Elements of Space Technology. For aerospace engineers

325 Pages·1999·7.85 MB·English

by  Rudolf X. Meyer (Auth.)

ECAFERP
This book is intended as a first introduction to space technology for aero-
space engineering students, at either the senior or graduate level. It is hoped
also to be useful to professional engineers who wish to become more familiar
with the aerospace aspects of space systems.
It would have been tempting to include all major disciplines that are re-
presented in space systems engineering, including electronics, space com-
munications, and even the civil engineering of launch site construction.
That such a broad, interdisciplinary approach would have merit is suggested
by the fact that quite often the effort (and cost) in developing a new spacecraft
and its ground support is about equally divided between the mechanical and
electronic systems. However, the division of most engineering curricula into
separate aerospace and electronic engineering disciplines would have made
such a collection quite impractical as a textbook. The concentration here
is therefore on the mechanical ("aerospace") aspects, omitting the equally
important electronic and communications disciplines. Nevertheless, in in-
stances in which the disciplines clearly overlap, as in the thermal control
of electronic components, an effort has been made to broaden the scope
sufficiently to make the student aware of the problems and solutions that
are addressed by the electronics and communications engineers.
Most of the material in this book can be covered by graduate students
in one semester and similarly by seniors at American universities. Several
parts, particularly in Chapter 4 on the chemistry of propellants are suited
for reading by the student, without need for class presentation. Major parts
have been taught by the author to engineering students at the University of
California, Los Angeles.
The book has been arranged such that, at the choice of the instructor,
entire sections can be omitted without losing continuity. These sections,
which sometimes contain more advanced topics, are indicated in the Table
of Contents by an asterisk. The problems at the end of each chapter that
relate to these sections are indicated in the same manner.
Chapters 1 and 2 are preliminary to the discussion of orbits and traject-
ories in Chapters 3 and 5. Some of this material may already be familiar
to the reader but may be used for review or may be skipped. Classical or-
bital mechanics is treated relatively briefly, forgoing the elegance gained
by Hamiltonian mechanics. The analysis of spacecraft orbits, in good part
borrowed from classical celestial mechanics, has become the specialty of
ix
x Preface
numerical analysts. Instead, more importance in this book has been placed
on the trajectories encountered in rendezvous and docking maneuvers and
on launch trajectories with thrust and aerodynamic drag.
For lack of space in the typical aerospace curriculum, more specialized
topics, such as spacecraft testing methods or reliability analyses, had to be
omitted or are mentioned only very briefly. Similarly, topics that are usually
dealt with elsewhere in the curriculum are introduced only in instances
in which their application to space technology is unique in some sense.
In spite of their importance, the design and analysis of launch vehicle and
spacecraft structures are therefore hardly touched upon in this book because
the principles involved are very similar to those of aircraft structures, a topic
that usually precedes this in the aerospace curriculum. Similarly, in the
chapter on attitude control, the emphasis is on sensors and actuators, not on
control theory, in which specialized courses are offered at most institutions.
The author may also have to apologize for not having included a chapter on
space mission design. The excuse is that at present mission designs and
their future capabilities change so rapidly that it seemed better to limit the
lecture course to the fundamentals that are likely to have more permanent
validity.
Because the aim is to emphasize the more fundamental aspects, the
treatment must necessarily be in good part theoretical. However, the author
hopes that he has resisted successfully the temptation to make the mathe-
matical arguments appear to be more sophisticated than warranted by the
subject matter. Where it seemed desirable, some brief mathematical com-
ments are made. In spite of this emphasis on the scientific and engineering
foundations and on analysis, an effort has been made to include the more
practical aspects encountered in the design of space vehicles and their com-
ponents whenever these topics were felt to be more than ephemeral, soon
to be replaced by more advanced designs. Emphasis has also been placed
on providing a "feel for numbers" so that the reader may develop a sense of
what is numerically important and what is not.
Each chapter is followed by some exercises, varying from the simple to
the more difficult. It is the author's contention, however, that such exercises,
which can be stated in a few lines, are less valuable to the engineering student
than assigned projects, be they mission analyses or hardware designs. oT
support such projects, space was found to include at least some of the more
important technical data in an appendix.
