Cont. Part 5. General Expressions of the Gravitational Kepler's Equations applied to Precession
Precession.
As we can realize, the new set of obtained equations of movement allow us to face in a simple way any kind of problems concerned with gravitational forces over particles with or without rest mass, in rectilinear or curvilinear motion. Say, they are, positively, general.
For example let's consider an isolated system of two masses so far apart that they are in a free rectilinear motion, and suddenly they mutually attract each other. Because the CMS follows a permanent rectilinear path with constant velocity (there are no other forces considered), they start approaching each other with a curvilinear and hyperbolical movement, until they cross themselves at their minimum distance, combined with the permanent rectilinear movement of the CMS. On the other hand, in all our development, we always have considered the CMS fixed. We will continue considering fixed the CMS by doing the following: The masses velocities we will use will be those produced by the substraction from the actual masses' velocity components the CMS's velocity components, in order to reduce the situation to that of the fixed CMS.
Let's separate this analysis in two periods: before ocurring the maximum approximation, in which the movement of each mass is hyperbolic, and that after, when movement is elliptic (we have chosen this last movement for simplicity, because it could occur any other types of motion). Let's analize the movement of the mass m1. By applying the convenient equation, we can write:
Integration from infinite to a moment before the maximum approximation is done by realizing that q1 at infinite equals the velocity V1¥ and that w = w1 = mo1.(Ro1)².Wo1/(m1.r1²). However, for the second term in the left side of the equation, we are going to obtain the integration of (dv1/v1).q1² by an indirect way, because its integration has become very difficult to obtain. Let's establish that:
(dv1/v1).q1² = d[q1².Ln(v1)] - 2.q1.dq1.Ln(v1)
By using the following integration property: òab [f(x).g(x)].dx = f(c).òab [g(x)].dx, where c is a constant value between a and b, we can put that:
With this equation we can know the value of V1¥ in function of V10 and Vq, which we will encounter later (we hope so!).
Now we are in the position of starting the derivation for the movement for the mass m1 after its maximum approximation. We will start the analysis of this movement with the initial conditions left by the previous movement at Ro. Now, the integration at the right side of the original equation, will be from a radius Ro1 (where q1o = 0) to a generic radius r1. Thus, the integration limits, at the left side of the equation will be (0,V10 : q1,v1). The equation for this new situation becomes:
where we see that at the starting point the right side term is null, and it also occurs with the left side. The new situation reproduces the initial conditions that there were before at Ro, and V1q will be a value between Vo and V1. Thus, proceeding as in Part 4, let's divide both members by (w1.r1².m1)² = (mo1.Ro1².Wo1)², multiplying by m1² (m1 behaves as a "constant" in the integration process), and remembering that q1 = dr1/dt and w1 = w = dq/dt, we have:
In which, the integration process obliges, as before, to consider h as a "constant". The factor [1 - 2.Ln(v1/Vq)]¹/² had been obtained as a constant factor for the integration process betweeen q = 0 and q = p.
Because in elliptical movement between the perihelio and the aphelio, the constant factor dividing q, is always a value greater than unity (v1 < Vq < V0), it will imply that when q becomes p, the argument of the function COS will be less than p. Say, the half roundtrip of the mass has not been completed yet, because radius has not arrived at its minimum value. This type of movement is called PRECESSION, which is seen in most of the stars and planets. The value of the constant factor Vq will be obtained later.
The angular difference between of the two extreme values of radius to the CMS, perihelio and aphelio, accomplished by the position of the mass m1 in its path around the CMS, is a manner to measure precession. In this case we only need to calculate when the cosine argument reaches p, and which is the angle q that meets it. After finding this value, we substract p to q, and so, we have the angular difference, relative to p, between the two extreme positions, perihelio and aphelio.
Doing this, the value of Precession, or angular difference, in a a half roundtrip of the mass, given by our equations, is:
Angular difference for m1 = [1 + 2.Ln(vq1/V1)]¹/² - 1].p.
In the same manner:
Angular difference for m2 = [[1 + 2.Ln(vq2/V2)]¹/² - 1].p.
Additionally, the condition of a fixed CMS of the two masses' system, imposes that the angular difference for the mass m1 must be equal to that of the mass m2, originating the following relationship:
{1 - 2.Ln(v1/Vq1)} = {1 - 2.Ln(v2/Vq2)}
v1/Vq1 = v2/Vq2
This result was indeed the expected one because of the proportionality of the ellipsed movements of the masses. Also, this result ratifies the Newtonian force definition also obtained in Part 4 and confirms the modification of Kepler's equation, both established before at the begining of this Part 5. Only, the task lacking is that of obtaining the value Vq, for both masses, to complete this analysis.
