Converting between orbital elements and state vectors (r, v)

3.2.17 · Physics › Orbital Mechanics & Astrodynamics

Chhe orbital elements (HAR ek ka MATLAB kya hai)

Woh foundations jinse hum SAB kuch DERIVE karte hain

Do conserved quantities saara kaam karti hain.

YEH kyun matter karte hain: $h$ orbit plane ke perpendicular hai, isliye yeh $i$ aur $Ω$ fix karta hai. Energy $ε$ size $a$ fix karti hai. Eccentricity vector (neeche) periapsis ki taraf point karta hai, $e$ aur $ω$ fix karta hai.

Direction A: State vectors → Orbital elements

Quadrant checks kyun? $cos^{- 1}$ sirf $[0, π]$ return karta hai. Ek companion quantity ki sign (jaise $r \cdot v$ : positive matlab periapsis se door ja raha hai, toh $ν < π$ ) poora $[0, 2 π)$ recover karta hai.

Direction B: Orbital elements → State vectors

Figure — Converting between orbital elements and state vectors (r, v)

Worked example 1 — State → Elements

Diya hua hai (units: km, km/s, $μ = 398600 km^{3} / s^{2}$ ): $r = (7000, 0, 0)$ , $v = (0, 7.5, 1.0)$ .

$h = r \times v = (0 \cdot 1 - 0 \cdot 7.5, 0 \cdot 0 - 7000 \cdot 1, 7000 \cdot 7.5 - 0) = (0, - 7000, 52500)$ . Kyun: cross product plane fix karta hai; $h = 52964.6$ .
$r = 7000$ , $v = 7. 5^{2} + 1^{2} = 7.566$ . $ε = \frac{7.56 6 ^{2}}{2} - \frac{398600}{7000} = - 28.31$ . Kyun: energy → $a = - μ /2 ε = 7040$ km.
$cos i = h_{z} / h = 52500/52964.6 = 0.9912 \Rightarrow i = 7.6 0^{\circ}$ . Kyun: chhota $v_{z}$ thoda sa tilt deta hai.
$e = \frac{1}{μ} [(v^{2} - μ / r) r - (r \cdot v) v]$ ; yahan $r \cdot v = 0$ , toh $e = \frac{v ^{2} - μ / r}{μ} r = (0.00579) (7000, 0, 0)$ ... giving $e \approx 0.0057$ . Kyun: $r \cdot v = 0$ matlab hum periapsis/apoapsis par hain, toh $ν = 0$ ya $18 0^{\circ}$ ; kyunki orbit yahan almost circular hai isliye $e$ bahut chhota hai.

Worked example 2 — Elements → State

Diya hua hai $a = 8000$ km, $e = 0.1$ , $i = 3 0^{\circ}$ , $Ω = 4 0^{\circ}$ , $ω = 6 0^{\circ}$ , $ν = 0^{\circ}$ (periapsis par).

$p = a (1 - e^{2}) = 8000 (0.99) = 7920$ km.
$r = p / (1 + e cos 0) = 7920/1.1 = 7200$ km. Kyun: periapsis distance $= a (1 - e) = 7200$ . ✓
$r_{pq w} = 7200 (cos 0, sin 0, 0) = (7200, 0, 0)$ .
$v_{pq w} = μ / p (- sin 0, e + cos 0, 0) = 7.094 (0, 1.1, 0) = (0, 7.803, 0)$ . Kyun: periapsis par velocity purely tangential hoti hai — koi radial component nahi, jo $- sin ν = 0$ se match karta hai.
Un angles ke saath $\tilde{R}$ apply karo $r, v$ inertial frame mein nikalne ke liye.

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ek race car ek oval track par hai. Ek tarika describe karne ka: "yeh map par yahan hai aur us speed se ja raha hai" (state vector). Doosra tarika: "track itna bada hai, itna squished, itna tilted, aur car start line se itni door hai" (orbital elements). Dono car ke baare mein sab kuch bata dete hain. Translate karne ke liye, tum do aisi facts use karte ho jo kabhi nahi badlti: car kitni tezi se area sweep karti hai (angular momentum) aur uski total energy kitni hai. Unse tum poora track rebuild karte ho aur car ki position — phir bas apna map north ke saath align karne ke liye ghuma dete ho.

Flashcards

State→element conversion mein kaunse do conserved vectors/scalars kaam aate hain?

Specific angular momentum

h = r \times v

(plane,

i

Ω

fix karta hai) aur specific energy

ε = v^{2} /2 - μ / r

(

a

fix karta hai).

Energy se semi-major axis ka formula?

a = - μ / (2 ε)

jahan

ε = v^{2} /2 - μ / r

Eccentricity vector kis taraf point karta hai?

Periapsis ki taraf; iska magnitude

e

ke barabar hota hai.

e = \frac{v \times h}{μ} - \frac{r}{r}

cos^{- 1}

use karne ke baad quadrant checks kyun chahiye?

cos^{- 1}

sirf

[0, π]

return karta hai; ek companion sign (jaise

ν

ke liye

r \cdot v

) poora

[0, 2 π)

recover karta hai.

True anomaly

ν

ke liye sign check?

Agar

r \cdot v < 0

ho (periapsis ki taraf ja raha hai), toh

ν = 2 π - cos^{- 1} (\dots)

Perifocal frame mein position vector?

r_{pq w} = r (cos ν, sin ν, 0)

jahan

r = p / (1 + e cos ν)

Perifocal frame mein velocity vector?

v_{pq w} = μ / p (- sin ν, e + cos ν, 0)

Perifocal se inertial frame mein rotation sequence?

R_{3} (- Ω) R_{1} (- i) R_{3} (- ω)

(swivel Ω, tilt i, spin ω),

r

aur

v

dono par apply hota hai.

p

(semi-latus rectum)

a, e

aur

h

ke terms mein kya hai?

p = a (1 - e^{2}) = h^{2} / μ

Circular orbit (

e = 0

) mein kaun se elements undefined ho jaate hain?

ω

(koi periapsis nahi); argument of latitude

u = ω + ν

use karo.

Connections

Two-body problem and the vis-viva equation
Angular momentum conservation in orbits
Kepler's equation and time-of-flight
Reference frames — perifocal (PQW) vs ECI
Orbital singularities and equinoctial elements
Hohmann transfers (state↔element conversions chahiye)