Environmental Engineering Reference
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U eff
∼1/ r 2
r 1
r 2
r
r 0
r 3
1/ r
Figure 9.8
The effective potential U eff .
the Sun. Scenario (b) occurs when the energy E< 0. In this case, the trajectory
is bound to lie in the region r 1 <r<r 2 . In the comet example, this corresponds
to a bound orbit with the comet orbiting the Sun in an ellipse, as illustrated in
Figure 9.9(b). The orbits of the planets around the Sun also correspond to this
type of E< 0 bound orbit. Finally, scenario (c) occurs when the total energy is
equal to the value of the effective potential at its minimum. In this case the orbit
is constrained to a single value of r
r 3 which corresponds to circular motion, as
illustrated in Figure 9.9(c). We see now why the 1 /r 2 contribution to the effective
potential was called the centrifugal barrier term. In the case of circular motion it
is the term which exactly balances the gravitational pull towards the centre. These
=
(a) hyperbola
(b) ellipse
(c) circle
Figure 9.9 The three types of orbit allowed for general two-body motion in a conservative
and central field of force.
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