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1 Introduction
The Sun, with all the planets revolving around it, and depending on it, can still ripen
a bunch of grapes as though it had nothing else in the Universe to do.
Galileo Galilei (1564-1642)
1.1 What is space mechanics?
Space mechanics is the branch of mechanics, which studies the motion of
objects traveling into space – subject to all possible forces. These objects
may be celestial or heavenly bodies created by God such as the Earth, the
Sun, and the Moon; or vehicles made by man such as satellites, space
shuttles and space probes. The branch of the subject studying the former
objects is usually referred to as celestial mechanics, whereas the branch
studying the latter is usually referred to as astrodynamics1. Orbital
mechanics may be considered as a synonym for space mechanics.
Astronomy is the scientific study of celestial objects (such as stars, planets,
comets, and galaxies) and phenomena that originate outside the Earth's
atmosphere (such as the cosmic background radiation). It is concerned with
the evolution, physics, chemistry, meteorology, and motion of celestial
objects, as well as the formation and development of the universe.
Therefore, in a correct sense, orbital mechanics may be considered as a
branch of astronomy. Old or even ancient astronomy is not to be confused
with astrology, the belief system that claims that human affairs are
correlated with the positions of celestial objects. Although the two fields
share a common origin and a part of their methods (namely, the use of
ephemerides), they are clearly distinct2.
1
Vallado, 2007: chapter 1.
2
Albert et al., 2001: through www.wikipedia.com
Galileo Galilei (1564-1642). Italian natural
Philosopher,
Astronomer
and
Mathematician who made fundamental
contributions to the development of the
scientific method and to the sciences of
motion, astronomy and strength of
materials.
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1.2 Why is space so attractive?
The Hubble Space Telescope's
launched in 1990. Its position above
the atmosphere, which distorts and
blocks the light reaching Earth, gives
it a view of the universe that far
surpasses that of ground-based
telescopes.3
Getting into space is tremendously expensive and associated with many
risks and expensive. So why do we bother the cost and risk? Space offers
human kind several compelling advantages which have been exploited in
modern societies. It provides a global perspective - Space provides the
highest post for surveillance and resource monitoring. Space also provides
a universal perspective - un-obscured view of the heavens. Earth telescopes
are susceptible to all kinds of obstacles whether they are natural or manmade. Finally, space offers a unique environment - a free-fall environment
which can be used to perform unprecedented material and biological
experiments. In addition, space has an abundant storage of resources
which can be used to replace depleting Earth resources.
1.3 Historical Background
1.3.1 Ancient Astronomy
Early cultures identified celestial objects with gods and spirits. They related
these objects (and their movements) to climate and weather changes such
as rain, drought, seasons, and tides.
Claudius Ptolemaeus (83–168), known in
English as Ptolemy. Greek-Egyptian
mathematician, astronomer, geographer
and astrologer. He was born, lived and
died in Roman Egypt. He was the author
of several scientific treatises, which had
great importance to later Islamic and
European science. His most famous
treastise on astronomy is now known as
the Almagest, or “The Great Treatise. The
second is the Geography, which is a
thorough discussion of the geographic
knowledge of the Greco-Roman world.4
Calendar systems were developed usually based on the Sun and Moon
apparent motion and were of great importance to agricultural societies.
Babylonian astronomy was the basis for much of the astronomical
traditions that later developed in Greek and Hellenistic astronomy, in
classical Indian astronomy, in Sassanid, Byzantine and Syrian astronomy, in
medieval Islamic astronomy, and in Western European astronomy.
Babylonians invented a sexagesimal (base 60) number system which is still
used in the modern practice of dividing a circle into 360 degrees, of 60
minutes each, began with the Sumerians.
Beginning around 600 BC, Greek philosophers and scientists developed a
number of important astronomical ideas. The early Greek astronomers
3
www.hubblesite.org/the_telescope/hubble_essentials/
4
www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html
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knew many of the geometrical relationships of the heavenly bodies.
