I thought about calling this thread "MacM's balls" but decided that might be too personal ;)

The purpose of this thread is to evaluate the UniKEF field inside and outside a spherical mass and the UniKEF force between two spherical masses. I plan to begin with uniform-density (but arbitrary density) spheres and will generalize later to spheres where density is an arbitrary function of the radial distance. Hopefully this will allow for unambiguous calculations of UniKEF orbital mechanics and stellar lifecycles, etc.

I plan to use the same procedure that I used in the Jello thread to calculate the interior field. Aer and Billy, since I am starting over I would be glad to use the more traditional spherical coordinates. The Mathematica default is {r,θ,φ} with r [0,Infinity), θ [0,π], and φ (-π,π] where θ is the angle with the positive z axis and φ is the projected angle in the x,y plane measured counter-clockwise from the positive x axis. What would you prefer I use for easier communication?

MacM, from our recent discussion on the main UniKEF thread you seemed to indicate that I can use a linear attenuation v. density. This means that I can do similar integrals as in the Jello thread and just multiply the length of each line by the density, ρ, to get the force (or integrate ρ along the line for non-constant densities). Of course, for exterior points the CoS will not be the whole sphere, but rather some smaller angle. However, this will only affect the limits of integration rather than the integrand itself. Basically, what I am asking you is for a simple "yes" or "no" on the integrand I used in the Jello thread and a simple "yes" or "no" on wether or not I can multiply that integrand by ρ to account for an arbitrary but constant density. This is basically the same as asking, "if the geometry does not change but the density doubles will the UniKEF field double?"

-Dale

...The purpose of this thread is to evaluate the UniKEF field inside and outside a spherical mass and the UniKEF force between two spherical masses. I plan to begin with uniform-density (but arbitrary density) spheres and will generalize later to spheres where density is an arbitrary function of the radial distance. Hopefully this will allow for unambiguous calculations of UniKEF orbital mechanics and stellar lifecycles, etc.This seems like a good plan, but I think even some considerations of stellar life cycles can be achieved with consideratition of two isolated sphere of the same total mass but different radii, R1 & R2, but each with uniform internal density, obviously not equal.

...I would be glad to use the more traditional spherical coordinates. The Mathematica default is {r,θ,φ} with r [0,Infinity), θ [0,π], and φ (-π,π] where θ is the angle with the positive z axis and φ is the projected angle in the x,y plane measured counter-clockwise from the positive x axis. What would you prefer I use for easier communication?Mathematica's unusual set is OK by me and I sure do not want you to get confused or need to convert when going to them. I do however, suggest that we frequently refer to θ as the "polar angle" and φ as the "azmuthial angle" so late comers do not assume they are the more conventionally defined ones. (I was there from the start in Jello etc and was confused by your choice.) I hope you will continue the convention that capital letters refer to surface points and lower case ones to interior point, but the r argument of a function is not any point so this is not really an exception and normally is lower case.

I will let MacM respond to remainder of your post, but I am reasonably confident that the density independent uniKEF is only one of his "options" I am suggesting that we call his other modle "uniKEF~" as it appears to contain a term I do not under stand but does depend explicitly upon density (or may be implictly as flux energy does so.) he has not yet replied to my most recent post in the uniKFE analysis thread where I asked about this.

In prior post I spoke of two isolated spheres of the same total mass but different radii, R1 & R2, each with uniform internal density, but obviously not the same, as a simple means to test uniKEF.

My idea only requires that the total force on a unit of surface area, F = P, the surface pressure, be evaluated. Thus, it may be wise to do this simple "surface force" calculation first for reasons that will be clear below.

Idea is the following:

In sphere 1's surface pressure equation (an expression we derived analytically, I hope), we set the value of the "uniKEF G" (Gu designated, or G if others prefer, as I do) or the "calibration constant", so that if R1 is the solar radius and the mass of sphere 1 is that of the sun we get with uniKEF the same surface gravity and pressure (P1) as Newtonian gravity gives for the surface of the sun.

Then we calculate P2 for sphere 2, (which has R2 < R1 and in the real world is hotter and denser), with the same G which made sphere 1 stable. We need to at least find that P2 > P1 if unikEF is to have any chance of containing the star 2 in later stages of fusion or even making it begin to contract when the first stage fusion is over.

I expect that when the calculation is done with even a slightly smaller radius for P2 it will turnout that P2 < P1 as the star 2 is intercepting less flux, but ihave been know to be wrong before. :(

I thought about calling this thread "MacM's balls" but decided that might be too personal ;)

LOL :D

MacM, from our recent discussion on the main UniKEF thread you seemed to indicate that I can use a linear attenuation v. density. This means that I can do similar integrals as in the Jello thread and just multiply the length of each line by the density, ρ, to get the force (or integrate ρ along the line for non-constant densities). Of course, for exterior points the CoS will not be the whole sphere, but rather some smaller angle. However, this will only affect the limits of integration rather than the integrand itself. Basically, what I am asking you is for a simple "yes" or "no" on the integrand I used in the Jello thread and a simple "yes" or "no" on wether or not I can multiply that integrand by &rho; to account for an arbitrary but constant density.

Yes.

This is basically the same as asking, "if the geometry does not change but the density doubles will the UniKEF field double?"

-Dale

I use different terminology here. "U" is the field and it has some intense but finite unknown strength. "~" is the attenuation factor, currently being considered to be linear with mass penetration but also unknown as to specific value.

Collectively U * ~ = UG which is a term comperable to "G" in Newtonian gravity but adjusted such that a specific case (i.e. congruent identical spheres in surface contact) calculated in Newtonian gravity will result in the same force of gravity when used in UniKEF's:

Fg = UG * PM1 *PM2

When you double the density of a sphere you double the attenuation or differential of strength in "U" in the shadow.

Yes.Excellent. Then I think I will be able to do the integrals analytically.
I use different terminology here. "U" is the field and it has some intense but finite unknown strength. "~" is the attenuation factor, currently being considered to be linear with mass penetration but also unknown as to specific value.

Collectively U * ~ = UG which is a term comperable to "G" in Newtonian gravity but adjusted such that a specific case (i.e. congruent identical spheres in surface contact) calculated in Newtonian gravity will result in the same force of gravity when used in UniKEF's:

Fg = UG * PM1 *PM2

When you double the density of a sphere you double the attenuation or differential of strength in "U" in the shadow.OK, I believe I understand your terminology. By "UniKEF field" I meant the force field around an object (or objects) due to the attenuation of the background UniKEF flux. I guess we could call my force field the attenuation field or the shadow field or something similar. It's your theory so you should probably pick the terminology and I will use it.

-Dale

My idea only requires that the total force on a unit of surface area, F = P, the surface pressure, be evaluated. Thus, it may be wise to do this simple "surface force" calculation first for reasons that will be clear below.
I think I will be able to calculate the gravitational force analytically. The pressure is simply the weight of a unit-area column of gas, so it should be pretty easy to calculate given an analytical expression for the force.

-Dale

Since beginning to read about UniKef, my beer bills have skyrocketed. My Fizzix book is in the pawnshop.

I don't remember anything about what Newton Gravity says concerning the field gradient inside a sphere of homogeneous density, or the gradient at its surface. Help?

And I don't remember anything about how attenuating ( weakening ) lines of force, which, however, do not diminish in quantity, can provide an inverse square field gradient. Help!

Maybe, if they DO diminish in QUANTITY also, it will make inverse square? Hey, A-OK!

Since beginning to read about UniKef, my beer bills have skyrocketed. My Fizzix book is in the pawnshop.

I don't remember anything about what Newton Gravity says concerning the field gradient inside a sphere of homogeneous density, or the gradient at its surface. Help?

And I don't remember anything about how attenuating ( weakening ) lines of force, which, however, do not diminish in quantity, can provide an inverse square field gradient. Help!

Maybe, if they DO diminish in QUANTITY also, it will make inverse square? Hey, A-OK!


You need to switch from beer to hard liquor. :D

I don't remember anything about what Newton Gravity says concerning the field gradient inside a sphere of homogeneous density, or the gradient at its surface.The classical field inside a spherical shell is zero and the classical field outside a sphere is exactly the same as though it were a point mass located at the center (G m/r^2). If you combine these two results then you can calculate the field at any point on the interior by splitting up the spherical mass into an exterior shell and a smaller internal sphere. In other words, if we want to know the force at a distance r from the center of a sphere of radius R with R > r then we split the mass into a sphere of radius r and a spherical shell of inner radius r and outer radius R. The shell has 0 field so the field is equal to that of a sphere of radius r. Basically, you wind up with a function of r^3/r^2 = r inside and 1/r^2 outside.

And I don't remember anything about how attenuating ( weakening ) lines of force, which, however, do not diminish in quantity, can provide an inverse square field gradient. Help!

Maybe, if they DO diminish in QUANTITY also, it will make inverse square? Hey, A-OK!I think that is basically what happens. The solid angle subtended by any object (MacM's CoS) decreases as a function of 1/r^2. Essentially a decrease in the quantity of lines.

-Dale

<font face=times new roman>Here are the results of the first integration. This is basically a repetition of a similar integration done in the Jello thread, except that now I have included the UG and density terms. First, I begin with the equation for a sphere in Cartesian coordinates:

(1) X² + Y² + Z² == R²

I then wrote the parametric equation for a line in Cartesian coordinates where bold indicates a vector quantity:

(2) x(t) == x0 + t v
where x0 == (0, 0, r)
and v == (cos[φ] sin[θ], sin[φ] sin[θ], cos[θ])

Note that by symmetry there is no loss in generality using this definition for x0. Also note that v is a unit vector and that as the polar angle, θ, varies over [0,π/2] and the azimuthal angle, φ, varies over (-π,π] we obtain the set of all lines passing through x0. Finally, note that t is the directed distance between x(t) and x0.

The simultaneous equations (1) and (2) reduce to a single second order polynomial in t. For any x0 on the interior of the sphere and for any v there are two real solutions, t1 and t2. Since t is a directed distance one solution will be positive and the other will be negative. Therefore the expression

(3) f[r,θ,φ] == UG &rho; (t1+t2) v

is the UniKEF force component in the v direction at x0. The total UniKEF force at x0 is given by the following integration over all v:

(4) f[r] == ∫ ∫ f[r,θ,φ] sin[θ] dφ dθ

The result of this integral is no big surprise, it is simply UG &rho; times the result already given on the Jello thread. Specifically:

f[r] == {0, 0, -4/3 &pi; UG &rho; r}

If we let m = 4/3 π ρ r&sup3; then f == {0, 0, -UG m/r²}. This is the same as the Newtonian inverse-squared result for the interior of a sphere. Note that, as before, the result is independent of R, again implying that the force inside any spherical shell is 0.

