covered the period 2011 Nov 22 - 2012 Jan 10, and employed the STOAG propagator and actual space weather; the results have been plotted on this graph: http://satobs.org/seesat_ref/phsrm/Fobos-Grunt_area_to_mass_ratio_evolution_v12.pdf Most of the points are at ~2 day intervals, and typically span the preceding ~2 day period. The following statistics are based on an assumed value of Cd of 2.2: A/m, m²/kg minimum 0.0010954 maximum 0.0013982 median 0.0012441 mean 0.0012523 std dev 0.0000808 points 28 STOAG was used to estimate the normal evolution of the orbit expected during Nov 10-21, by propagating the orbit represented by the epoch 11314.33481518 TLE, for the mean A/m of 0.00125 m^2/kg, Cd = 2.2, and actual space weather. The actual and normal evolution of perigee and apogee, argument of perigee, RAAN and inclination are shown below: http://satobs.org/seesat_ref/phsrm/Phobos-Grunt_orbit_evolution_during_manoeuvres_-_actual_and_normal.pdf That the manoeuvres were coplanar is confirmed by the negligible difference between actual and normal propagated RAAN and inclination. For the remaining elements, the approximate effect of the manoeuvres is the difference between the actual and normal rates of change: Element Actual minus Normal = Manoeuvres Apogee km/d -1.46 -1.75 +0.29 Perigee km/d +0.34 -0.69 +1.03 AOP deg/d +5.24 +3.92 +1.32 The estimated normal rate of decay of apogee and perigee were later refined to allow for the secondary effect of the increase in perigee height during this period, which progressively reduced the atmospheric density encountered, hence the rate of decay, which slightly reduced the estimated manoeuvre components: Element Actual minus Normal = Manoeuvres Apogee km/d -1.46 -1.60 +0.14 Perigee km/d +0.34 -0.65 +0.99 AOP deg/d +5.24 +3.92 +1.32 The near unity coefficients of determination in the above plots confirmed that the changes had been gradual, but their number and frequency remained to be determined. Did they occur daily or more often? 3. Ten Orbit Changes Identified Through Incremental Changes in TLEs If the manoeuvres had occurred roughly daily, then significant jumps in the perigee distance and argument of perigee should be discernible in the elements. The argument of perigee was analyzed first, since its evolution would have been only slightly affected by variations in drag. The method used was simple. For the 41 TLEs during the period of the manoeuvres, the argument of perigee was propagated from the epoch one TLE to the epoch of the next TLE in the chronological sequence, taking into account the precession due to Earth's oblateness. The actual and propagated values at each epoch were compared. In the absence of manoeuvres, the difference between them should have been negligible. Non-negligible changes were evaluated for the possibility that they were due to manoeuvres. Many of the differences were indeed negligible or very small, attributable to the normal uncertainty in the elements. Some were similar in magnitude to a manoeuvre, but were largely negated by changes of similar magnitude but opposite sign at the next TLE, indicative of temporary errors and their subsequent correction, which is a common occurrence. Ten (10) changes of ~1 deg of argument of perigee were found that were clearly the result of manoeuvres, totaling 11.5 deg. In many cases, adjacent TLEs revealed additional smaller changes, probably attributable to the same manoeuvre. It is not unusual for several TLE updates to be required before the results of a manoeuvre are fully and accurately reflected. Taking this into account, the effects of ten probable manoeuvres were quantified. The changes in argument of perigee ranged between +0.93 deg and +2.18 deg, averaging +1.41 deg, and totalling +14.09 deg. This was in excellent agreement with the statistically derived estimate of Section 2: +1.32 deg/d, over the 11 day TLE span, totalling +14.5 deg. The same method was applied to perigee distance. For the 10 manoeuvres identified on the basis of argument of perigee shift, the perigee distance changed between 0.34 km and 2.46 km, for a total of 10.93 km, in excellent agreement with the statistically derived value. As expected, this method failed to identify the apogee changes reliably, due to their small size in relation to the rate of decay, but this was not a serious problem for the analysis, because the changes to perigee distance and argument of perigee accounted for nearly all of the manoeuvre delta-V. The identified changes in argument of perigee and perigee distance are summarized below: Mnvr Date Äù ÄP UTC deg km Nov 10 1.06 0.39 Nov 11 0.98 1.27 Nov 11 0.93 0.70 Nov 13 1.17 0.47 Nov 13 1.49 1.27 Nov 15 1.57 1.65 Nov 16 1.73 1.02 Nov 18 1.95 2.46 Nov 19 1.03 0.34 Nov 20 2.18 1.35 Total 14.09 10.93 3.1 Manoeuvre Times Revealed Through Intersection of TLEs The exact time of manoeuvre was found through analysis of pairs of TLEs believed to represent the orbit before and after the manoeuvre, identified as part of the process described in the previous section. The analysis sought the time and place of intersection between the orbits, since that is where any manoeuvre must have occurred. The reliability of this method depends greatly on the