CIRCULATING TRANSPORTATION ORBITS BETWEEN EARTH AND M R S
.
1
2
Alan L. F r i e d l a n d e r and John C. N i e h o f f
Science A p p l i c a t i o n s I n t e r n a t i o n a l C o r p o r a t i o n
Schaumburg, I l l i n o i s
3
Dennis V. Byrnes and James M. Longuski
J e t Propulsion Laboratory
Pasadena, Cal if o r n i a
Abstract
.
s u p p l i e s i n s u p p o r t o f a s u s t a i n e d Manned Mars Sur-
T h i s paper d e s c r i b e s t h e b a s i c c h a r a c t e r i s t i c s
o f c i r c u l a t i n g ( c y c l i c a l ) o r b i t d e s i g n as a p p l i e d
f a c e Base sometime d u r i n g t h e second q u a r t e r o f t h e
next century.
t o r o u n d - t r i p t r a n s p o r t a t i o n o f crew and m a t e r i a l s
A c i r c u l a t i n g o r b i t may be d e f i n e d as a m u l t i -
between E a r t h and Mars i n support o f a s u s t a i n e d
manned Mars Surface Base.
The two main t y p e s o f
non-stopover c i r c u l a t i n g t r a j e c t o r i e s a r e t h e socalled
VISIT
orbits.
orbits
and
Up/Down
the
Escalator
Access t o t h e l a r g e t r a n s p o r t a t i o n f a c i l i -
t i e s placed i n t h e s e o r b i t s i s by way o f t a x i v e h i -
p l e - a r c t r a j e c t o r y b e t w e e n t w o o r more p l a n e t s
(1) does n o t s t o p a t t h e t e r m i n a l s b u t
which:
rather
involves gravity-assist
planet;
swingbys
of
each
( 2 ) i s r e p e a t a b l e o r p e r i o d i c i n some
sense; and ( 3 ) i s continuous, i.e.,
t h e process can
c l e s u s i n g h y p e r b o l i c rendezvous t e c h n i q u e s d u r i n g
be c a r r i e d on i n d e f i n i t e l y .
t h e s u c c e s s i v e e n c o u n t e r s w i t h E a r t h and Mars.
c i r c u l a t i n g o r b i t s b e t w e e n E a r t h and Mars were
1960s 1 i t e r a t ~ r e . l ' * ' ~ ' ~ Re-
S p e c i f i c examples o f r e a l t r a j e c t o r y d a t a a r e pre-
identified
sented i n e x p l a n a t i o n o f f l i g h t times,
examination o f t h i s i n t e r e s t i n g problem d u r i n g t h e
frequency,
hyperbol i c v e l o c i t i e s ,
distances,
and
encounter
c l o s e s t approach
A V maneuver requirements i n b o t h
i n the
D i f f e r e n t types o f
p a s t y e a r has produced new
solution^.^'^'^
p o r a t e Venus encounters i n t h e scenario.
i n t e r p l a n e t a r y and p l a n e t o c e n t r i c space.
Some
i n v o l v e o n l y these two b o d i e s w h i l e o t h e r s i n c o r Some have
more o r l e s s d e s i r a b l e c h a r a c t e r i s t i c s as measured
by f l i g h t t i m e s ,
b o l i c approach v e l o c i t i e s ,
I. I n t r o d u c t i o n
tances,
The
swingby
concepts
or
and
principles
"gravity-assist"
are
of
planetary
science
missions
gravity-assist
Mercury ( M a r i n e r l o ) ,
we1 1-estaflown
of
have
Venus
and
t h e Moon ( ISEE-3/ICE),
and
t h e g i a n t o u t e r p l a n e t s (Voyager).
T h i s energy-
saving technique can be expected t o see i n c r e a s e d
use i n f u t u r e automated and manned i n t e r p l a n e t a r y
travel.
this
The p o t e n t i a1 a p p l i c a t i o n o f i n t e r e s t i n
paper
is
the
transportation
of
crew
and
closest
hyper-
approach d i s -
and t h e amount o f midcourse p r o p u l s i v e 4 V
needed t o a d j u s t and m a i n t a i n t h e o r b i t c o n t i n uousl y
A number o f
a1 ready
swingbys
planetary
now
b l i s h e d i n b o t h t h e o r y and p r a c t i c e .
utilized
frequency o f encounters,
.
C i r c u l a t i n g o r b i t s share one p o t e n t i a l advant a g e f o r manned t r a n s p o r t a t i o n between E a r t h and
Mars.
They a l l o w a l a r g e o r b i t i n g f a c i l i t y ( h e r e i n
c a l l e d a "CASTLE") p r o v i d i n g a l l power, l i v i n g and
work space,
l i f e support, g r a v i t y environment, and
s o l a r storm s h e l t e r t o be "once-launched",
thereby
o b v i a t i n g t h e need t o c a r r y t h e s e massive elements
repeated1y t h r o u g h l a r g e p l a n e t o c e n t r i c A V maneuvers.
T r a n s p o r t a t i o n t o and from t h e s e CASTLES i s
c a r r i e d o u t by small e r Space T a x i s u s i n g hyperbol ic
System Analyst, Member AIAA and AAS
rendezvous techniques.
Department Manager, Member AIAA
Consultant, Member AIAA and AAS
Member o f Technical S t a f f , Member AIAA
T h i s paper w i l l d e s c r i b e t h e b a s i c c h a r a c t e r i s t i c s o f c i r c u l a t i n g o r b i t d e s i g n and s p e c i f i c examples o f r e a l t r a j e c t o r y d a t a t h a t have been gen-
T h i s work was performed by SAIC and JPL under JPL
C o n t r a c t No. 957404 and NASA C o n t r a c t No's. NAS7918 and NASW-3622.
This paper is declared a work of the U.S. Government and is
not subject to copyright protection in the United States.
e r a t e d i n a r e c e n t study o f t h i s problem.8
The two
main t y p e s o f c i r c u l a t i n g o r b i t s having d e s i r a b l e
c h a r a c t e r i s t i c s are the so-called VISIT o r b i t s and
t h a t t h e connecting h e l i o c e n t r i c o r b i t be n e a r l y
t h e Up/Down Escalator o r b i t s .
tangent t o t h e planetary o r b i t s ,
Each o f these candi-
i.e.,
the peri-
date o r b i t s w i l l f i r s t be discussed i n terms o f
h e l i o n o f t h e t r a n s f e r o r b i t must be near 1.0
t h e i r design p r i n c i p l e s i n t h e approximate model
and t h e aphelion near 1.52
AU.
