Ballistic Atmospheric Entry • State equations • Standard atmospheres • Orbital decay due to drag • Straight-line (no gravity) ballistic entry based on density and altitude • Planetary entries (at least a start) • Basic equations of planar motion

© 2020 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 Free-Body Diagram with Spherical Planet

v !

g r

✓

U N I V E R S I T Y O F Ballistic Entry MARYLAND 2 ENAE 791 - Launch and Entry Vehicle Design Orbital Planar State Equations

Inertial angular velocity v ! ! =˙ ✓˙ Sum of accelerations normal to velocity vector g g cos = !v r Sum of accelerations ✓ perpendicular to velocity vector g sin =˙v U N I V E R S I T Y O F Ballistic Entry MARYLAND 3 ENAE 791 - Launch and Entry Vehicle Design Orbital Planar State Equations (2) r˙ = v sin r✓˙ = v cos v ! =˙ ✓˙ =˙ cos r v g cos = ˙ cos v r v2⇣ ⌘ g cos =˙v r ✓ ◆ v2 1 g cos =˙v rg ✓ ◆ U N I V E R S I T Y O F Ballistic Entry MARYLAND 4 ENAE 791 - Launch and Entry Vehicle Design Canonical Orbital Planar State Equations 1 v2 ˙ = 1 g cos v v2 ✓ c ◆ v˙ = g sin r˙ = v sin v ✓˙ = cos r Coupled first-order ODEs r 2 g = g o o r U N I V E R S I T Y O F ⇣ ⌘ Ballistic Entry MARYLAND 5 ENAE 791 - Launch and Entry Vehicle Design Numerical Integration - 4th Order R-K Given a series of equations y¯˙ = f¯(t, x¯)

k¯1 =t f¯(tn, y¯n) t k¯ k¯ =t f¯ t + , y¯ + 1 2 n 2 n 2 ✓ ◆ t k¯ k¯ =t f¯ t + , y¯ + 2 3 n 2 n 2 ✓ ◆ k¯4 =t f¯ tn +t, y¯n + k¯3 k¯ k¯ k¯ k¯ y ¯ =¯y + 1 + 2 + 3 + 4 +O(t5) n+1 n 6 3 3 6 U N I V E R S I T Y O F Ballistic Entry MARYLAND 6 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 Entry MARYLAND 7 ENAE 791 - 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 Ballistic Entry MARYLAND 8 ENAE 791 - Launch and Entry Vehicle Design Energy Loss Due to Atmospheric Drag

1 Drag D v2Ac 2 D D v2 Ac Drag acceleration a = = D d m 2 m m <== Ballistic Coefficient ⌘ cDA ⇥v2 ad = 2 µ orbital energy E = ⇥ 2a dE µ da = dt 2a2 dt U N I V E R S I T Y O F Ballistic Entry MARYLAND 9 ENAE 791 - Launch and Entry Vehicle Design Energy Loss Due to Atmospheric Drag Since drag is highest at perigee, the first effect of atmospheric drag is to circularize the orbit (high perigee drag lowers apogee) dE drag = a v dt d

µ dE ⇤v2 µ v2 = drag = circ a dt 2 a

3 dE µ ⇤ µ µ 2 ⇤ drag = = dt a 2 a a 2 ⇥ U N I V E R S I T Y O F Ballistic Entry MARYLAND 10 ENAE 791 - Launch and Entry Vehicle Design Derivation of Orbital Decay Due to Drag Set orbital energy variation equal to energy lost by drag

