# Hyperbolic trajectory

Last updated The blue path in this image is an example of a hyperbolic trajectory A hyperbolic trajectory is depicted in the bottom-right quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the hyperbolic trajectory is shown in red. The height of the kinetic energy decreases as the speed decreases and distance increases according to Kepler's laws. The part of the kinetic energy that remains above zero total energy is that associated with the hyperbolic excess velocity.

In astrodynamics or celestial mechanics, a hyperbolic trajectory is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

## Contents

Under simplistic assumptions a body traveling along this trajectory will coast towards infinity, settling to a final excess velocity relative to the central body. Similarly to parabolic trajectories, all hyperbolic trajectories are also escape trajectories. The specific energy of a hyperbolic trajectory orbit is positive.

Planetary flybys, used for gravitational slingshots, can be described within the planet's sphere of influence using hyperbolic trajectories.

## Parameters describing a hyperbolic trajectory

Like an elliptical orbit, a hyperbolic trajectory for a given system can be defined (ignoring orientation) by its semi major axis and the eccentricity. However, with a hyperbolic orbit other parameters may be more useful in understanding a body's motion. The following table lists the main parameters describing the path of body following a hyperbolic trajectory around another under standard assumptions and the formula connecting them.

Hyperbolic trajectory equations 
ElementSymbolFormulausing $v_{\infty }$ (or $a$ ), and $b$ Standard gravitational parameter $\mu \,$ ${\frac {v^{2}}{(2/r-1/a)}}$ $bv_{\infty }^{2}\cot \theta _{\infty }$ Eccentricity (>1)$e$ ${\frac {\ell }{r_{p}}}-1$ ${\sqrt {1+b^{2}/a^{2}}}$ Semi-major axis (<0)$a\,\!$ $1/(2/r-v^{2}/\mu )$ $-\mu /v_{\infty }^{2}$ Hyperbolic excess velocity$v_{\infty }$ ${\sqrt {-\mu /a}}$ (External) Angle between asymptotes$2\theta _{\infty }$ $2\cos ^{-1}(-1/e)$ $\pi +2\tan ^{-1}(b/a)$ Angle between asymptotes and the conjugate axis
of the hyperbolic path of approach
$2\nu$ $2\theta _{\infty }-\pi$ $2\sin ^{-1}{\bigg (}{\frac {1}{(1+r_{p}v_{\infty }^{2}/\mu )}}{\bigg )}$ Impact parameter (semi-minor axis)$b$ $-a{\sqrt {e^{2}-1}}$ Semi-latus rectum$\ell$ $a(e^{2}-1)$ $-b^{2}/a=h^{2}/\mu$ Periapsis distance $r_{p}$ $-a(e-1)$ ${\sqrt {a^{2}+b^{2}}}+a$ Specific orbital energy$\varepsilon$ $-\mu /2a$ $v_{\infty }^{2}/2$ Specific angular momentum$h$ ${\sqrt {\mu \ell }}$ $bv_{\infty }$ Area sweeped up per time ${\frac {\Delta A}{\Delta t}}$ ${\frac {h}{2}}$ ### Semi-major axis, energy and hyperbolic excess velocity

The semi major axis ($a\,\!$ ) is not immediately visible with an hyperbolic trajectory but can be constructed as it is the distance from periapsis to the point where the two asymptotes cross. Usually, by convention, it is negative, to keep various equations consistent with elliptical orbits.

The semi major axis is directly linked to the specific orbital energy ($\epsilon \,$ ) or characteristic energy $C_{3}$ of the orbit, and to the velocity the body attains at as the distance tends to infinity, the hyperbolic excess velocity ($v_{\infty }\,\!$ ).

$v_{\infty }^{2}=2\epsilon =C_{3}=-\mu /a$ or $a=-{\mu /{v_{\infty }^{2}}}$ where: $\mu =Gm\,\!$ is the standard gravitational parameter and $C_{3}$ is characteristic energy, commonly used in planning interplanetary missions

Note that the total energy is positive in the case of a hyperbolic trajectory (whereas it is negative for an elliptical orbit).

