ASCI 512 Space Launch

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ASCI 512Space Launch Mission Operations

• Dr. Walter Goedecke
• Fall 2006

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Topics

• Orbital Mechanics
• Kinetic and Potential Energy
• Conic Sections
• Parabolic Trajectories.
• Circular Orbits.
• Elliptical Orbits.
• Hyperbolic Orbits and Flybys
• Lagrange Points

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Orbital MechanicsKinetic and Potential Energy

• An object in motion has two types of energy of
  importance in orbital mechanics kinetic energy
  and potential energy
• Kinetic energy energy due to the objects
  motion.
• Potential energy energy due to the objects
  position.
• The total energy is usually designated by E T
  U
• where T kinetic energy
• U potential energy

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Orbital MechanicsKinetic and Potential Energy

• An object in a uniform gravitational field will
  have mechanical energy as
• E T U
• where m particle mass
• v object velocity
• g gravitational acceleration m/s2
• h object height
• This means that the objects kinetic energy is
  proportional to the square of an objects
  velocity.
• Also, the potential energy is proportional to the
  height of an object above some reference plane.

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Orbital MechanicsKinetic and Potential Energy

• A conservative system is where the energy remains
  constant, an example is the falling ball.
• While the kinetic and potential energy may vary,
  the sum does not.
• As the ball is first dropped, the potential
  energy is at a maximum, and the kinetic energy is
  at a minimum.
• While falling, the potential energy decreases,
  and the kinetic energy increases.
• At the end, the potential energy is minimum, and
  the kinetic energy is maximum.

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Orbital MechanicsKinetic and Potential Energy

• In a non-uniform gravitational field, as
  experienced on a planet the gravitational force
  on an object of mass m is
• where
• G gravitational constant 6.67?10-11
  Newton?m2/kg2
• M a mass, such as the Earths ME
  5.98?1024kg
• r distance between center of masses
• The gravitational acceleration g is the force per
  mass
• This acceleration is typically 9.81 m/s2 on the
  Earths surface

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Orbital MechanicsKinetic and Potential Energy

• A falling objects velocity can be determined by
  the height it fell from, neglecting air
  resistance
• If a 10 gram particle is released, at 3 meters,
  the potential energy, U, is
• U mgh 0.01kg 9.81m/s2 3 m 0.2943
  Joules
• After falling, the objects kinetic energy, T,
  is
• T mv2/2 0.2943 J, so v 7.672 m/s, after
  falling for 3 meters.

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Orbital MechanicsKinetic and Potential Energy

• Since object mass varies considerably in
  mechanical analysis, a convenient means to
  quantify energy is by specific energy, or energy
  per mass
• In kinematics, only velocities matter, and
  consideration of masses is too cumbersome.

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Orbital MechanicsConic Sections

• There are four types of orbital paths that can be
  described by conic sections, or curves formed by
  a plane intersecting a cone, they are
• Circle
• Ellipse
• Parabola
• Hyperbola

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Orbital MechanicsConic Sections

• These curves are described by loci of points
  formed by a constant ratio, or eccentricity, of
  distance to a line, the directrix, to distance to
  a point, the focus.

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Orbital MechanicsConic Sections

• Note that two of the curves, which can be orbits,
  are closed, and the remaining two are open.
• Closed and open conic section curves represent
  bounded and unbounded orbits.
• Energies associated with the curves are
• Hyperbola -- E gt 0
• Parabola -- E 0
• Ellipse -- Vmin lt E lt 0
• Circle -- E Vmin

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Orbital Mechanics Parabolic Trajectories

• The trajectory of a particle thrown obliquely
  will form a parabola, as described by the x
  (horizontal) and z (vertical) displacements. The
  horizontal displacement or position is not
  influenced by the force of gravity, so x(t)
  vx0t
• However, the vertical position is influenced by
  the force of gravity.
• The acceleration g 9.81 m/s2, is considered a
  constant for applications on the Earths surface.
  
