Mecânica Orbital - Apostilas - Engenharia Aeronáutica, Notas de estudo de Engenharia Aeronáutica Unificada

Engenharia Aeronáutica Unificada

Descrição: Apostilas de Engenharia Aeronáutica sobre o estudo da Mecânica Orbital, Conic Sections, Orbital Elements, Types of Orbits, Newton's Laws of Motion and Universal Gravitation, Uniform Circular Motion.
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Conic Sections
Orbital Elements
Types of Orbits
Newton's Laws of Motion and Universal Gravitation
Uniform Circular Motion
Motions of Planets and Satellites
Launch of a Space Vehicle
Position in an Elliptical Orbit
Orbit Perturbations
Orbit Maneuvers
Escape Velocity
Orbital mechanics, also called ight mechanics, is the study of the motions of articial satellites and space
vehicles moving under the inuence of forces such as gravity, atmospheric drag, thrust, etc. Orbital
mechanics is a modern oshoot of celestial mechanics which is the study of the motions of natural
celestial bodies such as the moon and planets. The root of orbital mechanics can be traced back to the
17th century when mathematician Isaac Newton (1642-1727) put forward his laws of motion and
formulated his law of universal gravitation. The engineering applications of orbital mechanics include
ascent trajectories, reentry and landing, rendezvous computations, and lunar and interplanetary
Conic Sections
A conic section, or just conic, is a curve
formed by passing a plane through a right
circular cone. As shown in the gure to the
right, the angular orientation of the
plane relative to the cone determines whether
conic section is a circle, ellipse, parabola, or
hyerbola. The circle and the ellipse arise
when the intersection of cone and plane is a
bounded curve. The circle is a special case of
the ellipse in which the plane is
perpendicular to the axis of the cone. If the
plane is parallel to a generator line of
the cone, the conic is called a parabola.
Finally, if the intersection is an unbounded
curve and the plane is not parallel to a
generator line of the cone, the gure is a
hyperbola. In the latter case the plane will
intersect both halves of the cone, producing two
separate curves.
We can dene all conic sections in terms of the eccentricity. The type of conic section is also related to the
semi-major axis and the energy. The table below shows the relationships between eccentricity, semi-major
axis, and energy and the type of conic section.
Conic Section Eccentricity, e Semi-major axis Energy
Circle 0 = radius < 0
Ellipse 0 < e < 1 > 0 < 0
Parabola 1 infinity 0
Hyperbola > 1 < 0 > 0
Satellite orbits can be any of the four conic sections. In this section we will discuss bounded conic orbits,
i.e. circles and ellipses.
Orbital Elements
To mathematically describe an orbit one must dene six quantities, called orbital elements. They are
Semi-Major Axis, a
Eccentricity, e
Inclination, i
Argument of Periapsis,
Time of Periapsis Passage, T
Longitude of Ascending Node,
An orbiting satellite follows an oval shaped path known as an ellipse with the body being orbited, called
the primary, located at one of two points called foci. An ellipse is dened to be a curve with the following
property: for each point on an ellipse, the sum of its distances from two xed points, called foci, is
constant (see gure to right). The longest and shortest lines that can be drawn through the center of an
ellipse are called the major axis and minor axis, respectively. The semi-major axis is one-half of the major
axis and represents a satellite's mean distance from its primary. Eccentricity is the distance between the
foci divided by the length of the major axis and is a number between zero and one. An eccentricity of zero
indicates a circle.
Inclination is the angular distance between a satellite's orbital plane and the equator of its primary (or the
ecliptic plane in the case of heliocentric, or sun centered, orbits). An inclination of zero degrees indicates
an orbit about the primary's equator in the same direction as the primary's rotation, a direction called
prograde (or direct). An inclination of 90 degrees indicates a polar orbit. An inclination of 180 degrees
indicates a retrograde equatorial orbit. A retrograde orbit is one in which a satellite moves in a direction
opposite to the rotation of its primary.