AU
The o r b i t which
world o f c i r c u l ar and cop1 anar p l anetary motion.
satisfies
With t h i s background, q u a n t i t a t i v e r e s u l t s are then
model has a semi-major a x i s o f 1.26 AU and a period
t h i s exactly
i n t h e c i r c u l a r coplanar
presented and compared f o r t h e m u l t i - o r b i t propaga-
(P)
t i o n s i n t h e actual case o f eccentric and i n c l i n e d
quired t h a t the. t r a n s f e r o r b i t be resonant w i t h
planet o r b i t s .
both Earth and Mars, t h a t i s :
Also covered are o r b i t a l analysis
o f 1.415 years.
t o p i c s r e l a t e d t o t h e use o f the c i r c u l a t i n g t r a n s e.g.,
p o r t a t i o n mode,
Taxi
v e h i c l e maneuver re-
quirements i n launching t o ( o r b i t i n j e c t i o n ) and
separating from ( o r b i t
facilities.
capture)
the circulating
The reader i s r e f e r r e d t o a companion
paper i n t h e Conference Proceedings which describes
t h e o v e r a l l mass performance analysis o f c i r c u 1a t i n g o r b i t s compared t o t r a d i t i o n a l ,
energy
stopover
trajectories
9
sustained manned Mars Base.
in
minimum-
support
of
a
I f , i n addition,
i t i s re-
nP = m
and
(1)
j P = kPM
where n, m, j, and k are small integers, then t h e r e
are only two p o s s i b i l t i e s .
The f i r s t p o s s i b i l i t y
(denoted VISIT-1) has a period o f 1.25 years and a
semi-major a x i s o f 1.16
four
AU.
This o r b i t completes
r e v o l u t i o n s about the Sun while Earth com-
pletes five.
I t a1 so completes three r e v o l u t i o n s
about t h e Sun while Mars completes two.
4:5
resonance w i t h Earth and 3:2
Hence, t h e
resonance w i t h
Mars means t h a t Earth encounters w i l l occur once
every f i v e years and Mars encounters once every
11. Prelude t o C i r c u l a t i n g O r b i t Design
3.75
E a r t h and Mars r e v o l v e about t h e Sun w i t h
orbit
periods
respectively.
of
1.0
year
and
1.8808
years,
Earth t r a v e l s i n t h e e c l i p t i c plane
w i t h an o r b i t e c c e n t r i c i t y o f 0.0168
w h i l e Mars
t r a v e l s i n a plane i n c l i n e d by 1.85'
t o the ecl i p -
t i c w i t h an e c c e n t r i c i t y o f 0.0934.
As a help i n
understanding the p r o p e r t i e s o f c i r c u l a t i n g o r b i t s ,
ignore f o r the moment the non-circular,
planar motion o f t h e r e a l world.
non-co-
Additionally,
approximate the period o f Mars (PM) by 1.875 years,
years.
The
second
possibility
(denoted
VISIT-2) has a period o f 1.5 years and a semi-major
a x i s o f 1.31 AU.
This o r b i t completes two revolu-
t i o n s about t h e Sun w h i l e Earth completes three.
It also completes f i v e revolutions about t h e Sun
while Mars completes four.
The 2:3 resonance w i t h
Earth and 5:4 resonance w i t h Mars means t h a t Earth
encounters occur once every t h r e e years while Mars
encounters occur once every 7.5 years.
I n the c i r -
c u l a r coplanar world the VISIT-1 o r b i t must have a
p e r i h e l i o n o f l e s s t h a n 1.0
AU and t h e VISIT-2
so t h a t every 15 years Earth makes 15 r e v o l u t i o n s
o r b i t must have an aphelion o f greater than 1.52 AU
about the Sun while Mars makes exactly e i g h t r e v o l -
since t h e i r periods are respectively l e s s than and
utions.
Since the re1 a t i v e geometry between Earth
greater than t h e value o f 1.415 computed above.
In
and Mars repeats w i t h t h e synodic period o f 1517 =
the real world,
2.1429
makes t h e VISIT-1 o r b i t i d e a l f o r encountering Mars
years,
trajectories
connecting
re-occur w i t h t h i s period also.
metry repeats every 15 years.
have
important
ImpllcatSons
them w i l l
The i n e r t i a l geoThese r e l a t i o n s h i p s
in
the
design
and
t h e e c c e n t r i c i t y o f Mars'
orbit
near i t s p e r i h e l i o n and t h e VISIT-2 o r b i t i d e a l f o r
encountering Mars near i t s aphelion.
These o r b i t s
are i l l u s t r a t e d i n Figure 1.
r e p e a t a b i l i t y o f c i r c u l a t i n g o r b i t s between these
planets.
Due t o t h e resonant c h a r a c t e r i s t i c s o f t h e
VISIT o r b i t s w i t h t h e i r Earth and Mars encounters
The VISIT type o f c i r c u l a t i n g o r b i t ,
first
they remain f i x e d i n i n e r t i a l space i n t h e c i r c u l a r
~,
from consideration
proposed by ~ i e h o f f ~ ' evolved
coplanar world.
o f a class o f t r a j e c t o r i e s which might d i s p l a y low
t h e planetary encounters must have only a n e g l i g i -
relative velocities
b l e e f f e c t on t h e t r a n s f e r t r a j e c t o r y ,
minals.
a t both Earth and Mars t e r -
I n order f o r t h i s t o occur i t i s necessary
The i m p l i c a t i o n o f t h i s i s t h a t
but t h i s
implies i n t u r n t h a t the encounters while close on
a h e l i o c e n t r i c s c a l e must be d i s t a n t on a p l a n e t o -
t h e requirement o f encountering E a r t h and Mars.
T h i s leaves o n l y t h e e c c e n t r i c i t y , e, t o be d e t e r -
c e n t r i c scale.
mined.
