If we divide the distance from the Sun by the distance frbm Neptune for each satellite and square the result we get 162,000,000 for Triton and 655,000 for Nereid. We multiply each of these figures by the mass-ratio of Neptune to the Sun, and that gives us the tug-of-war value, which is:
Triton 8400
Nereid 34
The conditions differ markedly for the two satellites.
The gravitational influence of Neptune on its nearer satel lite, Triton, is overwhelmingly greater than the influence of the Sun on the same satellite. Triton is @y in Nep tune's grip. The outer satellite, Nereid, however, is at tracted by Neptune considerably, but not overwhelmingly, more strongly than by the Sun. Furthermore, Nereid has a highly eccentric orbit, the most eccentric of any satellite in the system. It approaches to within 800,000 miles of Neptune at one end of its orbit and recedes to as far as 6 million miles at the other end. When most distant from Neptune, Nereid experiences a tug-of-war value as low as For a variety of reasons (the eccentricity of Nereid's orbit, for one thing) astronomers generally suppose that Nereid is not a true satellite of Neptune, but a planetoid captured by Neptm on the occasion of a too-close ap, proach.
Neptune's weak hold on Nereid certainly seems to sup port this. In fact, from the long astronomic view, the asso ciation between Neptune and Nereid may be a temporary one. Perhaps the disturbing effect of the solar pull will eventually snatch it out of Neptune's grip. Triton, on the other hand, will never leave Neptune's company short of some catastrophe on tL System-wide scale.
There's no point in going through all the details of the calculations for all the satellites. I'll do the work on my own and feed you the results. Uranus, for instance, has five known satellites, all revolving in the plane of Uranus's equator and all considered true satellites by astronomers.
Reading outward from the planet, they are: Miranda, Ariel, UmbrieL Titania, and Oberon.
The tug-of-war values for these satellites are:
Miranda 24,600
Ariel 9850
Umbriel 4750
Titania 1750
Oberon 1050
All are safely and overwhelming in Uranus's grip, and the high of-war values fit Vith their status as true satellites.
We pass on, then, to Saturn, which has nine satellites:
Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Hyperion, ,Iapetus, and Phoebe. Of these, the eight innermost revolve in the plane of Satum's equator and are considered true satellites. Phoebe, the ninth, has a highly inclined orbit and is considered a captured planetoid.
The tug-of-war values for these satellites are:
Mimas 1 15,500
Enceladus 9800
Tethys 6400
Dione 4150
Rhea 2000
Titan 380
Hyperion 260
Iapetus 45
Phoebe 31/2
Note the low value for Phoebe.
Jupiter has twelve satellites and I'll take them in two installments. The first five: Amaltheia, lo, Europa, Gany mede, and Callisto all revolve in the plane of Jupiter's equator and all are considered true satellites. The tug-o amp; war values for these are:
Amaltheia 18,200 lo 3260
Europa 1260
Ganymede 490
Callisto 160 and all are clearly in Jupiter's grip.
Jupiter, however, has seven more satellites which have no official names (see Chapter 5), and which are com monly known by Roman numerals (from VI to XII) given in the order of their discovery. In order of distance from Jupiter, they are VI 'X, VII, XII, XI, VIII, and IX. All are small and with orbits that are eccentric and highly inclined to the plane of Jupitet's equator. Astronomers consider them captured planetoids. (Jupiter is far more massive than the other planets and is nearer the planetoid belt, so it is not surprising that it would capture seven planetoids.)
The tug-of-war results for these seven certainly bear out the captured planetoid notion, for the values are:
VI 4.4 x 4.3 vii 4.2 xii 1.3 xi 1.2
Vill 1.03 ix 1.03
Jupitees grip on these outer satellites is feeble indeed.
Mars has two satellites, Phobos and Deimos, each tiny and very close to Mars. They rotate in the plane of Mars's equator, and are considered true satellites. The tug-of-war values are:
Phobos 195
Deimos 32
So far I have fisted 30 satellites, of which 21 are con sidered true satellites and 9 are usually tabbed as (prob ably) captured planetoids. I would like, for the moment, to leave out of consideration the 31st satellite, which hap pens to be our own Moon (I'll get back to it, I promise) and summarize the 30 as follows:
Number of Satellites Planet true captured
Neptune I I
Uranus 5 0
Saturn 8 1
Jupiter 5 7
Mars 2 0
It is unlikely that any additional true satellites will be discovered (though, to be sure, Miranda was discovered as recently as 1948). However, any number of tiny bodies coming under the classification of captured planetoids may yet tum up, particularly once we go out there and actually look.
But now let's analyze this list in terms of tug-of-war values. Among the true satellites the lowest tug-of-war value is that of Deimos, 32. On the other hand, among the nine satellites listed as captured, the highest tug-of-war value is that of Nereid with an average of 34.
Let us accept this state of affairs and assume that the tug-of-war figure 30 is a reasonable mibimum for a true satellite and that any satellite with a lower figure is, in all likelihood, a captured and probably temporary member of the planet's family.
Knowing the mass of a planet and its distance from the
Sun, we can calculate the distance from the planet's center at which this tug-of-war value will be found. We can use
Equation 4 for the purpose, setting flf, equal to 30, put ting in the known values for m,, m,, and d,, and then solving for d,. That will be the maximum distance at which we can expect to find a true satellite. The only planet that can't be handled in this way is Pluto, for which the value of m, is very uncertain, but I omit Pluto cheer fully.
We can also set a minimum distance at which we can expect a true satellite; or, at least, a true satellite in the usual form. It has been calculated that if a true satellite is closer to its primary than a certain distance, tidal forces will break it up into fragments. Conversely, if fragments already exist at such a distance, they will not coalesce into a single body. This limit of distance is called the "Roche limit" and is named for the astronomer E. Roche, who worked it out in 1849. The Roche limit is a distance from a planetary center equal to 2.44 times the planet's radius.
So' sparing you the actual calculations, here are the results for the four outer planets:
Distance of True Satellite
(miles from the center of the primary)
Planet maximum minimum
(tug-of-war = 30) (Roche limit)
Neptune 3,700,000 38,000
Uranus 2,200,000 39,000
Saturn 2,700,000 87,000
Jupiter 2,700,000 106,000
As you see, each of these outer planets, with huge masses and far distant from the competing Sun, has ample room for large and complicated satellite systems within these generous limits, and the 21 true satellites all fall within them.
Saturn does possess something within Roche's limit-its ring system. The outermost edge of the ring system stretches out to a distance of 85,000 miles from the planet's center. Obviously the material in the rings could have been collected into a true satellite if it had not been so near Saturn.
The ring system is unique as far as visible planets are concerned, but of course the only planets we can see are those of our own Solar System. Even of these, the only ones we can reasonably consider in connection with satel lites (I'll explain why in a moment) are the four large ones.
Of these, Saturn has a ring system and Jupiter just barely misses one. Its innermost satellite, Amaltheia, is about 110,000 miles from the planet's center, with the