Pythagoras, who lived during the 500s BC, argued that the earth was round.
He also tried to explain the nature and structure for the universe as a
whole. He developed an early system of cosmology. In about 370 BC,
Euxodus of Cnidus had developed a mechanical system to explain the
motion s of the planets. Euduxos taught that the planets, sun, the moon,
and the stars revolved around the earth. In 300s BC, Aristotle incorporated
this earth centered or geocentric, theory into his philosophic system.
Ptolemy was the author of several scientific treatises including the
astronomical treatise, Almagest. His Planetary Hypotheses went beyond
the mathematical model of the Almagest to present a physical realization
of the universe as a set of nested spheres,[15] in which he used the
epicycles of his planetary model to compute the dimensions of the
universe. He estimated the Sun was at an average distance of 1210 Earth
radii while the radius of the sphere of the fixed stars was 20,000 times the
radius of the Earth.[16] (Wikipedia) (Valado, 2007: 1 – 6)
Naṣīr al-Dīn al-Ṭūsī, or Tusi (1201-1274).
Persian
astronomer,
philosopher,
physician, mathematician, physician,
physicist, and theologian. Tusi made very
accurate tables of planetary movements as
depicted in his book Zij-i ilkhani. Tusi
invented a geometrical technique called
Tusi-couple, which generates linear
motion from the sum of two circular
motions. He used this technique to replace
Ptolemy's problematic equant.
1.3.2 Islamic Astronomy
In the history of astronomy, Islamic astronomy or Arabic astronomy refers
to the astronomical developments made in the Islamic world, particularly
during the Islamic Golden Age (8th-16th centuries), and mostly written in
the Arabic language. These developments mostly took place in the Middle
East, Central Asia, Islamic Spain, North Africa, and later in China and India.
It closely parallels the genesis of other Islamic sciences in its assimilation of
foreign material and the amalgamation of the disparate elements of that
material to create a science that was essentially Islamic. These included
Indian, Sassanid and Hellenistic works in particular, which were translated
and built upon.[1] In turn, Islamic astronomy later had a significant
influence on Indian[2] and European[3] astronomy (see Latin translations of
the 12th century) as well as Chinese astronomy.[4] 5
It is really interesting to note that Muslims were in fact the first to
differentiate and separate the science of Astronomy from the pseudo
science of astrology. A significant number of stars in the sky, such as
5
Saliba, George (1999), Whose Science is Arabic Science in Renaissance Europe?, Columbia University,
http://www.columbia.edu/~gas1/project/visions/case1/sci.1.html, retrieved on 26 December 2008
Omar Khayyam (1048-1131). Persian
poet, mathematician, and astronomer.
Khayyam measured the length of the year
as 365.24219858156 days, which shows an
incredible confidence. For comparison the
length of the year at the end of the 19th
century was 365.242196 days, while today
it is 365.242190 days.
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Aldebaran and Altair, and astronomical terms such as alhidade, azimuth,
and almucantar, are still today recognized with their Arabic names.[5]
A large corpus of literature from Islamic astronomy remains today,
numbering approximately 10,000 manuscripts scattered throughout the
world, many of which have not been read or cataloged. Even so, a
reasonably accurate picture of Islamic activity in the field of astronomy can
be reconstructed.[6]
Astrolabe is a sophisticated tool for
observing the position of the stars
which was invented in ancient
Greece and vastly improved in early
Islam. Through Islamic Spain, the
new astrolabe was introduced to
Europe.
An important area in Astronomy is optics. It is very relevant in the
development of tools for observation like telescopes that employ lenses or
mirrors. Ibn al-Haytham (the Latin Alhazen) studied the property of lenses,
discovered the camera obscura, explained correctly the process of vision,
studied the structure of the eye, and explained for the first time why the
sun and the moon appear larger on the horizon (very simply put, it is
because the thicker layer of atmosphere at the horizon acts as magnifying
lens compared to overhead).
