In the Sun-Mercury closed gravitating system, from the viewpoint of the Sun, Mercury is oscillating back and forth, and from the viewpoint of Mercury it's the Sun that's oscillating back and forth. Beyond that, there is nothing else of any consequence within that system. Mercury's orbit trajectory is determined entirely from within that closed system.
The Sun is 5555556 times the mass of Mercury, and Mercury's orbit eccentricity from aphelion to perihelion is 2.4E+10 meters, so the Sun will oscillate over only 4320 meters.
When a change in the force of gravity is rapidly introduced into the system, i.e. during the fall or rise between the aphelion and perihelion, the change is necessarily directly added to or subtracted from the normal Newtonian gravity rate and the consequence will be as per diagrams.
But that has nothing to do with the gravity anisotropy generated by Mercury's motion to and from the Sun, which has slowly evolved to its current state over millions of years.
In the next diagram, the oscillatory motion of an object moving along the trajectory of the yellow line has been well established, and the straight line represents the plane where all forces will be zero. In order to halt the downward motion at point 1 and send it back to intersect with the line at point 2, a constant force is applied for the duration of the journey between points 1 and 2. The same applies for the journey between points 2 and 3, but the force direction is reversed. There's no other way the system could function. And it's permanently sustainable so long as the forces remain.
The force directions are exactly the same as those for the gravity anisotropy generated by Mercury's radial velocity relative to the Sun. In that case they act at 90 degrees to the line between the aphelion and perihelion. It's also exactly the same system as that for Mercury's natural elliptical orbit, where the forces are applied along the line through the aphelion and perihelion (not shown).
When the straight line graph is converted to an elliptical orbit, the forces all point in the same direction relative to the universe.
The next diagram was generated using the true anisotropy which is added to the Newtonian gravity rate. It's totally unsustainable and has nothing whatever to do with the gravity anisotropy. For a sustainable eccentric orbit based around Mercury's normal eccentric orbit, enroute to the perihelion, Mercury moves inside the normal trajectory, and enroute to the aphelion, it moves outside the normal trajectory, as shown in the above diagram.
This is the only thing of consequence resulting from the anisotropy. The whole thing oscillates back and forth as though it was all connected by invisible springs.
The black line represents the natural orbit.
Since both systems function in exactly the same way, they will proportionally play the same role in determining the rate of perihelion advance, if there is a role to play. The miniscule role played by the anisotropy can be ignored for this case.
The only apparent in-elasticity in each system is through variations in the time delay in the gravity link between the Sun and Mercury. At the perihelion of Mercury's orbit, the delay in its relationship with the Sun is (perihelion radius divided by the speed of light) 153 seconds. By the time it has reached its aphelion the delay has become 233 seconds. It has lost an additional 80 seconds in its relationship with the Sun.
The time delay has some consequence. It will cause Mercury's trajectory to point further away from the Sun enroute to the aphelion because centrifugal force is unaffected by gravity, while centripetal force has been anomalously reduced through the increasing delay time as Mercury moves further away from the Sun. Mercury is traveling faster toward the aphelion than normally expected if the delay was of no consequence.
Enroute to the perihelion, the relationship between the Sun and Mercury has gained an additional 80 seconds. Radial velocity will increase above that for a naturally flowing orbit where the time delay is of no consequence.
If the average radial velocity is around 5000 m/sec (it's more), multiplying that by 80 * 2 seconds for the complete orbit gives a 800000 meter advance for each orbit cycle, which is far too much. But the story doesn't end there.
The time delay in the gravity link between the Sun and Mercury is measured in the realm of light-time-gravity, where all linear measurements involve the dual planes of dimension perpendicular to the line along which the measurement is taken. But such a line doesn't exist in that realm because every measurement involves all dimensions.
The hypotenuse length of the imaginary right angle triangle scribed in space by a light ray emerging perpendicular to the line of motion, from a source which is moving relative to the local frame, is determined with the Pythagoras equation a^2 + b^2 = c^2. Measurements from the realm of matter are squared and thus elevated to the realm of light-time-gravity so that they can be properly added (in this case). The square root of the result returns it to the realm of matter.
The 80 second time shortfall difference between the aphelion and perihelion radii was determined assuming that time measurements can be determined as they are in the realm of matter, which is wrong. Converting the time measurements to the realm of matter by taking the square root of (aphelion radius divided by the speed of light) and subtracting from it the square root of (perihelion radius divided by the speed of light), results in a time shortfall equivalent in the realm of matter of 2.89 seconds. With the average radial velocity set at 5000 m/sec; 5000 * 2.89 * 2 = 28900 meters is the advance per orbit cycle for the mean orbit radius, which is 42.63 arc-seconds per century.
The forces acting to maintain a stable orbit are pointing in the opposite direction to those for the anisotropy. They are acting between Mercury and the orientation of its orbit eccentricity in the Sun's inertial frame. Hence the advance.