-Dale

</font face>

<font face=times new roman>Here are the results of the second integration. Steps (1) and (2) are the same as before. By the way, the solution to (1) and (2) is:

t1 = -r Cos[θ] - Sqrt[R² - r² Sin[θ]²]
t2 = -r Cos[θ] + Sqrt[R² - r² Sin[θ]²]

This is the t1 and t2 used here and in the previous integral. The difference is that for interior points the directed length of the line is equal to (t1+t2) while for exterior points the length of the line is equal to their difference. While (t1+t2) is unambiguous the difference could either be (t1-t2) or (t2-t1). In this case, (t1-t2) is geometrically determined to be the correct form, since (t2-t1) leads to a repulsive force.

In the previous integration we were integrating over the entire half-sphere, here we will only integrate over a smaller part of the sphere. Remember that x0 == (0, 0, r) so the azimuthal angle, φ, still varies over (-π,π]. What is restricted is the range of the polar angle, θ, which varies over some range [0,θmax]. Specifically, θmax must be such that t1 and t2 are always real. To find this θmax we solve the following expression:

(3) 0 == R² - r² Sin[θmax]²

The solution to (3) is θmax == ArcSin[R/r]. So for exterior points we have

(4) f[r,θ,φ] == UG ρ (t1-t2) v

Which we integrate over our restricted range to obtain the net UniKEF force field at an exterior point. Specifically

(5) f[r] == ∫ ∫ f[r,θ,φ] sin[θ] dφ dθ

The result of this integral is

f[r] == {0, 0, -4/3 π R³ UG ρ/r²}

If we let M = 4/3 π ρ R³ then f[r] == {0, 0, -UG M/r²}.

This is the same as the Newtonian inverse-square result for the exterior of a sphere. In UniKEF, as well as in classical gravity, the gravitation of a sphere is exactly the same as though it were a point mass located at the center of the sphere acting with an inverse-square law. Also, this result implies that the UniKEF force on the exterior of a spherical shell also acts as though the entire mass of the spherical shell were collapsed to that point. This means that any spherical object with any arbitrary ρ[r] will act as though the total mass were simply a point mass located at the center and acting with an inverse-squared UniKEF force.

-Dale

</font face>

....
So for exterior points we have
(4) f[r,θ,φ] == UG ρ (t1-t2) v
Which we integrate over our restricted range to obtain the net UniKEF force field at an exterior point. Specifically
(5) f[r] == ∫ ∫ f[r,θ,φ] sin[θ] dφ dθ
The result of this integral is
f[r] == {0, 0, -4/3 π R³ UG ρ/r²}
If we let M = 4/3 π ρ R³ then f[r] == {0, 0, -UG M/r²}.

This is the same as the Newtonian inverse-square result for the exterior of a sphere. ...If this is true, it seems MacM wins another round.

I admit to having some doubt still, particularlly with reguard to the right side of equation (4). For example, if the radius of the sphere is cut in half, &rho; increases eight fold and (t1 -t2) for the similar point in the smaller sphere (half as far from the center in the smaller sphere) is also only one half of the larger shere's &delta;t. So the product is four times greater in the smaller sphere. The flux intercepted is four times less. These two factors of four compenstate for each other. I think the Newtonian results follow from this.

I am trying to get an more intutive handel on this, but I am still troubled by the idea that a vanishingly small sphere, (of density going to infinity) can be generating the same surface gravity out of vanishing intercepted flux. I also like to understand the mechanistic details (Even MacM likes mechanistic details - the whole motivation for uniKEF instead of the much simplier Newtonian view) Thus, I want to think in terms of cross sections for absorption and impact parameters etc.

Suppose we had such data. I.e. suppose we knew the probability of absorption as funcion of the "miss distance" or impact parameter. Let me call it A(p) and imagine that the flux was passing thru a cubic lattice with edge length L of "absorption points" each l from it nearest neighbor.
Now my question is if L goes to L/2 and of course l to l/2 is it true that for every old impact parameter p1 against a lattice absorption point there are now in the smaller cube, eight impact parameters with that same value of p1?

I think that is what we are claiming in your equation 4 with a linear density factor. For this to be true for impact paramters greater than L/2 we need to be including flux that does not pass thru the lattice in the smaller cube case. (If that flux is parrallel with and edge. - Really I want to consider a sphere polished from a cubic lattice so L is really the daimeter of the larger sphere.)

I don't know the answer to my question, but there is nothing wrong in priciple with having absorption of a flux ray that passes outside of the smaller sphere, but most likely A(p) is a decreasing function of p.

Can anyone give a simple argument that shows that the half scale lattice does have eight times more cases of every impact parameter? If yes, this would justify the linear factor of density in Eq.(4).

MacM seems to have a viable model if it reproduces Newtonian gravity inside and outside a sphere, but ironically it seems not to be any better justified than that of Newton, as to why it should have a linear &rho; factor in Eq (4) unless it is true that the half size lattice does have eight fold more of every impact parameter. (And It may - I don't know how to evaluate this question.)

The real test may need to be empirical, not mathematical. As I understand uniKEF, and think MacM agrees, the only short-range difference is when three bodies are on the same line. His "eclipse effect." MacM's ratio of hemisphere to moon cylinder is pure numerology, but perhaps this "three in a row" case can be done properly to see what it predicts for the eclipse of the sun.

However, be careful not to fall in the common false idea that the Earth is the main gravitational force acting on the moon. It is not - the sun's gravity is stronger than the Earth's on the moon.

The real test may need to be empirical, not mathematical. As I understand uniKEF, and think MacM agrees, the only short-range difference is when three bodies are on the same line. His "eclipse effect." MacM's ratio of hemisphere to moon cylinder is pure numerology, but perhaps this "three in a row" case can be done properly to see what it predicts for the eclipse of the sun.

I have never claimed this as being different. But it may be. I rather think Newtonian gravity will result in the same thing. The point was that I predicted the affect and followers of Newton had not because of how they calculate gravity.

The differances in UniKEF and Newton are two fold:

1 - Where F = G * m1*m2 / r² goes toward zero as r approaches infinity; The PV goes toward V (the CoS goes toward zero degrees, a trig function = 1.000 as the seperation goes toward infinity.

Reducing the need for Dark Matter to account for star orbital velocity around galaxies.

2 - Over larger seperations sources inbetween two such spheres push them apart reducing the gravity affect. When the internal sources exceeds the external CoS sources then UniKEF becomes repulsive causing the acelerating expansion of the universe.

Eliminating the seperate Dark Energy.

I believe in fact that what they now call Dark Energy is UniKEF and is also causing gravity.

I am trying to get an more intutive handel on this, but I am still troubled by the idea that a vanishingly small sphere, (of density going to infinity) can be generating the same surface gravity out of vanishing intercepted flux. I also like to understand the mechanistic details (Even MacM likes mechanistic details - the whole motivation for uniKEF instead of the much simplier Newtonian view) My integration does not address this situation. It only applies in the low-density or large-flux limits. While this is obviously sufficient for orbital mechanics it will not be sufficient for black holes etc. I think UniKEF black holes will not be singularities because of exactly what you say. My opinion, based on the math I have done up to this point, is that UniKEF is simply an explanation for Newtonian gravity in the large-flux limit and that, if you want to see any differences, you are going to have to look at high-density situations like this.

Suppose we had such data. I.e. suppose we knew the probability of absorption as funcion of the "miss distance" or impact parameter. Let me call it A(p) and imagine that the flux was passing thru a cubic lattice with edge length L of "absorption points" each l from it nearest neighbor.
Now my question is if L goes to L/2 and of course l to l/2 is it true that for every old impact parameter p1 against a lattice absorption point there are now in the smaller cube, eight impact parameters with that same value of p1? Sorry Billy, I have no idea what you are talking about.

The real test may need to be empirical, not mathematical. As I understand uniKEF, and think MacM agrees, the only short-range difference is when three bodies are on the same line. His "eclipse effect." MacM's ratio of hemisphere to moon cylinder is pure numerology, but perhaps this "three in a row" case can be done properly to see what it predicts for the eclipse of the sun.I haven't quite gotten there yet. I have calculated the field inside and outside of a single sphere, now I need to use that to calculate the force between two spheres. Then you can start looking at orbital mechanics with UniKEF. That said, since the UniKEF force field is the same as the classical field I don't see how the total force can possibly be different. I think that the "eclipse effect" is consistent with both classical and UniKEF gravity.

-Dale

1 - Where F = G * m1*m2 / r² goes toward zero as r approaches infinity; The PV goes toward V (the CoS goes toward zero degrees, a trig function = 1.000 as the seperation goes toward infinity.

Reducing the need for Dark Matter to account for star orbital velocity around galaxies.Sorry MacM, the math doesn't bear this claim out. If you recall from the post on the 2nd integral the trig function of interest is ArcSin[R/r]. For any given spherical radius, R, as the separation r goes to infinity R/r goes to 0 and ArcSin goes to 0. What you don't seem to realize is that the PV is always V. In other words, for every point in M1 you can draw exactly one line through every point in M2, regardless of separation. Even in the integration done on your website you realize that the PV is a function of the angle that you are considering. As you integrate over all the angles you wind up hitting all of the volume.

-Dale

My integration does not address this situation. It only applies in the low-density or large-flux limits. While this is obviously sufficient for orbital mechanics it will not be sufficient for black holes etc. I think UniKEF black holes will not be singularities

Correct. UniKEF precludes singularities and predicts a breakdown in symmetry between inertia and gravity at super massive or dense objects.

My opinion, based on the math I have done up to this point, is that UniKEF is simply an explanation for Newtonian gravity in the large-flux limit and that, if you want to see any differences, you are going to have to look at high-density situations like this.

Partially correct. The external CoS integration does yield a Newtonian result however, it is the fact that UniKEF also has sources inbetween two objects that oppose gravity that makes one of the differances in large seperations.

Also the fact that Newtonain goes toward zero as r goes toward infinity and in UniKEF the PV goes toward V or a trig function of 1.0000 (zero degrees) as seperation goes toward infinity.

I haven't quite gotten there yet. I have calculated the field inside and outside of a single sphere, now I need to use that to calculate the force between two spheres. Then you can start looking at orbital mechanics with UniKEF. That said, since the UniKEF force field is the same as the classical field I don't see how the total force can possibly be different. I think that the "eclipse effect" is consistent with both classical and UniKEF gravity.

-Dale

I agree. I think the only differance here is the way gravity is calculated made it virtually impossible to envision the affect hence it was never predicted by Newtonian followers.

Partially correct. The external CoS integration does yield a Newtonian result however, it is the fact that UniKEF also has sources inbetween two objects that oppose gravity that makes one of the differances in large seperations.Yes, I haven't done the "internal source" calculation either.

