Fig[A-1]Let S be the rectilinear reference frame used for the description. As has been said, for the Ideal observer IO the aetherinos move at constant velocities in the rectilinear frames.
Fig[A-1]
Nevertheless the following results can be considered approximately valid if S is an inertial reference frame instead of a rectilinear one (see Section 5) as long as the distances involved are small because in this case the majority of the short trajectories of the aetherinos can not be easily distinguished from straight lines. (I.e. this approximation is valid for atoms rather than for planets). The aether drag force will be ignored in the calculus because, as shown in Section 5, this drag force cancels when translating the results to the Official observer.
Consider 1st the aetherinos that emerging from Q at a given epoch tE arrive at q at the epoch t :
Let v be the velocity in S of these aetherinos that emerging from Q at a given epoch t
E arrive to q at the (later) epoch t.Let V
Q be the velocity of Q in S at the epoch tE .Let u be the velocity of the selected aetherinos "relative to Q" , or more precisely, relative to the rectilinear reference frame associated with Q at the epoch t
E of their emergence (because after this epoch, Q may have varied its velocity). Therefore: [A-1] u = v - VQ or more explicitly u = v - VQ(tE)The aetherinical force exerted on q (at the epoch t) by these aetherinos will be calculated (like in preceding sections) from the knowledge of their number density in the vicinity of q at this epoch t. This density may on its turn be deduced from the knowledge of how many aetherinos of the pertinent type emerge from Q at time t
E and from the calculus of their "spread" at the epoch t when they pass by the position of q. This spread depends on its turn on the distance D travelled by the aetherinos relative to Q, or more precisely, on the distance travelled (between tE and t ) in the rectilinear frame associated with Q at the epoch tE of their emergence: [A-4] D = u . (t - tE )To calculate such density of pertinent aetherinos in the vicinity of q at time t , it will be assumed that the residual distribution r(u) that emerges from Q is independent of the movement of Q relative to its local aether. (This "principle of relativity" is believed to be approximately valid only for small speeds of Q but this statement is yet unproved). Furthermore in this first example of deduction of the force it will also be supposed that the residual distribution of Q is isotropic relative to Q (i.e. in the reference frame of Q emerges in all directions the same residual distribution of aetherinos). An expression of the wanted force could also be deduced supposing that from Q emerges a distribution that depends on its movement through the aether and/or that is not isotropic but such complication seems out of context and would only distract from the present purpose of explaining the general features of this model of the aether.
Let r(u).du = residual distribution of Q = excess (or deficit) of aetherinos with speeds relative to Q in the interval {u,u+du} that "emerge" from Q in unit time and by unit solid angle.
(Remember in what follows that the speeds u and the travelled distances D are referred to the inertial (or more precisely "rectilinear") reference frame defined by Q at the epoch t
E).The calculus of the force suffered at the epoch t by the particle q will be done adding the contributions of the pertinent aetherinos emerged from Q at all epochs t
E earlier than t. To find the contribution of the aetherinos emerged from Q during a time interval {tE,tE+ dtE} it will first be deduced the density of those aetherinos at the position of q at the epoch t. Suppose that all the aetherinos emerged from Q during the time interval {tE,tE+ dtE} do actually depart Q instantly at the beginning of such interval. At the epoch t those that have crossed a sphere of radius D centred at the extrapolated position of Q are: [A-5]
Those that have crossed a sphere of radius D+DD centred at the extrapolated position of Q are:
[A-6]
Therefore those that at the epoch t are between the 2 spheres:
[A-7]
Considering that the region of interest for the calculus of the density of the pertinent aetherinos is an infinitesimal volume swept by q at the epoch
t then it can be assumed that DD << D , and, since a typical redistribution r(u) is expected to vary smoothly with u, the integral [A-7] can be approximated by:
[A-8]
The volume between both spheres is:
Vol = 4p/3 [(D+DD)3 - D3] [A-10]
that for DD << D, neglecting the terms on DD3 and on DD2 , can be approximated by
Vol
@ 4p D2 DD [A-11]Finally the density (number per unit volume) of the pertinent aetherinos ( those that emerge Q during {t
E, tE+ dtE} and are at the epoch t at a distance D of the extrapolated position of Q at this epoch, which corresponds to the position of q ) is:
[A-12]
that using [A-4] can be written as:
[A-12b]
where the speed u is a function of
t and tE . For example if the law of movement of Q in the reference frame S is given by some known function pQ(tE) (where the letter p stands for "position vector") then, see Eq[A-1]:
[A-14]
Let Vq be the velocity of q in the reference frame S at the epoch t.
