Atmospheric drag — exponential atmosphere model, orbit decay: Question bank

3.2.34 · D5 · Physics › Orbital Mechanics & Astrodynamics › Atmospheric drag — exponential atmosphere model, orbit decay

Spot the error

"Kyunki

a_{d r a g} = \frac{1}{2} B ρ v^{2}

bahut chhota hai (~

1 0^{- 6}

m/s²), orbit lifetime ke liye drag negligible hai."

Magnitude har second ke liye chhota hai, lekin yeh mahino tak continuously act karta hai; cumulative energy loss enormous hai, isliye drag ultimately kisi bhi LEO satellite ki fate dominate karta hai.

"Main lifetime pane ke liye 400 km value par

ρ

fix karke decay integrate karunga."

Galat —

ρ (h) = ρ_{0} e^{- (h - h_{0}) / H}

altitude girne par orders of magnitude badal jaata hai; ek fixed

ρ

runaway final phase ko bahut badly underestimate karta hai. Tumhe

ρ (a)

use karna hoga.

"Drag speed churaata hai, aur kam speed ka matlab hai satellite ek higher, slower orbit mein climb karti hai."

Ulta hai — kam energy ka matlab hai ek chhoti orbit, aur chhoti orbit tezi hoti hai. Energy aur speed yahaan opposite directions mein move karte hain; sirf altitude aur energy girti hai.

H = k_{B} T / (m_{m o l} g)

, isliye jaise satellite neeche jaati hai aur

g

badhta hai,

H

bhi badhna chahiye."

H \propto 1/ g

, isliye bada

g

ek chhota

H

deta hai; lekin near-surface warming/cooling

T

ko bahut zyada badalta hai, isliye real profiles temperature structure se dominate hoti hain,

g

se nahi.

"Drag paradox energy conservation violate karta hai — satellite energy khokar speed up hoti hai."

Koi violation nahi — kinetic energy badhti hai lekin potential energy

- μ / a

do guni tezi se girti hai, isliye total

ε = - μ /2 a

decrease hoti hai. Khoyi hui energy atmosphere aur satellite ko heat karne mein jaati hai.

"Kyunki drag motion ka virodh karta hai, orbit zyada eccentric ho jaati hai jaise perigee drag down hota hai."

Near-circular orbit ke liye drag mainly use circularize karta hai — yeh perigee par sabse zyada bite karta hai (ghani hawa), apogee ko lower karta hai aur orbit ko rounder pull karta hai decay se pehle; yeh eccentricity pump up nahi karta.

"Ballistic coefficient

B = C_{D} A / m

sirf atmosphere ki ek property hai."

B

puri tarah satellite par depend karta hai — uska drag coefficient, cross-sectional area, aur mass. Atmosphere alag se

ρ

ke through enter karta hai. (Dekho Reentry Aerodynamics & Ballistic Coefficient.)

Why questions

ρ

ko altitude mein linear ki jagah exponential kyun model kiya jaata hai?

Kyunki hydrostatic balance aur ideal gas law milke

d ρ / ρ = - d h / H

dete hain, jiska ek hi solution exponential hai; density har fixed height par ek constant fraction se girti hai, constant amount se nahi.

Drag eccentric orbit ke liye almost entirely perigee ke paas kyun act karta hai?

Density exponential hai, isliye perigee aur apogee ke beech mein ek modest altitude difference bhi perigee par

ρ

ko enormously bada kar deta hai; drag

\propto ρ v^{2}

wahaan concentrate hota hai, jaise har pass par ek brief "brake tap."

Hum drag term

v v

kyun likhte hain

v^{2} \overset{v}{^}

ki jagah?

Yeh identical hain (

v = v \overset{v}{^}

), lekin

v v

equations of motion mein differentiate/integrate karna zyada clean hai aur

v^{2}

magnitude aur

- \overset{v}{^}

direction dono explicit karta hai.

Decay "catastrophically" end mein "accelerate" kyun karti hai?