3 µ da ⇤ µ 2 = 2a2 dt 2 a ⇥ da ⇤ = ⇥µa dt

h da dh = e hs a = h + rE = = o dt dt dh µ (h + rE) h = ⇤oe hs dt U N I V E R S I T Y O F Ballistic Entry MARYLAND 11 ENAE 791 - Launch and Entry Vehicle Design Derivation of Orbital Decay (2) This is a separable differential equation... 1 h ⇥µ e hs dh = ⇤odt ⇥rE + h h t 1 h ⇥µ e hs dh = ⇤o dt ⇥r + h ho E to Assume r + h ⇤r for r h E E E ⇥ h 1 h ⇥µ e hs dh = ⇤o (t to) ⇥r E ho U N I V E R S I T Y O F Ballistic Entry MARYLAND 12 ENAE 791 - Launch and Entry Vehicle Design Derivation of Orbital Decay (3)

hs h ho ⇥µ e hs e hs = ⇤o (t to) ⇥rE ⇥ h ho ⇥µrE e hs e hs = ⇤o (t to) hs

ho ⇥µrE h(t) = h ln e hs ⇤ (t t ) s h o o s ⇥ Note that some variables typically use km, and others are in meters - you have to make sure unit conversions are done properly to make this work out correctly! U N I V E R S I T Y O F Ballistic Entry MARYLAND 13 ENAE 791 - Launch and Entry Vehicle Design Orbit Decay from Atmospheric Drag

250

200

150 β=500 β=1500 100 β=5000 Altitude (km) 50

0 0 10000 20000 30000 40000 Time (sec)

U N I V E R S I T Y O F Ballistic Entry MARYLAND 14 ENAE 791 - Launch and Entry Vehicle Design Time Until Orbital Decay

h ho ⇥µrE e hs e hs = ⇤o (t to) hs

To find the time remaining (to=0) until the orbit reaches any given “critical” altitude, some algebra gives

hs ho hcrit t(hcrit)= e hs e hs pµrE⇢o ⇣ ⌘ t(h ) crit / U N I V E R S I T Y O F Ballistic Entry MARYLAND 15 ENAE 791 - Launch and Entry Vehicle Design Decay Time to r=120 km

600

500

400

300 β=500 β=1500 200 β=5000 Altitude (km) 100

0 0 20 40 60 80 100 Decay Time (yrs)

U N I V E R S I T Y O F Ballistic Entry MARYLAND 16 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 17 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 18 ENAE 791 - Launch and Entry Vehicle Design Ballistic Entry (3) m Let Ballistic Coecient 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 19 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 20 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 21 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 22 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 23 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 24 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 spacecraft parameters (i.e., ballistic coefficient). U N I V E R S I T Y O F Ballistic Entry MARYLAND 25 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 26 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 Entry MARYLAND 27 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 ⇥

⇤ohs ⇤ 1 = crit = crit sin ⇥ sin ⇥ U N I V E R S I T Y O F Ballistic Entry MARYLAND 28 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 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 29 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 30 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 Entry MARYLAND 31 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 32 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 33 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 34 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 35 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 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 36 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 37 ENAE 791 - Launch and Entry Vehicle Design Dimensional Deceleration, γ=-90°

120 ) 100 m k ( 80

60 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 38 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 39 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 Entry MARYLAND 40 ENAE 791 - Launch and Entry Vehicle Design Altitude of Maximum Deceleration

12

10

8

6

4

2 Altitude of Max h/hs Decel 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 Entry MARYLAND 41 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 ⇤n = ln e = sin ⇥ max ⇥ 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 Entry MARYLAND 42 ENAE 791 - Launch and Entry Vehicle Design Peak Ballistic Deceleration for Earth Entry

6000

5000

4000

3000

2000

1000 Peak Deceleration (m/sec^2) 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 Entry MARYLAND 43 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 ⇤n = ln e = sin ⇥ max ⇥ 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 Entry MARYLAND 44 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 27.7 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 Entry MARYLAND 45 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) 1E-18 1E-20 Altitude (km)

Earth Mars Venus

U N I V E R S I T Y O F Ballistic Entry MARYLAND 46 ENAE 791 - Launch and Entry Vehicle Design Planetary Entry Profiles

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

V

Ve

U N I V E R S I T Y O F Ballistic Entry MARYLAND 47 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 Entry MARYLAND 48 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 Entry MARYLAND 49 ENAE 791 - Launch and Entry Vehicle Design