Let b be the velocity of the pertinent aetherinos relative to the target q at the epoch t.
Recalling that v is the name given above to the velocity in S of the aetherinos that emerging from Q at a given epoch t
E arrive to q at the (later) epoch t, thenb = v - Vq or more explicitly b = v - Vq(t) [A-15]
The contribution of the presence of Q during {t
E, tE+ dtE} to the aetherinical force acting on q at the epoch t can then be calculated as follows:Ignoring at this stage the possible structure of q, let sq be the geometric cross section of q.
Since the pertinent aetherinos have a speed b relative to q and since their density in the vicinity of q is the dr (of [A-12]), then the number of pertinent aetherino collisions suffered by q in unit time is
dn = sq b dr [A-16]
Each of these collisions contributes by definition with an aetherinical impulse:
i
1 = k b [A-17]where the letter k is used here instead of the multiplicative constant q of Eq[1-1a] (Section 1) to avoid confusion with the name given here to the particle suffering the force being studied.
The aetherinical force (net aetherinical impulse in unit time) due to these collisions being considered is:
dF = i
1 dn = k b sq b dr [A-18]The total aetherinical force suffered by q at the epoch t due to the aetherinos emerging from Q in "all" passed epochs (and whose speed is of course such that they reach q at the epoch t ) is:
[A-19]
i.e. expliciting the time dependencies:
[A-20]
where in terms of their position vector time functions:
[A-21]
and
Autoforce.
It can be defined as the aetherinical force suffered at a given epoch by a body q due to the action of those aetherinos emerged in the past (i.e. in all earlier epochs in which q has existed) from q itself.
Note: From the material body q emerges at all epochs a specific aetherino distribution that is supposed to differ from that of the surrounding aether (being as it is believed, a redistribution of this latter) and therefore is able to produce aetherinical material forces on other bodies including, in most cases, the body q itself. In fact, if, in a rectilinear frame, q follows a curved or a zigzag trajectory, it will continuously receive impacts of aetherinos that emerged from q itself when it was passing through earlier positions. (The aetherinos travel in the rectilinear frames with constant velocities that can have any value).
Let R be the radius of the circular orbit (centred at P) followed by the body q. The orbit is circular in the rectilinear reference frame associated to P.
Let V
Fq be the OO speed of q in its orbit (i.e. the speed assigned by the Official Observer). Suppose that this speed VFq is constant at all epochs for the Official observer.Let V
q be the speed of q as seen by IO.Consider that the IO and OO speeds are related (as in Section 5, Eq[5-24]) by:
V
q = VFq f(t) [A-24]where f(t) is a yet undefined function of time relating the readings of the IO and OO clocks. More precisely (see sections 3 and 5 for details):
V
q , the IO speed of q, will therefore be a function of time.Suppose that at the epoch t where the autoforce is going to be evaluated the particle q is in position B (Fig A-26)
Fig [A-26]
Fig [A-26] represents the positions and velocities needed to calculate the contribution to the autoforce of those aetherinos that emerged from q itself at an earlier epoch t
E when q was passing by position A.The equation [A-20] is valid for this calculus having in mind that the role of Q used to obtain [A-20] is now played by q itself at the earlier epochs t
E .At the epoch t
E the angle a subtended by APB is given by:
[A-28]
where s(t
E) is the arc travelled by q between the epochs tE and t.Therefore from [A-24,25]:
The autoforce will be analyzed in 2 components x and y that are taken respectively tangent and perpendicular to the orbit.