Also the fact that Newtonain goes toward zero as r goes toward infinity and in UniKEF the PV goes toward V or a trig function of 1.0000 (zero degrees) as seperation goes toward infinity.Again, you are looking at the wrong trig function. The relevant trig function goes to zero as r gets very large relative to R. The PV is just part of the way you describe the integral. It gets integrated out. Again, for every point in M1 and every point in M2 you can draw exactly one line connecting the two. Since I approach the integration differently than you do I never need to use PV.

-Dale

...Sorry Billy, I have no idea what you are talking about.I will try again:

Imagine two identical sets of atoms, in similar but two-to-one scale configurations ("simliar" in the sense of "similar triangles" or Dij = 2 dij, where this is the distance between atoms i & j.) Now pass one flux ray thru each in similar paths, wrt the atoms. Assume there are 10E20 atoms and the average separation in the larger scale set is 1cm. Assume that in the large scale set, this ray has impact parameter bewteen 1.000 and 1.001 cm E11 times and 3E11 times the impact parameter is between 2.000 and 2.002. (Just numerical illustrations, not facts. I assume that you know the standard term "impact parqameter", but for others I explain it: If the trajectory far from a scattering or absorbing atom, is straight line extened and would miss the atom by "x" then x is the "impact parameter." It would also be the closest point of approach if the particle had no path bending interaction with the atom.)

When we assume that the absorption probability is the product of &rho;(t1 + T2) we are making assumptions about the number of events (atom passings) with the 1.000 t0 1.001 cm (and other) impact parameters in the smaller scale configuration. With the half-scale configuration of atoms, &rho; is eight times larger. I would think that if the number of impact parameters events in the smaller configuration with that same impact parameter is 3E11, (illustration only) then for the range 2.000 to 2.002 in the "full-scale" configuration it must have been 3E11 also.

Clearly for any similar internal point in the "half-scale" configuration, the quantity (t1 +t2) is exaclty half as large as it was in the "full-scale" configuration. The total absorption is the integral over all impact parameters of the flux times the probability of absorption as a function of impact parameter.


Now there are many different functional forms that will integrate to the same result, but clearly making the number of impact parameter events in the large scale configuration in the range 2.000 to 2.002 exactly twice the number of atom passing events in it in the range 1.000 to 1.001, etc. will guarentee that the absorption integral over all impact paramters and all atoms is same in the small scale configuration as it the large.

To make product &rho;(t1+t2) four times larger (&rho; up by 8 and the "t sum" down by 2) and thus get the Newtonian results, I think we need four times as many "atom passing events" in the full scale configuration's range 2.000 to 2.002 as in its 1.000 t0 1.001 range. I see no reason why this must be true.

That is, I am trying to look at the total absorption in the more conventional way as an integral of the differential absorptions in each differential range of impact parameters, to see if there is any justification for assuming that it is such that it gives the same result as assuming absorption is proportional to &rho;(t1+t2), which increases by 4 in the half-scale, equal mass (or same number of atoms) configuration. There may be, but I can not find it. The "justification" for this &rho;(t1+t2) does not seem to be one of principle, but that it reproduces the Newtonian results, if we do assume it.

I hope you now see what is troubling me. Each differential range of the full scale configuration's impact parameter ranges is "down shifted" by two in the smaller geometery and the total flux intercepted is also down by four. This seems to set an unwarrented restriction on the distribution of absorption events vs. impact parameter in the full scale configuration.

I haven't quite gotten there yet. I have calculated the field inside and outside of a single sphere, now I need to use that to calculate the force between two spheres. Then you can start looking at orbital mechanics with UniKEF. That said, since the UniKEF force field is the same as the classical field I don't see how the total force can possibly be different. I think that the "eclipse effect" is consistent with both classical and UniKEF gravity. -Dale In uniKEF there is force due to flux imbalance. the sun is the dominate gravitational pull on the moon, so calling it a Fs, I note that when the configuration is E-M-S in a line that by uniKEF the force Fs should be reduced as then the side of the moon towards the Earth is also in flux shadow instead of the full normal cosmic flux pushing the moon towards the sun.

In contrast Newtonian gravity passes thru mater without attenuation. I.e. I have the sun pulling on at midnight less than at noon, but only by my icreased distance from the sun, not by solar gravity being absorbed while passing thru the Earth. This is different from the uniKEF case and I think MacM agrees it is a difference.

It may seem strange to speak of the moon “being pushed towards the sun,“ as I did, but this is the case. The moon is always accelerating towards the sun, never towards the Earth. We only tend to think otherwise as we view moon from Earth and it appears to us as if it is going around the Earth, but moon is in orbit around the sun. The moon's orbit around the sun is always concave towards the sun, just as the Earth's is. The moon only "wiggles a little" from a 1AU distance from the sun as it orbits the sun, not the Earth, in a nearly circular orbit.

I assume that you know the standard term "impact parqameter", but for others I explain it: If the trajectory far from a scattering or absorbing atom, is straight line extened and would miss the atom by "x" then x is the "impact parameter." It would also be the closest point of approach if the particle had no path bending interaction with the atom.)Actually, I didn't know it so I am glad you included the explanation.

That is, I am trying to look at the total absorption in the more conventional way as an integral of the differential absorptions in each differential range of impact parameters, to see if there is any justification for assuming that it is such that it gives the same result as assuming absorption is proportional to &rho;(t1+t2), which increases by 4 in the half-scale, equal mass (or same number of atoms) configuration. There may be, but I can not find it. The "justification" for this &rho;(t1+t2) does not seem to be one of principle, but that it reproduces the Newtonian results, if we do assume it.

I hope you now see what is troubling me.I think I see what you are talking about. It is like in medical imaging, x-rays tend to attenuate exponentially, not linearly. Other particles have different depth/energy deposition curves that can be very useful in both diagnosis and therapy. I admit that this has troubled me a bit also, I don't think that MacM has given enough of a fundamental description of his idea in order to really determine from first principles what form the attenuation should take. However, my work here has shown that if is linear then he will get agreement with observation. I would suspect that if it is not linear then he will get disagreement. I think it is something he should address but in the end since it is his theory he is free to say "it attenuates linearly".

In contrast Newtonian gravity passes thru mater without attenuation. I.e. I have the sun pulling on at midnight less than at noon, but only by my icreased distance from the sun, not by solar gravity being absorbed while passing thru the Earth. This is different from the uniKEF case and I think MacM agrees it is a difference.So far all of my results to this point have been in the low-density or high-flux limit. Essentially I am assuming that U is not significantly attenuated. This assumption should be valid in the solar system, but may not be valid near a black hole or neutron star.

I mentioned previously that I think that if MacM wishes to distinguish his theory from classical gravity he needs to look at the high-density regime. I think you will find that for very dense objects the inertial mass is greater than the gravitational mass due to the significant attenuation. I think that would be the only way to measure his U. This is related to the linear attenuation question. Obviously, since the U is finite then the attenuation cannot be linear over the whole range, but it must be linear in the everyday range (everyday in the solar system). It would be good for him to derive one curve from first principles that covers the whole range.

-Dale

<font face=times new roman>OK, I finished with the last integral required to get the force between two spheres. So here is a graphic showing the 2-body situation.
http://img236.imageshack.us/img236/7598/unikef235xt.th.png (http://img236.imageshack.us/my.php?image=unikef235xt.png)

Here we have M1 (red) and M2 (blue). The radius and density of each sphere is independent as is the separation between the centers (except obviously that R > R1 + R2). Now, I want to calculate the total force on M2. Since I know the internal force (due to M2) on each differential element (green) as well as the external force (due to M1) I know the total force on each differential element. The net force on M2 is simply the integral of the total force on each differential element.

Now, by symmetry, the internal forces integrate to zero, so all I have to worry about is the force due to the red mass. Without loss of generality we place the center of the red mass at the origin and the center of the blue mass on the z axis at z=R. From the results of the second integration above we have that the force on each differential mass is:

(1) df[r,θ,φ] == -UG M1/r² v dm == -UG M1/r² v ρ2 Sin[θ] r² dφ dr dθ

where, again, v == (cos[φ] sin[θ], sin[φ] sin[θ], cos[θ]) is the unit vector pointing from the origin to the differential element.

This integral is slightly different from the previous ones because now we are integrating with respect to r as well. The azimuthal angle, φ, again varies over (-π,π]. The polar angle, θ, again varies over some range [0,θmax]. And the radial distance, r, varies over the range [t1,t2]. By substituting the proper radii into the right equations from the previous integral we have:

θmax == ArcSin[R2/R]
t1 == -R Cos[θ] - Sqrt[R2² - R² Sin[θ]²]
t2 == -R Cos[θ] + Sqrt[R2² - R² Sin[θ]²]

Then the final integral is:

(2) F == ∫ df == ∫ ∫ ∫ -UG M1/r² v ρ2 Sin[θ] r² dφ dr dθ

The result of (2) is F == (0, 0, -M1 4/3 π R2³ ρ2 UG/R²) == (0, 0, -M1 M2 UG/R²). Again, given linear attenuation and no significant attenuation UniKEF gives exactly the Newtonian result.

-Dale
</font face>

Actually, I didn't know it so I am glad you included the explanation.

I think I see what you are talking about. It is like in medical imaging, x-rays tend to attenuate exponentially, not linearly. Other particles have different depth/energy deposition curves that can be very useful in both diagnosis and therapy. I admit that this has troubled me a bit also, I don't think that MacM has given enough of a fundamental description of his idea in order to really determine from first principles what form the attenuation should take. However, my work here has shown that if is linear then he will get agreement with observation. I would suspect that if it is not linear then he will get disagreement. I think it is something he should address but in the end since it is his theory he is free to say "it attenuates linearly".

So far all of my results to this point have been in the low-density or high-flux limit. Essentially I am assuming that U is not significantly attenuated. This assumption should be valid in the solar system, but may not be valid near a black hole or neutron star.

I mentioned previously that I think that if MacM wishes to distinguish his theory from classical gravity he needs to look at the high-density regime. I think you will find that for very dense objects the inertial mass is greater than the gravitational mass due to the significant attenuation. I think that would be the only way to measure his U. This is related to the linear attenuation question. Obviously, since the U is finite then the attenuation cannot be linear over the whole range, but it must be linear in the everyday range (everyday in the solar system). It would be good for him to derive one curve from first principles that covers the whole range.

-Dale

Actually I am concerned about Billy T's continued presentation that the "Impact Parameter" somehow is a problem.

http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/impar.html

As you can see "Impact Parameter" deals with deflection of particles as a function of charge and proximity. None of this has anything to do with gravity attenuation.

He seems to view a change in diameter as a change in exposure to flux interaction. That simply is not the case. If you cut the diameter in half you don't merely reduce the cross section to 1/4. You reduce the volume to 1/8. That increases the density of mass in the CoS by a factor of 8. everything for a single object would appear to remain in balance.

