Thursday, December 17, 2015

Calculating Orbits: The Ellipse

The Ellipse

The ellipse is defined in the following way.  Let $F_1$ and $F_2$ be two distinct points on the plane.  Each will be referred to as a focus and together they will be referred to as foci (fo-sai).  An ellipse is a set of points $P$ such that the sum of the distance from $P$ to $F_1$ and the distance from $P$ to $F_2$ is a constant particular to the ellipse.   The language become less cumbersome when we introduce a coordinate system, so we shall.
Let the $x$-axis run through the foci so that the point $x=0$ is the midpoint between the foci. Choose orientation so that $F_1=(c,0)$ and $F_2=(-c,0)$.  Let $H$ be a point on the positive $x$-axis that is on the ellipse, say $H=(a,0)$. Note that the distance from $H$ to $F_1$ is $a-c$ and from $H$ to $F_2$ is $a+c$.  The sum of these two distances is $2a$. Note that $a$ is the length of the segment from the origin $O$, which is the center of the ellipse, and $H$.  This is called the semimajor axis of the ellipse.
Now let $P=(x,y)$ be a point on the ellipse; let $D_1$ be the distance from $P$ to $F_1$; let $D_2$ be the distance from $P$ to $F_2$.  Then
Now $D_1+D_2=2a$, so $D_1=2a-D_2$.  We can square both sides and rearrange so as to determine
As this difference is also
we can see that
This can be rearranged to
and squaring both sides will yield
which can be reduced and rearranged to
As $a^2-c^2>0$, we can choose $b>0$ so that $b^2=a^2-c^2$.  We now can define that important quantity $e$, referred to by the name eccentricity, by
For convenience sake, let us also quickly note that

Our aim now to put this into an appropriate polar form. In our application of this, we will want the sun to be at the focus $F_1$.  Consequently, we will perform a translation of the plane along the $x$-axis so that it is.  The foci $F_1$ and $F_2$ will have coordinates $(0,0)$ and  and $(-2c,0)$, respectively, while the center of the ellipse will be at $(-c,0)$. Taking the equation we obtained for the ellipse and replacing $a^2-c^2$ with $b^2$  we have
We now multiply both sides by $a^2b^2$, expand (x+c)^2, and distribute to obtain
In the coefficient for $x^2$, replace $b^2$ with $a^2-c^2$, carry out the multiplication, and rearrange the equation to be
We are aiming to put this into polar coordinates, so we replace $x^2+y^2$ with $r^2$.  A little algebra will allow us to replace $a^2b^2-b^2c^2$ with $b^4$. Therefore our equation becomes
That is to say
By taking the square root of both sides, we get that either $ar=cx-b^2$ or that $ar=b^2-cx$; that is $ar-cx=-b^2$ or $ar+cx=b^2$.  Let us now substitute $rcos (\theta)$ for $x$, $ae$ for $c$, and $a^2(1-e^2)$ for $b^2$.  In that case, the first alternative will give us
where the LHS is always positive and the RHS is always negative.  However, the second alternative will yield
and so

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