Inastrodynamics orcelestial mechanics, ahyperbolic trajectory orhyperbolic orbit (fromNewtonian theory:hyperbola shape) is the trajectory of any object around acentral body with enough velocity to escape the central object's gravitational field; expressed asorbital eccentricity designated by any number more than 1.

Under simplistic assumptions a body traveling along this trajectory will coast towards infinity, settling to a final excess velocity relative to the central body. As withparabolic trajectories, all hyperbolic trajectories are alsoescape trajectories. Thespecific energy of a hyperbolic trajectory orbit is positive.

Planetary flybys, used forgravitational slingshots, can be described within the planet'ssphere of influence using hyperbolic trajectories.

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]

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

The semi major axis (${\displaystyle a}$) is not immediately visible with a 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 thespecific orbital energy (${\displaystyle ϵ}$) orcharacteristic 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 (${\displaystyle v_{\mathrm{\infty }}}$).

${\displaystyle v_{\mathrm{\infty }}^2=2ϵ=C_3=-\mu /a}$ or${\displaystyle a=-\mu /v_{\mathrm{\infty }}^2}$

where:${\displaystyle \mu =Gm}$ is thestandard 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).

With a hyperbolic trajectory theorbital eccentricity 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 thetrue anomaly as distance tends to infinity (${\displaystyle {\theta }_{\mathrm{\infty }}}$), so${\displaystyle 2{\theta }_{\mathrm{\infty }}}$ is the external angle between approach and departure directions (between asymptotes). Then

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

Theimpact parameter is the distance by which a body, if it continued on an unperturbed path, would miss the central body at itsclosest 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 the planet's radius an impact should be expected. The distance of closest approach, or periapsis distance, is given by:

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

So if a comet approachingEarth (effective radius ~6400 km) with a velocity of 12.5 km/s (the approximate minimum approach speed of a body coming from the outerSolar 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 approachingJupiter (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.

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

In a hyperbolic trajectory thetrue anomaly${\displaystyle \theta }$ is linked to the distance between the orbiting bodies (${\displaystyle r}$) by theorbit equation:

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

The relation between the true anomalyθ and theeccentric anomalyE (alternatively the hyperbolic anomalyH) is:^[3]

${\displaystyle \cosh E=\frac{\cos \theta +e}{1+e\cdot \cos \theta }}$     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 anomalyE is related to themean anomalyM byKepler's equation:

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

The mean anomaly is proportional to time

${\displaystyle M=\sqrt{\frac{\mu }{-a^3}}.(t-\tau ),}$ whereμ is agravitational parameter anda is thesemi-major axis of the orbit.

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.

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

Under standard assumptions theorbital speed (${\displaystyle v}$) of a body traveling along ahyperbolic trajectory can be computed from thevis-viva equation as:

${\displaystyle v=\sqrt{\mu \left(\frac{2}{r}+\frac{1}{a}\right)}}$^[4]

where:

• ${\displaystyle \mu }$ isstandard gravitational parameter,
• ${\displaystyle r}$ is radial distance of orbiting body fromcentral body,
• ${\displaystyle a}$ is the absolute value (distance) of thesemi-major axis.

Under standard assumptions, at any position in the orbit the following relation holds fororbital velocity (${\displaystyle v}$), localescape velocity (${\displaystyle v_{esc}}$) and hyperbolic excess velocity (${\displaystyle v_{\mathrm{\infty }}}$):

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

Note that this means that a relatively small extradelta-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 theOberth 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-periodictrajectory on a straight line where the relative speed of the two objects always exceeds theescape 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.

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:

${\displaystyle \delta =2\arcsin \left(\frac{\sqrt{1-\frac{p_e}{R_{\text{SOI}}}}\sqrt{1+\frac{p_e}{R_{\text{SOI}}}-\frac{2\mu p_e}{v_{\mathrm{\infty }}^2{R}_{\text{SOI}}^2}}}{1+\frac{v_{\mathrm{\infty }}^2{p}_e}{\mu }-\frac{2p_e}{R_{\text{SOI}}}}\right)}$

In context of thetwo-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.

• Orbit
• Orbit equation
• Kepler orbit
• List of orbits
• Planetary flyby
• Hyperbolic asteroid
• List of hyperbolic comets

• Trajectories
• Orbits
• Hyperbolic

• Orbits

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