[287] 17.1 FERRARO ISOROTATION AND PARTIAL COROTATION
We have shown in sec. 16.3 that a difference in angular velocity between a magnetized central body and the surrounding plasma may lead to a transfer of angular momentum.
From a purely hydromagnetic point of view the final state would be a Ferraro isorotation with = . However, a transfer of angular momentum means an increase in rotational velocity of the plasma, with the result that it is centrifuged outwards. This will produce a region with low density between the central body and the plasma, and the density may decrease so much that anomalous resistance or the production of electrostatic double layers (see sec. 15.3.3) impedes a further transfer of angular momentum. In this way a state is established such that the rotational motion of an element of plasma is essentially given by the condition that the gravitational and the centrifugal forces balance each other. This state is called "partial corotation."
Partial corotation can be thought of as a transient state in the process of angular momentum transfer from the central body. This state is important if the time of transfer of angular momentum from the central body to a cloud of plasma is long compared to the time it takes for the cloud of plasma to find its equilibrium position on the magnetic field line; if the duration of transfer is much greater than the time needed to reach equilibrium, we can treat the partial corotation as a steady state.
We are especially interested in studying the state of motion of grains that are delivered from the plasma and put into a Kepler motion which is essentially independent of the plasma. Thus we should treat the transition through the size limit m_{Lm} of the grains which, according to sec. 5.4, controls whether the motion is essentially governed by electromagnetic forces (from the magnetized plasma) or by gravitation. The plasma is here (as in most regions in space) a "dusty plasma." A grain can pass the limit in three different ways. Its mass can increase due to condensation of refractory substances in the plasma or accretion of other grains. There can also [288] be a change in the electrostatic potential of the grain. As we have seen in ch. 5 such changes are known from space research to occur in an erratic way, sometimes resulting in a jump of two or three orders of magnitude (between a few volts and 1000 volts). It is quite likely that such changes would also occur under hetegonic conditions.
In the following we shall treat the simple case for which the transition from plasma motion to a collisionally perturbed Kepler motion takes place in a time which is short compared to one Kepler period.
If gas is falling in and becoming ionized at a constant rate, and the condensation products are also removed at a constant rate, a state of timeindependent partial corotation may be established. The condition for this is that the rate of transfer of angular momentum equals the angular momentum required to put the infalling gas into rotation. The transfer of angular momentum may be regulated by the density of the plasma in the depleted region between the central body and the plasma element to be accelerated. This density determines the maximum current which transfers the momentum.
In the next section we discuss the state of equilibrium motion of an element of plasma situated in a magnetic flux tube which we have earlier referred to as superprominence (see fig. 16.6.1).
17.2. PARTIAL COROTATION OF A PLASMA IN MAGNETIC AND GRAVITATIONAL FIELDS
We have found that it is important to study the fundamental behavior of a corotating plasma in the environment of a central body with mass M_{c} and a magnetic dipole moment µ, coaxial with the rotational axis (of the central body and of the plasma).
Consider a volume of plasma located at (r,) and in the state of partial corotation with angular velocity . We assume that the plasma temperature is so low that pressure effects and diamagnetic effects are negligible. The plasma is subject to three forces:
....where x is a unit vector perpendicular to the axis of rotation, B the magnetic field, and I the current in the plasma (fig. 17.2.1).
The condition for equilibrium is
The components of B along the r and axes are
and
[290] As thecomponents of F, f_{c} and f_{G} are zero we obtain:
showing that currents along the magnetic field lines are possible (under the condition that they do not perturb the dipole field too much). Further, = 0 gives:
or 0
Finally F_{r}= 0, and consequently
From eq. (17.2.9) follows:
Substituting eq. (17.2.11) into eq. (17.2.10) we see that the r component of the centrifugal force is twice the r component of the electromagnetic force and hence 2/3 of the gravitational force. From eqs. (17.2.10) and (17.2.11) follows a theorem for the partial corotation of a plasma: The gravitational force is balanced, 2/3 by the centrifugal force and 1/3 by the electromagnetic force.
[291] This law does not hold in the plane = 0 where eq. (17.2.8) allows any rotational velocity.
We now find the tangential velocity characteristic of the state of partial corotation. From eqs. (17.2.117.2.2) and (17.2.1017.2.11) follows
The state of rotation described in eq. (17.2.12) will be referred to a partiaI corotation.
