# Hyperbolic trajectory

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

### Semi-major axis, energy and hyperbolic excess velocity

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

${\displaystyle v_{\infty }^{2}=2\epsilon =C_{3}=-\mu /a}$ or ${\displaystyle a=-{\mu /{v_{\infty }^{2}}}}$

where: ${\displaystyle \mu =Gm\,\!}$ is the standard gravitational parameter and ${\displaystyle C_{3}}$ 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 (${\displaystyle e\,}$) 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 ${\displaystyle e={\sqrt {2}}}$ the asymptotes are at right angles. With ${\displaystyle e>2}$ 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 (${\displaystyle \theta _{\infty }\,}$), so ${\displaystyle 2\theta _{\infty }\,}$ is the external angle between approach and departure directions (between asymptotes). Then

${\displaystyle \theta {_{\infty }}=\cos ^{-1}(-1/e)\,}$ or ${\displaystyle e=-1/\cos \theta {_{\infty }}\,}$

### Impact parameter and the distance of closest approach

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:

${\displaystyle r_{p}=-a(e-1)={\frac {\mu }{v_{\infty }^{2}}}\left({\sqrt {1+\left(b{\frac {v_{\infty }^{2}}{\mu }}\right)^{2}}}-1\right)}$

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.

${\displaystyle \mu =bv_{\infty }^{2}\tan \delta /2}$ where ${\displaystyle \delta =2\theta _{\infty }-\pi }$ 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 ${\displaystyle \theta }$ is linked to the distance between the orbiting bodies (${\displaystyle r\,}$) by the orbit equation:

${\displaystyle r={\frac {\ell }{1+e\cdot \cos \theta }}}$

The relation between the true anomaly θ and the eccentric anomaly E (alternatively the hyperbolic anomaly H) is: [3]

${\displaystyle \cosh {E}={{\cos {\theta }+e} \over {1+e\cdot \cos {\theta }}}}$   or   ${\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {e+1}{e-1}}}\cdot \tanh {\frac {E}{2}}}$   or  ${\displaystyle \tanh {\frac {E}{2}}={\sqrt {\frac {e-1}{e+1}}}\cdot \tan {\frac {\theta }{2}}}$

The eccentric anomaly E is related to the mean anomaly M by Kepler's equation:

${\displaystyle M=e\sinh E-E}$

The mean anomaly is proportional to time

${\displaystyle M={\sqrt {\frac {\mu }{-a^{3}}}}.(t-\tau ),}$ 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.

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

### Velocity

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

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

where:

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

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

${\displaystyle v^{2}={v_{esc}}^{2}+{v_{\infty }}^{2}}$

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.

${\displaystyle {\sqrt {11.6^{2}-11.2^{2}}}=3.02}$

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.

## Deflection with finite sphere of influence

A more accurate formula for the deflection angle ${\displaystyle \delta }$ considering the sphere of influence radius ${\displaystyle R_{\text{SOI}}}$ of the deflecting body, assuming a periapsis ${\displaystyle p_{e}}$ is:

${\displaystyle \delta =2\arcsin \left({\frac {{\sqrt {1-{\frac {p_{e}}{R_{\text{SOI}}}}}}{\sqrt {1+{\frac {p_{e}}{R_{\text{SOI}}}}-{\frac {2\mu p_{e}}{v_{\infty }^{2}R_{\text{SOI}}^{2}}}}}}{1+{\frac {v_{\infty }^{2}p_{e}}{\mu }}-{\frac {2p_{e}}{R_{\text{SOI}}}}}}\right)}$

## 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 a hyperbola. Nonetheless, the term "hyperbolic trajectory" is still used to describe orbits of this type.

## Related Research Articles

In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. 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 celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction. The escape speed is independent of the mass of the escaping object, but increases with the mass of the primary body; it decreases with the distance from the primary body, thus taking into account how far the object has already traveled. Its calculation at a given distance means that no acceleration is further needed for the object to escape: it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop. On the other hand, an object already at escape speed needs slowing for it to be captured by the gravitational influence of the body.

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In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical two-body problem. It is the angular distance from the pericenter which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit.

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In celestial mechanics, the specific relative angular momentum of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative position and relative linear momentum, divided by the mass of the body in question.

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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.

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## References

• Vallado, David A. (2007). Fundamentals of Astrodynamics and Applications, Third Edition. Hawthorne, CA.: Hawthorne Press. ISBN   978-1-881883-14-2.
1. S.O., Kepler; Saraiva, Maria de Fátima (2014). Astronomia e Astrofísica. Porto Alegre: Department of Astronomy - Institute of Physics of Federal University of Rio Grande do Sul. pp. 97–106.
2. "Basics of Space Flight: Orbital Mechanics". Archived from the original on 2012-02-04. Retrieved 2012-02-28.
3. Peet, Matthew M. (13 June 2019). "Spacecraft Dynamics and Control" (PDF).