However, as I have pointed out in the case of two spheres where mass remains constant and center seperation distance remains constant, changes in diameter would have an affect because it alters the CoS.

However, as I have pointed out in the case of two spheres where mass remains constant and center seperation distance remains constant, changes in diameter would have an affect because it alters the CoS.Nope. Check out the previous post (10:37 PM). The UniKEF force between two spheres is independent of R1 and R2. It just depends on the masses and the separation.

As far as the attenuation thing goes. I am not as worried about it from a mechanistic standpoint as Billy is (although with your love of mechanistic explanations I would think that you would be eager to explain your flux the way he is talking about, in terms of UniKEF particles impacting atoms). My point is that in order to do any calculations in the high-density regime you will need to specify the attenuation function over the whole range of densities. It simply cannot be linear over all densities because at some point a line of any slope will hit 0 flux and you can't have less than that.

Basically, I think that I have done all of the math that can be done on your theory until you work on it some more yourself. So far with everything I have examined UniKEF is simply a mechanistic explanation for classical gravity. I have told you the next steps that you need to take if you want to get some non-classical results, but they are up to you.

-Dale

Nope. Check out the previous post (10:37 PM). The UniKEF force between two spheres is independent of R1 and R2. It just depends on the masses and the separation.

I know you have said this before. I do find it difficult to believe however since doubling diameter increases volume by a factor of eight (reduces density to 1/8) but changes the CoS at a different ratio depending on the seperation distance.

However, I'll not argue the point.

As far as the attenuation thing goes. I am not as worried about it from a mechanistic standpoint as Billy is (although with your love of mechanistic explanations I would think that you would be eager to explain your flux the way he is talking about, in terms of UniKEF particles impacting atoms). My point is that in order to do any calculations in the high-density regime you will need to specify the attenuation function over the whole range of densities. It simply cannot be linear over all densities because at some point a line of any slope will hit 0 flux and you can't have less than that.

I agree attenuation is linear in the range we are accustom to experience but I have said before that at some density or super massive level the symmetry between inertia and gravity will be broken.

Basically, I think that I have done all of the math that can be done on your theory until you work on it some more yourself. So far with everything I have examined UniKEF is simply a mechanistic explanation for classical gravity.
-Dale

Actually there are other calculations which you might find interesting. Take a case of star galatic rotation and compute the diameter of the universe to cause UniKEF to account for its orbit velocity.

If that figure ends up in the 60 B lyr range that would be most coincidental don't you think? 60 B lyr is the latest calculation made for the diameter of the universe (not 15 B lyr).

To do this calculation a good approximation is to merely divide the internal flux integration into the external flux integration (assuming a finite diameter of the universe and the orbit diameter of the star to the galaxy) such that the gravity is "X" times greater than Newton eliminating (mitigating) the need for Dark Matter. The CoS is approximately zero degrees but not precisely.

To be precise the affect of all the mass of other stars in the galaxy would need to be assessed.

Another might be to calculate the eclipse affect using precise mathematics rather than my general case. The result should be around 4.2E^-9 pertabation.

Take a case of star galatic rotation and compute the diameter of the universe to cause UniKEF to account for its orbit velocity.How is the diameter of the universe supposed to affect to galactic rotation? The UniKEF field is going to be irrotational, so I can see it moving galaxies out, but not making them rotate.

-Dale

Despite MacM’s childish name calling etc when I have suggested problems that may stress uniKEF or question its validity etc, I have always said that MacM is very intelligent, clever etc.

I think MacM has already covered all your {DaleSpam} concerns about exponential vs. linear absorption and about different materials normally having very different absorption cross sections, etc. He has also steadfastly ignored or refused to consider the flux ray that enters the first of two bodies from outside his CoS and scatters into the second of the two gravitational bodies. (His PVs and CoS are related in ways that I do not fully understand, so he may have some escape here also.) At present, I think, he simple claims he does consider “scattering” and has a unique, non-conventional, definition of “scattering,” which I forget, but is related to the ray emerging from the interaction with or without the same energy. I have not tried to get he to use the normal meaning of “scattering” for a long time or to admit that he is neglecting scattering from outside his CoS with no reasonable basis.)

MacM postulates that the uniKEF interaction with matter is very unlikely, so surely if it occurs with one ray, is only with one absorbing piece of matter that get momentum transferred to it. Thus, on a ray-by-ray point of view it really does not make any sense to speak either of linear or exponential attenuation. It is a “one time event.” Consequently, I think the standard approach to both scattering and absorption in terms of single interaction probability vs. “impact parameter” function as the ray passes the absorber point (atom?) is very appropriate and simple even if there are millions of such points near by with which the ray has no interaction, but with the modification discussed in next paragraph.

I think it is at least conceptually possible to postulate that in addition to some well defined interaction probability vs. impact parameter there is a statistical delta function of time type multiplier that is zero almost all the time but occasional is unity. (I suspect you know about delta functions, but roughly they integrate to unity and are zero in width except at one point of their argument where they are infinite, but well behaved (differential etc.). - they are one of the few useful math things physicists invented and given to mathematicians instead of the normal gifts mathematicians have provided to physics.)

Because uniKEF is very complex compared to Newtonian formulation of gravity there are only two reasons to prefer it.(1)If it does predict something different which is confirmed to be true. (By “predict” I do not mean, as MacM does, some individual making claims for it, but some calculated values that can be tested.) Or (2) because it provides a mechanistic explanation for gravity, even if it violates conservation of energy and momentum etc. as flux constantly is produced from nothing to make up for that being absorbed. (Really this is just trading one mystery for another like saying “god made the universe.”) I am concerned that our use of &rho; (t1 +t2) is not justified and, at least to me now, appears to unjustifiable from a more fundamental POV, like having a model of absorption vs. impact parameter. I do expect that if this assumption is made that uniKEF is able to compress the contracting star, so this challenge of mine will probably fail to disprove uniKEF because it seems to be producing the same results a Newtonian gravity in the single sphere case.

As for why the absorption only depends upon the mass and not the type of matter MacM has two answers: The obvious one being that it is a theory about gravity, not chemisty. He has on a few occasions also noted that perhaps it is the quarks in the nucleus that are the actual absorption agents.

...And the radial distance, r, varies over the range [t1,t2]. ...Very nice Dale.
I am 99+% sure but want to confirm that these t1 and t2 are not related as the old ones were to the flux ray but at the distances to the two intercepts along the line r.

MacM should throw away all that illegible inverse square between two disks and put your work in its place.

Actually I am concerned about Billy T's continued presentation that the "Impact Parameter" somehow is a problem.
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/impar.html
As you can see "Impact Parameter" deals with deflection of particles as a function of charge and proximity. None of this has anything to do with gravity attenuation....MacM you really need to learn a little about clasical mechanics, especially the "central force problem."
this is the second time you have done a search and concluded that some general thing is limited to the specific application you happened to find in your search. (First time was when you concluded that Taylor instabilities is always related to liquids and said it therefore did not apply to stars.) This time you think "impact parameters" have nothing to do with gravity. I am not sure but think that is where they were first applied about 200 years ago. They are more general than even all inverse square laws (gravity being one) They are a very standard approach to ANY form of force interaction between two bodies which can be characterized as acting along and depending only upon the separation between the bodies (not magnetic fields or velocity dependent force fields in general.) If you can only search then search: “Central Force Problem”

To make the mistake you are making more clear, assume I was as ignorant about electricity as you are about the 200 year old classic central force problem and all the mathematical machinery that has been developed for it. (for example the "reduced mass" reduction of it to a single body problem, which gives it its name: central force problem) then assume that to learn about "electricity" I did a search and found an article about "voltaic cells." Next thing I do is come and tell someone very well versed in all aspects of electricity that he can not calculate the force between two charges or relate it to radio waves because electricity only concerns batteries. This is exactly what you are doing. IMPACT PARAMETERS APPLY TO GRAVITATIONAL INTERACTIONS, mainly useful when the system energy is positive (Not a bound orbit), if you chose to look at them from this POV. It is a principled way to determine what happens if a cosmic body were to pass by another and you did not want to approach the problem in detail. It is how the nucleus was known to be much smaller than the atom even though there was no sure knowledge about the nature of the force causing the scattering. etc.

Until you have studied the subject instead of just searched it, you would be wise not to tell one who has where and where not a math technique widely used by professional physicists can be applied.

How is the diameter of the universe supposed to affect to galactic rotation? The UniKEF field is going to be irrotational, so I can see it moving galaxies out, but not making them rotate.

-Dale

The CoS goes toward zero degrees (where PV becomes equal to V) which is a greter gravity value than the inverse square from Newtonian gravity.

The amount of gravity is proportional to the internal flux pushing apart vs the external flux (CoS) pushing together. By adjusting the universal diameter (External CoS flux) vs the star seperation from the central galatic core )Internal repulsion flux) a gravity force which matches observation can be established.

Very nice Dale.
I am 99+% sure but want to confirm that these t1 and t2 are not related as the old ones were to the flux ray but at the distances to the two intercepts along the line r.

MacM should throw away all that illegible inverse square between two disks and put your work in its place.The simultaneous equations

(1) X² + Y² + Z² == R²

and

(2) x(t) == x0 + t v
where x0 == (0, 0, r)
and v == (cos[φ] sin[θ], sin[φ] sin[θ], cos[θ])

reduce to a single quadratic polynomial in t. The two solutions of that polynomial are t1 and t2. (1) is the equation of a sphere and (2) is the equation of a line, so the solution gives the two points where the line intersects the sphere. Since v is a unit vector then t is the directed distance from x0 in the v direction; I made extensive use of that property in the various integrals.

-Dale

Despite MacM’s childish name calling etc

Only childish if you are the recipient of the lables. You chose to ignore that you earned them by slander and false negative innuendos.

He has also steadfastly ignored or refused to consider the flux ray that enters the first of two bodies from outside his CoS and scatters into the second of the two gravitational bodies. (His PVs and CoS are related in ways that I do not fully understand, so he may have some escape here also.) At present, I think, he simple claims he does consider “scattering” and has a unique, non-conventional, definition of “scattering,” which I forget, but is related to the ray emerging from the interaction with or without the same energy. I have not tried to get he to use the normal meaning of “scattering” for a long time or to admit that he is neglecting scattering from outside his CoS with no reasonable basis.)

As with density the scattering issue has been made clear. Does anyone else have problems understanding that statistical scattering is the same as linear pass through?

That is statistically the angles at which particles could scatter are going to average out such that one can ignore the fact of scatter and merely look at the volume contained in the CoS.

MacM postulates that the uniKEF interaction with matter is very unlikely, so surely if it occurs with one ray, is only with one absorbing piece of matter that get momentum transferred to it.

Correct.

