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Chapter 10: Problem 36

Uranus occults a star at a time when the relative motion between Uranus and Earth is \(23.0 \mathrm{km} / \mathrm{s}\). An observer on Earth sees the star disappear for 37 minutes 2 seconds and notes that the center of Uranus passed directly in front of the star. a. On the basis of these observations, what value would the observer calculate for the diameter of Uranus? b. What could you conclude about the planet's diameter if its center did not pass directly in front of the star?

Short Answer

Expert verified
a. 51106 km. b. It would be an upper limit to the planet's diameter.

Step by step solution

01

- Convert Time to Seconds

First, convert the disappearance time from minutes and seconds to just seconds for easier calculations. The disappearance time is 37 minutes and 2 seconds.
02

- Compute Total Disappearance Time

As there are 60 seconds in a minute, multiply 37 minutes by 60 and then add the extra 2 seconds:\[ 37 \times 60 + 2 = 2220 + 2 = 2222 \text{ seconds} \]
03

- Determine Relative Motion

Given that the relative speed between Uranus and Earth is 23.0 km/s, use this to find the distance moved by Uranus in the time it occulted the star:
04

- Calculate Diameter of Uranus

Use the formula for distance: \[ \text{Distance} = \text{Speed} \times \text{Time} \] Hence, the diameter of Uranus can be found as;\[ \text{Diameter of Uranus} = 23.0 \text{ km/s} \times 2222 \text{ s} = 51106 \text{ km} \]
05

- Diameter if Center Did Not Pass Directly

If the center did not pass directly in front of the star, the computed value would only be an upper limit to the planet's diameter since the distance covered would be less for an offset path.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Planetary Science
Studying the diameter of Uranus provides valuable insights into planetary science. Planetary science, also known as planetology, involves the study of planets, moons, and planetary systems, particularly within our Solar System. Understanding a planet's size is crucial as it helps scientists learn about its formation, structure, and possible atmospheric components. Uranus, being one of the gas giants, has unique characteristics that set it apart in our Solar System. By determining its diameter, we can infer information about its mass distribution and gravitational influence, which are fundamental aspects of planetary science.
Celestial Mechanics
Celestial mechanics, the study of the movements and gravitational forces of celestial objects, plays a key role in calculating Uranus's diameter. This field utilizes the laws of motion and universal gravitation developed by Isaac Newton and further refined by others. To calculate Uranus's diameter, we observe its motion relative to Earth and measure the time it occults a star. From this, we apply formulas from celestial mechanics to determine the distance Uranus moves. This understanding helps scientists predict planetary motion, uncover the secrets of their orbits, and explore their interactions with other celestial bodies.
Occultation
Occultation occurs when one celestial body passes in front of another, obscuring it from view. In this exercise, Uranus occults a star, meaning Uranus moves across the star's line of sight from our perspective on Earth. The disappearance of the star signifies the beginning and end of the occultation. By recording the duration of this event, scientists can calculate the diameter of Uranus. Occultation provides a direct method for determining celestial sizes and distances, offering valuable data for both amateur and professional astronomers.
Relative Motion
Relative motion refers to the movement of one object relative to another. In this exercise, the relative motion between Uranus and Earth is crucial. The provided relative speed of 23.0 km/s allows us to compute how far Uranus travels while it occults the star. By knowing both the speed and the time duration, we can use the formula for distance to find the planet's diameter. Understanding relative motion helps in analyzing dynamic interactions between celestial bodies, ensuring accurate calculations and expanding our knowledge of the universe's mechanics.
Distance Calculation
Distance calculation in this context involves using the relative motion of Uranus and the duration of the occultation to determine its diameter. The key formula here is: \[ \text{Distance} = \text{Speed} \times \text{Time} \] Plugging in the values, we get: \[ \text{Diameter of Uranus} = 23.0 \text{ km/s} \times 2222 \text{ s} = 51106 \text{ km} \] This result gives us the diameter of Uranus. If Uranus's center did not pass directly in front of the star, this calculated distance would be an upper limit. This concept is essential for accurately measuring and understanding celestial distances, enhancing our overall grasp of celestial mechanics and planetary dimensions.

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