Specific orbital energy ε = −GM - 2a: Foundations

3.2.11 · D1 · Physics › Orbital Mechanics & Astrodynamics › Specific orbital energy ε = −GM - 2a

Isse pehle ki tum parent note mein woh boxed formula pe trust kar sako, us mein har letter ka koi aisa matlab hona chahiye jo tum picture kar sako. Yeh page unhe ek-ek karke, bilkul scratch se build karta hai. Kabhi bhi koi symbol use nahi karte jab tak woh apni jagah earn na kar le.

0. Stage: ek chhota body ek bade ke around loop kar raha hai

Figure — Specific orbital energy ε = −GM - 2a

Figure dekho: yellow blob central body $M$ hai. Blue curve orbit hai — woh path jo chhota body trace karta hai. Do special points, closest (red) aur farthest (green), wahi hain jahan saari interesting energy exchange peak karti hai. Hum neeche is picture ka har quantity naam karenge.

1. Distance $r$ — satellite kitna door hai

Picture: figure mein, $r$ us arrow ki length hai jo yellow centre se moving dot tak jaata hai. Jab satellite loop karta hai, yeh arrow bada aur chhota hota rehta hai — far point pe lamba, near point pe chhota.

Yeh topic ko kyun chahiye: gravity ka pull is baat pe depend karta hai ki bodies kitni door hain, isliye lagbhag har formula poochhta hai "kitna door?" — aur jawab hai $r$ .

2. Speed $v$ — kitni tezi se move karta hai

Picture: imagine karo ek tiny arrow moving dot se chipka hua hai, jahan woh ja raha hai wahan point karta hua. Uski length $v$ hai. Figure mein woh arrow red point ke paas lamba hota hai (fast) aur green point ke paas chhota (slow).

Yeh topic ko kyun chahiye: motion energy carry karta hai, aur kitni energy hai yeh speed pe depend karta hai. $v$ nahi, toh kinetic energy nahi.

3. Gravitational constant $G$ aur central mass $M$

Picture: $G$ ko ek "exchange rate" samjho jo masses-aur-distances ko ek pull mein convert karta hai. $M$ yeh hai ki pulling karne wala kitna stuff hai.

Yeh topic ko kyun chahiye: yeh hamesha product $GM$ ke roop mein chipke hue aate hain, aur orbit ko sirf yahi ek gravity-strength number ki parwah hai.

4. Kinetic energy per mass — "motion money"

Picture: velocity arrow (Section 2) jitna lamba, utna zyada "motion money." Red near-point pe arrow sabse lamba hota hai, isliye wahan yeh term peak karti hai.

$m$ se kyun divide karte hain? Taaki satellite ki khud ki mass drop out ho jaye — ek hi orbit mein ek feather aur ek boulder same budget share karte hain. Yahi toh specific word ka poora point hai (= per unit mass).

$v^{2}$ kyun, $v$ kyun nahi? Speed double karne se stored motion energy chaar guna ho jaati hai — yeh physics ka ek measured fact hai, square se capture hota hai.

5. Gravitational potential energy per mass — "height money"

Yeh subtle wala hai, isliye hum ise dheere-dheere build karte hain.

Hum potential energy ko zero kahan kehte hain? Hum choose karte hain ki yeh infinitely far away pe zero ho, kyunki wahi ek jagah hai jahan gravity ka pull khatam ho chuka hai. Phir har closer position ka energy us se kam hoga — isliye ek negative number aata hai.

Picture: figure mein $- GM / r$ ko ek hill (well) ke roop mein dikhaya gaya hai jo planet ke paas aate hi gehra ho jaata hai. Satellite is well mein roll karta hai: door pe woh top ke paas (near zero) baitha hai; paas mein woh pit mein deep (bahut negative) hota hai. Blue dot follow karo — jaise $r$ shrink karta hai, woh curve ke neeche slide karta hai.

Yeh topic ko kyun chahiye: yeh energy budget ka "height money" wala half hai. Motion money ke saath combine hokar total banta hai jo kabhi nahi badalta.

Shape $- 1/ r$ kyun hai aur kuch aur kyun nahi? Gravity distance ke square ke saath weak hoti hai ( $\propto 1/ r^{2}$ ). Jab tum door jaate ho toh us pull ko add up (integrate) karne se energy mein $1/ r$ dependence aati hai — yahi exact wajah hai ki yahan ek straight line ki jagah $1/ r$ shape aata hai.

