Orbit shape from eccentricity — circle (e=0), ellipse (0 - e - 1), parabola (e=1), hyperbola (e - 1): Question bank

This is a question bank of conceptual traps for the parent topic. No number-crunching here — every item probes whether you understand why one number, the eccentricity $e$ , splits "comes back" from "gone forever". Reveal each answer only after you have committed to a reason of your own.

Symbols used on this page (build the vocabulary first)

Before any trap, here is every symbol you will meet — each in plain words and anchored to the picture below. Nothing is used before it appears here.

Definition The cast of symbols

$r$ — the distance from the central mass to the orbiting body. Look at the length of the arrow from the dot (mass) to the small circle (body) in the figure.
$θ$ (theta) — the polar angle, measured at the central mass, from the closest-approach direction round to where the body currently is. It is the "how far around" angle in the figure.
$e$ — the eccentricity, a pure number (no units) telling how squashed the curve is. $e = 0$ round, bigger $e$ more stretched.
$c$ — the focal distance: how far one focus sits from the center of the ellipse (half the gap between the two foci). Marked as the short segment in the figure.
$a$ — the semi-major axis: half the length of the ellipse's longest diameter (its "size").
$p$ — the semi-latus rectum: the value of $r$ when $θ = 9 0^{\circ}$ (straight up from the focus). It is the orbit's "width scale". Drawn as the vertical bar in the figure.
$μ$ (mu) — the gravitational parameter $μ = GM$ ( $G$ = Newton's constant, $M$ = central mass). It bundles "how strong is the pull".
$h$ — the specific angular momentum $h = r^{2} \dot{θ}$ , i.e. angular momentum per unit mass. It measures the "sideways sweep" of the orbit and stays constant.
$ε$ (epsilon) — the specific orbital energy $ε = E / m$ , i.e. total energy per unit mass. Its sign is what decides bound vs unbound.

Figure — Orbit shape from eccentricity — circle (e=0), ellipse (0 - e - 1), parabola (e=1), hyperbola (e - 1)

The two master formulas that tie these together (from the parent note) are:

$r (θ) = \frac{p}{1 + e c o s θ}, p = \frac{h ^{2}}{μ}, e = 1 + \frac{2 ε h ^{2}}{μ ^{2}} .$

Where the focus-form equation comes from (so nothing below is a black box)

Intuition Why the solution is "constant + cosine", and why that IS a conic

The Binet equation $\frac{d ^{2} u}{d θ ^{2}} + u = \frac{μ}{h ^{2}}$ (with $u = 1/ r$ ) is the same shape of equation as a mass on a spring being pushed by a steady force. Such an equation always has a steady part plus a wobble part: $u (θ) = steady (particular) \frac{μ}{h ^{2}} + wobble (homogeneous) A cos θ .$ Factor the steady part out and rename $A = \frac{μ}{h ^{2}} e$ (this defines $e$ as the relative size of the wobble): $u (θ) = \frac{μ}{h ^{2}} (1 + e cos θ) .$ Now flip $u = 1/ r$ back into $r$ : $r = \frac{1}{u} = \frac{h ^{2} / μ}{1 + e c o s θ} = \frac{p}{1 + e c o s θ}, p = \frac{h ^{2}}{μ} .$ This exact form — " $r$ = (constant) over (1 + e·cos of the angle from the focus)" — is the textbook focus-directrix definition of a conic. That is why the orbit is guaranteed to be one of the four conics: the algebra can produce nothing else.

True or false — justify

Each item is a statement. Decide true/false, then give the mechanism — not just the verdict.

A larger semi-major axis

a

always means a larger eccentricity

e

False.

a

is a size and

e

is a shape ratio (

e = c / a

). A tiny near-circular orbit and a huge near-circular orbit can share

e \approx 0

; a small stretched orbit can have larger

e

than a giant round one.

An orbit with

e = 0.999

is a different kind of curve from one with

e = 1

True.

e < 1

is a closed ellipse (the body returns);

e = 1

is an open parabola (

a \to \infty

, never returns). Crossing

e = 1

is a bound→unbound phase change, not just "thinner".

