Apoapsis and Periapsis of an Ellipse

ellrad.html

Exploration

Show that the greatest, apoapsis, and least, periapsis, radial distance of a point on an ellipse as measured from a focus point is given by r=a(1+e) and r=a(1-e), respectively, where e is the eccentricity and a is the length of the semi-major axis of the ellipse.

Approach

Create a standard ellipse centered at the origin and create an expression representing the distance from a focus point to a point on the ellipse (in terms of the eccentricity and semi-major axis). Find the parameter value on the ellipse where the distance is a minimum or a maximum.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

The eccentricity is given by ; therefore, . Find the focus points and use this substitution.

Clear[a,b,e];
e1=Ellipse2D[{0,0},a,b,0];
fpts1=Foci2D[e1] /. Sqrt[a^2-b^2]->a*e

Find a general point on the ellipse in terms parameter t.

Clear[t];
pt=Point2D[e1[t]]

Solve for b in terms of a and e, where b>0.

ans=Simplify[Solve[Sqrt[a^2-b^2]/a==e,b],Assumptions->{a>0}]

Determine the distance, d, from the point to the focus.

d=Distance2D[fpts1[[1]],pt] /. ans[[2]] //Simplify

The maximum value of d occurs when cos (θ)=-1, or θ=π; the minimum value of d occurs when cos (θ)=1, or θ=0.

{apoapsis,periapsis}=Map[(d /. #)&,{Cos[t]->-1,Cos[t]->1}]

Since e<1, the sign must be reversed outside the radical.

Clear[E1];
{apoapsis,periapsis} /.
   Sqrt[a^2*E1_^2]->a*(-E1) //Factor