Pork chop plots — Δv vs launch - arrival date

3.2.27 · Physics › Orbital Mechanics & Astrodynamics

HUM yeh plot KYUN chahte hain?

Toh grid ka har cell ek Lambert problem hai, aur uska solution ek Δv value deta hai. Plot sirf Δv ko poore grid par draw karna hai.

AXES aur CONTOURS par EXACTLY kya hota hai?

HAR GRID CELL COMPUTE KAISE HOTA HAI? (Derivation scratch se)

Hum machinery ko first principles se banate hain.

Step 1 — Planets kahan hain?

Ek date dene par, ephemerides position vectors $r_{1}$ (launch par Earth) aur $r_{2}$ (arrival par target) deti hain, dono heliocentric.

Yeh step kyun? Transfer orbit $r_{1}$ par shuru aur $r_{2}$ par khatam honi chahiye; yeh boundary conditions hain.

Hum woh heliocentric conic dhundhte hain jo $r_{1} \to r_{2}$ ko chosen time of flight $Δ t$ mein join kare. Lambert's theorem kehta hai ki TOF sirf inhi par depend karta hai: $Δ t = f (r_{1} + r_{2}, c, a)$ jahan $c = ∣ r_{2} - r_{1} ∣$ chord hai aur $a$ semi-major axis hai.

Yeh step kyun? Do positions + ek time orbit ko fully constrain karte hain (short-way / long-way choice tak). Lambert's equation solve karne se $a$ milta hai, jisse poora conic milta hai.

Conic se hume required velocity vectors milte hain:

$v_{1}$ = departure par spacecraft velocity (transfer orbit par),
$v_{2}$ = arrival par spacecraft velocity.

Step 3 — Δv mein convert karna

Spacecraft pehle se hi Earth ki orbital velocity $V_{E}$ ke saath chal raha hai. Toh Earth ke relative jo velocity use gain karni hai woh hai: $v_{\infty}^{d e p} = v_{1} - V_{E}, C_{3} = v_{\infty}^{d e p}^{2}$

Yeh step kyun? Aap us speed ke liye pay nahi karte jo Earth aapko free mein deti hai; aap sirf difference ke liye pay karte hain. Woh difference, Earth ke gravity well se bahar nikalne ke baad, $v_{\infty}$ hai.

Arrival par target planet ke relative excess speed hai: $v_{\infty}^{a r r} = v_{2} - V_{t a r g e t}$

Step 4 — Parking orbits se Departure & arrival Δv

Earth ke around radius $r_{p}$ ki circular parking orbit chhodne par, $v_{esc}^{2} = 2 μ_{E} / r_{p}$ aur departure hyperbola par energy conservation use karke: $v_{p er i} = v_{\infty}^{2} + \frac{2 μ _{E}}{r _{p}}$

Kyun? Hyperbola par, specific energy $ε = v_{\infty}^{2} /2 = v_{p er i}^{2} /2 - μ_{E} / r_{p}$ . Rearrange karne par perigee speed milti hai.

Phir circular parking-orbit speed $v_{c} = μ_{E} / r_{p}$ se burn: $Δ v_{d e p} = v_{\infty}^{2} + \frac{2 μ _{E}}{r _{p}} - \frac{μ _{E}}{r _{p}}$

Isi tarah target par capture ke liye (radius $r_{a}$ , gravity $μ_{T}$ ): $Δ v_{a r r} = v_{\infty, a r r}^{2} + \frac{2 μ _{T}}{r _{a}} - \frac{μ _{T}}{r _{a}}$

Us cell ki total cost: $Δ v_{t o t} = Δ v_{d e p} + Δ v_{a r r}$

Yeh poora plot kyun hai: Steps 1–4 har (launch, arrival) pair ke liye run karo → har cell ke liye ek Δv value → usse contour karo. Ho gaya.

Figure — Pork chop plots — Δv vs launch - arrival date

Plot padhna

Worked examples

Common mistakes (Steel-manned)

Forecast-then-Verify

Active-recall flashcards

Pork chop plot ke do axes kya hain?

Launch date (x) aur arrival date (y).

Contours usually kaunsi quantity dikhate hain?

C₃ (v∞²) ya total Δv.

C₃ define karo.

Characteristic energy = hyperbolic excess speed ka square, C₃ = v∞², units km²/s².

Δv dates par kyun depend karta hai?

Kyunki dono planets chalte hain; launch+arrival fix karne se unki positions aur TOF fix hote hain, jo Lambert transfer conic fix karta hai aur isliye required speeds bhi.

Har grid cell ke liye kaunsa problem solve hota hai?

Lambert's problem (r₁→r₂ ko diye hue TOF mein join karne wala conic dhundho).

Circular parking orbit se departure Δv ka formula?

Δv = √(v∞² + 2μ/rₚ) − √(μ/rₚ).

v∞ paane ke liye planet ki velocity kyun subtract karte hain?

Planet pehle se apni orbital velocity free mein deta hai; aap sirf uske relative difference ke liye pay karte hain.

Synodic period formula kya hai?

1/T_syn = |1/T₁ − 1/T₂|.

Earth–Mars synodic period approximately kitna hai?

~780 days (~26 months).

Type I aur Type II transfers mein kya fark hai?

Type I < 180° heliocentric sweep karta hai (shorter TOF); Type II > 180° sweep karta hai (longer TOF).

Pork chop ko do lobes mein split karne wali ridge kyun hoti hai?

180° transfer par plane undefined hoti hai, bada plane change force karta hai → Δv spike hota hai.

Kya shortest-TOF transfer sabse sasta hota hai?

Nahi; minimum Δv near-Hohmann hota hai aur comparatively slow hota hai.

Constant-TOF lines plot par kaisi lagti hain?

Diagonal lines (arrival − launch = const).

Recall Feynman: 12-saal ke bachche ko samjhao

Socho tum ek merry-go-round (Earth) par ho aur apne ek dost ko ball throw karna chahte ho jo ek bade, slow merry-go-round (Mars) par hai. Tum apne dost ke aaj ke position par nahi phenko — tum wahan phenko jahan woh hoga jab ball pohonchegi. Exactly kab throw karte ho aur kitna time ball ko lagta hai, iske hisab se tumhe alag strength aur direction se throw karna padta hai. Kuch moments mein halka toss chahiye (sasta), kuch mein zor se maaro (mehenga). Agar tum ek map banao: "din jab throw kiya" left-to-right, "din jab land kiya" bottom-to-top, aur har spot ko kitni zor se maarna pada usse color karo, to saste spots ek blob mein jam jaate hain jo pork chop ki tarah dikhti hai. Woh map hi pork chop plot hai!

Connections

Lambert's Problem — har grid cell ke peeche ka solver.
Hohmann Transfer — minimum-energy limit jo chop ke center ke paas baithta hai.
Hyperbolic Excess Velocity & C3 — departure-cost quantity.
Oberth Effect — kyun $2 μ / r_{p}$ term deep-gravity burns ko efficient banata hai.
Synodic Period — kyun windows har ~26 months mein repeat hote hain.
Patched Conic Approximation — woh framework jo humein departure/cruise/arrival alag alag treat karne deta hai.
Tsiolkovsky Rocket Equation — kyun chhoti Δv savings badi payload gains mein translate hoti hain.