Pho.rs: Eclipses of Jupiter's Satellite

Eclipses of Jupiter's Satellite

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A1 ^2.50 Find the orbital radius $R_J$ of Jupiter. Find the ratio of the mass of Jupiter $M_J$ to the mass of the Sun $M_S$.

1 The 3rd Kepler's law is written for Jupiter and the Earth	0.40
2 \[ R_J = R_E \left( \frac{T_J}{T_E} \right)^{2/3}\]	0.40
3 Answer \[R_J = 778 \cdot 10^6~\text{km}\]	0.40
4 The 3rd Kepler's law in general form or 2nd Newton's law for circular motion is used.	0.50
5 \[ \frac{M_J}{M_S} = \frac{R_M^3}{R_E^3} \frac{T_E^2}{T_0^2} \]	0.40
6 Answer \[ \frac{M_J}{M_S} = 9.54 \cdot 10^{-4} \]	0.40

A2 ^1.00 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 $T_0$.

1 The geometrical viewpoint is described: sum of angles or difference in angular velocities.	0.50
2 \[T_\text{real} = \frac{T_J T_0}{T_J-T_0}\]	0.50

A3 ^1.50 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 $SJ$.

1 The angular velocity of the frame is obtained \[\omega_{SJ} = \frac{2 \pi }{T_J}\]	0.20
2 \[ \omega = \frac{2\pi}{T_E} - \omega_{SJ}\]	0.50
3 \[\omega = 2 \pi \frac{T_J-T_E}{T_J T_E}\]	0.50
4 \[ v = R_E \omega\]	0.30

A4 ^1.50 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.

1 The Pythagorean theorem is used	0.30
2 \[d^2 = R_E^2 \sin^2 \theta + (R_J - R_E \cos \theta)^2\]	0.40
3 After approximation it is found that \[d = R_J - R_E \cos \theta \]	0.40
4 Answer \[d = R_J - R_E \cos \omega t\]	0.40

A6 ^1.00 Estimate the speed of light $c$ using the given data.

1 \[T_\text{obs,max}-T_\text{obs,min} = \frac{2 \omega R_E}{c} T_\text{real}\]	0.50
2 \[c = 3.03 \cdot 10^8~\text{m}/\text{s}\]	0.50

1 The idea connected to changes of $d$ over a period is proposed, e.g. \[T_\text{obs} = T_\text{real} + \frac{\Delta d}{c},\]where $\Delta d$ - is change in $d$ over $T_\text{real}$	0.80
2 \[\dot{d} = \omega R_E \sin \omega t\]	0.20
3 \[ T_\text{obs} = T_\text{real} + T_\text{real} \frac{\omega R_E}{c} \sin \omega t\]	0.30
4 The graph is oscillating about $T_\text{real}$ function	0.20
5 The graph is a graph of sine	0.20
6 It's noticed that maximal and minimal values correspond to $\theta \approx \pm \pi/2$	0.20
7 The maximum of $T_\text{obs}$ corresponds to $\theta = \pi/2$.	0.10
8 The minimum of $T_\text{obs}$ corresponds to $\theta = \pi/2$.	0.10
9 The real period is observed when $\theta=0$	0.20
10 The real period is observed when $\theta=\pi$	0.20