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Topic: Hyperbolic trajectory

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	University of Bristol Laboratory for Advanced Computation in the Mathematical Sciences
		Hyperbolic or ``saddle type'' stagnation points for two-dimensional, incompressible, steady flows are often of great significance in a flow because they tend to be the ``origin'' of qualitatively different fluid particle motions.
		Numerical methods for locating hyperbolic trajectories that utilize the stretching and contraction properties to allow certain ``test regions'' to converge to the hyperbolic trajectory are not adequate for time-dependent velocity fields that are only known for a finite interval of time.
		However, for our purposes, we would not call the hyperbolic trajectories in the Cantor set ``distinguished'' as the transport of trajectories through this Cantor set is governed by the lobe dynamics associated with the hyperbolic trajectory whose transversely intersecting stable and unstable manifolds give rise to the hyperbolic Cantor set.
	lacms.maths.bris.ac.uk /projects/wiggins/net2 (1767 words)

	trajectory - Search Results - MSN Encarta
		Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e.g.
		In astrodynamics or celestial mechanics a hyperbolic trajectory is an orbit with the eccentricity greater than 1.
		Trajectory is an international award-winning branding marketing communicationsfirm working across a wide range of industry segments consumer, b2b, non-profits and associations.
	encarta.msn.com /encnet/refpages/search.aspx?q=trajectory (226 words)

	Hyperbolic trajectory
		In astrodynamics or celestial mechanics a hyperbolic trajectory is an orbit with the eccentricity greater than 1.
		Similarly to parabolic trajectory all hyperbolic trajectories are also escape trajectories.
		Specific energy of hyperbolic trajectory orbit is positive.
	www.algebra.com /algebra/about/history/Hyperbolic-trajectory.wikipedia (197 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		In the hyperbolic case the Hartman-Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J Â· x.
		Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems.
		In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Dynamical_system (3385 words)

	Rutherford Scattering
		The departure from the point-particle form of scattering has been an indicator of nuclear structure and then at higher energies, the structure of the proton.
		The scattering of the alpha particle by the central repulsive Coulomb force leads to a hyperbolic trajectory.
		From the scattering angle and momentum, one can calculate the impact parameter and closest approach to the target nucleus.
	230nsc1.phy-astr.gsu.edu /hbase/rutsca.html (378 words)

	Probe Trajectory Modelling
		Describing such a trajectory is inherently more complex than modelling the orbit of a spacecraft around the Earth, as the effects of the Earth, the Sun and the target body all have to be considered.
		Indeed, initial modelling of the probe's trajectory showed that rather than trying to set up a particular transfer orbit (the usual case for interplanetary flights) the actual aim was to reach a specified point in space as quickly as possible.
		This is in considerable contrast to most deep space trajectories, which are typically transfer orbits where the probe's position and velocity vectors relative to both the Earth and the target body very considerably over the course of the mission.
	homepage.mac.com /sjbradshaw/msc/traject.html (1945 words)

	Orbit equation (Site not responding. Last check: )
		Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance (such as gravity), has an orbit that is a conic section (i.e.
		circular orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory) with the central body located at one of the two foci, or the focus (Kepler's first law).
		If the conic section intersects the central body, then the actual trajectory can only be the part above the surface, but for that part the orbit equation and many related formulas still apply, as long as it is a freefall (situation of weightlessness).
	encyclopedia.vestigatio.com /Orbit_equation (857 words)

	Francois Lekien
		Figure 2 shows the hyperbolic trajectory at the South pole and its associated rotating invariant manifolds [Lekien et al., 2005d].
		Hyperbolic stretching near a boundary is best studied in terms of the existence of a separation profile emanating from the boundary [Sears and Telionis, 1975; Van Dommelen and Cowley, 1990].
		For example, the uniqueness of trajectory and the continuity of the flow can be used to prove that only anti-cyclonic rings are involved in transport across a meandering jet [Lekien, 2003].
	www.lekien.com /~francois/research (4653 words)

	Trajectories
		Trajectories are the planned paths which celestial bodies-whether they be star, planet, moon, or man made satellite- follow in their course through the sky.
		In mathematics, a hyperbola (singular form of hyperbolic) is a type of conic section (a curved locus of points, formed by intersecting a cone with a plane).
		In astrodynamics or celestial mechanics a parabolic trajectory is an orbit with the eccentricity equal to 1.
	users.desk003.com /carol/cporbits/cptrajectoryorbit.htm (944 words)

	gravity assist
		Also known as the slingshot effect, an important spaceflight technique used successfully on a number of interplanetary missions, including Voyager, Galileo, and Cassini, whereby the gravitational field of a planet is used to increase the speed and alter the course of a spacecraft without the need to expend fuel.
		One of the earliest, and most dramatic applications of the technique came in 1970 when the world watched as NASA used a lunar gravity-assist to rescue the Apollo 13 astronauts after an onboard explosion had severely damaged their spacecraft en route to the Moon.
		By using a relatively small amount of fuel to put the spacecraft onto a suitable trajectory, NASA engineers and the astronauts were able to use the Moon’s gravity to turn the ship around and send it back home.
	www.daviddarling.info /encyclopedia/G/gravityassist.html (1040 words)

