Specific orbital energy ε = −GM - 2a: Visual walkthrough

3.2.11 · D2 · Physics › Orbital Mechanics & Astrodynamics › Specific orbital energy ε = −GM - 2a

Step 0 — Kirdar ka parichay (har symbol kamao)

KYA. Kisi bhi equation se pehle, drawing mein pieces ko naam dete hain.

KYUN. Is poore page ka agreement yeh hai: koi bhi letter tab tak nahi aayega jab tak tum use kisi figure mein point na kar sako.

PICTURE.

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

Drawing dekho:

Bada dot planet hai, mass $M$ (kilograms mein ek number). $M$ = "puller kitna strong hai."
Chhota dot satellite hai. Uski apni mass $m$ hai, lekin dekho — $m$ cancel out ho jaayegi aur gayab ho jaayegi, aur yahi "specific" word ka poora point hai.
Planet-centre se satellite tak jaane wali chalk-blue line ki length $r$ hai — distance, metres mein measure ki gayi. Surface se nahi; centre se.
Pink arrow velocity $v$ hai: satellite abhi kitni tez chal raha hai (uski length) aur kis direction mein (uski disha).
$G$ gravitational constant hai, nature ka ek fixed number ( $6.674 \times 1 0^{- 11}$ SI units mein) jo gravity ki strength har jagah set karta hai.

Step 1 — Energy account kabhi nahi badlata

KYA. Hum claim karte hain ki $ε$ orbit ke har point par same number hai.

KYUN. Gravity ek conservative force hai: jo kaam woh karta hai woh sirf is baat par depend karta hai ki tum kahan se shuru kiye aur kahan khatam, kabhi bhi beech ke tedhe-medhe path par nahi. Yahi bilkul woh condition hai "kinetic + potential = constant" ke liye. Toh jaise satellite chakkar lagata hai, do pieces aage-peechhe trade karti hain lekin unka sum frozen rehta hai. Yeh gravitational PE apna hisaab-kitaab kar raha hai.

PICTURE.

Do bars dekho. Planet ke paas (left) pink motion bar lamba hai aur blue position bar gehra hai — satellite fast aur neecha hai. Door (right) motion bar shrink hota hai aur position bar zero ki taraf uthta hai — slow aur oopar. White line jo $ε$ mark karta hai — sum — poore trip mein flat rehti hai. Woh flat line hi conserved quantity hai.

Step 2 — Do special points jahan arrow "sideways" hai

KYA. Ellipse ke do turning points chuno: perigee (sabse karib, distance $r_{p}$ ) aur apogee (sabse door, distance $r_{a}$ ).

KYUN. Sirf inhi do points par, velocity arrow bilkul perpendicular hai distance line se. Isse yeh energy likhne ke liye sabse saaf jagah ban jaate hain — $v$ aur $r$ ke beech koi awkward angle nahi. Hum dono par $ε$ equate karte hain kyunki Step 1 kehta hai yeh same value hai.

PICTURE.

Ellipse: planet pale-yellow focus par. Left tip = perigee, pink arrow $v_{p}$ seedha upar point karta hai (chhoti blue line $r_{p}$ se perpendicular). Right tip = apogee, arrow $v_{a}$ seedha neeche point karta hai (lambi blue line $r_{a}$ se perpendicular). Har jagah $ε$ likhkar aur equal set karke:

$fast, close \frac{v _{p}^{2}}{2} - deep well \frac{GM}{r _{p}} = slow, far \frac{v _{a}^{2}}{2} - shallow well \frac{GM}{r _{a}}$

Har term labelled hai: left side perigee par account hai, right side apogee par — equal kyunki balance kabhi nahi badlata.

Step 3 — Second law: angular momentum do speeds ko bandha hai

KYA. Hume ek doosri equation milti hai $v_{p}$ aur $v_{a}$ ko jodhne wali: $r_{p} v_{p} = r_{a} v_{a}$ .

