Quantum and Classical Approach Applied to the Motion of a Celestial Body in the Solar System

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1. Introduction

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 [1] [2] .

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.

2. Quantization of the Motion of a Celestial Body

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 by the formula [1]

(1)

the index instead of n has been introduced for the sake of convenience, h is the Planck constant. In principle there is no limit for the integer number n.

If the path can be approximated by a circle of radius r and the body velocity on the circle is roughly constant, we obtain for (1) the equation

(2)

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 [5] . We have for the Earth planet [3] [4] the mass

(3)

the average velocity on the Kepler orbit

(4)

and the average distance from the Sun

(5)

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

(6)

the Formula (2) becomes

(7)

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

(8)

With S given in (7) there is connected the energy [6]

(9)

where denotes a larger semiaxis of the Kepler orbit, r (with subscript n + 1) replaces r presented in (5),

(10)

is the gravitational constant when the masses in (9) are in grams and the distances in centimeters [5] , and [3] [4]

(11)

is the mass of the Sun.

3. Spontaneous Emission of Energy by a Quantum System Applied to the Case of a Moving Earth

A quantum system being in the state can spontaneously emit its energy by going to a lower state n. In this case S in (7) is changed (decreased) by the interval

(12)

where

(12a)

(12b)

Respectively the energy is changed (decreased) by

(13)

The is a positive number because

(13a)

Finally due to the virial theorem [7] valid for any n we have

(14)

Here and are respectively the kinetic and potential energy averaged over the Kepler orbit, therefore

(15)

In effect of another representation of than (13), the formula

(16)

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

(17)

The equations can be easily solved by putting in the first step the calculated from (13) and from (16) into (12):

(18)

This gives

(19)

In obtaining the Formula (19) the approximate relation

(20)

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):

(20a)

valid because of the virial theorem (15). The validity of the Formula (20) is checked for different planets in Table 1 and for satellites of Jupiter in Table 2.

By having the result (19) for we can apply it in the formula representing a quantum aspect of the classical Joule-Lenz law for the dissipation of energy [8] - [13] :

(21)

A substitution of from (19) into (21) yields

Table 1. Check of validity of the formula (20) done for the planets; the gravitational constant G is taken from (10) and M is the solar mass from (11); v is the average planet velocity in km/sec, r is the average distance between the planet and Sun in 10^{6} km.

Table 2. Check of validity of the formula (20) done for the satellites of the Jupiter planet. G is the same as that taken in Table 1 [see (10)] but the mass of Jupiter is [3] [4] . The velocity v is calculated according to the formula where r is the average distance of the satellite from the Jupiter center (in km) and T is the time period of the satellite circulation about the Jupiter planet expressed as a multiple of 86,400 seconds.

(22)

from which

(23)

This holds because is the approximate path length of the Earth about the Sun and is the average speed of the Earth planet. In effect―in view of (22) and (23)― the energy by which the Earth planet energy is lowered is quite small:

(24)

Results (23) and (24) are similar to those attained before in the quantum-theoretical calculations [8] - [13] where the Formula (21) is established. In effect of that formula, as well as (23), the transition time is equal to the circulation time period of a moving particle and the transition energy is equal to h divided by that period of time.

4. Discussion

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. This is explained by the fact that for the emission energy from the quantum state to state n of the atom we have

(25)

where

(25a)

This time is much shorter than year characteristic for the circulation time of the Earth, and h entering the numerator of the Formula (25) does remain unchanged equally for the electron transition as well as for the transition of a planet from state to. The Formula (25a) gives

(26)

therefore for a sufficiently large n we can obtain

(27)

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 [14] . Evidently, the spontaneous emission of the kind of (24) makes irreversible the planetary, or satelitary, motion too.

5. Conclusions

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

(28)

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 [2]

(29)

―the time period of the planet circulation about the mass M of the Sun will be shortened too: because of (29) we have

(30)

so the shortening of will be stronger than that of.

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

(31)

will be slightly increased with a decrease of.

In calculating the angular momentum

(32)

of a planet which for the state is proportional to S in (2), the effects concerning and combine into

(33)

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.

References

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