homework and exercises - 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?

Answer

Rather than looking at one orbit of Io, consider observing Io and Jupiter for around 200 days, starting when the Earth is exactly between the Sun and Jupiter, and ending when the Earth is opposite Jupiter, with the Sun in between. In the 200 days, Io will make around 110 orbits of Jupiter. But, importantly, the light from that last orbit of Io will need to travel an extra distance equal to the diameter of the earth's orbit around the sun, making it arrive about 1000 seconds later than expected. This would add about 9 second to the average observed orbital period for Io over that 200 days. And in the next 200 days, as the earth caught up to Jupiter again, the average observed period of Io would be about 9 second less than expected.

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...