Mentum Planet 5 FULL Version 18
Motion and momentum transfer in the atmosphere are occurring at various scales simultaneously. Instabilities in the atmosphere and ocean are created by gradients of temperature, winds, humidity, and SSTs. Weather and climate phenomena are responses to these instabilities. Scales of atmospheric motion range from the short length and time scales of friction and turbulent motion to the decadal and planetary-scale circulations. Figure 1.29 illustrates some dynamical processes of the tropical atmosphere and their typical space and time scales. Note that most occupy a range of scales and that smaller scale features can occur within a larger scale circulation. For example, tropical cyclones consist of numerous thunderstorms; tropical cyclones can be spawned within the intraseasonal MJO; and both are modulated by the inter-annual ENSO.
Mentum Planet 5 FULL Version 18
2. Show that most of the angular momentum in the planets is associatedwith Jupiter? Show that most of the angular momentum in the solar systemis contained in Jupiter, not the Sun. [Click here,for some hints on how to proceed.]
so from this it is clear that Jupiter has 20 times more angular momentum(in its orbit) than does the sun (it its rotation) and Jupiter has verylittle angular momentum in its rotation, compared to the sun. Thus,unless the other planets have lots of angular momentum compared to Jupiter,the statement in the question is true.
Let's compare the angular momenta of the planets to that of Jupiter. From this, we quickly find that the other planets contribute very little,i.e., that Jupiter is dominant in the context of angular momentum. You can do these all from start-at-the-beginning calculations, like thoseabove, or you can do them as comparisons to Jupiter I(which is what I havedone below):
There are many quantization phenomena on cosmic scale in the solar system, such as the planetary distance, orbital energy and angular momentum, the distribution of regular satellites of giant planets, planetary ring system, and so on. Among these phenomena, the planetary distance law, called Titius-Bode law , is the most famous one. Although this law has been studied for over two hundred years, its physical explanation remains open so far. Titius-Bode law is generally denoted as
where and d are constants and are given different values in different articles in order to fit in with the observations. Although the two formulas are still imprecise, the quantization series of the planetary orbits is more obvious. So some hypotheses on quantum theory of gravitational field have been continually raised [11-21]. The common view of these hypotheses is that the quantum mechanics theory can be applied not only to the microscopic field, but also possibly to the gravitational field in the cosmic scale, with the large scale Planck constants. Among these researchers, some used the analogy between the planetary orbits and Bohr-Sommerfeld quantization orbits, while others applied the Schrödinger equation or Schrödingerlike equation to planetary system. Despite some exciting results, the phenomena are far from being completely explained due to the complexity of the formation and evolution of the solar system. In addition, some other phenomena, such as the planetary mass, energy, angular momentum, and the structure of rings, are rarely studied yet, and to explain them requires a uniform theory.
The deviation of the Jupiter and Saturn shown in Figure 1(a) can be explained by resonance mechanism. The resonance can cause changes of some planetary positions in later evolvement . Among all giant planets, since Jupiter and Saturn both have the largest mass and the shortest distance between them, their strong mutual attraction can easily draw themselves close to each other. It is known that the mean-motion resonance of 5:2 exists in the Jupiter-Saturn system. If the mass ratio of the Jupiter to Saturn remains a constant in evolvement, and the initial distances were 4.4 AU for the Jupiter and 10.3 AU for the Saturn, the ratio of the initial angular momentum of the Jupiter-Saturn system to modern that, 10.2/10.7, can be obtained. Obviously, the ratio approximately obeys the conservation law of angular momentum. The increase of 5% of modern angular momentum may be explained by the action of the solar wind or the other mechanism.
Whether the original nebular radial density really follows m(r) curves, there are further evidence according to the comparison of the energy and angular momentum in planets districts. Considering interaction between planets during their formation and growth, the system is divided into four regions as given in Table 1, which adopts Dai method . The inner boundary of the terrestrial region is the trough of the helium m(r) curve. Based on quantum mechanics, the energy of a particle with mass in n state is. From Equation (7), the radial number density of a certain particle is. The total energy of particles in a thin spherical shell with thickness dr is. If there are two kinds of particles, H and He atoms, the energy density in each of the regions can be expressed as, where and are the total energy and the total mass of a certain kind of particles respectively, and they can be written as
The angular momentum density in each region is calculated by. Since is unknown, we take the value of the Uranus region as unit. The approximate equation calculating of the angular momentum of the planet is, where e is the orbital eccentricity. The angular momentum density of a planet is, the unit is that of the Uranus. The calculated values of nebular and the actual values of the planets in each region are listed in Table 1. It is also seen that the model is consistent with observation, except the terrestrial region.
In conclusion, Table 1 shows a strong consistency between the theory and the observation in three giant planets regions, but the deviations in the terrestrial region are larger, especially for the angular momentum. Actually, this is reasonable as a result when most of the original gas materials have escaped from the terrestrial region. There are two reasons for the increase in the angular momentum of the unit mass of planets: 1) planets had captured particles with large angular momentum, 2) planets formation region had lost the materials with smaller angular momentum. In the terrestrial regionthe maximum amount of the primordial materials is H particles which are in state with zero angular momentum, and the next is smaller ratio of helium and other heavier particles being in any possible l state with larger angular momentum. During the earlier period of the Sun formation, the solar wind is very weak, so that H particles with zero angular momentum can be easily captured by the Sun. In the later stage, the strong solar wind takes away the most of H particles again and left some angular momentum in the collision with planets materials. Both the mechanisms can all make terrestrial region lose a lot of mass carrying small amount of angular momentum. Even the angular momentum taken away by solar wind might be smaller than what is brought in. As a result, the angular momentum of the unit mass of planets in the terrestrial region is increased. We can make a simple estimation. Let the original total mass of terrestrial region be m, the lost mass with zero angular momentum be, and is planetary mass. Since the ratio of original 0.074 and modern 0.214 (see Table 1), the total original angular momentum is . The ratio of lost mass can be obtained as %, which is reasonable. In addition, since the energy of the particle is proportional to the mass, the energy will naturally decreases with the decrease of the mass, and the ratio of energy and mass will be a constant approximately. So there is only small deviation of energy of the unit mass in the terrestrial region.
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