Planets and Satellites: Kepler's Laws — Question 1

Question 1

Apsis distances locate the focus and determine the ellipse

A negligible-mass satellite follows an ellipse about a fixed point source. Its minimum and maximum distances from the source are $r_p=3R$ and $r_a=9R$. Use $r_p=a(1-e)$, $r_a=a(1+e)$ and $b=a\sqrt{1-e^2}$.

1. Find semimajor axis $a$, eccentricity $e$ and semiminor axis $b$.

2. Find the center-to-focus distance and explain why averaging distances over time is not how $a$ is defined.

Question 1 - Solution

Apsis distances locate the focus and determine the ellipse

(a) Derivation and calculation

Add and subtract the apsis equations: $$r_p+r_a=2a,\qquad r_a-r_p=2ae.$$ Therefore $$\boxed{a=\frac{3R+9R}{2}=6R},\qquad \boxed{e=\frac{9R-3R}{9R+3R}=\frac12}.$$ $$\boxed{b=a\sqrt{1-e^2}=6R\sqrt{3/4}=3\sqrt3R}.$$

(b) Result and interpretation

The focus lies distance $$\boxed{c=ae=3R}=\sqrt{a^2-b^2}$$ from the ellipse center. Geometrically, $r_p=a-c$ and $r_a=a+c$, so $a=(r_p+r_a)/2$ exactly. A time average instead has the definition $$\langle r\rangle_t=\frac1T\int_0^T r(t)\,dt,$$ which weights each location by residence time.

Physical check

The inferred dimensions satisfy $b^2+c^2=a^2$ and reproduce both given apsis radii.