Re: [Phys-l] Rocket Science

["David Bowman" <David_Bowman@georgetowncollege.edu>]

On Jul 28, 2006, at 6:18 AM, John Mallinckrodt wrote:

> At least among power law central forces this is correct.  Other
> powers of the form n^2 - 3 (e.g., 6, 13, 22, etc.) give rise to
> approximately closed orbits in the "nearly circular" limit.  ....

In case anyone may be interested I thought I would explain how this
result that John mentioned for nearly circular orbits with attractive
power law potentials comes about using the closed orbit criterion I
recently gave, i.e. integer = n = [pi]/I where

I = Int{w_<, w_> | dw/sqrt(D(w))}

with

D(w) == sgn(E) - w^2 - V(L/(w*sqrt(2*m*|E|)))/|E|

and w_< and w_> being adjacent positive roots of D(w) when D(w) is
positive in the region of w between them.

First, suppose we have any generic attractive potential such that the
orbit is nearly circular.  If this is the case that means that the
values of the relevant adjacent roots (w_< & w_>) are very close to
each other.  Assuming that the potential function is a smooth
function in the vicinity of these roots we can fairly accurately
(i.e. to no more than cubic error terms) approximate the function
D(w) in the vicinity of these roots by an inverted parabola whose
vertex just barely pokes up above the D = 0 axis at a w-value that is
half way between the roots.  This nearly circular orbit case is a
case of the 2 real roots of interest being a nearly double root.  (An
actual circular orbit would actually have these roots being on top of
each other as a true double root.)

Making the requisite quadratic approximation to D(w) yields:

D(w) ~= D_0 -(|D''|/2)*(w - w_0)^2 + terms of order (w - w_0)^3

Here w_0 is the root of the equation that makes the first deriviative
of D(w) vanish, i.e. D'(w_0) = 0 where D'(w) == dD(w)/dw .  Also

D_0 == D(w_0)

and D'' is the second derivative evaluated at w_0, i.e.

D'' == D''(w_0) where D''(w) = d^2D(w)/dw^2 .  Note that for an

inverted parabola D'' is negative.  Substituting this approximation
for the D(w) function in our expression for I and factoring out the
positive constant |D''|/2 from inside the sqrt and factoring the
remaining quadratic polynomial in the sqrt gives an integral of the
form:

I = sqrt(2/|D''|)*Int{w_<, w_> | dw/sqrt((w_> - w)*(w - w_<))}

where the roots are w_> = w_0 + sqrt(2*D_0/|D''|) and
w_< = w_0 - sqrt(2*D_0/|D''|) .

Now it is a fairly straightforward matter to show that for *any* pair
of roots w_< & w_> (with w_< < w_>) the integral above is equal to
[pi] (or you can look it up in a table of integrals), i.e.

[pi] = Int{w_<, w_> | dw/sqrt((w_> - w)*(w - w_<))} .

This means that our value for I is given by

I = [pi]*sqrt(2/|D''|)

for any nearly circular orbit where our quadratic approximation to
D(w) is a good one.

Next we apply what we have here for nearly circular orbits to the
special case of a power law potential of the form:

V(r) = (k/p)*r^p

where the force constant k is positive (in order for the potential
to be attractive).  Note that the factor of 1/p out front changes
sign when p is negative and this automatically keeps the potential
attractive regardless of whether p is positive or negative.

Substituting in this form for V(r) into our expression for D(w)
gives:

D(w) == sgn(E) - w^2 - (k/p)*(L/(w*sqrt(2*m*|E|)))^p/|E|

Differentiating this expression w.r.t. w and setting the derivative
to zero and solving the resulting equation gives the value of w_0
which is the location of the peak of our D(w) function and the
peak of our quadratic approximation to it.  The result of this
process of locating the maximum of D(w) is

w_0 = (1/sqrt(2*|E|))*(k*(L/sqrt(m))^p)^(1/(p+2))

and the height of that peak is

D_0 = sgn(E) - (1 + 2/p)*(1/(2*|E|))*k^(2/(p+2))*(L^2/m)^(p/(p+2))

and

|D''| = 2*(p + 2) .

Since nearly circular orbits require that I = [pi]*sqrt(2/|D''|) this
means that for such orbits in a power law potential we have:

I = [pi]/sqrt(p + 2) .

Since the criterion for a closed orbit is [integer] = n = [pi]/I this
gives n = sqrt(p + 2) or

p = n^2 - 2

Since p is the power of r in the potential this means that p - 1 is
the power of r in the force law.  Thus an attractive power law force
has its exponent of the form n^2 - 3 for nearly circular closed
orbits.  The value of the integer n is the number or radial cycles in
one revolution.  So n = 1 for the inverse square law, n = 2 for the
spherical SHO potential, n = 3 for a r^6 force law, n = 4 for a r^13
force law, n = 5 for a r^22 force law, etc.

Using our expression relating the apapsis distance r_a to w_< and
the periapsis distance r_p to w_> and using the expressions for the
values of w_< & w_> in the nearly circular orbit approximation and
using the relevant parameters w_0, D_0, and |D''| for a power law
potential allows us to find the values of r_p & r_a in these
circumstances. The results are:

r_a = (1/(1 + sqrt(A))*(L^2/(m*k))^(1/(p + 2)) and

r_p = (1/(1 - sqrt(A))*(L^2/(m*k))^(1/(p + 2))

where we defined the parameter A as:

A == (2*E/(p + 2))*((m/L^2)^(p/(p + 2)))/k^(2/(p + 2)) - 1/p .

For the case of nearly circular orbits (where our approximation is
valid) the value of the dimensionless parameter A is just barely
positive but much much less than unity.  The criterion for an actual
circular orbit is the vanishing of A altogether.  Requiring that A
vanish allows us to find the value of E necessary to make a circular
orbit for given values of m, L, k, and p.  Note that when p is
positive E is positive, and when p is negative, so is E.

David Bowman