Some technicalities:
Throughout this text IS units are used. Only in a very few instances, and
only when sanctioned by almost universal usage, will other units, such as
electron volts for energy, appear. ESA and now also NASA publications have
been using exclusively the SI units, with only an occasional and parenthetical
mention of English units when dictated by very long usage, and as still occur
in major data collections.
For the convenience of the reader, the principal symbols used are sum-
marized at the end of each chapter, often with the number of the defining
equation.
ecaferP xi
Many of the subscripts are used uniformly throughout this book. They
are, with their meaning:
( )a ambient ( )h sun; solar
( )av average ( )pl planet
( le) electric ( rp) propellant
( xe) nozzle exit ( s) spacecraft
( g) earth; also gravitational ( is) sidereal
( )c Greenwich
To help distinguish the forest from the trees, equation numbers that are
printed in bold font indicate that the equation is either important, likely
to be referred to in a later chapter, or else the final result of a preceding
development. Simple corollaries or special cases of a preceding equation
are given the same number, but with a prime.
As would seem appropriate for a textbook of this type, journal papers or
monographs are referenced only when the text or illustrations are so close
to the original that it would be unprofessional not to refer to the author.
Instead, references are mainly to books, monographs, or data collections,
where the reader may find additional material or different presentations.
It remains for me to thank my colleagues and friends, far too many to
mention by name here, at The Aerospace Corporation and at the University
of California who have inspired much of this work. Space technology is a
national and international enterprise with many workers. To them go my
thanks.
Ad astra pontem fecerunt fabri, in majorem Dei gloriam.
11'
Reference Frames and Time
This chapter includes a discussion of the reference frames commonly used in
space vehicle dynamics. They are employed to describe and analyze orbits
and trajectories and also to describe the orientation in space ("attitude")
of the vehicle. Euler angles are introduced to characterize the rotation of
reference frames.
Because of the importance of relative motion in such problems as fluid
sloshing in the vehicle's propellant tanks or the effect of vehicle motion on
gyroscopes, the transformation equations for velocities and accelerations
are rederived. Although the reader will already be familiar with this topic
and other parts of this chapter, they are included because later chapters will
refer to them. A second purpose is to introduce notation that will remain
uniform for the remainder of this book.
The chapter concludes with a discussion of the various time measure-
ment systems that are in use in the design of space missions.
1.1 Reference Frames
A quantitative description of the motion of space vehicles, be they launch
vehicles, upper stage vehicles, artificial satellites, or deep-space probes, re-
quires one or several ecnerefer frames with respect to which the vehicle's
position, velocity, and acceleration as functions of time are defined. Refer-
ence frames are also needed to describe the vehicle's attitude relative, for
instance, to points on the earth, to the sun, or to guide stars. Similarly, the
time derivatives of the vehicle's attitude, particularly angular rates and an-
gular accelerations, are often needed. Implied in the use of reference frames
is also the existence of a precisely defined, "uniformly" evolving time.
Easily defined reference frames are those that are rigidly attached to
the earth, to some other astronomical body, or to the structure--assumed
rigid--of the space vehicle itself. This latter frame is useful, for instance,
in studying the dynamic effects on the vehicle resulting from the deploy-
ment of solar panels, antennas, or scientific probes. Although easily defined,
these reference frames will generally be accelerating; that is, they may be
rotating or may also accelerate by parallel translation. Table 1.1 contains
approximate values of the accelerations of several of such reference frames,
expressed by their ratios to the standard gravity og = 9.80665 m/s .2
The first few accelerations listed in Table 1.1 are large and must be taken
into account, together with other residual accelerations of comparable mag-
nitude, in the planning of typical space missions. Particularly sensitive to
small residual accelerations are missions in which a satellite is required to
2 C H A P T E ,P 1 Reference Frames and Time
Table 1.1 Magnitudes of Various Residual Accelerations
Acceleration at the equator due to the planet's rotation
Earth 3.44 10 -3 go
Mars 1.74 10 -3 go
Acceleration on the earth's surface
Due to earth-moon mutual attraction, with moon at zenith 1.13 10 -7 go
Due to sun-earth mutual attraction, with sun at zenith 0.53 10 -7 go
Acceleration (at the solar system location) due to rotation about
the galactic center 2.24 10 -11 go
execute, during its lifetime, a large number of orbits about a planet or moon.
