Transcript Document
Central Field Orbits Section 8.5
• We had, (for a GENERAL central potential!):
Radial velocity r vs. relative coordinate r
r(r) = (2μ)-1([E - U(r)] - [2(2μr2)])½
Time vs. r
t(r) = (2μ-1)∫dr([E - U(r)] - [2(2μr2)])-½
Derivative of θ with respect to r
(dθ/dr) = (r-2)(2μ)-½[E - U(r) -{2(2μr2)}]-½
The Orbit θ(r)
θ(r) = ∫(/r2)(2μ)-½[E - U(r) - {2(2μr2)}]-½dr
• All of these quantities have the common factor:
([E - U(r)] - [2(2μr2)])½
(1)
• (1) should remind you of 1d where we analyzed the particle
motion for various E using an analogous expression & found
turning points, oscillations, phase diagrams, etc.
– It is similar to conservation of energy in 1d, which gives:
(dx/dt) = x(x) = [(2m-1)(E - U(x))]½
• We now qualitatively analyze the RADIAL motion for a
“particle” of mass μ in the presence of a central potential U(r)
by methods similar to those used in Sect. 2.6 for 1d motion.
• Remember: The ACTUAL motion is 2 d! We need to
superimpose angular the motion (θ) on the results for the
radial motion, to get the ORBIT θ(r) or r(θ).
• Look at the radial velocity vs. r:
r(r) = (2μ)-1([E - U(r)] - [2(2μr2)])½ (a)
We could use this to construct the r vs. r phase diagram! See
Goldstein’s graduate mechanics text.
• Or, conservation of energy:
E = (½)μr2 + [2(2μr2)] + U(r) = const (b)
• Define: Radial turning points Points where r = 0
(the particle stops moving in the r direction!)
– Look at either (a) or (b) & find, at the radial turning points:
[E - U(r)] - [2(2μr2)] = 0
(1)
Turning Points
[E - U(r)] - [2(2μr2)] = 0 (1)
r(r) = (2μ)-1([E - U(r)] - [2(2μr2)])½ (2)
• The solutions r to (1) are those r where (2) = 0
Radial Turning Points are called Apsidal Distances
• Generally, (1) has 2 roots (the max & min r of the orbit)
rmax & rmin We know that rmax ≥ r ≥ rmin
The radial motion will be oscillatory between rmax & rmin.
• For some combinations of E, U(r), , (1) has only one root:
In this case, from (2) r = 0, for all t
r = constant
The orbit r(θ) will be CIRCULAR.
Closed & Open Orbits
[E - U(r)] - [2(2μr2)] = 0 (1)
r(r) = (2μ)-1([E - U(r)] - [2(2μr2)])½ (2)
• If the motion for a given U(r) is periodic in r, then
The orbit r(θ) will be closed. That is, after a
finite number of oscillations of r between rmax & rmin,
the motion will repeat itself exactly!
If the orbit does not exactly close
on itself after a finite number of
radial oscillations between
rmax & rmin, the orbit r(θ) will
be open. See figure
• Use the orbit eqtn θ(r) to find the change in θ due to one
complete oscillation of r from rmin to rmax & back to rmin:
– Angular change = 2 the change in going once rmin to rmax:
Δθ 2∫(/r2)(2μ)-½[E - U(r) - {2(2μr2)}]-½dr
(limits rmin r rmax)
Δθ |
|
Δθ = 2∫(/r2)(2μ)-½[E - U(r) - {2(2μr2)}]-½dr (rmin r rmax)
• Periodic motion & a closed orbit results if & only if the
angular change is a rational fraction of 2π. That is:
Δθ 2π(a/b) (a, b integers) periodic, closed orbit.
• If the orbit is closed, after b periods, the radius vector the of
particle will have made a complete revolutions & the particle
will be at its original position.
• It can be shown (Prob. 8.35) that if the potential is a power
law in r: U(r) = k rn+1 a closed NON-CIRCULAR path can
occur ONLY for n = -2 (the inverse square law force: gravity,
electrostatics; discussed in detail next!) & n = +1 (the 3d
isotropic, simple harmonic oscillator of Ch. 3).
– Footnote: Some fractional values of n also lead to closed
orbits. These aren’t interesting from the PHYSICS viewpoint.
Centrifugal Energy & Effective Potential
Section 8.6
• Consider again the common factor in r, Δθ, θ(r), t(r), etc.:
([E - U(r)] - [2(2μr2)])½
• Qualitatively analyze the RADIAL motion for a
“particle” of mass μ in a central potential U(r) by
methods similar to those used in Sect. 2.6 for 1d
motion.
– Remember that the ACTUAL motion is in 2d!
Must superimpose the angular motion (θ) on the
radial motion results, to get the ORBIT θ(r) or r(θ).
