Assume you want to deduce the radius of a planet in our Solar System as it occults a background star when the relative velocity between the planet and Earth is \(30 \mathrm{km} / \mathrm{s}\). If the star crosses through the middle of the planet and disappears for a total of 26 minutes, what is the planet's radius? a. \(3,000 \mathrm{km}\) b. \(23,000 \mathrm{km}\) c. \(15,000 \mathrm{km}\) d. \(5,000 \mathrm{km}\)

Relative velocity is the speed of one object as observed from another moving object. In this case, it's the speed at which the planet and Earth are moving towards or away from each other. The relative velocity between the planet and Earth is given as 30 km/s. This means that a background star would appear to cross the planet at this speed due to the combined motion of both the Earth and the planet.
The concept of relative velocity is essential when dealing with celestial bodies because their motion is not isolated. They are part of a dynamic system where multiple objects are simultaneously in motion. Calculating relative velocity helps us understand these interactions.

Occultation is an astronomical event that occurs when one object passes in front of another, blocking the latter from view. In our exercise, it refers to the planet obscuring a background star when viewed from Earth. The duration of this event helps in determining the size of the planet.
During the occultation, the background star disappears for a total of 26 minutes, which is crucial information to calculate the planet's radius. By observing how long the star is hidden, astronomers can deduce the size of the planet that is causing the obstruction. This duration, combined with the relative velocity, allows for precise calculations of planetary diameters and radii.

To find the radius of the planet, we need to understand the relationship between distance, speed, and time. The formula for distance is: \[ \text{Distance} = \text{Speed} \times \text{Time} \]In the problem, the planet's diameter can be found by multiplying the relative velocity (30 km/s) by the duration of the occultation in seconds (1560 seconds). This yields the diameter of the planet:
\[ \text{Diameter} = 30 \text{ km/s} \times 1560 \text{ seconds} = 46800 \text{ km} \]
To find the radius, we simply divide the diameter by 2:
\[ \text{Radius} = \frac{46800 \text{ km}}{2} = 23400 \text{ km} \]
The distance and speed measurements allow us to convert the observed occultation time into the physical dimensions of the planet, helping us to accurately deduce its radius. This is why understanding the concepts of distance, speed, and time is so critical in planetary science.