Where we stand right now.
1. I agree Tony your hollow sphere halo variation on my original solid sphere halo of exotic vacuum centered at Sun seems to evade the ephemeris constraint problem of 10^-6M(Sun) exotic vacuum inside the orbit of Uranus at 20AU from the Sun. Newton's potential theory explains that, i.e. neglecting post-Newtonian gravimagnetic "Lorentz forces" - the mass outside a sphere does not influence the interior of the sphere.
2. I also agree with you that the Sanchez Foucault's Pendulum idea applied to the Pioneer 10/11 Anomaly is a very pretty idea that we need to think about because of its possible connection to the Berry-Goldstone phase of the Macro-Quantum coherent rigid Higgs Ocean local order parameter that must be single-valued about any topological defect. In this case the topological defect appears to be an exotic vacuum fluid hedgehog with an effective inhomogeneous rotation with the acceleration anomaly a_P as a simple centripetal acceleration!
That is let f(r) be the local exotic vacuum rotation rate. The tangential speed about the axis of rotation is
v(r) = f(r)r
The inward pointing centripetal acceleration is
a_P(r) = v(r)^2/r = f(r)^2r
f(r)^2 ~ 1/r
Therefore a_P(r) is independent of r
The Einstein exotic vacuum tensor field equation is (Hal Puthoff's non-tensor PV model that conflicts with Einstein's battle-tested "GCT" theory, even Michael Ibison says so, is completely inadequate to deal with these real problems grounded in real observations)
Guv + /\zpfguv = 0
In this Newtonian limit we are essentially only dealing with the 00 component
Goo + /\zpfgoo = 0
Which in the well known way from Einstein's 1915 papers reduces to the Newtonian Poisson field equation
Grad^2V(Geometry) = 4pic^2/\zpf
i.e. G(ordinary mass density source) is replaced by c^2/\zpf
V is the vacuum gravity potential energy per unit test point particle mass.
Motional "Lorentz force" gravimagnetism is in the other neglected components that are here "small". Also no torsion here and no conformal boost gravity and no dilation gravity as yet. This is a gauge theory. Right now we only locally gauge the 4 infinitesimal generators of the 4D translation group i.e. the "momenergy" "space-time charge" in Wheeler's terminology. GCT is simply disguised local phase invariance on the single-valued macro-quantum coherenr local order parameter of the post-inflationary Higgs Ocean - the source of gravity and inertia unified in Einstein's principle of equivalence. I have only been using a U(1) Higgs Ocean, but it could be U(N) with more complex topological defects other than stringy "vortices" of disclination.
If the test particle has a gyroscopic "spin" like Pioneer 10-11's rotations around their centers of mass, then we will also need to check out the additional gravimagnetic components. For now I ignore these.
OK back to freshman physics 101
f(r)^2 ~ 1/r
We have two choices here. Both fit the present facts equally well.
I. fI(r,t) = cH(t)/r
H(t) = R(t)^-1dR(t)/dt in sense of FRW metric
R(t) is dimensionless
R(Now) = 10^61 using the Lp = 10^-33 cm measuring rod
Lp^2 = hG/c^3
cH(Now) = a_P ~ 10^-7cm/sec^2 = 10^-10g(Earth's surface)
II. fII(r,t) = (LpR(t)r)^-1
This choice gives same observational result.
We cannot choose which as yet.
In either case
c^2/\zpf(r,t) = f^2(r,t)
The two models differ in their cosmic time t dependence.
t ~ h/kT
Here t = 0 is the Big Bang.
If you believe quantum gravity foamy graniness of space-time then
"0" ~ 10^-44 sec
"infinite T" ~ hc/kLp
T is Kelvin temperature of the cosmic blackbody radiation that is maximally isotropic ~ 10^-5 or so from WMAP in the global Hubble flow of the physical vacuum, exotic or ordinary. The FRW cosmology metric breaks GCT symmetry of the Einstein field equations giving a global cosmic time with a preferred "rest" called the Hubble flow. This is like the ferromagnet whose spontaneous magnetization violates the rotational symmetry of its field equations - more precisely of the action out of which the field equations come as Lagrange equations from the extremal critical points of the action functional over Feynman histories.
On Oct 21, 2004, at 1:43 PM, Tony Smith wrote:
Bob, your idea "... Why could this slight perturbation not be explained
by some extra matter out there in the Kuiper belt
which the JPL navigators neglected to take into account? ..."
somewhat similar to the following suggestion that I made
to Jack Sarfatti in the context of his dark energy/dark matter model:
Consider a Feynman Lecture picture of a uniform mass distribution in
a spherical shell centered on the sun whose radius is at least
the 20 AU distance of Uranus.
NO MATTER THE SIZE OF THE MASS OR WHETHER IT IS POSITIVE OR NEGATIVE
the resulting potential in that part of the solar system
within 20 AU Uranus distance is CONSTANT and
the resulting force (and acceleration) is ZERO.
Therefore, ANY distibution of exotic dark energy/matter that
can be modelled by a set of such shells concentric on the sun
will have NO observable acceleration WITHIN 20 AU Uranus distance.
as Pioneer goes beyond Uranus and begins to enter the
set of concentric spheres,
it will then perceive the dark energy/matter mass that
is in the shells through which it has passed
as a point mass located at the sun (center of the spheres),
if the mass is positive dark matter, Pioneer will then experience
a corresponding acceleration toward the sun.
The next question is what type of distribution of mass shells
will give a constant acceleration as Pioneer progresses
outwardly through the shells.
[ NOTE - IN MY PREVIOUS MESSAGES I ERRONEOUSLY SAID THAT
URANUS WAS AT 30 AU - IN FACT IT IS ABOUT 20 AU ]
As you mention, shells of ordinary matter beginning at
the 20 AU radius of the orbit of Uranus would work just as
well if the ordinary matter followed a similar distribution
as mentioned in the suggestion above.
The main problem with using ordinary matter is that the
effect begins to be observed as far in as Uranus,
I don't know of any observed ordinary matter that far
inside the Kuiper belt (which conventionally starts beyond
the orbit of Neptune at about 30 AU)
that could account for the Pioneer anomaly.
In other words,
the search for the "new stuff" has to begin at Uranus
because that is where the acceleration change occurs.
It is also conventional that the Kuiper belt (unlike
the Oort cloud much further out) is in the ecliptic plane
rather than a spherical distribution (that could be
dealt with by using a proper planar mass distribution),
that Neptune and Uranus have swept up most of the Kuiper
material that may have been present in the Uranus-Neptune
region when the solar system formed (if true, that would
rule out a Kuiper belt model).
It is interesting that the important region to explore
to study this is the region around Uranus (20 AU)
rather than Neptune/Kuiper belt beginning (30 AU).
An odd fact is that Uranus (where the acceleration begins)
is the only planet whose spin axis is close to its orbital plane.
Maybe something about the acceleration transition region at Uranus
caused its rotation axis to lie almost in its orbital plane
when Uranus was formed.