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Gossamer ring c/R(1000) - sync orbit - - cR(2000) Jupiter

Lassell and Arago

Figure 2.14 Simplified diagram of the ring systems of the giant planets, scaled so that all four giants are the same radius. dR(1000) and dR(2000) are, respectively, the Roche limits of bodies with densities 1000 kg m- and 2000 kg m- . The synchronous orbits are also indicated.

### Lassell and Arago

Figure 2.14 Simplified diagram of the ring systems of the giant planets, scaled so that all four giants are the same radius. dR(1000) and dR(2000) are, respectively, the Roche limits of bodies with densities 1000 kg m- and 2000 kg m- . The synchronous orbits are also indicated.

rings consist of small bodies called ring particles. Very few are more than 1 m across, and the great majority are far smaller, down to less than 1 ^m. The rings are very thin - even in the case of Saturn they are no more than about 100 m thick, and their total mass is only of order 10-5 times the mass of the Earth!

### Rings and the Roche limit

An important concept relating to the origin of the rings is that of the Roche limit, named after the French scientist Edouard Albert Roche (1820-1883) who in 1847 derived the eponymous limit. It arises from the tidal force that one body exerts on another. Figure 2.15 shows the tidal distortion of a body of mass m a distance d from a body of mass M. If m is held together only

Mass M

Mass m tidally distorted by M

Figure 2.15 Tidal distortion, to illustrate the Roche limit.

by gravitational forces, and if both bodies are of uniform density, then it can be shown that m is torn apart if the distance d is less than dR, where

Rm is the radius of M, and pm and pM are respectively the densities of m and M. The second form is obtained from the first form using M = pM x (4^/3)RM. The quantity dR is the Roche limit. As one might expect, dR increases as M increases and as pm decreases.

Equation (2.3) applies to an initially uniform spherical body held together only by the gravitational attraction of one part on another. Bodies are also held together by non-gravitational forces. These operate at short range, binding molecule to molecule. By contrast, gravity, being proportional to 1/r2 (equation (1.5), Section 1.4.4), is a long-range force, and it is therefore the dominant cohesive force in large bodies. Therefore, the Roche limit applies only to bodies that are sufficiently large for gravitational cohesion to dominate - this explains why astronauts and satellites in Earth orbit are not torn apart by the tidal force of the Earth. For non-porous solid bodies made of icy or rocky materials, gravitational cohesion dominates only when they are more than a few hundred kilometres in radius. For poorly consolidated bodies, such as comets and loose aggregates, gravity dominates down to far smaller sizes.

Thus, any sufficiently large body that strays within the Roche limit will be torn apart. The resulting fragments will be numerous and in similar orbits, and therefore collisions among them will be frequent, resulting in further fragmentation. Any tendency to reassemble is counteracted by tidal disruption. After hundreds of millions of years the material evolves to a population of bodies that are predominantly smaller than a metre. This process is a plausible source of ring particles. But the Roche limit also provides a second source - the disruption of bodies already within this limit that are growing by accretion. □ How can this happen?

If such a body grows larger than the size at which gravitational cohesion predominates, it is then disrupted by tidal forces. Both processes continue today.

The importance of the Roche limit is illustrated by Figure 2.14, which shows that not much ring material exists outside the limits. The small quantity that does is readily explained by the inward spiralling of dust produced outside the limit.

Observations show that Jupiter's ring particles seem to be largely devoid of volatiles, and are probably composed mainly of silicates. Those of Saturn seem to be 'dirty snowballs' - mainly

water ice mixed with a trace of less volatile substances, including rocky materials. This can be explained by the higher temperatures expected in the inner protosatellite disc of Jupiter than in

Saturn's disc. This left Jupiter with ice-poor materials for its initial and subsequent populations of ring particles. The ring particles of Jupiter and Saturn also differ in size, with most of Saturn's particles being in the range 0.01-1 m, and most of Jupiter's being far smaller.

□ How can the greater average size of Saturn's ring particles be explained?

This can be put down to the survival around Saturn of water ice, which is an abundant substance.

Little is known about the composition of the ring particles of Uranus and Neptune. They are very dark and for an unknown reason seem to be less icy than Saturn's particles. Their low reflectivity might be the result of solar wind action on hydrocarbons (compounds of carbon and hydrogen). Silicates are presumably also present.