Unless corrected, small errors will tend to accumulate and will lead to large
deviations from the intended path. On the other hand, the effect produced
by the rotation of the solar system about the galactic center as a part of
a spiral arm of our galaxy is sufficiently small to be negligible for present
purposes.
Newton's second law refers only to inertial reference frames, that is,
nonaccelerated frames. This then raises two questions: How can an inertial
frame be found? And how is one to relate motions that are best described
relative to an accelerating frame--such as a frame attached to a spinning
spacecraft--to the same motion relative to an inertial frame?
An important postulate of Newtonian mechanics asserts that once an
inertial frame has been established, all other frames that move relative to
this frame with constant velocity and no rotation are also inertial frames
("Galilean invariance") ,1 2.
Attempts to define inertial frames and time as those for which the laws of
Newtonian mechanics are valid are in essence circulatory. Definitions that
attach inertial frames to the "fixed stars," taking advantage of the near-zero
apparent rotation of distant stars, leave open the question of acceleration
by parallel displacement.
The meaning of an inertial reference frame (often referred to as "inertial
space") in Newtonian mechanics can be made clear as a limiting case in the
theory of general relativity. Newtonian mechanics is a model of space and
time that postulates a homogeneous, isotropic space and a uniform time
proceeding at a uniform rate. It is a sufficiently accurate representation for
nearly all purposes that are of interest to space technology engineers.
Confining oneself to classical, Newtonian mechanics, one has at least
the heuristic fact that no contradiction has ever been found by postulat-
ing the existence of an inertial space. In particular, no such contradictions
have been found by observations of spacecraft paths. The assumption that a
frame of reference that is nonrotating with respect tothe most distant stars,
and is not accelerated with respect to the center of our galaxy, is for all
purposes a more than sufficient approximation to an ideal inertial space in
Newtonian mechanics. Similarly, no contradiction has been found by as-
suming that the time in Newtonian mechanics is the same as that measured
by the most reproducible timepieces, particularly by atomic clocks.
1.1 ecnerefeR semarF 3
erugiF 1.1 Ecliptic and celestial equatorial planes.
1.1.1 The Ecliptic and Celestial Equatorial Planes
The ecliptic (the path of the earth about the sun) is said to be in the eclip-
tic plane, which therefore contains the sun's and the earth's mass centers
(Fig. 1.1). One also speaks of the celestial equatorial plane, which is the
plane parallel to the earth's equatorial plane and through the sun's mass
center. The ecliptic and celestial equatorial planes intersect, as illustrated
in the figure, in a line referred to as the equinox line (because, when the
earth on its annual path crosses this line, day and night have equal length).
The crossing points are called the vernal equinox (the earth is at this point
on about the 21st of March) and the autumnal equinox (about the 22nd of
September). As discussed in Chapter 7, the vernal and autumnal equinox
points play an important role in the operation of geosynchronous and other
spacecraft. During two periods each year, centered around these points, and
centered around local midnight, the sun will be eclipsed for these satellites.
Special operational procedures are then needed to compensate for the lack
of solar radiation.
The direction of the equinox line that points from the earth at the vernal
equinox toward the sun is referred to, for historical reasons, as the first point
of Aries and is often designated by the symbol ~. sA( the name suggests, the
line was found by early astronomers some 2500 years ago to point toward the
constellation Aries. But since it moves by about 0.8' per year, it is presently
in Pisces, moving into Aquarius.)
4 C H A P T E R 1 Reference semarF and Time
The ecliptic plane is inclined to the equatorial plane at an angle ci called
the obliquity of the ecliptic. At present it is 23.44 ~ (Primarily as a conse-
quence ofthe gravitational attraction bythe sun and the moon on the earth's
equatorial bulge, the earth's spin axis, and hence also the equatorial plane,
precesses relative to the most distant stars with a period of 25,920 years.