The
orbit,
UpIDown
Escal a t o r
type
of
c ircula ting
From F i g u r e 2, i t i s seen t h a t t h e magnitude
,
from t h e
f i r s t proposed b y ~ l d r i n ~evolved
d e s i r e t o have more f r e q u e n t and r e g u l a r encounters
o f t h e t r u e anomaly a t E a r t h e n c o u n t e r ,
w i t h b o t h E a r t h and Mars.
e x a c t l y one ha1 f o f t h e r e q u i r e d advance o f p e r i -
Since encounter l o c a -
t i o n s on a 1517 synodic p e r i o d o r b i t r o t a t e 117 o f
h e l i o n , 101 = A 9/2.
a c i r c l e per r e v o l u t i o n ,
a t r u e anomaly o f 25.7"
t h e r e e x i s t s a synodic
0,
is
Thus, E a r t h i s encountered a t
and t h e f l y b y changes t h e
precession o f t h e l i n e o f apsides equal t o 51.4'
t r u e anomaly t o -25.7'.
The n e t e f f e c t i s t o r o -
p e r o r b i t w h i c h m u s t be accommodated b y e i t h e r
t a t e t h e o r b i t by 51.4O.
The v a l u e o f t r u e anomaly
p r o p u l s i v e maneuvers (which a r e n o t d e s i r a b l e ) or,
a t E a r t h encounter s p e c i f i e s t h e f i n a l o r b i t a l
t o t h e e x t e n t possible,
o f t h e p l a n e t s ( p r i m a r i l y E a r t h s i n c e i t i s more
element, e, through t h e c o n i c equation:
2
r = a(1-e )/(l+ecosQ)
massive).
where
by g r a v i t y - a s s i s t swingbys
Given t h i s e f f e c t i v e t r a j e c t o r y shaping,
b o t h E a r t h and Mars w i l l
be encountered sequen-
r = 1 AU
t i a l l y every 2-1/7 years.
The encounter speeds a t
a = ( 1 5 / 7 ) ~ / ~ - =1.662 AU
0 = -25.7
both p l a n e t s ( b u t p r i n c i p a l l y Mars), however, w i l l
(2)
degrees
be s i g n i f i c a n t l y h i g h e r t h a n those o f VISIT o r b i t s
Solving the quadratic equation i n e gives e =
because t h e 2-117 y e a r p e r i o d t r a n s f e r o r b i t has an
0.416.
aphelion
distance
of
about
2.32
AU,
thereby
Having s p e c i f i e d t h e he1 i o c e n t r i c
c r o s s i n g t h e o r b i t o f Mars a t steeper (non-tangen-
orbit,
t i a1 ) angles.
t h e c h a r a c t e r i s t i c s o f t h e E a r t h f l y b y can
be determined.
The 'Up"
Earth)
(Earth-to-Mars)
and "Down" (Mars-to-
E s c a l a t o r o r b i t s d e f i n e d here a r e essen-
Escalator
Figure 3 i l l u s t r a t e s the r o t a t i o n
o f t h e h e l i o c e n t i c v e l o c i t y v e c t o r f r o m Tin t o
-
VoUt
r e s u l t i n g i n t h e change i n he1 i o c e n t r i c v e l o -
c i t y , A V.
t r a n s f e r back, w h i l e t h e reverse i s t r u e f o r t h e
The f l i g h t p a t h angle, ,y, i s found as
2
tan Y = (erl(a(1-e ))sin0
(3)
S o l v i n g t h i s equation g i v e s y = 7.48 degrees. From
"DownH o r b i t .
F i g u r e 3 i t i s seen t h a t :
t i a l l y m i r r o r images o f each o t h e r .
The "Up" o r b i t
w i l l have a s h o r t t r a n s f e r t o Mars and a l o n g
F i g u r e 2 i l l u s t r a t e s t h e concept o f
rotating the orbit.
E a r t h a t E l and encounters Mars a t M2 ( f o r t h e "Up"
Escalator).
When t h e f a c i l i t y crosses t h e E a r t h ' s
o r b i t a g a i n a t 2-117 y e a r s l a t e r , E a r t h i s n o t
there, b u t i s re-encountered a t E3.
(4)
A V = 2Vsin r,
The c y c l i n g f a c i l i t y leaves
where V =
Iv.
I
=
~n
c i t y a t Earth, V,
I. The h e l i o c e n t r i c v e l o out
i s found from t h e energy equa-
tion:
At E3 E a r t h ' s
g r a v i t y r o t a t e s t h e o r b i t so as t o encounter Mars
a t M4 w h i l e m a i n t a i n i n g he1 i o c e n t r i c energy.
Simi-
l a r l y , t h e "Down" E s c a l a t o r i s d e f i n e d by t h e o r b i t
from E l t o M2' t o E3 t o M4'.
The advance o f t h e
p e r i h e l i o n p e r o r b i t i s d e f i n e d as
AY
= 2n 17
radians.
.,
F o u r o r b i t a l elements a r e r e q u i r e d t o s p e c i f y
an o r b i t i n a plane.
-
a r e chosen, t h e n
U,
I f t h e s e t (a, e,
and T)
For E a r t h VE = (,/r)112
= 29.785
kmls, where u i s
t h e g r a v i t a t i o n a l parameter o f t h e Sun, and so V =
35.2
km/s.
S u b s t i t u t i n g these values g i v e s A V =
9.17
km/s.
Again from F i g u r e 3 t h e asymptotic
v e l o c i t y magnitude i s given by
V, = ( v 2 + : V
2vvE cosy) 112
-
(6)
t h e argument o f p e r i a p s i s , can
be a r b i t r a r i l y s e t t o zero s i n c e t h e o r b i t s o f
E a r t h and Mars a r e c o n c e n t r i c c i r c l e s .
The semi-
major axis, a, i s determined by t h e synodic period,
and t h e t i m e o f p e r i a p s i s , T, w i l l be s p e c i f i e d by
S u b s t i t u t i n g g i v e s V,
= 6.88
kmls.
The f l y b y o f
E a r t h t u r n s t h e asymptotic v e l o c i t y v e c t o r by an
angle 26.
The equation r e l a t i n g t h i s t u r n a n g l e
and t h e r a d i u s o f c l o s e s t approach r
i s given by:
P'
4
Fig. 1 V I S T o r b i t d i a g r a .
-
Fig. 2 Escalator o r b i t diagram.
Fig. 3 Gravity-assist velocity diagraa.
34
2
s i n 6 = 1 / ( 1 + r , V /pE))
(7)
P
where pE i s the g r a v i t a t i o n a l parameter o f Earth.
approach d i s t a n c e s tend t o be f a i r l y l a r g e ,
par-
Since from F i g u r e 3
ticularly
relatively
weak
maintaining
the
A V = 2V=,sin &
(8)
a t E a r t h and 3.7
at
gravity-assist
t o 3.9
Earth,
kmlsec a t Mars.
indicating
requirements
for
VISIT t y p e o r b i t resonances.
