MARYLAND Orbital Mechanics

Orbital Mechanics • Orbital Mechanics, continued – Time in orbits – Velocity components in orbit – Deorbit maneuvers – Atmospheric density models – Orbital decay (introduction) • Fundamentals of Rocket Performance – The rocket equation – Mass ratio and performance – Structural and payload mass fractions – Multistaging – Optimal ΔV distribution between stages (introduction)

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Time in Orbit

• Period of an orbit a3 P = 2π µ • Mean motion (average angular velocity) µ n = a3 • Time since pericenter passage M = nt = E − esin E ➥M=mean anomaly E=eccentric anomaly U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Dealing with the Eccentric Anomaly

• Relationship to orbit r = a(1− e cosE) • Relationship to true anomaly θ 1+ e E tan = tan 2 1− e 2 • Calculating M from time interval: iterate

Ei+1 = nt + esin Ei until it converges

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Example: Time in Orbit

• Hohmann transfer from LEO to GEO

– h1=300 km --> r1=6378+300=6678 km

– r2=42240 km • Time of transit (1/2 orbital period)

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Example: Time-based Position

Find the spacecraft position 3 hours after perigee

E=0; 1.783; 2.494; 2.222; 2.361; 2.294; 2.328; 2.311; 2.320; 2.316; 2.318; 2.317; 2.317; 2.317

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Example: Time-based Position (continued)

Have to be sure to get the position in the proper quadrant - since the time is less than 1/2 the period, the spacecraft has yet to reach apogee --> 0°<θ<180°

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Velocity Components in Orbit

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Velocity Components in Orbit (continued)

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Atmospheric Decay

• Entry Interface – Altitude where aerodynamic effects become significant (acc~0.05 g) – Typically 400,000 ft ~ 80 mi ~ 120 km • Exponential Atmosphere – – Good for selected regions of atmosphere

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Atmospheric Density with Altitude

Ref: V. L. Pisacane and R. C. Moore, Fundamentals of Space Systems Oxford University Press, 1994

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Acceleration Due to Atmospheric Drag

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Orbit Decay from Atmospheric Drag

Ref: Alan C. Tribble, The Space Environment Princeton University Press, 1995

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Derivation of the Rocket Equation

Ve m-Δm v • Momentum at time t: m Δm v+Δv M = mv v-Ve • Momentum at time t+Δt:

M = (m − Δm)(v + Δv) + Δm(v − Ve ) • Some algebraic manipulation gives:

mΔv = −ΔmVe • Take to limits and integrate:

m final dm Vfinal  dv   = −   ∫m   ∫V initial m initial  Ve  U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design The Rocket Equation

• Alternate forms ΔV − m  m final  final Ve r ≡ = e ΔV = −Ve ln  = −Ve ln(r) minitial  minitial  • Basic definitions/concepts m m – Mass ratio r ≡ final orℜ = initial minitial mfinal – Nondimensional velocity change ΔV “Velocity ratio” Ve

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Rocket Equation (First Look)

1 1 Typical Range for Launch to ) Low Earth Orbit initial

M 0.1 / final M (

0.01 Ratio, Mass

0.001 0 1 2 3 4 5 6 7 8 Velocity Ratio, (ΔV/Ve)

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Sources and Categories of Vehicle Mass

Payload Propellants Inert Mass Structure Propulsion Avionics Mechanisms Thermal Etc.

