3.2.8
Ek orbit ek fixed ellipse hoti hai jo 3D space mein floating hai , jisme ek satellite move karta hai. Ise poori tarah se pin karne ke liye tumhe teen sawaalon ka jawab dena hoga:
Ellipse ki shape aur size kya hai? → a e Ellipse space mein kaise tilted/oriented hai? → i Ω ω Abhi satellite ellipse par kahan hai? → ν
Ye exactly 6 numbers hain, kyunki ek 3D position + 3D velocity bhi 6 numbers hoti hai. Keplerian elements sirf ( r , v ) geometrically meaningful repackaging hai.
Intuition Degrees of freedomEk satellite ki ek instant par poori state hoti hai position r = ( x , y , z ) v = ( v x , v y , v z ) 6 numbers . Newton ki equation r ¨ = − μ r / r 3 exactly 6 independent quantities hone chahiye. Keplerian elements wo choice hain jahan 5 constant hote hain (ideal two-body gravity mein orbit ki shape/orientation nahi badalti) aur sirf 1 (ν . Isliye yeh pasand kiye jaate hain: yeh unchanging geometry ko moving point se alag karte hain.
Definition Semi-major axisa sabse lambe diameter ka aadha . Yeh orbit ki size set karta hai aur, vis-viva relation ke through, total energy bhi.
a 1/ r 2 size lock ho jaati hai — chahe satellite kahaan bhi baithe.
HOW. Specific orbital energy (energy per unit mass):
ε = 2 v 2 − r μ . h = r v v = h / r ε 2 r p 2 h 2 − r p μ = 2 r a 2 h 2 − r a μ . r p = a ( 1 − e ) , r a = a ( 1 + e ) , r p + r a = 2 a ε = − 2 a μ ε = v 2 /2 − μ / r vis-viva equation :
v 2 = μ ( r 2 − a 1 )
Hum ek Earth-Centered Inertial (ECI) frame mein kaam karte hain: X Z X Y
i orbital plane ka tilt : orbit ke angular-momentum vector h Z i = 0 i = 9 0 ∘ i > 9 0 ∘
Ω X Ascending node woh jagah hai jahan satellite equator ko south→north jaate hue cross karta hai. RAAN batata hai tilted plane kis taraf ghuma hua hai .
Definition Argument of perigeeω Orbital plane mein angle, ascending node se perigee tak measure kiya gaya . Yeh batata hai plane ke andar ellipse kis taraf point kar rahi hai (closest approach kahan hai).
ν Perigee se satellite ki current position tak ka angle , focus (Earth's center) par measure kiya gaya, motion ki direction mein. Yeh akela element hai jo time ke saath badalta hai .
Intuition WHY ek rotation matrix chahiyeOrbit ke apne "perifocal" frame mein (P perigee ki taraf, Q 9 0 ∘ rotate karna hai — yahi Ω , i , ω
Perifocal position (P–Q–W axes):
r P Q W = r cos ν sin ν 0 , r = 1 + e c o s ν a ( 1 − e 2 ) . ECI mein transform teen elementary rotations se:
r E C I = R z ( Ω ) R x ( i ) R z ( ω ) r P Q W . R z ( θ ) Z R x ( i ) hai .
Worked example 1 — Apogee/perigee altitudes sea e 400 1600 R E = 6378 a e
Step 1: Earth's center se radii mein convert karo.
r p = 6378 + 400 = 6778 r a = 6378 + 1600 = 7978 Kyun? Orbital elements focus (Earth's center) se distance use karte hain, altitude nahi.
Step 2: a = 2 r p + r a = 2 6778 + 7978 = 7378 Kyun? Major axis = r p + r a = 2 a
Step 3: e = r a + r p r a − r p = 14756 1200 = 0.0813 Kyun? r a − r p = 2 a e r a + r p = 2 a
Worked example 2 — Perigee par speed (vis-viva)Wohi orbit, μ E = 3.986 × 1 0 5 v p
Step 1: r = r p v 2 = μ ( r p 2 − a 1 ) Kyun? vis-viva speed ko position se sirf a
Step 2: v p 2 = 3.986 × 1 0 5 ( 6778 2 − 7378 1 ) = 3.986 × 1 0 5 ( 2.951 × 1 0 − 4 − 1.355 × 1 0 − 4 )
Step 3: v p 2 = 3.986 × 1 0 5 ( 1.596 × 1 0 − 4 ) = 63.6 v p ≈ 7.98 Kyun yeh sensible hai? Circular LEO speed (~7.6 km/s) se thoda zyada, kyunki perigee sabse fast point hota hai.
