Hohmann Δv calculation — both maneuvers: Visual walkthrough

3.2.20 · D2 · Physics › Orbital Mechanics & Astrodynamics › Hohmann Δv calculation — both maneuvers

Hum ek running story chalate hain: ek chhoti craft inner circle par hai aur outer circle tak pahunchna chahti hai. Neeche har step ek naya idea aur ek nayi picture add karta hai.

Step 1 — Do circles aur ek oval jo dono ko kiss karta hai

KYA. Hum scene set karte hain. Teenh paths hain, do nahi:

inner circle jiska radius $r_{1}$ hai (jahan se hum shuru karte hain),
outer circle jiska radius $r_{2}$ hai (jahan hum pahunchna chahte hain, $r_{2} > r_{1}$ ke saath),
ek ellipse (ek stretched oval) jo inner circle ko uske sabse kareeb point par aur outer circle ko uske sabse door point par sirf touch karta hai.

KYUN. Tum ek circle se doosri par teleport nahi kar sakte. Ek freely falling object hamesha ek Kepler path (circle ya ellipse) follow karta hai. Sabse sasta bridge jo dono circles ko tangentially touch kare (jahan bhi touch kare wahan circle ke same direction mein move kare) yahi ek ellipse hai. Tangential touching isliye matter karta hai kyunki tab engine sirf speed change karta hai, kabhi direction nahi — koi bhi thrust velocity ko side mein modhne mein waste nahi hota.

PICTURE. Magenta inner circle, violet outer circle, aur orange transfer ellipse. Dekho ki ellipse ka sabse kareeb point (periapsis) inner circle par baitha hai, uska sabse door point (apoapsis) outer circle par baitha hai. Planet shared focus par orange dot hai.

Step 2 — Circle par speed (gravity = woh inward pull jo tumhe chahiye)

KYA. Pata karo ki radius $r$ ke circle par tum kitni tez chalte ho. Use $v_{circ}$ kaho.

KYUN. Dono burns ek circle-speed ko ek ellipse-speed se compare karte hain usi radius par. To pehle circle-speed hi chahiye. Yeh ek idea se aata hai: ek circle par, gravity exactly utna inward force supply karta hai jitna tumhare path ko circle mein modhne ke liye zarurat hai.

Un dono forces ko equal set karo (dekho Circular orbital velocity):

$gravity pulling in \frac{GM m}{r ^{2}} = inward force a circle needs \frac{m v ^{2}}{r}$

$G$ = gravitation constant, $M$ = planet mass, $m$ = craft mass — milkar pull bante hain.
$\frac{m v ^{2}}{r}$ = centripetal force: speed $v$ par radius $r$ par curve karte rehne ke liye tumhe kitni zor se inward kheencha jaana chahiye.

$m$ cancel karo, dono sides ko $r$ se multiply karo:

$v_{circ}^{2} = \frac{GM}{r} = \frac{μ}{r} \Rightarrow v_{circ} = \frac{μ}{r}$

$μ = GM$ standard gravitational parameter hai — hum $G$ aur $M$ ko bundle karte hain kyunki humhe inhe kabhi alag-alag nahi chahiye.

PICTURE. Inward gravity arrow aur "needed inward force" arrow ek circle par same length ke hain — wahi balance ek circular orbit hai. Radius jitna bada, $μ / r$ utna chhota, to outer circles slower hoti hain.

Recall Outer circle slow kyun hoti hai?

Kyunki $v_{circ} = μ / r$ $r$ badhne par ghatta hai ::: door jaane par gravity kamzor hoti hai, isliye andar girne se bachne ke liye tumhe kam speed chahiye — bahari merry-go-round dheere ghoomti hai.

Step 3 — Kisi bhi orbit par kahin bhi speed (vis-viva)

KYA. Ek master formula nikalo speed $v$ ke liye, kisi bhi distance $r$ par, kisi bhi diye hue size ke orbit ke liye.