17.2.1. Relation Between Ferraro Isorotation and Partial Corotation
If the conductivity of the central body and of the plasma is infinite, all parts of the plasma must rotate with the same angular velocity as the central body. Under these conditions eq. (17.2.4) is satisfied only at the surface given by eq. (17.2.12), where
and at the surface
If r < r_{s} gravitation dominates and the plasma will fall down on the central body.
If r > r_{s} centrifugal force dominates and the plasma will "fall down" to the equatorial plane.
Applying our model to cases of cosmic interest we will find under both conditions that the main opposing force is the pressure gradient which we have neglected. The result is that the plasma separates at r = r_{s}, the inner part becoming an ionosphere around the central body and the outer part forming a ring in the equatorial plane.
[292] 17.3. A PLASMA IN PARTIAL COROTATION
Comparing (17.2.12) with a circular Kepler motion with radius r characterized by
we can state as a general theorem: If in the magnetic dipole field of a rotating central body a plasma element is in a state of partial corotation, its kinetic energy is twothirds the kinetic energy of a circular Kepler motion at the same radial distance.
This factor 2/3 derives from the geometry of a dipole field and enters because the centrifugal force makes a smaller angle with a magnetic field line than does the gravitational force. The plasma element is supported against gravitation in part by the centrifugal force and in part by the current which interacts with the magnetic field to give a force. The above treatment, strictly speaking, applies only to plasma situated at nonzero latitudes. The equatorial plane represents a singularity. However, as this plane will be occupied by a disc of grains and gas with a thickness of a few degrees, the mathematical singularity is physically uninteresting.
Table 17.3.1 compares the energy and angular momentum of a circular Kepler motion and a circular motion of a magnetized plasma.
. 
Circular Kepler motion 
Partial corotation of magnetized plasma 
.  
Gravitational energy 


Kinetic energy 


Total energy 


Orbital angular momentum 


[293] If the plasma has considerable thermal energy, diamagnetic repulsion from the dipole gives an outward force having a component which adds to the centrifugal force. This makes the factor in eq. (17.2.12) smaller than twothirds. It can be shown that this effect is of importance if the thermal energy (where is the degree of ionization, k is Boltzmann's constant, and T_{e} and T_{i} are the electron and ion temperatures) is comparable to the kinetic energy of a plasma particle . Choosing arbitrarily the environment close to Saturn to give an example of the effect, we put m =10m_{H}_{ }= 1.7 X 10^{23} g, = 2 X 10^{6} cm/sec (=orbital velocity of Mimas) and =10 percent. We find that W_{T}/W = 1 percent, if T_{e} = T_{i} =15 000K. This indicates that the temperature correction is probably not very important in the case we have considered.
17.4. DISCUSSION
It is a wellknown observational fact that in solar prominences matter flows down along the magnetic flux tube to the surface of the Sun, presumably under the action of gravitation. The plasma cannot move perpendicular to the flux tube because of electromagnetic forces. The solar prominences are, however, confined to regions close to the Sun and this state of motion is such that the centrifugal force is unimportant. In contrast our superprominences would extend to regions very far away from the central body (see fig. 16.6.1), roughly to the regions where the resulting secondary bodies would be located. In these superprominences the components of the centrifugal force and the gravitational attraction parallel to the flux tube may balance each other, keeping the plasma in a state of dynamic equilibrium; i.e., the state of partial corotation. This state is analyzed in some further detail by De (1973).
17.5. CONDENSATION OF THE PLASMA: THE TWOTHIRDS LAW
If a grain in the plasma is transferred through the limit m_{Lm}, (sec. 5.4) its motion changes from the type we have investigated, and under certain conditions its trajectory will be a Kepler ellipse. We shall confine the discussion to the simple case of the grains which have grown large enough, or have had their electric charge reduced, so that they are influenced neither by electromagnetic forces nor by viscosity due to the plasma. Furthermore, this transition is assumed to be instantaneous so that the initial velocity of a grain equals the velocity of the plasma element from which it derives.
As the initial velocity of the grain is (2/3)^{1/2} of the circular Kepler velocity at its position, a grain at the initial position will move in an [294] ellipse with the eccentricity e = 1/3 (see fig. 17.5.1). Its apocenter A is situated at and its pericenter P at .
The ellipse intersects the equatorial plane = 0 at the nodal points and with
When the grain reaches its angular velocity equals the angular velocity of a body moving in a Kepler circle with radius in the orbital plane of the grain.