• The minus sign indicates gravity is downward.

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Orbital MechanicsParabolic Trajectories

• Integrating the last equation twice yields z(t)
  -gt2/2 vz(0)t z(0)
• This is a parabola, a trajectory that non-powered
  missiles and artillery take.

• Aircraft will have lift on the wings, as
  described by the Bernoullis principle, so
  parabolic flights on descent, can modify the
  gravitational acceleration g, simulating
  weightlessness, or reduced weight.
• Simulating a weightless environment altogether is
  also referred to as free fall.

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Orbital MechanicsParabolic Paths

• Eruption from hornito, Pu'u O'o, Hawaii, in late
  2003. Source http//www.decadevolcano.net/photos/
  hawaii_photos_1203_1.htm, by Tom Pfeiffer. Note
  the parabolic trajectories of the ejected
  material.

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Orbital MechanicsCentral Force Motion

• The concept of an orbiting mass as a satellite
  around a larger object such as a planetary mass
  can be solved analytically, or completely,
  ignoring other forces such a third body, drags,
  etc.
• This is called central force motion.
• Two bodies will orbit around their common center
  of mass.
• When one mass is very much larger that the other,
  i.e., M gtgt m, its center of mass is sufficiently
  close to the common center of mass for
  calculations, e.g., a satellite orbiting Earth.
• Also, the reduced mass of the system, used for
  orbital calculations, is very close to the
  smaller mass

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Orbital MechanicsCentral Force Motion

• The center of orbit of a large body is very close
  to its center of mass.
• If the body has concentric layers, no matter how
  big, this is still true.
• An example is the Earth, shown at the right.
• The crust, mantle, and core are very nearly
  concentric.
• The summation of gravitational forces of all the
  Earths parts on an orbiting body can essentially
  be concentrated as one mass located at the center
  of the Earth.

DINOSAURS AND THE HISTORY OF LIFE - GEOLOGY
V1001x site - Professor Paul Eric Olsen
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Orbital MechanicsCentral Force Motion

• Note, by this model, if the inner layers where
  empty, there would be no net gravitational
  forces all the force would cancel out as the
  concentric shells only have an attraction
  outward.

DINOSAURS AND THE HISTORY OF LIFE - GEOLOGY
V1001x site - Professor Paul Eric Olsen
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Orbital MechanicsCentral Force Motion

• Variations to the concentricity model will
  perturb the nominal orbital path, being the
  equivalent of additional forces rather than just
  a two body problem.

The Earths Structure http//www.lhs.sad49.k12.me.
us/ljhs/Website20Resources/earth's_structure.htm
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Orbital MechanicsCircular Orbits

• Most simple bounded orbits of an object acted
  upon by a central force are closed, that is,
  motion continues over the same path.
• This means that the object does not have enough
  energy to escape the central potential, and is
  trapped in a gravitational well.
• An open, or unbounded orbit, is where the path
  can extend on to infinity.
• The objects path may have been deflected, but is
  not trapped and continues away from the system.
• The simplest bounded orbit is circular.

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Orbital Mechanics - Circular Orbits
A circular orbit is characterized by uniform
circular motion. This is constant speed at a
constant radius from the center of mass. As seen
from the diagram, In the limit, we get the
acceleration
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Orbital MechanicsCircular Orbits

• Objects in circular orbits must have a constant
  velocity or energy to stay at a constant radius.
• As can be generalized with all elliptical orbits,
  the angular momentum of a circular orbit is
  constant
• L mr2? mrv constantwhere ? angular
  velocity, in radians per second
• The orbital energy of the object or satellite
  is
• Note how the potential energy, U, is now
  negative.
• This implies the object must expend energy to
  break orbit from the central mass, M.

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Orbital MechanicsCircular Orbits

• Figure from MatLab.

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Orbital MechanicsCircular Orbits

• This figure shows both the velocity of a
  satellite in circular orbit, and revolutions per
  day, verses the orbital radius in Earth radii,
  RE, from the Earths center.

