Ballistic Atmosperic Entry Straigt-line (no gravity) ballistic entry based on density and altitude Planetary entries (at least a start) Basic equations of planar motion 206 David L. Akin - All rigts reserved ttp://spacecraft.ssl.umd.edu
(no lift) s = distance along te fligt pat dt = g sin D m dt = ds 2 2 sin 2 d(v 2 ) ds d(v 2 ) ds ds dt = V ds = 2 = g sin = g sin D m d(v 2 ) ds v 2 2m Ac D d(v 2 ) d = g sin v 2 2m Ac D D Drag D ds mg 2 v2 Ac D d ds = orizontal d sin v, s 2
(2) Exponential atmospere = o e s d o = e s d s = o e o s d s = o d s d = s d sin 2 sin 2 d(v 2 ) d d(v 2 ) d = g sin v 2 2m Ac D v 2 = g sin 2 s Ac D m d(v 2 ) d = 2g s 3 + sv 2 sin Ac D m
(3) Let m c D A Ballistic Coe cient d(v 2 ) d s sin v2 = 2g s Assume mg D to get omogeneous ODE Use d(v 2 ) d s sin v2 = 0 v 2 as integration variable d(v 2 ) v 2 = s sin d v v e d(v 2 ) v 2 = s sin 0 d v e = velocity at entry 4
(4) Note tat te effect of ignoring gravity is tat tere is no force perpendicular to velocity vector constant fligt pat angle γ straigt line trajectories ln v2 v 2 e = 2 ln v v e = s sin v v e = exp s 2 sin v v e = exp s o 2 sin o Ceck units: m kg m 3 kg m 2 5
Eart Entry, γ=-60 v/v e 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 0 0.2 0.4 0.6 0.8 / o Beta=00 kg/m^3 300 000 3000 0000 6
Wat About Peak Deceleration? To find n max, set d dt d 2 v dt 2 = 2 d 2 v dt 2 = 2 n dt = dt v2 2 = d2 v dt 2 = 0 2v dt 2 2 v 3 2 + v2 d dt + v 2 d dt = 0 = 0 2 v 3 = v 2 d dt 2 v = d dt 7
Peak Deceleration (2) From exponential atmospere, d dt = From geometry, d dt = v sin d dt = o s e v s sin 2 v = s d dt = d s dt 2 v = v sin s d dt Remember tat tis refers to te conditions at max deceleration nmax = s sin 8
Critical β for Deceleration Before Impact crit = o s sin At surface, = o Value of at wic veicle its ground at point of maximum deceleration How large is maximum deceleration? dt = v2 2 = v2 dt max 2 = n v2 max dt max 2 sin s = 2 v 2 s sin Note tat tis value of v is actually v nmax 9
Peak Deceleration (3) From page 4, v nmax v e dt max = 2 v v e = exp 0 s 2 sin s = exp sin 2 sin s 2 v e e 2 = e 2 s sin = v2 e sin 2 s e Note tat te velocity at wic maximum deceleration occurs is always a fixed fraction of te entry velocity - it doesn t depend on ballistic coefficient, fligt pat angle, or anyting else! Also, te magnitude of te maximum deceleration is not a function of ballistic coefficient - it is dependent on te entry trajectory (v e and γ) but not spacecraft parameters (i.e., ballistic coefficient).
