_{1}

^{*}

A updated numerical evaluation of the Newtonian component of Mercury’s perihelion advance over more than two centuries starting from about the year 2000 is made using the Euler’s algorithm as well as a modified Euler algorithm. Results are given for about the last 30 years of this interval which show that the precession rate may be substantially higher than what it is believed to be.

Recently [

where the time step of integration is

on the computation of the terms ^{8} meters/day^{2}. If this is multiplied by

day^{2} we obtain a quantity which can be added to

contribution to

the solar contribution is entirely lost. Thus this is a neglect of a corresponding planetary contribution to some solar one and hence it will show a lower value of the perihelion precession in the final result as planets are solely responsible for Mercury’s perihelion advance in the Newtonian theory. We have noted in our previous paper (Ref. [

Euler algorithm which neglects the term

in the position coordinates for a particular value of

In our previous paper [

From the previously determined values of

At the present moment the author does not have access to quadruple precision computing systems equipped with ODE solvers and so in order to make the best use of the available facilities we tried to apply a modified form of the Euler algorithm that is a modified form of Equations (1) and (2). These are

and are identified as the “Euler algorithm” in certain textbooks instead of Equations (1) and (2). Although this neglect of the term proportional to ^{th} century) we just make a sober assessment that there is an urgent need to calculate the perihelion precession rate of Mercury using more sophisticated techniques. The actual value may well exceed 540 arc sec/cy.

First of all I would like to thank the anonymous referee for his constructive criticism. In his report he has referred to this link http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Pollock.pdf from which I quote the following: “It is possible to model, very precisely, the Newtonian force each planet was expected to exert on Mercury. An elegant approximation, however, was described by Price and Rush in their paper entitled Non-relativistic contribution to Mercury’s perihelion precession. They replaced each of the outer planets by a ring of uniform linear mass density. Since Mercury’s precession is slow compared with the ortits of even Uranus, their approach yields a fairly accurate time-averaged effect of the moving planets’ effects. An sketch of their calculations follows. First, let’s replace each planet by a ring with linear mass density given by the following

equation:

where _{i} is the mass of the ith planet, and R_{i} is the radius of the ith planets orbit, which is assumed to be circular.” The present author feels that the ring like mass concept has been more precisely introduced by Stewart [

where all notations are explained in figure 1 of this reference. If we take the gradient of this quantity it will be found to be zero at

shows that all the perturbing terms including the pseudo force terms are present in the summation. Numerical integration does not necessitate any ring like assumption as each of the terms above is integrated as it is by the integration algorithm. The only error that is introduced is by the use of finite quantities instead of infinitesimals. These errors are larger for the simple algorithms like the one used here necessitating smaller time steps. Of a different nature is the error introduced by machine precision which affects all algorithms although to varying extents as we noted earlier. It is difficult to quantify the extent of the error for any given algorithm. The referee has raised an issue which is quoted as “Another problem comes from the integration method. Finite precision methods introduce a perturbation by themselves and this gives rise to an extra precession of the perihelion (this is a general consequence of Bertrand’s theorem).” to which I should say that finite precision is a property of the computing system or the language and is dependent on the machine architecture for example the normal Matlab is double precision. By the phrase “finite precision method” (of integration?), I feel a certain amount of confusion is being introduced as it is not clear what is being referred to that is to the algorithm or to Matlab. Now the question as to whether such computations are overestimating the rate of perihelion advance is difficult to answer (it may well be underestimating). I would just like to state that if Prof. M.G. Stewart’s calculations are correct then perturbation theory should predict no more than 529 arc sec/cy as the node of Mercury precess at a rate of

−451 arc sec/cy and a term

definition of longitude of perihelion as is used for example by Clemence. All these have been explained in one of our previous papers that is Ref. [

Rajat Roy, (2015) Newtonian Computation of Perihelion Precession of Mercury—An Update. Open Access Library Journal,02,1-5. doi: 10.4236/oalib.1101665