If we assume that grains are released only from a ring element (r_{O}, ) of plasma, all of them will then cross the equatorial plane at the circle = 2r_{O}/3. Suppose that there is a small body (embryo) moving in a circular Kepler orbit in the equatorial plane with orbital radius . It will be hit by grains, and we assume for now that all grains hitting the embryo are retained by it. Each grain has the same angular momentum per unit mass as the embryo. However, the angular momentum vector of the embryo is parallel to the rotation axis, whereas the angular momentum vector of the grain makes an angle with the axis. In case is so small that we can put cos = 1, the embryo will grow in size but not change its orbit. (If cos < 1, the embryo will slowly spiral inward while growing.)
Seen from the coordinate system of the embryo, the grains will arrive with their velocity vectors in the meridional plane of the embryo. These velocities have a component parallel to the rotation axis of the central body, equal to (2 GM_{c}/3r_{0})^{1/2} sin, and a component in the equatorial plane of, and directed toward, the central body, equal to (GM_{c}/12r_{O})^{1/2}.
The existence of an embryo in the above discussion is assumed merely to illustrate the importance of the circular orbit with radius 2r_{0}/3 in the equatorial plane. All the grains which are released at a distance r_{0} from the center will cross the equatorial plane at the circumference of this circle, irre....
....spective of the value of (under the condition that we can put cos = 1). These grains will collide with each other and coalesce to form increasingly larger embryos until these are large enough to accrete smaller grains. The large bodies thus produced will move in a circular orbit in the equatorial plane with radius 2r_{0}/3.
17.5.1. Conclusions
Summarizing our results, we have found that a plasma cloud in the dipole field of a rotating central body need not necessarily attain the same angular velocity as the central body. If in the region between the plasma cloud and the central body the density is so low that the parallel electric [296] field may differ from zero, a steady state characterized by a partial corotation described in table 17.3.1 is possible. If at a central distance r_{0} grains condense out of such a plasma, they will move in ellipses with a semimajor axis 3r_{0}/4 and an eccentricity e = 1/3. Mutual collisions between a population of such grains will finally make the condensed matter move in a circle in the equatorial plane with the radius 2r_{0}/3 (see fig. 17.5.1).
In the more general case, when condensation takes place over a wider range of latitudes and central distances in an extended region, one would expect that each condensate grain will ultimately be moving in a circle at a distance of 2/3 times the distance where the condensation has taken place. This may occur under certain conditions, but is not generally true because collisions between the grains are no longer restricted to the equatorial plane. There will he competitive processes through which grains accrete to become larger embryos moving in eccentric orbits. However, the semimajor axes of these orbits are 2/3 the weighted mean of the radius vector to the points of condensation (see ch. 18).
17.6. ENERGY RELEASE DURING ANGULAR MOMENTUM TRANSFER
The transfer of angular momentum from the central body to the surrounding plasma is accompanied by a conversion of kinetic energy into heat. Suppose that a central body with a moment of inertia is decelerated from the spin angular velocity to  by accelerating a mass m, at orbital distance r, from rest to an angular velocity . Then we have
The energy released by this process is
Assuming << we have
[297] and with eq. (17.6.1) we find
As has been studied previously in detail (Alfvén, 1954), the ionized gas will fall toward the central body along the magnetic lines of force, but at the same time its value is increased because of the transfer of momentum from the central body. When the velocity has reached approximately the Kepler velocity, the gas will move out again. The bodies that are formed out of such a nebula move in Kepler orbits. Hence, the final result is that equals the Kepler velocity, so that
This gives
If to this we add the kinetic energy of the falling gas, GM_{c}m/r, we obtain the total available energy,
This energy is dissipated in the plasma in the form of heat. In fact, this may have been the main source of heating and ionizing of the circumsolar and circumplanetary nebulae during the hetegonic era.
Equation (17.6.7) gives the energy release which necessarily accompanies any process by which a mass m initially at rest is put into orbit by transfer of spin angular momentum from a central body. If the transfer is effected by electromagnetic forces, the energy is normally released by electric currents which ionize and heat the plasma. As typically W is much larger than [298] the sum of ionization energies for all the atoms in m, often several hundred times (see ch.23), the energy released in the process of putting a mass into orbit is amply sufficient for producing a high degree of ionization. This emphasizes the conclusion in sec. 15.6 that hydromagnetic processes necessarily must control the formative processes in the solar system.