**Introduction**

As readers here know, we have considered the possibility that MH370 turned to the south later than 18:40 UTC and crashed along the 7th arc to the north of the seabed that was searched. What led investigators in the past to believe that MH370 turned at some point between 18:28 and 18:40 are the satellite data obtained at times that bookend this time interval. If we assume that MH370 was flying at nearly constant altitude, the BFO value at 18:28 is consistent with a plane flying at 500 knots and on a track of 296°T, which puts it parallel to airway N571 around waypoint NILAM. Similarly, the BFO value at 18:40 is consistent with a plane flying at on a track of 180°T at about 462 knots. That means a turn to the south must have occurred during this time interval if the plane was flying at constant altitude. However, the BFO values at 18:40 also match a plane descending and maintaining a northwest track, which would imply a later turn to the south. Here we consider this possibility. In particular, we consider whether:

- The combinations of groundspeed and vertical speed required to match the BFO at 18:40 are typical of a B777 in a descent
- The variation in the BFO values recorded at 18:40 are what we would expect if MH370 was descending

**BFO Values at 18:40 UTC**

A log-on sequence to Inmarsat’s satellite network occurred between 18:25 and 18:28. The BTO and BFO data obtained during the log-on suggests that MH370 was flying parallel but to the right of airway N571, near waypoint NILAM. Commenter @Andrew, a former B777 pilot, advised us that in the event that an aircraft does not have clearance to fly an assigned airway, a 15-NM offset from the airway is recommended to avoid other traffic. It is therefore possible that MH370’s pilot, knowing that the transponder was inoperative, chose to fly at 15 NM to the right of N571 to avoid other traffic.

At the time of the call at 18:40, if MH370 was flying offset from N571, it would have been just past waypoint IGOGU in the Andaman Sea and flying towards the Nicobar Islands on a track of 294°T. The call at 18:40 produced 49 BFO values that were recorded over a period of about one minute. The values ranged between 86 Hz and 90 Hz, and averaged 88 Hz. Using this average value of 88 Hz, we can determine the values of groundspeed and vertical speed that would produce a BFO value of 88 Hz for a plane flying along a track of 294°T. Knowing the groundspeed (GS) and vertical speed (VS), the BFO is calculated from the equation

BFO(Hz) = 128.8 + 0.0372 GS(kn) + 0.0228 VS(fpm)

A table of selected values of groundspeed and vertical speed that result in a BFO value of 88 Hz is shown below. Also shown in the table is the calculated flight path angle (FPA), which is the negative of the descent angle, and gives some indication as to whether drag or thrust is required to maintain a particular descent rate. In general, a descent angle greater than around 3.0° will require additional drag (by deploying the spoilers, for instance), and a descent angle less than around 3.0° will require thrust from the engines. A plane gliding with a descent angle of 3.0° has a lift-to-drag ratio (L/D) of around 19. We see in the table that for groundspeed between 450 and 500 kn, the descent angle is between 2.9° and 3.2°, and the vertical speed is between -2500 and -2600 fpm. This translates to typical descent angles.

Descent Conditions to Match the BFO Values at 18:40

**Automated Descent of MH370**

Because the BFO varies with groundspeed and vertical speed, any variation in either of these two parameters would be represented as a variation in the BFO values that were recorded. Using the BFO equation, we find that either a ±54-knot variation in ground speed or a ±88-fpm variation would produce a ±2-Hz variation in BFO. For an automated descent, once established, it would be rare to see a 54-knot change in groundspeed over the course of a minute. However, a 88-fpm change in vertical speed is very possible, especially for descent modes in which the elevator is used to control either Mach number (M) or indicated airspeed (IAS). For this reason, we consider that MH370 was descending in an automated mode that minimizes the variation in vertical speed. In particular, we consider a “V/S” descent in which the elevator is used to maintain the selected vertical speed and the autothrottle is used to maintain the selected airspeed, i.e., the selected value of M or IAS.

