Using void regions trailing orbiting bodies to conserve propellant of untethered modules flying in formation with mother spacecraf

Butcher, J. R.

Stafford, W. H.

English

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NASA TECHNICAL
MEMORANDUM
Report No. 53949
V
I
PRELIMINARY ANALYSIS OF AWAKE TRAILING A
SPACECRAFT
By Walter H. Stafford and John R. Butcher
Advanced Systems Office
September 26, 1969
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NASA
^eorge C. Marshall space Flight Center
Manhall Space Flight Center, Alabama
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(PAGES)
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01
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Changed to TM X-53949, September 26, 1969
PRELIMINARY ANALYSIS OF A WAKE
TRAILING A SPACECRAFT
By
WALTER H. TAFFORD
and
JOHN R. BUTCHER
George C. Marshall Space Flight Center
Huntsville, Alabama
ABSTRACT
It was formulated that a region existed behind an orbiting
spacecraft which was relatively drag free, Due to the interest in
conserving propellant on untethered modules flying in formation
with a "mother" spacecraft, consideration was given to inserting
these free modules inside this region. Calculations were made to
determine the dimensions of the region trailing a cylinder 600 inches
long and 260 inches in diameter. Regions were alsp calculated for
minimum diameters ranging from 200 inches to 500 inches. The results
of this study are presented graphically and in pictorial form.
tNASA-GEORGE C. MARSHALL SPACE FLIGHT CENTER
PRELIMINARY ANALYSIS OF A WAKE TRAILING A , ► PACECRAFT
BY
WALTER H. STAFFORD
and
JOHN R. BUTCHER
TABLE OF CONTENTS
Page
INTRODUCTION •00.6.6.•••.♦.•4•••••••.6.66•••0•♦•0•••4••
	
1
ASSUMPTIONS
	 3
ANALYS I5 0 0 • 6 4 • • • 6 • 0 . • • . • 4 . • • 0 0 0 6 • 0 .. • • • • 4 • • 4 • 0 0 6 4 • • • 0 4	 4
CONCLUSION ..............••••.••..4...••••.••....•••••.
	
22
BIBLIOGRAPHY	 23
LIST OF ILLUSTRATIONS
Figure	 Title	 Page
1	 Pictorial Illustration of the Trailing
Void Region of an Orbiting Cylinder . * 64oa .	 2
2	 Density (KM/M3) Versus Geometric Altitude
from 100 km through 450 km with Winter and
Summer Exospheric Temperatures as a Para--
meter .......... 00 00. s•.±..:•••-. ••••.. ••S•9	 5
2-A
	 Density (KM/M3) Versus Geometric Altitude.
from 250 km through 1000 km with Winter. and
Summer Exospheric Temperatures as a para-
meter •s•••• ••o••••...•.••s•...•v•••r•e4.o•
	
6
3 Log Dumber Density (parts/0) Versus Geometric.
Altitude from 0 km through 1000 km with Winter
and Summer Exospheric Temperatures as a
parameter ..........••••4c•.••••••••..•..••	 7
4	 Diurnal Survey of the N 2 Number Density*
Versus Geometric Altitude from 150 km through
230 km
	 8
4-A	 Diurnal Survey of N2 Number Density Versus
Geometric Al^itude from 230 km through
300 km ..•..00..a 0 0 0 a... 0 	 . 0 a 6 6	 Q
4-B 	Diurnal Survey of N2 Number Density Versus
Local Solar Time from 0 hours through
21 hours 0000.. ....•.......:..	 .........	 10
PRECEDING PAGC. BLAWK NOT FiL&4ce.:.
--	 •.:,-... ^. _ .max.-.., h^
	
