Astronomy Principles and Practice Fourth Edition.pdf

Special perturbation theories 191
Figure 14.4. Theeffectofasmallerrorɛ in the desired angle θ in which a billiards player wants the cue-ball to
travel. In n rebounds, the distance D between the intended and actual places of the ball on the opposite side of the
table of width d,isgivenbyD ∼ ndɛ/cos 2 θ, still a small quantity.
limit of resolution of the human eye). Let the billiard table have width d and let the player try to push
the ball, C, from one side of the table to the other at an angle θ degrees to the normal to that side the
ball is resting against. Then the ball rebounds repeatedly from side to side of the table making impacts
on alternate sides at points A 1 , A 2 ,...,A n (see figure 14.4). Suppose that the player has projected the
ball at angle (θ + ɛ) degrees instead of θ, ɛ being a small error of one arc minute. The ball’s impact
points would now be B 1 , B 2 ,...,B n . It is obvious that the error between the directions of travel of the
intended shot at θ and the actual shot at (θ +ɛ) degrees is still ɛ after n rebounds. The distance between
the ‘intended’ and ‘actual’ points where the ball strikes the table side (A n − B n ) at the nth rebound
will be ∼(ndɛ)/cos 2 θ which remains a small distance even for reasonably large n. For example, if
n = 10, d = 1·5m,θ = 30 ◦ and ɛ = 1 arc minute, A 10 − B 10 ≃ 0·0058 m.
Consider now the situation as in figure 14.5(a) where the player, using the same cue-ball at the
same table, tries to strike a second ball B 1 fixed against the opposite side of the table so that the cueball
rebounds from B 1 and strikes a second ball B 2 fixed to the opposite side. It is easily shown that
if the radius of the balls is R, an error ɛ in the direction the cue-ball is projected becomes an error
after rebounding off the fixed ball B 1 of (2dɛ)/R (see figure 14.5(b)). For d = 1·5 m,R = 0·025 m,
ɛ = 1 arc minute, the new error is almost two degrees. The linear error of the ball when it reaches the
second fixed ball is about 0·06 m. If, indeed, it does strike the second ball and rebounds, the angular
error can be larger than 90 degrees. One could safely wager money against even an expert billiards
player in such circumstances managing to strike a third fixed ball B 3 .
If we now consider a small body in the Solar System (the cue-ball) to pursue an orbit that does
not involve a close encounter with a massive planet (the fixed ball), its orbit will be relatively stable
and its predictability horizon will be measured in millions of years. But if the small body makes a
close encounter, even a slight change in the circumstances of the encounter will produce large changes
in the post-encounter orbit. A subsequent encounter with the same or another planet (which may
not have taken place if the first close encounter had occurred under slightly different conditions) will
again drastically alter the small body’s orbit. It is obvious that the body’s orbit is chaotic and has a
predictability horizon far closer to the present than the body that suffers no close encounters.
A current concern of humanity is the possibility of comets and asteroids that cross the Earth’s