3.2.3 · D5Orbital Mechanics & Astrodynamics
Question bank — Orbit equation r = p - (1 + e·cos θ) — derivation from equations of motion
Before we hunt traps, look at the picture every question secretly depends on. This one diagram names every symbol used on the page — pin it in your mind.
True or false — justify
Every answer begins with True or False and then why — a bare verdict earns nothing.
TF1 — In , the angle is measured from the center of the ellipse.
False. is the true anomaly, measured from perihelion as seen from the focus (the Sun), not the geometric center; the angle measured at the center is the eccentric anomaly, a different angle entirely.
TF2 — Setting turns the equation into , a constant, i.e. a circle.
True. With the term vanishes, so for all — a fixed radius is precisely a circle, and here too.
TF3 — For a bound ellipse, is largest when is largest.
False. is a fraction with on the bottom; is largest when the denominator is smallest, i.e. when at (aphelion).
TF4 — The semi-latus rectum equals the semi-major axis for every orbit.
False. In general ; they only coincide when (circle). For any real ellipse .
TF5 — Specific angular momentum is conserved because gravity does no work on the planet.
False. is conserved because gravity is a central force (always along the radial direction ), giving zero torque about the focus — that is a statement about direction, not about work.
TF6 — A parabolic orbit () has a finite aphelion distance.
False. At the denominator as , so ; the body escapes and never returns, so there is no aphelion.
TF7 — Two orbits with the same but different pass through the same point at .
True. At , so regardless of ; all conics sharing a cross the semi-latus-rectum point together.
TF8 — Increasing while holding fixed makes the perihelion move closer to the focus.
True. shrinks as grows, so a more eccentric orbit at fixed dives nearer the Sun at closest approach.
TF9 — The orbit equation was derived assuming the two masses are equal.
False. It assumes one mass dominates so ; it also holds exactly for the relative coordinate of any two-body system with .
Spot the error
Each line contains a plausible-looking but flawed statement. The reveal names the flaw.
SE1 — "Since repeats every , the orbit equation gives two different radii for and , so the orbit isn't closed."
The error: repeating is exactly what makes the orbit retrace itself each revolution; different giving different is the oval shape, not a break in closure. A bound orbit closes precisely because is periodic in over .
SE2 — "Perihelion is where the planet moves slowest, since it's turning the sharp corner."
The error: perihelion is where the planet moves fastest. By , small forces large ; the corner is sharp because the planet is whipping through it, not crawling.
SE3 — "Because , a heavier orbiting satellite follows a wider orbit for the same launch conditions."
The error: since depends only on the central body, dividing by removes the satellite's own mass. Same launch conditions ⇒ identical path regardless of the small mass.
SE4 — "Writing instead of describes a completely different orbit."
The error: it's the same orbit, just with perihelion relabelled to instead of . The sign only rotates the angle origin; the geometric shape is untouched.
SE5 — ", so a faster-spinning orbit (bigger ) is always a smaller orbit."
The error: bigger increases (it's in the numerator), making the orbit larger, not smaller. More angular momentum flings the body wider.
SE6 — "The substitution is just algebraic bookkeeping with no physical reason."
The error: it's chosen because plotting against turns the orbit's curve into a plain cosine wave — the force becomes and the ODE becomes the linear oscillator , whose solution is . The whole derivation hinges on that tool choice.
SE7 — "At the planet is at the end of the minor axis of the ellipse."
The error: at we are at measured from the focus, which is the end of the latus rectum, not the minor-axis tip. That tip is measured from the center and its distance there equals the semi-minor axis — a different point and a different length.
Why questions
WQ1 — Why does the derivation switch from time derivatives to derivatives?
Because we want the shape , not the timing , and increases monotonically over , so it's a clean, single-valued independent variable — while goes back and forth in time.
WQ2 — Why is the driving term a constant in the oscillator equation?
Both (central-body property) and (conserved specific angular momentum) are fixed once the orbit is set, so the right-hand side never varies with , giving a simple constant-forced harmonic oscillator.
WQ3 — Why does conserved angular momentum immediately imply a flat (planar) orbit?
is perpendicular to both position and velocity; if never changes direction, position and velocity are forever confined to the single plane perpendicular to .
WQ4 — Why does the eccentricity come out as a free constant rather than being fixed by the physics?
When we solve the oscillator , the wave part carries an amplitude constant (the height of the cosine, set by initial conditions); bundling it as makes free. The equations permit any shape, and it is where you launch and how fast that selects circle, ellipse, parabola or hyperbola.
WQ5 — Why does exactly at regardless of ?
Because kills the entire -dependent term, leaving ; this is why is literally the distance "sideways" from perihelion for any conic.
WQ6 — Why can the same formula describe both a captured planet and an escaping comet?
The sign of in the denominator changes behaviour: for the denominator stays positive for all (closed ellipse), while for it can reach zero, sending (open escape) — one equation, all conics.
Edge cases
EC1 — What does the formula predict as for the aphelion distance ?
It blows up: . The ellipse stretches without bound and, at exactly , becomes an open parabola with no far turning point.
EC2 — What happens to for at the angle where ?
The denominator , so : these two angles are the asymptote directions of the hyperbola, beyond which the formula would give unphysical negative (that branch is not travelled).
EC3 — Is (aphelion) meaningful for a hyperbolic orbit?
No. For , gives negative , which is unphysical; the body never reaches because it has already escaped along an asymptote.
EC4 — What is the orbit if exactly?
A degenerate radial "orbit" — zero specific angular momentum means , so the body falls straight in (or straight out) along a line; the polar formula collapses since .
EC5 — In the circle limit , where is perihelion?
Everywhere and nowhere — with constant there is no unique closest point, so the choice of becomes arbitrary; perihelion is only well-defined once .
EC6 — For a bound ellipse, can the denominator ever reach zero?
No. Since , the smallest value is at ; the denominator stays strictly positive, so is always finite — exactly why the orbit is closed.
Active recall
Recall One-line reflexes (cover the answers!)
- is measured from what point? → the focus (the Sun), not the center.
- Which of the three anomalies appears in the orbit equation? → the true anomaly (measured at the focus).
- Largest occurs at which for an ellipse? → (aphelion), where the denominator is smallest.
- Does the small orbiting mass affect the path shape? → No; depends only on the central body.
- What makes possible? → denominator , needing .
See also: Kepler's Laws, Conic Sections, Specific Angular Momentum h, Eccentricity and Orbital Energy, Central Force Motion, Vis-viva Equation.