-^-^	 _
n
K1
._	
T
,^m4^•••:	 .^_.^^	 ___0000_	 - _ 	 Y,	 -	 . ^^.,....	 :.	 1_atx.rti ^_t-	 ..d	 000:0..	 i :."",.._.Gp.. .	 ...._	 _.	 .m .^	 a	 o
rti^Mirinw"^.`1
LIST OF ILLUSTRATIONS	 Concluded
Figure Title Page
5 Pressure (mb) Versus Geometric Altitude from
100 km through 1000 km with Winter and Summer
Exospheric Temperature as a parameter ....:..••••• 11
6 Winter and Summer Models of the Exospheric
Temperature Versus Geometric Altitude from
100 km through 1000 km with Solar Flux Heating
as	 a Parameter	 ................................... 13
7 Speed Ratio(s) Versus Geometric Altitude from
200 km through 500 km with Winter and Summer
Exospheric Temperature as a parameter ............ 15
8 Divergence Angle, A, Versus Speed Ratio;; S ....... 17
9 Length of railing Void Region Behind Orbiting
Cylinder (inches) Versus Minimum Diameter of
Cylinder (inches) with Divergence Angle as a
Parameter 18
10 Pictorial Illustration of the Trailing Void
Region of an Orbiting Cylinder with Angle of
Attack equaling 90°	 .............................. 19
11 Pictorial Illustration of the. Trailing Void
Region of an Orbiting Cylinder with Angle of
Attack equaling
	
45'°	 . . ............................ 20
12 Pictorial Illustration of the Trailing Void
Region of an Orbiting Cylinder with Angle of
Attack equaling 0°
	 ............................... 21
Tab le
I	 Speed Ratio, S. and Divergence Angles at
Various Circular Orbit Altitudes	 16
SYMBOLS AND ABBREVIATIONS
Symbol
	
Definition
i
cm	 Centimeter i
I
erf	 Probability integral
i
3
F10.7	 10.7 Solar flux
ge	 Gravity at Earth Surface; km/sect
h	 height from earth surface, km
H	 Hydrogen Atom
HE	 Helium Atom
oK	 Degrees, in thermodynamic Kelvin scale
i
Km.	 Kilometer
M Meter
max. Maximum
mb Millibar
min. Minimum
N2 Nitrogen
n Number density
0 (ht;ygen Atom
02 Diatomic Oxygen
R Universal Gas Constant
Re Mean radius of earth, km
S Speed Ratio,	 circular satellite velocity
most probable particle
velocity
S
SYMBOLS AND ABBREVIATIONS
	 Concluded
Symbol	 Definition
T«
	Exospheric temperature
VP	 Probable particle velocity
Ve	 Circular satellite velocity
Z	 Geometric Altitude
A	 Divergence ,angle
Y[S]	 Se-s2 + tf w - (S2+1/2) (1 + er£s)
X(S)	 e-s2 + fT S(l + erfs)
FOREWORD
The high velocity motion of a space station through the
f
	
	 rarefied atmosphere of high a ptitudes causes complex gas flaw
patterns around the orbiting station. Well known are the re-
sulting drag forces. Little if any effort has been devoted
to the wake which trails a space station.
This idea of an orbital wake behind a space station came
to 'mind when the .concept of a space station was developed flying
formation with several separate sub-satellites or modules. The
higher drag forces on those modules as compared to the space
station would require constant expenditures of propellants in
order for them to maintain their position, Over a period of a
year or two this would amount to sizeable quantities.
This combination of orbital wake and independent module
resulted in the idea to place the independent module in the wake
of a spate .station and keep it there. This could result in size-
able savings of propellants, and therefore weights and complexities.
The authors undertook an investigation of this problem.
Dr. R. Oman of the Grumman Aircraft Company gave encourage-
ment to this idea and calculated drag reductions of over 50% for
bodies in the wake. We will. continue this effort.
	 ;
r^ o tU
Irg n Tiesgenhausen
INTRODUCTION
A study of void regions trailing orbiting bodies could
be of fundamental importance in conserving the propellant of
smaller utntethered modules flying in formation with a "mother"
spacecraft.
The purpose of this study was to provide dimensions of
void regions for typical cylinderical bodies of various sixes.
These objects were ccnsidered to be in circular orbits from
200 km thtvugh 500 km about the earth with a SO" inclination.
The environment of these orbiting bodies was assumed to
be of a'hyperthermal free molecular origin. Thus the approach
used was to determine, through theoretical, assumptions, the
speed rat°,ios $
 divergence angles, and the configurations
as ociated with these orbiting bodies from the most probable
molecular particle velocities. (See Figure 1)
fi
t
1
FLOW
8<
TRAILING VOID REGIONS OF AN ORBITING CYLINDER
FREESTREAM REGION
°
MINIMUM
DIAMETER
ol
°
— -- -- -- -- -- DISPERSED REGION
FLOW
ago-
CYLINDER	 DIVERGENCE ANGLE
	
