# Rømer's determination of the speed of light

I am trying to understand Rømer's determination of the speed of light ($c$). The geometry of the situation is shown in the image below. The determination involves measuring apparent fluctuations in the orbital period of Io. (Jupiter's moon)

The Earth starts from point A. $r(t)$ is the distance between the Earth and Jupiter. $r_e$ is the radius of the (assumed) circular orbit of the Earth around the Sun, while $r_0$ is the same for Jupiter. $T$ is the period of the Earth's orbit.

Under the assumption that the Jupiter-Io system is stationary, $r(t)$ can be expressed as

$$r(t) = \sqrt{r_E^2 + r_0^2 -2r_0 r_E \cos \left(\frac{2\pi t}{T}\right)}$$

If we further assume that the period of Io's orbit around Jupiter, $\Delta t$ is much smaller that $T$, then it can be shown that the distance the Earth moves, $\Delta r$ when Io completes one orbit is:

$$\Delta r = \frac{2\pi r_E \Delta t}{T} \sin\left( \frac{2\pi t}{T} \right)$$

The point I am stuck is about why is there an apparent fluctuation in Io's orbit as observed on the Earth? And how can we derive the observed delay using these expressions?

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–  Qmechanic Apr 28 '13 at 14:45
The earth's orbital velocity is about 30 km/sec. During one orbit of Io around Jupiter, anout 1.8 days, the Earth-Io distance could at most increase by $\frac{30000\times86400\times 1.8}{c}$ light-seconds; adding 15 seconds to Io's apparent period; still an observable amount...