Earth-Comet Encounter
المؤلف:
Sidney B. Cahn And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 12
28-7-2016
1396
Earth-Comet Encounter
Find the maximum time a comet (C) of mass m following a parabolic trajectory around the Sun (S) can spend within the orbit of the Earth (E). Assume that the Earth’s orbit is circular and in the same plane as that of the comet (see Figure 1.1).
Figure 1.1
SOLUTION
The total energy of the comet is zero since its trajectory is parabolic. In general,
(1)
where r is the comet’s distance from the Sun, is its angular momentum, and U(r) is its potential energy (see Figure 1.2). U(r) = -GmMsun/r,
Figure 1.2
where G is the gravitational constant. Find the total angular momentum defined at the perihelion, where
Therefore,
(2)
where α = -GmMsun. From (1)
so the time the comet spends inside the Earth’s trajectory is
But l2 = 2mαp, so from (2)
(3)
where a is the radius of the Earth’s orbit. The expression 
can be easily integrated by parts, yielding
Substituting this result back into (3) gives
We know that the period of the Earth’s revolution about the Sun equals one year, and noting that 
we can rewrite (4) in the form
Denoting we find the maximum of 
given that 
Therefore tmax= 2TE/3π ≈ 77days = 11 weeks.
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