Orbit determination — Gauss's method, Gibbs method: Foundations

3.2.18 · D1 · Physics › Orbital Mechanics & Astrodynamics › Orbit determination — Gauss's method, Gibbs method

Yeh page assume karta hai ki tumne kuch nahi dekha. Hum har symbol ko brick by brick banate hain, hamesha ek picture ke saath, aur ek brick rakhne ke baad hi hum agle ko uske upar stack karne dete hain.

1 — Ek vector: ek arrow jiske paas length aur direction hoti hai

Figure s01 dekho. Earth ka centre ek dot par baitha hai jise hum origin kehte hain. Arrow wahan tak pahunchta hai jahan satellite abhi hai. Woh arrow hi satellite ki position hai.

Hume arrow kyun chahiye aur sirf ek plain number nahi? Kyunki space mein "kahan" ek number nahi hota — usse teen chahiye (left–right, forward–back, up–down). Ek arrow teeno ko ek object mein bundle karta hai. Hum unhe teen numbers ke ek triple ke roop mein likhte hain: $r = (r_{x}, r_{y}, r_{z}) .$ $r_{x}, r_{y}, r_{z}$ mein se har ek ek component hai — arrow teen fixed directions mein se ek ke saath kitna dur tak pahunchta hai. Woh teen fixed directions ek coordinate frame ki axes hain; Earth ke around orbits ke liye hum asmaan se anchored ek frame use karte hain (ECI frame).

2 — Arrows ko add aur subtract karna: vector addition (tip-to-tail)

Figure s02 dekho. Yeh "tip-to-tail" hop poore topic ka workhorse hai: har ek satellite position arrows ko chain karke banai jaati hai — known hops aur unknown hops — taaki tum likh sako ki satellite kahan hai chahे journey ka ek leg abhi bhi unknown ho.

Subtraction abhi kyun mention karein? Kyunki baad ki orbit geometry ko lagaataar do points ke beech ka arrow chahiye — jaise ek observation se doosre tak position kaise badli — aur woh "beech wala" arrow exactly subtraction $r_{i} - r_{j}$ hai.

3 — Ek arrow ko scale karna: scalar multiplication

Toh "direction $\overset{ρ}{^}$ ke along $ρ$ distance travel karna" $ρ \overset{ρ}{^}$ likha jaata hai: length-1 direction arrow lo aur use length $ρ$ tak stretch karo. Hum exactly yahi §8 mein satellite ki position rakhne ke liye use karenge. Scaling aur addition milakar wo sab kuch hai jo tumhe kuch reference arrows se koi bhi arrow banane ke liye chahiye — woh fact Lagrange blend $r_{i} = f_{i} r_{2} + g_{i} \dot{r}_{2}$ ko §10 mein underlie karta hai.

4 — Vector ki length: $r = ∣ r ∣$

Triple $(r_{x}, r_{y}, r_{z})$ ke liye length 3D mein Pythagoras se aati hai: $r = ∣ r ∣ = r_{x}^{2} + r_{y}^{2} + r_{z}^{2} .$

Worked example mein parent $r_{1} = 7360.5$ km seedha is rule se compute karta hai (teen numbers ko upar wale formula mein plug karo aur tum wahan pahunch jaoge). Chhota subscript ( $1$ , $2$ , $3$ ) sirf kaun si observation label karta hai; arrow aur uski length har ek ke liye same tarah kaam karti hai.

5 — Velocity $v$ aur upar ke dots ( $\dot{r}$ , $\ddot{r}$ )

Figure s03 dekho. Satellite apne curved path par baitha hai. Green arrow $r$ Earth ke centre se use reach karta hai; blue arrow $v$ satellite se path ke along (usse tangent hokar) nikalti hai. Pink arrow $\ddot{r}$ wapas Earth ki taraf point karta hai — gravity hamesha inward pull karti hai.

