SP-345 Evolution of the Solar System

 

5. FORCES ACTING ON SMALL BODIES

 

5.1. INTRODUCTION

[73] Even if a hetegonic theory were restricted to explaining the origin of the planets and satellites alone, the study of the motion of smaller bodies is of basic importance because the large bodies once accreted from small bodies. A large number of small bodies asteroids, comets, and meteoroids move in interplanetary space. The latter category includes micrometeoroids (interplanetary dust). Although generally not included in the discussion of small bodies there are also the constituents of the interplanetary plasma: atoms, molecules, ions, and electrons. The total mass spectrum excluding the Sun covers about 57 orders of magnitude from electrons (10-27 g) to Jupiter (2 X 1030 g). The dynamic behavior of bodies in space depends in a decisive way on their mass. The bodies at the upper end of the mass spectrum obey the laws of celestial mechanics, whereas the particles at the lower end must be dealt with in the framework of plasma physics. See fig. 5.1.1.

 

5.2. GRAVITATIONAL EFFECTS

A body of mass M is subject to the Newtonian gravitation

 

mathematical equation(5.2.1)

 

where G is the gravitational constant, Mn is the mass of other bodies, and rn is the position vector of Mn with respect to M.

The motions of the large bodies (e.g., the planets) are exclusively dominated by fG, and hence obey the laws of celestial mechanics. (The tiny perturbation of the motion of Mercury, which is attributed to general relativity effects, is not significant for this discussion.)

 


[
74]

FIGURE 5.1.1. Survey of forces governing the motion of bodies in space.

FIGURE 5.1.1. Survey of forces governing the motion of bodies in space.

 

5.2.1. Kepler Motion

As we have seen in ch. 3, the motion of planets and satellites can be very accurately described by Kepler's laws. The motion of the asteroids follows the same laws. No exception to this rule has been observed, even for the smallest observed asteroids (of the size of one kilometer) However, because of the large number of asteroids in the main belt, we should expect that they sometimes collide, with the result that a discontinuous change in their orbits takes place. Because the collisional cross section per unit mass of all [75] asteroids increases with decreasing size of the asteroids (see ch. 7), the influence of collisions must be greater the smaller the bodies. Thus the motion of subvisual asteroids from kilometer size down to the small particles now known to exist (Kinard et al., 1974; Soberman et al., 1974), is likely to be affected both by collisional processes and by other nongravitational forces which we discuss below.

The motion of comets also obeys the Kepler laws, but only to a first approximation. Deviations from Kepler motion are ascribed to nongravitational effects (Marsden, 1968), which we will discuss later.

 

5.2.2. Collision-Perturbed Kepler Motion

As seen from fig. 5.1.1, an important type of motion is the Kepler motion perturbed by collisions with other bodies or particles. A motion of this type is difficult to treat by celestial mechanics in its traditional formulation. In fact, celestial mechanics can handle the two-body problem very well, and, if sufficiently computerized, also the few-body problem. For example, the motion of a planet is treated as a two-body problem with perturbations caused by several other bodies. In contrast to this simplifying description of planetary motion, the mutual interaction among asteroids and among meteoroids constitutes a many-body problem, of the same general type as is treated in theoretical plasma physics. Indeed, the collisions among asteroids and among meteoroids are analogous to the collisions among par-ticles in a plasma and can be treated by the same general formalism. The somewhat unconventional presentation of celestial mechanics in ch. 3 is designed to facilitate synthesis of celestial mechanics and the formalism of plasma physics.

As we shall discuss in more detail in ch. 6, the collisions between bodies in Kepler orbits lead, under certain conditions, to a focusing effect that concentrates the bodies into jet streams. The formation of jet streams seems to be a very important intermediate phase in the accretion of small bodies into large bodies.

 

5.3. ELECTROMAGNETIC EFFECTS

If a body has an electric charge q, it is subject to an electromagnetic force...

 

mathematical equation
(5.3.1)

 

[76] ...where E is the electric field; B, the magnetic field; v, the velocity of the body; and c, the speed of light.

Let us consider the constituents of an ordinary plasma in space: atoms, molecules, ions, and electrons. The motion of charged particles in such a plasma is governed by electromagnetic forces. We will not discuss the properties of a plasma in detail until later (Part C), but we introduce plasma effects here because of their primary influence on the motion of very small particles. As will be shown in sec. 5.4, plasma effects delineate the lower limit of applicability for collision-perturbed Kepler motion.

In addition to the plasma constituents of atoms, molecules, ions, and electrons, there is likely to be a population of dust grains. These grains, if small enough, may form part of the plasma. Because, initially, they are preferentially hit by plasma electrons, they normally have a negative electric charge. This charge might change into a positive charge ;f for example, an intense radiation produces a photoelectric emission. Both negative and positive grains can be considered as plasma constituents as long as their Larmor radius is small enough, which essentially means that fq must be much larger than fG.