Periapsis is the point in an orbit closest to the primary. The opposite of periapsis, the farthest point in an
orbit, is called apoapsis. Periapsis and apoapsis are usually modied to apply to the body being orbited,
such as perihelion and aphelion for the Sun, perigee and apogee for Earth, perijove and apojove for
Jupiter, perilune and apolune for the Moon, etc. The argument of periapsis is the angular distance between
the ascending node and the point of periapsis (see gure below). The time of periapsis passage is the time
in which a satellite moves through its point of periapsis.
Nodes are the points where an orbit crosses a plane, such as a satellite crossing the Earth's equatorial
plane. If the satellite crosses the plane going from south to north, the node is the ascending node; if
moving from north to south, it is the descending node. The longitude of the ascending node is the node's
celestial longitude. Celestial longitude is analogous to longitude on Earth and is measured in degrees
counter-clockwise from zero with zero longitude being in the direction of the vernal equinox.
In general, three observations of an object in orbit are required to calculate the six orbital elements. Two
other quantities often used to describe orbits are period and true anomaly. Period, P, is the length of time
required for a satellite to complete one orbit. True anomaly, v, is the angular distance of a point in an orbit
past the point of periapsis, measured in degrees.
Types Of Orbits
For a spacecraft to achieve Earth orbit, it must be launched to an elevation above the Earth's atmosphere
and accelerated to orbital velocity. The most energy ecient orbit, that is one that requires the least
amount of propellant, is a direct low inclination orbit. To achieve such an orbit, a spacecraft is launched in
an eastward direction from a site near the Earth's equator. The advantage being that the rotational speed
of the Earth contributes to the spacecraft's nal orbital speed. At the United States' launch site in Cape
Canaveral (28.5 degrees north latitude) a due east launch results in a "free ride" of 1,471 km/h (914 mph).
Launching a spacecraft in a direction other than east, or from a site far from the equator, results in an
orbit of higher inclination. High inclination orbits are less able to take advantage of the initial speed
provided by the Earth's rotation, thus the launch vehicle must provide a greater part, or all, of the energy
required to attain orbital velocity. Although high inclination orbits are less energy ecient, they do have
advantages over equatorial orbits for certain applications. Below we describe several types of orbits and
the advantages of each:
Geosynchronous orbits (GEO) are circular orbits around the Earth having a period of 24 hours. A
geosynchronous orbit with an inclination of zero degrees is called a geostationary orbit. A spacecraft in a
geostationary orbit appears to hang motionless above one position on the Earth's equator. For this reason,
they are ideal for some types of communication and meteorological satellites. A spacecraft in an inclined
geosynchronous orbit will appear to follow a regular gure-8 pattern in the sky once every orbit. To attain
geosynchronous orbit, a spacecraft is rst launched into an elliptical orbit with an apogee of 35,786 km
(22,236 miles) called a geosynchronous transfer orbit (GTO). The orbit is then circularized by ring the
spacecraft's engine at apogee.
Polar orbits (PO) are orbits with an inclination of 90 degrees. Polar orbits are useful for satellites that
carry out mapping and/or surveillance operations because as the planet rotates the spacecraft has access
to virtually every point on the planet's surface.
Walking orbits: An orbiting satellite is subjected to a great many gravitational inuences. First, planets
are not perfectly spherical and they have slightly uneven mass distribution. These uctuations have an
eect on a spacecraft's trajectory. Also, the sun, moon, and planets contribute a gravitational inuence on
an orbiting satellite. With proper planning it is possible to design an orbit which takes advantage of these
inuences to induce a precession in the satellite's orbital plane. The resulting orbit is called a walking
orbit, or precessing orbit.
Sun synchronous orbits (SSO) are walking orbits whose orbital plane precesses with the same period as
the planet's solar orbit period. In such an orbit, a satellite crosses periapsis at about the same local time
every orbit. This is useful if a satellite is carrying instruments which depend on a certain angle of solar
illumination on the planet's surface. In order to maintain an exact synchronous timing, it may be
necessary to conduct occasional propulsive maneuvers to adjust the orbit.