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Basic Vehicle Parameters • Basic mass summary

m0 = mL + mp + mi m0 = initial mass

• Inert mass fraction mL = payload mass m m m = propellant mass δ = i = i p m = inert mass m0 mL + mp + mi i • Payload fraction m m λ = L = L m0 mL + mp + mi • Parametric mass ratio r = λ +δ U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design

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€ Rocket Equation (including Inert Mass)

1 Typical Range ) for Launch to

initial Low Earth Orbit M

/ Inert Mass 0.1 Fraction δ 0.00 payload 0.05

M 0.10 ( 0.15 0.20

0.01 Fraction,

0.001

Payload 0 1 2 3 4 5 6 7 8 Velocity Ratio, (ΔV/Ve)

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design The Rocket Equation for Multiple Stages

• Assume two stages   mfinal,1 ΔV1 = −Ve,1 ln  = −Ve,1 ln(r1)  minitial,1    mfinal,2 ΔV2 = −Ve,2 ln  = −Ve,2 ln(r2 )  minitial,2 

• Assume Ve,1=Ve,2=Ve

ΔV1 + ΔV2 = −Ve ln(r1 )−Ve ln(r2 ) = −Ve ln(r1r2 )

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design

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€ Continued Look at Multistaging

• Converting to masses   m final,1 m final,2 ΔV1 + ΔV2 = −Ve ln(r1r2 ) = −Ve ln   minitial,1 minitial,2 

• Keep in mind that mfinal,1~minitial,2   m final,2 ΔV1 + ΔV2 = −Ve ln(r1r2 ) = −Ve ln  = −Ve ln(r0 )  minitial,1 

• r0 has no physical significance!

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design

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€ Multistage Vehicle Parameters

• Inert mass fraction m n stages j−1 ∑ i, j   δ0 = = ∑ δ j ∏ λl m0 j=1  l =1  • Payload fraction n stages mL λ0 = = ∏λi m0 i=1 • Payload mass/inert mass ratio λ0

δ 0

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Effect of Staging

Inert Mass Fraction δ=0.2 ) 0.35 initial

M 0.3 /

0.25

payload 1 stage

M 0.2 ( 2 stage 0.15 3 stage 4 stage 0.1 Fraction, 0.05

Payload 0 0.5 1 1.5 2 2.5

Velocity Ratio, (ΔV/Ve)

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Effect of ΔV Distribution

1st Stage: LOX/LH2 2nd Stage: LOX/LH2

1.000

0.800

0.600

lambda0 0.400 delta0 lamd0/del0

0.200

0.000 0 2000 4000 6000 8000 10000

-0.200

1st Stage Delta-V (m/sec)

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Lagrange Multipliers

• Given an objective function y = f (x) subject to constraint function z = g(x) • Create a new objective function y = f (x) + λ[g(x) − z] • Solve simultaneous equations ∂y ∂y = 0 = 0 ∂x ∂λ U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Optimum ΔV Distribution Between Stages

• Maximize payload fraction (2 stage case)

λ0 = λ1λ2 = (r1 − δ1 )(r2 −δ 2 ) subject to constraint function

ΔVtotal = ΔV1 + ΔV2 • Create a new objective function ΔV ΔV − 1 − 2 Ve1 Ve2 λ0 = (e −δ1)(e − δ2 ) + K[ΔV1 + ΔV2 − ΔVTotal ] ➥Very messy for partial derivatives!

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Optimum ΔV Distribution (continued)

• Use substitute objective function

max(λ0 ) ⇔ max[ln(λ0 )] • Create a new constrained objective function

ln(λ0 ) = ln(r1 − δ1 ) + ln(r2 −δ 2 )

+K[ΔVTotal + Ve1 ln(r1 ) + Ve2 ln(r2 )] • Take partials and set equal to zero ∂ ln(λ ) ∂ ln(λ ) ∂ ln(λ ) 0 = 0 0 = 0 0 = 0 ∂r1 ∂r2 ∂K

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Optimum ΔV Special Cases • “Generic” partial of objective function ∂ ln(λ ) 1 V 0 = + K ei = 0 ∂ri ri − δi ri

• Case 1: δ1=δ2 Ve1=Ve2

r1 = r2 ⇒ ΔV1 = ΔV2

• Case 2: δ1≠δ2 Ve1=Ve2 r r 1 = 2 δ1 δ 2 • More complex cases have to be done numerically