Acceleration woh hai jo Earth ki taraf point karta hai kyun? Kyunki yahan sirf gravity ek hi force hai, aur gravity mass ki taraf pull karti hai. Newton ka gravity ka law kehta hai ki pull ki strength $μ / r^{2}$ hai (jitna dur jaao utna weaker — famous inverse-square), aur uski direction seedha wapas Earth ki taraf hai, yani $- \overset{r}{^}$ ke along jahan $\overset{r}{^} = r / r$ outward pointing unit arrow hai. Strength ko direction se multiply karo: $\ddot{r} = - \frac{μ}{r ^{2}} \overset{r}{^} = - \frac{μ}{r ^{2}} \cdot \frac{r}{r} = - \frac{μ}{r ^{3}} r .$

6 — Gravity number $μ = GM$

$G$ aur $M$ ko ek symbol mein kyun bundle karein? Kyunki har ek orbit formula mein woh sirf ek product ke roop mein saath aate hain. Nature tumhe orbit se $G$ aur $M$ ko alag feel nahi karne deta — sirf unka combined pull. Us product ko $μ$ naam dena ink bachata hai aur physics ko reflect karta hai. Yeh ek akela knob hai jo sab kuch ka pace set karta hai: bada $μ$ ⇒ faster orbits, tighter curves.

7 — Cross product $a \times b$

Figure s04 dekho. Do arrows $a$ aur $b$ board par flat lie karte hain; unka cross product $a \times b$ seedha board se bahar upar uthta hai (right-hand rule), sheet se right angle par. Shaded parallelogram jo woh span karte hain uska ek area hai, aur woh area exactly length $∣ a \times b ∣$ hai: do arrows ko apart stretch karo aur parallelogram (aur cross product) bade ho jaate hain; unhe line up karo aur parallelogram kuch nahi ho jaata.

Is topic mein cross products par itna lean kyun? Kyunki hume lagaataar yeh answer karna hai ki "orbit plane kis taraf face karta hai?" Ek plane ko us arrow se naam diya jaata hai jo usse bahar nikalta hai (uska normal). Cross product woh machine hai jo tumhe woh normal deta hai. Do facts jo hum reuse karenge:

$a \times b$ reverse ho jaata hai agar tum order swap karo: $b \times a = - (a \times b)$ .
Agar $a$ aur $b$ same direction mein point karte hain, parallelogram flat hai, area $= 0$ , toh cross product zero arrow hai.

Yeh angular-momentum vector $h = r \times v$ aur Gibbs ke helper vectors $N, D, S$ mein har ek $r_{i} \times r_{j}$ term ka engine hai.

8 — Dot product $a \cdot b$ aur coplanarity check

Figure s05 dekho. Dot product $b$ ki length times us shadow ki length ke barabar hai jo $a$ $b$ ke along dalta hai (uska projection). Jab $a$ $b$ ki same direction mein jhukta hai toh shadow lamba aur positive hota hai; $a$ ko right angle par ghuma do aur shadow zero ho jaata hai; $9 0^{\circ}$ se aage dhakelo aur shadow doosri taraf girega, ek negative number dega.

Hume parent note mein ek kaam ke liye dot product chahiye: coplanarity check. Teen unit position vectors ke liye $\overset{r}{^}_{1}, \overset{r}{^}_{2}, \overset{r}{^}_{3}$ likho (har ek $\overset{r}{^}_{i} = r_{i} / r_{i}$ ). Teen arrows ek flat plane mein lie karte hain sirf tab jab teesre ki doosre do ke plane ke upar zero height ho. Recipe: $\overset{r}{^}_{1} \cdot (\overset{r}{^}_{2} \times \overset{r}{^}_{3}) \approx 0.$ Ise inside-out padho. Pehle $\overset{r}{^}_{2} \times \overset{r}{^}_{3}$ woh arrow deta hai jo vectors 2 aur 3 ke plane se bahar nikalti hai (§7). Phir $\overset{r}{^}_{1}$ ke saath dot karna poochhta hai "kya vector 1 ki us poking-out direction ke along koi height hai?" Agar jawaab zero hai, vector 1 same plane mein flat lie karta hai — sab teen coplanar hain, aur ek single orbit unhe thread kar sakta hai.

