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Re: [Phys-l] Rocket Science



I have worked out the answer to Carl's question about the criterion
for closed orbit motion in a bound state central force problem.

In the discussion below I will not consider the trivial case of
circular orbits which is an allowed closed orbit for any attractive
radial potential with the right combination of total energy and
orbital angular momentum. All considered motions will have the
orbit's radius change in some way over the course of the orbit.

If the motion has a closed bound state orbit that means the orbit
has at least one periapsis and at least one apapsis. Some thought
ought to convince one that if the orbit has more than one each of
these turning point apsides then a) the number of perispses equals
the number of apsides, b) the radial distance of all the periapses
is teh same common value, c) the radial distance of all the apapses
is another common value (but larger than the one for the periapses),
d) the orbit has mirror reflection symmetry across a plane that is
perpendicular to the orbital plane and also contains the center of
force and an apside, and e) directly across the orbit on a line
containing any apside and the force center is another apside. This
last property may either pair up a periapsis with an apapsis across
the orbit (like the case of the attractive inverse square law
potential) or pair up pairs of periapses and pairs of apapses (like
the case of the spherically symmetric SHO potential).

We can decide whether a given combination of potential function,
energy E, and orbital angular momentum L corresponds to a closed
orbit or not by considering the value of a particular definite
integral I whose integrand depends on these things. The integral
in question is given below:

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

The lower limit of the integral is w_<, the upper limit is w_>
and the function D(w) in the sqrt() in the denominator of the
integrand is defined as:

D(w) == 1 - w^2 - V(L/(w*sqrt(2*m*E)))/E .

Here the function V(r) is the central potential as a function
of the radial distance r. The dimensionless integration variable
w is (for a given fixed combination of L, E and particle mass m)
related to this distance according to:

r = L/(w*sqrt(2*m*E)) .

The orbit's periapsis distance r_p is given in terms of the upper
limit of the integral according to:

r_p == L/((w_>)*sqrt(2*m*E))

and the orbit's apapsis distance r_a is, likewise, given in terms of
the lower limit of the integral according to:

r_a == L/((w_<)*sqrt(2*m*E)) .

In order for a bound orbit to exist the function D(w) must have a
pair of positive real roots with the value of D(w) being positive
between these two roots. If this is the case then w_< is the lesser
of these roots and w_> is the greater of these roots. This means
that for a bound orbit the domain of integration of the integral I is
from the lower such root to the greater such root, and the integrand
blows up (in an integrable way) at both integration limits.

If the motion under consideration is not a bound state, say the
particle escapes to (and/or comes in from) infinity then the value of
D(0) > 0 and w = 0 (i.e. r = [infinity]) is included in the radial
allowed domain of the orbit (which is also the integration domain of
the integral).

If the function D(w) has no positive roots and has a positive value
for all positive w then the allowed motion for the orbit has the
particle's motion moving all the way between the force center and
infinity (without turning around at either). Any region of r having
a negative value of D(w) is forbidden region. Since we only care
about positive distances r we only consider posistive w values.

If the function D(w) has only one real root where D(w) is negative
for w greater than that root and positive for w less than this root
then the motion has a single periapsis and comes in from infinity,
turns around at the periapsis, and later escapes back to infinity.

If the function D(w) has only one real root where D(w) is positive
for w greater than that root and negative for w less than this root
then the motion has a single apapsis and climbs out from the force
center, turns around at the apapsis, and later falls back into the
force center.

If D(w) has many positive roots then there are multiple allowed
bound orbital bands that are disconnected from each other all having
the same values of E and L and are distinguished from each other
only by the initial condition r-value of the allowed r-band the
particle ends up orbiting in. The domains of the allowed bands
are regions of positive D(w) and the interleaved forbidden radial
bands are the regions of negative D(w).

Now for the criterion for closed orbits that Carl asked about:

For any allowed bounded radial orbital band (where D(w) is
positive between an adjacent pair of positive roots) that orbit is
a closed figure that doesn't cross itself if and only if

[pi]/I = [positive integer] .

If the value of [pi]/I is, instead of an integer, a positive
rational number (whose rational denominator is not unity) then
the orbit is closed but it *does* cross itself and looks like
some sort of polar Lissajous plot or Spirograph (TM) shape.

If the value of [pi]/I is irrational then the orbit is not
closed at all will never retrace itself at all.

Armed with the above info we can apply this to some familiar
special cases. If we use an attractive inverse square law
potential then the value of [pi]/I = 1 for all noncircular
bound orbits having a nonzero amount of angular momentum.
And if we use a spherically symmetric SHO potential the value
of [pi]/I = 2 for all noncircular orbits having a nonzero amount
of angular momentum.

Pretty much for any other potential the value of [pi]/I depends on
the details of the E & L values and thus such potentials have the
vast majority of bound orbits being unclosed figures with maybe a
small subset (of measure zero) of them being Spirograph-shaped with a
smaller subset of them being actual closed figures that don't cross
themselves.

BTW, for all circular orbits for any potential the value of I is
undefined because then the integrand is infinity and width of the
domain of integration is zero.

David Bowman