• Note how the velocity needed to maintain a
  circular orbit increases the closer to Earth.
• To maintain a geostationary satellite (GSO
  geostationary orbit), the revolutions/day 1,
  and this occurs at

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Orbital MechanicsElliptical Orbits

• Elliptical orbits are a more general form of
  closed orbits.
• The revolving body describes the ellipse about
  the central mass located at one of the foci.
• Both orbital radius and velocity are not
  constant, although the angular momentum and
  energy are constant.
• Because the angular momentum is constant, the
  area swept out from orbit with respect to time is
  also constant

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Orbital MechanicsElliptical Orbits
Equation for ellipse

• Important elliptical parameters are
• Focus location of central mass.
• a semi-major axis.
• b semi-minor axis.
• e eccentricity
• a latus rectum
• rmin pericenter
• rmax apocenter

• Both pericenter (perigee/perihelion) apocenter
  (apogee/aphelion) are the apsides.
• Perigee apogee refer to the minimum and maximum
  of an Earth orbit.
• Perihelion aphelion refer to the minimum and
  maximum of an Sun orbit.

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Orbital MechanicsElliptical Orbits
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Orbital Mechanics Elliptical Orbits

• Kinetic Energy upper dashed line.
• Potential Energy lower dashed line.
• Total Energy V(r)

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Orbital Mechanics Elliptical Orbits
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Orbital Mechanics Elliptical Orbits

• This energy diagram shows the possible orbits
  allowed, for a specific angular momentum.
• r3 designates a pure circular orbit with energy
  E3.
• Adding energy, E2 to this orbit allows an
  elliptical orbit with pericenter r2 and apocenter
  r4.

• An energy of zero allows escape from the central
  mass and the orbit becomes parabolic.
• A body with positive energy E1 will become
  hyperbolically deflected and leave orbit
  unbounded.

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Orbital Mechanics Elements of an Elliptical
Orbit
http//en.wikipedia.org/wiki/Longitude_of_ascendin
g_node
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Orbital Mechanics Elements of an Elliptical
Orbit
? right ascension of the ascending node i
inclination e eccentricity a semi-major axis
? argument of perigee M mean anomaly
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Orbital Mechanics Elements of an Elliptical
Orbit
? right ascension of the ascending node i
inclination e eccentricity a semi-major axis
? argument of perigee M mean anomaly
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Orbital MechanicsChanging Orbits

• The force to move mass, m, from orbit R1 to R2
  requires energy of

• Note that if R2 ? 8, the energy potential is
  null.
• This is done so that the objects potential
  energy is zero when it is no longer under the
  influence of the central masss gravitational
  pull.
• U is taken as a negative value, and is zero when
  at infinity

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Orbital Mechanics Changing Orbits

• Orbital Escape Energy and Velocity
• Escape from a central mass occurs when the
  orbiting bodys energy is greater than the
  planetary potential energy.
• For example, on Earth, to move from rest at the
  surface to an unbounded position at infinity

• Solving for v yields

• This means that a projectile must be fired at
  least 7 miles per second from the Earths surface
  to escape into space.

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Orbital Mechanics Changing Orbits

• Hohmann Transfers
• This is an energylean means to transfer between
  orbits.
• A forward firing at v1 accelerates the spacecraft
  from a circular orbit to a higher energy
  elliptical orbit.
• The spacecraft moves from perihelion to aphelion.
  
• There the spacecraft fires forward again, adding
  more energy to form a circular orbit at Mars.

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Orbital Mechanics Hyperbolic Flybys

• Hyperbolic acceleration is a means for a
  spacecraft to steal energy from a larger
  celestial body.
• Also known as
• the sling-shot effect
• a flyby
• When a body moves near the gravitational field of
  a much larger mass, but the body has sufficient
  energy to avoid being gravitationally trapped, it
  can either accelerate or decelerate as its orbit
  is hyperbolically deflected.

