# Kepler's first law; mathematical way of thinking

I have Kepler's first law of planetary motion:

$$\text{r}=\frac{\text{p}}{1+\epsilon\cos\left(\theta\right)}\tag1.$$

Now, for $\epsilon$ I have:

$$0<\epsilon=\sqrt{1+\frac{2\cdot\eta\cdot\text{h}^2}{\mu^2}}<1\tag,$$

because it is an ellipse.

Now, for $\mu$:

$$\mu=\text{G}\cdot\text{M}\tag3,$$

and for $\eta$:

$$\eta=-\frac{\mu}{2\text{a}}\tag4,$$

where $\text{a}=\frac{\text{r}_\text{min}+\text{r}_\text{max}}{2}$.

Questions:

1. For the Earth's orbit around the sun, which mass ($\text{M}$) should I pick, the mass of the sun or the mass of the earth?
2. What is $\text{h}$ in the equation for $\epsilon$ and how can I find the value for it?
3. What are the values of $\text{r}_\text{min}$ and $\text{r}_\text{max}$?

1. $M$ stands for the mass of the Sun.

2. $h$ is the angular momentum. It is defined as $\vec h=\vec r\times\vec p$, where $\vec p$ is the linear momentum. For central forces (such as the gravitational) it is conserved and perpendicular to the plane of orbit. Therefore if you know the velocity $\vec v$ of the planet as well as its radius vector $\vec r$ at a some instant, you know the angular momentum everywhere in the orbit. You can also compute its magnitude from Kepler's second law if you know the area $A$ and period $T$ of the orbit. They are related as $$A=\frac{hT}{2m},$$ where $m$ is the mass of the planet.

3. $r_{min}$ and $r_{max}$ are minimum and maximum distance from the planet to the Sun. It can be computed to be $$r_{\pm}=\frac{r_0}{1\pm\sqrt{1-\frac{E}{E_0}}},$$ where the sign plus (minus) refers to the minimum (maximum) distance. The constant $E$ is the mechanical energy of the planet. It is a free parameter, you can write the eccentricity of the orbit in terms of it, $$\epsilon=\sqrt{1-\frac{E}{E_0}}.$$ The constant $E_0$ is the smallest energy a orbit can have (circular motion), $$E_0=\frac{-G^2M^2m^3}{2h^2}.$$ Finally the constant $r_0$ is the radius of the circular orbit, $$r_0=\frac{h^2}{GMm^2}.$$

• First of all, thanks for your answer. How can I find $\text{r}_0$, $\text{E}$ and $\text{E}_0$? And what is $\text{r}_0$
– treq
Mar 3 '17 at 13:29
• And what are $\text{A}$ and $\text{T}$ for values?
– treq
Mar 3 '17 at 13:44
• Oke, thanks! But what do you mean by 'free parameter'? Second what is $\text{A}$? Third is $\text{T}$ the number of seconds in a year?