accuracy of the before and after orbits, which was somewhat in doubt in this case, for two reasons. First, the time between manoeuvres was short, leaving little time for the TLEs to converge to accurate solutions after a manoeuvre. Second, the effects of the manoeuvres were small, so normal errors in the elements could cause considerable uncertainty in the point of intersection. The best approach would have been to perform a retrospective analysis of the observations to identify the coasting arcs between manoeuvres and fit new elements, which would have enabled finding the points of intersection with greater confidence. Lacking the observations, the best alternative was to use the existing TLEs with caution. The following table lists the epoch of the selected pre and post-manoeuvre TLEs, their time of intersection, and information derived: http://satobs.org/seesat_ref/phsrm/Phobos-Grunt_manoeuvre_time_and_deltaV_from_TLE_intersections.pdf The first orbit change occurred about 48 h after launch; the remainder followed at intervals ranging between 9 h and 43 h. Mean and median interval was ~27 h. Standard deviation was ~10 h. The latitude at intersection was within about 20 deg of the apex, mainly the northern. The true anomaly of the northern hemisphere intersections ranged between 8 deg and 43; in the southern hemisphere between 165 deg and 204 deg. Not shown in the table, but perhaps worth noting is that the northern hemisphere manoeuvres occurred in eclipse, the southern in sunlight. The delta-V was estimated as the difference between osculating velocity vectors at the intersection of the pre and post-manoeuvre TLEs. The reliability of this method is in doubt because the two orbits seldom intersected exactly, and because the real manoeuvre occurred over a period of time; therefore, I consider the results to be a rough estimate. Values ranged between 1.3 m/s and 3.0 m/s. Mean and median were 2.0 m/s. Total delta-V was 20.3 m/s. 3.2 Theoretical Location of Intersection and Delta-V Theoretical calculations were performed to aid in evaluating the location (true anomaly and latitude) and delta-V determined from the intersection of the TLEs. A manoeuvre occurs at the point of intersection between the orbits, the location of which is a function of the pre and post-manoeuvre semi-major axis, eccentricity and argument of perigee. The delta-V is a function of the circular orbit velocity, pre and post-manoeuvre semi-major axis, eccentricity, and flight path angle at the point of intersection. The calculations and results are in the following spreadsheet: http://satobs.org/seesat_ref/phsrm/Phobos-Grunt_manoeuvre_deltaV_estimate_10_single.xls The 10 orbit changes are evaluated across columns 4-13. Rows 4-8 define the evolution of the orbit over time, including the rate of change of perigee and apogee due to drag, derived from the STOAG analysis discussed in Section 2, and the change of perigee, apogee and argument of perigee due to the manoeuvres. The perigee and argument of perigee changes are as reported in Section 3.0. The apogee change due to manoeuvre is modelled as the mean daily rate reported in Section 2, due to the inability to extract reliable daily values from the TLEs, as discussed in Section 3.0. Rows 9-14 state the pre-manoeuvre elements. The values in column 4 (red font) are estimated from a TLE propagated to that time. Values in the subsequent columns add the results of the previous manoeuvre plus the effects of drag in the interim. Rows 15-20 apply the orbit changes due to the manoeuvres. The remaining rows calculate the two points of intersection between the pre and post-manoeuvre orbit, where the manoeuvre could have been made, assuming that the orbit change was due to a single manoeuvre. One set of intersections occurred for true anomaly between 18 deg and 63 deg, with corresponding latitude between 33 N and 51 N. The other set were near true anomaly ~175 deg, with latitude between 20 S and 51 S. Delta-V at both points was nearly identical. The magnitude of the flight path angle change was nearly identical, differing only in the sign. The small magnitude, ~0.01 deg, reveals the posigrade thrust vector. The theoretical manoeuvre location and delta-V are in reasonable agreement with the estimates of the TLE intersection method of Section 3.1: http://satobs.org/seesat_ref/phsrm/Phobos-Grunt_manoeuvre_time_and_deltaV_from_TLE_intersections_and_theory.pdf The theoretical total delta-V, 18.3 m/s, is 90 percent of the value from TLE intersections, but most of the individual values are closer. The three largest differences, which involve orbit changes #1, #2 and #10, account for 70 percent of the total difference. Due to the limitations of the intersection method, I accepted the theoretical estimates as probably closer to reality. The remainder of the analysis was to attempt to determine which engines performed the manoeuvres, the resulting fuel consumption, and the function of the burns. 