9 — Line of sight $\overset{ρ}{^}$ , slant range $ρ$ , observer $R$

Ab jo scalar multiplication (§3) aur vector addition (§2) haath mein hain, hum satellite position ko honestly bana sakte hain. Length-1 sight-line $\overset{ρ}{^}_{i}$ ko true distance tak scale karke stretch karo: $ρ_{i} \overset{ρ}{^}_{i}$ . Phir Earth ke centre se telescope tak tip-to-tail hop karo aur us stretched sight-line ke along continue karo: $r_{i} = R_{i} + ρ_{i} \overset{ρ}{^}_{i} .$ Figure s06 dekho: known hop $R_{i}$ (blue) observer tak, phir stretched sight-line $ρ_{i} \overset{ρ}{^}_{i}$ (pink) yellow satellite tak bahar, total arrow $r_{i}$ (yellow) deta hua. Gauss ka poora kaam missing distances $ρ_{i}$ nikalna hai. Ek baar nikal gaye, har ek $r_{i}$ known hai aur Gibbs aage le leta hai.

10 — Loop par ek spot ko naam dene wale angles: $α, δ, θ, e, p, h$

Pehle, ek word jo har conic ko chahiye.

Figure s07 dekho: ellipse Earth ek focus par, perigee (closest point) mark kiya gaya, true anomaly $θ$ perigee se satellite tak khulta hua, semi-major axis $a$ (long way across ka aadha), aur $p$ focus se seedha upar drawn.

Yeh sabki sabki ek equation mein milte hain jo har orbit define karti hai, conic (orbit) equation: $r = \frac{p}{1 + e c o s θ} .$ Padho: focus se distance $r$ sirf is par depend karta hai ki tum loop par kahan ho ( $θ$ ), $p$ se scaled aur $e$ se squashed. Jab $θ = 0$ (perigee) bottom sabse bada hai, toh $r$ sabse chhhota hai — closest approach, exactly jaisa "perigee" hona chahiye.

In numbers se orbit ke naam-tag tak ka bridge classical orbital elements ka set hai, aur time ko position se pin karna Kepler's equation use karta hai.

11 — Lagrange coefficients $f$ aur $g$ , aur time-gaps $τ_{i}$

Aisa blend exist kyun karta hai? Kyunki orbit ek flat plane mein rehti hai aur $r_{2}$ , $v_{2}$ do arrows hain jo us plane ko span karte hain — plane mein har ek aur point unhi do ka kuch scaled-and-added combination hai (§2 addition aur §3 scaling, exactly). Toh $f$ measure karta hai "starting position ka kitna" aur $g$ measure karta hai "starting velocity ka kitna" tumhe chahiye.

Formula Taylor's idea se short-arc series banana

Ek Taylor series kehti hai: kisi cheez ko predict karne ke liye ki woh short time $τ$ baad kahan jaayegi, abhi se start karo aur corrections add karo — pehle velocity times $τ$ , phir acceleration times $\frac{1}{2} τ^{2}$ , phir acceleration ke change ki rate times $\frac{1}{6} τ^{3}$ , aur aage bhi ( $\frac{1}{2}, \frac{1}{6}$ $\frac{1}{2 !}, \frac{1}{3 !}$ se aate hain). Middle time ke baare mein position expand karo: $r (t) = r_{2} + \dot{r}_{2} τ + \frac{1}{2} \ddot{r}_{2} τ^{2} + \frac{1}{6} r ..._{2} τ^{3} + \dots$ Ab motion ka law $\ddot{r}_{2} = - \frac{μ}{r _{2}^{3}} r_{2}$ substitute karo. $τ^{2}$ term ban jaata hai $- \frac{μ}{2 r _{2}^{3}} τ^{2} r_{2}$ — yeh $r_{2}$ ka ek multiple hai, toh yeh $f$ recipe mein jaata hai. Law ko ek baar differentiate karne se $r ...$ term mein $\dot{r}_{2}$ ke proportional ek piece aata hai, jo $- \frac{μ}{6 r _{2}^{3}} τ^{3}$ ko $g$ recipe mein daal deta hai. $r_{2}$ -parts aur $\dot{r}_{2}$ -parts collect karke: $f_{i} \approx 1 - \frac{μ}{2 r _{2}^{3}} τ_{i}^{2}, g_{i} \approx τ_{i} - \frac{μ}{6 r _{2}^{3}} τ_{i}^{3} .$ Leading $f \approx 1$ aur $g \approx τ$ sirf "jagah par raho plus constant velocity par drift karo" hain; $μ$ -terms gravity ke path ko bend karne wale hain. Short arc ⇒ chhhota $τ$ ⇒ yeh kuch terms kaafi hain. Yeh Gauss ka doosra key idea hai, aur yeh $r_{2}$ ke liye 8th-degree polynomial mein collapse ho jaata hai.

