To solve this problem, we need to find the distance $d$ between Earth and Jupiter. From the information provided:

• The orbital radius of Europa around Jupiter is $r = 670,900 \, \text{km}$.
• The maximum angle $\theta$ formed by Europa, Earth, and Jupiter is $0.056^\circ$.

This forms a small-angle triangle with Jupiter at the vertex, Europa at one point on the circumference, and Earth far away. The relation between the radius of Europa's orbit and the distance from Earth to Jupiter can be derived using the small angle approximation in radians.

Small Angle Approximation:

For small angles measured in radians, we use the approximation: $\theta \approx \frac{r}{d}$ Where:

• $\theta$ is in radians.
• $r$ is the distance from Jupiter to Europa.
• $d$ is the distance between Earth and Jupiter.

First, we need to convert $\theta = 0.056^\circ$ to radians: $\theta (\text{radians}) = \theta (\text{degrees}) \times \frac{\pi}{180}$ $\theta = 0.056^\circ \times \frac{\pi}{180} = 0.000977 \, \text{radians}$

Now, apply the small-angle formula: $d = \frac{r}{\theta}$ Substitute the values: $d = \frac{670,900 \, \text{km}}{0.000977 \, \text{radians}} = 686,781,689 \, \text{km}$

Round to the nearest million kilometers:

$d \approx 687,000,000 \, \text{km}$

Thus, the distance between Earth and Jupiter is approximately 687 million kilometers.

Would you like more details on this solution, or do you have any questions? Here are some related questions to explore further:

1. What would the distance be if the angle $\theta$ were doubled?
2. How does the small angle approximation work, and when can it be used?
3. What is the maximum distance between Earth and Jupiter during their orbits?
4. How can we calculate the actual position of Europa using its orbital radius?
5. How does the distance between Earth and Jupiter change during their respective orbits?

Tip: Small angle approximations are useful when dealing with celestial bodies because their angular separations are often tiny when viewed from Earth.