Question bank — Orbit determination — Gauss's method, Gibbs method

True. A two-body orbit lives entirely in one plane (because $\vec{h}=\vec{r}\times \vec{v}$ is constant), so three points off that plane can never lie on one orbit — no velocity choice can fix it.

False. Gibbs is pure geometry: three coplanar points already overdetermine the conic's shape, so velocity comes out without any clock. Times only enter later if you want the true anomaly or epoch.

False. Gauss propagates $\vec{r}$ with the $f,g$ series which depend on ${\tau }_i=t_i-t_2$; without the times the Lagrange coefficients are undefined and the whole slant-range solve collapses.

True. Algebra doesn't know about physics, so multiple positive roots can appear; you keep only the one giving $r_2$ above Earth's surface with all slant ranges ${\rho }_i>0$.

True. Coplanarity plus lying on one conic pins a single tangent direction and speed at the middle point, so ${\vec{v}}_2$ is uniquely determined by $\vec{N},\vec{D},\vec{S}$.

False. $\vec{S}$ points toward perigee (it's tied to the eccentricity vector $\vec{e}$); the size/scale $p=N/D$ comes from $\vec{N}$ and $\vec{D}$.

False. The three vectors must be genuine points of the same real orbit; a fabricated midpoint won't be coplanar-and-conic-consistent, so $\vec{N},\vec{D},\vec{S}$ will not describe a valid orbit.

True. This identity is exactly why the Gibbs prefactor rearranges to $\sqrt{\mu /(ND)}$; $p$ is the semi-latus rectum recovered from the helper vectors.

Error: the Gibbs formula is derived at the middle point. The symmetric sums $\vec{N},\vec{D},\vec{S}$ are centred so only ${\vec{v}}_2$ emerges clean; using ${\vec{r}}_1$ gives a wrong number.

Error: real measurements carry noise, so a small nonzero value (rule of thumb under $\sim 1^{\circ }$) is acceptable. Demanding exact zero throws out perfectly usable data.

Error: ${\rho }_i=0$ puts the satellite at the observer, which is nonsense. Gauss instead solves the polynomial for $r_2$ first, then back-substitutes to get the ${\rho }_i$ — it never assumes them.

Error: the truncated series is only a short-arc approximation. You must iterate with the exact Lagrange coefficients until ${\rho }_2$ converges, especially for wide-angle arcs.

Error: $p$ uses the magnitudes $N=|\vec{N}|$ and $D=|\vec{D}|$. Dividing the vectors component-wise is meaningless; $\vec{N}$ and $\vec{D}$ are actually parallel (both normal to the orbit plane).

Error: Gauss outputs the three position vectors (via slant ranges). Getting the velocity still requires handing those positions to Gibbs (or a Lagrange-coefficient step).

Error: two points and a focus leave a one-parameter family of conics; you need a third constraint (a third point in Gibbs, or times in Gauss) to pin the unique orbit.

Cross products build vectors normal to the orbit plane, which is exactly where $\vec{h}$ lives; the symmetric cross-product sums let the unknown true anomalies cancel, leaving clean geometry.

It's the expansion centre. Gauss Taylor-expands $\vec{r}(t)$ about $t_2$, and Gibbs's symmetric sums are balanced around point 2, so the algebra is cleanest and errors smallest there.

Each observation supplies two angles but hides one distance. Two observations give four knowns for six unknowns plus three ranges; the third observation plus coplanarity closes the count and lets the polynomial in $r_2$ be formed.

$\vec{h}$ fixes the orbit plane and its size ($\sqrt{\mu p}$); $\vec{e}$ fixes the shape and orientation inside that plane. Together they carry every geometric element, so reconstructing them is reconstructing the orbit.

Kepler's time law (equal areas in equal times) is separate from the orbit's shape. Three points already lock the shape; you only invoke time when you later want where on the orbit at a given instant.

A negative slant range would place the target behind the observer along the line of sight — physically impossible, since you actually saw it in front. Only positive ranges correspond to a real sighting.

The series is a Taylor truncation in ${\tau }_i$; larger ${\tau }_i$ means the dropped higher-order terms grow, so the approximation degrades and iteration (or exact coefficients) becomes essential.

Collinear points give a vanishing $\vec{D}$ and $\vec{N}$, so $\sqrt{\mu /(ND)}$ blows up — the geometry is degenerate and no unique orbit can be extracted. You need genuinely spread-out points.

The coefficients become ill-conditioned: small angle differences get divided into large ranges, amplifying measurement noise wildly. Very short arcs give unreliable $r_2$ even though the math formally runs.

The $f,g$ relations degenerate because two observations carry the same propagated $\vec{r}$; the linear system for the ranges loses rank and Gauss cannot separate the points. Distinct times are required.

Gibbs still works — it uses the conic equation $r=p/(1+e\cos \theta )$ which covers all conics, and $p=N/D$, $|\vec{h}|=\sqrt{\mu p}$ stay valid. The method never assumed a bound ellipse.

Either the data is badly noisy or the three sightings aren't of the same object. Proceeding gives a meaningless "orbit"; you should reject or re-observe rather than force Gibbs through.

That root is unphysical (the satellite would be underground) and must be discarded, even though it solves the polynomial. Pick the other positive real root that clears the surface.