Other Muslim mathematicians such as Khayyam and al-Tusi examined
Euclidean geometry that is the geometry of flat surfaces. The Muslim
mathematicians, especially al-Battani, Abu'l-Wafa', Ibn Yunus and Ibn alHaytham, also developed spherical Astronomy. Euclidean and spherical
geometry are particularly useful in studying the overall geometry of the
Universe in the study of cosmology.
Muhammad ibn Jabir al-Batani (858–929)
Latinized as Albategnius, Albategni or
Albatenius. Arab astronomer, astrologer,
and mathematician. One of his bestknown achievements in astronomy was
the determination of the solar year as
being 365 days, 5 hours, 46 minutes and
24 seconds.
The Muslims also applied their astronomical knowledge to questions of
time-keeping and the calendar in making almanacs, this word too being
Arabic in origin. The most exact solar calendar existing to this day is the
Jalali calendar which was developed under the direction of Omar Khayyam
in the 12th century. This is still in use in Persia and Afghanistan6.
Many people think that “the Arabs gave us (the) zero”. It is more
appropriate to look at the phenomenon of Islamic science within its own
cultural context. Then, if we want to compare, we discover that actually, as
far as astronomy is concerned, little was achieved in Europe until ca. 1550
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www.moonsighting.com/articles/roleofislam.html
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that had not been achieved previously by Muslim scholars at some time
between the 9th and the 15th century.7
1.3.3 Medieval European Astronomy
Nicholas Copernicus (1473 – 1543)
Copernicus was the first astronomer to formulate a scientifically-based
heliocentric cosmology that displaced the Earth from the center of the
universe. His epochal book, De revolutionibus orbium coelestium (On the
Revolutions of the Celestial Spheres), is often regarded as the starting point
of modern astronomy and the defining epiphany that began the Scientific
Revolution. The book was published the year of Copernicus’ death, 1543,
though he had arrived at his theory several decades earlier.
Nicholas Copernicus (1473-1543). Polish
astronomer and mathematician. He began
to believe that the earth went round the
sun about 1507 and from that time until
his death worked, more or less
intermittently, on his exposition of his
theory. He delayed the publication of this
exposition because of fear of being
accused of heresy.
Although Greek, Indian and Muslim savants had published heliocentric
hypotheses centuries before Copernicus, his publication of a scientific
theory of heliocentrism, demonstrating that the motions of celestial objects
can be explained without putting the Earth at rest in the center of the
universe, stimulated further scientific investigations and became a
landmark in the history of modern science that is known as the Copernican
Revolution.
Galileo Galelei (1564-1642)
Galileo Galelei provided the crucial observations that proved the
Copernican hypothesis, and also laid the foundations for a correct
understanding of how objects moved on the surface of the earth
(dynamics) and of gravity. (Valado, 2007: 7 – 9)
Johanns Kepler (1571-1630)
The greatest achievement of Kepler was his discovery of the laws of
planetary motion. There were such three laws, but here we shall deal only
with the first two - those that govern the motion of an individual planet.
These are found in Astronomia Nova (New Astronomy, 1609), underpinned
by important work in Epitome (of Copernican Astronomy) Book V (1621).
The laws are:
Prof. David A. King in his lecture titled “ASTRONOMY IN THE BAGHDAD OF THE CALIPHS”
at the Institute of the History of Science, Johann Wolfgang Goethe University.
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www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html
Johannes Kepler (1571 - 1630). German
mathematician and astronomer who
discovered that the Earth and planets
travel about the sun in elliptical orbits. He
gave three fundamental laws of planetary
motion. He also did important work in
optics and geometry.8
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First Law:
The orbit of a planet/comet about the Sun is an ellipse with
the Sun's center of mass at one focus.
Second Law:
A line joining a planet/comet and the Sun sweeps out equal
areas in equal intervals of time.
Third Law:
The squares of the periods of the planets are proportional
to the cubes of their semimajor axes.
