Orbit determination — Gauss's method, Gibbs method

3.2.18 · Physics › Orbital Mechanics & Astrodynamics

DO methods kyun? Kyunki tumhara data type alag hota hai. Agar tumhare paas pehle se position vectors hain (jaise radar/GPS se), seedha Gibbs pe jaao. Agar sirf telescope se angles hain, toh pehle Gauss chalao taaki position vectors milo, phir unhe Gibbs ko do.

Part 1 — Physics jo dono methods use karte hain

YE DO kyun? $h$ plane fix karta hai; $e$ us plane ke andar shape aur orientation fix karta hai. Dono milke saari geometry contain karte hain — Gibbs basically "teen points se $e$ aur $h$ reconstruct karo" hai.

Part 2 — Gibbs's Method (positions → velocity)

Step 1 — Coplanarity check (pehle kyun?)

Teeno ek hi plane mein hone chahiye, warna koi single orbit exist nahi karega. $\overset{u}{^}_{r 1} \cdot (\overset{u}{^}_{r 2} \times \overset{u}{^}_{r 3}) \approx 0 (should be < 1^{\circ} off) .$ Ye step kyun? $r_{2} \times r_{3}$ , 2 aur 3 ki plane ka normal hai; $r_{1}$ se dot karne se check hota hai ki 1 bhi usi mein hai ya nahi.

Step 2 — Teen helper vectors banao

Kyunki conic hai $r = \frac{p}{1 + e cos θ}$ , dikhaya ja sakta hai ki point 2 pe velocity in vectors ke through express hoti hai: $N = r_{1} (r_{2} \times r_{3}) + r_{2} (r_{3} \times r_{1}) + r_{3} (r_{1} \times r_{2})$ $D = (r_{1} \times r_{2}) + (r_{2} \times r_{3}) + (r_{3} \times r_{1})$ $S = r_{1} (r_{2} - r_{3}) + r_{2} (r_{3} - r_{1}) + r_{3} (r_{1} - r_{2})$ jahaan $r_{i} = ∣ r_{i} ∣$ .

Ye combinations kyun? Ye is tarah se engineer ki gayi hain ki $N / D$ scale carry kare ( $p$ , angular momentum) aur $S$ perigee ki direction carry kare ( $e$ ). Derivation sketch: har $r_{i}$ ko orbit equation se likho aur unknown true anomalies eliminate karo — jo bacha hua symmetric sums hain woh $N, D, S$ mein collapse ho jaate hain.

Step 3 — Velocity formula

$v_{2} = \frac{μ}{N D} (\frac{D \times r _{2}}{r _{2}} + S)$ jahaan $N = ∣ N ∣, D = ∣ D ∣$ .

Ab tumhare paas $(r_{2}, v_{2})$ hai → classical elements mein convert karo. Ho gaya.

Figure — Orbit determination — Gauss's method, Gibbs method

Part 3 — Gauss's Method (teen angles → teen positions)

Key idea A — sector/triangle f & g series

Do nearby times ke beech, $r$ ko Lagrange coefficients se propagate karo: $r_{i} = f_{i} r_{2} + g_{i} \dot{r}_{2}, f_{i} \approx 1 - \frac{μ}{2 r _{2}^{3}} τ_{i}^{2}, g_{i} \approx τ_{i} - \frac{μ}{6 r _{2}^{3}} τ_{i}^{3}$ jahaan $τ_{i} = t_{i} - t_{2}$ . Series kyun? Abhi orbit nahi pata, toh hum $r (t)$ ko middle observation ke around Taylor-expand karte hain $\ddot{r} = - μ r / r^{3}$ use karke. Short arc ⇒ kuch terms kaafi hain.

Key idea B — $ρ_{2}$ solve karo

Coplanarity + f,g relations milke $r_{2}$ mein ek scalar 8th-degree polynomial dete hain: $r_{2}^{8} + a r_{2}^{6} + b r_{2}^{3} + c = 0$ Physically valid positive real root $r_{2}$ ke liye (numerically) solve karo. Back-substitute karke saare $ρ_{i}$ milo, aur hence saare $r_{i}$ .

Step 4 — Gibbs ko hand off karo (ya improve karo)

Teen $r_{i}$ pata hone ke baad, $v_{2}$ ke liye Gibbs chalao. Phir iterate karo: behtar orbit use karke f,g coefficients sharpen karo (ab exact wale, truncated series nahi) aur tab tak re-solve karo jab tak $ρ_{2}$ converge na ho jaaye.

