Lambert's problem — connecting two positions in given time: Foundations

3.2.28 · D1 · Physics › Orbital Mechanics & Astrodynamics › Lambert's problem — do positions ko given time mein connect karna

Yeh ek prerequisite page hai. Parent note Lambert's problem mein $r_{1}$ , $c$ , $s$ , $μ$ , $Δ θ$ , $α$ , $β$ , $a$ , $e$ , aur $E$ jaise symbols udate rehte hain. Agar inme se koi bhi alphabet soup lagta hai, toh yahan se shuru karo. Hum har ek ko simple shabdon mein define karenge, uska picture banaenge, aur batayenge ki topic ko uske bina kaam kyun nahi chalta.

1. Ek position vector $r$ — "ek jagah ki taraf ek arrow"

Neeche di gayi figure dekho. Beech ka dot central body hai — woh cheez jiske gravity mein hum orbit karte hain (Earth, Sun...). Isse hum do arrows khinchte hain: $r_{1}$ us jagah point karta hai jahan hum shuru karte hain, $r_{2}$ us jagah point karta hai jahan hume pahunchna hai.

Figure — Lambert's problem — connecting two positions in given time

Arrow $r_{1}$ ki length $r_{1}$ likhi jaati hai (bina arrow ke) aur ise "the magnitude of $r_{1}$ " padha jaata hai. Toh:

$r_{1}$ = arrow (woh jagah jahan se hum shuru karte hain).
$r_{1}$ = plain number = woh jagah centre se kitni door hai.

Yehi $r_{2}$ aur uski length $r_{2}$ ke liye bhi.

Topic ko iske zaroorat kyun hai

Lambert ka poora sawaal hai "inhe dono jagahon ko connect karo" — iska statement karna hi impossible hai jab tak jagahon ko name karne ka koi tarika nahi, aur position vector wohi naam hai.

2. Vectors ko subtract karna → chord $c$

Agar $r_{1}$ aur $r_{2}$ do jagahon ki taraf arrows hain, toh jagah 1 se jagah 2 tak jaane wala arrow unka difference hai, jo $r_{2} - r_{1}$ likha jaata hai.

Us shortcut ki length chord hai, jise $c$ kaha jaata hai: $c = ∣ r_{2} - r_{1} ∣.$

Topic ko iske zaroorat kyun hai

Lambert's Theorem kehta hai time-of-flight sirf

a

c

, aur

r_{1} + r_{2}

par depend karta hai — toh

c

un teen numbers mein se ek hai jo sab kuch decide karte hain.

3. Transfer angle $Δ θ$ aur law of cosines

Do arrows $r_{1}$ aur $r_{2}$ ek pair of scissors ki tarah khulte hain. Unke beech ka angle transfer angle hai, jo $Δ θ$ likha jaata hai (Greek letter "theta," aur $Δ$ matlab "change/difference in").

Ek sundar shortcut hai jo $c$ , do lengths, aur is angle ko link karta hai — law of cosines. Yeh us triangle centre → tip 1 → tip 2 se aata hai: $c^{2} = r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} cos Δ θ .$

Topic ko iske zaroorat kyun hai

Δ θ

se hi tum Lambert ko batate ho "short way jaao" ya "long way around jaao" — alag answers, same endpoints.

4. Semiperimeter $s$

Jab hmare paas triangle ki teen sides ( $r_{1}$ , $r_{2}$ , $c$ ) ho jaati hain, toh hum unhe ek number mein bundle karte hain, semiperimeter $s$ ("semi" = aadha, "perimeter" = ghoomne ki distance): $s = \frac{1}{2} (r_{1} + r_{2} + c) .$

Topic ko iske zaroorat kyun hai

Auxiliary angles

α, β

s

aur

s - c

se bane hain, aur minimum possible orbit

a_{m i n} = s /2

hai — toh

s

woh gateway number hai.

5. Gravitational parameter $μ$

Topic ko iske zaroorat kyun hai

Time-of-flight mein

μ

ka factor hota hai — gravity ki strength orbit ki ghadi ki speed set karti hai.

6. Ellipse: $a$ aur $e$

Har gravity orbit jo wapas apne pe aati hai woh ek ellipse hai — ek squashed circle. Iske shape ko do numbers describe karti hain.

