Newton's Law of Gravity for Solar System Planets (visualization)

According to Newtonian gravity theory \[ v(r) =\sqrt{ \frac{G M}{r}} \] If the orbit is circular, the speed is simply proportional to $1/\sqrt{r}$. For general cases, however, after some derivation, the average speed, \[ \bar{v} = \frac{1}{2\pi}\int_0^{2\pi} \mathrm{d}\theta v(\theta) = \sqrt{\frac{GM}{a} } \frac{2 \mathrm{E}(\frac{2\epsilon}{1+\epsilon})}{\pi \sqrt{1-\epsilon}}, \] where $\mathrm{E}(z) = \int_0^{\frac{\pi}{2}} (1-z \sin^2\theta)^{1/2} \mathrm{d}\theta$ is the elliptic function. Note that $\mathrm{E}(0) = \frac{\pi}{2}$, restoring the circular motion result. So, the average speed is proportional to $\frac{1}{\sqrt{a}}$ where $a$ is the semi-major axis.

Fig. 1: the semi-major axis vs. average orbital speed for solar system planets in linear coordinates

Fig. 2: the semi-major axis vs. average orbital speed for solar system planets in logarithmic coordinates

The best fit of the slope gives 29.779763 km/s/AU, which is about the earth average orbital speed. Using the data of solar mass and gravitational constant, the average eccentricity is about 0.0195386. This is the absolute value.