^ a	 1r
3
r
e	 E
-	
••":: ••:: ••::: •:: •^ ^
•r+a•^ i•: •"i:: i::: ii: ii: i •iiiiy^'. •
	 iPy,y
r
FIGURE I
o	 _	 t^	 a	 ^"?"""'rd^' ""	
_ '#TIC'=..
ASSUMPTIONS
The following is a summary of the basic assumptions
used in this study:
1. Hyperthermal free molecular flow
environment.
r	
2. Cylindrical configuration 600 inches long and
260 inches in diameter. (Approximates
shape of S-IVB stage.)
3. Flow is to be steady with uniform density
upstream of the surface.
4. Gas has a maxwellian velocity distribution
superimposed on the mass flow velocity.
5. Dynamic and kinetic atmosphere criterion.
i
p
it
i
,r.
t
Fi
r
1 &
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i
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.^Y"y^' ..	 ^MR}^4^1]RW^+.^dLL yry	 ..	 ".	 .^	 ^'
_
45
ANALYSIS
	
1.1	 Atmosphere
The atmospheric shell or region is determined by four
basic criterions. The first criterion, being a temperature
distribution, is one of the most common ways of defining the
layers of the earth atmosphere. The second criterion is the
physicochemical process which defines the atmosphere layers
through molecular, reactions caused by the absorption of
radiation. The third criterion defines the distinction of
atmospheric layers according to photo-dissociation and gravi^
tational separation of mean molecular weights. The fourth
criterion is the dynamic and Kinetic processes which are re-
presented by the model atmosphere used in the study. In this
model, the exosphere, from which all calculations are based,
is the outer-most portion of the atmosphere. The lower boundary
of this model atmosphere is the critical level of escape,
variously estimated at 500 km to 1000 km about the earth surface.
In the exosphere, the air density is so low that the mean free
path or free molecular path of individual particles is completely
undirectional and depends upon their direction with respect to
the local vertical, being greatest for upward-moving particles.
It is only from the exosphere that atmospheric gasses can, to
any appreciable extent, escape into outer space.
The data used in this study which pertains to the
exosphere originates from talc, 1966 SupYleiu^i"st to v • S • a Laiidisrd
Atmosphere 1962.
	
1.2	 Exospheric Density
Above. 120 km of the exosphere there is one basic para-
meter in the 1966 Supplementary Atmosphere, a set of exponential
temperature curves, which were empirically derived to provide
density - altitude profiles that coincide with satellite - drag
densities derived for various degrees of solar and geomagnetic
activity and varying solar angles. Three sets of density models
above 120 km were developed from the 1966 Supplement. At lower
altitudes eight supplementary atmospheres converged at 120 k%, to
form these three sets. A single set is applicable to.spring and
fall conditions, a set for winter, and a set for summer. Tempera-
ture and density data were taken from these winter and summer sets.
Density and number density profiles are presented in Figures 2-,
2-A, and 3. Based on satellite data, variations in nitrogen
number densities as a function of solar time are presented in
Figure 4, 4-" and 4-B. Variations of pressure_ between summer and
winter models ire shown in Figure 5.
4
r,
m a a
^^	 , -	 °--^"'	 •^	 ^,. w.^ru'.r,-„ rs 	 ^	 rw	 ,..^......^ ^.,,^...
	 „^,_.-^,M.,.,,_..,
	
^. a	
^2
^r	 -,^.stl'f'f^	 =..;	 ._..,,	 ..+	 s	
...a;.	 -C.:.. •. -.	 w.a^^f"^'^'`.
	
