How did the Doppler effect help narrow the search for MH370?

The final fate of Malaysia flight MH370 is still a terrible, unresolved mystery, and I can’t imagine the anguish the relatives of those on-board must be going through, but last week some work by the British satellite communications firm, Inmarsat, helped to narrow down the search.

As I understand it, nearly all of the usual aircraft communication systems became inoperable shortly after the plane left Malaysian airspace, so the normal means by which flights are tracked were not functioning. One system that remained working throughout the flight, was a “handshake” ping that continued responding to Inmarsat’s satellite every hour. In effect, the satellite, sat in geosynchronous orbit over the Indian Ocean, yelled out “Hey, is there anybody out there?” every hour at eleven minutes past the hour. Each time MH370 was hailed it replied “Yep, I’m here.” MH370 did this at 04:11, 05:11, 06:11, 07:11, 08: 11, and then it stopped responding.

Now, these “handshake” pings contained next to no information. No location, no speed, no flight status details, nothing except probably just the plane’s identifier. (I would imagine the satellite is in “contact” with several thousand planes in any given hour, so identity tags would be essential. This could be having assigned frequencies for each plane, perhaps). So, how could these pings help locate MH370?

Well, when Inmarsat’s satellite sends out a ping, an electromagnetic wave hurtles out at the speed of light. The time taken for a return ping to arrive back at the satellite would then tell you how far away the replier is. Remember that the satellite is sat roughly 36,000km above the surface of the Earth, so the problem is actually a three-dimensional one. All the satellite can know is that the replier is somewhere on the surface of a sphere which has a radius equal to the distance away, and is centred on the satellite’s position. Given that planes tend to fly at roughly constant heights, we can see that the three-dimensional sphere surface can be reduced to the circumference of a circle–which is what you see in the red lines below. It’s important to understand that those bold red lines indicate where MH370 could’ve been based on the final “handshake” at 08:11, nothing more.

We can actually do the maths fairly easily to work out the rough timescales involved. Lets imagine that a plane is directly below the satellite, exactly 36000km away. Then, assuming the plane’s reply is instantaneous, the time difference between when the satellite sends its ping and receives a reply will be equal to time = distance/speed = 36,000,000 x 2 / 300,000,000 = 0.24 seconds (note we need to double the distance between plane and satellite as the signal goes there and back). As the plane moves further from the satellite this time length will increase, which means if you can track the time differences to a high degree of accuracy, you will know how far the plane is. Of course, you would also need to know how much “delay” there is between MH370 receiving the satellite’s ping and sending its response. This is probably one of the reasons why Inmarsat looked at the communications profiles of other similar planes in Malaysia Airlines stable, so that had a good idea of how long this delay would’ve been.

On the diagram above along the southern route you can see part of the circle arcs for the earlier pings, and if you assume that the flight is roughly proceeding at constant velocity and direction, you can narrow down the likely final position substantially. Given that there was no reply at 09:11, Inmarsat concluded that the flight must’ve ended before this time. But couldn’t it have landed rather than crashed sometime between 08:11 and 09:11?

Well, if MH370 had proceeded along the northern corridor, which remember, coincides with a solution to the ping return times, then that would be a possibility. The northern corridor proceeds over vast tracts of central Asia, so places to land must be numerous. Two facts, sadly, undermine this thesis. Firstly, a plane flying over Indian, Pakistani, Kazakh, and other countries’ airspace, would likely be picked up by military radar. Conspiracy theorists might point out that this information might’ve been deliberately withheld, but to the second fact these theorists are silent. The second fact being that by analysing the Doppler effect of the pings, Inmarsat was able to categorically state that MH370 headed south. South into the endless blue of the Indian Ocean where there are no runways. And hence, the plane must’ve crashed.

The Doppler effect? What’s that again? The Doppler effect is a phenomenon that effects any wave motion when there is relative motion between the source and the “observer”. When you’re stood on a street corner and an ambulance passes you by at full pelt, sirens blaring, you will hear a sudden change in the pitch of its sirens. As the ambulance approaches you the waves are compressed, the wavelength slightly shorter than they would be if the ambulance was stationary. When the ambulance has passed, the wavelength is stretched. These shifts in the wavelengths correspond to inverse shifts of the frequency, and hence to the pitch of the sound that you hear. The animation below shows the effects–the wavefront compressed ahead of the source, and stretched behind it.

The magnitude of the shift in frequency is given by a simple equation:

frequency shift = original frequency x (speed of source/speed of wave)

For a sound wave, the speed of sound is approx. 340m/s, so a typical frequency shift for an ambulance travelling at a speed of 30m/s (roughly 60mph) with a siren of frequency 1000Hz, would be 88Hz. In other words, when this ambulance was heading towards you would hear its frequency as 1088Hz and when it headed away you would hear it as 912Hz. A distinct difference.

The Doppler effect that Inmarsat would’ve been working with wasn’t the sound wave version, but the electromagnetic wave version. In this case the percentage change of the frequency of the radio waves would be substantially smaller than the case of sound waves because the ratio (speed of plane/speed of light) is that much smaller. However, with radio frequencies in the region of 1.9GHz, the shift would still be on the order of hundreds of Hz, presumably enough to be detected by the satellite receiver. With slightly different relative motion between MH370 and the satellite in the northern and southern corridors, Inmarsat was able to look at the detected Doppler shifts and determine which way the plane went. The maths isn’t especially complex–its necessary to do some 3D trignometry to take the appropriate component of the plane’s velocity–and the conclusion was clear. MH370 headed south.

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