KYUN. Ellipse par speed badlati rehti hai — planet ke paas fast, door slow. Ek akela circle formula yeh handle nahi kar sakta. Humhe ek aisa rule chahiye jo position ke function ke roop mein speed de. Woh rule vis-viva hai, aur yeh seedha energy bookkeeping se aata hai (dekho Specific orbital energy aur Vis-viva equation).

Craft ki energy per kilogram (uski specific energy $ε$ ) kinetic minus ek gravity term hai:

$ε = motion energy \frac{v ^{2}}{2} - depth in the well \frac{μ}{r} = - \frac{μ}{2 a}$

$\frac{v ^{2}}{2}$ = kinetic energy per kg.
$- \frac{μ}{r}$ = gravitational potential per kg (negative: tum ek "well" mein ho, deeper = zyada negative).
$a$ = semi-major axis, orbit ki lambi width ka aadha — yeh akela total energy fix karta hai. Yeh ek Kepler result hai: same $a$ ⇒ same energy, shape chahe koi bhi ho.

$v^{2}$ ke liye solve karo:

$v^{2} = μ (\frac{2}{r} - \frac{1}{a})$

$a = r$ plug karo (ek circle) aur tumhe milta hai $μ (\frac{2}{r} - \frac{1}{r}) = \frac{μ}{r}$ — Step 2 wapas aa jaata hai. ✔

PICTURE. Ellipse par teen spots par energy ka ek bar chart. Total height ( $ε$ ) har jagah same hai (flat dashed line), lekin split shift hoti hai: planet ke paas, bahut saari kinetic + bahut gehri potential; door, thodi kinetic + shallow potential.

Step 4 — Transfer ellipse ka size

KYA. Hamare specific bridge ellipse ka semi-major axis $a_{t}$ nikalo.

KYUN. Vis-viva ko $a$ chahiye. Hamare ellipse ki lambi axis seedhi periapsis (planet se $r_{1}$ door) se apoapsis (planet se $r_{2}$ door) tak jaati hai, dono same focus se — planet ke center se — measure kiye gaye.

$full long axis 2 a_{t} = near side r_{1} + far side r_{2} \Rightarrow a_{t} = \frac{r _{1} + r _{2}}{2}$

$2 a_{t}$ ellipse ki lambi taraf ki poori length hai; use aadha karne par $a_{t}$ milta hai, do radii ka average.

PICTURE. Poori lambi axis ek seedhi line ke roop mein planet se guzarti hui draw ki gayi hai, chhoti side par $r_{1}$ label hai aur lambi side par $r_{2}$ , aur $a_{t}$ poore ka aadha mark kiya gaya hai.

Step 5 — Chaar speeds jinhein hum compare karenge

KYA. Woh chaar speeds line up karo jo matter karti hain, do per burn.

KYUN. Har burn ek fixed radius par hota hai, wahan ki circle speed ko ellipse speed se compare karte hue. Yeh kul chaar numbers hain.

Kahan	Speed	Kaun sa formula	Role
Inner circle	$v_{c 1} = μ / r_{1}$	Step 2 at $r_{1}$	burn 1 se pehle
Ellipse periapsis	$v_{p} = μ (\frac{2}{r _{1}} - \frac{1}{a _{t}})$	Step 3 at $r = r_{1}$	burn 1 ke baad
Ellipse apoapsis	$v_{a} = μ (\frac{2}{r _{2}} - \frac{1}{a _{t}})$	Step 3 at $r = r_{2}$	burn 2 se pehle
Outer circle	$v_{c 2} = μ / r_{2}$	Step 2 at $r_{2}$	burn 2 ke baad

Inner radius par, ellipse inner circle se faster hai ( $v_{p} > v_{c 1}$ ): uske paas extra energy hai khud ko bahar fling karne ke liye.
Outer radius par, ellipse outer circle se slower hai ( $v_{a} < v_{c 2}$ ): woh apoapsis tak rengta hua aaya hai aur charhte waqt speed kho chuka hai.