Closest
T h i s geometry char-
s o l v i n g gives r
a c t e r i s t i c has somewhat adverse i m p l i c a t i o n s on t h e
2 6 = 83.4'.
hyperbolic
= 4,221 km, and t h e t u r n angle
P
Since t h e E a r t h ' s r a d i u s i s 6,371 km,
t h e value o f r
corresponds t o a f l y b y o f E a r t h
P
beneath t h e surface.
I n o r d e r t o have a p h y s i c a l l y
rendezvous t r a j e c t o r y o f t h e outgoing
Taxi v e h i c l e s .
That i s , a rendezvous
A V
penalty
o f up t o several hundred meters/sec (depending on
r e a l i z a b l e f l y b y , t h e c y c l i n g f a c i l i t y must t h e r e -
t h e t i m e allowed from Taxi launch t o rendezvous)
f o r e perform a p r o p u l s i v e maneuver somewhere on t h e
w i l l have t o be accounted f o r t o compensate f o r t h e
orbit.
A r e l a t i v e l y small maneuver b f about 230
m/sec performed a t a p h e l i o n o f t h e E s c a l a t o r o r b i t
non-aligned asymptotes o f t h e CASTLE and Taxi t r a j e c t o r i e s (see Section V).
can r o t a t e t h e argument o f p e r i a p s i s by an amount
s u f f i c i e n t t o make t h e f l y b y o f E a r t h o c c u r a t
The VISIT o r b i t s o l u t i o n i s n o t t o t a l l y unique
can be found f o r neigh-
i s t h a t e t h e r "free o r b i t s '
1,000 km above t h e surface;
boring variational trajectories.
Although
other
types
of
circulating orbits
Table 2 1 i s t s one
such s o l u t i o n having s l i g h t l y d i f f e r e n t encounter
have been i d e n t i f i e d , these t y p i c a l 1y
have some
d a t e s and e n c o u n t e r speeds, b u t v e r y d i f f e r e n t
major drawback such as much h i g h e r
,V
values a t
c l o s e s t approach distances.
has
therefore
reduce t h e E a r t h swingby distance.
Earth
and/or
Mars.
This
study
The aim here was t o
focused on t h e VISIT and E s c a l a t o r o r b i t concepts
i n t a k i n g t h e necessary n e x t s t e p o f v e r i f y i n g
their
existence
world
of
and c h a r a c t e r i s t i c s
non-ci r c u l a r ,
i n the real
non-coplanar
planetary
Another important p o i n t i s noted w i t h r e f e r ence t o F i g u r e 4.
There i s a r e t r o g r a d e s h i f t o f
b o t h E a r t h and Mars encounter l o n g i t u d e s amounting
t o about 38' over 20'years.
orbits.
I f l e f t uncompensated,
t h i s d r i f t away from t h e d e s i r e d l o n g i t u d e (e.g.,
near p e r i h e l i o n a t Mars) w i l l e v e n t u a l l y r e s u l t i n
111.
VISIT Orbit Characteristics
Optimal t r a j e c t o r y searches using a p o i n t - t o -
Therefore,
over
p o i n t conic model were made f o r several successive
extended times, i t w i l l be necessary t o " r e s e t " t h e
launch o p p o r t u n i t i e s near t h e end o f t h i s century.
o r b i t p e r i o d i c a l l y (perhaps every 15 years o r so)
T h i s i s by way o f example v e r i f i c a t i o n o n l y , s i n c e
by u t i l i z i n g t h e excess g r a v i t y - a s s i s t c a p a b i l i t y
i t became c l e a r t h a t s i m i l a r r e s u l t s would be ob-
t a i n e d f o r a l a t e r t i m e p e r i o d i n t h e 21st c e n t u r y
when a sustained Mars Base may become a p r a c t i c a l
reality.
F i g u r e 4 shows a 20-year propagation o f a
VISIT-1 o r b i t launched i n 1996.
o r b i t " meaning t h a t no midcourse
This i s a "free
AV
i s required
o t h e r than n a v i g a t i o n t a r g e t i n g maneuvers which
w i l l be necessary t o compensate f o r expected small
errors.
Table 1 l i s t s t h e key parameters o f t h i s
c i r c u l a t i n g o r b i t , namely, t h e h y b e r b o l i c approach
-
h i g h e r values o f t h e encounter speed.
t o m a i n t a i n t h e VISIT o r b i t c h a r a c t e r i s t i c
approach speeds and t h e c l o s e s t approach d i s t a n c e s
a t E a r t h and Mars swingby.
Note t h a t p e r i h e l i o n
and aphelion d i s t a n c e s a r e maintained n e a r l y cons t a n t a t a b o u t 0.94
AU and 1.39
AU.
Encounter
speeds a r e a l s o f a i r l y constant; 4.2 t o 4.5 km/sec
t h a t i s a v a i l a b l e from c l o s e r E a r t h swingbys.
The
nominal encounter t i m e sequence may be i n t e r r u p t e d
b r i e f l y during t h i s resetting revolution.
extended times,
Over
t h e Mars encounter l o c a t i o n s w i l l
be seen t o o s c i l l a t e o r rock about Mars' p e r i h e l i o n
longitude.
T r a j e c t o r y runs have been made t o v e r i -
f y the a b i l i t y t o reset the orbit.
Table 1
VISIT-I Orbit Encounter Conditions
Encounter
Date
08
23
30
01
22
16
13
06
30
01
May
Jan
Apr
Nov
Jul
Apr
Apr
Apr
Dec
Apr
1996
1998
2001
2001
2005
2006
2009
2011
2012
2016
Approach V,
km/sec
Closest approach distance
planet r a d i i
4.50 (launch)
3.69
4.33
3.71
3.70
4.20
3.72
4.21
3.85
4.45
a Sphere-of-infl uence, 270 Earth r a d i i
Table 2
VISIT-1 Orbit Encounter Conditions ( v a r i a t i o n )
Encounter
Date
08
05
21
28
20
13
18
06
30
06
May
Jan
Mar
Oct
Jun
Apr
Apr
Apr
Dec
Apr
1996
1998
2001
2001
2005
2006
2009
2011
2012
2016
Approach V,
km/sec
4.76
4.04
4.20
4.07
4.08
4.45
3.77
4.24
3.84
4.45
(launch)
Closest approach d i s t a n c e
planet radi i
Fig. 4
20-year propagation o f VISIT-1 orbit.
