Lifting Ballistic Coefficient”

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Ballistic/Lifting Atmospheric Entry • Straight-line (no gravity) ballistic entry based on altitude, rather than density • Planetary entries (at least a start) • Basic equations of planar motion (with lif) • Lifing equilibrium glide

© 2014 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 1 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 and Lifting Atmospheric Entry MARYLAND 2 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 (kg/m2) 3938 1532 89

ρmd (kg/m3) 0.555 0.216 0.0125

hmd (m) 5600 12,300 32,500

Vimpact (m/s) 1998 355 0*

Vterm (m/sec) 251 156 38 *Artifact of assumption that D mg U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 3 ENAE 791 - Launch and Entry Vehicle Design Atmospheric Density with Altitude

Pressure=the integral of the atmospheric density in the column above the reference area

1 1 h h h 1 Po = ⇢gdh = ⇢og e hs dh = ⇢oghs e s = f(h) o o o Z Z h i = ⇢ gh [0 1] o s

Po = ⇢oghs kg Earth: = 1.226 ; h = 7524m; o m3 s

Po(calc) = 90, 400 Pa; Po(act) = 101, 300 Pa

o, Po U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 4 ENAE 791 - Launch and Entry Vehicle Design Nondimensional Ballistic Coefficient

v h ⇤ ⇤ P ⇢ = exp s o =exp o v 2 sin ⇥ ⇤ 2g sin ⇢ e o ⇥ ✓ o ◆ g Let = (Nondimensional form of ballistic coecient) ⌘ ⇢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 ⇥

⇤ h ⇤ 1 = o s = crit sin ⇥ crit sin ⇥ U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 5 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 and Lifting Atmospheric Entry MARYLAND 6 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 = Drag D v2Ac dt m 2 D 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 and Lifting Atmospheric Entry MARYLAND 7 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 2v 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 and Lifting Atmospheric Entry MARYLAND 8 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 and Lifting Atmospheric Entry MARYLAND 9 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 and Lifting Atmospheric Entry MARYLAND 10 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 ve 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 and Lifting Atmospheric Entry MARYLAND 11 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 and Lifting Atmospheric Entry MARYLAND 12 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 and Lifting Atmospheric Entry MARYLAND 13 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 and Lifting Atmospheric Entry MARYLAND 14 ENAE 791 - Launch and Entry Vehicle Design Dimensional Deceleration, γ=-90°

120 ) 100 m ( k 80

60 Altitude Altitude

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 and Lifting Atmospheric Entry MARYLAND 15 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 and Lifting Atmospheric Entry MARYLAND 16 ENAE 791 - Launch and Entry Vehicle Design Altitude of Maximum Deceleration

1 + 2Be = 0 e = 2B ⇥ ⇥ = ln ( 2B) nmax 1 ⇤nmax = ln sin ⇥ ⇥ Converting from parametric to dimensional form gives ⇤ ⇤ h h = h ln o s nmax s sin ⇥ ⇥ Altitude of maximum deceleration is independent of entry velocity!

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 17 ENAE 791 - Launch and Entry Vehicle Design Altitude of Maximum Deceleration

2 Altitude of Max Decel h/hs Altitude of 0 0 0.2 0.4 0.6 0.8 1 Ballistic Coefficient

-15 -30 -45 -60 -90 U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 18 ENAE 791 - Launch and Entry Vehicle Design Magnitude of Maximum Deceleration Start with the equation for acceleration -

1 1 ⇤ = e exp e 2 sin ⇥ ⇥ and insert the value of at the point of maximum deceleration ⇤ ⇤ 1 ⇤ = ln e = sin ⇥ nmax ⇥ sin ⇥ ⇥ 1 sin ⇥ sin ⇤nmax = ⇤ sin ⇥ exp ⇥nmax = 2 ⇤ sin ⇥ ⌅ 2e ⇥ ⇧ 2 ⇧ ve sin nmax = ⇧ ⇧ hs 2e Maximum deceleration is not a function of ballistic coecient! U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 19 ENAE 791 - Launch and Entry Vehicle Design Peak Ballistic Deceleration for Earth Entry

6000

5000

4000

3000

2000

1000 Peak Deceleration (m/sec^2) Peak Deceleration 0 0 5 10 15 Entry Velocity (km/sec)

-15 -30 -45 -60 -90

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 20 ENAE 791 - Launch and Entry Vehicle Design Velocity at Maximum Deceleration Start with the equation for velocity

v 1 = exp e v e 2 sin ⇥ ⇥ and insert the value of at the point of maximum deceleration ⇤ 1 ⇤ = ln e = sin ⇥ nmax ⇥ sin ⇥ ⇥ v sin ⇥ v ⇤ v = e 0.606v = exp nmax = e ve 2 sin ⇥ ⇥ ⇥ ⇤e ⇤