Thus, on a ray-by-ray point of view it really does not make any sense to speak either of linear or exponential attenuation. It is a “one time event.”

Incorrect. The amount of energy (momentum) transfer can vary for the individual impact.

Consequently, I think the standard approach to both scattering and absorption in terms of single interaction probability vs. “impact parameter” function as the ray passes the absorber point (atom?) is very appropriate

Incorrect assessment.

and simple even if there are millions of such points near by with which the ray has no interaction, but with the modification discussed in next paragraph.

I think it is at least conceptually possible to postulate that in addition to some well defined interaction probability vs. impact parameter there is a statistical delta function of time type multiplier that is zero almost all the time but occasional is unity. (I suspect you know about delta functions, but roughly they integrate to unity and are zero in width except at one point of their argument where they are infinite, but well behaved (differential etc.). - they are one of the few useful math things physicists invented and given to mathematicians instead of the normal gifts mathematicians have provided to physics.)

Unjustified complexiety.

Because uniKEF is very complex compared to Newtonian formulation of gravity there are only two reasons to prefer it.(1)If it does predict something different which is confirmed to be true. (By “predict” I do not mean, as MacM does, some individual making claims for it, but some calculated values that can be tested.)

I still disagree with your use of the terms. Calculation is calculation, not prediction. Prediction is just that a statement about what is to be discovered. Calculations can either verify or invalidate the prediction.

Or (2) because it provides a mechanistic explanation for gravity, even if it violates conservation of energy and momentum etc. as flux constantly is produced from nothing to make up for that being absorbed.

Less of a mystery than the Big Bang which seems to have done the same. Unless of course you subscribe to the view that the universe has always existed hence therefore never came into existance. :bugeye:

In UniKEF we have the fact that (N)othing and (S)omething are related in conservation by:

N -------> (+S) + (-S)

That is on balance the universe is comprised of bifurcated nothing, hence no violation of conservation has occured.

(Really this is just trading one mystery for another like saying “god made the universe.”)

Only when you do not understand the full concept (not just gravity) of UniKEF.

As for why the absorption only depends upon the mass and not the type of matter MacM has two answers: The obvious one being that it is a theory about gravity, not chemisty. He has on a few occasions also noted that perhaps it is the quarks in the nucleus that are the actual absorption agents.

Only partially correct. I have never excluded any matter from being an absorber of UniKEF. I have suggested that at the Quark level one will find an enhaned gravity which we have labeled the strong nuclear force.

Very nice Dale.
I am 99+% sure but want to confirm that these t1 and t2 are not related as the old ones were to the flux ray but at the distances to the two intercepts along the line r.

MacM should throw away all that illegible inverse square between two disks and put your work in its place.

I would only if Dale chooses to have his work posted and it was fully illustrated step by step.

(By “predict” I do not mean, as MacM does, some individual making claims for it, but some calculated values that can be tested.) :D Hehe. I have had that same conversation with him. He wants to consider his theory scientific while using the definition of "prediction" that applies to horoscopes.

-Dale

MacM you really need to learn a little about clasical mechanics, especially the "central force problem."
this is the second time you have done a search and concluded that some general thing is limited to the specific application you happened to find in your search. (First time was when you concluded that Taylor instabilities is always related to liquids and said it therefore did not apply to stars.)

You tell only part of the story here. The issue had to do with the injection of high velocity streams, turblulance, etc. It had nothing to do with quiesent pull or push forces forming a sphere.

This time you think "impact parameters" have nothing to do with gravity.

And you are to quick to spout criticisim. I posted one link. There are hundreds. This one for example:

http://scienceworld.wolfram.com/physics/ImpactParameter.html

Which I passed over since it is a discussion of "Particles" being deflected by gravity, not energy attenuation causing gravity. I do not envision UniKEF being subjected to gravity anymore than kenetic energy of a particle.

Do you? :bugeye: You continue to repeat the same error in assumptions about what I know or understand. Why? Over confident in yourself or misjudging your opponent?

If you can only search then search: “Central Force Problem”

Done. And SO? Why do you think you can always pick some other subject and make unsupported claims that the other person doesn't understand anything about it?

To make the mistake you are making more clear, assume I was as ignorant about electricity as you are about the 200 year old classic central force problem and all the mathematical machinery that has been developed for it. (for example the "reduced mass" reduction of it to a single body problem, which gives it its name: central force problem) then assume that to learn about "electricity" I did a search and found an article about "voltaic cells." Next thing I do is come and tell someone very well versed in all aspects of electricity that he can not calculate the force between two charges or relate it to radio waves because electricity only concerns batteries. This is exactly what you are doing. IMPACT PARAMETERS APPLY TO GRAVITATIONAL INTERACTIONS, mainly useful when the system energy is positive (Not a bound orbit), if you chose to look at them from this POV.

Stop being a jerk or jackass. See above and stop assuming you know what I know.

It is a principled way to determine what happens if a cosmic body were to pass by another and you did not want to approach the problem in detail. It is how the nucleus was known to be much smaller than the atom even though there was no sure knowledge about the nature of the force causing the scattering. etc.

Tell me something I am not familiar with.

Until you have studied the subject instead of just searched it, you would be wise not to tell one who has where and where not a math technique widely used by professional physicists can be applied.

And you would be wise to understand that part of nuclear engineering deals with barns cross-section, scattering, etc. What a quack.

I would only if Dale chooses to have his work posted and it was fully illustrated step by step.Feel free to copy my work and/or modify it as you see fit. It is in the public domain now, there is no intellectual property, and I will exercise no copyright. Let me know if you need any of the intermediate results that I may not have posted, I can certainly put those up too. You can develop your own illustrations and you can copy the illustration I put in the 3rd integral post.

I will not post it under my own name as I already have a good track record publishing in the scientific literature in my own field. Anyone with a semester of Vector Calculus (or Mathematica) could have done the integrals I did.

-Dale

Feel free to copy my work and/or modify it as you see fit. It is in the public domain now, there is no intellectual property, and I will exercise no copyright. Let me know if you need any of the intermediate results that I may not have posted, I can certainly put those up too. You can develop your own illustrations and you can copy the illustration I put in the 3rd integral post.

I will not post it under my own name as I already have a good track record publishing in the scientific literature in my own field. Anyone with a semester of Vector Calculus (or Mathematica) could have done the integrals I did.

-Dale

I appreciate the offer and will seriously consider it. However, there are some points I need to clarify. While I appreciate not only your efforts but your findings, I am concerned that you must have an error in your process.

You have repeatedly insisted that Geometry of the sphere (i.e. the value of r) has no bearing. It simply must.

I agree that the form remains the same but the specific integrated value must change.

For example given two identical spheres where the centers are seperated by a distance of two spherical diameters, the CoS is 60 degrees. Enlarge the sphere by a factor of two and the volume increases by a factor of 8 and given the same mass the density would be reduced by a factor of 8. Those two affects cancel and gravity would be the same. But if the center distance between two such spheres is retained the CoS goes from 60 degrees to 180 degrees.

This increase in UniKEF source exposure MUST increase gravity. As you have pointed out the CoS change is a function of distance of seperation. What you didn't point out is that it is also a function of geometry (diameter).

Given the same two initial spheres at a seperation of 3 sphere diameters the CoS is about 39 degrees. Given the same spheres and a change in distance where the only change is the CoS, the change in gravity force is clearly a function of the CoS.

So geometry (diameter) simply must affect gravity between two or more objects where gravity is a function of the CoS.

:D Hehe. I have had that same conversation with him. He wants to consider his theory scientific while using the definition of "prediction" that applies to horoscopes.

-Dale

Keep it technically correct. I said nothing about my theory or it being scientific. In fact if you have read the "Introduction" I do just the opposite and state up front that it is mostly "By-Way-Of-Example".

However, your response does not address my issue. Calculation is not prediction. Calculation can verify or invalidate prediction. Prediction is primarily verbal. What you like to call prediction is nothing more than calculation, the results of mathematical manipulation of a pre-established formula.

Computing gamma at 0.866c and determining gamma = 2.000 is not a prediction. It is a calculation. Formulating the formula for gamma and claiming that observation will follow the formula is predicting.

Do you really believe formulating the gamma function is the equivelent of doing horoscopes? :D

Very nice Dale.
I am 99+% sure but want to confirm that these t1 and t2 are not related as the old ones were to the flux ray but at the distances to the two intercepts along the line r.

MacM should throw away all that illegible inverse square between two disks and put your work in its place.

The simultaneous equations

(1) X² + Y² + Z² == R²

and

(2) x(t) == x0 + t v
where x0 == (0, 0, r)
and v == (cos[φ] sin[θ], sin[φ] sin[θ], cos[θ])

reduce to a single quadratic polynomial in t. The two solutions of that polynomial are t1 and t2. (1) is the equation of a sphere and (2) is the equation of a line, so the solution gives the two points where the line intersects the sphere. Since v is a unit vector then t is the directed distance from x0 in the v direction; I made extensive use of that property in the various integrals.

For those of us without Mathematica:

equation 2 is simply:

(2) [X, Y, Z] = [0, 0, r] + t*[cosφ sinθ, sinφ sinθ, cosθ]

(2) into (1):

(t cosφ sinθ)² + (t sinφ sinθ)² + (r + t cosθ)² = R²

which reduces:

t² (cos²φ sin²θ + sin²φ sin²θ + cos²θ) + t (2 r cosθ) + r² = R²

t² + t (2 r cosθ) + r²-R² = 0

thus:

t = (-2 r cosθ +/- &radic;(4 r² cos²θ - 4(r²-R²)) ) / 2

&nbsp; = -r cosθ +/- &radic;(R² - r² sin²θ)

Just for clarification on:


t1 and t2 are not related as the old ones were to the flux ray but at the distances to the two intercepts along the line r.

It is my understanding that r is the distance from the center of the sphere to the point. Furthermore the z axis is defined along this line. The intercepts are for various &phi; and &theta;. I am very troubled by the fact that t is not a function of &phi; when the integration is done over &phi;. If you conceptualize exactly what is being integrated: Imagine a sphere with radius R and a point at r>R. A line from the center of the sphere to the point defines the z axis. By symmetry it doesn't matter where you visualize the x and y axes as long as you have a cartesian coordinate system with the z axis. Now imagine a random &theta; that is between 0 and &theta;_max. t1 and t2 are the distances from r to the two intercepts on the surface of the sphere. Now imagine a random &phi;: t1 should increase while t2 should decrease until t1=t2 which would be &phi;_max. Perhaps I missed it, but I do not see where this is applied in the integration (I haven't really gone over the integration stuff yet).

