Kepler's first law; mathematical way of finding the eccentricity We know that Kepler's first law of planetary motion is defined as:
$$\text{r}=\frac{\text{p}}{1+\epsilon\cos\left(\theta\right)}\tag1$$
Now, for $\epsilon$ I have (see wikipedia):
$$0\space<\space\epsilon=\sqrt{1+\frac{2\cdot\text{E}\cdot\text{h}^2}{\mu^2}}\space<1\space\tag2$$
Now, we have also that:
$$\text{E}=-\frac{\text{G}\cdot\left(\text{M}+\text{m}\right)}{2\text{a}},\text{h}=\frac{2\text{A}\text{m}}{\text{T}},\mu=\text{G}\cdot\text{M}\tag3$$
The known constants for the Earth's orbit about the Sun are:
$$\text{G}\approx6.6740831\times10^{-11}\space\text{m}^3\text{kg}^{-1}\text{s}^{-2},\text{M}\approx1.98855\times10^{30}\space\text{kg},$$
$$\text{T}\approx365.25636\cdot24\cdot60\cdot60\space\text{s},\text{m}\approx5.972\times10^{24}\space\text{kg}\tag4$$
Now, at wikipedia I found: $\epsilon\approx0.0167086$, But in order to find that I need to find $\text{A}$ and $\text{a}$ and my question is how can I find those?
 A: Mass, energy, angular momentum? These are not observable quantities. Those quantities that are observable do not provide a complete picture of a body's state. Welcome to the wonderful world of orbit determination!
Until rather recently (the 1950s), the only observations available for determining the orbit of a body in the solar system were the body's angular position in the sky as seen from the Earth. Each such observation provided but two parameters, the azimuth and elevation of the body as measured by an observer on the surface of the Earth. Compare that against the (more than) twelve degrees of freedom involving the Earth's and the observed body's motions about the Sun (or six+ degrees of freedom in the case of the Earth's orbit about the Sun). More than one observation is required.
That one observation did not suffice drove a lot of mathematical development. That each observation wasn't precise drove even more development, starting with Kepler. Newton, Laplace, Lagrange, Gauss, and many others added to this significant body of knowledge.
