Ballistic Atmospheric Entry (Part II) News updates Straight-line (no gravity) ballistic entry based on altitude, rather than density Planetary entries (at least a start) 1 2010 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu
Notable News Notices No class on Tuesday, Feb. 9 - download lecture video from Blackboard site, and slides from spacecra.ssl.umd.edu Special bonus value for term project - winning team will be entered in NASA RASC-AL design competition All expenses paid trip to RASC-AL conference in Cocoa Beach, FL June 7-9 15-page summary paper due May 26; presentation slides due May 28 Requirement for outreach activities in competition rules Let me know if you want to opt-out 2
Ballistic Entry (no lift) s = distance along the flight path dv dt = g sin γ D m D Again assuming D g, dv Drag D 1 2 ρv2 Ac D dt = D m dv dt = ρc DA 2m v2 mg v, s γ horizontal Separating the variables, dv v 2 = ρ 2β dt 3
Calculating the Entry Velocity Profile dh dt = v sin γ dt = dh v sin γ dv v 2 = ρ 2βv sin γ dh dv v = ρ 2β sin γ dh dv v = ρ o 2β sin γ e h hs dh v v e ln v v e = dv v = ρ o 2β sin γ ρ oh s 2β sin γ h e h hs dh h e h e 1 h hs = h e 2β sin γ e h hs e h e hs 4
Deriving the Entry Velocity Function Remember that e h e hs v = exp v e ρ e = 0 ρ o 1 2 β sin γ e h hs We have a parametric entry equation in terms of nondimensional velocity ratios, ballistic coefficient, and altitude. To bound the nondimensional altitude variable between 0 and 1, rewrite as v v e = exp 1 2 β sin γ e h he he hs h e h s and β are the only variables that relate to a specific planet 5
Earth Entry, γ=-90 Altitude Ratio 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Velocity Ratio 0.03 0.1 0.3 1 3 6
Deceleration as a Function of Altitude Start with v v e = exp v = exp v e d v = exp dt v e 1 h 2β sin γ e Be h hs he Be h hs dv dt = v e exp Be h hs dh dt = v sin γ 7 he hs B d dt Be h hs h s e h hs dh dt Let B = v e sin γ exp Be h hs 1 2 β sin γ
Parametric Deceleration dv dt = v e exp Be h hs dv dt = Bv2 e h s sin γ dv dt = v2 e 2h s β e h hs ν = 1 2 β B e h hs exp h s exp e h hs v e sin γ exp 2Be h hs 1 β sin γ e h hs Let n ref v2 e, ν dv/dt, ϕ h e h s n ref h s e ϕ h 1 he exp h β sin γ e ϕ he Be h hs 8
Nondimensional Deceleration, γ=-90 Altitude Ratio h/he 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 β Deceleratio 0.03 0.1 0.3 1 3 ν 9
Deceleration Equations Nondimensional Form ν = 1 2 β e ϕ h he Dimensional Form n = ρ ove 2 2β e h hs exp exp 1 β sin γ e ϕ h he ρo h s β sin γ e h hs Note that these equations result in values <0 (reflecting deceleration) - graphs are absolute values of deceleration for clarity. 10
Dimensional Deceleration, γ=-90 120 (km) Altitude 100 80 60 40 20 0 β 0 500 1000 1500 2000 kg m 2 Deceleration 277 922 2767 9224 27673 11 m sec 2
Altitude of Maximum Deceleration Returning to shorthand notation for deceleration ν = B sin γ e h hs exp 2Be h hs ν = B sin γ e η exp 2Be η dν dη = B sin γ d dη Let η e η exp 2Be η + e η d dη exp 2Be η dν dη = B sin γ e η exp 2Be η + e η 2Be η exp 2Be η dν dη = B sin γe η exp 2Be η 1 + 2Be η = 0 h h s 12
Altitude of Maximum Deceleration 1 + 2Be η = 0 e η = 2B η nmax η nmax = ln ( 2B) 1 = ln β sin γ Converting from parametric to dimensional form gives ρo h s h nmax = h s ln β sin γ Altitude of maximum deceleration is independent of entry velocity! 13
Altitude of Maximum Deceleration 12 Altitude of Max Decel h/hs 10 8 6 4 2 0 0 0.2 0.4 0.6 0.8 1 γ Ballistic Coefficient -15-30 -45-60 -90 β 14
Magnitude of Maximum Deceleration Start with the equation for acceleration - ν = 1 1 2β e η exp β sin γ e η and insert the value of η at the point of maximum deceleration 1 η nmax = ln e η = β β sin γ sin γ ν nmax = 1 2β β β sin γ exp sin γ ν nmax = sin γ β sin γ 2e n max = v2 e sin γ h s 2e Maximum deceleration is not a function of ballistic coefficient! 15
Peak Ballistic Deceleration for Earth Entry Peak Deceleration (m/sec^2) 6000 5000 4000 3000 2000 1000 0 0 5 10 15 Entry Velocity (km/sec) γ -15-30 -45-60 -90 16
Velocity at Maximum Deceleration Start with the equation for velocity v 1 = exp v e 2β sin γ e η and insert the value of η at the point of maximum deceleration 1 η nmax = ln e η = β β sin γ sin γ v = exp β sin γ v e 2β sin γ v nmax = v e e = 0.606v e Velocity at maximum deceleration is independent of everything except v e 17
Planetary Entry - Physical Data µ ρ o Radius (km) (km 3 /sec 2 ) (kg/m 3 ) hs (km) vesc (km/sec) Earth 6378 398,604 1.225 7.524 11.18 Mars 3393 42,840 0.0993 27.70 5.025 Venus 6052 325,600 16.02 6.227 10.37 18
Comparison of Planetary Atmospheres 100 Atmospheric Density (kg/m3) 1 0.01 1E-04 1E-06 1E-08 1E-10 1E-12 1E-14 1E-16 1E-18 1E-20 0 50 100 150 200 250 300 Altitude (km) Earth Mars Venus 19
Planetary Entry Profiles γ = 15 o v e = 10 km sec β = 300 kg m 2 V V e 20
Planetary Entry Deceleration Comparison γ = 15 o v e = 10 km sec β = 300 kg m 2 21
Check on Approximation Formulas γ = 15 o v e = 10 km sec β = 300 kg m 2 22