Two orbits with the same

e

must have the same shape but not the same size.

True. Same

e

means geometrically similar conics (one is a scaled copy of the other);

a

(or

p

) sets the absolute size independently.

The central mass sits at the center of the orbital ellipse.

False. It sits at a focus. Only for the special case

e = 0

(circle) do the two foci merge onto the center.

For any conic orbit,

r

reaches infinity somewhere.

False. For

e < 1

the denominator

1 + e cos θ

never hits zero (since

- 1/ e < - 1

is unreachable), so

r

stays finite everywhere — a bound loop. Only

e \geq 1

lets

r \to \infty

A parabolic comet returns to the Sun after an extremely long period.

False.

e = 1

means

ε = 0

: it reaches infinity with zero speed and never comes back. There is no period — the orbit is open.

Increasing the launch speed at a fixed perihelion always raises

e

True, but only a tangential boost is this clean. A tangential (sideways) burn at perihelion adds speed without changing the pull-line, so it raises both

ε

and

h

; plugging into

e = 1 + 2 ε h^{2} / μ^{2}

pushes

e

up. A radial (along-

r

) burn raises

ε

but changes

h

differently and shifts where perihelion even is — so "at fixed perihelion" no longer holds. The clean monotone rise assumes the tangential case.

A hyperbolic orbit exists for all angles

θ

from

0

36 0^{\circ}

False. Only for

cos θ > - 1/ e

r

positive and physical. Beyond the asymptote angles the formula gives

r < 0

, which the body never occupies — it exists on just one open arc.

Spot the error

Each line contains a flawed claim or reasoning step. Find it and correct it.

"Eccentricity equals the distance between the two foci."

Error:

e

is that distance divided by

a

— a dimensionless ratio

e = c / a

(with

c

= half the focus-to-focus gap), not a length. Doubling both foci separation and the orbit size leaves

e

unchanged.

"Since a circle has no focus offset, the conic formula

r = p / (1 + e cos θ)

can't describe it."

Error: Set

e = 0

and the formula gives

r = p

(a constant, since

p = h^{2} / μ

is fixed by the orbit), exactly a circle. The circle is the smooth

e = 0

limit, not an exception.

"At

e = 1

the ellipse is just maximally stretched, so its semi-major axis is smallest."

Error: At

e \to 1

the semi-major axis

a \to \infty

, not smallest. The orbit opens, it doesn't shrink.

r_{m i n} = a (1 + e)

and

r_{m a x} = a (1 - e)

for an ellipse."

Error: They're swapped. Closest approach uses the minus:

r_{m i n} = a (1 - e)

; farthest uses plus:

r_{m a x} = a (1 + e)

, since

1 - e < 1 + e

"For a hyperbola the mass sits outside the curve, so there's no focus."

Error: A hyperbola still has a focus (see the overlay figure above), and the attracting mass sits at the focus of the branch the body travels. The formula

r = p / (1 + e cos θ)

is a focus-origin equation for all four conics — same origin, only

e

changes.

"Because seasons happen, Earth's orbit must be strongly eccentric."

Error: Earth's

e \approx 0.0167

is nearly circular (

r_{m a x} / r_{m i n} \approx 1.034

). Seasons come mainly from axial tilt, not the tiny distance change.

"A body with total energy

E > 0

can still be captured into a closed orbit."

Error:

E > 0

forces

e > 1

(hyperbola), which is unbound. A closed orbit requires

E < 0

. (Capture would need energy loss, e.g. drag or a third body.)

Why questions

Answer the mechanism, not the label.

Why does the sign of the energy, not its magnitude, decide bound vs unbound?

Because

e = 1 + 2 ε h^{2} / μ^{2}

: the crossover

e = 1

happens exactly at

ε = 0

. Negative

ε

→

e < 1

(trapped in the potential well); positive →

e > 1

(leftover kinetic energy at infinity).

Why is

e = 1

called the "marginal" or "knife-edge" escape orbit?

It's the exact boundary where the body reaches infinity with zero speed (

ε = 0

). Any less energy and it falls back; any more and it escapes with speed to spare.

