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According to the classical mechanics the energy of a celestial body circulating in the solar system is a constant term. This energy is defined by the masses product of the larger and smaller body entering into a mutual attraction as well as the size of the major semiaxis characteristic for the corresponding Kepler orbit. A special situation concerns the planet interaction with the Sun because of a systematic decrease of the Sun mass due to the luminosity effect. The aim of the paper is to point out that even in the case of perfectly constant interacting masses the energy of the moving body should decrease when a quantum treatment of the body motion is considered. The rate of the energy decrease is extremely small, nevertheless it gives a shortening of the distance between the interacting bodies leading to a final effect of a touch of the larger body and a smaller one.

A selection of physical objects to the quantum or classical kind of a study is well known. In principle the microscopic bodies representing mainly the atomic and molecular systems are the objects belonging to the quantum kind of approach. On the other side there do exist the celestial bodies for which the classical treatment of their motion is a well-established theory.

Paradoxally, in spite of a different kind of forces and an extremely different size of the geometrical parameters entering the microscopic and celestial physical world, respectively, the mathematical treatment developed in the Bohr atomic theory and Newton mechanics of the solar system are much similar [

The energy of the moving body-which is an electron in the hydrogen atom and a planet or satellite in the solar system-is a constant number, and the same property concerns the angular momentum. In fact both theories represent the two-body problem of the interaction between a stable heavy body with a much less heavy moving body. An essential difference between the atomic and solar systems is that a stationary electron energy in an atom-excepting for the situation characteristic for the atomic ground state-can be spontaneously changed, but it seems that no similar change can apply to the energy of a planet or satellite.

The aim of the present paper is to examine a possibility of a spontaneous change of the constant energy attributed to a celestial body moving in the solar system.

To the best knowledge of the author this problem has been never investigated before and no similar study has been raised in the past.

If the trajectory of the body is a definite closed path, the old quantum theory refers the quantum number n of the body state to its momentum

the index

If the path can be approximated by a circle of radius r and the body velocity

m is the mass of the body.

Henceforth let us specialize the calculations to the motion of the Earth planet taken as an example. In fact no essential difference does concern the Earth case and cases represented by other planets or satellites interacting with their gravitational centers, on condition the luminosity effect of the mass decrease of the Sun in neglected [

the average velocity on the Kepler orbit

and the average distance from the Sun

With the data in (3)-(5) and

the Formula (2) becomes

In the sense of the quantum theory the Earth is on a high quantum level

With S given in (7) there is connected the energy [

where

is the gravitational constant when the masses in (9) are in grams and the distances in centimeters [

is the mass of the Sun.

A quantum system being in the state

where

Respectively the energy is changed (decreased) by

The

Finally due to the virial theorem [

Here

In effect of another representation of

is also valid. As a result we obtain three Equations [(12), (13) and (16)] for three unknown parameters

The equations can be easily solved by putting in the first step the

This gives

In obtaining the Formula (19) the approximate relation

has been applied. The (20) is descending from the ratio of the kinetic energy of the body to the absolute value of the body energy represented by the approximate Formula (9):

valid because of the virial theorem (15). The validity of the Formula (20) is checked for different planets in

By having the result (19) for

A substitution of

Planet | v | r | v^{2}r (in 10^{20} m^{3}∙s^{−2}) | GM (in 10^{20} m^{3}∙s^{−2}) |
---|---|---|---|---|

Mercury | 47.89 | 57.91 | 1.328 | 1.327 |

Venus | 35.03 | 108.20 | 1.328 | 1.327 |

Earth | 29.79 | 149.60 | 1.328 | 1.327 |

Mars | 24.13 | 227.94 | 1.327 | 1.327 |

Jupiter | 13.06 | 778.33 | 1.328 | 1.327 |

Saturn | 9.64 | 1426.98 | 1.326 | 1.327 |

Uranus | 6.81 | 2870.99 | 1.331 | 1.327 |

Neptune | 5.43 | 4497.07 | 1.326 | 1.327 |

Pluto | 4.74 | 5913.52 | 1.329 | 1.327 |

Satellite | r | T | v^{2}r (in 10^{16} m^{3}∙s^{−2}) | GM_{J} (in 10^{16} m^{3}∙s^{−2}) |
---|---|---|---|---|

