Foundations — Hohmann Δv calculation — both maneuvers

3.2.20 · D1 · Physics › Orbital Mechanics & Astrodynamics › Hohmann Δv calculation — both maneuvers

Parent note Hohmann Δv calculation — both maneuvers padhne se pehle, tumhe kuch symbols mein fluent hona padega. Yeh page har ek ko bilkul zero se banata hai — pehle plain words mein, phir ek picture, phir yeh bhi ki topic uske bina kyun kaam nahi kar sakta.

1. Orbit, circle, ellipse — woh shapes jisme hum udte hain

Figure s01 dekho. Dono blue circles starting orbit (radius $r_1$) aur target orbit (radius $r_2$) hain. Dono ko chhooti coral oval transfer ellipse hai. Dhyan do ki yeh inner circle ko apne sabse paas wale point par chhooti hai aur outer circle ko apne sabse door wale point par — yeh chhoona hi woh cheez hai jo sirf do burns ko zaruri banata hai.

• $r_1$ ::: inner (starting) circular orbit ka radius — ellipse ka near point.
• $r_2$ ::: outer (target) circular orbit ka radius — ellipse ka far point.

Topic ko yeh kyun chahiye: har speed formula poochta hai "abhi centre se kitni door ho?" Woh distance hamesha $r_1$, $r_2$, ya beech ke kisi point mein se ek hoti hai.

2. Periapsis aur apoapsis — do special points

Ek comet ki picture karo: woh Sun ke paas se tezi se nikalta hai (periapsis) aur door wale end par dheere dheere chalta hai (apoapsis). Yahan bhi wahi idea hai.

• Symbol $v_p$ ::: spacecraft ki periapsis par speed (ellipse par sabse fast point).
• Symbol $v_a$ ::: spacecraft ki apoapsis par speed (sabse slow point).

Topic ko yeh kyun chahiye: dono burns bilkul periapsis aur apoapsis par hote hain, isliye $v_p$ aur $v_a$ woh "before/after" speeds hain jo tum circle speeds ke saath compare karte ho.

3. Semi-major axis $a$ — orbit ki "size"

Figure s02 dikhata hai ki transfer ellipse ke liye $2a=r_1+r_2$ kyun hai. Periapsis se chalo, planet ke centre (the focus) se hote hue, poore raaste apoapsis tak: tum poora lamba axis cross karte ho. Woh poori length $r_1+r_2$ hai, focus se maapi gayi. Uska aadha $a$ hai:

$a_t=\frac{r_1+r_2}{2}$

• Symbol $a$ ::: semi-major axis, kisi bhi orbit ka "size" number.
• Symbol $a_t$ ::: khaas taur par transfer ellipse ka semi-major axis.

Topic ko yeh kyun chahiye: master speed formula (vis-viva, neeche) jaanna chahta hai orbit kitni badi hai, aur $a$ woh number hai jo orbit size ko capture karta hai.

4. $G$, $M$, aur $\mu$ — planet kitna kheenchta hai

Hum $G$ aur $M$ alag alag ki jagah $\mu$ use karte hain kyunki gravity ka orbit par asar hamesha sirf product $GM$ ke roop mein aata hai. Dono ko saath leke chalna bekar mehnat hai, aur planet ke liye $M$ ek bada messy number hai — $\mu$ directly aur clearly measure kiya jaata hai.

• Symbol $\mu$ ::: $=GM$, central body ki pull-strength, units ${\text{km}}^3/{\text{s}}^2$.

Poori baat ke liye Standard gravitational parameter dekho.

5. Velocity vs. speed, aur $\Delta v$ — woh cheez jiske liye hum pay karte hain

Greek letter $\Delta$ (delta) ka matlab hamesha "change in" hota hai. Toh $\Delta v=|v_{\text{after}}-v_{\text{before}}|$ — velocity arrow kitna shift hua.

Yeh ek lucky break hai, Figure s03 mein dikhaya gaya: Hohmann transfer mein, dono burns direction of motion ke saath (prograde) hote hain. Before-arrow aur after-arrow ek hi taraf point karte hain, isliye unhe subtract karna mushkil vector arithmetic nahi rehta aur do numbers ki plain subtraction ban jaata hai:

$\Delta v=|v_{\text{after}}-v_{\text{before}}|=v_{\text{after}}-v_{\text{before}}(\text{same direction})$

• Symbol $v$ ::: ek speed (velocity arrow ki length), units km/s.
• Symbol $\Delta v$ ::: engine dwara di gayi speed change — yahi fuel khareedta hai.

Topic ko yeh kyun chahiye: $\Delta v$ hi pura point hai — yeh trip ki "cost" hai. Kam $\Delta v$ matlab kam fuel.

6. Centripetal force — circle ke liye andar ki taraf pull kyun chahiye

Orbit ke liye, andar ki taraf kheenchne wali sirf ek cheez hai — gravity. "Gravity supplied" ko "centripetal needed" ke barabar set karna hi circular speed ko janm deta hai:

$\underset{\text{gravity pulls in}}{\underbrace{\frac{\mu m}{r^2}}}=\underset{\text{circle needs}}{\underbrace{\frac{m{v}^2}{r}}}\Rightarrow v_{\text{circ}}=\sqrt{\frac{\mu }{r}}$

Yeh poori tarah Circular orbital velocity mein derive kiya gaya hai.

• Symbol $v_{c1}$ ::: radius $r_1$ par circular speed, $=\sqrt{\mu /r_1}$.
• Symbol $v_{c2}$ ::: radius $r_2$ par circular speed, $=\sqrt{\mu /r_2}$.

Topic ko yeh kyun chahiye: yeh dono circles ki speeds hain — burn 1 ka "before" aur burn 2 ka "after."

7. Energy aur vis-viva — kahin bhi speed ke liye ek formula

Us equality ko $v$ ke liye rearrange karne par woh tool milta hai jis par poora topic tika hai — vis-viva equation:

$v^2=\mu \left(\frac{2}{r}-\frac{1}{a}\right)$

Vis-viva equation aur Specific orbital energy dekho.

Topic ko yeh kyun chahiye: yeh $v_p$ deta hai ($r=r_1$, $a=a_t$ rakhne par) aur $v_a$ ($r=r_2$, $a=a_t$ rakhne par) — dono elliptical speeds jo kisi aur tarike se nahi milti.

8. Impulsive burn — ek burn ko instant maanna

Topic ko yeh kyun chahiye: yeh humein har burn ko exactly ek radius par hone wala treat karne deta hai, toh hum us spot par ek single clean speed use kar sakte hain rather than ek smeared-out thrust arc track karne ke bajay.

(Baad ke liye ek aside: gravity well mein fast aur neecha burn karne se extra energy nikalta hai — Oberth effect. Aur agar $r_2$, $r_1$ se bahut bada hai, toh teen-burn Bi-elliptic transfer Hohmann ko beat kar sakta hai.)

Foundations topic ko kaise feed karte hain

Top-down padho: $\mu$ aur radii dono circle speeds aur vis-viva ko feed karte hain; woh speeds subtract hoti hain (impulsive burns ke roop mein) har $\Delta v$ mein; dono $\Delta v$'s milke total ban jaate hain.

Equipment checklist

Khud test karo — dayi taraf cover karo.