The velocity v appearing in [A-20] (see also [A-21,22]) was defined at the beginning of this Annex as the velocity in S of those aetherinos that emerging from Q at a given epoch t
E arrive to q at the (later) epoch t. Therefore (see Fig[A-26]) :
[A-30]
has the components:
[A-31]
The components of Vq(t
E) , see [A-24], are:VqX (tE) = VFq f(tE) Cos a
VqY (tE) = VFq f(tE) Sin a [A-32]
The components of Vq(t) , considering that the x and y axes are respectively the tangent and the perpendicular (to the orbit) at this epoch t of observation, are :
[A-33]
The 2 components of the autoforce, see [A-20] are:
[A-34]
[A-35]
where, using above equations and remembering that the role of Q in [A-21,22] is now played by q itself:
[A-36]
[A-37]
b = ( [b
X]2 + [bY]2 ) 1/2 [A-38]
u2(t
E,t) = [uX]2 + [uY]2 = [A-39]
and where the a dependency on time (that must be accounted for in the integration) is:
For example, in the case of an electron orbiting a proton, the net force acting on the electron at a given epoch t, is the vector sum of the electron "autoforce", whose components are given in [A-34,35] , and the centripetal force of the proton, whose components may be deduced from [A-20..] . Care must be taken in the use of the residual distribution r(u) appearing in the above equations since this r(u) is not the same for the electron and the proton. The simplest plausible choice is (see Section 1) to take both residual distributions as equal but of opposite sign, i.e. r
Q(u) = - rq(u).-----------------------------------------------------
An example of central force in 3D.
Consider a system made of two charged particles A and B. Let A be much more massive than B (e.g. A is a proton and B is an electron). Let the reference frame of description be associated with the body A. This reference frame can be treated (in a first approximation) as a rectilinear reference frame (in which the aetherinos travel in straight lines at constant speeds) because, in a strictly rectilinear frame, the body A (due to its much greater mass) would suffer, due to the force F
BA, negligible accelerations compared to those suffered by the body B due to the force FAB.The first goal of the present calculus is to deduce the force F
AB suffered by B as a function of its velocity u and its position in the reference frame of description.By the moment it will also be supposed that (1) A is at rest in the aether, and (2) the aetherinical redistribution of A is the same in all directions (the particle A has no preferential axis of redistribution). With those suppositions, due to symmetry, the force F
AB should only depend on the distance AB and on the velocity of B relative to A (or more precisely on the modulus "u" and on the angle between u and the semi-direction AB). Nevertheless the system of coordinates chosen for the description will be the standard 3D Cartesian system due to its simplicity (in spite that a more natural choice in this case would be a polar coordinate system). The body A is supposed to be located at all epochs at the origin of coordinates (0,0,0) while the body B (at the epoch of determination of the force FAB) is supposed to be passing the generic position (x,y,z) with the generic velocity (uX, uY, uZ).Since in this example the aetherinical redistribution that origininates of the force does not change with time it is simpler to integrate for all the pertinent aetherinos in consideration of their speed instead of their emission epoch.
In the vicinity of B, the density of aetherinos with speeds in the interval {v, v+dv} that have been redistributed by A is:
[A-71]
where:
dr(v) is the number of such speed {v, v+dv} aetherinos in unit volume at the vicinity of B.
r
A(v) is the redistribution caused by A, i.e. the excess/deficit number of speed v aetherinos "emerging" from A by unit time, by unit speed interval and by unit solid angle.In the reference frame chosen for the description (associated with A) the velocity direction of all the aetherinos that, coming from A, collide with B at any given epoch is equal to the direction AB of the vector joining A and B at that given epoch.