PICTURE. Chaar dots ke saath ek speed-vs-radius plot. Magenta circle-speed curve aur orange ellipse-speed curve ka behaviour aapas mein cross karta hai: $r_{1}$ par orange magenta ke upar baithta hai; $r_{2}$ par orange magenta ke neeche baithta hai.

Step 6 — Burn 1: periapsis kick

KYA. Pehla velocity change, $r_{1}$ par.

KYUN. Tum inner circle par $v_{c 1}$ par ho; ellipse mein enter karne ke liye $v_{p}$ tak pahunchna zaroori hai, jo bada hai. To tum engine prograde (aage, apni motion ke saath) fire karte ho — pure speed-up, koi turn nahi (velocities parallel hain), isliye vector change speeds ke subtraction mein collapse ho jaata hai.

$Δ v_{1} = v_{p} - v_{c 1} = \frac{μ}{r _{1}} (\frac{2 r _{2}}{r _{1} + r _{2}} - 1)$

$μ / r_{1}$ = inner circle speed, hamaari baseline.
$\frac{2 r _{2}}{r _{1} + r _{2}}$ = boost factor; door wala radius $r_{2}$ upar hai kyunki tum bahar pahunch rahe ho. Kyunki $r_{2} > r_{1}$ yeh factor $1$ se zyada hai, isliye bracket positive hai.
" $- 1$ " woh piece hai jo hmare paas pehle se tha (circle speed jisse hum shuru karte hain).

PICTURE. $r_{1}$ par: ek chhota magenta arrow (current $v_{c 1}$ ) aur ek lamba orange arrow (target $v_{p}$ ) dono same direction mein point kar rahe hain; extra orange length $Δ v_{1}$ hai, violet mein added thrust ke roop mein draw ki gayi.

Step 7 — Burn 2: upar circularize karo

KYA. Doosra velocity change, $r_{2}$ par.

KYUN. Tum ellipse par coast karte hue apoapsis tak pahunchte ho aur $v_{a}$ par aate ho, jo outer circle ke $v_{c 2}$ se slower hai. Agar tum kuch nahi karte to tum ellipse par wapas gir jaate. Isliye tum phir se prograde fire karte ho $v_{a}$ se $v_{c 2}$ tak top up karne ke liye — ek aur pure speed-up.

$Δ v_{2} = v_{c 2} - v_{a} = \frac{μ}{r _{2}} (1 - \frac{2 r _{1}}{r _{1} + r _{2}})$

$μ / r_{2}$ = outer circle speed, hamaara target baseline.
$\frac{2 r _{1}}{r _{1} + r _{2}}$ = apoapsis-speed factor; ab kareeb wala radius $r_{1}$ upar hai (Step 6 ka mirror). Kyunki $r_{1} < r_{2}$ yeh $1$ se neeche hai, isliye bracket positive hai — tum speed add karte ho.
Order $v_{c 2} - v_{a}$ hai (circle minus ellipse) kyunki apoapsis par ellipse slower wali hai.

PICTURE. $r_{2}$ par: ek chhota orange arrow ( $v_{a}$ ) aur ek lamba magenta arrow ( $v_{c 2}$ ), same direction; violet gap $Δ v_{2}$ hai.

Step 8 — Edge aur limiting cases (koi gap mat chhodho)

KYA. Corners check karo taaki koi scenario surprise na kare.

KYUN. Ek derivation jis par tum trust karo, use apne extremes mein survive karna chahiye.

