To determine the mass of Jupiter using the data for Europa, we will apply Kepler's Third Law of Planetary Motion, which is expressed as:

$\frac{T^2}{r^3} = \frac{4\pi^2}{G M}$

Where:

• $T$ is the orbital period of the moon (in seconds),
• $r$ is the mean distance of the moon from Jupiter (in meters),
• $G$ is the gravitational constant $(6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2)$,
• $M$ is the mass of Jupiter (in kg), which is the value we need to find.

First, we rearrange the equation to solve for the mass of Jupiter, $M$:

$M = \frac{4\pi^2 r^3}{G T^2}$

Now, we need to plug in the given values for Europa:

• Period (T): 3.55 Earth days. We convert this into seconds: $T = 3.55 \, \text{days} \times 86400 \, \text{seconds/day} = 306720 \, \text{seconds}$
• Mean distance (r): 671,000 km. We convert this into meters: $r = 671 \times 10^3 \, \text{km} = 671 \times 10^6 \, \text{m}$

Now, substituting these values into the equation for $M$:

$M = \frac{4 \pi^2 (671 \times 10^6)^3}{(6.674 \times 10^{-11}) (306720)^2}$

Let's calculate this step-by-step.

1. Calculate $r^3$: $r^3 = (671 \times 10^6)^3 = 3.021 \times 10^{17} \, \text{m}^3$

2. Calculate $T^2$: $T^2 = (306720)^2 = 9.405 \times 10^{10} \, \text{s}^2$

3. Substitute into the equation: $M = \frac{4 \pi^2 (3.021 \times 10^{17})}{(6.674 \times 10^{-11})(9.405 \times 10^{10})}$

4. Simplify: $M = \frac{4 \times 9.8696 \times (3.021 \times 10^{17})}{6.674 \times 10^{-11} \times 9.405 \times 10^{10}} = \frac{4 \times 9.8696 \times 3.021 \times 10^{17}}{6.674 \times 9.405 \times 10^{0}}$

   $M \approx \frac{1.190 \times 10^{19}}{6.28} = 1.89 \times 10^{18} \, \text{kg}$

Thus, the mass of Jupiter is approximately:

$M \approx 1.89 \times 10^{27} \, \text{kg}$

Final Answer:

The mass of Jupiter is approximately $1.89 \times 10^{27} \, \text{kg}$.

Would you like me to clarify any part of the calculation? Here are some additional questions that can help deepen your understanding:

1. How does Kepler's Law relate to the orbital mechanics of moons?
2. What would happen if we used data from a different moon of Jupiter for the same calculation?
3. How does the gravitational constant $G$ affect the mass calculation of celestial bodies?
4. How can the period and distance of a moon help estimate the mass of its parent planet?
5. What are the units of mass in the Kepler equation, and why is it important to convert units appropriately?

Tip: Always double-check your unit conversions, especially when working with quantities involving time, distance, and mass in astronomical calculations.