The following figure shows the Earth from a vantage point above the west coast of Australia. Three lines (semi-ellipses) of longitude are shown in colour. The red one passes through the point on the surface directly below the Inmarsat-3F1 satellite. The green one passes through the last known position (last secondary radar contact) of flight MH370. The red one passes through the two large pieces of debris spotted several days ago by the Australian government. The three points on the Earth's surface are marked with dots in the corresponding colours. The left and right limbs of the Earth do not correspond exactly to lines of longitude. In order to approximately define the visible horizon, I have shown the western-most and eastern-most lines of longitude which are almost fully visible from the selected vantage point.
The co-ordinates of the lines of longitude and the coloured dots are as follows:
Western-most (leftmost) visible longitude = 15 deg E
Co-ordinates of Inmarsat-3F1 = 64.4804 deg E, 0.7495 deg N
Co-ordinates of last known position = 103.5786 deg E, 6.9208 deg N
Co-ordinates of debris sighting = 43.9761 deg S, 90.9603 deg E
Eastern-most (rightmost) visible longitude = 175 deg E
Note the following distances:
Radius of the Earth, to sea level = 6,371 km (kilometers)
Altitude of Inmarsat-3F1 = 35,786 km
Estimated altitude from during silent flight = 5 km
Using trigonometry, one can calculate the distance from the satellite to the airplane:
Distance from Inmarsat-3F1 to last known position = 37,455 km
Distance from Inmarsat-3F1 to debris field = 42,515 km
Now, using a value of 298,000 km/sec for the speed of light, the time taken for a radio signal to make a round-trip from the satellite to MH370 can be calculated as:
... to last known position = 0.251 seconds
... to debris sighted = 0.285 seconds
Several days ago, a spokesman for Inmarsat said that they had observed "the pings lengthening" as the flight progressed. The round-trip times I have just calculated suggest that the round-trip ping duration increased by 34 milliseconds during the silent part of the flight.
One can also calculate the average speed of flight MH370. To do this, one must know the following times:
Time at last known position = 17:21 UTC on March 8, 2014
Time of last satellite ping = 24:11 UTC, just after midnight
Subtracting the latter from the former gives the time of silent flight as 6.83 hours
The distance flown by MH370 during its silent flight can be calculated using the angle enclosed by two of the rays shown in the figure. The first is the ray from the Earth's center (the black dot) to the last known position (the green dot). The second ray extends from the Earth's center to the debris field (the red dot). The enclosed angle is 52.16 degrees. Assuming flight MH370 flew a great circle route, this enclosed angle represents 52.16 / 360 = 14.49% of a complete circle. Since the Earth's radius is 6,371 km, the Earth's circumference is 2 times pi times the radius, or 40,030 km. MH370 flew 14.49% of this circumference, or a distance of 5,800 km. It flew this distance in 6.83 hours, at an average speed of 5,800 km / 6.83 hours = 849 kilometers per hour. This can be converted into knots at a rate of 0.54 knots per km, to obtain 460 knots.
References:
a.
Telecom, Media and Finance Associates, Inc. reports that the pings from MH370 were exchanged with the Inmarsat-3F1 satellite at 64E longitude.
b.
Home - Inmarsat states that their satellites are in geostationary orbit at an altitude of 35,786 km (22,236 miles) above sea level.
c.
Online Satellite Calculations states that the location of Inmarsat-3F1 at 22:51 UTC on March 22, 2014, was longitude 64.4804 deg E and latitude 0.7495 deg N. [As the tilt of the Earth changes during the seasons, geostationary satellites drift above and below the equatorial plane. Since we are quite close to the Equinox now, the satellite is quite close to the Equator. I have neglected the drift which has occurred during the last 14 days.]
d.
Wikipedia reports that the last known position of MH370, defined as its last secondary radar contact, was longitude 103 deg 34 min 43 sec E and latitude 6 deg 55 min 15 sec N at 17:21 Zulu on March 8, 2014. The same site reports that the final satellite ping was received at 00:11 Zulu on March 9, 2014.
e. A satellite photograph has been copied onto many sites. It is labeled as "Australian Government, Department of Defence, Mar 16/14" and the source of the photograph (Digital Globe) is given as well. Figures overlaid on the photograph show the debris at longitude 90 deg 57 min 37 sec E and latitude 43 deg 58 min 34 sec S.
f.
Wikipedia states that the mean radius of the Earth is 6,371 km (3,959 statute miles).
g. I have assumed that the airplane was at an altitude of 5,000 meters both at its last known position and at the start of its descent into the Australian debris field.