DaleSpam, I was attempting to verify the excellent work you have done but I unfortunately ran into a bit of a snag. Can you explain your equation:


(5) f[r] == ∫ ∫ f[r,θ,φ] sin[θ] dφ dθ


I'll give an example, the derivation for the volume of a sphere is:

Imagine a differential volume in a sphere. The sides of this volume are r d&theta;, r d&phi;, and dr sin&theta;, so:

dv = r d&theta; * r d&phi; * dr sin&theta;

V = &int;dv

V = &int; &int; &int; r&sup2; sin&theta; d&phi; d&theta; dr

Can you provide a similar derivation of: f[r] == ∫ ∫ f[r,θ,φ] sin[θ] dφ dθ

Edit: also, what is your lower limit on &theta; and what are the limits on &phi;? If the lower limit of &theta; is 0, then I understand the sin, but I do not understand why you don't multiply by 2.

While I appreciate not only your efforts but your findings, I am concerned that you must have an error in your process.I have posted my methods publicly so that you could point out any errors. Show me the error and I will fix it.

You have repeatedly insisted that Geometry of the sphere (i.e. the value of r) has no bearing. It simply must.

I agree that the form remains the same but the specific integrated value must change.By specific integrated value do you mean the integrand or the result of the integral. I can tell you that Mathematica will not give an incorrect result for the integral, so if you think the result is wrong you need to look at the integrand and tell me what you want integrated.

For example given two identical spheres where the centers are seperated by a distance of two spherical diameters, the CoS is 60 degrees. Enlarge the sphere by a factor of two and the volume increases by a factor of 8 and given the same mass the density would be reduced by a factor of 8. Those two affects cancel and gravity would be the same. But if the center distance between two such spheres is retained the CoS goes from 60 degrees to 180 degrees.

This increase in UniKEF source exposure MUST increase gravity. As you have pointed out the CoS change is a function of distance of seperation. What you didn't point out is that it is also a function of geometry (diameter).I have gone over this explanation before, but I will do it again. Look at a single differential element of sphere A. As you double the radius of sphere B the solid angle subtended by B quadruples (your CoS) leading to a four-fold increase in UniKEF flux. At the same time the length of each line doubles. This results in a net 8-fold increase in the UniKEF flux attenuation volume which is exactly compensated by the 8-fold decrease in the density. There is no change in force.

So geometry (diameter) simply must affect gravity between two or more objects where gravity is a function of the CoS.You are wrong as I demonstrated above. If you don't believe me then do the math yourself and prove it to yourself.

-Dale

...And you are to quick to spout criticisim. I posted one link. There are hundreds. This one for example:
http://scienceworld.wolfram.com/physics/ImpactParameter.html

Which I passed over since it is a discussion of "Particles" being deflected by gravity, not energy attenuation causing gravity...

As you can see "Impact Parameter" deals with deflection of particles as a function of charge and proximity. ...To which I responded:

This time you think "impact parameters" have nothing to do with gravity.Al I was trying to do was point out to you that the use of "impact parameters in the analyisof two body interaction is:

(1) not limited to: "deflection of particles as a function of charge and proximity"

(2) not limited to electric forces, but does infact have good applications to gravittional force interactions.

(3) not even limited to inverse square law forces, like gravity and electrostatic forces

(4) that the two body problem can be reduced to a one body and traveling under the force that appears to come from the cooridinate origin (or any fixed point you want top chose, but it is alsway the orign in pratice.)

That is why I suggested you search "central force problem" and you have and now seem to understand this better as you now give a link to a gravitational application of the impact parameter which previously said only applied only to charged particles.

For those of us without Mathematica:

equation 2 is simply:Thanks for presenting the detail!

I am very troubled by the fact that t is not a function of &phi; when the integration is done over &phi;.By symmetry, and without loss of generality, I have chosen x0 to be a point on the z axis. I chose this location specifically because the z-axis symmetry would eliminate the dependence on &phi;. Remember, I am still using the Mathematica definition where &phi; is the azimuthal angle which defines rotation about the z-axis, the axis of symmetry in this case. You are probably just confused because the Mathematica convention, &phi; == azimuthal angle and &theta; == polar angle, is backwards from yours.

-Dale

Can you explain your equation:

(5) f[r] == ∫ ∫ f[r,θ,φ] sin[θ] dφ dθ
...
Edit: also, what is your lower limit on &theta; and what are the limits on &phi;? If the lower limit of &theta; is 0, then I understand the sin, but I do not understand why you don't multiply by 2.Sure, recall that the force field at a point on the exterior of the sphere is given by:

f[r,θ,φ] == UG ρ (t1-t2) v
where
v == (cos[φ] sin[θ], sin[φ] sin[θ], cos[θ])
and t1 and t2 are the solutions to the intersection of the line and sphere.

Notice that v is a unit vector which can be written in spherical coordinates (1,θ,φ ) where the radius is 1, the polar angle is θ, and the azimuthal angle is φ. The Jacobian determinant for the spherical coordinate system is r² sin[θ] as you derived above. Since r==1, this reduces to sin[θ].

So (5) is an integral over all possible v. In other words, we are looking at force due to the flux in all possible directions. So even though f is a Cartesian vector the integration variables are in spherical coordinates. Again, we are integrating v over a unit sphere for a fixed x0, so the r is not a variable of integration.

You also asked about the limits of the variables of integration. The polar angle θ is integrated over the range [0,θmax] where θmax == asin(R/r). The azimuthal angle φ is integrated over the range (-π,π] due to the same z-axis symmetry as I described above.

-Hope this helps
Dale

I have gone over this explanation before, but I will do it again. Look at a single differential element of sphere A. As you double the radius of sphere B the solid angle subtended by B quadruples (your CoS) leading to a four-fold increase in UniKEF flux. At the same time the length of each line doubles. This results in a net 8-fold increase in the UniKEF flux attenuation volume which is exactly compensated by the 8-fold decrease in the density. There is no change in force.

Given two identical spheres the CoS is the total angle of a tangent line at the upper surface of sphere A which projects downward to be tangent at the bottom of sphere B. The vertex of this angle is midway between the seperation of the two spheres.

Making that distance 10 sphere diameters between centers results in a CoS of 23.07 degrees. Doubling the diameter of the spheres increases the CoS to 47.15 degrees. It does not quadruple it.

Infact the magnitude of such change is a function of the initial seperation and diameters.

It appears we are talking about something different as being the CoS.

You are wrong as I demonstrated above. If you don't believe me then do the math yourself and prove it to yourself.

-Dale

I have and the CoS (integrated PV including trig angle) affects the force of gravity.

To which I responded:

Al I was trying to do was point out to you that the use of "impact parameters in the analyisof two body interaction is:

(1) not limited to: "deflection of particles as a function of charge and proximity"

(2) not limited to electric forces, but does infact have good applications to gravittional force interactions.

(3) not even limited to inverse square law forces, like gravity and electrostatic forces

(4) that the two body problem can be reduced to a one body and traveling under the force that appears to come from the cooridinate origin (or any fixed point you want top chose, but it is alsway the orign in pratice.)

That is why I suggested you search "central force problem" and you have and now seem to understand this better as you now give a link to a gravitational application of the impact parameter which previously said only applied only to charged particles.

And that is why I suggested you read my reply before your reply. I read the impact parameter in relation to gravity and chose to ignore it since it involves the motion of particles being accelerted by gravity and has nothing to do with gravity being produced by attentuation of energy in mass.

Again you attempt to introduce some new issue and make claims which have absolutely no bearing on the subject matter.

The vertex of this angle is midway between the seperation of the two spheres.Only for equal-radius spheres.

Making that distance 10 sphere diameters between centers results in a CoS of 23.07 degrees. Doubling the diameter of the spheres increases the CoS to 47.15 degrees. It does not quadruple it.Sigh. Note the term solid angle (http://en.wikipedia.org/wiki/Solid_angle). You are thinking in 2D here.

I have and the CoS (integrated PV including trig angle) affects the force of gravity.Please post it.

-Dale

MacM I think that I should justify my approach and show you why it is mathematically the same as using the pseudo-volume (PV) approach that you suggest. I think this may clear up some of what you percieve to be errors in my approach.

I am sure you are familiar with the associative property of addition which states that the order of addition is not important:

(a + b) + c == a + (b + c)

Well, an integration is essentially just an infinite number of additions, so there is a similar property for integrals which states that the order of integration is not important:

∫ ∫ f[x,y] dx dy == ∫ ∫ f[x,y] dy dx

Now, it is well known that, although the order of integration will not affect the result, it may have a strong impact on the difficulty of the problem. Often the most important part of solving a difficult problem is selecting the easiest coordinate system and/or the easiest order of integration.

For the case of UniKEF we have 5 variables of integration θ, φ, x, y, z. In other words we need to consider every flux direction and every spatial location. Now, the approach shown in your website is to determine all of the x, y, z for a given θ, φ first and integrating over θ, φ last. Mathematically this is:

∫ ∫ ∫ ∫ ∫ f[x,y,z,θ,φ] dx dy dz dθ dφ

In other words, first fix the direction and then find all of the volume where the projection of the two objects overlap (the PV). Then, to obtain the final force you integrate over all directions, each with its own PV.

However, this is mathematically exactly the same as:

∫ ∫ ∫ ∫ ∫ f[x,y,z,θ,φ] dθ dφ dx dy dz

In other words, first fix the location and then find all of the directions where you can draw a line to the other mass (your CoS or my solid angle). Then, to obtain the final force you integrate over all locations, each with its own CoS.

Now, there are two reasons for calculating the integral in the order I suggest. The first and most important is that it is easier to evaluate. The second reason is that the intermediate result of my integral is useful by itself. Specifically, once you have integrated over θ and φ you have the UniKEF force field for a particular spatial location. On the other hand, with your approach once you integrate over x, y, and z you have the PV for a particular flux direction. The force field is quite useful, e.g. it can be used to determine if UniKEF is conservative and if so you can determine the related scalar potential field. On the other hand I don't know of any particular use of the PV.

I hope you now see how the two approaches are mathematically the same, just like (a + b) + c == a + (b + c). I hope you can also see that the PV is just an intermediate result and that you agree that not only is my approach easier to solve, but it also provides a more useful intermediate result.

-Dale

By symmetry, and without loss of generality, I have chosen x0 to be a point on the z axis. I chose this location specifically because the z-axis symmetry would eliminate the dependence on φ. Remember, I am still using the Mathematica definition where φ is the azimuthal angle which defines rotation about the z-axis, the axis of symmetry in this case. You are probably just confused because the Mathematica convention, φ == azimuthal angle and θ == polar angle, is backwards from yours.

You are right, by symmetry the point is always on the z axis and the x & y axes are irrelevant.


Notice that v is a unit vector which can be written in spherical coordinates (1,θ,φ ) where the radius is 1, the polar angle is θ, and the azimuthal angle is φ. The Jacobian determinant for the spherical coordinate system is r² sin[θ] as you derived above. Since r==1, this reduces to sin[θ].