Fig. 6
15-year propagation o f Dawn Escalator o r b i t
been
m o t i o n o f Mars i s n o t c o n s t a n t a l o n g i t s r e a l
generated t o v e r i f y t h e t h e o r e t i c a l p r e d i c t i o n , b u t
e l l i p t i c a l p a t h (and s i m i l a r l y f o r E a r t h t o a
t h e s e d a t a a r e o m i t t e d due t o space l i m i t a t i o n s i n
l e s s e r e x t e n t ) , t h e l o n g i t u d i n a l d i f f e r e n c e between
The main r e s u l t i n terms o f encounter
encounters i s l i k e w i s e n o t c o n s t a n t n o r a r e t h e
VISIT-2
t h i s paper.
circulating
orbits
have
a1 so
speeds compared t o VISIT-1 i s a s l i g h t r e d u c t i o n a t
E a r t h t o t h e range 3.7
t o 4.0
km/sec and a l a r g e r
r e d u c t i o n a t Mars t o t h e range 2.6
t o 2.8
km/sec.
Earth-Mars and Mars-Earth f l i g h t times.
This v a r i -
a b i l i t y i s r e l a t e d t o d i f f e r e n t bending r e q u i r e ments
of
the
trajectory
during
the
successive
VISIT-2 o r b i t s have t h e advantage o f more f r e q u e n t
p l a n e t a r y swingbys.
E a r t h encounters ( e v e r y 3 y e a r s ) b u t t h e s i g n i f i -
n o t l a r g e : 147-170 days f o r E a r t h t o Mars t r a n s i t s
cant
disadvantage of
infrequent
Mars
encounters
Thus, t h i s t y p e o f c i r c u l a t i n g
( e v e r y 7.5 y e a r s ) .
The f l i g h t t i m e v a r i a t i o n i s
on t h e Up E s c a l a t o r and f o r Mars t o E a r t h t r a n s i t s
on t h e Down E s c a l a t o r .
o r b i t does n o t appear t o be v e r y u s e f u l f o r t r a n s p o s s i b l y , i n combination
R e c a l l t h a t i n t h e c i r c u l a r coplanar problem,
w i t h o t h e r c y c l i n g f a c i l i t i e s having more f r e q u e n t
more bending i s r e q u i r e d o f E a r t h t h a n i s a v a i l -
Mars encounters.
abl e ,
p o r t a t i o n t o Mars except,
so a p r o p u l s i v e maneuver i s
r e q u i r e d on each o r b i t .
IV.
Up/Dorm E s c a l a t o r O r b i t C h a r a c t e r i s t i c s
W i t h an understanding o f t h e E s c a l a t o r o r b i t
from t h e c i r c u l a t i n g c o p l a n a r problem, a search was
made t o f i n d t h e e x i s t e n c e o f t r a j e c t o r i e s based
upon a c t u a l p l a n e t ephemerides data.
The t r a j e c -
t o r i e s were begun i n t h e 1995196 t i m e p e r i o d .
tially,
a point-to-point
Ini-
c o n i c model was used t o
i d e n t i f y t h e o p p o r t u n i t i e s and v e r i f y t h e i r e x i s t a n c e i n t h e r e a l world.
I n t h i s model h e l i o c e n -
t r i c c o n i c s connect t h e f l y b y s of
E a r t h and Mars
w i t h t h e t i m e s o f f l y b y v a r i e d t o o b t a i n a match o f
V,
magnitude a t each f l y b y .
Thus a sequence o f
f l y b y s s i m i l a r t o those o f the c i r c u l a r coplanar
I n t h e r e a l world,
the
o p t i m a l t r a j e c t o r y c o v e r i n g t h i s 15 y e a r c y c l e
r e q u i r e s p r o p u l s i v e maneuvers on o n l y t h r e e o f t h e
seven o r b i t s .
I n t e r e s t i n g l y enough,
t h e sum o f
t h e s e t h r e e maneuvers i s a p p r o x i m a t e l y seven t i m e s
t h e p e r o r b i t requirement i n t h e c i r c u l a r c o p l anar
problem.
The p r o p u l s i v e maneuvers a r e made near
aphelion o f the c i r c u l a t i n g o r b i t ,
which o c c u r s
about e i g h t months a f t e r Mars encounter on t h e 'Up"
o r b i t and about e i g h t months b e f o r e Mars encounter
on t h e "Down"
orbit.
Note t h a t these maneuvers
occur on t h e o r b i t s which encounter Mars i n t h e
general r e g i o n of i t s p e r i h e l i o n passage when i t i s
moving t h e f a s t e s t .
model a r e o b t a i n e d (El-M2-E3-M4- ...) ; t h e d i f f e r ence b e i n g t h a t t h e r e a l w o r l d geometry causes t h e
p a t t e r n not t o repeat exactly.
V
These sequences
. P l a n e t o c e n t r i c Maneuvers
o v e r a 15 y e a r p e r i o d w e r e t h e n used as f i r s t
I n t r a d i t i o n a l i n t e r p l a n e t a r y f l i g h t , a space-
guesses f o r a computer program which uses t h e more
c r a f t i s i n j e c t e d from low E a r t h o r b i t o n t o a t a r -
accurate
Mu1t i -Conic
p r o p a g a t i o n t e c h n i q u e along
w i t h o p t i m i z a t i o n techniques1'
t o generate a t r a -
geted
hyperbolic
departure
trajectory
having
a
p e r i g e e d i s t a n c e e s s e n t i a l l y equal t o t h e low o r b i t
A s i m i l a r i n j e c t i o n geometry
j e c t o r y w i t h minimum p r o p u l s i v e maneuvers s u b j e c t
injection point.
t o c o n s t r a i n t s on f l y b y a l t i t u d e s .
m i g h t be e f f e c t e d a t Mars f o r t h e r e t u r n f l i g h t t o
The i d e a l ( o p t i m a l ) c o p l a n a r i n j e c t i o n
Earth.
F i g u r e s 5 and 6 i l l u s t r a t e t h e c i r c u l a t i n g
maneuver i s g i v e n by
o r b i t s f o r b o t h E s c a l a t o r s as t h e y e v o l v e over a
c o m p l e t e 15 y e a r c y c l e .
h y p e r b o l i c approach speed,
C o r r e s p o n d i n g d a t a on
c l o s e s t approach d i s -
tance, and p r o p u l s i v e AV requirements a r e l i s t e d i n
i n t h e case o f d e p a r t i n g from a c i r c u l a r o r b i t .
