The designated 7th Arc search area was based in part on the assumption of a straight flight path after flight MH370 turned south. The timing of the ping rings indicates that the plane appeared to fly on either a linear or curved path that approached and then receded from satellite. It would have been closest to the satellite and flying tangent to it shortly after the 2nd arc ping timing. New candidate search areas need to be consistent with the tangent heading at that time. Improving the estimate of the tangent timing could be important in finding the plane.
Early analysis of ping timing on the INMARSAT 3F1 satellite communications with the plane throughout its flight showed that the almost hourly Burst Transfer Offset (BTO) delay samples over time fit onto a smooth curve. The BTO represents distance from the satellite, and the timing of the minimum on the curve is when the plane would be flying tangent to it. The smooth curve could be representative of a constant flight heading, a steadily changing heading, or possibly other more complex paths that still match the BTO timing.
The radio frequency Doppler shift would also be at a minimum at the tangent point, which should be useful for calibrating the BFO (Beat Frequency Oscillator) measurement variations.
The timing is somewhat complicated by the movement of the satellite, as its eccentric orbit takes it hundreds of kilometers from its average geostationary parking spot over the equator, but that movement was factored into the calculation of the seven ping arc locations.
Early estimates of the BFO did not take the satellite movement into account, partly because the plane itself uses the average geostationary spot above the equator when compensating for Doppler shift based on instrument readings of its movement. BFO calculations will not be addressed here, beyond noting that other researchers have used BFO to conclude the the plane was flying between waypoints before a final turn south around waypoint IGOGU about 70 km NE of Nicobar Islands.
Researcher Brian Anderson published BTO research reports early on that detail the math of finding the BTO minimum by fitting a third order polynomial to all of the BTO samples. He found a minimum time at 19:52:12 UTC. Using a simple straight line flight path and a starting point after the the turn south resulted in a tangent point heading of 186 degrees, which then gave a 7th arc crossing location and ground speed estimate of 494 kt. He also showed a geometric method for finding the flight speed given that the tangent is between the 2nd and 3rd arcs.
A Slightly Different Tangential Approach
Without prior benefit of Brian’s insights, fitting a third order polynomial curve to the BTO samples in MATLAB produced similar results. To incorporate the satellite movement, the curve was then fit to the derived radius of each arc. The time had moved to 19:52:46 UTC. To find the heading at the tangent for a straight line path, an iterative method was used to find where two great circle paths intersected at 90 degrees, one to the sub-satellite position at the fitted minimum radius, and one to the last turn. The results are very sensitive to the turning point, but using the post-turn 18:40:56 Final Major Turn (FMT) marker 7.2732N 94.3516E (per Richard Godfrey’s v13.1 spreadsheet model) resulted in a ground track at the tangent of 185.2 degrees at -1.04725S 93.5979E. The average ground speed to get there was 772 kph = 416 kt.
Seeking a Better Fit
A third order polynomial through all the points requires some least-squares fitting to the data. Using fewer data point should get a more precise fit. Fitting data points before the turn south violates the assumption of a straight line flight path. Fitting points late in the flight also would affect the accuracy at the tangent. This leaves fitting at arcs 2 through 5. A third order polynomial still fits the pings around 18:28 in the Straits of Malacca, but not the FMT mark at 18:40:56. The tangent point has moved a minute later to 19:53:12 UTC, which gives a 7th arc crossing even farther south.
By fitting a simpler second order polynomial to the four mid-flight arc rings, the fit is still very good and the curve is now a nearly exact match with the 18:40:56 turn.
The tangent point also moved significantly to 19:45:39 with a radius to the sub-satellite point of 3253 km. A match with the 18:40:56 turn gives a similar heading 185.07 south of the equator at lat: -0.974183S 93.6212E. The speed is still very high to reach this point at 849 kph = 457 kt. The 7th arc crossing from bearing 185 puts it in an area already searched, near the shipwreck site scanned in May 2015.
It is notable that the plane was flying between waypoints just before it turned south. If the turn at IGOGU was to the next waypoint to the SSE, it would likely to have been BEDAX. Closer due south would have been ANOKO and NOPEK, then SSE to BEDAX. The next waypoint beyond BEDAX is just south of the equator at ISBIX, where the straight line tangents are near the minimum. It is practical to assume that the plane may not have gone on a direct or random heading past ISBIX, but instead continued to fly between waypoints if they better match the timing.
The second ping ring just grazes the long BEDAX-ISBIX path segment, which is very close to it being tangent. This seems like more than a coincidence. The 19:45:39 tangent timing is just 4:36 after the second arc timing of 19:41:03 UTC.
Finding the iterative tangent using BEDAX (5.364330N 93.78758E) as the starting point gets a heading of 181.651 at a distance of 425.26 km from BEDAX. This is close to the BEDAX-ISBIX heading of 181.295. The tangent point with this method at 1.51866N 93.7447E is just 5 km from the waypoint path. This is a good match given that the waypoints are only approximating a curve. The error is just 0.15% of the predicted minimum radius distance of 3253 km.
Adding the distance from the 18:40:56 FMT to BEDAX of 220.13 km gives 645.4 km. Using the time from FMT to the tangent time of 19:45:39 gives an average ground speed of 598 kph = 322.3 kt. The second arc timing at 19:41:03 on this waypoint path and speed would be 45.8 km north on the flight path. That point is 6.6 km from the second arc ring. The 2nd arc ring was previously calculated with an assumption of flight level of 35K ft, which is too high for a speed of 322 kt. Adjusting the BTO calc for a sub-satellite distance at 19:41:03 of 3248.07 on the waypoint path for an altitude below 15K ft gives a nearly exact match for the second arc and the waypoint path.
Matching the plane precisely on the waypoint path begs the question of where the plane went after ISBIX. If that was the last waypoint entry, it has been reported that the flight management computer would have continued on the last heading of 181.30 degrees. A flight speed of 322 kt is nearer the best holding speed, but reaching the 7th Arc on that heading would require higher LRC (Long Range Cruise) efficiency speeds, and thus a high altitude.
The previous details about a possible detection of a Cocos Island flyby at 22:22:22 make it likely waypoint destination, especially since it has a runway that handles commercial jets. The slower average speed to Cocos is compatible with the “tangent” estimate for a second arc crossing. With some speed variations, the arc timing for a flight path along waypoints ISBIX-POSOD-COCO is a close match to the BTO and ping rings, including the incidental curve fit.
If Cocos Island was the final destination, but with no pilot ability to land or further navigate by then, the plane would have continued on the POSOD-COCO bearing of 163.28 degrees. This heading would have carried it over an early search area east of Batavia sea mount where underwater pings like those of the black box were detected by a Chinese search vessel near 25S 101E (later dismissed). The endpoint happens to match early calculations here of an intersection point at the 7th arc with the Curtin event on bearing 301.6 from the hydrophone array H01. That would still be a possibility if the detected Curtin event were from a late underwater implosion near the SOFAR deep sound channel at about 1000m depth.
Using the tangent point on the assumption of a constant heading points to areas already searched, and perhaps out of reach. A waypoint path is a much better fit for the ping rings and speed to Cocos Island, with a closer northern 7th arc endpoint.
Flying at a lower speed and altitude is also compatible with the crew of a disabled plane attempting to supply the passengers with air after the oxygen masks ran out. If the plane was unable to dump fuel but was responding to autopilot entries, a decision might have been made to run most of the fuel out flying to Cocos Island before ditching, rather than running a holding pattern and then attempting to land near a more populated area.