The figure below shows the calculated values of BFO for a descent in which VS=-2600 fpm is held constant and the airspeed was selected as M=0.8. The calculations assume a headwind of 3 kn, a temperature of ISA+10.8K, and a ratio of 1.06 between the geometric and pressure altitudes, which is all based on the appropriate GDAS meteorological data. The assumed descent is between FL340 to FL200, and lasts for about 5.3 min. The BFO values are calculated assuming a bias of 151 Hz, i.e., the oscillator in the SATCOM has drifted up by 1 Hz in frequency from the time it was at KLIA. This small drift allows the calculated BFO to match the measured values of BFO with a value of VS that is a multiple of 100 fpm, which is the resolution of VS that is selectable by the thumbwheel on the Mode Control Panel (MCP) of a B777. In fact, we don’t precisely know the true value of the BFO bias, but a 1-Hz drift is possible.

Also shown in the preceding figure are all 49 recorded values of BFO at 18:40. The best-fit line of the BFO data indicates a trend with a slope of 0.4 Hz/min, corresponding to a variation of ±0.2 Hz about the mean over that 1-minute interval, while the measured variation in BFO is ±2 Hz about the mean of 88 Hz. From this, we conclude that there is no discernable trend in the BFO data over this 1-minute interval, i.e., the BFO values do not appear to vary with time. The ±2 Hz variation in the BFO data is therefore treated as noise.

For the first two minutes of the calculated descent, the airspeed is held at a constant M=0.8, and the IAS increases during the descent from its initial value of 278 kn. As the outside air temperature increases during the descent, the true airspeed (TAS) increases, which causes the BFO to rise at a rate of 0.2 Hz/min. This corresponds to a variation of BFO of ±0.1 Hz over the 1-minute interval of the recorded BFO data. This variation is much smaller than the value of ±2 Hz due to BFO noise.

After about two minutes of descent at M0.8, the plane reaches a pressure altitude of 29,000 ft (FL290), and the airspeed has increased to 310 kn. At this speed, the autothrottle automatically changes modes and begins to maintain a constant airspeed of 310 KIAS during the descent. The Mach number also begins to fall. Increasing air density during the descent causes a reduction of true airspeed with a corresponding reduction in groundspeed, and the calculated BFO changes at a rate of -0.6 Hz/min. If the descent was timed so that BFO data was recorded during the descent at 310 KIAS, the variation in BFO due to the descent would be ±0.3 Hz. Although higher than the variation in BFO during a descent at M0.8, the variation is still much lower than the ±2 Hz due to BFO noise.

It should be noted that when 310 KIAS is displayed in the speed window of the MCP, this value represents the minimum airspeed during the descent at VS=-2600 fpm. If the airspeed falls below 310 KIAS, the autothrottle would increase the thrust to restore the airspeed to 310 KIAS. However, if the speed increases past 310 KIAS and the thrust is already at idle, the speed can only be maintained if the pilot adds drag by operating the speed brake lever. If the speed brake is not used, then the airspeed will increase past 310 KIAS. The net effect of this increase in airspeed will be to reduce the rate of BFO from falling at -0.6 Hz/min. Since we are concerned here with the variation of BFO during the descent, modeling the descent at a constant value of 310 KIAS provides a worst case estimate of this variation.

**Conclusion**

The timing of MH370’s final turn to the south has an important impact on the estimated end point along the 7th arc. The later the timing of the turn, the further north the end point is predicted to be. Although the BFO values at 18:40 UTC recorded during a 1-minute interval suggest the plane was flying at constant altitude and had already turned south, an alternative interpretation is the plane was still traveling northwest but was descending. Here, we find that the combination of groundspeed and vertical speed that is required to match the BFO at 18:40 also corresponds to a typical descent rate of around 3°. We also find that over the 1-minute interval, the change in BFO that is expected due to this descent is small compared to the noise in the BFO values that were recorded. An autopilot mode that minimizes the variation in BFO is V/S at -2600 fpm.

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