The obliquity fluctuates between 21.5 ~ and 24.5 ~ with a period of about
41,000 years and is presently decreasing by about 0.5" per year. The eccen-
tricity of the earth's orbit about the sun also has a small fluctuation, with a
period of about 100,000 years. All these effects, however, are too small to be
significant for most space operations.)
1.1.2 Reference Frames and Coordinate Axes
The following reference flames and coordinates are in frequent use 3. They
are arranged in order of increasing closeness to ideal inertial frames.
(la) Earth-fixed geocentric equatorial frame: As shown in Fig. 1.2, this is the frame
of reference conventionally used in cartography. Longitude is measured
eastward from the Greenwich meridian (the former site of the Royal Obser-
vatory in Greenwich, a suburb of London). The latitude is measured positive
northward, negative southward, starting at the equator. These two spherical
coordinates determine the location, for instance, of a ground station. They
are also useful in stating the instantaneous location of a near-earth space
vehicle, by radially projecting its location in space on the earth's surface (a
htumizA
3gX
raloP sixA B
noitavelE
edutitaloC
/ I 3tX/
hciwneerG i
naidireM ~ Xtl
f
.. " f V ~ ~ /--/----- rotauqE
gX 2
gX "1 ~ /
edutignoL j tinU erehpS
edutitaL
erugiF 1.2 Earth-fixed geocentric equatorial (Xgl, ,zgX )3gX and
topocentric ,ltX( ,2tx )3tx reference frames. ,P space vehicle loca-
tion; ,O ground station.
1.1 ecnerefeR semarF 5
spherical approximation to the geoid is assumed here). The curve obtained
by this projection of a space vehicle moving relative to the earth is often
referred to as the vehicle's ground track.
The same system of spherical coordinates is also useful for other planets
and for moons. The reference frames are then referred to as "planetocentric"
and "selenocentric," respectively. Because longitude and latitude are angles
independent of the radius of the astronomical body, it is convenient to think
of them as arc lengths on the "unit sphere" (the sphere of radius one).
(lb) Topocentricframe: The line of sight to the instantaneous position of a space
vehicle is given by its azimuth and by its elevation (Fig. 1.2). The azimuth
angle is measured in the local horizon plane ~ that is, the plane tangent to
the (spherical) earth at the observer's location~ and starts at local north,
positive for the direction toward local east. The elevation angle is zero at the
horizon and positive for lines of sights above it.
With the possible exceptions of short-range sounding rockets or tactical
missiles, the two reference systems (la) and (lb) listed here are hardly ever
suitable in space technology as approximations to an inertial frame. They
are needed, however, for specifying space vehicle locations in coordinates
that are directly observable. Coordinate transformations, to be discussed
later, will therefore often be required to convert from one system to another.
)2( Geocentric equatorial reference frame: In contrast to frame (la), this frame
does not co-rotate with the earth. The origin of the spherical coordinate
system is in the earth's mass center (or, correspondingly, in the mass center
of other planets or moons). As shown in Fig. 1.3, the longitude is again
naidireM
~.~x / / P
raloP sixA
,'X / , - eclinat,on
I /
I /
I /
I /
I/
..J.
-/--~ rotauqE
xoniuqE eniL
thgiR "noisnecsA ~ ~ ~
Xg, 2
tinU erehpS
erugiF 1.3 Geocentric, equatorial, nonrotating reference
frame ,~,gX( ,2,gX .)3,gX For many space missions this frame
is a sufficient approximation to the ideal inertial reference
frame. ,P location of space vehicle.
6 C H A P T E R 1 Reference Frames and Time
laitseleC
North eloP
citpilcE
North eloP
3cX
2cX
citpilcE
\ Latitude
\
\
2q
citpilcE
Longitude
laitseleC
Equatorial enalP
Ecliptic enalP
Right Ascension
Declination
Xcl
Unit erehpS
Equinox Line Pq
Figure 1.4 Heliocentric equatorial ,~qX( ,2qX )3qX and ecliptic
,~cX( ,2cX )acX reference flames. ,P location of space vehicle.
taken positive eastward along the equator, but starts at the first point of
Aries (equinox line). It is referred to as the right ascension of a point. The
latitude is now referred to as the declination. This reference frame, which
is very nearly nonrotating, is a suitable approximation to an inertial frame
for the analysis of near-earth space vehicle trajectories and for their attitude
control.
