Question

In the Hafele–Keating experiment in 1971, two commercial planes are flying at the same speed V , one towards the East and the other towards the West. They circled the Earth twice during a flight of ∆T = 41 hours. Two ultra-precise atomic clocks on board of each plane measured the duration of the flight. It was compared to the duration measured by ultra-precise clocks that stayed on Earth. The measured differences were:

∆TE = −59 ns

∆TW = +273 ns

The total shift was, with the precision of the measurement:

∆Tmeasured = −332 ± 17 ns

1. Express the speeds VAW and VAE of the planes going towards West and East respectively as a function of the flight speed V and the Earth speed VEarth. The latter is measured in a reference frame R aligned with the Earth’s rotation axis and pointing to distant stars (Earthcentered inertial). Illustrate with one or two schematics.

2. Express the delays of the clocks ∆TE and ∆TW as a function of the durations measured in the planes ∆TAE, ∆TAW , the duration measured on Earth ∆T and the duration that would have been measured from the reference frame ∆TR.

3. Express and calculate the total delay ∆Tpredicted = ∆TE − ∆TW as a function of V , VEarth, ∆TR. Compare with ∆Tmeasured. Are the measured data matching the model prediction?

Data: use the approximation for x 1 and stop the expansion at the second order. √ 1 − x 2 ≈ 1 − x 2 2

Answer 1