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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 © 2004 David L. Akin - All rights reserved Launch and Entry Vehicle Design MARYLAND http://spacecraft.ssl.umd.edu Calculating Time in Orbit 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 € € 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 € € € 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 € € Multistage Vehicle Parameters • Inert mass fraction n stages j−1 mi, j δ = ∑ = δ λ 0 m ∑ j ∏ l 0 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 0 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 n mo,k = mL + m p,k + mi,k + ∑m p, j + mi, j j = k+1 n 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 € € € € 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 ∑Ve,l − l =1 m0,l mf ,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 ∑Ve,l − l =1 m0,l mf ,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 ∑Ve,l − l =1 m0,l mf ,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
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