Velocity at maximum⇤ deceleration is independent of everything except ve

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 21 ENAE 791 - Launch and Entry Vehicle Design Planetary Entry - Physical Data

Radius µ o hs vesc (km) (km3/sec2) (kg/m3) (km) (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

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 22 ENAE 791 - Launch and Entry Vehicle Design Comparison of Planetary Atmospheres

100 1 0.01 0 50 100 150 200 250 300 1E-04 1E-06 1E-08 1E-10 1E-12 1E-14 1E-16 Atmospheric Density (kg/m3) Density Atmospheric 1E-18 1E-20 Altitude (km)

Earth Mars Venus

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 23 ENAE 791 - Launch and Entry Vehicle Design Planetary Entry Profiles

= 15o kg = 300 km m2 v = 10 e sec

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 24 ENAE 791 - Launch and Entry Vehicle Design Planetary Entry Deceleration Comparison

= 15o km v = 10 e sec kg = 300 m2

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 25 ENAE 791 - Launch and Entry Vehicle Design Check on Approximation Formulas

= 15o km v = 10 e sec kg = 300 m2

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 26 ENAE 791 - Launch and Entry Vehicle Design Lifting Entry – Free-Body Diagram

mv2 v, s L r D horizontal mg

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 27 ENAE 791 - Launch and Entry Vehicle Design Dynamics of Lifting Entry

Along the velocity vector, dv v2 m = m sin mg sin D dt r Perpendicular to the velocity vector, d v2 mv = L + m cos mg cos dt r (unbalanced lift rotates ﬂight path angle)

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 28 ENAE 791 - Launch and Entry Vehicle Design Equations of Planar Lifting Entry

dv v2 D = g sin dt r m ✓ ◆ d L v2 v = + g cos dt m r ✓ ◆

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 29 ENAE 791 - Launch and Entry Vehicle Design Equilibrium Glide • Forces perpendicular to velocity vector are balanced d = = 0; = constant ) dt • Typically very shallow glide

= assume 0; sin() 0; cos() 1 ) ! ! !

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 30 ENAE 791 - Launch and Entry Vehicle Design Equilibrium Glide Equations dv D = 1 dt m D = ⇢v2c A 2 D h ⇢ = ⇢oe hs

dv 1 2 cDA h = ⇢ v e hs dt 2 o m L v2 0= + g m r ✓ ◆ 2 v 1 2 cLA h g = ⇢ v e hs r 2 o m U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 31 ENAE 791 - Launch and Entry Vehicle Design Dynamics Perpendicular to Velocity c L L = cD D L/D set by vehicle aerodynamics, ﬂight velocity, and angle of attack (assumed constant) 2 v 1 2 L cDA h g = ⇢ v e hs r 2 o D m 2 v 1 2 L/D h = ⇢ v e hs + g r 2 o

2 1 2 L/D h v = ⇢ rv e hs + gr 2 o U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 32 ENAE 791 - Launch and Entry Vehicle Design More Lift-Direction Dynamics

h ⇢ Let e hs = density ratio ⌘ ⌘ ⇢ ✓ o ◆ 1 L/D v2 + ⇢ rv2 = gr 2 o

1 L/D v2 1+ ⇢ r = gr 2 o ✓ ◆ gr v = ⇢or(L/D) s1+ 2

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 33 ENAE 791 - Launch and Entry Vehicle Design Velocity during Entry µ v = = pg r µ = g r2 co r o o o o r o v 1 = ⇢ r (L/D) vc o o o s1+ 2 v v (within 1-2% for Earth) co ⇠= e 1 2 v ⇢oro L h = 1+ e hs v 2 D e  Entry trajectory as a function of altitude, ratio L/D ✓ ◆ U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 34 ENAE 791 - Launch and Entry Vehicle Design Equilibrium Glide Velocity Trends

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 35 ENAE 791 - Launch and Entry Vehicle Design Trends with “Lifting Ballistic Coefficient”

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 36 ENAE 791 - Launch and Entry Vehicle Design Nondimensional Form of Equation 1 2 v ⇢oro L h = 1+ e hs v 2 D e  Remember 1 ⌘ ⇢ohs 2 v ⇢ohs ro L h = 1+ e hs v 2 h D b e  s 1 2 v 1 ro L h he = 1+ e he hs v h D e  2 s

b U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 37 ENAE 791 - Launch and Entry Vehicle Design Deceleration

