We must begin by realising what sort of curves a satellite which
disturbs the surface of a planet would leave behind it after its
demise. You might think that the satellite revolving round and
round the planet must simply describe a circle upon the spherical
surface of the planet: a "great circle" as it is called; that is
the greatest circle which can be described upon a sphere. This
great circle can, however, only be struck, as you will see, when
the planet is not turning upon its axis: a condition not likely
to be realised.
This diagram (PI. XXI) shows the surface of a globe
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covered with the usual imaginary lines of latitude and longitude.
The orbit of a supposed satellite is shown by a line crossing the
sphere at some assumed angle with the equator. Along this line
the satellite always moves at uniform velocity, passing across
and round the back of the sphere and again across. If the sphere
is not turning on its polar axis then this satellite, which we
will suppose armed with a pencil which draws a line upon the
sphere, will strike a great circle right round the sphere. But
the sphere is rotating. And it is to be expected that at
different times in a planet's history the rate of rotation varies
very much indeed. There is reason to believe that our own day was
once only 2½ hours long, or thereabouts. After a preliminary rise
in velocity of axial rotation, due to shrinkage attending rapid
cooling, a planet as it advances in years rotates slower and
slower. This phenomenon is due to tidal influences of the sun or
of satellites. On the assumption that satellites fell into Mars
there would in his case be a further action tending to shorten
his day as time went on.
The effect of the rotation of the planet will be, of course, that
as the satellite advances with its pencil it finds the surface of
the sphere being displaced from under it. The line struck ceases
to be the great circle but wanders off in another curve—which is
in fact not a circle at all.
You will readily see how we find this curve. Suppose the sphere
to be rotating at such a speed that while the satellite is
advancing the distance _Oa_, the point _b_ on the
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sphere will be carried into the path of the satellite. The pencil
will mark this point. Similarly we find that all the points along
this full curved line are points which will just find themselves
under the satellite as it passes with its pencil. This curve is
then the track marked out by the revolving satellite. You see it
dotted round the back of the sphere to where it cuts the equator
at a certain point. The course of the curve and the point where
it cuts the equator, before proceeding on its way, entirely
depend upon the rate at which we suppose the sphere to be
rotating and the satellite to be describing the orbit. We may
call the distance measured round the planet's equator separating
the starting point of the curve from the point at which it again
meets the equator, the "span" of the curve. The span then depends
entirely upon the rate of rotation of the planet on its axis and
of the satellite in its orbit round the planet.
But the nature of events might have been somewhat different. The
satellite is, in the figure, supposed to be rotating round the
sphere in the same direction as that in which the sphere is
turning. It might have been that Mars had picked up a satellite
travelling in the opposite direction to that in which he was
turning. With the velocity of planet on its axis and of satellite
in its orbit the same as before, a different curve would have
been described. The span of the curve due to a retrograde
satellite will be greater than that due to a direct satellite.
The retrograde satellite will have a span more than half
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