Looking back at the equation of trajectory

$$ \frac{X^2}{a^2}+\frac{Y^2}{b^2}=1 $$

where $a=r_0/(1-\epsilon^2)$ and $b=r_0/\sqrt{1-\epsilon^2}$

🧠 Remember: $\epsilon$ is the eccentricity $\epsilon^2=1-(b/a)^2$

Consider when $\epsilon \ge 1$

  1. $\epsilon=1$: We get $y^2=r^2_0-2r_0x$ which gives us a parabola where $(X,Y)=(\frac 12 r_0-x,y)$

  2. $\epsilon> 1$: We get the following

    $$ \frac{X^2}{a^2_1}-\frac{Y^2}{b^2_1}=1 $$

    where $a_1=r_0/(\epsilon_1^2-1)$, $b_1=r_0/\sqrt{\epsilon_1^2-1}$ and $\epsilon^2_1=1+(b_1/a_1)^2$

Properties

The equation of trajectory of motion in Newtonian gravity describes conic sections

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/ff68ffc1-7004-4267-9747-60358b6417d1/Conic_sections.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/ff68ffc1-7004-4267-9747-60358b6417d1/Conic_sections.png" width="40px" /> Conic sections: are curves creates by slicing through a cone in various ways

Conic_Sections.svg.png

</aside>

Directrix

Untitled

We define a focus point $F$ at $(f,0)$, a directrix $X=d$ and a series of points $D=(d,y)$ where $y\in \R$.

💡 The conic section with eccentricity $\epsilon$ is generated by defining $P$ which is a point in the conic section to be at position $(x,y)$ such that $|PF|=\epsilon|PD|$ which can be written:

$$ \begin{aligned} (x-f)^2+y^2&=\epsilon^2(x-d)^2 \\ (1-\epsilon^2)x^2+y^2-2(f-\epsilon^2d^2)x&=\epsilon^2d^2-f^2 \end{aligned}

$$

  1. if $f=\epsilon^2 d$ and $\epsilon\ne 1$.

    $$ \frac{x^2}{(\epsilon d)^2}+\frac{y^2}{(\epsilon d)^2(1-\epsilon^2)} $$

    And $d=\pm a/\epsilon$ and $b^2=\pm a^2(a-\epsilon^2)$ from which we can deduce that $f=\pm\epsilon a$

If $\epsilon^2=1-\left ( \frac ba \right )^2$ then it is an ellipse

Untitled

if $a=b$ then circle

if $\epsilon^2=1+\left ( \frac ba \right )^2$ then it is a hyperbolae

Untitled

  1. if $\epsilon=1$ and $f=-d$ we get $y^2=4fx$ which is a parabola

    Screenshot 2023-04-10 152458.png

🗒️ The semi-latus rectum is the value of $y$ when $x=f$ is given by: