Ballistic Atmospheric Entry • Straight-line (no gravity) ballistic entry based on density and altitude • Planetary entries (at least a start) • Basic equations of planar motion
© 2016 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu U N I V E R S I T Y O F Ballistic Entry MARYLAND 1 ENAE 791 - Launch and Entry Vehicle Design Ballistic Entry (no lift) s = distance along the flight path v, s dv D = g sin dt m D horizontal dv dv ds dv 1 d(v2) mg = = V = dt ds dt ds 2 ds 2 1 d(v ) D 1 2 = g sin Drag D v AcD 2 ds m 2 2 2 1 d(v ) ⇥v ds dh = g sin AcD 2 ds 2m dh ds = sin d(v2) ⇥v2 sin = g sin Ac 2 dh 2m D
U N I V E R S I T Y O F Ballistic Entry MARYLAND 2 ENAE 791 - Launch and Entry Vehicle Design Ballistic Entry (2)
h Exponential atmosphere = e hs o h d h dh oe hs dh dh = e hs = = h h h o s ⇥ o s ⇥ o s ⇥ h dh = s d sin d(v2) ⇥v2 = g sin Ac 2 dh 2m D sin d(v2) ⇥ ⇥v2 Ac = g sin D 2 d⇥ h 2 m s ⇥ d(v2) 2gh h v2 Ac = s + s D d⇥ ⇥ sin m U N I V E R S I T Y O F Ballistic Entry MARYLAND 3 ENAE 791 - Launch and Entry Vehicle Design Ballistic Entry (3) m Let Ballistic Coe cient cDA ⇥ d(v2) h 2gh s v2 = s d⇤ sin ⇥ ⇤ Assume mg D to get homogeneous ODE d(v2) h d(v2) h s v2 = 0 = s d⇤ d⇤ sin ⇥ v2 sin ⇥ Use v2 as integration variable v d⇥(v2) h = s d⇤ v = velocity at entry v2 sin ⇥ e ve 0
U N I V E R S I T Y O F Ballistic Entry MARYLAND 4 ENAE 791 - Launch and Entry Vehicle Design Ballistic Entry (4)
Note that the effect of ignoring gravity is that there is no force perpendicular to velocity vector ⇒ constant flight path angle γ ⇒ straight line trajectories 2 v v hs⇤ ln 2 = 2 ln = ve ve sin ⇥
v h ⇤ = exp s v 2 sin ⇥ e ⇥
kg v h ⇤ ⇤ m 3 = exp s o Check units: m v 2 sin ⇥ ⇤ kg e o ⇥ m2 ⇥ U N I V E R S I T Y O F Ballistic Entry MARYLAND 5 ENAE 791 - Launch and Entry Vehicle Design Earth Entry, γ=-60°
v/ve 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 / o Beta=100 kg/m^3 300 1000 3000 10000
U N I V E R S I T Y O F Ballistic Entry MARYLAND 6 ENAE 791 - Launch and Entry Vehicle Design What About Peak Deceleration?