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Orbital Mechanics Hyperbolic Flybys

• Consider diagram
• B is large body
• vi and vf are the initial and final small body
  velocities respectively.
• If B was motionless, vf vi
• Since B moves to right, vf gt vi
• If B moved to left, vf lt vi

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Orbital Mechanics Hyperbolic Flybys
Below, the spacecrafts speed is increased by
Jupiters motion.

• Another depiction of a spacecrafts flight being
  hyperbolically deflected by large motionless
  Jupiter, without change of speed.

Wikipeadia Gravitational slingshot
http//en.wikipedia.org/wiki/Gravitational_slingsh
ot
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Orbital Mechanics Hyperbolic Flybys
Voyager 2s journey, with several flybys to
accelerate the spacecraft with the slingshot
effect. Source Classical Dynamics, by J.
Marion, and S. Thornton
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Orbital Mechanics Hyperbolic Flybys
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Orbital Mechanics Hyperbolic Flybys
(Prior Slide) International Sun-Earth Explorer 3
(ISEE-3) Launched in 1978 Mission to monitor the
solar wind between the Sun and Earth. Redeployed
in 1985 to fly a billion miles on through
Giacobini-Zinner Comet September 1985. Also flew
only 75 miles from lunar surface. Source
Classical Dynamics, by J. Marion, and S. Thornton
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Orbital Mechanics Hyperbolic Flybys
Wikipedia, http//en.wikipedia.org/wiki/Gravitatio
nal_slingshot
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Orbital Mechanics Hyperbolic Flybys
A Hohmann transfer to Saturn would require a
total of 15.7 km/s delta velocity (?V) which is
not within the capabilities of current spacecraft
boosters.
A trip using multiple gravitational assists was
used on the Cassini probe, which was sent past
Venus twice, Earth, and finally Jupiter on the
way to Saturn. The 6.7-year transit is slightly
longer than the six years needed for a Hohmann
transfer, but cut the total amount of ?V needed
to about 2 km/s, so much that the large and heavy
Cassini was able to reach Saturn even with the
small boosters available.
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Orbital Mechanics Hyperbolic Flybys
Messenger launch to Mercury, using several
flybys. Courtesy LASP
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Orbital Mechanics Lagrange Points a Three
Body Interaction

• Joseph Louis Lagrange (1736-1813) showed that
  three bodies, consisting of two large ones and a
  third smaller one, can orbit about the common
  center of mass in the same plane.
• The Lagrange points are equilibrium positions
  where the gravitational pull of the two large
  masses equals the centripetal force of the third
  smaller mass, required to rotate with them.

Jet Propulsion Laboratory, Basics of Space
Flight, http//www2.jpl.nasa.gov/basics/
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Orbital Mechanics Lagrange Points Continued

• Consider a system with two large bodies being the
  Earth orbiting the sun, or the Moon orbiting the
  Earth.
• The third body, such as a smaller spacecraft or
  satellite, can occupy any of five Lagrange or
  libration points.
• In line with the two large bodies are the L1, L2
  and L3 points.
• These points are unstable, and thus spacecraft in
  these positions must compensate for orbit drift.

Jet Propulsion Laboratory, Basics of Space
Flight, http//www2.jpl.nasa.gov/basics/

• The leading apex of the triangle is L4 the
  trailing apex is L5.
• These two are also called the Trojan points, and
  are stable positions.

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Orbital Mechanics Lagrange Points Continued

• To illustrate concepts about the Lagrange points,
  consider the larger mass as the Sun, and the
  smaller mass the Earth
• L1 has equal gravitational attraction between the
  two larger masses, thus this orbits period
  within the Earths is slower
• Even though L2, and L3, are at greater orbit
  distances from the Sun than the Earths, the
  combined attraction of the larger masses in one
  direction reduces the orbits period
• L4 and L5 are at 600 from the Sun Earth forming
  an equilateral triangle pair.