7
r. m
xxw+
10-8
MODEL ATMOSPHERE
DENSITY VS. ALTITUDE
10-9
m
E0
r
W 10-10
V
W
0
H
a
10-11
X0--12
0
10 —12
MODEL ATMOSPHERE
DENSITY VS. ALTITUDE
WINTER MODEL
EXOSPHERIC TEMP. 21050° K (MAX.)
SUMMER MODEL
EXOSPHERIC TEMP. 600°K (MIN.)
WINTER MODEL
10-13
10-14
9*!	 I
10—'15
f
10-16
200	 400
	
600	 800	 1000	 x
1x
ALTITUDE VKm )
REF.1966 SUPPLEMENT TO U.S. STANDARD ATMOSPHERE 1962
FIGURE 2a
6
F
..	 .. _m...... -i._fW.P _: ._3+s...4....	 .._'"` .^+.!^.r^
	
.4.. 3^ 	 ^^	 iury-+r.
	 ^i!	 -	 "'^GSr4^..	 R	 4`.^re.
n.i.1s.-L	 a*	 a.!	 w.^.,,w:	 ^^.	 .1a	 ^^ a.^	 ^ }r^R
 ^^ •,. ^^
-^^yC^yKty,+^^..^rx.e•	 ^:.^ia^N#SCS.^'y^.;cw.c'®
	 'G^.^	 ^	 ia^nHA°^_^	
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yy
	
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M
E
0
X
F-
t/f
zW
W
lwW
Q.
Nn
c0C
a
MODEL ATMOSPHERE
17,0 NUMBER DENSITY VS. ALTITUDE
WINTER EXOSPHERIC TEMP. 2100°K (MAX.)
16.0 SUMMER EXOSPHERIC TEMP. 600 0 K (MIN.)
• 15.0
14.0 ^•
13.0 , _ LOG n O WINTER
12.0 LOG n HE WINTER^`	 ^'^	 ^'`t	 — —` LOG n H SUMMER
M 11.0
 '= -^_s LOG n N 2
 WINTER
E LOG n HE SUMMER
10.0 \
z \ LOG n O Z WINTERo 9.0 ._._...._..._.. 	 ..`.... ,_\ LOG n H WINTER
oc
Lu 8.0 \m\
7.0
c
• 6.0
LOG n O SUMMER
O \
u
•	 u 5.0 ^
oc
= 4.0
c^ \
O 3.0
2.0 \
1.0
0.0
—1.0 LOG n N2 SUMMER
—2.0
A.
1010
DIURNAL SURVEY OF THE THERMOSPHERE
N2 DENSITY PARTS/cen 3 VS.
ALTITU QE
CAPE KENNEDY, FLA.
JAN. 24, 1967	 LAT.28° 27N
REF. RESEARCH ACHIEVEMENTS REVIEW
VOL. II REPORT NO. 10
M
Ea
aG.
to
W
G
Ei
W
kn
O
a
LOCAL SOLAR TIME
0326
—^ 0618 SUNRISE
---- — 2126
-	 0935
1712 SUNSET
- - — 140► 0
109
140	 160	 180	 200	 220	 240	 8FIGURE 4	 ALTITUDE Z,(Km)	 I
i
....	 # s	 t7lcx	 ^	 :tea
220
FIGURE 4a
240	 260
ALTITUDE Zj(Km)
108
200 280
	 0
109
DIURNAL SURVEY OF THE THERMOSPHERE
N 2
 DENSITY PARTS/cm3
`^	 \ vs.
`;	 \	 ALTITUDE
o	 CAPE KENNEDY, FLA.
% 
V
JAN. 24,1967 LAT. 28 0 27'N
\ \" \\ \ REF. RESEARCH
VOL. I REPORT  NO
ElJT^S REVIEW
sn
E
Q
o.
F-
Z
Lu
NZ
W
C.hO
a
i
i111
1	 ^	 ^
1
i
1
LOCAL SOLAR TIME	 \ \
1	 ^
^---- 0618 SUNRISE
	