KYUN. Gravity hamesha planet ki taraf jaane wali line ke saath point karti hai — seedha centre ki taraf. Centre se jaane wali force zero twist (torque) exert karti hai. Koi twist nahi matlab "spin quantity" — angular momentum per kilogram — conserved hai. Yeh distance times sideways speed ke barabar hai. Perigee aur apogee par arrow pehle se hi perpendicular hai, toh poori speed sideways hai, aur woh spin quantity simply $r_{p} v_{p}$ ek tip par aur $r_{a} v_{a}$ doosri par hai. Equal spin deta hai $r_{p} v_{p} = r_{a} v_{a}$ .

PICTURE.

Do shaded "sweep" wedges equal area ke — ek patla-aur-lamba perigee par, ek mota-aur-chhota apogee par. Equal time mein equal areas same law ka Kepler ka version hai. Rearrange karke:

$v_{a} = v_{p} \cdot small / big < 1 \frac{r _{p}}{r _{a}}$

Label dikhata hai kyun $v_{a}$ chhota hai: apogee par lever arm $r_{a}$ lamba hai, toh product fixed rakhne ke liye speed chhoti honi chahiye.

Step 4 — Perigee speed solve karo (seedha algebra, imandaari se dikhaya)

KYA. Step 2 ki energy equation mein $v_{a} = v_{p} r_{p} / r_{a}$ dalo aur $v_{p}^{2}$ isolate karo.

KYUN. Hamare paas do equations hain aur do unknown speeds. $v_{a}$ eliminate karne se $v_{p}^{2}$ sirf geometry mein likh jaati hai ( $r_{p}$ , $r_{a}$ , $GM$ ) — chase karne ke liye koi speed nahi bachti.

PICTURE.

Figure cancellation ka "flow" hai. Substitute karne par milta hai $\frac{v _{p}^{2}}{2} (1 - \frac{r _{p}^{2}}{r _{a}^{2}}) = GM (\frac{1}{r _{p}} - \frac{1}{r _{a}}) .$ Left bracket factor hota hai $\frac{( r _{a} - r _{p} ) ( r _{a} + r _{p} )}{r _{a}^{2}}$ aur right $\frac{r _{a} - r _{p}}{r _{p} r _{a}}$ ke roop mein. Figure mein red slash dono sides par common factor $(r_{a} - r_{p})$ ko khatam karta hai, bachta hai

$v_{p}^{2} = \frac{2 GM r _{a}}{r _{p} ( r _{a} + r _{p} )}$

Har symbol kamaya: $2 GM$ = strength, $r_{a}$ upar / $r_{p} (r_{a} + r_{p})$ neeche = ellipse ki geometry.

Step 5 — Wapas daalo aur speed ko gayab hote dekho

KYA. Woh $v_{p}^{2}$ ko $ε = \frac{v _{p}^{2}}{2} - \frac{GM}{r _{p}}$ mein substitute karo.

KYUN. $ε$ wahi hai jo hum actually chahte hain, aur Step 4 ne $v$ hata diya. Plug in karne par sab kuch sirf pure distances mein collapse ho jaana chahiye.

PICTURE.

$ε = v_{p}^{2} /2 \frac{GM r _{a}}{r _{p} ( r _{a} + r _{p} )} - PE at perigee \frac{GM}{r _{p}} = \frac{GM}{r _{p}} \cdot \frac{r _{a} - ( r _{a} + r _{p} )}{r _{a} + r _{p}} = \frac{GM}{r _{p}} \cdot \frac{- r _{p}}{r _{a} + r _{p}} .$

$r_{p}$ cancel ho jaata hai (chalk mein crossed out dikhaya gaya), bachta hai clean intermediate result:

$ε = - \frac{GM}{r _{a} + r _{p}}$

Labels padho: do extreme distances ke sum par numerator. Almost aa gaye.

Step 6 — Ellipse ki apni geometry isko khatam karti hai

KYA. $r_{a} + r_{p}$ ko $2 a$ se replace karo.

KYUN. Ellipse ko end-to-end dekho: perigee distance plus apogee distance poora major axis hai, jo definition se $2 a$ hai (semi-major axis $a$ ka double). Yeh pure shape hai, koi physics nahi.

PICTURE.