4. Engines Used and Fuel Consumed Detailed technical specifications of P-G were found in documents obtained by Novosti Kosmonavtiki, scans of which are available here: http://www.novosti-kosmonavtiki.ru/phpBB2/viewtopic.php?t=12378 Allen Thomson greatly assisted my analysis by translating significant portions of those documents, in search of relevant information. Thank you, Allen! The attitude control system (ACS) thrusters and their bipropellant fuel supply were located on the cruise stage. A total of sixteen (16) 11D457F thrusters, rated at 53.9 N, 290 s Isp, were mounted in groups of four, on booms extending from the cruise stage. Each boom also held a single 17D58EF thruster, rated at 12.45 N, 260 s ISP. The 10 orbit changes of Nov 10-20 almost certainly are attributable to the four 11D457F (one per boom) that faced -X (aft) in the vehicle's coordinate system, that were used for propellant settling (ullage burns) prior to main engine firings, as well as attitude control. Close inspection of drawings and photos reveals they were canted. The cant angle does not appear to be stated in the documents on NK; I estimate it to have been at least 15 deg, probably closer to 30 deg, which I used in my calculations. Total ACS fuel plus oxidizer was 529 kg. A table in the NK documents allocates a minimum of 350 kg for the large cruise stage engine, 100 kg for the set of sixteen 11D457F thrusters, and 100 kg for the four 17D58EF thrusters, for a total of 550 kg. Whether this allocation was enforced in some way, either physically or through software is unclear, but seems unlikely. Relative the commonly reported s/c mass of 13,200 kg, the 18.3 m/s total delta-V of the 10 manoeuvres evident in the TLEs, represents fuel mass of 97.7 kg. With all four 11D457F thrusters firing, the total fuel flow rate = 4 X 53.9 / 290 / 9.806 = 0.0758 kg/s, resulting in burn duration of 1289 s. 5. Probably Ullage Burns In the planned mission, the main engine would have performed two lengthy burns 127 min apart to exit LEO and head for Mars. Both burns would have been preceded by 58 s ullage burns, that would each have consumed 4.4 kg of fuel plus oxidizer and imparted delta-V of 0.82 m/s. The total manoeuvre delta-V during Nov 10-20 is equivalent to approximately 22 single ullage burns. Since that is roughly double the 10 orbit changes evident in the TLEs, it is reasonable to speculate whether they could have been due to pairs of ullage burns 127 min apart. In the nominal mission, both main engine firings would have occurred in the southern hemisphere. The first one would have raised the apogee to about 4100 km, and the period to 131 min. The 127 min interval between burns was to allow the spacecraft to complete one revolution and return to vicinity of the perigee of the new orbit in the southern hemisphere. However, when stranded in LEO with a period of 90 min, the 127 min spacing would be equal to about 1.41 revolutions, which would result in one burn in both hemispheres. Reviewing the spreadsheet of Section 3.2 reveals that the true anomaly of the northern hemisphere burns would have ranged between 18 deg and 63 deg; mean value 38 deg. For the southern hemisphere burns, the true anomaly was within a few degrees of the mean value of 175 deg. Since the eccentricity was low, the true anomaly approximates the mean anomaly; therefore, the difference between 38 deg and 175 deg is equivalent to about 0.38 rev. Allowing for one complete revolution plus the fraction, yields 1.38 revs, which is sufficiently close to the aforementioned 1.41 revs to be interesting. The 1.38 rev spacing occurs only when the first burn is in the northern hemisphere. The spreadsheet of Section 3.2 has been modified to evaluate this possibility: http://satobs.org/seesat_ref/phsrm/Phobos-Grunt_manoeuvre_deltaV_estimate_10_pair.xls The single manoeuvre in either the northern or southern hemispheres, has been replaced by two manoeuvres half the size; the first in the northern hemisphere; the second in the southern hemisphere. The total delta-V of 18.4 m/s is nearly identical to the single-burn version, but it is now divided between a northern and a southern hemisphere burn. Row 52 shows that the time of flight for this scenario would average about 2h04m24s, or 124.4 min - within 2 percent of the nominal 127 min. Ullage burns best fit the magnitude of the manoeuvres, their precisely in-plane, posigrade thrust vector, and their frequency, so I am 95 percent confident of that explanation. Although I have shown that that they could have occurred in pairs 127 min apart, with the first one always in the northern hemisphere, I have not proved it, so I am only 50 percent confident of that part of the explanation. Given the lack of telemetry, I would expect the Russian investigators to have performed a similar, but more sophisticated version of my analysis, in order to determine the likely state of the hardware and software that would account for such behaviour. That state may be more symptom than cause of the failure to ignite the main engines, but it could offer clues, or help narrow down the list of possible causes. I look forward to read their report, to see how they approached these mysterious manoeuvres, how they explained them, and whether and how it aided their investigation. Ted Molczan _______________________________________________ Seesat-l mailing list http://mailman.satobs.org/mailman/listinfo/seesat-l

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