-- 2126	 \
--
	
0935	 i
---- 1712 SUNSET t
------- 1400
:sr..w.w.........^+.ux	 f14::...,....>..,..... .=,..Y..-tom€	 >•-_"x+ 	 ..-,	 .,: ^ .........s.^.^'.^^^= M .,	 ,r+...»r..:•^,..^.. .+..,...^.
`^..^yr
DIURNAL SURVEY OF THE THERMOSPHERE
N 2
 NUMBER DENSITY
VS.
TIME
	
O	 p
	
1501(m
	
150 Km
1010
O
O
M
E	 200Km
tn 2001(m
Q	 109
CL
N
W
°C	p 	 250Km250Km
z
108
4
3001(m3001(m
CAPE KENNEDY, FLA.	 MUMP DATA28 ° 27' N. LATITUDE
JAN. 24r
 1967	 0 CIRA 1965 MOD. 4
107 1
05	 10	 15	 20	 01 ` f
LOCAL SOLAR TIME, hrs.
REF. RESEARCH ACHIEVEMENTS REVIEW VOL. II REPORT NO. 10 f
FIGURE 4b
io 3
x
..
^yI
,
.o
4
a z.	 r_
.Xt'
MODEL ATMOSPHERE
PRESSURE VS. ALTITUDE
10-b
AAX.)
AIN.)
10-7
E
W
HNW
CC
D,.
10-8aW
a
O
10-9
10-10
I.
1
	
1.3
	
Exospheric Temperature
].your types of temperature variations have been
recognized at heights, greater than 200 ism, namely:
1. A variation with solar activity
2. A semiannual variation
3. A diurnal variation, and
4. A variation with geomagnetic activity.
Empirical formulas have been constructed to compute the
exospheric temperature when these parameters are known. Once
the exospheric temperature T oo is computed, atmospheric densities
and related quantities can be found for any given height.
Temperatures used in the calculations of the most
probable velocity are:
1. Winter exospheric temperature at
a. 600° K minimum, model
b. 21,00° K maximum model
2. Summer exospheric temperature at
a. 600° K minimum mo iel
b. 2100° K maximum model
Temperatures from the winter model (max) and summer model (min)
as a function of altitude are presented in Figure 6.
	
1.4	 Computations
Most probable velocitlAs were calculated from 200 km to
500 km with 30 km increments The following equation form was
used:
3.
r.
2200
2000
1800
he
0	 1600
W
oC
a
W	 1400
1200
1000
800
600
400
MODEM ATMOSPHERE
TEMPERATURE VS. ALTITUDE
2400 r	 WINTER MODEL EXOSPHERIC TEMP. 2100° K (MAX.)
SUMMER MODEL EXOSPHER.IC TEMP. 600°K (MIN.)
IA speed ratio for each altitude of 50 km increments
between 200 km and 500 km was calculated as shown in Figure 7.
The following equation was used and the results are tabulated
in Table 1.
Speed Ratio S .. Circular satellite velocity/most
probable particle velocity.
Divergency angles were determined for altitudes
corresponding to the altitudes used in the previous calculations.
Angles were determined from the equation.
8 - tan -1 1 X(5W.. 
WISJ
Divergence angles determined from this equation are presented
in Fig. 8 as a function of speed ratio, S. The results are
tabulated in Table 1.
Since the minimum diameter and divergence angles are known,
trigonometric relations were used to find the length of the void
regions. The lengths are presented in Figure 9 as a function of
the minimum diameter with divergence angles as a parameter.
Figures 10, 11 and 12 il1ug trate the shape of the void
regions with angles of attack 90% 45° and 0° respectively. For
zero angles of attack the void region is a cone of 8 a half angle
and for all other angles of attack greater than zero the void
region is wedge shaped.
^` I
MODEL ATMOSPHERE
SPEED RATIO
F S.
ALTITUDE (KM)
WINTER & SUMMER E:XOSPHERIC TEMP. 600°K (MIN.)
WINTER & SUMMER EXOSPHERIC TEMP. 2100° K (MAX.)
12.0
^^,. ,._...,,,, W INTER EXO.
	