Terminal Velocity Full form of ODE - d v 2 d s sin v2 = 2g s At terminal velocity, v = constant v T s sin v2 T = 2g s v 2 T = 2g sin
Cannon Ball γ=-90 6.75 diameter spere, c D =0.2, V E =6000 m/sec Iron Aluminum Balsa Wood Weigt 40 lb 5.6 lb 4.5 oz β 3938 532 89 ρ 0.555 0.26 0.025 5600 2,300 32,500 V 998 355 0* V 25 56 38 *Artifact of assumption tat D 2 mg
Nondimensional Ballistic Coefficient v v e = exp s o 2 sin o Po =exp 2 g sin Let b = g (Nondimensional form of ballistic coe cient) o s P o Note tat we are using te estimated value of P o = o g s, not te actual surface pressure. v = exp v e 2 sin o o crit = o s sin crit = sin 3
Entry Velocity Trends, γ=-90 Density Ratio 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Velocity Ratio 0 0.2 0.4 0.6 0.8 0.03 0. 0.3 3 4
, Again s = distance along te fligt pat dt = g sin D m Again assuming D g, Drag D dt = D m dt = c D A 2m v2 Separating te variables, v 2 = 2 dt D 2 v2 Ac D mg orizontal v, s 5
Calculating te Entry Velocity Profile d dt = v sin v 2 = dt = d v sin d 2 v sin v = 2 sin d v = v v e v = ln v v e = o 2 sin e s d o 2 sin e e o s 2 sin e s = e s d 2 sin e s e e s 6
Deriving te Entry Velocity Function Remember tat e e s = e v v e = exp o 2 sin e s 0 We ave a parametric entry equation in terms of nondimensional velocity ratios, ballistic coefficient, and altitude. To bound te nondimensional altitude variable between 0 and, rewrite as v v e = exp 2 sin e e e s e s and are te only variables tat relate to a specific planet 7
Eart Entry, γ=-90 Altitude Ratio 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 0 0.2 0.4 0.6 0.8 Velocity Ratio 0.03 0. 0.3 3 8
Deceleration as a Function of Altitude Start wit v v e = exp 2 sin e e v = exp Be s v e d v = exp Be dt v e dt = v e exp Be s 9 s e s d dt B s e Be s s Let B d dt d dt = v sin = v e sin exp Be s 2 sin
Parametric Deceleration dt = v e exp dt = dt = Be s Bv2 e s sin e v2 e 2 s e s s exp B s e exp s v e sin exp Be 2Be sin e v 2 e Let n ref, s = 2 e e exp e sin s s /dt, n ref e e s s 20
Nondimensional Deceleration, γ=-90 Altitude Ratio /e 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 0 0.05 0. 0.5 0.2 Deceleratio 0.03 0. 0.3 3 2
Deceleration Equations Nondimensional Form = 2 e e Dimensional Form n = ove 2 2 e exp s exp sin e o s sin e Note tat tese equations result in values <0 (reflecting deceleration) - graps are absolute values of deceleration for clarity. e s 22
Dimensional Deceleration, γ=-90 20 (km) Altitude 00 80 60 40 20 0 0 500 000 500 2000 kg m 2 Deceleration 277 922 2767 9224 27673 23 m sec 2
Altitude of Maximum Deceleration Returning to sortand notation for deceleration = B sin e s exp 2Be s = B sin e d d = B sin d d = B sin e exp 2Be Let d d e exp 2Be + e d d exp 2Be + e 2Be d d = B sin e exp 2Be + 2Be = 0 s exp 2Be exp 2Be 24
Altitude of Maximum Deceleration + 2Be = 0 e = 2B n max = ln ( 2B) nmax = ln sin Converting from parametric to dimensional form gives o s nmax = s ln sin Altitude of maximum deceleration is independent of entry velocity! 25
Altitude of Maximum Deceleration 2 Altitude of Max Decel /s 0 8 6 4 2 0 0 0.2 0.4 0.6 0.8 Ballistic Coefficient -5-30 -45-60 -90 26
Magnitude of Maximum Deceleration Start wit te equation for acceleration - = 2 e exp e sin and insert te value of at te point of maximum deceleration nmax = ln e = sin sin nmax = sin 2 sin exp nmax = sin sin 2e n max = v2 e s sin 2e Maximum deceleration is not a function of ballistic coe 27 cient!
Peak Ballistic Deceleration for Eart Entry Peak Deceleration (m/sec^2) 6000 5000 4000 3000 2000 000 0 0 5 0 5 Entry Velocity (km/sec) -5-30 -45-60 -90 28
Velocity at Maximum Deceleration Start wit te equation for velocity v = exp v e 2 sin e and insert te value of at te point of maximum deceleration nmax = ln e = sin sin v sin = exp v e 2 v nmax = v e = 0.606v e sin e Velocity at maximum deceleration is independent of everyting except v e 29
Planetary Entry - Pysical Data Radius (km) (km µ o (kg/m (km) v (km/sec) Eart 6378 398,604.225 7.524.8 Mars 3393 42,840 0.0993 27.7 5.025 Venus 6052 325,600 6.02 6.227 0.37 30
Comparison of Planetary Atmosperes 00 Atmosperic Density (kg/m3) 0.0 E-04 E-06 E-08 E-0 E-2 E-4 E-6 E-8 E-20 0 50 00 50 200 250 300 Altitude (km) Eart Mars Venus 3
Planetary Entry Profiles = 5 o v e = 0 km sec = 300 kg m 2 V V e 32
Planetary Entry Deceleration Comparison = 5 o v e = 0 km sec = 300 kg m 2 33
Ceck on Approximation Formulas = 5 o v e = 0 km sec = 300 kg m 2 34