Steps to Calculate MH370’s Endpoint
The following steps are intended to be used with the embedded spreadsheet data and formulas shown below.
- First, we need to create six ping rings. Those ping rings, in turn, are necessary to tell us how far the plane was from its satellite on each ping (i.e., ring radius). Ping rings DO NOT tell us where the plane was on each 360 degree ping ring; that is a separate step below. (We will not be creating a 7th ping ring. There were only six viable pings between MH370 and its 3-F1 satellite.)
- Ping ring radii come from BTO values (Column B above), which are simply the number of microseconds (μs) between cockpit and satellite. Microseconds are easily converted to more common measures like meters and kilometers, and they are the only distance data available for MH370. For the 6th ping, add the BTO of 18040 μs to 495,679 μs, and divide by 2 to get 256,860 μs; that is the one-way line-of-sight microsecond distance between plane, satellite, and Perth Ground Station on the 6th ping; we will come back to it in Step 4;
- The entry in column G is a lookup and is given as 39,221 for the 6th ping;
- Now convert the 256,860 microseconds from step 2 to kilometers by multiplying it by the distance light travels in one microsecond, as shown in column E (0.299792458 km per second); you should get 77,005 km (not shown as a separate step); that is the one-way-line-of-sight distance between plane, satellite, and Perth Ground Station on the 6th ping in kilometers;
- The altitude of the satellite is a lookup and is given as 35,794 km on the 6th ping;
- For column K, we combine the radius of earth (6,378.1 km) with the satellite’s altitude (35,795 km) to get 42,173 km;
- Use a spreadsheet to calculate the Arc Cosine, which will give you the radius of the 6th ping, and the central angle;
- As shown, the radius for Ping 6 is 4,818 km. The central angle (CA) is 43.281°.
- This central angle calculation is the angle between the satellite, earth’s center, and a ping ring of interest. It does not tell us where the plane was on the ping ring; that is a different central angle calculation;
- We now know that the plane was somewhere on the 6th ping when it sent the last good ping, and we also know it was 4,818 line-of-sight kilometers from the satellite that tracked it. How do we find out exactly where the plane was on the final ping, which has a circumference of 30,272 km?
- The easiest calculation is to simply make use of orthogonal relationships by calculating the length of the red vertical Radical Line below, and dividing by 2.
- An alternative and the preferred way to find the length of the Radical line is to use the GPS coordinates for the northern and southern intersections between the orange rings. The screen capture below shows a distance of 5,340.2 km using GPS Visualizer and the coordinates shown:
- While the image above gives us a nice overview of MH370’s surface geometry, it is also useful at times to see the internal relationships, as shown in the image below.
- To double-check math like this, I recommend the online calculator at http://www.1728.org/circsect.htm. For example, let’s feed the length of the Radical Line and earth’s radius to that calculator to see what else we can learn. One of the goodies shown below is a central angle of 47.97° between earth’s center and the ends of the Radical Line. Plus, it’s just good practice to double-check. One of the first ‘whole earth’ checks I make is circumference. If it isn’t 40,075 for a radius of 6,378.1, something’s wrong and I need to find out why.
- The image above illustrates that in a profile view, the Radical Line is actually an Arc, and the top portion of an imaginary wedge extending to earth’s center.
- Incidentally, this is where Boeing is said to have placed the plane shortly after it crashed. If true, Boeing got it exactly right.