Tables 3 and 4.
The s i t u a t i o n i s somewhat d i f f e r e n t f o r c i r c u l a t i n g
The o r b i t p l o t s show t h e charac-
t e r i s t i c r o t a t i o n of t h e encounter l o c a t i o n s caused
orbits
by t h e p l a n e t a r y f l y b y s .
facility"
Because t h e o r b i t a l
s i n c e rendezvous must occur w i t h a " r e a l
f l y i n g by t h e l a u n c h p l a n e t on a s p e c i -
Table 3
Up Escalator Encounter Conditions
Encounter
Earth-1
Mars-2
Earth-3
Mars-4
Earth-5
Mars-6
Maneuver
Earth-7
Mars-8
Maneuver
Earth-9
Mars-10
Maneuver
Earth-11
Mars-12
Earth-13
Mars-14
Earth-15
Approach ,V
km/sec
Date
19
01
01
28
08
06
13
16
12
17
07
13
23
06
16
10
28
13
Nov
May
Jan
May
Feb
Jul
Mar
Apr
Sep
May
Jul
Dec
Jul
Sep
Feb
Oct
Mar
Nov
1996
1997
1999
1999
2001
2001
2002
2003
2003
2004
2005
2005
2006
2007
2008
2009
2010
2001
Closest approach d i s t a n c e
planet radi i
--
6.19 ( l a u n c h )
10.69
5.94
11.74
5.67
10.22
(AV~)
0.54
5.67
7.28
(hV2)
0.74
5.87
6.05
0.45
(AV3)
5.87
7.43
5.89
8.66
5.81
5.8
1.3
29.1
1.2
1.3
-1.2
1.3
--
1.2
3.4
--
1.8
6.4
1.9
5.0
1.9
Table 4
Down Escalator Encounter Conditions
Encounter
Earth-1
Mars-2
Earth-3
Mars-4
Earth-5
Maneuver
Mars-6
Earth-7
Maneuver
Mars-8
Earth-9
Maneuver
Mars-10
Earth-11
Mars-12
Earth-13
Mars-14
Earth-15
Approach ,V
km/sec
Date
05
20
09
07
17
28
15
08
04
07
02
02
10
12
19
16
13
22
Jun
Jan
Jul
Mar
Aug
Sep
May
Oct
Dec
Aug
Jan
Feb
Oct
Mar
Nov
Apr
Dec
May
1995
1997
1997
1999
1999
2000
2001
2001
2002
2003
2004
2005
2005
2006
2007
2008
2009
2010
5.88 ( l a u n c h )
8.52
5.95
7.35
6.01
0.27
(aV1)
6.60
5.88
1.11 (AV2)
7.30
5.39
0.66
(AV3)
9.96
5.48
11.59
5.96
10.55
5.93
Closest approach d i s t a n c e
planet r a d i i
-5.5
1.8
9.4
1.4
--
5.2
1.2
--
1.3
1.4
--
1.3
1.5
8.4
1.5
5.0
1.8
Generally, t h e p e r i a p s i s d i s t a n c e o f t h i s hyperbola
encounter c o n d i t i o n s .
6
For asymptotic displacement up t o 10 km, t h e A V
may n o t be v e r y c l o s e t o t h e p l a n e t and c e r t a i n l y
p e n a l t y c o u l d b e as h i g h as 1.65
f i e d h y p e r b o l i c t r a j e c t o r y a t a s p e c i f i e d time.
varies
from encounter t o encounter as discussed
previously.
T h i s i m p l i e s t h e n e c e s s i t y o f a two-
tabu1 a r d a t a
for
specific
rendezvous t i m e i s l i m i t e d t o 7 days.
Allowing 21
days f o r rendezvous reduces t h e p e n a l t y c o r r e -
impulse maneuver sequence, a t t h e 1east, r e q u i r i n g
spondingly by a f a c t o r o f three.
a f i n i t e amount o f t i m e t o accomplish.
AB = 10
Limiting
km/sec i f t h e
This example w i t h
6 km represents one o f t h e worst cases o f
t h e rendezvous t r a n s f e r t i m e i s an i m p o r t a n t con-
t h e VISIT-1 encounters w i t h E a r t h (Earth-8 i n Table
s i d e r a t i o n s i n c e t h e crew i n t h e small Taxi vehi-
1).
Another example i s t h e Mars-4 encounter i n
c l e s w i l l n o t have t h e same environmental l u x u r i e s
Table 1:
o f t h e l a r g e t r a n s p o r t a t i o n f a c i l i t y t o which t h e y
0.223 km/sec f o r A T = 7 days o r 0.074 km/sec f o r AT
a r e destined.
= 21 days.
Furthermore, t h e i n j e c t i o n maneuvers
w i t h A B = 135,000 km, t h e A V p e n a l t y i s
w i l l i n c u r a A V p e n a l t y compared t o t h e i d e a l v a l u e
o f e q u a t i o n ( 9 ) inasmuch a s t h e t w o h y p e r b o l i c
asymptotes (i.e.,
o f t h e Taxi and CASTLE) w i l l n o t
be a l i g n e d but,
rather,
w i l l cross a t some angle
depending on t h e t i m e i n t e r v a l allowed.
The l o n g e r
t h e time, t h e s m a l l e r t h e angle and, t h e r e f o r e , t h e
Up/Down E s c a l a t o r encounter c o n d i t i o n s 1 is t e d
i n Tables 3 and 4 show g e n e r a l l y s m a l l e r c l o s e s t
approach
penalties.
distances
translates
to
smaller
o f t h e Up E s c a l a t o r which has AB = 95,000 km f o r a
p e n a l t y o f 0.157
s m a l l e r t h e qV penalty.
which
The worst case i s t h e Mars-4 encounter
km/sec w i t h rendezvous i n 7 days.
A more t y p i c a l case i s t h e Earth-11 encounter o f
T h i s problem has been examined q u a n t i t a t i v e l y
i n some d e t a i l u s i n g b o t h an approximate two-body,
Table 3 which has AB = 5,400 km and a 7-day p e n a l t y
o f o n l y 9 m/sec.
The c l e a r i m p l i c a t i o n i s t h a t
Earth-centered model and an accurate N-body i n t e -
s h o r t e r Taxi r i d e s c o u l d e a s i l y be a f f o r d e d i n t h e
g r a t i o n model.