Chalk ruler tip se tip tak chalta hai: sabse karib gap $r_{p}$ aur dur gap $r_{a}$ milke poori lambi axis span karte hain $= 2 a$ . $r_{a} + r_{p} = 2 a$ substitute karke:

$ε = - \frac{GM}{2 a} ■$

Woh result jo parent ne promise kiya tha — energy sirf $a$ par depend karti hai ( $GM$ se multiply hokar). Shape, speed, aur satellite ki mass saab raaste mein cancel ho gayi.

Step 7 — Har case, koi gap nahi

KYA. Saare orbit shapes mein $ε$ ka sign check karo, degenerate wale bhi.

KYUN. Sirf ellipses ki picture reader ko escape ya fly-by par atka degi. Hum saari conics cover karte hain.

PICTURE.

Ellipse (bound): $a > 0 \Rightarrow ε = - \frac{GM}{2 a} < 0$ . Nested, wapas aata hai. Circle special case hai $r_{p} = r_{a} = a$ .
Parabola (escape edge): $a \to \infty \Rightarrow ε \to 0$ . Infinity tak coast karne ke liye bilkul enough, zero speed se pahunchta hai. vis-viva mein $ε = 0$ set karo to escape speed $v_{esc} = 2 GM / r$ padh sako.
Hyperbola (unbound fly-by): $ε > 0 \Rightarrow a = - GM /2 ε < 0$ . Negative $a$ hamesha ke liye jaane ka sachcha signature hai — koi galti nahi.

Ek-picture summary

Ek frame sab chain karta hai: (1) flat conserved line $ε$ → (2) perigee & apogee par equate karo → (3) angular momentum speeds ko link karta hai → (4–5) algebra $v$ erase karta hai giving $- GM / (r_{a} + r_{p})$ → (6) ellipse geometry $r_{a} + r_{p} = 2 a$ → (7) box $ε = - GM /2 a$ , sign strip ke saath ellipse/parabola/hyperbola dikhata hua. Dekho Kepler's Third Law aur Hohmann transfer orbit jahan yeh single number real kaam karta hai.

Recall Poore walkthrough ki Feynman-style retelling

Ek satellite ke paas ek bank balance hota hai jo speed-money aur height-money se bana hota hai. Gravity ek imaandaar banker hai — woh kabhi money add ya remove nahi karta, sirf dono tarah ki money ko swap karne deta hai, toh total poore trip ke liye ek flat line hai. Main do aasaan jagahon par — sabse karib aur sabse door wale points par — balance dekh leta hoon, jahan velocity arrow cleanly sideways point karta hai, aur main dono balances equal set karta hoon. Ek doosra niyam — spin leak nahi ho sakti kyunki gravity dead-centre pull karti hai — mujhe batata hai ki dur wala point exactly usi proportion mein slower move karta hai jitna door hai. In do facts se main speeds ko equations se poori tarah chase kar sakta hoon; woh cancel ho jaati hain, aur main sirf do extreme distances ke saath bachta hoon. Phir main notice karta hoon ki woh do distances, end to end rakhne par, loop ki poori length hain, $2 a$ . Toh balance simply $- GM /2 a$ hai: bada loop, ameer (kam negative) account. Agar account kabhi zero tak chadh jaaye, toh satellite bye keh ke chalaa jaata hai aur kabhi wapas nahi aata.

Active-recall

Kin do points par

v ⊥ r

hota hai, jisse hum energy saaf se likh sakein?

Perigee aur apogee (ellipse ke turning points).

Satellite ki apni mass

m

kyun gayab ho jaati hai?

Hum energy per kilogram ("specific") use karte hain, toh

m

cancel ho jaata hai.

Kaun sa conservation law

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

deta hai?

Conservation of angular momentum (central force ⇒ zero torque).

Kaun sa geometric fact

r_{a} + r_{p}

2 a

mein badalta hai?

Sabse karib aur sabse door ki distances milke poori major axis

= 2 a

span karti hain.

Negative

a

kya signal karta hai?

Ek unbound hyperbolic trajectory, kyunki

a = - GM /2 ε

mein

ε > 0

hai.