☰ Year plan · Science

Upper layers, escape velocity and ozone

Week 10 · 2-1 Components and Layers (B): ionosphere, aurora, exosphere, escape velocity and Mercury; 2-2 Chemical Reactions in the Atmosphere: ultraviolet rays, ozone formation, ozone depletion and its effects, protection measures, ozone in the troposphere · 3 lessons
2

Escape velocity: why Earth keeps its air

Integrated Science Sec 1 Term 1, Unit 2: pp. 73–83
GoalExplain escape velocity and use the simple rule to say why Earth keeps most of its air and Mercury does not.
New words
escape velocity · the speed an object needs to leave a planet’s gravity; about 11.2 km/s for Eartheffective speed (Vrms) · a typical speed of the molecules of a gasgravity · the pull of a planet on everything near it
Explanation

Gas molecules move very fast. Their effective speed, Vrms, is larger when the molecules are lighter and when the temperature is higher. An object of any mass must reach the escape velocity, Ve, to leave a planet, and Ve has one fixed value for each planet. For Earth, Ve is about 11.2 km/s, which is about 40 300 km/h. The book’s simple rule: a planet keeps a gas whose Vrms is below Ve, and a gas whose Vrms reaches Ve or goes above it drifts off into space. This is only a first model. In reality a planet loses a gas over a long time even when Vrms is well below Ve, which is why light gases such as hydrogen and helium do leave Earth. Mercury, the smallest planet, has weak gravity and strong heating by day, so it holds almost no atmosphere; with nothing to hold the heat it also cools quickly at night. A gas does not escape because it is “light” alone: at the same temperature lighter molecules just move faster.

CaseVrmsVePlanet keeps the gas?
A2 km/s11.2 km/syes
B12 km/s11.2 km/sno
C11.2 km/s11.2 km/sno, by the book’s rule
Worked examples
A gas has Vrms = 0.5 km/s on a planet with Ve = 4 km/s. Compare: 0.5 < 4, so by the simple rule the planet keeps the gas. If the same gas had Vrms = 5 km/s, then 5 > 4 and the gas would drift away.
Change units. Earth’s escape velocity is 11.2 km/s. In one hour there are 3600 s, so 11.2 × 3600 = 40 320 km/h. In 10 s an object at this speed covers 11.2 × 10 = 112 km.
Why is a ball thrown outdoors, away from windows, not enough to leave Earth? Even a very hard throw reaches only a few tens of metres per second, thousands of times less than 11 200 m/s, so gravity pulls the ball back. The harder the throw, the higher it goes, but it still comes down.
Class activity

Outdoors, away from windows and people, with a grown-up: throw a soft ball gently, then harder, straight upward, and note which goes higher and how long each stays in the air. Write that the harder throw goes higher but both fall back. Then write why no ordinary throw can reach 11.2 km/s.

Practice
1
What is escape velocity?
2
A gas has Vrms = 9 km/s and the planet has Ve = 6 km/s. By the book’s rule, does the planet keep it?
3
Which gas molecules move faster at the same temperature, light or heavy ones?
4
Give two reasons why Mercury keeps almost no atmosphere in the book’s account.