E s c a l a t o r t r a n s p o r t a t i o n mode.
Suprisingly
l i t t l e error i n the
Recall , however,
two-body r e s u l t s were found even f o r l a r g e swingby
t h a t t h e i d e a l i n j e c t i o n A V's a r e much l a r g e r f o r
d i s t a n c e o f t h e CASTLE and f o r rendezvous p o i n t s
t h e E s c a l a t o r mode t h a n t h e VISIT mode because o f
o u t s i d e t h e E a r t h ' s sphere-of-influence.
t h e h i g h e r values o f ,V,
more,
a very
simpl i f i e d "guidance
Further-
formula"
also
propel 1a n t
y i e l d s an accurate measure o f t h e AV p e n a l t y :
AVpenalty
-
p a r t i c u l a r l y a t Mars.
performance t r a d e o f f measured
requirement
is
The
i n terms o f t o t a l
therefore
not
so
simpl i s t i c b u t must be s u b j e c t t o c a r e f u l a n a l y s i s .
AB
-
AT
(10)
There a r e o t h e r penal t i e s i n p l a n e t o c e n t r i c
where AT i s t h e t i m e allowed from i n j e c t i o n t o rendezvous and AB i s t h e a b s o l u t e magnitude d i f f e r e n c e
between t h e two h y p e r b o l i c asymptotes,
t r a d i t i o n a l aim p o i n t o r miss distance.
i.e.,
the
The miss
d i s t a n c e i s c a l c u l a t e d by t h e expression:
maneuvers r e 1ated t o plane changes,
and launch delays.
time-phasi ng ,
These problems have been exa-
mined b u t o n l y i n a v e r y c u r s o r y manner t o date.
S p e c i f i c r e s u l t s o r c h a r a c t e r i s t i c s o f t h e plane
change and time-phasing penal t i e s depend very much
on t h e p a r t i c u l a r s t a g i n g "spaceportii
c e n t r i c space t h a t i s assumed.
i n planeto-
A1 so, a p p r o p r i a t e
software t o o b t a i n optimal s o l u t i o n s has y e t t o be
where r i s t h e p e r i a p s i s o r c l o s e s t approach d i s P
tance o f t h e a p p r o p r i a t e hyperbola.
A typical
p e r i a p s i s i n j e c t i o n p o i n t a t e i t h e r E a r t h o r Mars
developed.
Some p r e l i m i n a r y c a l c u l a t i o n s assuming
an Earth-Moon L1 p o i n t s t a g i n g base have i n d i c a t e d
t h a t p l a n e change/timing p e n a l t i e s c o u l d p o s s i b l y
be l i m i t e d t o 250 m/sec b u t may r e q u i r e e l a b o r a t e
encounter i s 500 km a l t i t u d e .
and l e n g t h l y maneuver s t r a t e g i e s .
A graph o f equation (10) shown i n F i g u r e 7 i s
a more convenient and g e n e r a l i z e d way t o present
the
main
characteristic
rather
than
presenting
F i g u r e s 8 and 9 i l l u s t r a t e Taxi maneuvers made
a t t h e Mars t e r m i n a l assuming a s t a g i n g spaceport
i n t h e v i c i n i t y o f Phobos (cryogenic p r o p e l l a n t
AV
PENALTY
KWIC
.
Fig 7
Hyberboli c rendezvous requirements
\
TARGETING TO AEROCAF'TIRE
FROM INCW -
'4-
-
I(
I
\
6- = -23.5
L
a_ = 286.6
\
~b = 16,130 km
'4-'
.
= 9.97 kmlsec
Limit
\
\
0.557 km/se
POST-INSERTION ORBIT
i
n
= 23.7'
= 23.7'
o = 180'
ra = 10.68 RM
Fig. 8
-
Taxi manuovers to Phobos Up Escalator
Mars encounter a t 5 , + 0 . 4 5 ~
t
TARGETING TO OUTGOING ASYMPTOTE
V_ = 6.602 km/sec
6- = 6.5'
Ab
= 13,949 km
AV4 = 0 . 0 2 3 km/s
0-
= 177. l o
C---------
t
15.0 RM
12.5
10.0
7.5
PRE-INJECTION
ORB lT
INITIAL
TRANSFER
ORBIT
i = 6.6'
n = 75.P
0
r
= o0
= 1.147 R,,
ra = 10.68 R,,
Fig. 9
t3.43. - D
Taxi maneuvers f r a Phobo
ILrs CMUntCl i t
m Escalator
The example cases correspond
production f a c i l i t y ) .
r e t u r n s t o E a r t h on t h e Down Escalator.
Use of
t o t h e Mars-2 encounter o f t h e Up E s c a l a t o r and t h e
VISIT o r b i t s would r e q u i r e a t l e a s t t w i c e as many
Mars-6 encounter o f t h e Down Escalator.
o p e r a t i n g CASTLES t o achieve comparable encounter
Mars o r b i t
capture i s assumed t o be accomplished by a combina-
frequencies and crew t r a n s i t times.
tion
maneuvers
t h e making o f an i n t e r e s t i n g performance t r a d e o f f
e f f e c t e d by t h e Taxi v e h i c l e d e p a r t i n g t h e CASTLE.
p r o b l e m between t h e s e t w o t y p e s o f c i r c u l a t i n g
of
aerocapture
and
propulsive
LID a e r o s h i e l d i s assumed t o
The low-to-moderate
o r b i t s and,
f o r t h a t matter,
Thus, t h e r e i s
between t h e c i r c u -
have an u p p e r l i m i t on e n t r y v e l o c i t y o f 11.1
l a t i n g mode o f t r a n s p o r t a t i o n compared t o t h e more
km/sec (Vm l e s s than 9.97 km/sec).
t r a d i t i o n a l minimum-energy r o u n d - t r i p missions w i t h
F i g u r e 8 shows
t h a t t h e f i r s t p r o p u l s i v e maneuver o f 0.721 km/sec
l e n g t h y stopovers.
provides t h e v e l o c i t y r e d u c t i o n i n combination w i t h
t r a d e o f f i s t h e s u b j e c t o f t h e companion paper by
t h e A6 t a r g e t i n g .
