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

True. Ek two-body orbit poori tarah ek hi plane mein rehti hai (kyunki $\vec{h}=\vec{r}\times \vec{v}$ constant hai), toh us plane se bahar teen points kabhi ek orbit pe nahi ho sakte — koi bhi velocity choice ise fix nahi kar sakti.

False. Gibbs pure geometry hai: teen coplanar points pehle se hi conic ki shape overdetermine kar dete hain, toh velocity bina kisi clock ke nikal aati hai. Times baad mein tabhi aati hain jab true anomaly ya epoch chahiye ho.

False. Gauss $\vec{r}$ ko $f,g$ series se propagate karta hai jo ${\tau }_i=t_i-t_2$ pe depend karti hai; times ke bina Lagrange coefficients undefined hain aur poora slant-range solve collapse ho jaata hai.

True. Algebra ko physics ki parwah nahi, toh multiple positive roots aa sakte hain; sirf woh root rakhna hai jo $r_2$ ko Earth ki surface ke upar rakhta ho aur jisme saare slant ranges ${\rho }_i>0$ hon.

True. Coplanarity plus ek conic pe hona middle point pe ek single tangent direction aur speed pin kar deta hai, toh ${\vec{v}}_2$ uniquely $\vec{N},\vec{D},\vec{S}$ se determine hoti hai.

False. $\vec{S}$ perigee ki taraf point karta hai (yeh eccentricity vector $\vec{e}$ se tied hai); size/scale $p=N/D$, $\vec{N}$ aur $\vec{D}$ se aata hai.

False. Teen vectors genuine points of the same real orbit hone chahiye; ek fabricated midpoint coplanar-and-conic-consistent nahi hoga, toh $\vec{N},\vec{D},\vec{S}$ koi valid orbit describe nahi karenge.

True. Yahi identity exactly hai jis wajah se Gibbs prefactor $\sqrt{\mu /(ND)}$ mein rearrange hota hai; $p$ semi-latus rectum hai jo helper vectors se recover hota hai.

Error: Gibbs formula middle point pe derive hoti hai. Symmetric sums $\vec{N},\vec{D},\vec{S}$ point 2 ke around centred hain, toh sirf ${\vec{v}}_2$ cleanly nikalti hai; ${\vec{r}}_1$ use karne se galat number aayega.

Error: real measurements mein noise hoti hai, toh ek small nonzero value (rule of thumb $\sim 1^{\circ }$ se kam) acceptable hai. Exact zero demand karna perfectly usable data throw out kar deta hai.

Error: ${\rho }_i=0$ satellite ko observer ke paas rakh deta hai, jo nonsense hai. Gauss instead pehle polynomial solve karke $r_2$ nikalti hai, phir back-substitute karke ${\rho }_i$ get karta hai — yeh kabhi assume nahi karta.

Error: truncated series sirf ek short-arc approximation hai. Tumhe exact Lagrange coefficients ke saath iterate karna hoga jab tak ${\rho }_2$ converge na ho, especially wide-angle arcs ke liye.

Error: $p$ magnitudes $N=|\vec{N}|$ aur $D=|\vec{D}|$ use karta hai. Vectors ko component-wise divide karna meaningless hai; $\vec{N}$ aur $\vec{D}$ actually parallel hain (dono orbit plane ke normal hain).

Error: Gauss teen position vectors output karta hai (slant ranges ke zariye). Velocity nikalne ke liye unhe positions ko Gibbs ko (ya ek Lagrange-coefficient step ko) dena padta hai.

Error: do points aur ek focus conics ki ek one-parameter family chhodte hain; unique orbit pin karne ke liye ek teesra constraint chahiye (Gibbs mein teesra point, ya Gauss mein times).

Cross products orbit plane ke normal vectors banate hain, exactly wahan jahan $\vec{h}$ rehta hai; symmetric cross-product sums unknown true anomalies ko cancel karne dete hain, clean geometry chhodke.

Yeh expansion centre hai. Gauss $\vec{r}(t)$ ko $t_2$ ke around Taylor-expand karta hai, aur Gibbs ke symmetric sums point 2 ke around balanced hain, toh algebra cleanest hai aur errors smallest wahan.

Har observation do angles deti hai lekin ek distance chhupa leta hai. Do observations chaar knowns deti hain six unknowns plus teen ranges ke liye; teesri observation plus coplanarity count close karta hai aur polynomial in $r_2$ form karne deta hai.

$\vec{h}$ orbit plane fix karta hai aur uska size ($\sqrt{\mu p}$); $\vec{e}$ us plane ke andar shape aur orientation fix karta hai. Saath mein yeh har geometric element carry karte hain, toh unhe reconstruct karna orbit ko reconstruct karna hai.

Kepler's time law (equal areas in equal times) orbit ki shape se alag hai. Teen points pehle se shape lock kar dete hain; time sirf tab invoke hoti hai jab tumhe kisi given instant pe orbit pe kahan jaanna ho.

Ek negative slant range satellite ko line of sight ke saath observer ke peeche rakh dega — physically impossible, kyunki tumne use actually saamne dekha tha. Sirf positive ranges ek real sighting se correspond karti hain.

Series ek Taylor truncation hai ${\tau }_i$ mein; bade ${\tau }_i$ ka matlab hai dropped higher-order terms badh jaate hain, toh approximation degrade hoti hai aur iteration (ya exact coefficients) zaruri ho jaata hai.

Collinear points ek vanishing $\vec{D}$ aur $\vec{N}$ dete hain, toh $\sqrt{\mu /(ND)}$ blow up ho jaata hai — geometry degenerate hai aur koi unique orbit extract nahi ho sakta. Tumhe genuinely spread-out points chahiye.

Coefficients ill-conditioned ho jaate hain: chote angle differences bade ranges mein divide ho jaate hain, measurement noise ko wildly amplify karte hue. Bahut short arcs unreliable $r_2$ dete hain chahe math formally run ho.

$f,g$ relations degenerate ho jaate hain kyunki do observations same propagated $\vec{r}$ carry karti hain; ranges ke liye linear system rank lose karta hai aur Gauss points separate nahi kar sakta. Distinct times required hain.

Gibbs tab bhi kaam karta hai — yeh conic equation $r=p/(1+e\cos \theta )$ use karta hai jo saare conics cover karta hai, aur $p=N/D$, $|\vec{h}|=\sqrt{\mu p}$ valid rehte hain. Method ne kabhi bound ellipse assume nahi ki.

Ya toh data bahut zyada noisy hai ya teen sightings same object ki nahi hain. Aage badhne se ek meaningless "orbit" aayegi; tumhe reject karna chahiye ya re-observe karna chahiye instead of Gibbs ko force karne ke.

Woh root unphysical hai (satellite underground hogi) aur discard karna zaroori hai, chahe woh polynomial solve kare. Dusra positive real root choose karo jo surface clear kare.