Worked examples — Kepler's connection — orbits are ellipses (motivation)

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The scenario matrix

Every Kepler-orbit problem reduces to juggling four numbers: the semi-major axis $a$, the eccentricity $e$, the two extreme distances $r_{\min }$ (perihelion) and $r_{\max }$ (aphelion), plus the period $T$. The relations are:

$r_{\min }=a(1-e),r_{\max }=a(1+e),a=\frac{1}{2}(r_{\min }+r_{\max }),e=\frac{r_{\max }-r_{\min }}{r_{\max }+r_{\min }},T^2\propto a^3$

Below, each cell is a distinct situation. The examples that follow are tagged by cell.

Cell	Situation	Why it is its own case	Example
A	$e=0$ (circle)	limiting case: focus = centre, $r_{\min }=r_{\max }$	Ex 1
B	small $e$ (near-circle)	subtraction of tiny fraction; sign traps	Ex 2
C	large $e$ (elongated)	focus far off-centre; big spread	Ex 3
D	$e\to 1$ (parabolic limit)	orbit no longer closes — degenerate	Ex 4
E	given the two extremes → recover $a,e$	reverse direction of the formulas	Ex 5
F	period law $T^2\propto a^3$ (forward)	predict $T$ from $a$	Ex 6
G	period law (backward)	recover $a$ from $T$	Ex 7
H	word problem (real satellite)	translate English → the four numbers	Ex 8
I	exam twist (mix + Law 2 speed)	combine geometry with equal-areas	Ex 9

We now hit each cell.

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The figure above shows the same orbit skeleton every example uses: centre $C$, focus $F$ (the Sun), the perihelion end (nearest, distance $r_{\min }$) and the aphelion end (farthest, distance $r_{\max }$). Keep glancing back at it — every symbol below lives on this picture.

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Cell A — the circle ($e=0$)

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Cell B — near-circular (small $e$)

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Cell C — highly elongated (large $e$)

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Cell D — the degenerate limit ($e\to 1$)

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Cell E — reverse: extremes → $a$ and $e$

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Cell F — period law forward ($a\to T$)

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Cell G — period law backward ($T\to a$)

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Cell H — the word problem

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Cell I — the exam twist (geometry + Law 2 speed)

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Active recall

Recall Cover and answer

• In Cell A ($e=0$), what happens to $r_{\min }$ and $r_{\max }$? ::: They become equal to $a$ — a circle.
• As $e\to 1$ with $r_{\min }$ fixed, what happens to $a$? ::: It blows up to infinity — the ellipse degenerates to a parabola.
• How do you get $a$ and $e$ from two altitudes above a planet's surface? ::: Add the planet's radius first (focus = centre), then average and take spread/total.
• Speed ratio perihelion:aphelion from extremes? ::: $v_{\text{peri}}/v_{\text{aph}}=r_{\max }/r_{\min }$.
• Recover $a$ from a period $T$ (Earth units)? ::: $a=T^{2/3}$.

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Connections

• 3.4.06 Kepler's connection — orbits are ellipses (motivation) (Hinglish)
• Ellipse — standard equation and focus
• Eccentricity of a conic
• Focus–directrix definition of conics
• Conic sections as slices of a cone
• Parabola — projectile motion connection
• Hyperbola — unbound orbits and escape
• Angular momentum conservation