Editing Hyperbolic trajectory (section)

==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.

{| class="wikitable"
|-
|+ Hyperbolic trajectory equations <ref>{{Cite book|last1=S.O.|first1=Kepler|title=Astronomia e Astrofísica|last2=Saraiva|first2=Maria de Fátima|publisher=Department of Astronomy - Institute of Physics of Federal University of Rio Grande do Sul|year=2014|location=Porto Alegre|pages=97–106}}</ref>
! Element!! Symbol !! Formula!! using <math>v_\infty</math> (or <math>a</math>), and <math>b</math> 
|-
| [[Standard gravitational parameter]] || |<math>\mu\,</math> || <math>\frac{v^2}{(2/r-1/a)}</math> || <math>b v_\infty^2 \cot \theta_\infty </math>
|-
| [[Orbital eccentricity|Eccentricity]] (>1) || <math>e</math> || <math>\frac{\ell}{r_p} -1 </math>|| <math>\sqrt{1+b^2/a^2}</math>
|-
| [[Semi-major axis]] (<0)|| <math>a\,\!</math> || <math>1/(2/r-v^2/\mu)</math>|| <math>-\mu/v_\infty^2</math>
|-
| Hyperbolic excess velocity || <math>v_\infty</math> || <math>\sqrt{-\mu/a}</math> ||
|-
| (External) Angle between asymptotes || <math>2\theta_\infty</math> || <math>2 \cos^{-1}(-1/e)</math> || <math>\pi + 2 \tan^{-1}(b/a)</math><ref>{{Cite web |url=http://www.braeunig.us/space/orbmech.htm#hyperbolic |title=Basics of Space Flight: Orbital Mechanics |access-date=2012-02-28 |archive-url=https://web.archive.org/web/20120204054322/http://www.braeunig.us/space/orbmech.htm#hyperbolic |archive-date=2012-02-04 |url-status=dead }}</ref>
|- 
| Angle between asymptotes and the conjugate axis <br /> of the hyperbolic path of approach || <math>2\nu</math> || <math> 2\theta_\infty - \pi </math>  || <math> 2\sin^{-1}\bigg(\frac{1}{(1 + r_pv_\infty^2/\mu)}\bigg)</math>
|- 
| [[Impact parameter]] ([[semi-minor axis]]) || <math>b</math> || <math> -a \sqrt{e^2-1}</math> || <math></math> 
|-
| [[Semi-latus rectum]] || <math>\ell</math> || <math>a (1-e^2)</math> || <math> -b^2/a = h^2/\mu</math>
|- 
| [[Apsis|Periapsis distance]] || <math>r_p</math> ||<math>-a(e-1)</math> || <math>\sqrt{a^2+b^2}+a</math> 
|-
| [[Specific orbital energy]] || <math>\varepsilon</math> || <math>-\mu/2a</math> ||<math>v_\infty^2/2</math> 
|-
| [[Specific angular momentum]] || <math>h</math> || <math>\sqrt{\mu  \ell}</math> || <math>b v_\infty</math> 
|-
|[[Kepler's laws of planetary motion|Area swept up per time]]
|<math>\frac{\Delta A}{\Delta t}</math>
|<math>\frac{h}{2}</math>
|
|}

===Semi-major axis, energy and hyperbolic excess velocity===
{{see also|Characteristic energy}}

The semi major axis (<math>a\,\!</math>) 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 the [[specific orbital energy]] (<math>\epsilon\,</math>) or [[characteristic energy]] <math>C_3</math> of the orbit, and to the velocity the body attains at as the distance tends to infinity, the hyperbolic excess velocity (<math>v_\infty\,\!</math>). 
:<math>v_{\infty}^2=2\epsilon=C_3=-\mu/a</math>  or <math>a=-{\mu/{v_\infty^2}}</math>
where: <math>\mu=Gm\,\!</math> is the [[standard gravitational parameter]] and <math>C_3</math> 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]] (<math>e\,</math>) 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 <math>e=\sqrt 2</math> the asymptotes are at right angles. With  <math>e>2</math> 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 (<math>\theta_\infty\,</math>), so <math>2\theta_\infty\,</math> is the external angle between approach and departure directions (between asymptotes). Then

:<math>\theta{_\infty}=\cos^{-1}(-1/e)\,</math> or <math>e=-1/\cos\theta{_\infty}\,</math>

=== Impact parameter and the distance of closest approach {{anchor|Impact parameter|Closest approach}}===
[[File:Hyperbolic trajectories with different impact parameters.png|thumb|upright=1.5|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:
 
:<math>r_p = -a(e-1)= \frac{\mu}{v_\infty^2} \left(\sqrt{1 + \left(b \frac {v_\infty^2}{\mu}\right)^2} - 1\right)</math>

So if a comet approaching [[Earth]] (effective radius ~6400&nbsp;km) with a velocity of 12.5&nbsp;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&nbsp;km, or 34% more than the Earth's radius. A body approaching [[Jupiter]] (radius 70000&nbsp;km) from the outer Solar System with a speed of 5.5&nbsp;km/s, will need the impact parameter to be at least 770,000&nbsp;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.

:<math>\mu=b v_\infty^2 \tan \delta/2</math> where <math> \delta = 2\theta_\infty - \pi </math> is the angle the smaller body is deflected from a straight line in its course.