dv ⇥v2 n = ⇥ dt 2 d dv d2v To find n , set = = 0 max dt dt dt2 ⇥ d2v 1 dv d⇥ = 2⇥v + v2 = 0 dt2 2 dt dt ⇥ d2v 1 2⇥2v3 d⇥ = + v2 = 0 dt2 2 2 dt ⇥ ⇥2v3 d⇥ d⇥ = v2 ⇥2v = dt dt
U N I V E R S I T Y O F Ballistic Entry MARYLAND 7 ENAE 791 - Launch and Entry Vehicle Design Peak Deceleration (2) From exponential atmosphere,
d o h dh dh = e hs = dt hs dt hs dt dh From geometry, = v sin dt d⇥ ⇥v d⇥ = sin ⇥2v = dt hs dt ⇤v ⇤2v = sin ⇥ h s ⇥ Remember that this refers to the conditions at max deceleration ⇤nmax = sin ⇥ hs U N I V E R S I T Y O F Ballistic Entry MARYLAND 8 ENAE 791 - Launch and Entry Vehicle Design Critical β for Deceleration Before Impact
At surface, = o ⇤ h = o s Value of at which vehicle hits crit sin ⇥ ground at point of maximum deceleration How large is maximum deceleration? 2 2 dv ⇥v dv ⇥n v = = max dt 2 dt 2 max 2 dv v 1 v2 = sin ⇥ = sin dt 2 h max ⇥ s ⇤ 2 hs Note that this value of v is actually vnmax
U N I V E R S I T Y O F Ballistic Entry MARYLAND 9 ENAE 791 - Launch and Entry Vehicle Design Peak Deceleration (3)
From page 14, v h ⇤ = exp s v 2 sin ⇥ e ⇥
vnmax hs 1 = exp sin ⇥ = e 2 v 2 sin ⇥ h e ⇤ s ⇥⌅ 1 2 dv 1 vee 2 v2 sin = sin = e dt 2 ⇥ h ⇤ 2h e max s s Note that the velocity at which maximum deceleration occurs is always a fixed fraction of the entry velocity - it doesn’t depend on ballistic coefficient, flight path angle, or anything else! Also, the magnitude of the maximum deceleration is not a function of ballistic coefficient - it is dependent on the entry trajectory (ve and γ) but not spacecraf parameters (i.e., ballistic coefficient). U N I V E R S I T Y O F Ballistic Entry MARYLAND 10 ENAE 791 - Launch and Entry Vehicle Design Terminal Velocity Full form of ODE - d v2 h 2gh s v2 = s d⇤ ⇥ sin ⇥ ⇤ At terminal velocity, v = constant v T h 2gh s v2 = s sin ⇥ T ⇤
2 2g sin ⇥ vT = ⇤
U N I V E R S I T Y O F Ballistic Entry MARYLAND 11 ENAE 791 - Launch and Entry Vehicle Design “Cannon Ball” γ=-90° Ballistic Entry
6.75” diameter sphere, cD=0.2, VE=6000 m/sec
Iron Aluminum Balsa Wood Weight 40 lb 15.6 lb 14.5 oz β 3938 1532 89 ρ 0.555 0.216 0.0125 h 5600 12,300 32,500 V 1998 355 0* V 251 156 38 *Artifact of assumption that D mg U N I V E R S I T Y O F Ballistic Entry MARYLAND 12 ENAE 791 - Launch and Entry Vehicle Design Nondimensional Ballistic Coefficient
v h ⇤ ⇤ P ⇢ = exp s o =exp o v 2 sin ⇥ ⇤ 2 g sin ⇢ e o ⇥ ✓ o ◆ g Let = (Nondimensional form of ballistic coe cient) ⌘ ⇢ohs Po Note that we are using the estimated value of P = ⇢ gh , b o o s not the actual surface pressure. v 1 ⇤ = exp v ⇤ e 2 sin ⇥ o ⇥
⇤ohs ⇤ 1 = crit = crit sin ⇥ sin ⇥ U N I V E R S I T Y O F Ballistic Entry MARYLAND 13 ENAE 791 - Launch and Entry Vehicle Design Entry Velocity Trends, γ=-90°
Velocity Ratio 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6
Density Ratio 0.7 0.8 0.9 1 0.03 0.1 0.3 1 3
U N I V E R S I T Y O F Ballistic Entry MARYLAND 14 ENAE 791 - Launch and Entry Vehicle Design Ballistic Entry, Again s = distance along the flight path v, s dv D = g sin dt m D horizontal Again assuming D g, mg dv D 1 2 = Drag D v AcD dt m 2 dv c A = D v2 dt 2m Separating the variables, dv ⇥ = dt v2 2