ν a = 7378 e = 0.0813 ν = 9 0 ∘ r
Step 1: p = a ( 1 − e 2 ) = 7378 ( 1 − 0.00661 ) = 7329 Step 2: r = 1 + e cos ν p = 1 + 0.0813 ⋅ 0 7329 = 7329 Kyun? ν = 9 0 ∘ cos ν = 0 r = p
Common mistake "Inclination batata hai ki plane kis direction mein rotate hua hai."Kyun sahi lagta hai: i Ω Fix: i tilt hai (plane equator se kitna jhuka hua hai). Ω swing hai (tilted plane ka node kis longitude par hai). Ek jhukta hai, doosra Z
Common mistake "Semi-major axis = orbit ki altitude."Kyun sahi lagta hai: circular LEO ke liye a ≈ Fix: a Earth's center se distance hai, aur yeh apogee aur perigee radii ka average hai, altitude nahi. Altitude ke liye R E
Common mistake "True anomalyν Kyun sahi lagta hai: yeh obvious "main kahan hun" angle hai.
Fix: ν perigee par sabse fast, apogee par sabse slow badalta hai (Kepler's 2nd law). Uniformly-increasing angle mean anomaly M ν M E
ω ν Kyun sahi lagta hai: dono in-plane angles hain focus par measure kiye gaye.
Fix: ω ascending node se perigee tak hai (fixed); ν perigee se satellite tak hai (moving). Unka sum, argument of latitude u = ω + ν
Recall Feynman: ek 12-saal ke bachchey ko explain karoSocho ek hula-hoop ek tabletop par tili hui hai, aur ek bead hoop ke around slide kar rahi hai.
Hoop ki size aur kitni squashed hai = a e Tumne hoop ko table se kitna tilaya = i Poori tili hui hoop ko kis taraf ghuma diya = Ω Hoop par "candle ke sabse paas wala point" kahan hai = ω Abhi is second hoop par bead kahan hai = ν ν
Mnemonic Order aur meaning"A Eccentric Italian Opera, Omitting Nothing."
a e i Ω ω ν shAPE (a , e fRAME (i , Ω , ω plAce (ν
Semi-major axis a Orbit ki size aur (via ε = − μ /2 a
Specific orbital energy aur a ε = − μ / ( 2 a )
Vis-viva equation batao. v 2 = μ ( r 2 − a 1 )
Apogee aur perigee radii se a a = ( r a + r p ) /2
Apogee aur perigee radii se e e = ( r a − r p ) / ( r a + r p )
Inclination i Orbital plane (
h vector) aur equatorial plane (
Z -axis) ke beech ka angle; orbit ka tilt.
RAAN Ω Equatorial plane mein vernal equinox (X
Ascending node kya hota hai? Woh point jahan satellite equator ko south→north jaate hue cross karta hai.
Argument of perigee ω In-plane angle ascending node se perigee tak.
True anomaly ν Focus par perigee se satellite ki current position tak ka angle, motion ki direction mein.
Ideal two-body problem mein kaun sa orbital element time ke saath badalta hai? Sirf true anomaly ν
r ( ν ) r = 1 + e cos ν a ( 1 − e 2 ) = 1 + e cos ν p
Semi-latus rectum p h p = h 2 / μ = a ( 1 − e 2 )
Perifocal se ECI coordinates mein map karne ka rotation sequence kya hai? R z ( Ω ) R x ( i ) R z ( ω )
Exactly 6 elements kyun chahiye? Kyunki full state
( r , v ) 6 numbers hota hai; 6 elements equivalent information carry karte hain.
Argument of latitude u u = ω + ν
Position r and velocity v, 6 numbers
Keplerian elements, 6 numbers
Vis-viva, energy = -mu/2a
Ellipse geometry, rp and ra
