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7.7 Why Giant Planets are Giant

A set of general rules can explain the important properties of planetary atmospheres. They also define the subject of comparative planetology. We can apply these rules to the giant planets to explain their massive atmospheres. One rule, for example, notes that bigger planets have stronger gravity, so they are better able to keep gas from drifting away into space.

The temperature of an ideal?monatomic?gas?is related to the average?kinetic energy?of its?atoms. In this animation, the?size?of?helium?atoms relative to their spacing is shown to scale under 1950?atmospheres?of pressure. These atoms have a certain, average speed (slowed down here two?trillion?fold from room temperature). Click here for original source URL.

Temperature is a measure of particle kinetic energy: the higher the temperature of a gas, the higher the kinetic energy, and therefore the velocity, of the molecules that make up the gas. Therefore, cold planets that are far from the Sun will generally have low gas particle velocities. Among all molecules at any given temperature, the lightest ones will be the fastest-moving ones. Hydrogen as the lightest gas will have the fastest moving molecules, on average. Helium will rank second fastest. Combining these ideas, we can predict that hydrogen and helium are the most likely gases to escape from a planet, and can only be retained by the largest and coldest planets.

Applying these principles of gravity and the kinetic theory of gases, we can understand why giant planets became so large in contrast to the terrestrial planets. On small Earth, hydrogen molecules that found themselves in our atmosphere moved fast enough to escape into space. As a result, Earth has virtually no hydrogen left in its atmosphere. Instead, heavy gases such as oxygen, nitrogen, and carbon dioxide make up the atmospheres of the terrestrial planets. In contrast, on the four cold giant planets, hydrogen and helium moved slower due to the lower temperature. The planets’ more massive cores also helped them retain the lightest gases. So hydrogen and helium were retained, along with all the other gases that were in that area of the solar nebula. Since hydrogen was originally more abundant than all other gases put together, hydrogen dominates on all four of these planets. Their hydrogen-rich atmospheres today are very massive and very deep.

The retention of planetary gas depends on the mass of the gas atom or molecule and the temperature and escape velocity of the planet. Various planets and large solar system bodies are plotted. The lines show the highest typical velocities (10 times the average velocity) for different atoms and molecules. The giant planets have high escape velocities and can hold onto even the lightest gases such as hydrogen and helium. Terrestrial planets have lost hydrogen and helium but can retain heavier gases. Click here for original source URL.

Information on the gravity and atmospheric composition of a planet can be combined on one plot, where the vertical axis is the escape velocity of a planet, and the horizontal axis is the temperature. Sloping lines show the highest typical velocities of various gas atoms and molecules. If the line falls below a particular planet, those gas molecules never reach the escape velocity, and the gas is retained. This is how we use gravity and the thermal motions of atoms to explain the composition of planet atmospheres. The physical principles used in the explanation are completely general. In other words, they would apply to any planet in any solar system. All we need to know is the mass of the planet, which determines the escape velocity, and the distance from the central star, which determines the temperature.

How does a giant planet form? We can actually predict roughly how big such a body should be. First, we can estimate how the temperature of a planet varies with distance from the Sun. Jupiter is 5.2 AU from the Sun, so a planet at that position receives (1/5.2)2 = 1/27 of the radiation that the Earth receives. According to the Stefan-Boltzmann law, temperature is related to the radiation received by S ∝ T4. Therefore, T ∝ S1/4, so the temperature at Jupiter’s position is (1/27)1/4 = 0.44 times the Earth’s surface temperature of 293 K. This gives a temperature of 293 × 0.44 = 128 K. The kinetic theory of matter tells us that the kinetic energy of an average atom is 1/2 mv= 3/2 kT. So atom velocity is proportional to the square root of temperature, v ∝ T1/2. Now we use the result that the velocity of hydrogen in the Earth’s atmosphere would be 16.2 kilometers per second. Therefore, the velocity of hydrogen around an Earth-sized object at Jupiter distance would be 16.2 (0.44)1/2 = 10.7 kilometers per second. Since this is close to the escape velocity, no hydrogen would be retained.

So if there were an Earth-sized planet at the distance of Jupiter, it would stay an Earth-sized planet. But a slightly larger rocky planet, just several times the mass of the Earth, will retain a hydrogen atmosphere. There would then be a feedback effect: as it trapped hydrogen, its mass would increase, thus increasing its gravity and its escape velocity. So it would trap still more hydrogen. If there were a rocky planet in the outer solar system several times bigger than the Earth, its gravity would eventually be so strong that it would not only retain hydrogen in its atmosphere, but also attract and trap the hydrogen from the space around it. This is how giant planets are built. Interestingly, Jupiter is about the maximum size a planet can grow. If it were to attract more gaseous material, its self-gravity would just compress it a bit more, and it would remain about the same diameter.