• Consider: ([E - U(r)] - [2(2μr2)])½
• Recall the physics of the term [2(2μr2)]! From previous
discussion (conservation of angular momentum; = μr2θ):
[2(2μr2)] (½)μr2θ2
The angular part of the kinetic energy of mass μ.
• When it is written in the form [2(2μr2)], this
contribution to the energy depends only on r.
Because of this, when analyzing the r part of the
motion, we can treat this as an additional term in
the potential energy. It is often convenient to call
it another potential energy term
THE “CENTRIFUGAL” POTENTIAL ENERGY
Centrifugal Potential
[2(2μr2)] “Centrifugal” Potential Energy Uc(r)
– As just discussed, this is really the angular
part of the Kinetic Energy!
Consider the “Force” associated with Uc(r):
Fc(r) - (∂Uc/∂r) = [2(μr3)]
Or, using = μr2θ :
Fc(r) = [2(μr3)] = μrθ2 “Centrifugal Force”
Centrifugal Force
Fc(r) = [2(μr3)] “Centrifugal Force”
• Fc(r) = A fictitious “force” arising due to fact that the
reference frame of the relative coordinate r (of the “particle”
of “mass” μ) is not an inertial frame!
– Its not a force in the Newtonian sense! (It doesn’t come from any
interaction of the “mass” μ with its environment!) It’s a part of the “μa”
(right!) of Newton’s 2nd Law, rewritten to appear on the “F” (left) side.
For more discussion, see Ch. 10.
• “Centrifugal Force” is an unfortunate terminology! Its
confusing to elementary (& also sometimes to advanced!) physics
students. Direction of Fc: Outward from the force center!
– I always tell students that there is no such thing as centrifugal force!
• For a particle moving in a circular arc, the actual, physical force in an
Inertial Frame is directed INWARD TOWARDS THE FORCE
CENTER Centripetal Force [F = μrθ2 = μrω2 = μ(v2/r)]
Effective Potential
• Consider again: [E - U(r) - {2(2μr2)}]½
• For both qualitative & quantitative analysis of the
RADIAL motion for the “particle” of mass μ in the
central potential U(r), Uc(r) = [2(2μr2)] acts as an
additional potential & we can treat it as such!
– Recall that physically, it comes from the Kinetic Energy of
the particle!
Its convenient to lump U(r) & Uc(r) together into
an Effective Potential V(r) U(r) + Uc(r)
V(r) U(r) + [2(2μr2)]
• Effective Potential V(r) U(r) + [2(2μr2)]
• Consider now: r [E - V(r)]½
(1)
• Given U(r), we can use (1) to qualitatively (&
quantitatively) analyze the RADIAL motion
for the “particle”. Get (radial) turning points,
(radial) oscillations, etc.
– Very similar to 1d where we analyzed
particle motion for various E using an
analogous expression.
• Consider now, an important special case (in the
mathematical sense) which is the MOST important
case in the physics sense!
• Consider the Inverse Square Law central force:
F(r) = - kr-2
U(r) = - kr-1 = - (k/r)
– For convenience, take U(r) 0
Gravitational Force: k = GmM
Coulomb Force: (SI Units): k = (q1q2)/(4
• The Effective Potential for this case is:
V(r) -(k/r) + [2(2μr2)]
0)
• Effective Potential, Inverse Square Law
force. (see figure): V(r) = -(k/r) + [2(2μr2)]
• Lets now qualitatively analyze the radial motion using V(r) for
the Inverse Square Law force (figure).
E = (½)μr2 + V(r) E - V(r) = (½)μr2
r = 0 at turning points (where E = V(r))
• Consider E ≥ 0
(E1 in figure): This
case corresponds to
E = E1
unbounded radial
motion. μ moves in
r = r1
(½)μr2
from r = towards
the force center (at
r = 0), “hits” the
barrier at (turning
point) r1 ( r = 0), and is “reflected” back to r = .
• Qualitative analysis of the radial motion using V(r) for the
Inverse Square Law force (figure).
E = (½)μr2 + V(r) E - V(r) = (½)μr2
r = 0 at turning points (where E = V(r))
• Consider
Vmin < E < 0 (E2 in
figure): This case
corresponds to
oscillatory radial
r = r2 E = E2 r = r4
motion. μ moves
back & forth between
(½)μr2
(turning points) r2 &
Vmin
r4 ( r = 0). These turning points are the Apsidal Distances of
the orbit (the rmax & rmin from before).
• Qualitative analysis of the radial motion using V(r) for the
Inverse Square Law force (figure).
E = (½)μr2 + V(r) E - V(r) = (½)μr2
r = 0 at turning points (where E = V(r))
• Consider E = Vmin
= -(μk2)/(22) (E3 in
the figure): This case
corresponds to no
radial motion at all
r = r3
because r = 0 for all
E = E3
time. That is, the orbit
radius is constant
Vmin
(r = r3 in the figure). That is, μ moves in a Circular Orbit!
• Note: E < Vmin is Not Allowed! (r would be imaginary!)