)3( Heliocentric equatorial ecnerefer frame: This reference flame, illustrated in
Fig. 1.4, is the same as the one in item (2), except that the origin of the coordi-
nate system is now in the mass center of the sun and the celestial equatorial
plane now replaces the usual equatorial plane. This flame is an even closer
approximation to an inertial frame than is the geocentric equatorial refer-
ence flame.
)4( Heliocentric ecliptic reference frame: Again, as is also illustrated in Fig. 1.4,
the origin of the spherical coordinate system is in the sun's mass center. The
ecliptic plane now replaces the celestial equatorial plane. Longitude and
latitude are now referred to as the ecliptic longitude and ecliptic latitude,
respectively. Again, the ecliptic longitude is measured eastward from the
first point of Aries. The ecliptic latitude is taken positive extending toward
the ecliptic north pole, negative when pointing toward the ecliptic south
pole. This reference flame is most often used as a suitable approximation to
an ideal inertial flame when analyzing space missions to the planets.
)5( Barycentric reference frame: This frame, which is a still closer approximation
to an ideal inertial frame, has its coordinate origin in the center of mass of
the solar system. It accounts, therefore, for the acceleration of the sun by
2.1 Motion in detareleccA ecnerefeR Frames 7
the planets, principally by Jupiter, and to a lesser extent by Saturn. Because
of the large mass of the sun relative to the planets (Sun/Jupiter mass ra-
tio approximately 1047), the barycenter stays at all times close to the sun,
at a distance from the sun's center that is comparable to the solar radius.
A nearly perfect inertial frame can then be constructed by using in place
of the ecliptic plane the Laplaceinuariable plane, which is the plane through
the barycenter and perpendicular to the orbital angular momentum of all
the masses in the solar system. The inclination of this plane is intermediate
between the orbital planes of Jupiter and Saturn. Its inclination relative to
the ecliptic plane is 1.65 ~ . Although it is important in astronomy when cal-
culating planetary positions over long time spans, either forward or back-
ward in time, for the much shorter times of flights of spacecraft within the
solar system, the slight deviation of the nonrotating heliocentric ecliptic ref-
erence frame from an ideal inertial frame can almost always be neglected.
1.2 Motion in Accelerated Reference Frames
The need to consider two or more reference flames that are in relative mo-
tion to each other arises frequently in all branches of dynamics. It is also
important in space technology. An example is the relation between an in-
ertial flame and a reference frame attached to a spin-stabilized spacecraft.
A second example, in which both frames are accelerated, is the relation
between an earth-fixed reference flame and the spacecraft-fixed flame. In
analyzing, for instance, the effect of the sloshing motion of propellant in a
vehicle tank, it will be advantageous to formulate the equations of motion
of the fluid and the boundary conditions in a reference flame that is rigidly
attached to the propellant tank, hence to the vehicle, yet taking into account
the space vehicle's own translation and rotation.
We consider here velocities and accelerations, hence derivatives of the
position vector of a point with respect to time. Of interest here are reference
spaces that are in motion relative to each other.
As illustrated in Fig. 1.5, the reference spaces are indicated by their
Cartesian coordinate axes. The coordinate flame (Xl, ,2x )3x translates and
rotates relative to the (X1, ,2X )3X flame. The first flame's motion relative to
the second one is determined by the time derivative of the position vector
R0(t) and by the relative angular velocity vector ,~(t). The position vector
r(t) defines the point P in the (Xl, ,2x )3x coordinate flame. In applications
to space technology, this point is often the instantaneous center of mass of
a space vehicle. (The fact that the mass itself may be a function of time as
a consequence of expenditure by the vehicle of propellant is immaterial for
what follows.) Time derivatives in the space spanned by ,1X ,2X 3X will be
designated by d/dt, in the space spanned by xl, ,2x 3x by the dot notation (').
Let ui, i = ,1 ,2 3 designate the orthonormal base vectors of the ,~x( ,2x
)3x coordinate flame and b(t) an arbitrary vector with components bi(t) in
this flame. Hence
b= ~ biui
i