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Orbital Mechanics Lagrange Points Continued

• An L1, L4, or L5 space station between the Earth
  and Moon is desirable since this may be a
  stepping stone for future missions.
• Since L4 and L5 are stable orbital points, cosmic
  debris accumulates there, some examples
• The Sun Jupiter L4 L5 points contain
  thousands of Trojan Asteroids, hence Trojan
  points.
• The Saturn moon Tethys has two smaller moons in
  its L4 L5 points, Telesto and Calypso.
• Some dust was found in the Sun Earth Trojan
  points, and the very faint Kordylewski clouds in
  the L4 and L5 points of the Earth Moon system.

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Orbital Mechanics Lagrange Points Continued

• The Solar and Heliospheric Observatory (SOHO),
  launched in 1995 to study the Sun, orbits about
  L1
• Its six month orbit perpendicular to the Sun
  Earth line about the L1 point keeps it from
  blocking the Sun on Earth.
• It is about 1.5 million km from the Earth.

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Orbital Mechanics Lagrange Points Continued

• Advanced Composition Explorer (ACE) is an
  explorer satellite mission to study the solar
  wind, interpanetary magnetic field (IMF), and
  other cosmic energetic particles
• It was launched in 1997 and is currently
  operating at the L1 Lagrange point.

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Orbital Mechanics Lagrange Points Continued

• The International Cometary Explorer (ICE)
  spacecraft, originally known as International
  Sun/Earth Explorer 3 (ISEE-3) satellite, was
  launched Aug. 12, 1978.
• The constellation of ISEE-1, ISEE-2, and it were
  designed to study the interaction between the
  Earth's magnetic field and the solar wind.

• On June 10, 1982, ISEE-3 was diverted to
  intercept the Giacobini-Zinner comet.
• By gravitational instabilities of the Earth/Moon
  and Earth/Sun Lagrange points, the spacecraft
  made a series of lunar orbits

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Orbital Mechanics Lagrange Points Continued

• On Dec. 22, 1983 it made its closest pass to the
  comet and was renamed the International Cometary
  Explorer.
• Then on June 5, 1985, ICE passed through the
  plasma tail of the Giacobini-Zinner comet.

• It made measurements of energetic particles,
  waves, plasmas, and fields.
• In March 1986 it rendezvoused with spacecrafts
  Giotto, Vega 1 and 2, Suisei and Sakigake, near
  Halleys Comet.
• ICE was shutdown in May 1997.

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Orbital Mechanics Lagrange Points Continued

• The spacecraft may be captured in 2014 when it
  again makes a close approach to Earth.
• If the craft is recovered, it will end up in the
  Smithsonian Institute.
• http//en.wikipedia.org/wiki/International_Cometar
  y_Explorer

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Orbital Mechanics Lagrange Points the Song!

• Home on Lagrange - The L5 Song
• by William S. Higgins and Barry D. Gehm
• 1978
• Sung to the tune of Home on the Range
• Home, home on Lagrange,  Where the space debris
  always collects,  We possess, so it seems, two
  of Man's greatest dreams  Solar power and
  zero-gee sex.

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Orbital Mechanics References

• Classical Dynamics, by J. Marion, and S. Thornton
• http//www.decadevolcano.net/photos/hawaii_photos_
  1203_1.htm, by Tom Pfeiffer.
• Dinosaurs and the History of Life Geology
  V1001x site - Professor Paul Eric Olsen
• The Earths Structure, http//www.lhs.sad49.k12.me
  .us/ljhs/Website20Resources/earth's_structure.htm
  
• Wikipeadia Gravitational slingshot
  http//en.wikipedia.org/wiki/Gravitational_slingsh
  ot
• Jet Propulsion Laboratory, Basics of Space
  Flight, http//www2.jpl.nasa.gov/basics/
• Wikipedia Lagrange points, http//en.wikipedia.or
  g/wiki/Lagrange_points