It is important to notice that the angular difference resulting from the analisys of the mass trip between perihelio and aphelio, positive, or in advance, is different to that going from the aphelio to the perihelio, negative, or in retrocession. Say, the total precession, given by the addition of the two angles q1 = Ap.p = p + qq1, from perihelio to aphelio, and q2 = Aa.p = p - qq2, from aphelio to perihelio again, minus 2.p, originates that the total precession qP becomes the substraction of the two angular differences: qP = qq1 - qq2, but not the double of the first angular difference, as Einstein's Theory of Relativity predicts in its development. Say, Einstein's precession doesn't take in account this effect.
Let's obtain other relationships. From the equation previously obtained:
Which is an indetermination, we obtain that: in aphelio V1 = w1².r1² = 2.(m0/m1).G.Me/[(1/Ro1 + 1/r1).r1²].
And, from the equation for the mass going from aphelio to perihelio:
Which is another indetermination, we obtain that: in perihelio V0 = w0².r0² = 2.(m1/m0).G.Me/[(1/Ro1 + 1/r1).r0²]. These two relationships will be very useful, because with them we have both velocities in function of radius.
On the other hand, let's obtain the value of Vq, by looking for the value of A². For that, we are going to derive another independent relationship, in the following manner:
We know that the variation of the kinetic energy must be compensated by the variation of the potential energy, in order to preserve conservation of the total energy. Let's drop the subindex 1, referred to the mass m1, because it repeats everywhere, in order to simplify the notation. Similarly when we put M, it will be referred to Me. In Part 2, we had obtained that:
K - Ko = m(2.V² - c²) - mo(2.Vo² - c²)
The expression of the variation of the potential energy can be calculated from the expression:
dP = -(G.M.m/r²).[v²/(w².r²)].dr
We know that v²/(w².r²) = (q² + w².r²)/(w².r²) = 1 + q²/(w².r²)
By recalling that the factor 1/{1 - 2.Ln(v1/Vq)} is constant, we can obtain finally the expression of the variation of the potential energy, in the following way:
Thus, in this way we have determinated, at aphelio, the value of A², which was our main interest.
The total angular difference (TAD) or precession from perihelio to aphelio (qP = AP.p) and from aphelio to perihelio (qA = AA.p) for a complete roundtrip of the mass around Me will be given by the addition of the two half rountrips minus 2.p:
TAD = (AP + AA - 2).p
It is also noticeable that the phenomenon of PRECESSION, or the complete roundtrip angular advance of a mass in its movement around the CMS, was obtained thanks the additional term introduced in the last correction to the Kepler's equation done in this Part, which implicitly contains the effect of the movement of the mass inside the force exerted by the fictitious mass Me, which in turn takes in account the existence of other masses in the system. Also it is noticeable that, independently of the number of constituent masses in the system, each one can be treated separately, which simplifies enormously the calculations, all thanks the definition of the fictitious mass Me.
With the previous obtained relationships, by applying the suitable equations, we can determine the whole movements of both masses. Now, if this is a correct analysis, we can calculate the precession, the values of Vq before and after the maximum approximation, and from the first one and Vo the value of V¥ for both masses (previous history). By now, this is enough for us because the whole set of magnitudes around this problem can become known. However, the application of these equations to the case of the Solar System, is simplified by the fact that Sun is almost in the CM of the total system, and then it is not necessary to make the calculations of the Me located at the CMS, because sun mass can be considered fixed at the CM of such isolated system, and we would obtain, with a very good approximation, all the Precessions of those planets we know their radiuses at perihelio and at aphelio.
The solution for precession given by Einstein's general theory of relativity, depends only in a simple way upon the maximum and minimum radiuses of the elliptical motion of the mass [Angular difference = (3.G.M.p/c²)/(1/Ro1 + 1/R'o1), where R'o1 is radius at aphelio], independently of other variables, as initial velocities, which depend on the previous history of the movement, and also on the different angular differences between the two half roundtrips, perihelio to aphelio and aphelio to perihelio, observed in our equations.
May be, is in these aspects that Einstein's precession fails. Really, I don't say that his theory is incorrect. What I believe is that it is very difficult to apply, when you have more than two masses (although in this case, it is very difficult!). However, I know that many people do not agree, and they say that it is the "simplest" and "beautiful" theory never created. But, I believe that this work and many others are attempts to search for simpler ways of understanding the universe.
CONCLUSION. We believe that the divergences with reality that Einstein's General Theory of Relativity has, observed by Edward Guinan and Daniel Popper with binary stars Di Herculis and As Camelopardalis (see article "Was Einstein wrong?" Astronomy, Nov. 1995, pages 54-59, written by Robert Naeye), could be given to inaccuracies in the statement of the problem, in the sense of not all the aspects had been taken in account for the completeness of the Einstein's solution to the problem of gravitation.
We also believe that the treatment given in this Part 5 to the Gravitation problem is simpler than that of the Einstein's general theory, because, as it can be realized, simple math, logical assumptions and simple Physics were used in here.
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