So (5) is an integral over all possible v. In other words, we are looking at force due to the flux in all possible directions.


OK so given (4) and (5):

(4) f[r,θ,φ] == UG ρ (t1-t2) v

(5) f[r] == ∫ ∫ f[r,θ,φ] sin[θ] dφ dθ

What we have is:

Since t1-t2 = -2&radic;(R²-r²sin²θ)

fx[r] = UG ρ ∫ ∫ -2 sin²θ cosφ &radic;(R²-r²sin²θ) dφ dθ

fy[r] = UG ρ ∫ ∫ -2 sin²θ sinφ &radic;(R²-r²sin²θ) dφ dθ

fz[r] = UG ρ ∫ ∫ -2 sinθ cosθ &radic;(R²-r²sin²θ) dφ dθ

Since ∫ cosφ dφ = ∫ sinφ dφ = 0 with limits -&pi; and &pi; fx and fy are both zero

And fz = -4/3 UG ρ R&sup3;/r&sup2;

And if r >> R, the force goes to zero which I believe is contrary to MacM's claim.

A little bit of searching turned up this relic (http://en.wikipedia.org/wiki/LeSage_gravity) from ~16th century which seems to parallel UniKEF. It too predicts a deviation from the inverse-square law but not for the same reason that MacM has claimed. The reason it claims is "Yet another prediction of this theory is a deviation from the inverse-square law for very large masses. As a body approaches a certain critical size, all of the LeSage particles incident upon it are absorbed and/or scattered by the body. Beyond this size no greater screening can thus occur."

Only for equal-radius spheres.

So?

Sigh. Note the term solid angle (http://en.wikipedia.org/wiki/Solid_angle). You are thinking in 2D here.

If you actually read my abstract you would have noted that I am not "thinking" in 2D but described the angular surface as that of a frostrum.

And if r >> R, the force goes to zero which I believe is contrary to MacM's claim.Yes, he was using the wrong trig function.

-Dale

If you actually read my abstract you would have noted that I am not "thinking" in 2D but described the angular surface as that of a frostrum.Don't try and pull that on me. I read your whole confusing, disorganized, incoherent, rambling abstract. I spent much more time reading that terrible piece of work than you have any right to expect anyone to spend.

I was refering specifically to your post that I quoted. In that post you were only talking about 2D angles and you therefore reached an erroneous conclusion which I pointed out. What you put in your "abstract" was irrelevant to my comments. If you can't keep your own presentation of your own theory coherent and consistent then don't complain to me when I point out your current logical mistake that lead to your current incorrect conclusion. I see no need for me to remind you of parts of your own theory that you may forget.

-Dale

So?So I was attempting to solve the general problem.

-Dale

Don't try and pull that on me. I read your whole confusing, disorganized, incoherent, rambling abstract. I spent much more time reading that terrible piece of work than you have any right to expect anyone to spend.

Illustrations do not "Cut & Paste" but here is the text. Your assertion that I am talking about a 2D affect is simply irresponsible. The angle is clearly defined by the tangent lines but they are clearly rotated to provide the 3D volume calculation.

******************** Extract **************************
http://www.unikef-gravity.com/UniKV2/page2.htm

[025] The length of each penetration line represents some force which also has a trigometric function that limits the effective force as gravity due to its angle of push relative to the line of gravity between the masses.

[026] Further there are rays from sources at any given angle from the line of gravity (zero degrees) to the CoS (the maximum angle tangent to the masses) which must be considered from all 360 degrees around the line of gravity.

[027] It can be seen in Fig # 7 that the numbered parallel rays inscribe a circle on the surface of the sphere as the 360 degree rotation of the penetration angle is considered.

[028] The geometry being considered actually forms the surface of a right circular cone or a "Frustum of Right Circular Cone". The formula for the area is:

s = (h1^2 + (rB -rb)^2)^.5

Fig 7b: Frustum of Right Circular Cone.

[029] As you move out radially from the Center of Mass (COM) from ray #1 toward #4 along the parallel rays, the length of the penetration decreases but the circle (cone) it enscribes increases. The same affect may be seen in Fig #6 where the example of penetration angles from zero to the CoS is shown for two different separation distances.

[030] A 3D integration of the mass penetration volume from 0 degrees to the CoS angle produces a pseudo inverse square result in relation to the separation distance between the masses, if you do not include counter source integrations that are effective at repulsion which exist in between the masses with separation. This integration is a UniKEF Pseudo Volume (UPV).
************************************************** ******

I was refering specifically to your post that I quoted. In that post you were only talking about 2D angles and you therefore reached an erroneous conclusion which I pointed out. What you put in your "abstract" was irrelevant to my comments. If you can't keep your own presentation of your own theory coherent and consistent then don't complain to me when I point out your current logical mistake that lead to your current incorrect conclusion. I see no need for me to remind you of parts of your own theory that you may forget.

-Dale

I have forgot nothing and you are acting childish.

Yes, he was using the wrong trig function.

-Dale

Sorry but you are both wrong. The angle I am talking about is the angle of penetration relative to the line of gravity between centers. The trig function (effective force) is the Cosine of the angle and it goes to 1.000 when the CoS is zero degrees which happens at the seperation = infinity.

At that point the pseudo volume penetration equals the physical volume and the trig function = 1.000. At the same point Newton has gone to zero.

Making that distance 10 sphere diameters between centers results in a CoS of 23.07 degrees. Doubling the diameter of the spheres increases the CoS to 47.15 degrees. It does not quadruple it.The 23.07º and 47.15º are 2D angles. This is specifically what I refered to. The fact that you mention cones in your abstract is irrelevant to the fact that in the quoted post you were talking only about 2D angles.

You should stop telling me I am wrong all the time with the math here. It is obvious that you are very weak in math. It feels like an English-as-a-second-language student correcting my grammar. Instead you should try to actually evaluate the math itself, the three integrations I posted. If I made a mistake point it out. Which step is wrong and why. If you need more detail on a particular step, please ask. If you got different results post them. Put your math where your mouth is.

-Dale

Sorry but you are both wrong. The angle I am talking about is the angle of penetration relative to the line of gravity between centers. The trig function (effective force) is the Cosine of the angle and it goes to 1.000 when the CoS is zero degrees which happens at the seperation = infinity.

At that point the pseudo volume penetration equals the physical volume and the trig function = 1.000. At the same point Newton has gone to zero.Yes MacM, cos(0)==1, we all know that.

Look up at the fz[r] integral that Aer was kind enough to post, do you see any cosines there by themselves? What does sin(0) cos(0) equal? You are using the wrong trig function. You are only looking at one part of the problem (the angle of penetration) and not thinking about the whole (including the amount of flux). The force is dependent on all the factors, not just the angle of penetration. Think it through a bit instead of reflexively quoting yourself all the time.

-Dale

Sorry but you are both wrong. The angle I am talking about is the angle of penetration relative to the line of gravity between centers. The trig function (effective force) is the Cosine of the angle and it goes to 1.000 when the CoS is zero degrees which happens at the seperation = infinity.

At that point the pseudo volume penetration equals the physical volume and the trig function = 1.000. At the same point Newton has gone to zero.

Yes MacM, cos(0)==1, we all know that.

Look up at the fz[r] integral that Aer was kind enough to post, do you see any cosines there by themselves? What does sin(0) cos(0) equal? You are using the wrong trig function. You are only looking at one part of the problem (the angle of penetration) and not thinking about the whole (including the amount of flux). The force is dependent on all the factors, not just the angle of penetration. Think it through a bit instead of reflexively quoting yourself all the time.

I assume the "angle of penetration" is &theta;_max, that is with a sphere of radius R and at a point of separation r from the center of the sphere:

|
|
R
|
|____________r____________

The angle of penetration is sin^-1(R/r) = &theta;_max

For r < R, &theta;_max &gt; 45&deg; and the point is inside the circle so our analysis in this thread doesn't apply.

But for &theta;_max &lt; 45&deg;, the force is going to decrease as r increases and &theta;_max decreases. Since cos&theta;_max increases as &theta;_max goes to zero, to say that the effective force is proportional to the cosine function is clearly wrong.


Do we all agree that this is the proper integral for the force in the z direction where the z axis is defined along the distance of seperation, r:

fz[r] = UG ρ ∫ ∫ -2 sinθ cosθ √(R²-r²sin²θ) dφ dθ

<table border=0 cellspacing=0 cellpadding=0><tr><td>_&pi;</td><td> </td></tr><tr><td>&int;&nbsp; </td><td> -2 sinθ cosθ √(R²-r²sin²θ) dφ = -4&pi; sinθ cosθ √(R²-r²sin²θ) </td></tr><tr><td>_-&pi;</td><td> </td></tr></table>

Now I am going to show the result of the integral with &theta; without the limits and plug in the limits to show where everything comes from:

∫ -4&pi; sinθ cosθ √(R²-r²sin²θ) dθ = 4/3 &pi; (R²-r²sin²θ)^3/2 / r²

The lower limit is 0:

4/3 &pi; (R²-r²sin²0)^3/2 / r² = 4/3 &pi; R³/r²

The upperlimit is sin^-1(R/r):

4/3 &pi; (R²-r²(R/r)²)^3/2 / r² = 0

So we get that the integral is equal to:

0 - 4/3 &pi; R³/r²

And if we multiply by the constants infront of the integral, we finally get:

- 4/3 &pi; UG ρ R³/r²

Note the correction, I left out the constant &pi; in the other post.

The angle of penetration is sin^-1(R/r) = &theta;_max

For r < R, &theta;_max &gt; 45&deg; and the point is inside the circle so our analysis in this thread doesn't apply.

But for &theta;_max &lt; 45&deg;, the force is going to decrease as r increases and &theta;_max decreases. Since cos&theta;_max increases as &theta;_max goes to zero, to say that the effective force is proportional to the cosine function is clearly wrong.That is correct (except for the very minor point that you meant 90º instead of 45º). The problem is that MacM has been saying "the force goes to 1 as r goes to infinity" for so long that he forgot that he never derived the force.

Do we all agree that this is the proper integral for the force in the z direction where the z axis is defined along the distance of seperation, r:

fz[r] = UG ρ ∫ ∫ -2 sinθ cosθ √(R²-r²sin²θ ) dφ dθI agree.
Now I am going to show the result of the integral with &theta; without the limits and plug in the limits to show where everything comes from:I really appreciate this derivation. Mathematica is great for evaluating the integrals and getting the right answers quickly, but it can't give me a step-by-step explanation like this.

-Thanks
Dale

The 23.07º and 47.15º are 2D angles. This is specifically what I refered to. The fact that you mention cones in your abstract is irrelevant to the fact that in the quoted post you were talking only about 2D angles.