U N I V E R S I T Y O F Ballistic Entry MARYLAND 15 ENAE 791 - Launch and Entry Vehicle Design Calculating the Entry Velocity Profile
dh dh = v sin dt = dt v sin dv ⇤ dv ⇤ = dh = dh v2 2 v sin ⇥ ⇥ v 2 sin ⇥
dv ⇤o h = e hs dh v 2 sin ⇥ v h dv ⇤o h = e hs dh v 2 sin ⇥ ve he h v ⇤ohs h 1 h he ln = e hs = e hs e hs ve 2 sin ⇥ he 2 sin ⇥ ⇥ ⇥ ⇤ U N I V E R S I T Y O F Ballistic Entry MARYLAND 16 ENAE 791 - Launch and Entry Vehicle Design Deriving the Entry Velocity Function
he e Remember that e hs = 0 o
v 1 h = exp e hs v e 2 sin ⇥ ⇥ 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 1 h he = exp e he hs v e 2 sin ⇥ ⇥ he and are the⇤ only variables that relate to a specific planet hs U N I V E R S I T Y O F Ballistic Entry MARYLAND 17 ENAE 791 - Launch and Entry Vehicle Design Earth Entry, γ=-90°
1 0.9 0.8 0.7 0.6 0.5 0.4
Altitude Ratio 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
U N I V E R S I T Y O F Ballistic Entry MARYLAND 18 ENAE 791 - Launch and Entry Vehicle Design Deceleration as a Function of Altitude Start with
v 1 h he 1 = exp e he hs Let B v e 2 sin ⇥ ⇥ 2 sin ⇥ v h = exp B⇤e hs ve ⇥ d v h d h = exp Be hs Be hs dt ve dt ⇤ ⌅ ⇥ ⇥ dv h B h dh = ve exp Be hs e hs dt hs dt ⇥ ⇥ dh h = v sin = v sin exp Be hs dt e U N I V E R S I T Y O F Ballistic Entry MARYLAND 19 ENAE 791 - Launch and Entry Vehicle Design Parametric Deceleration
dv h B h h = ve exp Be hs e hs ve sin exp Be hs dt hs ⇥ ⇥ ⇥ 2 dv Bve h h = sin e hs exp 2Be hs dt hs ⇥ ⇥ 2 dv ve h 1 h = e hs exp e hs dt 2h sin ⇥ s ⇥ ⇤ ⌅ 2 ve dv/dt he ⇧ Let nref ⇧, , ⇥ hs nref hs
1 h 1 h ⇤ = e he exp e he 2 sin ⇥ ⇥ ⇤ ⌅ U N I V⇧ E R S I T Y O F ⇧ Ballistic Entry MARYLAND 20 ENAE 791 - Launch and Entry Vehicle Design Nondimensional Deceleration, γ=-90°
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3
Altitude Ratio h/he Altitude Ratio 0.2 0.1 0 0 0.05 0.1 0.15 0.2 Deceleratio 0.03 0.1 0.3 1 3
U N I V E R S I T Y O F Ballistic Entry MARYLAND 21 ENAE 791 - Launch and Entry Vehicle Design Deceleration Equations
Nondimensional Form 1 h 1 h ⇤ = e he exp e he 2 sin ⇥ ⇥ ⇤ ⌅ Dimensional Form ⇧ 2 ⇧ ⇤oVe h ⇤ohs h n = e hs exp e hs 2 sin ⇥ ⇥ ⇤ ⌅ Note that these equations result in values <0 (reflecting deceleration) - graphs are absolute values of deceleration for clarity.
U N I V E R S I T Y O F Ballistic Entry MARYLAND 22 ENAE 791 - Launch and Entry Vehicle Design Dimensional Deceleration, γ=-90°
120 ) 100 m ( k 80
60 Altitude Altitude
40
20
0 0 500 1000 1500 2000 m Deceleration kg 2 sec m2 277 922 2767 9224⇥ 27673 ⇥ U N I V E R S I T Y O F Ballistic Entry MARYLAND 23 ENAE 791 - Launch and Entry Vehicle Design Altitude of Maximum Deceleration Returning to shorthand notation for deceleration h h ⇥ = B sin e hs exp 2Be hs h ⇥ ⇥ Let hs ⇥ = B sin e exp 2Be
d⇤ d⇥ ⇥ d = B sin e exp 2Be + e exp 2Be d⇥ d⇥ d⇥ ⇤ ⌅ ⇥ ⇥ ⇥ ⇥ d⇤ = B sin e exp 2Be + e 2Be exp 2Be d⇥ ⇤ ⇥ ⇥ ⇥ ⇥ ⇥⌅ d⇤ = B sin e exp 2Be 1 + 2Be = 0 d⇥ ⇥ ⇤ ⇥⌅ U N I V E R S I T Y O F Ballistic Entry MARYLAND 24 ENAE 791 - Launch and Entry Vehicle Design Altitude of Maximum Deceleration