A long time ago, before scientists could measure the speed of light accurately, Ole Römer, a Danish astronomer, studied the eclipses of Jupiter's satellite Io. He was able to determine the speed of light from the observed periods of the satellite orbiting the planet Jupiter. Römer observed the time interval between two successive emergences of Io from behind the shadow of Jupiter.

Figure 1 shows the orbit of the Earth $E$ around the Sun $S$ and the orbit of Io (denoted by $M$) around the planet Jupiter. A long series of observations of the eclipses permitted an accurate evaluation of the orbital period $T_0=42~\text{h}~28~\text{min}~16~\text{s}$ of $M$.

The average distance of the Earth $E$ to the Sun is $R_E = 149.6 \cdot 10^6~\text{km}$. The period of revolution of the Earth is $T_E = 365~\text{days}$ and of Jupiter is $T_J = 11.86~\text{years}$. The distance of the satellite $M$ to the planet Jupiter is $R_M = 422 \cdot 10^3~\text{km}$.

A1 Find the orbital radius $R_J$ of Jupiter. Find the ratio of the mass of Jupiter $M_J$ to the mass of the Sun $M_S$.

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A2 The satellite $M$ orbits Jupiter, periodically entering and emerging from its shadow. Express the period $T_\text{real}$ between successive emergences in terms of $T_J$ and $T_0$.

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A3 Consider a reference frame ($SJ$) in which Jupiter is at rest with respect to the Sun. Determine the relative angular velocity $\omega$ of the Earth in $SJ$ frame. Calculate the speed $v_E$ of the Earth in $SJ$.

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Due to the finite speed of light, the period $T_\text{obs}$ between successive emergences observed from the Earth differs from $T_\text{real}$. An observer saw $M$ begin to emerge from the shadow at the moment $t$ when his position was at $\theta$ and the next emergence at the moment $t+T_\text{obs}$ when he was at $\theta+\Delta \theta$. Let us assume that $\theta=0$ for $t=0$.

A4 Derive the observed period $T_\text{obs}$ as the function of time. Sketch the graph of $T_\text{obs}$ vs $t$. Find the positions of the Earth at which the maximum period, the minimum period, and the true period of the satellite $M$ were observed.

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A5 Derive the relation between $d(t_k)$ and $T(t_k)$. Plot the period $T(t_k)$ as a function of the time of observation $t_k$. Find the positions of the Earth at which the maximum period, the minimum period, and the true period of the satellite $M$ were observed.

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Difference between maximal and minimal values of $T_\text{obs}$ is $\Delta T = 30~\text{s}$.

A6 Estimate the speed of light $c$ using the given data.

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