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Sensitivity to Inert Mass

ΔV for multistaged rocket n stages n  m   o,k  ΔVtot = ∑ΔVk = ∑Ve,k ln  k=1 k=1  m f ,k 

mo,k = mL + m p,k + mi,k + ∑m p, j + mi, j j = k+1

mf ,k = mL + mi,k + ∑m p, j + mi, j j= k+1

∂ΔVtot ∂ΔVtot dmL + dmi, j = 0 ∂mL ∂mi, j

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design

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€ Trade-off Ratio: Payload<–>Inert Mass

k  1 1  −∑Ve, j  −  ∂m j=1  m0, j mf , j  L = N ∂mi,k  1 1  ∂ΔVTotal =0 V  −  ∑ e,l m m l =1  0,l f ,l 

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Trade-off Ratio : Payload<–>Propellant

k  1  V −∑ e, j   ∂m j =1  m0, j  L = N ∂mp,k  1 1  ∂ΔVTotal =0 V  −  ∑ e,l m m l =1  0,l f ,l 

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Trade-off Ratio: Payload<–>Exhaust Velocity

k  m  ∑ln 0,k  ∂m j=1  mf ,k  L = N ∂Ve,k  1 1  ∂ΔVTotal =0 V  −  ∑ e,l m m l =1  0,l f ,l 

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Trade-off Ratio Example: Gemini-Titan II

Stage 1 Stage 2

Initial Mass (kg) 150,500 32,630

Final Mass (kg) 39,370 6099

Ve (m/sec) 2900 3097

dmL/dmi,j -0.1164 -1

dmL/dmp,j 0.04124 0.2443

dmL/dVe,j (kg/m/sec) 2.870 6.459

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Parallel Staging

• Multiple dissimilar engines burning simultaneously • Frequently a result of upgrades to operational systems • General case requires “brute force” numerical performance analysis

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Parallel-Staging Rocket Equation

• Momentum at time t: M = mv • Momentum at time t+Δt: (subscript “b”=boosters; “c”=core vehicle)

• Assume thrust (and mass flow rates) constant

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Parallel-Staging Rocket Equation

• Rocket equation during booster burn  m   m + m + χ m + m   final   i,b i,c p,c o,2  ΔV = −Ve ln  = −Ve ln   minitial   mi,b + m p,b + mi,c + mp,c + mo,2 • where

Ve,b m˙ b +Ve,c m˙ c Ve,bm p,b +Ve,c (1− χ)m p,c Ve = = m˙ b + m˙ c m p,b + (1− χ)m p,c

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design

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€ Analyzing Parallel-Staging Performance

Parallel stages break down into pseudo-serial stages: • Stage “0” (boosters and core)  m + m + χ m + m   i,b i,c p,c o,2  ΔV0 = −Ve ln   mi,b + m p,b + mi,c + m p,c + mo,2 • Stage “1” (core alone)  m m   i,c + o,2  ΔV1 = −Ve,c ln   mi,c + χ m p,c + mo,2 • Subsequent stages are as before

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design

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€ Modular Staging

• Use identical modules to form multiple stages • Have to cluster modules on lower stages to make up for nonideal ΔV distributions • Advantageous from production and development cost standpoints

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design Module Analysis

• All modules have the same inert mass and propellant mass • Because δ varies with payload mass, not all modules have the same δ! • Introduce two new parameters m m ε = i σ = i mi + m p m p

δ δ • Conversions ε = σ = 1− λ 1−δ − λ

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design

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€ € Rocket Equation for Modular Boosters

• Assuming n modules in stage 1, m nε + o,2 n(mi ) + mo,2 mo,mod r1 = = n m m m mo,2 ( i + p ) + o,2 n + mo,mod

• If all 3 stages use same modules, nj for stage j,

n1ε + n2 + n 3 + ρL € r1 = n1 + n2 + n 3 + ρL

mL • where ρL = ; mmod = mi + mp mmod

U N I V E R S I T Y O F Orbital Mechanics MARYLAND Launch and Entry Vehicle Design

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