The expression [A-71] of the density of aetherinos should be evident considering that, according to the definition of r
A(v)dv as a flux by unit solid angle, rA(v)dv/D2 is the number of aetherinos (of speeds in {v, v+dv}) crossing in unit time (at all epochs) a unit area surface oriented perpendicular to their velocity and located at a distance D from A. The number of aetherinos of speed (approximately) v having crossed such unit surface in a unit time interval can therefore be found in an imaginary cylinder of base 1 and length v whose volume 1*v has therefore a magnitude equal to that of the speed v.When an aetherino proceeding along the semi direction AB collides with B it gives to this particle an aetherinical impulse q vR (where vR = v - u is the velocity of the aetherino relative to B). Therefore for an aetherino of speed v, the Cartesian components of this elementary aetherinical impulse are (see Fig[A-70]):
iX = q (v - u)X = q (vX – uX) = q (v x/D – uX)
[A-72] iY = q (v - u)Y = q (vY – uY) = q (v y/D – uY)
iZ = q (v - u)Z = q (vZ – uZ) = q (v z/D – uZ)
where it has been acknowledged that when the particle B is at the position (x,y,z) the components of the velocity v of an aetherino traveling along AB are:
vX = v x/D
[A-73]
vY = v y/D
vZ = v z/D
Notice also that the components of the velocity vR of an aetherino relative to B are (v x/D–uX, v y/D–uY, v z/D–uZ) and therefore the modulus vR of such relative velocity is:
[A-74]
The number of collisions in unit time between B and the pertinent aetherinos (those whose density is given in [A-71]) can again be calculated considering that in the specific reference frame where these pertinent aetherinos (of velocity v) are at rest the particle B of geometric cross section sB sweeps in unit time a cylindrical volume of length vR. Hence this rate of collisions is:
[A-75]
The aetherinical impulse given to B by all these collisions occurring in unit time has therefore the components:
[A-76]
where vR is given in [A-74] and the distance D is equal to:
[A-77]
The components of the aetherinical force FAB are finally obtained adding for the pertinent aetherinos of all speeds.
For example if it is supposed that the above expression of the elementary aetherinical i1 = q vR is true whatever the relative speed vR of the aetherino, then the Cartesian components of the force FAB would be:
[A-78] 
where the integration limits include the aetherinos of all speeds.
But in other sections of this work it has been found that the model makes more reasonable predictions if it is assumed that "only the aetherinos whose speed relative to the detector is bigger than c are able to produce an impulse" (hypothesis c+) on the electron. According to this hypothesis c+ the elementary aetherinical impulse would be:
i1 = q vR for vR > c
[A-79]
i1 = 0 for vR £ cTo implement this hypothesis in the calculus of the force F
AB, the integration limits must be revised, for example as follows:From [A-74], solving for v, an aetherino that has a speed v
R=c relative to B has in the reference frame of description a speed: [A-80]
Note: Since v
C must be positive, only the solution with the positive sign before the square root sign makes it possible and will be considered. Hence in what follows it must be understood that:[A-80b]
Therefore assuming the hypothesis c+ and supposing that the speed u of the particle B is smaller than c, the revised expressions of the force components F
X, FY, FZ (see A-78 and notice the new integration limits) are:[A-81]
where v
R is given in [A-74] and D is given in [A-77].Note: The reason that the validity of the force F
AB given in [A-81] is restricted to speeds of u<c is that for higher speeds there can be situations (e.g. when B moves straight away from A faster than c) in which aetherinos of slow speed v (having "departed" A in a remote past) are caught from behind by B with a relative speed higher than c. Those aetherinos, that would also produce a non-zero impulse on B, are nevertheless not accounted for by the expressions [A-81]. Therefore in such scenario the expressions [A-81] would no longer be valid but should be replaced by more complicated ones.Quantitative examples.
In the following examples it will be supposed that the hypothesis c+ (see [A-79]) applies to the collisions between the aetherinos and the target particle B. Therefore the Cartesian components of the force F
AB are those given in [A-81].It will be supposed that the particle A produces an isotropic redistribution of aetherino speeds given by:
[A-82]
This function is qualitatively similar to others that have been tried (postulated) in this work for the redistribution of an electric charge at rest in the aether. It is also the type of function suggested by the model of redistribution analysed in the Annex D. It is nevertheless quantitatively different from earlier redistributions in which the speed v appeared raised to lower powers.
Fig[A-82] is a plot of r
A[v] for an example set of values of the constants k1, k2, k3
Fig[A-82]
Fig [A-81] corresponds to k
0=1; k1=1.08; k2 = 0.72; They are in the "order of values" that gives some reasonable predictions (see for example a model of the Compton effect in Section 6).
- Example (1). The particle B moves along the straight line AB.
