Loose Leaf For Explorations:  Introduction To Astronomy
Loose Leaf For Explorations: Introduction To Astronomy
9th Edition
ISBN: 9781260432145
Author: Thomas T Arny, Stephen E Schneider Professor
Publisher: McGraw-Hill Education
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Chapter 8, Problem 3P

Look up the masses and radii of Mercury and Jupiter and calculate their escape velocities, using the equation in chapter 3. Does this help you see why the one body has an atmosphere but the other doesn’t? (Note: Be sure to convert kilometers to meters as necessary.)

Expert Solution & Answer
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To determine

The escape velocities of Mercury and Jupiter. And explain why the one body has an atmosphere but the other doesn’t.

Answer to Problem 3P

The escape velocity of Jupiter is 60 km/s and the escape velocity of Mercury is 4.25 km/s. The escape velocity of Mercury is much lesser than the escape velocity of Jupiter. Hence Mercury does not have an atmosphere.

Explanation of Solution

The mass of Jupiter is MJup=1.90 × 1027 kg and its radius is RJup=7.1492 × 107m. The mass of Mercury is MMer=3.30 × 1023 kg and its radius is RMer=2.44 × 106 m.

Write the formula to find the escape velocity

    vesc=2GMR        (I)

Here, vesc is the escape velocity, G is the Universal gravitational constant, M is the mass of the planet and R is the radius of the planet.

Substitute 6.67 × 1011 m3kg1s2 for G, 1.90 × 1027 kg for MJup and 7.1492 × 107m for RJup in equation (I) to find the value of vesc(Jup)

vesc(Jup)=2(6.67 × 1011 m3kg1s2)(1.90 × 1027 kg)7.1492 × 107m=3.5453×109 m2s2=5.95×104 m/s6×104 m/s

The escape velocity of Jupiter is 60 km/s.

Substitute 6.67 × 1011 m3kg1s2 for G, 3.30 × 1023 kg for MMer and 2.44 × 106 m for RMer in equation (I) to find the value of vesc(Mer)

vesc(Mer)=2(6.67 × 1011 m3kg1s2)(3.30 × 1023 kg)2.44 × 106 m=1.80×107 m2s2=4.25×103 m/s

The escape velocity of Mercury is 4.25 km/s.

Mercury is closer to the Sun and therefore, its gas molecule in its surface will have high temperature and the molecules will have higher velocity. Jupiter is far away from the Sun and hence, it is cold and has gas molecules with lower velocity. Also the escape velocity of Mercury is must lesser than Jupiter, the gases will escape from Mercury easily. Therefore, Mercury does not have an atmosphere.

Conclusion:

Thus, the escape velocity of Jupiter is 60 km/s and the escape velocity of Mercury is 4.25 km/s. The escape velocity of Mercury is much lesser than the escape velocity of Jupiter. Hence Mercury does not have an atmosphere.

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