6. Total energy per mass — woh budget jo kabhi nahi hilti

Picture: figure orbit ke saath dono contributions stack karta hai. Jab satellite andar dive karta hai (red), height bar drop karta hai (zyada negative) lekin motion bar exactly utni hi amount se rise karta hai — total line flat rehti hai. Woh flat line $ε$ hai.

"Conserved" kyun? Gravity ek conservative force hai: woh jo kaam karta hai woh sirf start aur end pe depend karta hai, kabhi bhi path pe nahi. Yeh guarantee karta hai ki motion-money aur height-money sirf trade karte hain, kabhi leak nahi hote.

7. Semi-major axis $a$ — "loop ki size"

Upar sab kuch ek single instant pe rehta hai. Aakhri symbol orbit ko poori tarah describe karta hai.

Picture: figure mein near point (perigee, distance $r_{p}$ ) aur far point (apogee, distance $r_{a}$ ) opposite ends pe hain. Across ka full span $r_{p} + r_{a}$ hai, aur $a = \frac{r _{p} + r _{a}}{2} .$

Yeh topic ko kyun chahiye: headline result yeh hai ki poora energy budget sirf is size pe depend karta hai: $ε = - \frac{GM}{2 a} .$ Bada loop $\Rightarrow$ bada $a$ $\Rightarrow$ $ε$ zero ke qareeb $\Rightarrow$ "zyada loosely bound."

8. Angular momentum per mass $h$ (derivation mein use hota hai)

Picture: perigee aur apogee pe velocity arrow radius arrow ke straight across point karta hai — koi in/out component nahi. Yahi woh exact jagah hai jahan clean product $r \times v$ hold karta hai.

Yeh topic ko kyun chahiye: parent ki derivation do points pe energy equate karti hai aur do speeds ko link karne ke liye ek doosra equation chahiye. Kyunki central pull zero twist (torque) exert karta hai, $h$ conserved hai — wahi doosra equation hai.

Poori kahaani ke liye dekho Conservation of angular momentum ki swirl kyun conserved hoti hai.

Prerequisite map

Yeh map flow dikhata hai: raw ingredients ( $G$ , $M$ , $r$ , $v$ ) se dono energy halves bante hain, jo $ε$ mein sum hote hain; phir derivation ( $h$ use karke) $r$ aur $v$ ko pure size $a$ ke liye trade karti hai, jisse $- \frac{GM}{2 a}$ milta hai, jo Vis-viva equation aur Conic sections in orbits ke zariye orbit-type rule unlock karta hai.

Equipment checklist

Khud test karo — right side cover karo aur out loud jawab do.

r

kya measure karta hai, aur kahan se?

Chhote body ki bade body ke centre se distance, metres mein.

v

kya hai?

Orbiting body ki speed (velocity arrow ki length), m/s mein.

G

aur

M

hamesha single product

GM

ke roop mein kyun aate hain?

Yeh orbital formulas mein kabhi alag act nahi karte;

GM

woh ek gravity-strength number hai jis ki orbit ko parwah hai.

Kinetic energy per unit mass likho aur uske units batao.

\frac{v ^{2}}{2}

, units

m^{2} / s^{2} = J/kg

Gravitational PE per mass negative kyun hai?

Hum ise zero at infinity set karte hain; har closer position wahan se downhill hai, isliye negative hai.

PE per mass formula mein likho.

- \frac{GM}{r}

Total

ε = \frac{v ^{2}}{2} - \frac{GM}{r}

orbit ke saath constant kyun hai?

Gravity conservative hai, isliye motion-energy aur height-energy sirf trade karte hain — sum conserved rehta hai.

"Specific" ka matlab kya hai aur hum iska bother kyun karte hain?

Per unit mass;

m

se divide karne se result satellite ki khud ki mass se independent ho jaata hai.

a

r_{p}

aur

r_{a}

ke terms mein define karo.

a = \frac{r _{p} + r _{a}}{2}

, ellipse ke major axis ka aadha.

h = r_{p} v_{p} = r_{a} v_{a}

sirf perigee aur apogee pe kyun hota hai?

Wahan velocity perpendicular hoti hai radius ke, isliye

r v_{⊥}

simplify hokar

r v

ban jaata hai.

Bada orbit —

ε

zyada negative hoga ya kam?

Kam negative (zero ke qareeb), kyunki

ε = - \frac{GM}{2 a}

size mein shrink karta hai jab

a

badhta hai.