12 — Har ek foundation topic ko kaise feed karta hai

Neeche ke do boxes un do methods ke naam batate hain jo parent note inhi foundations ke upar build karta hai; yeh page woh arrows, lengths, products aur series supply karta hai jo woh consume karte hain.

Equipment checklist

Khud ko test karo — reveal karne se pehle answer zor se bolo.

Arrow

r

kis se kis ko point karta hai?

Earth ke centre se satellite tak.

Do arrows kaise add karte ho?

Tip-to-tail: doosre ka tail pehle ki tip par; sum start se end tak jaata hai (components add karo).

a - b

kaise subtract karte ho, aur result kaisa picture hota hai?

Flipped arrow add karo

a + (- b)

; yeh

b

ki tip se

a

ki tip tak arrow hai.

Scaling

ρ \overset{ρ}{^}

ek unit arrow ke saath kya karta hai?

Use (direction rakhte hue) length

ρ

tak stretch karta hai.

∣ r ∣

(ya plain

r

) ka kya matlab hai?

Arrow ki length — ek single non-negative number.

Ek symbol ke upar dot ka kya matlab hai?

Uski rate of change per second;

\dot{r} = v

\ddot{r} =

acceleration.

Law

\ddot{r} = - μ r / r^{3}

mein

r^{2}

ki jagah

r^{3}

kyun hai?

Kyunki

\overset{r}{^} = r / r

mein ek extra

r

hai;

- μ \overset{r}{^} / r^{2} = - μ r / r^{3}

Ek orbit ke liye

\ddot{r}

kis taraf point karta hai, aur kyun?

Earth ki taraf inward, kyunki gravity ek hi force hai.

μ

kya hai aur uski Earth value kya hai?

μ = GM

, gravity strength;

398600 km^{3} / s^{2}

Cross product

a \times b

tumhe kis type ka object deta hai, aur uski direction kaise choose karte ho?

Dono ke perpendicular ek arrow, length = parallelogram area; direction right-hand rule se.

Cross product zero arrow kab hota hai?

Jab do arrows parallel hों (flat parallelogram).

Dot product ke liye component formula do.

a \cdot b = a_{x} b_{x} + a_{y} b_{y} + a_{z} b_{z}

Dot product

a \cdot b

tumhe kya deta hai, aur kab zero hota hai?

Ek single number; zero jab arrows perpendicular hों.

Coplanarity test

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

kya confirm karta hai?

Ki sab teen position arrows ek plane mein lie karte hain.

Hat (jaise

\overset{ρ}{^}

mein) kya signify karta hai?

Ek unit vector — sirf direction, length exactly 1.

Slant range

ρ

kya hai aur yeh mushkil part kyun hai?

Telescope se satellite tak distance; ek telescope ise directly measure nahi kar sakta.

Ek observation se satellite position likho.

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

Central body orbit ellipse par kahan baitha hai?

Ek focus par (centre par nahi).

True anomaly

θ

kya measure karta hai?

Focus par perigee se current position tak angle.

Semi-major axis

a

kya hai?

Ellipse ka sabse lamba diameter ka aadha — uski overall size.

Conic equation batao aur

θ = 9 0^{\circ}

par

p

kya hai.

r = \frac{p}{1 + e cos θ}

;

θ = 9 0^{\circ}

par,

r = p

Char eccentricity cases do.

e = 0

circle,

0 < e < 1

ellipse,

e = 1

parabola,

e > 1

hyperbola.

f

aur

g

tumhe kya likhne dete hain, aur

τ_{i}

kya hai?

r_{i} = f_{i} r_{2} + g_{i} \dot{r}_{2}

;

τ_{i} = t_{i} - t_{2}

middle observation se time gap hai.

f, g

series mein

\frac{1}{2}

aur

\frac{1}{6}

kahan se aate hain?

Taylor-series factorials

\frac{1}{2 !}

aur $\tfr