### Eccentricity and angle between approach and departure

With a hyperbolic trajectory the orbital eccentricity ($e\,$ ) is greater than 1. The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At $e={\sqrt {2}}$ the asymptotes are at right angles. With $e>2$ the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion approaches a straight line.

The angle between the direction of periapsis and an asymptote from the central body is the true anomaly as distance tends to infinity ($\theta _{\infty }\,$ ), so $2\theta _{\infty }\,$ is the external angle between approach and departure directions (between asymptotes). Then

$\theta {_{\infty }}=\cos ^{-1}(-1/e)\,$ or $e=-1/\cos \theta {_{\infty }}\,$ ### Impact parameter and the distance of closest approach Hyperbolic trajectories followed by objects approaching central object (small dot) with same hyperbolic excess velocity (and semi-major axis (=1)) and from same direction but with different impact parameters and eccentricities. The yellow line indeed passes around the central dot, approaching it closely.

The impact parameter is the distance by which a body, if it continued on an unperturbed path, would miss the central body at its closest approach. With bodies experiencing gravitational forces and following hyperbolic trajectories it is equal to the semi-minor axis of the hyperbola.

In the situation of a spacecraft or comet approaching a planet, the impact parameter and excess velocity will be known accurately. If the central body is known the trajectory can now be found, including how close the approaching body will be at periapsis. If this is less than planet's radius an impact should be expected. The distance of closest approach, or periapsis distance, is given by:

$r_{p}=-a(e-1)=\mu /v{_{\infty }}^{2}({\sqrt {1+(bv{_{\infty }}^{2}/\mu )^{2}}}-1)$ So if a comet approaching Earth (effective radius ~6400 km) with a velocity of 12.5 km/s (the approximate minimum approach speed of a body coming from the outer Solar System) is to avoid a collision with Earth, the impact parameter will need to be at least 8600 km, or 34% more than the Earth's radius. A body approaching Jupiter (radius 70000 km) from the outer Solar System with a speed of 5.5 km/s, will need the impact parameter to be at least 770,000 km or 11 times Jupiter radius to avoid collision.

If the mass of the central body is not known, its standard gravitational parameter, and hence its mass, can be determined by the deflection of the smaller body together with the impact parameter and approach speed. Because typically all these variables can be determined accurately, a spacecraft flyby will provide a good estimate of a body's mass.

$\mu =bv_{\infty }^{2}\tan \delta /2$ where $\delta =2\theta _{\infty }-\pi$ is the angle the smaller body is deflected from a straight line in its course.

## Equations of motion

### Position

In a hyperbolic trajectory the true anomaly $\theta$ is linked to the distance between the orbiting bodies ($r\,$ ) by the orbit equation:

$r={\frac {\ell }{1+e\cdot \cos \theta }}$ The relation between the true anomaly θ and the eccentric anomaly E (alternatively the hyperbolic anomaly H) is: 

$\cosh {E}={{\cos {\theta }+e} \over {1+e\cdot \cos {\theta }}}$ or   $\tan {\frac {\theta }{2}}={\sqrt {\frac {e+1}{e-1}}}\cdot \tanh {\frac {E}{2}}$ or  $\tanh {\frac {E}{2}}={\sqrt {\frac {e-1}{e+1}}}\cdot \tan {\frac {\theta }{2}}$ The eccentric anomaly E is related to the mean anomaly M by Kepler's equation:

$M=e\sinh E-E$ The mean anomaly is proportional to time

$M={\sqrt {\frac {\mu }{-a^{3}}}}.(t-\tau ),$ where μ is a gravitational parameter and a is the semi-major axis of the orbit.

### Flight path angle

The flight path angle (φ) is the angle between the direction of velocity and the perpendicular to the radial direction, so it is zero at periapsis and tends to 90 degrees at infinity.