Molniya orbits are highly eccentric Earth orbits with periods of approximately 12 hours (2 revolutions per
day). The orbital inclination is chosen so the rate of change of perigee is zero, thus both apogee and
perigee can be maintained over xed latitudes. This condition occurs at inclinations of 63.4 degrees and
116.6 degrees. For these orbits the argument of perigee is typically placed in the southern hemisphere, so
the satellite remains above the northern hemisphere near apogee for approximately 11 hours per orbit.
This orientation can provide good ground coverage at high northern latitudes.
Hohmann transfer orbits are interplanetary trajectories whose advantage is that they consume the
least possible amount of propellant. A Hohmann transfer orbit to an outer planet, such as Mars, is
achieved by launching a spacecraft and accelerating it in the direction of Earth's revolution around the sun
until it breaks free of the Earth's gravity and reaches a velocity which places it in a sun orbit with an
aphelion equal to the orbit of the outer planet. Upon reaching its destination, the spacecraft must
decelerate so that the planet's gravity can capture it into a planetary orbit.
To send a spacecraft to an inner planet, such as Venus, the spacecraft is launched and accelerated in the
direction opposite of Earth's revolution around the sun (i.e. decelerated) until it achieves a sun orbit with
a perihelion equal to the orbit of the inner planet. It should be noted that the spacecraft continues to move
in the same direction as Earth, only more slowly.
To reach a planet requires that the spacecraft be inserted into an interplanetary trajectory at the correct
time so that the spacecraft arrives at the planet's orbit when the planet will be at the point where the
spacecraft will intercept it. This task is comparable to a quarterback "leading" his receiver so that the
football and receiver arrive at the same point at the same time. The interval of time in which a spacecraft
must be launched in order to complete its mission is called a launch window.
Newton's Laws of Motion and Universal Gravitation
Newton's laws of motion describe the relationship between the motion of a particle and the forces acting
on it.
The rst law states that if no forces are acting, a body at rest will remain at rest, and a body in motion will
remain in motion in a straight line. Thus, if no forces are acting, the velocity (both magnitude and
direction) will remain constant.
The second law tells us that if a force is applied there will be a change in velocity, i.e. an acceleration,
proportional to the magnitude of the force and in the direction in which the force is applied. This law may
be summarized by the equation
where F is the force, m is the mass of the particle, and a is the acceleration.
The third law states that if body 1 exerts a force on body 2, then body 2 will exert a force of equal
strength, but opposite in direction, on body 1. This law is commonly stated, "for every action there is an
equal and opposite reaction".
In his law of universal gravitation, Newton states that two particles having masses m1 and m2 and
separated by a distance r are attracted to each other with equal and opposite forces directed along the
line joining the particles. The common magnitude F of the two forces is
where G is an universal constant, called the constant of gravitation, and has the value 6.67259x10 -11 N-
m2/kg2 (3.4389x10-8 lb-ft2/slug2).
Let's now look at the force that the Earth exerts on an object. If the object has a mass m, and the Earth
has mass M, and the object's distance from the center of the Earth is r, then the force that the Earth exerts
on the object is GmM /r2 . If we drop the object, the Earth's gravity will cause it to accelerate toward the
center of the Earth. By Newton's second law (F = ma), this acceleration g must equal (GmM /r2)/m, or
At the surface of the Earth this acceleration has the valve 9.80665 m/s2 (32.174 ft/s2).
Many of the upcoming computations will be somewhat simplied if we express the product GM as a
constant, which for Earth has the value 3.986005x1014 m3/s2 (1.408x1016 ft3/s2). The product GM is often
represented by the Greek letter .
For additional useful constants please see the appendix Basic Constants.
For a refresher on SI versus U.S. units see the appendix Weights & Measures.
Uniform Circular Motion
In the simple case of free fall, a particle accelerates toward the center of the Earth while moving in a
straight line. The velocity of the particle changes in magnitude, but not in direction. In the case of uniform
circular motion a particle moves in a circle with constant speed. The velocity of the particle changes
continuously in direction, but not in magnitude. From Newton's laws we see that since the direction of the
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