Jaise

a

shrink hota hai satellite bahut denser hawa mein girta hai;

ρ

exponentially badhta hai jabki

μ a

barely change hota hai, isliye

∣ d a / d t ∣ = B ρ μ a

run away karta hai — last kuch kilometers minutes mein hote hain. Dekho Perturbations in Orbital Mechanics.

Drag decay describe karne ke liye humein vis-viva / two-body relations ki zaroorat kyun hai?

Drag sirf humein energy loss ki rate batata hai; use altitude change mein convert karne ke liye hume

ε = - μ /2 a

aur

v = μ / a

chahiye Two-Body Problem aur Vis-Viva Equation se, jo energy, speed, aur orbit size ko link karte hain.

Thermosphere ka scale height (~60 km) ground ke (~8 km) se itna zyada kyun hai?

H = k_{B} T / (m_{m o l} g)

; thermosphere bahut hot hai (

T \sim 1000

K) aur light atomic species se bana hai, isliye yeh "puff up" karta hai — bada

T

aur chhota

m_{m o l}

dono

H

badhate hain.

Orbit se energy remove karne se satellite simply kyun nahi rukta?

Kyunki ek bound orbit ki energy negative hai; use aur zyada negative karne se orbit shrink hoti hai aur (

v = μ / a

ke zariye) body speed up hoti hai — yeh sirf "ruk" sakti hai agar atmosphere itni dense ho ki reenter ho, coast karke rest mein nahi.

Edge cases

d a / d t = - B ρ μ a

perfect vacuum (

ρ = 0

) mein kya deta hai?

Zero — koi atmosphere nahi hai toh koi drag nahi aur orbit hamesha ke liye stable hai; poori decay story

ρ > 0

se driven hai.

a \to R_{\oplus}

(surface grazing) limit mein decay rate ka kya hota hai?

ρ

bahut saare

e

-foldings se badh gayi hai aur enormous hai, isliye

∣ d a / d t ∣

enormous hai; physically satellite pehle hi dense lower atmosphere mein pahunch chuka hai aur burn up ho jaata hai — near-circular model yahaan break down kar deta hai.

Agar

C_{D} = 0

(ek perfectly slippery hypothetical body) ho, toh orbit lifetime kya hai?

Infinite —

B = C_{D} A / m = 0

a_{d r a g} = 0

banata hai, isliye koi energy lost nahi hoti aur orbit kabhi decay nahi karti, density regardless.

Bahut high altitude par jahan

ρ \approx 0

lekin nonzero hai, model ko kya limit karta hai?

Decay astronomically slow hai lekin exactly zero nahi; aur extreme altitudes par mean free path huge hai (free-molecular flow), isliye continuum drag formula

\frac{1}{2} B ρ v^{2}

sirf ek approximation ban jaata hai.

Agar satellite tumble kare, apna cross-section

A

time ke saath badle, toh kya hoga?

Tab

B = C_{D} A / m

constant nahi hai; effective drag tumble par average hota hai, aur ek unpredictable spinning body ki lifetime ek stable, fixed-attitude wali se predict karna bahut mushkil hota hai.

Drag paradox ke exact center par — satellite ki kinetic energy badhti hai ya girti hai?

Kinetic energy badhti hai (

v

badhti hai jaise

a

girta hai), even though total energy girti hai; extra KE potential energy

- μ / a

mein bahut badi drop se aati hai.

Recall Yahaan har trap ki ek-line summary

Drag energy lower karta hai → $a$ lower karta hai → speed badhata hai; density exponential hai isliye decay nonlinear aur perigee-focused hai; $B$ satellite ki property hai, $ρ$ atmosphere ki; aur kuch bhi $ρ > 0$ aur $C_{D} > 0$ ke bina decay nahi karta.

Question bank — Atmospheric drag — exponential atmosphere model, orbit decay

True or false — justify

Spot the error

Why questions

Edge cases