Hohmann transfer — derivation, minimum energy transfer: Foundations

3.2.19 · D1 · Physics › Orbital Mechanics & Astrodynamics › Hohmann transfer — derivation, minimum energy transfer

The parent note mein jo bhi hai woh kuch symbols par tikaa hai. Neeche, har ek ko zero se build kiya gaya hai: plain words → picture → yeh topic isko kyun use karta hai. Upar se neeche padho; koi bhi cheez use nahi hogi jab tak woh introduce na ho jaaye.

1. Central body aur uski pull: $M$ , $G$ , $μ$

Inhe bundle kyun karein? Har orbit formula mein $G$ aur $M$ saath chipke huye aate hain, hamesha product $GM$ ke roop mein. Toh hum pair ko ek naam dete hain:

2. Position: $r$ (radius) aur orbit ki picture

"Centre se" kyun? Gravity aise kaam karti hai jaise saari $M$ central point par baithi ho, isliye distances us point tak measure ki jaati hain. Yahi reason hai ki Earth ke around ek "7000 km" orbit (radius $\approx 6371$ km) surface se sirf $\sim 629$ km upar hai.

3. Circles vs ellipses: $r_{1}$ , $r_{2}$ , aur paths ki shape

Ab woh shape jo inhe connect karti hai:

4. Orbit ki size: $a$ (semi-major axis)

$a$ ke baare mein sabse useful fact: closest aur farthest distances add hokar full long diameter dete hain, jo $2 a$ hai: $r_{peri} + r_{apo} = 2 a$

5. Speed: $v$ , $v_{circ}$ , aur subscripts

Parent mein kaafi saare flavoured $v$ 's use hote hain; yahan poora dictionary hai taaki koi subscript tumhe surprise na kare:

Symbol	Plain words
$v_{c 1}$	inner circle par speed
$v_{c 2}$	outer circle par speed
$v_{p}$	transfer ellipse ke periapsis par speed
$v_{a}$	transfer ellipse ke apoapsis par speed

(square root) kyun? Square root woh tool hai jo squaring ko undo karta hai. Orbital energy naturally humein $v^{2}$ deti hai (kinetic energy $v^{2}$ use karti hai), isliye actual speed $v$ recover karne ke liye humein jo energy equation deta hai uska square root lena padta hai.

6. Master trade-off: $v^{2} = μ (\frac{2}{r} - \frac{1}{a})$

Yeh Vis-viva equation hai, woh ek law jo $v$ , $r$ , aur $a$ ko saath baandhta hai.

Humein sirf $v_{circ}$ nahi, yeh kyun chahiye? Ellipse par, $r$ fly karte waqt change hota hai, lekin $v_{circ}$ sirf tab kaam karta hai jab $r = a$ ho. Vis-viva woh general tool hai jo kisi bhi orbit ke kisi bhi point par $v$ deta hai — dekho Orbital energy & semi-major axis.

7. Jo cheez hum minimise karna chahte hain: $Δ v$

Yeh currency kyun hai: rocket fuel almost directly $Δ v$ par map hoti hai (rocket equation ke through, dekho Delta-v budget). Kam $Δ v$ = kam fuel = sasti mission. "Minimum energy transfer" ka matlab sach mein hai minimum total $Δ v$ .

8. Greek aur symbol quick-reference

Recall Poora symbol table (right side cover karo)

$M$ ::: mass of the central body
$G$ ::: universal gravitational constant
$μ = GM$ ::: gravitational parameter of the central body
$r$ ::: distance from the planet's centre to the craft
$r_{1}, r_{2}$ ::: inner / outer circular orbit radii
$a$ ::: semi-major axis = half the ellipse's long diameter = orbit "size"
$a_{t}$ ::: semi-major axis of the transfer ellipse $= (r_{1} + r_{2}) /2$
$v$ ::: orbital speed (magnitude)
$v_{circ} = μ / r$ ::: speed to hold a circle of radius $r$
$v_{p}, v_{a}$ ::: speeds at periapsis / apoapsis of the transfer ellipse
$Δ v$ ::: change in speed produced by a burn (our fuel currency)
$ε$ ::: specific orbital energy $= \frac{1}{2} v^{2} - μ / r$
$π$ ::: 3.14159..., appears in the transfer-time (half-orbit) formula

Yeh topic ko kaise feed karte hain

Equipment checklist

Khud ko test karo — answer zor se bolo, phir reveal karo:

μ

kya stand karta hai aur

G

aur

M

ko glue kyun karein?

μ = GM

; yeh hamesha saath aate hain, isliye ek symbol central body ki gravity strength capture karta hai.

Kya

r

ground se measure hota hai ya centre se?

Planet ke centre se:

r = R_{planet} + altitude

a

kya hai ek phrase mein?

Half the ellipse's longest diameter — orbit ki "size" aur energy fingerprint.

a_{t} = (r_{1} + r_{2}) /2

kyun hai?

r_{1}

par periapsis plus

r_{2}

par apoapsis long diameter

2 a_{t}

ke barabar hai.

Vis-viva state karo aur har term ka matlab batao.

v^{2} = μ (2/ r - 1/ a)

;

2/ r

= paas-hai-fast,

- 1/ a

= size-set tax.

v_{circ}

kya hai aur kab valid hai?

μ / r

, radius

r

ka circle hold karne ki speed — sirf jab

r = a

ho.

Δ v

kya measure karta hai aur hum ise minimise kyun karte hain?

Ek burn se speed jump ki size; yeh fuel par map hoti hai, isliye kam

Δ v

= sasti mission.

Hohmann burns plain numbers ki tarah kyun add ho sakte hain?

Burns tangent hote hain, isliye old aur new velocities collinear hoti hain — magnitudes subtract karo, koi vectors nahi chahiye.