Foundations — Patched conic method — interplanetary trajectory design

3.2.26 · D1 · Physics › Orbital Mechanics & Astrodynamics › Patched conic method — interplanetary trajectory design

Is parent note ko padhne se pehle patched conic method par, tumhe har woh symbol jo woh tum par fenkta hai pehle se obvious lagna chahiye. Yeh page har ek ko kuch nahi se build karta hai, ek aisi order mein jahan har idea usse pehle wale par tikaa hota hai.

---

0. "Central body" aur "conic" ka matlab kya hota hai

Figure dekho: wahi cone, chaar slices, chaar curves. Ek closed slice (ellipse) matlab object bound hai aur wapas aata rahega. Ek open slice (hyperbola) matlab woh hamesha ke liye ud jaata hai. Poori trip mein ek ellipse (Sun-bound cruise) use hoti hai jo do hyperbolas (har planet ke paas fly-away/fly-in) se glued hoti hai. Yeh mental image rakhna — neeche har symbol inhi curves mein se ek ko decorate karta hai.

---

1. $r$ — distance, sabse basic symbol

---

2. $a$ — semi-major axis, orbit ka "size"

Subscripts jo milenge: $a_p$ = ek planet ke orbit ka size (Sun se uski distance), $a_t$ = transfer ellipse ka size.

---

3. $m$, $M$, $G$ aur shortcut $\mu$

---

4. $v$ — speed, aur special speeds

Parent note kai speeds name karta hai — yeh sab sirf $v$ hai jo kisi khaas jagah ya kisi khaas body ke relative measure kiya gaya:

Symbol	Seedha matlab
$v_c$	kisi radius par circular orbit hold karne ki speed
$v_{esc}$	escape speed — bas itna kaafi ki hamesha ke liye nikal jao
$v_p$	perigee par speed (closest approach)
$v_{\infty }$	bahar nikalne ke baad infinitely door jaane par jo speed bachi, woh
$v_t$	transfer ellipse par speed

---

5. Energy: $\frac{1}{2}{v}^2$ aur $-\frac{\mu }{r}$ pieces

Yeh single conserved quantity isliye vis-viva formula (agle mein) exist karta hai, aur isliye well mein deep burn karna sasta hota hai.

---

6. Vis-viva equation (energy aur speed ke beech glue)

Do useful special cases seedhe nikalte hain:

• Circle ($r=a$): $v_c=\sqrt{\mu /r}$.
• Escape ($a\to \infty$, isliye $1/a\to 0$): $v_{esc}=\sqrt{2\mu /r}$ — $2$ isliye hai ki escape speed circular speed ka $\sqrt{2}\times$ hoti hai.

---

7. Powers aur roots: $2/5$,  ​, aur $r^5$

---

8. Circular-orbit geometry aur $\pi$

---

9. $\Delta v$ — spaceflight ki currency

---

Foundations topic ko kaise feed karte hain

Har foundation box ek symbol hai jo tumne abhi seekha; arrows dikhate hain ki woh parent-note ka kaunsa result power karta hai.

---

Related vault topics

Jab yeh symbols automatic feel hone lagein, parent directly inhi par lean karta hai: Sphere of influence, Hohmann transfer orbit, Hyperbolic escape trajectories, Oberth effect, aur timing ke liye Launch windows & synodic period.

---

Equipment checklist

Answers cover karo; tum ready ho jab har ek instant ho.

• $\mu$ matlab... ::: $G$ times central body ka mass — "pulling strength," isliye use karte hain taaki $G$ aur $M$ alag alag na carry karne paden.
• $r$ hamesha pehle ek question demand karta hai, yani... ::: "kis body se distance?" (cruise mein Sun, fly-by mein planet).
• Hohmann transfer ka semi-major axis hai... ::: $a_t=(r_1+r_2)/2$, near aur far distances ka average.
• Specific orbital energy ke do parts hain... ::: kinetic $v^2/2$ minus potential $\mu /r$; unka sum coast ke dauran constant hai.
• $\epsilon <0$, $=0$, $>0$ kaunse curves dete hain... ::: ellipse, parabola, hyperbola respectively.
• $v_{\infty }$ woh speed hai... ::: planet se infinitely door jaane par jo bachi, us planet ke relative measure ki gayi.
• Vis-viva equation hai... ::: $v=\sqrt{\mu \left(2/r-1/a\right)}$.
• Escape speed circular speed se ek factor se related hai... ::: $\sqrt{2}$ (escape $=\sqrt{2\mu /r}$, circular $=\sqrt{\mu /r}$).
• $x^{2/5}$ matlab... ::: fifth root lo aur square karo (ek $r^5$ equation ko undo karta hai).
• $\Delta v$ matlab... ::: ek burn se speed mein badlaav — ek manoeuvre ka fuel "cost."
• Ek elliptical period ka aadha hai... ::: $t=\pi \sqrt{a^3/\mu }$ (full period ka $2$ ka factor drop ho jaata hai).