	LCS Tutorial: How the FTLE is used
		However, let us just think of hyperbolic trajectories as ones about which there is a direction of expansion and a direction of compression.
		Exponential separation between trajectories is to be expected, however if the domain of the system is compact, which is typically the case, these trajectories must mix together.
		Heteroclinic trajectories are referred to as separatrices because they separate distinct regions in the flow.
	www.cds.caltech.edu /~shawn/LCS-tutorial/FTLE-interp.html (821 words)

	Hyperbolic trajectory
		Under standard assumptions the body traveling along along hyperbolic trajctory will attain in infinity an orbital velocity called hyperbolic excess velocity () that can be computed as:
		A square of hyperbolic excess velocity() is also known as characteristic energy and denoted by :
		Under standard assumptions, specific orbital energy () of parabolic trajectory is greater than zero and the orbital energy conservation equation for this kind of trajectory takes form:
	www.xasa.com /wiki/en/wikipedia/h/hy/hyperbolic_trajectory.html (216 words)

	NPG - Abstract
		While the notion of a hyperbolic trajectory is central to dynamical systems theory, much of the theoretical developments for Lagrangian transport proceed under the assumption that such a special hyperbolic trajectory exists.
		This is not surprising since hyperbolicity is a "linearized" notion.
		To extend the analytical formula to more general nonlinear time-dependent velocity fields, we develop a series of coordinate transforms including a type of linearization that is not typically used in dynamical systems theory.
	www.copernicus.org /EGU/npg/9/237.htm (411 words)

	Press Release
		Rosetta was placed into a hyperbolic liberation orbit - the first time an Ariane 5 has been used for this type of mission.
		It was able to carry out the mission thanks to the delayed ignition capability of Ariane 5's storable propellant upper stage, allowing it to perform a 1 hour 46 minute ballistic "coast" phase that optimized the stage's performance to place Rosetta on the correct trajectory.
		Hyperbolic excess velocity: 3544 m/s for a target of 3545 m/s (±32)
	www.arianespace.com /site/news/releases/presrel04_3_2.html (419 words)

	Springer Online Reference Works (Site not responding. Last check: )
		Under a small perturbation a separatrix loop can turn into a limit cycle (this is one of the basic types of bifurcation for flows on the plane; see [2], [3]).
		2) In the multi-dimensional case separatrices (or separatrix manifolds) are most often stable or unstable manifolds of a hyperbolic equilibrium position or periodic trajectory (cf.
		The formulations that have been suggested are fairly complicated (see [4]), and one should not expect a complete description of different types of separatrices and of the sets formed from them.
	eom.springer.de /s/s084540.htm (479 words)

	Mars Trajectories
		The difference between the transfer velocity at Earth and Earth's orbital velocity is hyperbolic excess velocity, which is the speed that must be attained by the spacecraft upon departure from Earth's sphere of influence.
		After determining the lunar hyperbolic excess velocity and the speed necessary to overcome the lunar gravity, the DV for a launch from the moon can be calculated to be 3.25 km/s, which is even more advantageous than a launch from a parking orbit.
		If the same trajectory were to be used on the return leg, the planets would align for the return transfer only twelve days after arrival; therefore, the whole mission would be only 152 days.
	www.stanford.edu /~klynn/mars_paper.htm (4294 words)

	This weeks seminar
		Filamentation, from the Lagrangian point of view, occurs when a stable manifold of a hyperbolic trajectory enters the vortex, and this can occur without a saddle point entering the vortex.
		By looking at the hyperbolic trajectories and their stable and unstable manifolds, calculated from the numerically generated time evolving velocity field, we make several observations regarding the tripole.
		When the stable manifolds fold and wrap around the areas of negative vorticity, thereby forming a barrier for their mixing, the flow is seen to develop a quasi-steady tripole.
	www.jhu.edu /ceafm/Seminars-Spring-06/L-Barba.htm (315 words)

	Parabolic Elliptical Hyperbolic Trajectory
		For a parabolic orbit this equation simplifies to: For a hyperbolic trajectory this specific orbital energy...
		a two-periodic orbit is elliptical, parabolic or hyperbolic, the evolution...
		In real-world cases, the trajectory of a body is more likely to be elliptical (if it lacks escape velocity) or hyperbolic...
	www.elliptical-exercise.com /elliptical-exercise-8/parabolic-elliptical-hyperbolic-trajectory.html (724 words)

