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{{Unreferenced|date=August 2009}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
[[File:OrbitalEccentricityDemo.svg|thumb|right|The green path in this image is an example of a parabolic trajectory.]]
[[File:Gravity Wells Potential Plus Kinetic Energy - Circle-Ellipse-Parabola-Hyperbola.png|thumb|250px|A parabolic trajectory is depicted in the bottom-left quadrant of this diagram, where the [[gravity well|gravitational potential well]] of the central mass shows potential energy, and the kinetic energy of the parabolic trajectory is shown in red. The height of the kinetic energy decreases asymptotically toward zero as the speed decreases and distance increases according to Kepler's laws.]]
{{about|a class of Kepler orbits|a free body trajectory at constant gravity|Ballistic trajectory}}
In [[astrodynamics]] or [[celestial mechanics]] a '''parabolic trajectory''' is a [[Kepler orbit]] with the [[Orbital eccentricity|eccentricity]] equal to 1. When moving away from the source it is called an '''escape orbit''', otherwise a '''capture orbit'''. It is also sometimes referred to as a '''C<sub>3</sub>&nbsp;=&nbsp;0 orbit'''.


Under standard assumptions a body traveling along an escape orbit will [[orbital coast|coast]] along a [[Parabola|parabolic]] shaped trajectory to infinity, with velocity relative to the [[central body]] tending to zero, and therefore will never return. Parabolic trajectory are minimum-energy escape trajectories, separating positive-[[characteristic energy|energy]] [[hyperbolic trajectory|hyperbolic trajectories]] from negative-energy [[elliptic orbit]]s.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


==Velocity==
Registered users will be able to choose between the following three rendering modes:  
Under standard assumptions the [[orbital velocity]] (<math>v\,</math>) of a body travelling along parabolic trajectory can be computed as:
:<math>v=\sqrt{2\mu\over{r}}</math>
where:
*<math>r\,</math> is the radial distance of orbiting body from [[central body]],
*<math>\mu\,</math> is the [[standard gravitational parameter]].


At any position the orbiting body has the [[escape velocity]] for that position.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


If the body has the escape velocity with respect to the Earth, this is not enough to escape the Solar System, so near the Earth the orbit resembles a parabola, but further away it bends into an elliptical orbit around the Sun.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


This velocity (<math>v\,</math>) is closely related to the [[orbital velocity]] of a body in a [[circular orbit]] of the radius equal to the radial position of orbiting body on the parabolic trajectory:
'''source'''
:<math>v=\sqrt{2}\cdot v_o</math>
:<math forcemathmode="source">E=mc^2</math> -->
where:
*<math>v_o\,</math> is [[orbital velocity]] of a body in [[circular orbit]].


==Equation of motion==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
Under standard assumptions, for a body moving along this kind of [[orbit|trajectory]] an [[orbital equation]] becomes:
:<math>r={{h^2}\over{\mu}}{{1}\over{1+\cos\nu}}</math>
where:
*<math>r\,</math> is radial distance of orbiting body from [[central body]],
*<math>h\,</math> is [[specific angular momentum]] of the [[orbiting body]],
*<math>\nu\,</math> is a [[true anomaly]] of the orbiting body,
*<math>\mu\,</math> is the [[standard gravitational parameter]].


==Energy==
==Demos==
Under standard assumptions, [[specific orbital energy]] (<math>\epsilon\,</math>) of parabolic trajectory is zero, so the [[orbital energy conservation equation]] for this trajectory takes form:
:<math>\epsilon={v^2\over2}-{\mu\over{r}}=0</math>
where:
*<math>v\,</math> is [[orbital velocity]] of orbiting body,
*<math>r\,</math> is radial distance of orbiting body from [[central body]],
*<math>\mu\,</math> is the [[standard gravitational parameter]].


