Meaning of passages by Gauss on the "convergence of expansions (in infinite series) of the (elliptical) equation of the center"?

The "midpoint equation" is another name for the "equation of the centre", related to kepler's equation. What Gauss wanted to determine in this investigation is the asymptotic mathematical expression for the coefficients of the trigonometric series (Fourier series) for the difference between true anomaly $v$ and mean anomaly $M$:

$$v - M = sum_{n=1}^infty frac{{1}}{{n}}C_n sin(nM)$$

where $C_n = f(epsilon)$. The problem of determination of this trigonometric series is essentially a modern equivalent of the ancient method of epicycles - this was a model of the celestial positions of the planets which used a series of "wheels" ("epicycles") with different radiuses and angular speeds to trace the location of the planet. The coefficient $frac{C_n}{n}$ is essentially the radius of the $n$th wheel, while it's angular frequency is $frac{nM}{t}$ ($t$ is the time of a given mean anomaly measurement).

Gauss's final asymptotic formula for the coefficients is written in the last line of p.423 of volume 10-1 as:

$$C_n sim gamma^n(1+frac{sqrt{8}}{3cos^{frac{3}{2}}varphi cdot sqrt{npi}})$$,

where $e$ is the base of natural logs and $varphi$ is the arcsine of the eccentricity $varepsilon$ ($varepsilon = sinvarphi$), while $gamma$ is defined through a new angle $theta$ as $gamma = tan(frac{theta}{2})$ which itself is connected to angle $varphi$ by the relation: $tan(frac{theta}{2}) = tan(frac{{varphi}}{{2}})e^{cosvarphi}$. Note that this formula descibes the asymptotic behaviour of $C_n$ - it's not an explicit formula for the coefficients! Gauss's formula for $C_n$ is therefore identical with Jacobi's formula from his 1849 memoir - $C_n sim (tan(frac{{varphi}}{{2}})e^{cosvarphi})^n(1 + frac{{4}}{{3}}frac{{1}}{{sqrt{{2npi cos^3varphi}}}})$.

It's worth remarking that $C_n$ tends to zero as $n$ tends to infinity only when $gammale1$ (otherwise, it tends to infinity); this means that the trigonometric series converges only when $tan(frac{{varphi}}{{2}})e^{cosvarphi}le 1$. If we recall the definition of $phi$ - that $varepsilon = sinvarphi$ - we get after using several trigonometric identities that the critical value of the eccentricity $varepsilon$ for which the series begins to diverge is the numerical solution of the following equation : $frac{epsilon e^{sqrt{1-epsilon^2}}}{1+sqrt{1-epsilon^2}} = 1$.

This is exactly the definition of "Laplace limit" (whose value is approximately $0.662$); it's a very significant point since Laplace's discovery of the divergence of this series for sufficiently large eccentricity as well as the calculation of this limit was published only in 1827 (so Gauss preceeded him by more than 20 years). On p. 424 of the same manuscript appears a table of relevant quantities which Gauss calculated for 25 different values of mean anomaly $M$ (between 0 and 180 degrees) by cutting the infinite trigonometric series after several terms. The whole table is calculated for a large value of eccentricity - $0.655$ - a value which is very close to Laplace limit. This table is probably the numerical aspect of the investigation Gauss conducted on "the degree of convergence of trigonometric series"; i.e the rate of convergence of such series.

In order to derive the closed form expression for the coefficients of the series Gauss applied a cetain imaginary transformation for the eccentric anomaly $E$ (the imaginary transformation is $E = icdot logcot(frac{{varphi}}{{2}}) + epsilon$. Gauss doesn't expain how he arrived at this method, as he is operating entirely with series representations, but later the mathematician Jakob Horn, who was known for his contributions to the theory of asymptotic expansions, showed that one can arrive at this susbtitution by operating with integrals in the complex plane. The following is an excerpt from a letter of Jakob Horn in which he explains the meaning of Gauss's substitution:

For the developement $$C_n = frac{1}{pi}int_{-pi}^{+pi}frac{dv}{dM}e^{inM}dM = frac{cosvarphi}{pi}int_{-pi}^{+pi}e^{inM}frac{dE}{1-f cosE}$$ by introducing $z = e^{iE}$ we get $$C_n = -ifrac{cosvarphi}{pi}int F(z)(Phi(z))^ndz$$ in which $$F(z) = frac{1}{z-frac{f}{2}z^2-frac{f}{2}}, Phi(z) = ze^{-frac{f}{2}(z-frac{1}{z})}$$ is integrated over the unit circle $|z| = 1$. The function $F(z)$ has the real singular points $$z_0 = frac{1-sqrt{1-f^2}}{f} = mathbb{tan(frac{varphi}{2})} = e^{imathfrak R}<1$$ $$z_1 = frac{1+sqrt{1-f^2}}{f} = mathbb{cot(frac{varphi}{2})} = e^{-imathfrak R}>1$$ which are also zeroes of $Phi'(z)$. The integration path $$|z| = 1$$ can be replaced by the integration path $|z| = z_0$, which only has to avoid the singular point $z_0$. One has to set accordingly $z = z_0e^{iepsilon}$, or what is the same $$E = mathfrak R +epsilon = imathbb{logcot(frac{varphi} {2})}+epsilon$$ which is precisely the substitution used by Gauss. If one takes the circle $|z| = z_1$ as the integration path, bypassing the singular point $z_1$, one has $$z = z_1e^{itheta}$$ or $$E = theta - imathbb{logcot(frac{varphi} {2})}$$ a substitution which is used in a paper by Wilhelm Scheibner.

Therefore, it seems that the significance of this investigation of Gauss lies in the fact that his method of derivation is based on his ideas on the foundations of complex analysis. In his historic reconstruction of Gauss's line of thought, Schlesinger also mentiones some preliminary research done by Gauss and recorded in several diary entries. These entries are concerned with the divergent series $0!-1!+2!-3!+...approx 0.5963$ (Euler's famous summation) and the asymptotic behaviour of Bessel functions and the location of their zeroes (see Gauss's werke, volume 10, p.382-389).

Historical aspects:

Very late in his life (1850 and on) Gauss mentioned in several letters to his friend Schumacher his wish to write a fundamental treatise on the convergence of trigonometric series, which will include the connections to the doctrines of complex quantities (complex functions) and analysis situs (topological ideas about connectivity of surfaces). He claims priority for Jacobi's 1849 researches on kepler's equation, stating that he possesed the solution to the problem of coefficients of the trigonometric series for about $pm 50$ years. The exact year at which he made his relevant researches cannot be dated exactly, but Schlesinger speculates that it was around the year 1805.

Actually the first mathematical astronomer who tried to solved this problem was Francesco Carlini, who gave a remarkable but flawed solution in 1817, so Gauss's solution predates even him. For a survey article on Carlini's work, see the article; "Francesco Carlini: Kepler’s equation and the asymptotic solution to singular differential equations". Carlini's solution was essentially different from Gauss's, and involved the earliest use of the WKB approximation of quantum mechanics, while Gauss's solution is rather similar to that W. Scheibner has given in 1856, which used Cauchy's theory of residues.

The significance of Gauss's way of deriving the solution is that it's incomparably shorter than that of Jacobi. If i'm not mistaken, one of the possible subjects Gauss suggested to Riemann for his doctoral dissertation was trigonometric series, so i suspect that (since it was around the same year of his letters to schumacher) it might be connected somehow with that, but this is entirely a speculation of mine.

Finally, it's worth remarking that the famous mathematician Vladimir Arnold, had the following things to say about Laplace's limit:

In attempting to explain the origin of this constant, Augustin Cauchy created complex analysis. Numerous fundamental mathematical notions and results, such as Bessel functions, Fourier series, the topological index of vector fields, and the argument principle in the theory of complex functions, also made their first appearence in investigations connected with Kepler's equation.

This quotation is taken from Arnold's article "Kepler's second law and the topology of Abelian Integrals (According to Newton)", which isn't really about Laplace's constant (it's actually about "Newton's theorem on ovals"), but i mentioned it just to bring the circle of ideas surrounding Gauss's unpublished results to a certain historic closure.

Remaining questions

I still don't grasp the basic idea behind Gauss's substitution $E = icdot logcot(frac{{varphi}}{{2}}) + epsilon$, and i don't even understand how well defined quantities like $E,varphi,epsilon$ can be related by this equation. Anyway, the meaning of this substitution is the subject of a post on Astronomy stack exchange, and i just keep updating this HSM post as well to increase the chances of getting a response on the meaning of Gauss's substitution.