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Chapter 10: Problem 41

A small cloud in Jupiter's equatorial region is observed to be at a longitude of \(122.0^{\circ}\) west in a coordinate system rotating at the same rate as the deep interior of the planet. (West longitude is measured along a planet's equator toward the west.) Another observation, made exactly 10 Earth hours later, finds the cloud at a longitude of \(118.0^{\circ}\) west. Jupiter's equatorial radius is \(71,500 \mathrm{km}\). What is the observed equatorial wind speed, in kilometers per hour? Is this wind from the east or west?

Short Answer

Expert verified
The equatorial wind speed is 498.64 km/h eastward.

Step by step solution

01

- Calculate the change in longitude

Determine how much the longitude of the cloud has changed over the 10-hour period. The initial longitude is given as \(122.0^{\text{°}}\) west and the final longitude is \(118.0^{\text{°}}\) west. The change in longitude is the initial longitude minus the final longitude: \(\Delta \text{Longitude} = 122.0^{\text{°}} - 118.0^{\text{°}} = 4.0^{\text{°}}\)
02

- Convert the change in longitude to distance

To find the distance traveled along the equator, convert the change in longitude from degrees to kilometers. Since Jupiter’s equatorial radius is 71,500 km, the circumference of Jupiter at the equator is given by: \[ \text{Circumference} = 2 \pi \times 71,500 \text{ km} \] Using the circumference, calculate the distance corresponding to the change in longitude: \[ \text{Distance} = \text{Circumference} \times \frac{4.0^{\text{°}}}{360^{\text{°}}} = 2 \pi \times 71,500 \text{ km} \times \frac{4.0}{360} \] This simplifies to: \[ \text{Distance} = \frac{2 \pi \times 71,500 \text{ km} \times 4.0}{360} = 4986.42 \text{ km} \]
03

- Calculate the wind speed

The wind speed is calculated by dividing the distance traveled by the time taken. The cloud traveled the calculated distance over 10 Earth hours. \[ \text{Wind Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{4986.42 \text{ km}}{10 \text{ hours}} = 498.64 \text{ km/h} \]
04

- Determine the direction of the wind

To determine whether the wind is from the east or west, consider the direction of movement. If the longitude decreases from 122.0° to 118.0°, the cloud is moving eastward, against the planet’s rotation. Therefore, the wind is an easterly wind.

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Key Concepts

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

planetary atmospheres
Jupiter's atmosphere is a complex and dynamic system, with various layers and phenomena that vary significantly from those on Earth. It is mainly composed of hydrogen and helium, with traces of other gases like methane, ammonia, and hydrogen sulfide. The rapid rotation of Jupiter, combined with its composition and large size, results in strong atmospheric motions and distinct cloud bands. These bands are formed due to variations in the chemical composition, temperature, and cloud density at different latitudes. Understanding Jupiter's atmosphere is crucial for calculating wind speeds and predicting weather patterns on the planet.
rotational dynamics
Rotational dynamics deals with the behavior of rotating bodies, such as planets. Jupiter spins much faster than Earth, with one rotation taking about 10 hours. This swift rotation significantly affects its atmospheric dynamics. The planet’s rotation creates a system of zonal winds – winds that flow parallel to the equator. These winds can reach very high speeds, contributing to the banded appearance of the planet. Using principles of rotational dynamics helps to understand and predict how objects move on or near rotating planets, influencing the calculations of wind speeds based on observed movements over time.
longitude and latitude measurements
Longitude and latitude are critical for positioning and navigation on planets. Longitude measures positions east or west of a prime meridian, while latitude measures positions north or south of the equator. In the context of Jupiter, longitude measurements allow us to track the movement of clouds and other atmospheric features. The exercise uses the change in longitude to determine how far a cloud has traveled over time. Understanding how these measurements work is essential to converting angular positions into distances along the planet's surface, especially when calculating wind speed from observed movements.
wind speed calculation
Calculating wind speed on any planet involves determining the distance traveled by an atmospheric feature over a certain period. To find the wind speed on Jupiter's equatorial region:
  • First, observe the change in longitude – in this case, the cloud moved from 122.0° to 118.0° west, a change of 4.0°.
  • Next, convert this angular change to a linear distance. This involves calculating Jupiter's equatorial circumference using the formula \(2 \pi \times 71,500\) km, and then finding the proportion of this circumference corresponding to the 4.0° movement. The result is 4986.42 km.
  • Finally, divide the distance traveled by the time taken (10 hours) to get the wind speed: \(\frac{4986.42\, \text{km}}{10 \text{ hours}} = 498.64\, \text{km/h}\).
This strong wind speed indicates an easterly wind, as the cloud moved westward (or from 122.0° to 118.0° west), moving against Jupiter’s rotation.

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Most popular questions from this chapter

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. \(5,000 \mathrm{km}\)

Uranus and Neptune are different from Jupiter and Saturn in that a. Uranus and Neptune have a higher percentage of ices in their interiors. b. Uranus and Neptune have no rings. c. Uranus and Neptune have no magnetic field. d. Uranus and Neptune are closer to the Sun.

The chemical compositions of Jupiter and Saturn are most similar to those of a. Uranus and Neptune. b. the terrestrial planets. c. their moons. d. the Sun.

Jupiter's equatorial radius \(\left(R_{\text {Jup }}\right)\) is \(71,500 \mathrm{km}\), and its oblateness is \(0.065 .\) What is Jupiter's polar radius \(\left(R_{\text {Polar }}\right) ?\) (Oblateness is given by \(\left[R_{\text {Jup }}-R_{\text {Polar }}\right] / R_{\text {Jup }}\).)

Describe how the giant planets differ from the terrestrial planets.
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