Dusty plasma with these general properties is likely to have been of decisive importance during the formation of a solar system, when the concentration of plasma as well as of condensed grains must have been high in the circumsolar region. Development of a detailed theory for dusty plasmas is highly desirable. When we discuss the behavior of hetegonic plasmas, we generally assume them to be dusty. A particularly important point is that the charged dust particles add to the plasma a component of nonvolatile substances (see further ch. 19).

 

5.4. LIMIT BETWEEN ELECTROMAGNETICALLY AND GRAVITATIONALLY CONTROLLED MOTION

We have seen that due to their electric charge very small grains may form part of a plasma, whereas for large grains gravitation rules the motion. The limit between these two types of motion can be estimated by comparing the period of gyration of a grain spiraling in a magnetized plasma,

 

 

mathematical equation
(5.4.1)

 

with the Kepler period TK of its orbital motion. If the grain is a sphere of radius R, density [Greek letter] capital theta, and electrostatic potential V (in esu), we have m =mathematical equation
, q = RV, and

 

[77] mathematical equation
(5.4.2)

 

Solving eq. (5.4.2) for R we have

 

 

mathematical equation
(5.4.3)

 

 

To estimate the limiting value of the grain radius RLm at which Tgy becomes comparable to TK, we set V= 10-2 esu, B = 3 X 10-5 G (the present magnetic field in interplanetary space), [Greek letter] capital theta
=1 g/cm3 (a typical density for interplanetary grains), and Tgy =3 X 1O7 sec (1 yr) to obtain

 

RLm = 0.3 X 10-5 cm (5.4.4)

 

This limiting radius corresponds to a limiting grain mass mLm of 10-16 g. If R << RLm , the period of gyration is small compared to the Kepler period and the grain forms part of the plasma. If R << RLm, the grains move in a Kepler orbit only slightly perturbed by plasma effects.

In the hetegonic era B could very well have been 104 times larger, corresponding to an increase of RLm to 0.3 X 10-3 cm and mLm = to 10-10 g. On the other hand, for a plasma producing particle streams around a planet we may, for example, have Tgy smaller by a factor of 100, and, hence, RLm = 0.3 X 10-4, and mLm = 10-13 g. Hence, the transition between the dominance of electromagnetic and of gravitational forces may be anywhere in the range 10-10 ggreater or equal to mLm greater or equal to
10-16 g, depending upon what spatial environment is being discussed.

In our numerical examples, we have assumed the electrostatic potential of a grain to be a few volts. This is a normal value for a charged solid body in a laboratory plasma. However, a cosmic plasma usually contains high-energy particles such as Van Allen radiation and cosmic rays. It is known that spacecrafts often acquire a potential of some thousand volts due to the charge received at impact by high-energy particles (Fableson, 1973). This is especially the case if some part of the surface consists of an electrically insulating material. It seems quite likely that the grains we are discussing should behave in a similar way under hetegonic conditions. This would increase the value of RLm by one order of magnitude, and the limiting mass by a factor of 1000. As the charging of the grain may take place in an erratic way, RLm may often change rapidly.

 

[78] 5.5. RADIATION EFFECTS

The motion of small bodies may also be affected by radiation. Under present conditions solar radiation has a great influence on bodies the size of a micron (10-4 cm) or less and may also perturb the motion of bodies as large as a meter in size. The effect is due to radiation pressure, the Poynting-Robertson effect, and the ionization and photoelectric effects produced by solar radiation

There is no certain indication that solar radiation had a decisive influence during the formative era of the solar system. As we shall find, the solar system could very well have acquired its present structure even if the Sun had been dark during the hetegonic period. However, it is also possible that solar radiation effects, as we know them today, were important, particularly after the hetegonic era; hence they are discussed below (secs. 5.5.1-5.5.2).

Similarly, there seems to be no reason to attribute any major role to the solar wind; the observed irradiation of grains before their ultimate accretion (sec. 22.9.5) could as well be caused by particles accelerated in the circumsolar structures as in the Sun (sec. 16.8). A very strong solar wind, a "solar gale," is sometimes hypothesized to occur late in the hetegonic era (after accretion). This is done in order to achieve various aims such as to remove gas or excess solids, to provide additional heating of bodies, or to blow away planetary atmospheres. As we will find later, none of these effects are needed to explain the present structure of the solar system and no indication of such a postulated enhancement is found in the early irradiation record (sec. 22.9.5). Hence the "solar-gale" hypothesis appears unnecessary and counterindicated (see sec. 16.2).

 

5.5.1. Radiation Pressure

If a grain of mass m with the cross section [Greek letter]  sigmais hit by radiation with energy flux [Greek letter] capital psi, it will be acted upon by the force

 

mathematical equation
(5.5.1)

 

if the body is black and absorbs all the radiation. If the body is a perfect mirror reflecting all light in an antiparallel direction, the forcemathematical symbol is doubled. If the energy is reemitted isotropically (seen from the frame of reference of the body), this emission produces no resultant force on the body.