$\tan(\phi )={\frac {e\cdot \sin \theta }{1+e\cdot \cos \theta }}$ ### Velocity

Under standard assumptions the orbital speed ($v\,$ ) of a body traveling along a hyperbolic trajectory can be computed from the vis-viva equation as:

$v={\sqrt {\mu \left({2 \over {r}}-{1 \over {a}}\right)}}$ where:

• $\mu \,$ is standard gravitational parameter,
• $r\,$ is radial distance of orbiting body from central body,
• $a\,\!$ is the (negative) semi-major axis.

Under standard assumptions, at any position in the orbit the following relation holds for orbital velocity ($v\,$ ), local escape velocity (${v_{esc}}\,$ ) and hyperbolic excess velocity ($v_{\infty }\,\!$ ):

$v^{2}={v_{esc}}^{2}+{v_{\infty }}^{2}$ Note that this means that a relatively small extra delta-v above that needed to accelerate to the escape speed results in a relatively large speed at infinity. For example, at a place where escape speed is 11.2 km/s, the addition of 0.4 km/s yields a hyperbolic excess speed of 3.02 km/s.

${\sqrt {11.6^{2}-11.2^{2}}}=3.02$ This is an example of the Oberth effect. The converse is also true - a body does not need to be slowed by much compared to its hyperbolic excess speed (e.g. by atmospheric drag near periapsis) for velocity to fall below escape velocity and so for the body to be captured.

A radial hyperbolic trajectory is a non-periodic trajectory on a straight line where the relative speed of the two objects always exceeds the escape velocity. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1 this is not a parabolic orbit.

## Relativistic two-body problem

In context of the two-body problem in general relativity, trajectories of objects with enough energy to escape the gravitational pull of the other, no longer are shaped like an hyperbola. Nonetheless, the term "hyperbolic trajectory" is still used to describe orbits of this type.

## Related Research Articles In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit. In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. In physics, escape velocity is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a massive body, that is, to eventually reach an infinite distance from it. Escape velocity rises with the body's mass and falls with the escaping object's distance from its center. The escape velocity thus depends on how far the object has already traveled, and its calculation at a given distance takes into account the fact that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop. Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control. In gravitationally bound systems, the orbital speed of an astronomical body or object is the speed at which it orbits around either the barycenter or, if one object is much more massive than the other bodies in the system, its speed relative to the center of mass of the most massive body. Projectile motion is a form of motion experienced by a launched object. Ballistics is the science of dynamics that deals with the flight, behavior and effects of projectiles, especially bullets, unguided bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance. In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C3 = 0 orbit (see Characteristic energy).

In astrodynamics, the characteristic energy is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length2 time−2, i.e. velocity squared, or energy per mass. In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler's orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In celestial mechanics the specific angular momentum plays a pivotal role in the analysis of the two-body problem. One can show that it is a constant vector for a given orbit under ideal conditions. This essentially proves Kepler's second law. In the gravitational two-body problem, the specific orbital energy of two orbiting bodies is the constant sum of their mutual potential energy and their total kinetic energy, divided by the reduced mass. According to the orbital energy conservation equation, it does not vary with time:

In astrodynamics an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time. 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, has an orbit that is a conic section with the central body located at one of the two foci, or the focus. Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag. In astronautics and aerospace engineering, the bi-elliptic transfer is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer maneuver.

The perifocal coordinate (PQW) system is a frame of reference for an orbit. The frame is centered at the focus of the orbit, i.e. the celestial body about which the orbit is centered. The unit vectors and lie in the plane of the orbit. is directed towards the periapsis of the orbit and has a true anomaly of 90 degrees past the periapsis. The third unit vector is the angular momentum vector and is directed orthogonal to the orbital plane such that:

The gravitational two-body problem concerns the motion of two point particles that interact only with each other, due to gravity. This means that influences from any third body are neglected. For approximate results that is often suitable. It also means that the two bodies stay clear of each other, that is, the two do not collide, and one body does not pass through the other's atmosphere. Even if they do, the theory still holds for the part of the orbit where they don't. Apart from these considerations a spherically symmetric body can be approximated by a point mass. In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination. In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.

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