Metis | 127,360 | 0.295 | 12.73 | 12.67 |

Adrastea | 128,980 | 0.298 | 12.78 | 12.67 |

Amaltea | 181,300 | 0.498 | 12.71 | 12.67 |

Tebe | 221,900 | 0.675 | 12.68 | 12.67 |

Io | 421,600 | 1.769 | 12.66 | 12.67 |

Europa | 670,900 | 3.551 | 12.66 | 12.67 |

Ganimedes | 1,070,000 | 7.155 | 12.66 | 12.67 |

Callisto | 1,883,000 | 16.689 | 12.68 | 12.67 |

Leda | 11,094,000 | 238.72 | 12.67 | 12.67 |

Himalia | 11,480,000 | 250.57 | 12.74 | 12.67 |

Lysithea | 11,720,000 | 259.22 | 12.67 | 12.67 |

Elara | 11,737,000 | 259.65 | 12.68 | 12.67 |

Ananke | 21,200,000 | 631 | 12.66 | 12.67 |

Carme | 22,600,000 | 692 | 12.75 | 12.67 |

Pasifae | 23,500,000 | 735 | 12.70 | 12.67 |

Sinope | 23,700,000 | 758 | 12.25 | 12.67 |

from which

This holds because

Results (23) and (24) are similar to those attained before in the quantum-theoretical calculations [

A peculiar result is that the quantum of energy (24) emitted by the Earth is much smaller than that emitted by a hydrogen atom for the excited states n having the quantum numbers equal to about

where

This time is much shorter than

therefore for a sufficiently large n we can obtain

also for the hydrogen atomic state. In this case, however, the electron energy in the atom would be very close to zero giving a situation corresponding to the positive hydrogen ion.

All processes the effect of which can be completely annulled, are called reversible [

In the framework of the classical mechanics a celestial body entering the solar system is circulating incessibly about its gravitational center without any loss of energy. In other words in the classical theory we cannot indicate a parameter existent in the system which will cause any slowdown of the motion.

A different situation is obtained in the scheme of the old quantum theory. Here a planet, say the Earth, is located on a very high quantum level, but the energy distant of that level from the nearest lower level is very small. It is not compulsory for a planet to make a step to this lower level, nevertheless such possibility does exist. According to the quantum aspect of the Joule-Lenz law the time period necessary for such step is rather long: for example for the Earth planet it is equal to one year. Next the situation of a planet circulating about the Sun is repeated, but now the planet energy is slightly smaller than before its first step. The second step can be done to the next lower energy level in course of the time period which is close to that necessary for the first step.

In effect the rate of decrease of the planet energy is very small. For the Earth it is about

per one year.

Correspondingly to the lowering of energy, the distance between a planet and the Sun will be shortened.

A limiting situation will be attained when these two bodies-a planet and the Sun-will touch together. Such a picture is fully absent in the framework of the classical mechanics.

But beyond of the shortage of the distance between two masses-in virtue of the formula [

―the time period

so the shortening of

On the other hand, in virtue of (30), the planet velocity about the Sun, i.e.

will be slightly increased with a decrease of

In calculating the angular momentum

of a planet which for the state

It should be noted here that the effect of reduction of the mass M of the Sun [see (11)] due to the energy emission has been fully neglected. This effect leads evidently to an increase of the energy of a moving planet, so it acts in direction opposite to the quantum decrease of energy presented in the paper. Physically this means that the energy decrease discussed above (Section 3 and Section 4) is more sound for the satellites of non-radiating planets than for the planets themselves.

Olszewski, S. (2016) Quantum and Classical Approach Applied to the Motion of a Celestial Body in the Solar System. Journal of Modern Physics, 7, 1901-1908. http://dx.doi.org/10.4236/jmp.2016.714168