Taking k
0=1; k1=1.08; k2 = 0.72; as the constants of the redistribution rA[v], the force FAB has been evaluated for several values of uX supposing:u
Y=0; uZ=0; x=1; y=0; z=0;q=1; s
B=1;It is evident that with those suppositions the components F
Y and FZ of the force are zero.The following Fig[A-83] plots the results of the evaluation of the component F
X (or more precisely of –FX) as a function of uX/c 
Fig [A-83]
The evaluations (and the plot) have been made with Mathematica 5.1 of Wolfram Research, performing numerical integrations (with NIntegrate) of the expression [A-81] of FX obtained above.
- An interesting feature of F
X(uX) is that it approaches zero when uX approaches either c or –c. This behaviour is adequate to explain the "relativistic" features of high speed particles. If a force law of the type represented in Fig[A-83] is correct but mainstream Physics fails to acknowledge it and postulates instead in its theory that such forces do not depend on the velocity of the target particle, then it is not surprising that to explain the experimental facts, mainstream Physics needs a theory (like Special Relativity) in which the effective (relativistic) "mass" of the moving body suffering the force increases with speed. (See more in Section 12).- Another interesting feature is that, for |u
X| << c, the force FX[uX] can be approximated by: [A-84]
( for |uX | << c )
- Example (2).
The particle B is moving "abeam" A (i.e. the velocity of B is perpendicular to AB).The forces acting in this scenario can be evaluated taking:
x=D; y=0; z=0;
u
X=0; uY=u; uZ=0;In this example, an aetherino of speed v has a speed v
R relative to the electron given simply by (see A-74):
and the slower aetherinos to be considered in the integration (in application of the hypothesis c+) have now a speed v
C (see A-80b):
The centripetal force (i.e. the force component along the direction AB) acting on the electron is now given by (see A-81):
[A-86]
No exact integration of F
X has been obtained but it has been guessed that, for |u| significantly smaller than c, this centripetal force FX can be approximated by the following function: [A-86b]
for |u| < c
Taking k
0=1; k1=1.08; k2 = 0.72 as the constants of the redistribution rA[v], the constant k4 must be fitted to approximately k4 = 0.190The transverse force (i.e. the force component along a direction perpendicular to AB that in this example acts along the direction of the velocity u of the electron) is now given by (see A-81):
[A-87]
for |u| < c
No exact integration of F
Y has been obtained but it has been guessed that, for |u| significantly smaller than c, this transverse force FY can be approximated by the following function: [A-87b]
Taking again k
0=1; k1=1.08; k2 = 0.72 as the constants of the redistribution rA[v], the constants must be fitted to approximately k5 = 0.125 and k6 = 0.205Taking for example:
u = 0.1 c; D=1;
the predictions are (approximately):
F
X = -0.189; FY = 0.011; FZ = 0;And therefore the angle between AB and F
AB is (in degrees):a
= ArcTan[FY/FX] 180/p = -3.43ºThis "aberration angle of the force" is much greater than the one predicted in Section 5 in which the hypothesis c+ was not applied. In Section 5 it was calculated that, if the target particle B is moving "abeam" A with a speed 0.1 c, the aberration angle is only of about 0.05º. That result of Section 5 was "more" consistent with the predictions of Special Relativity for which the angle is zero (i.e. the electric force F
AB always acts along the instantaneous (non delayed) direction AB). But the model needs an hypothesis c+ (or alike) to make reasonable predictions in other contexts.A possible way out is to assume that the hypothesis c+ only applies to aetherinos incident on electrons but not on protons. In fact, the main reason to adopt the hypothesis c+ was to allow the model to predict the stability of the features of radiation when it travels along large distances. But the features of radiation (intensity, frequency, modulation, polarization,…) are "read" and replicated by the electrons of the detector (and not by its protons). Furthermore it seems reasonable to assume that a proton is a composite particle made of inner particles of high speed in relation to the whole. It also seems reasonable to assume that if the hypothesis c+ applies to these inner particles the result is that it can no longer be assumed that the hypothesis c+ applies to the proton as a whole. It seems therefore plausible at first sight to make a consistent description assuming that the hypothesis c+ applies only to electrons but not to protons, but the issues commented here have not yet been analysed in detail in this work.
- Example (3).
An electron B orbiting a proton A.(3a) Supposing that the proton's redistribution is isotropic (i.e. the
same in all directions).
