ballistic atmospheric entry (part ii) - umd...ballistic atmospheric entry ii enae 791 - launch and...

Ballistic Atmospheric Entry IIENAE 791 - Launch and Entry Vehicle Design

U N I V E R S I T Y O FMARYLAND

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)

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© 2010 David L. Akin - All rights reservedhttp://spacecraft.ssl.umd.edu



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 7-9– 15-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

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Ballistic Entry (no lift)s = distance along the flight path

dv

dt= −g sin γ − D

m

Drag D ≡ 12ρv2AcD

D

mg

horizontal

v, s

γ

3

Again assuming D � g,

dv

dt= −D

m

dv

dt= −ρcDA

2mv2

dv

v2= − ρ

2βdt

Separating the variables,



Calculating the Entry Velocity Profile

4

⇒ dt =dh

v sin γ

dv

v2= − ρ

2βv sin γdh ⇒ dv

v= − ρ

2β sin γdh

dv

v= − ρo

2β sin γe−

hhs dh

� v

ve

dv

v= − ρo

2β sin γ

� h

he

e−h

hs dh

lnv

ve=

ρohs

2β sin γ

�e−

hhs

�h

he

=1

2�β sin γ

�e−

hhs − e−

hehs

�

dh

dt= v sin γ



Deriving the Entry Velocity Function

5

Remember that e−hehs =

ρe

ρo≈ 0

v

ve= exp

�1

2�β sin γe−

hhs

�

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

ve= exp

�1

2�β sin γe−

hhe

hehs

�

he

hsand �β are the only variables that relate to a specific planet



Earth Entry, γ=-90°

6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Velocity Ratio

Alt

itu

de R

ati

o

0.03 0.1 0.3 1 3



Deceleration as a Function of Altitude

7

v

ve= exp

�1

2�β sin γe−

hhe

hehs

�Start with

v

ve= exp

�Be−

hhs

�

Let B ≡ 12�β sin γ

d

dt

�v

ve

�= exp

�Be−

hhs

� d

dt

�Be−

hhs

�

dv

dt= ve exp

�Be−

hhs

� −B

hs

�e−

hhs

� dh

dt

dh

dt= v sin γ = ve sin γ expBe−

hhs



Parametric Deceleration

8

dv

dt= ve exp

�Be−

hhs

� −B

hs

�e−

hhs

�ve sin γ exp

�Be−

hhs

�

dv

dt=−Bv2

e

hssin γ

�e−

hhs

�exp

�2Be−

hhs

�

dv

dt=−v2

e

2hs�β

�e−

hhs

�exp

�1

�β sin γe−

hhs

�

Let nref ≡v2

e

hs, ν ≡ dv/dt

nref, ϕ ≡ he

hs

ν =−12�β

�e−ϕ h

he

�exp

�1

�β sin γe−ϕ h

he

�



Nondimensional Deceleration, γ=-90°

9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2

Deceleratio

Alt

itu

de R

ati

o h

/h

e

0.03 0.1 0.3 1 3

ν�β



Deceleration Equations

10

ν =−12�β

�e−ϕ h

he

�exp

�1

�β sin γe−ϕ h

he

�

n =−ρoV 2

e

2β

�e−

hhs

�exp

�ρohs

β sin γe−

hhs

�

Nondimensional Form

Dimensional Form

Note that these equations result in values <0 (reflecting deceleration) - graphs are absolute values of deceleration for clarity.



Dimensional Deceleration, γ=-90°

11

0

20

40

60

80

100

120

0 500 1000 1500 2000

Deceleration

Alt

itu

de

277 922 2767 9224 27673

(km

)

� m

sec2

�

β

�kg

m2

�



Altitude of Maximum Deceleration

12

ν = −B sin γ�e−

hhs

�exp

�2Be−

hhs

�Returning to shorthand notation for deceleration

ν = −B sin γ�e−η

�exp

�2Be−η

�Let η ≡ h

hs

dν

dη= −B sin γ

�d

dη

�e−η

�exp

�2Be−η

�+

�e−η

� d

dηexp

�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




13

1 +�2Be−η

�= 0 ⇒ eη = −2B

ηnmax = ln (−2B)

ηnmax = ln�

−1�β sin γ

�

hnmax = hs ln�−ρohs

β sin γ

�

Altitude of maximum deceleration is independent of entry velocity!

Converting from parametric to dimensional form gives




14

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

Ballistic Coefficient

Alt

itu

de o

f M

ax D

ece

l h

/h

s

-15 -30 -45 -60 -90

�βγ



Magnitude of Maximum Deceleration

15

and insert the value of η at the point of maximum deceleration

ηnmax = ln�

−1�β sin γ

�⇒ e−η = −�β sin γ

νnmax =−12�β

�−�β sin γ

�exp

�−�β sin γ�β sin γ

�⇒ νnmax =

sin γ

2e

Start with the equation for acceleration -

ν =−12�β

e−η exp�

1�β sin γ

e−η

�

nmax =v2

e

hs

sin γ

2e

Maximum deceleration is not a function of ballistic coefficient!



Peak Ballistic Deceleration for Earth Entry

16

0

1000

2000

3000

4000

5000

6000

0 5 10 15

Entry Velocity (km/sec)

Peak D

ece

lera

tio

n (

m/

sec^

2)

-15 -30 -45 -60 -90γ



Velocity at Maximum Deceleration

17

Start with the equation for velocity

and insert the value of η at the point of maximum deceleration

ηnmax = ln�

−1�β sin γ

�⇒ e−η = −�β sin γ

v

ve= exp

�1

2�β sin γe−η

�

v

ve= exp

�−�β sin γ

2�β sin γ

�⇒ vnmax =

ve√e∼= 0.606ve

Velocity at maximum deceleration is independent of everything except ve



Planetary Entry - Physical Data

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Radius(km) (km3/sec2) (kg/m3)

hs(km)

vesc(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

µ ρo



Comparison of Planetary Atmospheres

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1E-20

1E-18

1E-16

1E-14

1E-12

1E-10

1E-08

1E-06

1E-04

0.01

1

100

0 50 100 150 200 250 300

Altitude (km)

Atm

osp

heri

c D

en

sity

(kg

/m

3)

Earth Mars Venus



Planetary Entry Profiles

20

V

Ve

γ = −15o

ve = 10km

sec

β = 300kg

m2



Planetary Entry Deceleration Comparison

21

γ = −15o

ve = 10km

sec

β = 300kg

m2



Check on Approximation Formulas

22

γ = −15o

ve = 10km

sec

β = 300kg

m2

ballistic atmospheric entry (part ii) - umd...ballistic atmospheric entry ii enae 791 - launch and...

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