Planets and Satellites: Kepler's Laws — Question 8

Question 8

An algebraically valid ellipse can intersect the central body

A proposed exterior satellite orbit around a spherical planet of radius $R$ has semimajor axis $a=2R$ and eccentricity $e=.60$. Ignore atmospheric drag but require the trajectory not enter the solid planet.

1. Test whether the proposed orbit is physically admissible.

2. Find the largest eccentricity allowed at fixed $a=2R$, and explain the grazing equality case.

Question 8 - Solution

An algebraically valid ellipse can intersect the central body

(a) Derivation and calculation

The closest source distance is $$r_p=a(1-e)=2R(1-.60)=\boxed{.8R}.$$ Since $$r_p-R=-.2R<0,$$ the proposed trajectory penetrates the solid planet and is not an admissible exterior satellite orbit.

(b) Result and interpretation

At fixed $a=2R$, impose clearance: $$a(1-e)\ge R\quad\Longrightarrow\quad e\le1-\frac Ra.$$ $$\boxed{e\le.50}.$$ Equality gives $r_p=R$, grazing the ideal surface. Strictly positive clearance requires $e<.50$; period alone does not constrain this geometry.

Physical check

The forbidden orbit is a valid point-source ellipse mathematically, but its exterior-field and collision assumptions fail inside the planet.