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From: George Huxtable (no email)
Date: Wed Nov 05 2003 - 15:00:07 EST
>Herbert Prinz wrote:
>
>> On a spheroidal earth, if you proceed on a rhumb line with constant
>>speed, you
>> will arrive at a pole after a finite time. You won't be able to stop
>>your vessel
>> at this very moment, because of your inertia . This raises the puzzling
>> question: Where will you be a second after you will have passed through the
>> pole? Neither Dutton nor Bowditch has the answer.
Trevor Kenchington responded-
>Following a rhumb line on a spheroidal Earth results in a path which
>follows a loxodrome. A loxodrome only reaches either pole after
>_infinite_ time, gradually spiralling in towards the pole but never
>quite getting there. Hence, you cannot pass through, or even reach, a
>pole while on a rhumb line, though you could get very, very close if you
>were really determined to try.
Walter Guinion thought the same.
They are both wrong, and Herbert is correct; though I'm sure he is capable
of defending his corner without my aid.
The presumption is that you are steering a rhumb-line course, with a
Northerly component, and that all the Polar ice has melted.
As you spiral in toward the pole on a rhumb-line course, you travel a
FINITE distance to get there. If you can maintain a constant speed, then
that can be done in a finite time. The snag is, you have to travel in
ever-decreasing circles, as the size of the spiral shrinks. To get exactly
to the pole, you have to make an infinite number of such gyrations, so the
vessel has to be spinning at an infinite rate. This is rather an unphysical
state of affairs, to say the least: a "singularity".
The pole will be reached at a predictable moment, at which the ship will be
spinning round at infinite speed. Just after that moment, it will escape
from the pole, still spinning with infinite speed, and the radius of the
spiral then increases, and the spinning slows, until the vessel reaches its
original latitude. What will its longitude be then?
Of all the unrealistic questions we have considered on this list, this is
perhaps the most unrealistic of all. But why should we let that deter us
from playing such games?
The picture above, of a vessel spinning infinitely fast at the pole,
applies to all incoming courses except 0deg and 90 deg, as Trevor points
out. At 90deg, the pole is never reached at all: the vessel sticks to the
equator. At 0deg, the vessel approaches the pole along a certain line of
longitude, then emerges along a line of longitude 180deg different. In that
case, no spinning of the vessel occurs. One might argue, by analogy with
that 0deg case, that a vessel spiralling into the pole on any other course
might emerge from its traumatic spinning at the pole at an instantaneous
meridian exactly opposite (180 deg away) to the way it went in. I don't
know how that would hold up mathematically. In that case, it would spiral
back toward the equator, always 180deg in longitude from its incoming track
at the same latitude, and with a course that differs by 180d.
However, in view of the unphysical infinities involved, I would prefer to
say that the longitude (at a particular latitude) of the outgoing spiral is
indeterminate, and perhaps the course is also. The coward's way out?
George.
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