Using "Hat Size" to determine the Speed of Light

Experiment by Geoff Hitchcox, Christchurch, New Zealand ---- Submitted 2nd December 2002

My two previous experiments proposed ways of improving the classic Roemer Speed of Light experiment, and the limitations of using a Jupiter Moon as a clock. However I had not yet offered my own numeric value for the Speed of Light (SOL). From the beginning of 2002 I had often pondered if I could measure the SOL using inexpensive equipment that employed a method that had not (to my knowledge) previously been used.

To set the scene for my experiment, let us first consider a method to find the Speed of Sound (SOS) by using lightning. We use 3 observers scattered over a distance of several kilometres looking at the sky with a stopwatch in hand. They "start" the stopwatch when they see the lightning flash, and press "stop" when they first hear the sound of thunder. However in our haste to set up the experiment we have not looked up the "accepted" value for the Speed of Sound. Someone "thinks" it is 400 metres per second, so we plot the range from each observer, multiplying the number of elapsed seconds each observer recorded from flash to bang by 400! [Cocked Hat]

Assuming our timings are accurate, we can see that the SOS must be "less" than 400 metres per second. The range circles do not meet at a single point but overlap enclosing an area that navigators have long called a "Cocked Hat" - a three-sided hat. In navigation, a "Cocked Hat" is the result of observation and instrument error; the more experienced the navigator using good equipment, the smaller the "hat" size.

To find the SOS for the experimental temperature and humidity, we can reduce the 400 metre "guess" until the three circles meet at a single point. So although we did not have the SOS "constant", it could be determined by making the size of the "cocked hat" as small as possible, so the 3 range circles (spheres if you consider 3 dimensions) meet at one single point - the location of the "flash".

Can this same technique be used for the SOL? Galileo tried with lanterns and came to the conclusion that light was "instant". Using technology available in 2002 can we do better than Galileo?

The answer, and a very good bargain, arrived for me in mid 2002. A mail order firm in the U.S. was selling Trimble SveeSix GPS receivers for only $25. They had spent their working life on long haul trucks, but were in excellent operational condition.

The GPS (Global Positioning System) has now become a consumer product like cell phones that any person can purchase quite cheaply. Just switch it on, and in a few minutes you know your latitude and longitude about a hundred times more accurate than an expert navigator can find with a sextant (assuming he has a clear view of the sun, moon or stars). The computer in the GPS does all the required computations.

GPS operates in a surprising similar manner to the above SOS experiment. In simple terms the GPS is a stopwatch that times the transmission delay from 3 or more satellites to the receiver. By multiplying these delta times by the SOL gives the distance to each satellite, trilateration then gives the position of the receiver. The software in each GPS has the SOL as a built in constant. The GPS user does not see all the steps in the solution, only the final position of the receiver is presented. However many GPS modules can give the raw timing information or the "pseudorange" to each satellite. The $25 Trimble gives the raw timing information to each satellite.

With the GPS placed on my house roof for good reception (I have many large trees around the house) I used a PC to record the Trimble SV6 raw timing data for a few hours. Later I used software I have written to convert the raw data into what is known as a RINEX observation file. RINEX (Receiver INdependant EXchange) files can then be fed to a post-processing program to calculate antenna position. If precise satellite orbital information is used this can give position solutions that are 10 times or more accurate than the GPS itself manages.

At the stage in my software where I converted raw delta time into "pseudorange" I needed to multiply by the SOL, and this is where I was able to "manipulate" the value. Using different values for the SOL (with the same raw timing data) I was able to measure the effect (size of the Cocked Hat) that "displaced" values of the SOL gave to the trilateration. I used the GPS standard value for SOL equal to:

GPS (WGS84) Speed of Light = 299,792,458 (metres per second)

I used the FREE GPSPACE Canadian Post Processing Software to find the XYZ location of a particular tile on my house roof. In the following table, rather than show the actual XYZ ECEF (Earth Centred Earth Fixed) location, I have shown the difference from the "accepted" SOL derived position. In this table notice the symmetry of the magnitude and sign either side of zero. This symmetry points to the central value (of the SOL) as having the only solution that allows the radio waves to converge on my roof and not elsewhere!

In a similar way to the above Speed of Sound experiment, the speed of the GPS signal is varied (on paper) and used to determine position. From the tabulated results, convergence to a single "point" from the satellites ( rather than enclosing a volume of doubt caused by a "Cocked Hat ) can be observed.

Signal_Speed    dX     dY      dZ   (all table values in metres)

SOL + 1000  -14.75   2.22  -12.90
SOL + 100    -1.48   0.23   -1.29
SOL + 10     -0.15   0.03   -0.13
SOL + 1      -0.02   0.01   -0.01
SOL + 0       0.00   0.00    0.00
SOL - 1       0.01   0.00    0.01
SOL - 10      0.15  -0.02    0.13
SOL - 100     1.47  -0.22    1.29
SOL - 1000   14.75  -2.21   12.90

            WGS84 XYZ POSITION
    X (m)       Y (m)         Z (m)
-4594883.35   588439.52   -4369572.13

LATITUDE    43 31' 10.932"  SOUTH
LONGITUDE  172 42'  7.889"  EAST
HEIGHT      32.980 metres

Because tropospheric delay is accounted for in the post processing software, the SOL value is for a vacuum.

In conclusion this experiment allows "confirmation" of the value of the Speed of Light ( 299,792,458 metres per second ), down to the one metre level.

Summary:

This experiment provides a low cost way to measure the SOL to one part in 300 million.

The experiment can be carried out in any weather, day or night, from any location on earth with a clear view of the sky. Low cost GPS modules are now available for under $100 that can provide the necessary raw data to conduct the experiment.

Although the GPS module used in the experiment only cost $25, many thanks to the US taxpayer for the $20 billion cost for the GPS system that makes the experiment possible.

Internet Links:

GPSPACE Canadian FREE Post Processing Software used in this experiment.

Photos of my $25 Trimble SV6 GPS, with interface electronics.

A Freeware WWV emulator and precision timer I have developed using the $25 Trimble SV6 module.