Different-sized ring particles are affected differently by a variety of processes acting on them. One of several gravitational processes arises from the slight departure from spherical symmetry of the giant planet's gravitational field. The outcome depends on whether the orbital period of the particle is greater or less than the giant planet's rotation period. If these two periods are equal then the particle (or any other orbiting body) is said to be in a synchronous orbit (Figure 2.14). In such an orbit there is zero effect. In a closer orbit the outcome is a slow spiralling towards the giant, whereas in a larger orbit the outcome is a slow spiralling outwards. This effect tends to clear the rings of bodies of all sizes, but the replenishment rate is higher for small particles, and so the net effect is a downward trend in the size distribution.

Another gravitational effect occurs in close encounters between particles in nearly identical orbits. After the encounter is over the inner particle is in an even smaller orbit, and the outer one is in an even larger one. This effect is greater, the larger the mass of the particles, and thus it also causes a downward trend in the size distribution of the ring particles. The observed scarcity of ring bodies larger than a metre or so can be explained by these two gravitational effects.

Two other effects are greater, the smaller the body. As a body is swept by solar radiation it encounters the photons rather in the manner that you encounter raindrops when you are running - the front of you collects more raindrops than your back. The effect of the extra photon bombardment on the leading face of a body is to decelerate it. This is the Poynting-Robertson effect, named after the British physicist John Henry Poynting (1852-1914) and the US cosmologist Howard Percy Robertson (1903-1961). For a ring particle the effect is to cause it to spiral towards the giant. The effect is greater, the smaller the particle, because the magnitude F of the net force exerted by the bombardment is proportional to the surface area of the particle, whereas the magnitude of the deceleration (or acceleration) is given by F/m where m is the mass of the particle (equation (1.4), Section 1.4.4). The area, and hence F, are proportional to the square of the particle's mean radius rm, and m is proportional to its cube, so F/m is proportional to r-1. The Poynting-Robertson effect explains the sparseness of ring particles within the inner edge of the rings - particles of sizes that typify the rings traverse this inner region rapidly in their downward spiral.

The second effect is really a group of effects involving electromagnetic forces. A proportion of ring particles is electrically charged through the action on them of electrons and ions in the vicinity of the giant planet. These charged ring particles are then susceptible to electromagnetic forces exerted not only by the planet's magnetic field, but also by the electric and magnetic forces exerted by the ions and electrons that charged the ring particles. As for the Poynting-Robertson effect, small bodies suffer greater accelerations, and therefore electromagnetic forces are particularly important at micrometre sizes and below. Additional removal mechanisms that affect small particles are collisions, including collisions with micrometeorites sweeping in from interplanetary space. Collisions fragment or remove small particles. Bodies of all sizes are removed by the gravitational effects of satellites.

Ring particle lifetimes of 10-100 Ma have been estimated, which is much shorter than the 4600 Ma age of the Solar System. For Saturn, evidence of such a relatively short lifetime is the brightness of the rings, which darken under micrometeorite bombardment. The narrowness of Uranus's rings indicates youth, because rings tend to spread with age. In the case of Jupiter the particles are so small that they spiral into the planet in no more than about 1000 years.

Persistent sources of ring particles are therefore needed. Disruption within the Roche limit has already been described. The fragments from this disruption will collide and produce ring particles. A relatively recent disruption might explain why Saturn's rings are the most massive. Today, all the ring systems have small satellites interspersed among them, and perhaps100-1000 greater than 1 km in size await discovery. These small bodies are ground down, partly by existing ring particles, partly by micrometeorite bombardment, which is though to provide the major source of fresh ring particles. Micrometeorites themselves can become ring particles. In the case of Jupiter, a significant contribution comes from the volcanic emissions of Io, which consist mainly of silicates. The likely composition of the small satellites matches what we know about the composition of the rings.

### Ring structures

The ring systems are structures of exquisite complexity (Plate 18, Figure 2.14). Electromagnetic forces and gravitational forces are responsible for this fine structure too. Of particular note are the gravitational effects of satellites, not only the large satellites well outside the rings, but also small satellites embedded within the rings. Their gravitational effects sustain much of the fine structure. The rings are a playground for modellers. Here, we merely list some of the types of structure seen. Further Reading contains publications where the rings are discussed in much greater detail.

• Narrow gaps between rings, either containing a small satellite or cleared by an mmr with a satellite.

• Narrow rings confined by small satellites.

• Dark radial rings where ring particles have been raised by electrostatic forces.

• Eccentric and inclined rings.

• Density variations around a ring.

And so on, including waves, kinked, and braided rings. What a feast! Question 2.11

Discuss whether, at some time in the future, compared with today

(a) the ring system of Saturn could be much less extensive;

(b) the ring system of Jupiter could be much more extensive.

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