Worked Example — Gibbs (numbers)

Diya hua (Earth, $μ = 398600 km^{3} / s^{2}$ ): $r_{1} = (- 294.32, 4265.1, 5986.7)$ , $r_{2} = (- 1365.5, 3637.6, 6346.8)$ , $r_{3} = (- 2940.3, 2473.7, 6555.8)$ km.

Step 1 — magnitudes. $r_{1} = 7359.6, r_{2} = 7433.5, r_{3} = 7473.6$ km. Kyun? $N, S$ mein scalar weights ke liye chahiye.

Step 2 — cross products. $r_{1} \times r_{2}$ , etc. compute karo. Kyun? Ye $N, D$ banate hain; aur inke triple product se coplanarity check hoti hai (≈0 ✓).

Step 3 — assemble $N, D, S$ , phir $N = ∣ N ∣, D = ∣ D ∣$ . Kyun? $p = N / D$ orbit scale set karta hai.

Step 4 — velocity. $v_{2} = \frac{μ}{N D} (\frac{D \times r _{2}}{r _{2}} + S) \approx (- 6.218, - 4.014, 1.599) km/s .$ Aakhri kyun? Ye missing 3 numbers hain; $(r_{2}, v_{2})$ = full state.

Common Mistakes

Recall Ek 12-saal ke bachche ko samjhao (Feynman)

Socho ek ball ek bade loop mein ud rahi hai. Tum teen photos lete ho aur mark karte ho ki ball exactly kahan thi har ek mein. Stopwatch ke bina bhi, teen dots poora loop trace karne aur yeh pata karne ke liye kaafi hain ki ball middle photo mein kitni tez ja rahi thi — woh hai Gibbs. Lekin agar dots ki jagah tumne sirf woh direction likhi jis taraf tum dekh rahe the (ball kitni door thi nahi), toh pehle tum clever ban ke distances guess karte ho — woh hai Gauss — aur phir woh dots Gibbs ko dete ho.

Flashcards

Gibbs method input data kya hai?

Teen position vectors

r_{1}, r_{2}, r_{3}

(coplanar), koi times nahi chahiye.

Gauss method input data kya hai?

Teen angle-only observations (time,

α

δ

) aur observer positions

R_{i}

Teen Gibbs helper vectors kaunse hain?

N = \sum r_{i} (r_{j} \times r_{k})

D = \sum (r_{i} \times r_{j})

S = \sum r_{i} (r_{j} - r_{k})

(cyclic).

Gibbs velocity formula kya hai?

v_{2} = μ / (N D) (D \times r_{2} / r_{2} + S)

Gibbs mein semi-latus rectum kaise milta hai?

p = N / D

, phir

∣ h ∣ = μ p

Gibbs ke liye coplanarity test kya hai?

\overset{u}{^}_{r 1} \cdot (\overset{u}{^}_{r 2} \times \overset{u}{^}_{r 3}) \approx 0

Gauss observation se position kaise milti hai?

r_{i} = R_{i} + ρ_{i} \overset{ρ}{^}_{i}

, unknown slant range

ρ_{i}

Gauss ka 8th-degree polynomial kis quantity ke liye solve karta hai?

Middle radius

r_{2}

Gauss polynomial ka kaunsa root valid hai?

Positive real root jo Earth ki radius se upar ho aur positive slant ranges de.

Lagrange f,g short-arc approximations kya hain?

f \approx 1 - \frac{μ}{2 r _{2}^{3}} τ^{2}

g \approx τ - \frac{μ}{6 r _{2}^{3}} τ^{3}

τ = t - t_{2}

Orbit reconstruction ke neeche do conserved vectors kaunse hain?

Angular momentum

h = r \times v

aur eccentricity vector

e

Gauss ko iterate kyun karna padta hai?

f,g series truncated hai; exact coefficients ke saath refine karo jab tak

ρ_{2}

converge na ho.

Telescope angles se orbit tak ka full pipeline kya hai?

Angles → Gauss (

r_{i}

dhundo) → Gibbs (

v_{2}

dhundo) → classical elements.

Connections

Two-body Problem — $\ddot{r} = - μ r / r^{3}$ supply karta hai jo har jagah use hota hai.
Angular Momentum & Eccentricity Vector — woh invariants jo Gibbs rebuild karta hai.
Lagrange Coefficients (f and g) — Gauss ke short-arc propagation ka engine.
Classical Orbital Elements — jahan $(r_{2}, v_{2})$ aakhirkar jaata hai.
Kepler's Equation — orbit pata hone ke baad propagate karne ke liye.
Coordinate Frames (ECI, topocentric) — $\overset{ρ}{^}_{i}$ aur $R_{i}$ kaise bante hain.