11.0 	 TEMP. 6000 K
10.0 t SUMMER EXO.
TEMP. 600° K
0 9.0
dt
ac0WW0.
	
8.0
WINTER EXO.
Z	 TEN! P. 2100? K
7.8
SUMMER EXO.
TEMP. 2100° K
6.0
200	 300	 400	 500
AI TITI MC 7 /Vm%
5.0
Speed Ratio, S. and Divergence angles, at various circular Orbit Altitudes
Altitudes, km
19on	 12- go	 lino	 1.,150	 1400	 1450	 1500
210+9°K Summer Divergence 4.2 4.7 5.1 5.3 ,. 5.5 5.7 5.8
Exo. Temp. (max,) Speed Ratio, S 7.63 6.75 6.32 6.03 ;5.82 5.63 5.48
r.^.rwr.r.^. 1r4uw^.
-
Y wr+w+rwrt+..wr
-
w'W.i. wMM.MM
-
aM1.uiM!M •.
-2100" K Winter	 Speed patio, S 7.89	 6.81 -
Exo. Temp. (max.) Divergence Q! 4.1 4.7 - - -- - -
600° K Summer Divergence ¢ 2.8 3.1 3.3 3.5 3.9 4,8 6.1
Exo. Temp. (min.) Speed Ratio, S 11.50 10.45 9 . 80 9.23 8.16 6.64 5.17
600° K Winter Speed Ratio, S 11.61 - - - - - -
Exo. Temp. (min.) Divergence 0 2.75
t
A
16
vtWW
W
WJ
Z
Q
W
V
zW
W
a
7.0
6.0
5.0
10.0
9.0
4.0
3.0
2.0
;r
8.0
DIVERGENCE ANGLE 0 AS A FUNCTION
OF SPEED RATIO, S
0	 1	 2	 3	 4	 5	 b	 7	 8 9
,,	
10	 11	 12
SPEED RATIO, S
r
FIGURE 8
17	 i
s
^`isu^_....._
	 , s....^,..i..	 ,.	 .,.^. .. •	 _	 ^	 _.	 .. N^. v` ., ^ e	 ^	 ..^	 _	 f ^.tm ^. .^	 i
....	
_	 xrW	 u.	 -.	 ...	 _ .. bneu-wr.1S4	 .,	 ^^.
2300
2200
2100
2000
DISPERSION
ANGLE 0 6°
LENGTH OF "OID VS. MINIMUM DIAMETER
2400s-
2u
zK
O
Ls-
O
z
OZ
LU
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
DISPERSION
ANGLE 15 80
DISPERSION
ANGLE 0 10 o
DISPERSION
ANGLE 0 12 °
DISPERSION
ANGLE 0 140
500
•	 1
400
e
FLOW
TRAILING `v011) REGIONS OF AN ORBITING CYLINDER
ANGLE OF ATTACK c- 900
FREESTREAM REGION
ulvtKGtNCE ANGLE
L
FREESTREAM REGION
TRAILING VOID R EGIONS 01= AN ORBITING CYLINDER
ANGLE OF ATTACK ac= 45 
I FREESTREAM REGION
-- -
TRAILING VOID REGION OF AN ORBITING CYLINDER
ANGLE OF ATTACK a,-" O 0
TRAILING VOID REGION
MOTION OF CYLINDER
7
DIVERGENCE ANGLE
FREESTREAM REGION
CONCLUSION
There exists a sizable wake trailing a typical
orbiting body which is dependent, to a large extent, on
the speed ratio and minimum diameter of the module. The
wake is considered large enough to accommodate reasonably
large modules.
ti
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22
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BIBLIOGRAPHY
U. S. Standard Atmosphere Supplements, 1966
Robertson, J. S.: Free Molecular Wake of an Orbiting Cylinder,
April 1968. Lockheed Missiles and Space Company, Huntsville
Research and Engineering Center, Huntsville Research Park,
Huntsville, Alabama
Robertson, J. S.: Thermal Diaper-ion of Free Molecular Shading,
December 1967., Lockheed Missiles and Space Company, Huntsville
Research and Engineering, Huntsville Research Park, Huntsville
Alabama
n 	 P
I	 '


Preliminary analysis of a wake trailing a spacecraft

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