"Max Keon" wrote in message
...

Max, there are two major difficulties in what you are
saying. I'll trim the rest until we resolve those.

"George Dishman" wrote in message
oups.com...
Max Keon wrote:
...
... The
moving matter will be slowed in the direction of motion according
to a combination of the two equations.

The equations describe slowing for an inward moving
mass but increasing speed for an outward moving
mass.

I understand what you are saying George, but your description
is invalid in this case.

What I wrote is simply applying your equations and that
was the point, one of your equations must be invalid.
Let's look at the details.

((c+v)^2/c^2)^.5*G*M/r^2-(G*M/r^2) ...

--- an aside ---

Try applying the laws on this page

http://www.mathsisfun.com/associativ...tributive.html

To start, look at the very last example and see if you
can replace their "16" with "(G*M/r^2)" in your equation.
Then using the other laws, see if you can reach:

(v/c) * (G*M/r^2) if you take the positive root, or

-(2 + v/c) * (G*M/r^2) if you take the negative root.

Note that the latter result, when added to the normal
Newtonian acceleration would produce triple the usual
value hence I assume you intended the positive root.

--- end aside ---

.. applies for motion away from
a gravity source, which naturally increases velocity relative to
the in-moving dimension, and consequently increases the pull
toward the gravity source.

Newton's acceleration due to gravity is -GM/r^2. Note
the negative sign which indicates the force is inward,
or more accurately in the direction of reducing r, that
is towards the mass M. Note that well Max, it will also
be important later, the acceleration acts towards the
body of mass M which is producing it.

The action of gravity, only, has
increased. The acceleration is not applied in the direction of
the outward moving mass, it's applied inward.

For a body moving away from mass M, the value of v is
positive and your equation gives a positive value so it
_decreases_ the effect of gravity.

The opposite of
course applies for motion toward the gravity source. (c+v) is
replaced with (c-v).

For a body moving towards the mass, v is negative. Since
the sign of v has changed and you have replaced (c+v) with
(c-v), the anisotropic force acts in the same direction.
Two negatives make a positive.

I thought you were telling me that motion of the object
towards M reduced the Newtonian force while motion away
from M increased it. If you use this equation

a = -GM/r^2 * (1 + v/c)

then the positive value of v for motion away from M
increases the acceleration while the negative value
of v for motion towards M decreases it.

Of course for v=0 you get the classical result.

Pioneer

For Pioneer, the two directions are similar so we can look
at Max's numbers:

... The Universe alone is responsible for the
anomalous acceleration which appears to be directed toward the
Sun.

((c+v)^2/c^2)^.5 * (G*M/r^2) - (G*M/r^2) = 8.34E-10 m/sec^2.

However, the anomalous acceleration of Pioneer 10 is
-8.74E-10 m/s^2 hence in the opposite direction. This
has already been pointed out to Max.

You still have it all wrong. For Pioneer's 12500m/sec velocity
away from the Sun, relative to the Sun, at the radius of e.g.
Neptune, ((c+v)^2/c^2)^.5*(G*M/r^2)-(G*M/r^2) = 2.626E-10m/sec^2
added pull to the Sun. That's an acceleration toward the Sun,
not away from it.

It is a positive number so it is away from the Sun. If it
was towards the Sun it would be -2.626E-10m/s^2.

GMsun = 1.33e20

For Neptune, r= 4.5e12

So -GM/r^2 = -6.57e-6

If you add 2.626e-10 to -6.57e-6, it gives a _smaller_
acceleration. I think you want a bigger acceleration.

But it's Pioneer's motion relative to the universe that's
responsible for the anomaly because that's the bit which can't
be concealed within any error relating to local gravity (1/r^2).

Pioneer is traveling away from the universe in one direction
while traveling at that same rate toward the universe in the
opposite direction. ((c+v)^2/c^2)^.5 * (G*M/r^2) - (G*M/r^2)
applies for the retreat direction (moving away from the gravity
source) while ((c-v)^2/c^2)^.5 * (G*M/r^2) - (G*M/r^2) applies
for the advancing direction. You can use either one, or more
correctly, both, with half the effective mass of the universe
placed at each end.

Yes, I agree that but we have to sort out some basics
before we can use it.

I didn't specify which formula I was using this time around, but
I may have got it wrong in the past. Anyway, it might be best if
you stick with the un-simplified version of the formula because
your version is only causing confusion.

No, what is causing confusion is that you haven't realised
that positive numbers represent motion and acceleration
_away_ from the mass. You are trying to use a positive
speed for Pioneer to mean its motion away from the Sun but
then use the positive acceleration to mean towards the Sun.
That contradiction is the first problem. If you just stick
with a single equation regardless of the direction of
motion, then the change of sign of v will automatically
change the direction of the acceleration the way you want.

I'm pretty sure if you just say the Newtonian acceleration
is modified to be

a = -GM/r^2 * (1 + v/c)

then you will get all the numbers you are expecting. It
increases the gravitational acceleration for a body
moving outwards, decreases it for a body moving inwards
and gives -2.74e-10 m/s^2 for a craft moving at 12.5 km/s
at the radius of Neptune.

The second problem relates to the direction of the
acceleration. All the discussion above has the effect as
a weakening or a strengthening of the Newtonian force so
obviously it has the usual direction, towards the body M.

The next paragraph addresses that but is full of
contradictions. I'll extract the parts relevant to the
direction:

Pioneer 10's trajectory is 11 degrees off a line through the Sun.

Yes, that is correct.

Its motion relative to the universe is generating a force
.. also along that line
and pointing back to the Sun.

You just noted that "along that line" differs from
"pointing back to the Sun" by 11 degrees.

Which of course sets the anomalous
acceleration in the direction of the Sun along its trajectory
path.

Again "in the direction of the Sun" differs from
"along its trajectory path" by 11 degrees.

Which is it to be Max? All the stuff above says it is
a modification of GM/r^2 which is "in the direction of
the Sun", not "along its trajectory path" so please
decide which it is to be.

Then there's the reaction to the velocity slowing and
consequent loss of momentum, which emerges in the perpendicular
plane as a velocity increase.

That is complete rubbish, any motion other than in the
direction of the force violates conservation of momentum.

Overall, momentum is conserved because the Pioneer craft
is exerting an equal and opposite force on the Sun which
produces a tiny acceleration. That would have the value

a = -Gm/r^2 * (1 + v/c)

where m is the mass of Pioneer, about 241kg from memory.

George