Evaluations have been made of the trajectory {x(t), y(t), z(t)} followed by the electron B
in presence of the aetherinical redistribution created by the proton A. It has again been
supposed that the redistribution of the proton is of the type [A-82]. Only one
non-trivially distinct stable orbit has been found. This case of isotropy has actually
been studied as a particular case of the redistribution described in the following (3b)
paragraph.
(3b) Supposing (1) that the proton's redistribution along any given
direction depends on the angle that such direction makes with a privileged axis of the
proton that will be called its redistribution axis (RA) ...
(3b-1) In this first example it will be supposed that the proton has no intrinsic rotation. i.e. the axis RA points at all epochs in the same direction of space. Let z be such direction. Suppose that the proton’s redistribution varies in intensity (not in form) as a function only of the angle j that the direction of emergence of the aetherinos forms with the axis z. And suppose that the redistribution has an "axial-type" symmetry such that (for example):
[A-90]
where rA[v] is the redistribution given in [A-82] and a is some "distortion" numerical constant such that –1£ a £1 (that will allow to test different anisotropies in the redistribution).
The centre of the proton is located at (x=0, y=0, z=0) and the RA is coincident with the z-axis of the Cartesian coordinate frame of description. Therefore when the target electron B is passing the position (x,y,z), the direction AB makes an angle j with the RA given by:
[A-91]
and for the evaluation of the force at this position the above expressions [A-81] may be used but replacing in them r
A[v] by the R[v,j] given in [A-90], because at all epochs the proton has been "emitting" the same redistribution R[v,j] towards this position.Some example simulations have been done. A few of them have been detailed in a Mathematica 5.1 notebook that can be downloaded right-clicking the following link AOE.nb (52 KB) and choosing "Save destination As".
Here is a brief summary of some observations from the simulations:
- No stable orbits have been obtained assuming that the electron is acted only by the proton force (Eqs[A-80]). Assuming only such attraction force, trajectories of ever increasing distance to the proton are, in general, obtained.
- Assuming that, together with the proton force, the electron suffers an aether drag force tending to decrease its speed (see Section 2 of this work), then, stable (repetitive) orbits have been found if the drag force depends on the speed u of the electron according to some possible functions. That is consistent with the model since it must be realized that the calculus of the spatial trajectory of a particle acted by aetherinical forces must be made taking into account all aetherinical influences, including the drag force. Drag force functions of the following type have been tested:
[A-92] FDRAG = - (K1 u + K2 u2 + K3 u3 + K4 u4) u/u (with the constants Ki bigger or equal than zero).With the following 3 drag force function types it has been observed that stable orbits are possible:
[A-92a] K3>0 K1=K2=K4=0 [A-92b] K4>0 K1=K2=K3=0 [A-92c] K2>0,K4>0 K1=K3=0But the aether drag force predicted in Section 2 for the electron (for which the hypothesis c+ has been assumed that applies), can be approximated by [A-92] but with K
1>0,K3>0, K2=K4=0 and with this type of drag force (together with the example assumptions about the proton's redistribution) no stable orbits are obtained.- The three alternate assumptions about the drag force expression, (i.e. A-92a, A-92b and A-92c) give qualitatively similar results. Nevertheless the simulations of AOE.nb (52 KB) have been done assuming [A-92b] (F
DRAG = - K4 u4 u/u) because in this case an electron set with random initial conditions seems to converge faster to a final stable orbit.- Assuming a = 0 in [A-90] (i.e. assuming isotropy of the proton’s redistribution) only one non-trivially distinct stable orbit has been found.
- Most simulations have been done assuming a = -0.5 i.e. supposing that the proton redistribution has a "moderate" anisotropy with an axial symmetry. In this case 4 non-trivially different stable orbits have so far been found. They are shown together in the following 3D plot (Fig A-95).
Fig [A-95]
The following plot (Fig A-96) represents instead the trajectory of an electron dropping towards a stable orbit:
Fig [A-96]
All the calculi of the simulations mentioned here and all the graphical plots have been done with Mathematica 5.1 of Wolfram Research. (Wolfram’s Mathematica is a wonderful tool that has been very useful for these types of evaluations).
