Neutrinos through the earth

Neutrinos are interesting little fundamental particles. These particles interact very weakly with everything else, so they essentially pass right through everyday stuff. In fact there are tremendous numbers of neutrinos passing through your body at this very second! In 2011, two experiments in Italy measured the time it took for neutrinos generated at a particle accelerator at CERN in Geneva, Switzerland to get to a pair of detectors built into the Gran Sasso mountain in central Italy. One of them mistakenly measured the speed as faster than the speed of light, which caused a furor.

The straight line distance through the earth from \(x_0\), the point of production of the neutrinos at CERN, to \(x_1\), the point of measurement at Gran Sasso was roughly \(732~\mbox{km}\). Now let \(D\) be the shortest distance between \(x_0\) and \(x_1\) over the surface of the earth, which is the path we'd have to take if we wanted to get from CERN to Gran Sasso without heavy machinery.

How much longer is \(D\) than the neutrino path, i.e. what is \(D-732~\mbox{km}\) in meters?

Details and assumptions

You may assume that \(x_0\) and \(x_1\) are on the surface of the earth and that the earth is a perfect sphere of radius \(6,370~\mbox{km}\).


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