Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: TIDES, was Asteroid Problem



At 04:49 PM 9/1/01 -0400, Ludwik Kowalski wrote:

Stephen Muray referred to the the 1/r^3 tidal effect; I was
under the impression that this is true only when the tidal
source (asteroid) is far away from us (with respect to the
earth diameter).

Write the asteroid's gravitational field g(X) as a Taylor series. Write
X = R + x
where R is the vector from the asteroid to the center of the earth, and x
is the vector from the center of the earth to the point where we want to
evaluate g(X). Do not expand the Taylor series around g(0), but rather
around g(R+0) i.e. x=0.

The lowest-order term is the steady gravitational pull. This is
unobservable or at best uninteresting because we (the observers) and our
frame of reference (the earth) are both freely falling under the influence
of this steady pull. This term scales like
|x|^0 |R|^-2

The next term, the first-order term, is what drives the usual tides. This
term scales like
|x|^1 |R|^-3

We can keep playing this game. There will be a third term, which we can
call the hyper-tide driving term. It scales like
|x|^2 |R|^-4
This term will be smaller than the previous term by a factor of
|x|/|R|. This is small unless the asteroid is nearly touching the surface
of the earth.

There will be yet-higher-order terms, but they will be even smaller.