& Tornado Alley
The purpose of this material is to derive the shape of the equipotential surfaces of a rotating gas cloud held together by gravitational attraction to a central mass. The equipotential surfaces describe the shape of a body because the gradient of these surfaces gives the force which counterbalances the gradient of the pressure in the body.
Let the y-axis be the axis of rotation. For a particle at the point (x,y) the components of the forces on it are:
where r²=x²+y², G is the gravitational constant and M is the central mass.
The equation of the equipotential surface is given by:
For the components of force given above this equation reduces to:
A numerical solution of this equation for arbitrary values of the variables and the parameter is shown below. This shows the general shape of the solutions to the equation.
For x<<y this is approximately
Let y0 be the value of y at x=0. Then
On the other hand, where y<<x the equation for the equipotential surfaces reduces to
The equation for the profile of an equipotential surface has a far simpler form in polar coordinates; i.e., r=r(θ).
Equating these two expressions for dy/dx and cross multiplying to clear fractions gives
With the elimination of the common term r'sin(θ)cos(θ) and the combining of rsin²(θ) and rcos²(θ) the result is
The quadrature (integration) of this last equation gives
Imposing a boundary condition at θ=0 is inappropriate. A more convenient point is θ=π/4. If r(π/4) = r0 then C=(α/3)r0² so the polar form of the equation for the profile of the equipotential surface is
(To be continued.)
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