Why does the orbit equation automatically come out as a conic and not some other curve?

The Binet equation is a simple-harmonic-plus-forcing equation, whose solution is "constant +

A cos θ

". Renaming

A

\frac{μ}{h ^{2}} e

and inverting

u = 1/ r

gives exactly

r = p / (1 + e cos θ)

— the focus-form of a conic (worked out above).

Why does

r \to \infty

require

e \geq 1

r

diverges when

1 + e cos θ = 0

, i.e.

cos θ = - 1/ e

. A cosine can only equal a number in

[- 1, 1]

, so

- 1/ e

is reachable only when

1/ e \leq 1

, i.e.

e \geq 1

Why does angular momentum set the semi-latus rectum

p

but not directly the eccentricity?

p = h^{2} / μ

depends only on

h

(the "width" of the orbit), while

e

mixes both energy and

h

. You can change

e

by changing energy while holding

h

fixed.

Why does a comet on a parabola visit the Sun exactly once?

Parabolas are open curves — after perihelion the comet recedes toward infinity along the single arm and never re-closes, so there is no return pass.

Why can a hyperbolic flyby deflect an object without capturing it?

With

e > 1

the object has

E > 0

; gravity bends its path (deflection) but cannot remove the surplus kinetic energy, so it swings through perihelion and escapes along the outgoing asymptote.

Edge cases

Boundary and degenerate scenarios — these are where intuition usually breaks.

What happens to the two foci as

e \to 0

They merge into a single point at the center (the focal distance

c = e a \to 0

), and the ellipse becomes a circle with constant

r = p

What happens to the semi-major axis

a

e \to 1

from below?

a \to \infty

. The far end of the ellipse runs off to infinity, and the closed loop opens into a parabola.

e

allowed to be negative?

No.

e \geq 0

by definition; it's a magnitude of deviation from roundness. A "negative

e

" just relabels the perihelion direction (a

18 0^{\circ}

rotation of

θ

What is

p

physically, and does it exist for all four shapes?

p

is the semi-latus rectum, the value of

r

θ = 9 0^{\circ}

. It's positive and well-defined for circle, ellipse, parabola, and hyperbola alike, since

p = h^{2} / μ

needs only nonzero

h

What if the angular momentum

h = 0

(a purely radial drop)?

Then

p = h^{2} / μ = 0

and the conic degenerates into a straight line through the focus — a radial fall/escape, not a curved orbit.

For a hyperbola, what does a "negative

r

" at forbidden angles actually mean?

In polar form,

r < 0

at angle

θ

means "point

∣ r ∣

away in the opposite direction, i.e. at

θ + 18 0^{\circ}

" — those points belong to the other branch of the hyperbola. The physical body sits on the near-focus branch only, so it never occupies those angles; they mark space beyond the asymptotes.

At the exact asymptote angle of a hyperbola, what is happening physically?

The denominator

1 + e cos θ \to 0^{+}

, so

r \to \infty

: the body is infinitely far away, moving along the straight-line asymptote — its incoming or outgoing direction at infinity.

e \to \infty

, what does the orbit approach?

The asymptote angle from

cos θ = - 1/ e \to 0

tends to

9 0^{\circ}

, and the orbit flattens toward a nearly straight-line flyby — an almost undeflected pass.

Recall One-line summary of every trap

$e$ is a shape ratio, the mass sits at a focus, the sign of $ε$ (not its size) picks bound vs unbound, and $e = 1$ is a genuine open/closed phase change — not a very thin ellipse.

Connections

Conic Sections — where "same $e$ = similar curve" comes from
Specific Orbital Energy — the $ε$ sign behind every bound/unbound trap
Angular Momentum in Orbits — why $p = h^{2} / μ$ and the $h = 0$ degenerate case
Vis-viva Equation — speed at any $r$ , the energy bookkeeping
Escape Velocity — the $e = 1$ , $ε = 0$ knife-edge
Kepler's First Law — ellipse, Sun at a focus ( $0 \leq e < 1$ )