Is english a second language for you? I have made it clear that these angles define the tangent lines (2D view) of a 3D cone. Now address the change in CoS and stop playing games.

If you got different results post them. Put your math where your mouth is.

-Dale

Simply address the fact that a change in diamter with the same center distance increases the exposure.

Yes MacM, cos(0)==1, we all know that.

Look up at the fz[r] integral that Aer was kind enough to post, do you see any cosines there by themselves? What does sin(0) cos(0) equal? You are using the wrong trig function. You are only looking at one part of the problem (the angle of penetration) and not thinking about the whole (including the amount of flux). The force is dependent on all the factors, not just the angle of penetration. Think it through a bit instead of reflexively quoting yourself all the time.

-Dale

The issue is very much the Cos of the penetration angle since my jpoint is that at zero degrees the Cos = 1.000 and the flux penetrating the spheres is such that the PV = V, not zero as in Newtonian.

This does not require calculus or integrals. It is a circular beam with a cross-section equal to the diameter of the spheres.

That is correct (except for the very minor point that you meant 90º instead of 45º). The problem is that MacM has been saying "the force goes to 1 as r goes to infinity" for so long that he forgot that he never derived the force. Dale

Wrong. I never once said force = 1.000. I said the Cosine of the angle of penetration goes to 1.000 as the seperation goes toward infinity. This causes the CoS to become merely a circular beam which has a cross-section equal to the diameter of the spheres and PV = V. Force does not go to zero in that case as it does in Newtonian but is based on a trig function of 1.000, PV = V and the other UniKEF factors.

That is correct (except for the very minor point that you meant 90º instead of 45º)

Ahh yes, It should be 90&deg;. My simple character drawing in the previous post is misleading. I made a new one to show a couple of things. This is more for MacM and Billy T to show visually how the method works.

http://img206.imageshack.us/img206/6414/spherept8zo.th.png (http://img206.imageshack.us/my.php?image=spherept8zo.png)

I set R=1 and r=3 which gives &theta;_max = 19.5&deg;
The black circles are the t1 and t2 locations (t1 on the left and t2 on the right) where length = t1-t2.

I also supplied the equations used to make the graph. &theta; is a column vector from 0 to &theta;_max with intervals of 1&deg; (except for the last element which is simply just &theta;_max).

t is a matrix comprised of 2 column vectors corresponding to t1 and t2 for each angle. As you can see, I plotted every third degree.

Wrong. I never once said force = 1.000. I said the Cosine of the angle of penetration goes to 1.000 as the seperation goes toward infinity.

Can you clarify what you mean by angle of penetration?

I believe your definition of the CoS angle is in this case 2&theta;_max.

But the analysis given above is only for a sphere and a point.

Can you clarify what you mean by angle of penetration?

It would be your theta. However, when I speak of the Cosine I am projecting a line parallel to z from the tangent point on the circle. The law of parallelgrams the angle is the same as theta but the z gravity component is some fraction of the penetration force value by the Cosine of theta.

I believe your definition of the CoS angle is in this case 2&theta;_max.

That is correct. However, unlike Dale's claim I know this is not a 2D angle but is the vertex angle of a cone.

But the analysis given above is only for a sphere and a point.

Correct again. However, the pt is precisely in the mid point between two identical spheres and escribes the CoS

Aer,

I have noticed something interesting. When you post drawings the link does not show in the text of your post on this forum. I do see it and can access it when your post arrives in my e-mail?

Anybody know why?

I believe your definition of the CoS angle is in this case 2θmax.
That is correct.The force goes to 0 as θmax goes to zero.

-Dale

Now address the change in CoS and stop playing games.

Simply address the fact that a change in diamter with the same center distance increases the exposure.I have already done so twice, but you are obviously a little slow on the uptake. For your convenience I have re-posted the second time I addressed this.I have gone over this explanation before, but I will do it again. Look at a single differential element of sphere A. As you double the radius of sphere B the solid angle subtended by B quadruples (your CoS) leading to a four-fold increase in UniKEF flux. At the same time the length of each line doubles. This results in a net 8-fold increase in the UniKEF flux attenuation volume which is exactly compensated by the 8-fold decrease in the density. There is no change in force.

MacM, the problem you are having is that you are closed-minded and mathematically-inept. Mathematically-inept because you can't even follow the math enough to realize that I already did include all of the geometry and other concerns that you are bringing up. Closed-minded because you know the result you want and refuse to accept the fact that the math says differently. As evidence I quote you:You have repeatedly insisted that Geometry of the sphere (i.e. the value of r) has no bearing. It simply must."It simply must" doesn't reflect anything other than your expectation of what the answer should be and your prejudice against any other result. To be honest, this result is not at all what I was originally expecting either. However, unlike you, I was willing to adapt my understanding to match the evidence. This willingness to change your mind in the face of evidence is the chief difference between scientists and pseudo-scientists, it is your choice which camp you want to join.

MacM, I have mentioned several times that I am doing this for entertainment. You are rapidly becoming tiresome. I will not answer any more questions from you regarding the results. I will answer questions from you regarding the methods. If you think that one of my steps is incorrect then we can discuss that, but if you can find nothing wrong with my steps then you can find nothing wrong with the results.

-Dale

The force goes to 0 as θmax goes to zero.

-Dale

Incorrect. At theta = 0 you still have a circular beam of flux penetrating the entire volume of mass along lines parallel to the z. How on earth do you claim gravity has gone to zero?

MacM, the problem you are having is that you are closed-minded and mathematically-inept. Mathematically-inept because you can't even follow the math enough to realize that I already did include all of the geometry and other concerns that you are bringing up. Closed-minded because you know the result you want and refuse to accept the fact that the math says differently. As evidence I quote you:"It simply must" doesn't reflect anything other than your expectation of what the answer should be and your prejudice against any other result. To be honest, this result is not at all what I was originally expecting either. However, unlike you, I was willing to adapt my understanding to match the evidence. This willingness to change your mind in the face of evidence is the chief difference between scientists and pseudo-scientists, it is your choice which camp you want to join.

MacM, I have mentioned several times that I am doing this for entertainment. You are rapidly becoming tiresome. I will not answer any more questions from you regarding the results. I will answer questions from you regarding the methods. If you think that one of my steps is incorrect then we can discuss that, but if you can find nothing wrong with my steps then you can find nothing wrong with the results.

-Dale

And you chose to ignore my response which indicates that your assertion that the CoS quadruples is simply totally false. That the change in the CoS is a function not only of the change in diameter but also as a function of the seperation distance.

Hint: " The change in CoS is not and cannot be a fixed quantity relative to the diameter alone"

Hint: "You are wrong".

Hint: "One major issue your post ignores is even if doubling the diameter quadrupled the CoS (which it doesn't) that does not result in equal net force in the end because the CoS is an angle and as the angle changes the effective force changes. You have stated above that the increase in the CoS and the increase in penetration length still equal the same force. (i.e. density decrease by a factor of 8 and a quadruple CoS * double penetration length = 8, etc) but if they were equal the change in the CoS angle, hence effective force would be less. :D

How on earth do you claim gravity has gone to zero?Like this:<font face=times new roman>(1) X² + Y² + Z² == R²

(2) x(t) == x0 + t v
where x0 == (0, 0, r)
and v == (cos[φ] sin[θ], sin[φ] sin[θ], cos[θ])

t1 = -r Cos[θ] - Sqrt[R² - r² Sin[θ]²]
t2 = -r Cos[θ] + Sqrt[R² - r² Sin[θ]²]

x0 == (0, 0, r)

φ varies over (-π,π].
θ varies over some range [0,θmax]

(3) 0 == R² - r² Sin[θmax]²
θmax == ArcSin[R/r]

(4) f[r,θ,φ] == UG ρ (t1-t2) v

(5) f[r] == ∫ ∫ f[r,θ,φ] sin[θ] dφ dθ

f[r] == {0, 0, -4/3 π R³ UG ρ/r²}

If we let M = 4/3 π ρ R³ then f[r] == {0, 0, -UG M/r²} </font face>That's the nice thing about doing the math before making claims. I don't just have to claim, I can prove. :)

You should try it sometime. Oh, sorry, I forgot. That would mean that you would actually have to admit you don't know everything and make the effort to learn something instead of just telling yourself how brilliant and right you think you are.

-Dale

Hint: "You are wrong". :D Oh MacM, you are so funny. Your ignorance is superceded only by your arrogance. I can just envision a tribe of savages haughtily telling the engineer that he is foolish and wrong, the helicopter can't fly because it is obviously too heavy.

Look at the methods. If you can't find anything wrong with the methods then you can't find anything wrong with the results.

-Dale

Like this:That's the nice thing about doing the math before making claims. I don't just have to claim, I can prove. :)

It is even nicer to recognize that at CoS = 0 there is a circular cross-section beam of flux still penetrating the masses and producing gravity. Sorry your math may be correct for some other function but it cannot be correct and also be describing UniKEF flux patterns and the CoS function.

You should try it sometime. Oh, sorry, I forgot. That would mean that you would actually have to admit you don't know everything and make the effort to learn something instead of just telling yourself how brilliant and right you think you are.

-Dale

You may be well educated and currently fluent in mathematics. However your selfappointed superior attitude is not justified. A few decades ago I was your superior and working in the frontiers of science developing nuclear power.

I once did calculus, etc but no longer have the need nor ambition. Big whoop. I don't believe it is I that seems to think they know everything. In fact I have been enjoying your little selfcentered "Listen to me I'm right and you don't know anything" diatribes.

Why because it has become funny. I began here over two years ago with the claim that UniKEF showed the PV concept to produce gravity in keeping with the inverse square locally.

Others insisted I was wrong and didn't know anything and also choose to post page after page claiming to show I was wrong. Well it seems now several times I have been shown right on that issue.

But what is even more funny is that you have concurred with that but then have failed to recognize what I claim (and I am correct) about UniKEF and geometery actually disqualifies UniKEF as being valid as currently presented since that feature does not match observation.

I have been waiting for one of you smart guys to realize that fact but you are so busy patting yourselves on the back and slamming me that you have missed this very important point. UniKEF as currently presented with a fixed density vs attenuation, does not match observation because of the geometry affect. :o

Oh well back to the drawing board.

But what is even more funny is that you have concurred with that but then have failed to recognize what I claim (and I am correct) about UniKEF and geometery actually disqualifies UniKEF as being valid as currently presented since that feature does not match observation.Well, even if we can't agree on the reason, we can at least agree on the conclusion.
Oh well back to the drawing board.Good luck with your reformulation.

-Dale

Well, even if we can't agree on the reason, we can at least agree on the conclusion.
Good luck with your reformulation.

-Dale

Actually I intend to do more testing to see if this geometric function has simply been missed historically. Some testing already done supports the geometric view but I need