Isaac Newton (1643 – 1727)
According to the well-known story, it was on seeing an apple fall in his
orchard at some time during 1665 or 1666 that Newton conceived that the
same force governed the motion of the Moon and the apple. He calculated
the force needed to hold the Moon in its orbit, as compared with the force
pulling an object to the ground. He also calculated the centripetal force
needed to hold a stone in a sling, and the relation between the length of a
pendulum and the time of its swing. These early explorations were not
soon exploited by Newton, though he studied astronomy and the problems
of planetary motion.
Sir Isaac Newton (1642-1727). British
mathematician and physicist, one of
the foremost scientific intellects of all
time. He was elected a Fellow of
Trinity College in 1667 and Lucasian
Professor of Mathematics in 1669.
During two to three years of intense
mental effort he prepared Philosophiae
Naturalis
Principia
Mathematica
(Mathematical Principles of Natural
Philosophy) commonly known as the
Principia, although this was not
published until 1687.9
Correspondence with Hooke (1679-1680) redirected Newton to the
problem of the path of a body subjected to a centrally directed force that
varies as the inverse square of the distance; he determined it to be an
ellipse, so informing Edmond Halley in August 1684. Halley's interest led
Newton to demonstrate the relationship afresh, to compose a brief tract on
mechanics, and finally to write the Principia.
Book I of the Principia states the foundations of the science of mechanics,
developing upon them the mathematics of orbital motion round centers of
force. Newton identified gravitation as the fundamental force controlling
the motions of the celestial bodies. He never found its cause. To
contemporaries who found the idea of attractions across empty space
unintelligible, he conceded that they might prove to be caused by the
impacts of unseen particles.
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www.newton.ac.uk/newtlife.html
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Book III shows the law of gravitation at work in the universe: Newton
demonstrates it from the revolutions of the six known planets, including
the Earth, and their satellites. However, he could never quite perfect the
difficult theory of the Moon's motion. Comets were shown to obey the
same law; in later editions, Newton added conjectures on the possibility of
their return. He calculated the relative masses of heavenly bodies from
their gravitational forces, and the oblateness of Earth and Jupiter, already
observed. He explained tidal ebb and flow and the precession of the
equinoxes from the forces exerted by the Sun and Moon. All this was done
First Law of Motion:
Every object remains in its state of rest or motion
in a straight line unless a force is exerted upon
which.
Second Law of Motion:
The rate of change of (linear) momentum of
a body is proportional to the force exerted on
which and is in the same direction.
Third Law of Motion:
To every action there is always an equal and
opposite reaction.
by exact computation.
Newton’s second law can be expressed mathematically as
∑𝐅 =
𝑑
(𝑚𝐕)
𝑑𝑡
( 1-1 )
Universal Gravitational Law: Any two bodies attract one another with
a force proportional to the product of their
masses and inversely proportional to the
square of the distance between them.
Newton’s universal law of gravitation can be expressed as follows. Figure
𝐅𝑔 = −𝐺
𝑀𝑚 𝐫
𝑟2 𝑟
( 1-2 )
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Z
F
g
r
m
M
X
Fig. 1-1. Gravitational force.
Y
y
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1.3.4 Modern Astrodynamics
We need to write here about modern astrodynamics.
1.4 Where are we in space?
The following plate illustrates the relative sizes of the planets belonging to
our solar system. Jupiter is the largest and Ceres is the smallest.
Fig. 1-2. The Solar system.
1.5 References
Kit, S. T. (2007). Astrodynamics Premier.
Unsöld, A., Baschek, B., & Brewer, W. (2001). The New Cosmos: An Introduction to
Astronomy and Astrophysics. Berlin: Springer.
Albrecht Unsöld; Bodo Baschek, W.D. Brewer (translator) (2001). The New Cosmos: An
Introduction to Astronomy and Astrophysics. Berlin, New York: Springer.
Satellite Tool Kit Tutorial. STK version 8.0
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