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Chapter 12: Problem 36

Earth's Moon has a diameter of \(3,474 \mathrm{km}\) and orbits at an average distance of \(384,400 \mathrm{km}\). At this distance. it subtends an angle just slightly larger than half a degree in Earth's sky. Pluto's moon Charon has a diameter of \(1,186 \mathrm{km}\) and orbits at a distance of \(19,600 \mathrm{km}\) from the dwarf planet. a. Compare the appearance of Charon in Pluto's skies with the Moon in Earth's skies. b. Describe where in the sky Charon would appear as seen from various locations on Pluto.

Short Answer

Expert verified
Charon appears about 7 times larger in Pluto's sky than the Moon in Earth's sky. It appears almost stationary due to synchronous orbit.

Step by step solution

01

Identify the subtended angle for the Moon in Earth's sky

The subtended angle in Earth's sky is slightly larger than half a degree, given as approximately 0.5 degrees.
02

Calculate the angular diameter formula

The formula to find the angular diameter is: \[ \theta = 2 \cdot \arctan \left( \frac{d}{2D} \right) \] where \(d\) is the diameter of the object and \(D\) is the distance to the object.
03

Calculate the subtended angle for Charon in Pluto's sky

Using \(d_{Charon} = 1,186 \, \mathrm{km}\) and \(D_{Charon} = 19,600 \, \mathrm{km}\), we have: \[ \theta_{Charon} = 2 \cdot \arctan \left( \frac{1,186}{2 \cdot 19,600} \right) \].
04

Simplify the expression

Simplify the expression inside the arctan: \( \frac{1,186}{2 \cdot 19,600} = \frac{1,186}{39,200} \approx 0.0302 \). Thus: \[ \theta_{Charon} = 2 \cdot \arctan (0.0302) \].
05

Calculate the arctan value

Using a calculator: \( \arctan (0.0302) \approx 0.0302 \) radians. Therefore: \[ \theta_{Charon} \approx 2 \cdot 0.0302 = 0.0604 \] radians.
06

Convert radians to degrees

Convert the angle from radians to degrees: \[ 0.0604 \, \text{radians} \times \frac{180}{\pi} \approx 3.46 \] degrees.
07

Compare the appearance

Compare the angles: The Moon appears as around 0.5 degrees in Earth's sky, while Charon appears as approximately 3.46 degrees in Pluto's sky. Hence, Charon appears significantly larger.
08

Describe Charon's position in Pluto's sky

Charon orbits close to Pluto, so it would appear very large and prominently in the sky, almost stationary due to the synchronous rotation and revolution with Pluto.

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Key Concepts

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

Moon and Charon comparison
The comparison between Earth's Moon and Pluto's moon Charon involves looking at their apparent sizes in the skies.
The Moon has a diameter of 3,474 km and orbits at a distance of 384,400 km from Earth. In contrast, Charon has a diameter of 1,186 km and orbits 19,600 km from Pluto.
Using the formula for angular diameter, we can calculate how they appear. For Earth's Moon, the angle subtended is about 0.5 degrees. For Charon, this angle turns out to be 3.46 degrees.
This means that, when viewed from the surface of Pluto, Charon appears significantly larger in the sky compared to how the Moon appears from Earth.
Subtended angle
The subtended angle is a key concept in understanding how large an object appears in the sky. It is the angle formed by lines drawn from the observer to the two opposite edges of the object.
To calculate the angular diameter, use the formula: \[ \theta = 2 \, \arctan \left( \frac{d}{2D} \right) \] where \( d \) is the diameter of the object and \( D \) is the distance to the object.
For example, Charon's angular diameter is calculated by substituting Charon's diameter and its distance from Pluto into the formula. This results in an angle of approximately 3.46 degrees.
Satellite observation in planetary skies
Observing satellites like moons in planetary skies can vary widely depending on their size and distance from their parent planet.
On Earth, we observe our Moon, which has an average subtended angle of about 0.5 degrees. This makes it appear moderately large in our skies.
On Pluto, observing Charon would be different, as it appears about 3.46 degrees in the sky, making it look significantly larger than the Moon appears from Earth.
The appearance and size of a satellite in the sky are influenced by its orbital distance and physical size.
Astronomical calculations
Performing astronomical calculations like angular diameter helps us understand how celestial bodies appear to us.
Here’s a step-by-step example for such a calculation:
  • Identify the values needed (for Charon, the diameter is 1,186 km, and the distance is 19,600 km).
  • Use the angular diameter formula: \( \theta = 2 \, \arctan \left( \frac{1,186}{2 \cdot 19,600} \right) \).
  • Simplify the expression inside the arctan: \( \frac{1,186}{39,200} \approx 0.0302 \).
  • Calculate arctan value: \( \arctan (0.0302) \approx 0.0302 \) radians.
  • Convert radians to degrees: \( 0.0604 \times \frac{180}{\pi} \approx 3.46 \) degrees.

These steps are critical for accurately determining how celestial bodies look from different vantage points.

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Most popular questions from this chapter

The three types of meteorites come from different parts of their parent bodies. Stony-iron meteorites are rare because a. they are hard to find. b. the volume of a differentiated body that has both stone and iron is small. c. there is very little iron in the Solar System. d. the magnetic field of the Sun attracts the iron.

Meteorites can provide information about all of these except a. the early composition of the Solar System. b. the composition of asteroids. c. the composition of comets. d. the Oort Cloud.

What are the differences between a comet and a meteor in terms of their size, distance, and how long they remain visible?

Comets have two types of tails. Describe them and explain why they sometimes point in different directions.

Meteor showers occur because Earth passes through the path of a. another planet. b. a planetesimal. c. a comet. d. the Moon.
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