Hoffman, e t a1 appearing i n t h i s Proceedings.
i s -23.5",
Since t h e approach d e c l i n a t i o n
As m e n t i o n e d e a r l i e r , t h i s
t h e p o s t - i n s e r t i o n o r b i t has t h i s value
o f i n c l i n a t i o n which must then be a d j u s t e d t o t h e
Mars e q u a t o r i a l plane i n which Phobos moves.
second maneuver o f 0.322
The
km/sec provides t h e com-
The
potential
application
of
circulating
o r b i t s i n s u p p o r t o f a s u s t a i n e d , manned Mars
exploration
program appears t o have promise b u t
b i n a t i o n o f plane change and p e r i a p s i s r a i s i n g t o
c l e a r l y r e q u i r e s much more extensive study.
Phobos o r b i t distance.
The t h i r d and f i n a l maneu-
area o f o r b i t a l mechanics, t h e main problem n o t y e t
km/sec a c h i e v e s rendezvous a t t h e
addressed thoroughly i s t h e a n a l y s i s (and o p t i m i -
v e r o f 0.557
Phobos spaceport.
I n the
F i g u r e 9 describes t h e r e v e r s a l
z a t i o n ) o f p l a n e t o c e n t r i c maneuver s t r a t e g i e s f o r
o f t h i s process f o r t h e crew d e p a r t i n g f o r E a r t h on
i n j e c t i o n t o and departure from t h e i n t e r p l a n e t a r y
t h e Down E s c a l a t o r f o u r years l a t e r .
face-to-Phobos
The Mars sur-
spaceport s h u t t l e maneuvers a r e n o t
shown here, b u t t h e round t r i p AV of t h e aers-pro-
p u l s i v e s h u t t l e i s about 7.5.
stations.
T h i s i s a f e r t i l e area o f i n v e s t i g a t i o n
s i n c e t h e r e a r e a number o f p o s s i b l e o p t i o n s f o r
l c c s t i n g t h e p l a n e c r e n t r ; ~ s t a g i n g base.
one example,
As j u s t
A l d r i n has p r o p o s e d an Earth-Moon
c y c l e r as t h e a p p r o p r i a t e stepping o f f base f o r
VI.
i n t e r p l a n e t a r y t r a v e l due t o i t s p r o x i m i t y t o b o t h
Comparison o f R e s u l t s and Conclusions
sources o f
T h i s f i n a l s e c t i o n o f t h e paper c o n s o l i d a t e s
p r o p e l l a n t and i t s high-energy,
perigee o r b i t .
low-
Others have proposed t h e Earth-Sun
t h e study r e s u l t s by comparing t h e main c h a r a c t e r -
L1 p o i n t and t h e Earth-Moon L2 p o i n t , and s i m i l a r
i s t i c s o f t h e two t y p e s o f c i r c u l a t i n g o r b i t s
v a r i a t i o n s on t h e theme apply t o t h e Mars t e r m i n a l .
between E a r t h and Mars.
I t i s hoped t h a t t h i s p a p e r w i l l i n s p i r e much
Table 5 summarizes t h e key
parameters o f t h e VISIT and E s c a l a t o r o r b i t s .
In
i n t e r e s t i n g new work.
a d d i t i o n t o t h e midcourse p r o p u l s i v e A V r e q u i r e ment, two key d i f f e r e n c e s o f E s c a l a t o r o r b i t s com-
References
pared t o VISIT o r b i t s a r e h i g h e r values o f encount e r approach speed a t E a r t h b u t m a i n l y a t Mars, and
much lower values o f c l o s e s t approach d i s t a n c e a t
b o t h p l a n e t s b u t m a i n l y a t Earth.
The v e l o c i t y
c h a r a c t e r i s t i c i s a c l e a r disadvantage i n terms of
t h e Taxi p r o p u l s i o n requirements f o r launch t o t h e
c i r c u l a t i n g CASTLES.
The d i s t a n c e c h a r a c t e r i s t i c
i s an advantage ( b u t n o t n e c e s s a r i l y an o f f s e t t i n g
one) i n terms o f t h e Taxi rendezvous t i m e and AV
requirement.
O f course,
the Escalator-type o r b i t
o f f e r s more r e g u l a r , more frequent encounters w i t h
E a r t h and Mars t h a n does VISIT o r b i t s .
Also they
have t h e d e s i r a b l e c h a r a c t e r i s t i c o f s h o r t t r a n s i t s
(150-170 d a y s ) t o Mars on t h e Up E s c a l a t o r and
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o f Non-stop
I n t e r p l a n e t a r y Round T r i p s ,"
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AAS Meeting), Los Angeles, CAY January 1963.
H o l l i s t e r , W.
n a u t i c a Acta.,
M.,
"Castles i n Space,"
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Astro-
P 3 l l i s t e r , W. M., " P e r i o d i c O r b i t s f o r I n t e r p l a n e t a r y F l i g h t , " AAS Paper No. 68-102, AASI
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1969.
--.
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N a t i o n a l Academy o f Sciences,
Washington, DC, J u l y 1985.
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-
Table 5
Key Parameter Canparison o f VISIT and E s c a l a t o r O r b i t s
Parameter
Frequency o f E a r t h encounters ( y r s )
Frequency o f Mars encounters ( y r s )
VISIT-2
5.0
3.0
Down E s c a l a t o r
2.14
2.14
7.5
2.14
2.14
0.5-3.0
1.0-2.4
0.43
1.71
Mars-to-Earth f l i g h t t i m e ( y r s )
0.7-3.3
0.6-2.1
1.71
E a r t h encounter V,
4.2-4.8
3.7-4.0
5.7-6.2
5.4-6.0
3.7-4.1
2.6-2.8
6.1-11.7
6.6-11.6
E a r t h encounter d i s t a n c e (RE)
6.9-SO1
8.3-SO1
1.2-1.9
1.2-1.8
Mars encounter d i s t a n c e (RM)
1.5-40.7
2.0-18.5
1.3-29.1
1.3-9.4
(km/sec)
(km/sec)
Midcourse adjustment, 15 y e a r s (km/sec)
Max. E a r t h access AV,
Max. Mars access AV,
*
14 days (km/sec)*
14 days (km/sec)*
3.75
Up E s c a l a t o r
Earth-to-Mars f l i g h t t i m e ( y r s )
Mars encounter ,V
I-
VISIT-1
0.43
0
0
1.7
2.0
5.5
5.2
4.8
4.7
2.9
2.2
9.4
9.2
Sum o f i d e a l i n j e c t i o n and rendezvous maneuvers