Ballistic Atmospheric Entry (Part II)

Transcription

1 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) David L. Akin - All rights reserved

2 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 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

3 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

4 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

5 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

6 Earth Entry, γ=-90 Altitude Ratio Velocity Ratio

7 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

8 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

9 Nondimensional Deceleration, γ=-90 Altitude Ratio h/he β Deceleratio ν 9

10 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

11 Dimensional Deceleration, γ= (km) Altitude β kg m 2 Deceleration m sec 2

12 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

13 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

14 Altitude of Maximum Deceleration 12 Altitude of Max Decel h/hs γ Ballistic Coefficient β 14

15 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

16 Peak Ballistic Deceleration for Earth Entry Peak Deceleration (m/sec^2) Entry Velocity (km/sec) γ

17 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

18 Planetary Entry - Physical Data µ ρ o Radius (km) (km 3 /sec 2 ) (kg/m 3 ) hs (km) vesc (km/sec) Earth , Mars , Venus ,

19 Comparison of Planetary Atmospheres 100 Atmospheric Density (kg/m3) E-04 1E-06 1E-08 1E-10 1E-12 1E-14 1E-16 1E-18 1E Altitude (km) Earth Mars Venus 19

20 Planetary Entry Profiles γ = 15 o v e = 10 km sec β = 300 kg m 2 V V e 20

21 Planetary Entry Deceleration Comparison γ = 15 o v e = 10 km sec β = 300 kg m 2 21

22 Check on Approximation Formulas γ = 15 o v e = 10 km sec β = 300 kg m 2 22