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.
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