	The Math Forum - Math Library - Hyperbolic Geom.
		Hyperbolic tessellations shown in various stages of truncation, and represented by their Schlafli symbols.
		Krickl's diploma thesis: a list of all pentahedra that tessellate hyperbolic 3-space in the sense that they are a fundamental domain to their discrete reflection group.
		One illustrates that hyperbolic reflection in the Poincare disk corresponds to Euclidean inversion.
	mathforum.org /library/topics/hyperbolic_g (1547 words)

	CalendarHome.com - Orbit - Calendar Encyclopedia (Site not responding. Last check: )
		A comet in a parabolic or hyperbolic orbit about a central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system.
		Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about that planet.
		At a faster velocity called escape velocity, again dependent on the firing height and mass of the object, an infinite orbit such as (E) is produced — first a range of parabolic orbits, and at even faster velocities a range of hyperbolic orbits.
	encyclopedia.calendarhome.com /cgi-bin/encyclopedia.pl?p=Orbit (3226 words)

	Pluto New Horizons (Site not responding. Last check: )
		Achieving a thorough understanding of a planned trajectory is a vital portion of any satellite mission.
		The stipulation is that for a valid trajectory, the velocity at the end of the Earth to Jupiter portion of the mission and the initial velocity of the Jupiter to Pluto portion of the mission must be related such that there is a valid hyperbolic orbit about Jupiter in between the two.
		This is again a hyperbolic escape trajectory with respect to the Sun.
	ccar.colorado.edu /~nerem/zipfiles/hunkins (4237 words)

	Analysis of Trajectories (Site not responding. Last check: )
		This hyperbolic orbit around the Earth is an elliptical orbit around the Sun, with perihelion at a distance equal to Earth’s distance from the sun.
		Many fundamental concepts, as the trajectories described above, are already known; however, by analyzing the actual trajectory of the satellite and its data, scientist and engineers can further their knowledge on the subject of interplanetary trajectories.
		Some parameters for the new hyperbolic orbit with respect to the Earth are helpful in small corrections that may be needed during the transfer.
	ccar.colorado.edu /asen5050/projects/projects_2000/lemos/analysis1.htm (1390 words)

	Proxima Centauri (Gliese 551)
		It may be gravitationally bound to Alpha Centauri A and B, 0.29 light-years (13,000 AU) away, with an orbital period of 0.5 to 2 million years.
		Alternatively, it may be on an open (hyperbolic) trajectory past Alpha Centauri that will eventually take it away from the system.
		According to this idea, Proxima is an independent member of a moving group that includes Alpha Cen A and B and a number of other nearby stars.
	www.daviddarling.info /encyclopedia/P/ProximaCen.html (294 words)

	Ranger 6-7-8-9
		Two solar panel wings, each 73.9 cm wide by 153.7 cm long, extended from opposite edges of the base with a full span of 4.6 m, and a pointable high gain dish antenna was hinge mounted at one of the corners of the base away from the solar panels.
		The spacecraft encountered the lunar surface in direct motion along a hyperbolic trajectory, with an incoming asymptotic direction at an angle of -5.57 degrees from the lunar equator.
		The accuracy of the initial trajectory enabled delay of the planned mid-course correction from 22 March to 23 March when the manoeuvre was initiated at 12:03 GMT.
	www.friends-partners.org /partners/mwade/craft/ranr6789.htm (2199 words)

	Hyperbolic trajectory (Site not responding. Last check: )
		Under standard assumptions the body traveling along along hyperbolic trajctory will attain in infinity an orbital velocity called hyperbolic excess velocity ( $v_\infty\,\!) that can be computed as:$
		A square of hyperbolic excess velocity( $v_\infty\,\!) is also known as characteristic energy and denoted by C_3\,\! :$
		Under standard assumptions, specific orbital energy ( $\epsilon\,) of parabolic$ trajectory is greater than zero and the orbital energy conservation equation for this kind of trajectory takes form:
	www.infobadger.com /articles/Hyperbolic_trajectory (243 words)

	Planetary orbit - ExampleProblems.com
		Any comet in a parabolic or hyperbolic orbit about the central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system.
		Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about that planet.
		As the firing velocity is increased, the cannonball will hit the ground further (B) and further (C) away from the cannon, because while the ball is still falling towards the ground, the ground is curving away from it (see first point, above).
	www.exampleproblems.com /wiki/index.php?title=Planetary_orbit&printable=yes (2518 words)

	BIS JBIS Issue
		This paper explores what cyclic trajectories are possible within the limits of known laws of physics and practical propulsions systems for a spacecraft or world ship that could travel between the Sun and the near stars periodically.
		The minimum period found for this class of trajectory was about 41,000 years for a trip around three stars with a 16 light-year closed path and about 57,000 years for four stars with a 22 light-year path length.
		If an order of magnitude more heat flux could be handled somehow, then the spacecraft could just skim the surface of the Sun (200 km/s or 0.0007c) giving a minimum cycle time of 24,000 years for three stars and 33,000 years for fours stars.
	www.bis-spaceflight.com /sitesia.aspx/page/358/id/420/l/en (299 words)

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