This is entirely equivalent to the [[characteristic energy]] (square of the speed at infinity) being 0:
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


:<math>C_3 = 0</math>


==Barker's equation==
* accessibility:
Barker's equation relates the time of flight to the true anomaly of a parabolic trajectory.<ref>{{cite book
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
| last1 = Bate
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
| first1 = Roger
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
| last2 = Mueller
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
| first2 = Donald
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
| last3 = White
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
| first3 = Jerry
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
| title = Fundamentals of Astrodynamics
| publisher = Dover Publications, Inc., New York
| year = 1971
| ISBN = 0-486-60061-0}} p 188</ref>


<math>
==Test pages ==
t - T = \frac{1}{2}\sqrt{\frac{p^{3}}{\mu}}\left (D + \frac{1}{3}D^{3}\right )
</math>


Where:
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*D = tan(ν/2), ν is the true anomaly of the orbit
*[[Displaystyle]]
*t is the current time in seconds
*[[MathAxisAlignment]]
*T is the time of periapsis passage in seconds
*[[Styling]]
*μ is the standard gravitational parameter
*[[Linebreaking]]
*p is the [[Conic section#Features|semi-latus rectum]] of the trajectory ( p = h<sup>2</sup>/μ )
*[[Unique Ids]]
*[[Help:Formula]]


More generally, the time between any two points on an orbit is
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
<math>
==Bug reporting==
t_{f} - t_{0} = \frac{1}{2}\sqrt{\frac{p^{3}}{\mu}}\left (D_{f} + \frac{1}{3}D_{f}^{3} - D_{0} - \frac{1}{3}D_{0}^{3}\right )
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
</math>
 
 
Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit r<sub>p</sub> = p/2:
 
<math>
t - T = \sqrt{\frac{2r_{p}^{3}}{\mu}}\left (D + \frac{1}{3}D^{3}\right )
</math>
 
Unlike [[Kepler's equation]], which is used to solve for true anomalies in elliptical and hyperbolic trajectories, the true anomaly in Barker's equation can be solved directly for t. If the following substitutions are made<ref>
{{cite book
| last1 = Montenbruck
| first1 = Oliver
| last2 = Pfleger
| first2 = Thomas
| title = Astronomy on the Personal Computer
| publisher = Springer-Verlag Berlin Heidelberg
| year = 2009
| ISBN = 978-3-540-67221-0}} p 64</ref>
 
<math>
A = \frac{3}{2}\sqrt{\frac{\mu}{2r_{p}}}(t-T)
</math>
 
<math>
B = \sqrt[3]{A + \sqrt{A^{2}+1}}
</math>
 
then
 
<math>
\nu = 2\arctan(B - 1/B)
</math>
 
==Radial parabolic trajectory==
A radial parabolic trajectory is a non-periodic [[Radial_trajectory|trajectory on a straight line]] where the relative velocity of the two objects is always the [[escape velocity]]. There are two cases: the bodies move away from each other or towards each other.
 
There is a rather simple expression for the position as function of time:
 
:<math>r=(4.5\mu t^2)^{1/3}\!\,</math>
 
where
* μ is the [[standard gravitational parameter]]
* <math>t=0\!\,</math> corresponds to the extrapolated time of the fictitious starting or ending at the center of the central body.
 
At any time the average speed from <math>t=0\!\,</math> is 1.5 times the current speed, i.e. 1.5 times the local escape velocity.
 
To have <math>t=0\!\,</math> at the surface, apply a time shift; for the Earth (and any other spherically symmetric body with the same average density) as central body this time shift is 6 minutes and 20 seconds; seven of these periods later the height above the surface is three times the radius, etc.
 
==See also==
* [[Kepler orbit]]
 
==References==
{{Reflist}}
 
<!-- there is a navbox, such list was redundant -->
{{orbits}}
 
{{DEFAULTSORT:Parabolic Trajectory}}
 
[[Category:Orbits]]
 
[[it:Traiettoria parabolica]]
[[ja:放物線軌道]]
[[pt:Trajetória parabólica]]
[[tr:Parabolik yörünge]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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