L v2 gv2 v 2 = g = g = g 1 m r gr v " ✓ co ◆ #

dv D L 1 1 L = = = dt m L/D m L/D m

dv g v 2 = 1 dt L/D v " ✓ co ◆ #

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 38 ENAE 791 - Launch and Entry Vehicle Design Deceleration 1 dv Let n (deceleration in g’s) ⌘ g dt 1 v 2 n = 1 L/D v " ✓ co ◆ # 1 1 n = 1 ⇢ r h L/D 2 o o L hs 3 1+ 2 D e 4 5 1 1+K 1 K Note: 1 = = 1+K 1+K 1+K 1+K ✓ ◆ U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 39 ENAE 791 - Launch and Entry Vehicle Design Deceleration

h ⇢oro L 1 e hs n = 2 D ⇢ r h L/D 2 o o L hs 3 1+ 2 D e 4 5 ⇢ r h o o e hs n = 2 ⇢ r h o o L hs 1+ 2 D e 1 n = 2 + h L e hs + ⇢oro D n monotonically increases with decreasing altitude U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 40 ENAE 791 - Launch and Entry Vehicle Design Deceleration Trends with Altitude

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 41 ENAE 791 - Launch and Entry Vehicle Design Limiting Deceleration 1 nlimit = 2 + L ⇢oro D 3 6 2 3 O(10 ); ⇢o O(1); ro O(10 )= O(10 ) ⇠ ⇠ ⇠ ) ⇢oro ⇠ 1 n = limit ⇠ L/D

Lif signiﬁcantly moderates peak g’s on entry (L/D=0.25 --> nlimit=4 g‘s)

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 42 ENAE 791 - Launch and Entry Vehicle Design Time for Entry

dv g v 2 = 1 dt L/D v " ✓ co ◆ # (L/D)dv dt = 2 g 1 v vco  ⇣ ⌘ t 0 (L/D)dv dt = 2 Z0 Zve g 1 v vco  ⇣ ⌘ U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 43 ENAE 791 - Launch and Entry Vehicle Design Time for Entry

2 1+ v 1 r L vc t = o ln o 2 g D ⇣ ⌘2 r o 1 v vco ⇣ ⌘ L Time for entry / D Not a function of

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 44 ENAE 791 - Launch and Entry Vehicle Design Time From Entry Interface

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 45 ENAE 791 - Launch and Entry Vehicle Design Distance Along Flight Path ds For shallow entry, = v ds = vdt dt ⇠ (L/D)vdv ds = 2 g 1 v vco  ⇣ ⌘ 2 v 2v v2 Let u 1 du = dv vdv= co du ⌘ vc v2 2 ✓ o ◆ co L/D v2 du ds = co 2g u Z Z U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 46 ENAE 791 - Launch and Entry Vehicle Design Distance Along Flight Path

(L/D)v2 (L/D)g r v 2 s = co ln u = o o ln 1 2g 2g v o " ✓ co ◆ #

r L v 2 s = o ln 1 2 D v " ✓ co ◆ #

Not a function of or g

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 47 ENAE 791 - Launch and Entry Vehicle Design Distance from Entry

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 48 ENAE 791 - Launch and Entry Vehicle Design Bank Angle

L L cos () D horizontal Bank Angle mg ⌘ Level turn: L cos = mg

L 1 = n mg cos ⌘ bank

(g’s you pull in the banked turn)

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 49 ENAE 791 - Launch and Entry Vehicle Design Crossrange (without derivations) • Use of lif to deviate laterally from planar groundtrack • Optimum bank angle to maximize crossrange 2 1 L = cot 1+0.106 opt ⇠ D s ✓ ◆ • Maximum achievable crossrange 2 ro L 1 ymax = ⇠ 5.2 D L 2 ✓ ◆ 1+0.106 D q U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 50 ENAE 791 - Launch and Entry Vehicle Design Crossrange Based on L/D

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 51 ENAE 791 - Launch and Entry Vehicle Design Early Shuttle Design Configurations • Delta wing conﬁguration – High hypersonic lif – High landing velocity – High crossrange • Straight-wing conﬁguration – High subsonic lif – Low(er) landing velocity – Low crossrange

U N I V E R S I T Y O F Ballistic and Lifting Atmospheric Entry MARYLAND 52 ENAE 791 - Launch and Entry Vehicle Design