[79] Corpuscular radiation such as the solar wind results in a force of the same kind. Under the present conditions in the solar system this is usually negligible because the energy flux is much smaller than the solar radiation, and there is no compelling reason why it ever should have produced very significant dynamic effects.

A black body moving with the radial and tangential velocity components mathematical equation
in the environment of the Sun, and reradiating isotropically, is acted upon by radiation pressure with the components

 

mathematical equation
(5.5.2)

 

mathematical equation
(5.5.3)

 

The effect of the tangential component mathematical symbol is called the Poynting-Robertson effect; this effect is due to the motion of the body in relation to the radiation field of the Sun.

Because [Greek letter] capital psi
decreases in the same way as the gravitational force,

mathematical equation

we put

 

mathematical equation
(5.5.4)

Solving for[Greek letter] gamma we find

 

mathematical equation
(5.5.5)

 

At the Earth's orbital distance we have mathematical equation
for solar radiation. As in the cases of interest to us vr/c <<1, we have approximately

[80] mathematical equation
(5.5.6)

and

mathematical equation
(5.5.7)

For a black sphere with density [Greek letter] capital theta
and radius R we have

mathematical equation
(5.5.8)

 

For [Greek letter] capital theta
of the order of 1 g/cm3, mathematical symbol=1 if R = 0.6 X 10-4cm. From eq. (5.5.6) we conclude that the Sun will repel particles with R < 0.6 X 10-4 cm. (See Lovell, 1954, p. 406.) The corresponding mass is of the order of 10-12 g. This is one of the effects putting a limit on the dominance of gravitation. It so happens that the size of the particles at this limit is of the same order as the wavelength of maximum solar radiation. The existence of such particles today is inferred from the study of the zodiacal light and micrometeorites. From a theoretical point of view, not very much can be said with certainty about their properties.

 

5.5.2. The Poynting-Robertson Effect

Although comparable to gravitation effects for micron-size grains, radiation effects decrease with 1/R as R increases. For [Greek letter] capital theta
= 1 g/cm3 and R = 1 cm, mathematical symbol is 0.6 X 10-4. This is usually unimportant for the radial force, but not for the tangential component, because when applied for a long time it may change the orbital momentum C. As

 

mathematical equation
(5.5.9)

we can write

mathematical equation
(5.5.10)

[81] with

mathematical equation
(5.5.11)

 

where TK is the Kepler period. During a time Te, the orbital momentum decreases by a factor of e. For a grain with R = 1 cm and [Greek letter] capital theta
= 1 g/cm3 near the Earth (mathematical symbol
/c = 10-4, mathematical symbol
= 0.6 X 10-4, and TK = 1 yr) we have

 

Te = 25 X 106 yr (5.5.12)

 

To make this e-folding time equal to the age of the solar system, the body must have R = 150 cm (m = 107 g).

It is generally concluded that the Poynting-Robertson effect causes all small bodies (as we have found, "small" means m < 107 g) to spiral slowly into the Sun. This is not necessarily true. As we shall find in ch. 8, resonances are a characteristic feature of the solar system. If a body once is trapped into resonance with another body, it is very difficult to break this resonance locking. Hence, when a small grain, slowly spiraling inward due to the Poynting-Robertson effect, reaches an a value such that it is in resonance with one of the planets, it may be trapped there forever.

Consider, for example, a small body that is a member of the Hilda family and thus in 2/3 resonance with Jupiter. If the body is so small that the Poynting-Robertson effect would make it spiral inward, this has the same effect as a viscosity. Hence, the drag is compensated by a resonance transfer of angular momentum, with the result that the body remains in resonance. The only net effect is that the eccentricity of the orbit decreases.

Even high-order resonances may be efficient. For example, Jupiter may produce a series of close barricades in the asteroidal belt that prevent bodies, including grains, from changing their periods (and, hence, give them locked a values). The remarkable fact that the present structure of the asteroid belt appears to be directly related to the hetegonic processes may be due to such effects (ch. 18.8).

 

5.6. CONCLUSIONS

(1) Planets and satellites move in Kepler orbits determined solely by gravitation.

[82] (2) For asteroids and all smaller bodies (including single grains), the Kepler motion is perturbed by collisions (viscosity). This type of motion has a tendency to focus the bodies into jet streams. The smaller the bodies, the more pronounced the effect becomes.

(3) Due to their electric charge, very small grains behave as ions and form part of a plasma. Such a "dusty plasma" may contain grains with molecular weights as high as 106 and, under certain conditions, even 1012 or higher.

(4) Under present conditions, solar radiation produces light pressure that completely dominates the motion of micron-size (10-4 cm) and smaller grains. The Kepler motion of larger grains, with sizes up to a centimeter and meter, may be perturbed by the Poynting-Robertson effect. It is doubtful whether these effects were of any importance during the formative period of the solar system, during which period solar radiation may or may not have been significant. The influence of these effects today is also uncertain.

(5) There is no certain indication that the solar wind has had any major influence on the solar system in the formative era.


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