The gravity anisotropy subtracts around .02 arc-seconds from that result because it's acting to retard the orbit ellipse within the Sun's inertial frame.
Mercury's average radial velocity is, (aphelion radius - perihelion radius) 2.4e+10 divided by the half cycle time of 3801600 sec = 6313 m/sec. But that isn't correct for the above purpose because 100% of the radial velocity is not effective in advancing the eccentricity. The 100% only occurs when Mercury is aligned with the Sun along a line which is perpendicular to the line through the aphelion and perihelion. I don't know what the true balance is, but 5000 m/sec was chosen because it gave the correct result, and it wasn't going to be too far out. 5000 m/sec is 79% of the average.
15-11-2008
I'm now using a coordinate system to generate the results, and that's
resulting in a greater degree of advance than before. For the 100%
effective radial velocity, the advance is 41690 meters per orbit,
requiring only 70% of the total radial velocity to be active to achieve
the correct result. (6313 * .7) * 2.89 * 2 = 25542 meters is the simple
advance, which is too little, but that's not what eventuates when the
orbit cycle is indexed around in discrete steps. The final result, 46190,
multiplied by .7 is 29183, which is near enough for the purpose. The .7
figure may seem a little low perhaps, but there will be a reason for that,
as there always is.
The result is almost identical using more conventional logic. For the missing 80 seconds in the Sun-Mercury gravity link enroute to the aphelion, where the Sun and Mercury have been further separated in time, with the average Newtonian gravity rate for the orbit cycle of .04m/sec^2 and an average radial velocity of 6313m/sec, the advance per half orbit is .04 * 80 * 6313 = 20202 meters, or 40404 meters for the complete orbit. 40404 * .7 = 28283 meters. The results are similar because each system tells much the same story.
18-11-2008
The program now addresses the radial velocity per angle problem and
the results are better than expected. The orbit eccentricity advance at
the mean orbit radius in the Sun's frame is 31637 meters per orbit, which
is 46.67 arc-seconds per century. That's around 4 arc-seconds more than
is predicted by conventional logic, which doesn't address the time delay
problem at all. What other effects might the time delay have?
The program is attached to the end of the page.
Mercury's perihelion advance within the Sun-Mercury closed gravitating system indicates that the system is not entirely elastic. If the varying time delay is the cause, it will also account for the lack of elasticity in each closed gravitating system formed between the Sun and every individual component of matter in the universe. The average elasticity in all Sun-universe systems is .36%, as is demonstrated here.
' -----Program start------------
' The program is designed to demonstrate that the advance of
' Mercury's orbit eccentricity within the Sun's inertial frame
' is due to the time delay variations between the Sun and
' Mercury as they move between the aphelion and perihelion of
' their orbits.
SCREEN 12
CLS
CIRCLE (230, 240), 8, 14 'Sun.
c = 299792458
G = 6.67E-11
M = 1.99D+30
pi = 3.14159
'Mercury data --------------------------------
x = 6.982E+10: vy = 38850 ' Aphelion start.
'x = 4.5961E+10: vy = 59017 ' Perihelion start.
multi = 1E-09 ' Multiplier for the graphics.
dt = 1000
progend = 7603200 ' orbit cycle time.
' GOSUB ab ''' Remove the ' switch for Pluto.
' --------------------------------------------
lastradius = x
angle = 8.2638E-07 ' pi / half orbit time of 3801600 sec.
aa:
ryx = x * x + y * y
radius = SQR(ryx)
' ----These three equations are explained here ----
' http://members.optusnet.com.au/maxkeon/sun-merc.html
tim1 = (radius / c) ^ .5
tim2 = (lastradius / c) ^ .5
timdif = tim1 - tim2
' --------------------------------------------
vr = (radius - lastradius) / dt
lastradius = radius
newton = -G * M / ryx
acceleration = newton
ax = acceleration * (x / radius)
ay = acceleration * (y / radius)
vx = vx + dt * ax
vy = vy + dt * ay
x = x + dt * vx
y = y + dt * vy
v = (vx ^ 2# + vy ^ 2#) ^ .5# ' Orbital speed.
CIRCLE (230 + x * multi, 240 + y * multi), 0, 13
LOCATE 20, 1
PRINT vr; "m/sec radial velocity. "
PRINT v; "orbital speed. "
PRINT radius; "radius. "
PRINT
vrangle = ATN(y / x) / (pi / 2)
IF vrangle < 0 THEN vrangle = -ATN(y / x) / (pi / 2)
' The advance is always positive.
PRINT vrangle * 90; "radius angle. "
adv = timdif * vr * vrangle
PRINT adv; "current advance per"; dt; "seconds. "
store1 = store1 + adv
PRINT store1; "total advance. "
dtstore = dtstore + dt
IF dtstore > progend THEN END
GOTO aa
ab: 'Pluto
x = 7.395E+12
vy = 3640
multi = 1E-11
dt = 1000000
progend = 7.8E+09 'orbit cycle time.
RETURN
' ------Program end----------------------