$r_{2} = r_{1}$ (koi transfer nahi). Tab $a_{t} = r_{1}$ , "ellipse" circle hai, $v_{p} = v_{a} = v_{c 1} = v_{c 2}$ , isliye $Δ v_{1} = Δ v_{2} = 0$ . Jahan tum already ho wahan "move" karne ka zero cost. ✔
$r_{2} \to \infty$ (escape-like reach). Burn 1 ka factor $\frac{2 r _{2}}{r _{1} + r _{2}} \to 2$ , isliye $Δ v_{1} \to v_{c 1} (2 - 1)$ — exactly woh extra kick jo circular speed se escape speed tak jaata hai. Iske saath $v_{c 2} \to 0$ aur $v_{a} \to 0$ , isliye $Δ v_{2} \to 0$ : infinity par circularize karne ke liye koi orbit nahi hai. Saara cost burn 1 mein hai. ✔
Andar jaana ( $r_{2} < r_{1}$ ). Wohi algebra chalti hai, lekin ab dono brackets sign flip kar lete hain — burns retrograde ho jaate hain (tum do baar slow down karte ho). Magnitudes $Δ v$ abhi bhi wohi hain jo formulas dete hain (absolute value lo); direction ulta ho jaata hai. Isliye Hohmann dono taraf kaam karta hai.
Impulsive assumption. Ye sab har burn ko ek single point par instantaneous treat karta hai — Impulsive maneuver approximation. Real engines ek arc par burn karte hain, jo actual cost thodi badhata hai; Oberth effect explain karta hai ki deep aur fast burn karna (jaise burn 1) energetically efficient kyun hota hai. Bahut bade ratios $r_{2} / r_{1}$ ke liye, ek Bi-elliptic transfer Hohmann ko bhi beat kar sakta hai.

PICTURE. Do mini-panels: left, woh degenerate case jahan teeno paths ek circle par collapse ho jaate hain ( $Δ v = 0$ ); right, $r_{2} \to \infty$ case jahan ellipse near-parabola mein khul jaata hai aur burn 2 gayab ho jaata hai.

Ek-picture summary

Sab kuch compressed: do circles, transfer ellipse, dono prograde arrows unke radii par, aur dono speed-differences unke formulas ke saath label kiye gaye. Ek baar trace karo aur tumne poori cheez re-derive kar li.

Recall Poore walkthrough ki Feynman retelling

Tum ek chhote, tez loop par ek planet ka chakkar kaat rahe ho. Ek oval draw karo jiska kareeb ka end tumhara loop kiss kare aur door ka end ek bada loop kiss kare jahan tum jaana chahte ho. Ek baar aage push karo jahan tum ho: yeh describe karne mein sasta hai lekin sabse bada push hai, kyunki neeche sab kuch tezi se move karta hai aur tumhe oval ke saath bahar fling hone ke liye real speed add karni padti hai. Coast karo — tum oval climb karte ho, poore raste slow hote hue, jab tak uske door waale end tak nahi pahunch jaate, ab itna slow ki wahan bahar reh nahi sakte. Dobara aage push karo apni speed ko bade loop ke saath top up karne ke liye, aur tum usme settle ho jaate ho. Do forward pushes, unke sizes add karo, ho gaya. Pehla push bada isliye hota hai kyunki yeh itni door pahunchne ki energy inject karne ka bhaari kaam karta hai, wahin deep jahan gravity sabse strong hoti hai.

Active Recall

Exactly ek ellipse kyun, seedha hop kyun nahi?

Free-fall paths Kepler conics hain; dono circles ke tangent ellipse sabse sasta two-impulse bridge hai.

r_{1}

par, ellipse circle se faster hai ya slower?

Faster —

v_{p} > v_{c 1}

, isliye burn 1 ek speed-up hai.

r_{2}

par, ellipse circle se faster hai ya slower?

Slower —

v_{a} < v_{c 2}

, isliye burn 2 bhi ek speed-up hai.

r_{2} \to \infty

par

Δ v_{1}

ka kya hota hai?

Yeh

v_{c 1} (2 - 1)

tak pahunchta hai, circular-to-escape kick.

r_{2} = r_{1}

par

Δ v_{2}

kya hota hai?

Zero — koi transfer nahi chahiye.

LEO→GEO ke liye kaun sa burn dominate karta hai aur kyun?

Burn 1, kyunki yeh gravity well mein deep rehte hue door apoapsis tak pahunchne ki energy inject karta hai.