Central body ek focus par hota hai — ek special off-centre point, middle mein nahi. Isliye orbit ka ek side close hota hai (fast) aur doosra far (slow).

Topic ko iske zaroorat kyun hai

Lambert ka answer ek orbit hai, aur

a, e

uski shape hain. Poori time-equation hai "woh

a

dhundo jo sahi time deta hai."

7. Ek orbit ki ghadi: $E$ , mean anomaly, aur $Δ t$

Ek orbit constant speed se nahi chalti — focus ke paas fast, door slow. Timing ka hisaab rakhne ke liye, astronomers ek clever helper angle use karte hain jise eccentric anomaly $E$ kehte hain.

Is angle aur real clock time ke beech ka link Kepler's equation hai: $M = E - e sin E, M = \frac{μ}{a ^{3}} (t - t_{p}) .$ Yahan $M$ mean anomaly hai: ek fake angle jo time ke saath perfectly evenly tick karta hai (jaise ek clock hand), aur $t_{p}$ closest approach ka time hai.

$Δ t$ simply safar ka elapsed time hai: arrival time aur departure time ka difference.

Recall

$E$ skip karke real angle kyun nahi use kar sakte? Kyunki real angle vs time relation ka koi closed form nahi hai — $E$ ke saath Kepler's equation sabse clean bridge hai, aur yahi woh hai jo parent ka Step 3 arc ke across subtract karta hai.

Topic ko iske zaroorat kyun hai

Parent ki Lambert equation literally Kepler's time law hai jo dono ends par apply ki gayi hai aur simplify ki gayi hai —

E

M

μ

a

sab wahan appear karte hain.

8. Auxiliary angles $α$ aur $β$

Aakhir mein, parent do mysterious angles introduce karta hai: $sin^{2} \frac{α}{2} = \frac{s}{2 a}, sin^{2} \frac{β}{2} = \frac{s - c}{2 a} .$

Topic ko iske zaroorat kyun hai

Yeh ugly Kepler subtraction ko clean boxed formula

μ Δ t = a^{3/2} [(α - sin α) - (β - sin β)]

mein convert karte hain.

Yeh cheezein topic ko kaise feed karti hain

Yeh bhi dekho, jab tum yahan comfortable ho jaao: Kepler's Equation and Time of Flight, Lagrange Coefficients (f and g functions), aur Universal Variable Formulation, jinhe parent note aage use karta hai.

Equipment checklist

Right side cover karo aur khud ko test karo — jab har reveal obvious lagne lage, tum parent note ke liye ready ho.

Arrow

r_{1}

ka kya matlab hai aur

r_{1}

kya hai?

r_{1}

= central body se start place tak arrow;

r_{1}

= uski length (centre se distance).

Chord

c

ko ek phrase mein kya kehte hain?

do jagahon ke beech ki straight-line distance,

∣ r_{2} - r_{1} ∣

— flown path NAHI.

c

ke liye law of cosines likho.

c^{2} = r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} cos Δ θ

Δ θ

kya measure karta hai aur short vs long way kaun choose karta hai?

r_{1}

r_{2}

tak sweep kiya gaya angle;

Δ θ < 18 0^{\circ}

short way hai,

> 18 0^{\circ}

long way hai.

s

define karo.

semiperimeter

s = \frac{1}{2} (r_{1} + r_{2} + c)

, space-triangle ke perimeter ka aadha.

μ

kya hai aur ise bundle kyun karein?

μ = GM

, central body ki gravity strength; accuracy ke liye ek product ke roop mein measure kiya jaata hai.

a

aur

e

kya describe karte hain?

a

= size (long axis ka aadha);

e

= squash (0 circle, near-1 cigar).

E

aur

M

kyun introduce karein?

M

time ke saath evenly tick karta hai;

E

ek helper angle hai; Kepler's equation

M = E - e sin E

unke beech translate karta hai.

α, β

kya hain?

s

s - c

, aur

a

se bane endpoint angles;

β

ka sign short/long way encode karta hai.

Kaun se teen scalars akele time of flight fix karte hain?

a

, chord

c

, aur radius sum

r_{1} + r_{2}

(Lambert's Theorem).