en>Tom.Reding: +{{Solar mass}}

2014-12-02T03:19:24Z

+{{Solar mass}}

en>Klenze197: /* Energy transport */

2014-02-10T09:46:48Z

Energy transport

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	~~{{Astrodynamics \|Equations}}~~

	In [[astrodynamics]], the '''''vis viva'' equation''', also referred to as '''orbital energy conservation equation''', is one of the fundamental and useful equations that govern the [[motion (physics)\|motion]] of [[orbit]]ing bodies.
	~~It is the direct result of the law of conservation of energy, which requires that the sum of kinetic and potential energy is constant at all points along the orbit.~~

	~~''[[Vis viva]]'' (Latin for "live force") is a term~~ from the ~~history of mechanics~~, ~~and it survives~~ in ~~this sole context. It represents the principle that the difference between the aggregate [[Mechanical work\|work]]~~ of ~~the [[acceleration\|accelerating]] [[force]]s~~ of ~~a [[system]]~~ and ~~that of~~ the ~~retarding forces is equal~~ to ~~one half~~ the ~~''vis viva'' accumulated or lost~~ in ~~the system while the work is being done~~.		You could download from the below hyperlink, if you're in need of clash of families no charge gems, elixir and unwanted watches. You'll get the greatest secret text to get accessibility assets and endless diamonds by downloading from the subsequent links.<br><br>Interweaving social styles form that strong net in and we are all encased. When The Tygers of Pan Tang performed 'It's lonely at its top. Everybody's struggling to do you in', these people borrowed much more from clash of clans hack tool no [http://Www.Bbc.Co.uk/search/?q=feedback feedback]. A society with out having clash of clans get into tool no survey is ordinarily like a society and no knowledge, in which often it is quite awesome.<br><br>If you liked this short article and you would certainly like to obtain additional facts regarding [http://prometeu.net clash of Clans hack android] kindly check out our web page. A great clash of clans hack equipment and as a result tricks could be exceptionally handy in the labor that you are an enthusiastic player of the diversion from unwanted feelings. The finest hack apparatuses tackle just about any doing work platform - Android and Mac products iOS. You can exploit the obtain that can be used on your apple new iphone 4 or Android telephones.<br><br>Unswervingly check several distinct boutiques before purchasing a gaming. Be sure which will look both online not to mention in genuine brick moreover mortar stores in your amazing region. The costs of a video game may differ widely, particularly if a game is not brand new. By performing a hardly any additional leg work, understand it is possible to gain clash of clans.<br><br>Several ways you can variety of participants what individuals perform Clash of Clans across the world which offers you the chance and crew up with clans that have been invented by players from different nations around the world and can also have competitive towards other clans. This will produce the game considerably more interesting as you will look for a great deal of assorted strategies that might be employed by participants and this fact boosts the unpredictability section. Getting the right strategy to win is where the gamblers skills are tested, though the game is simple to play and understand.<br><br>Make sure that clients build and buy a little bit of new laboratory so you are able to research improved barbarians. Eventually, in proceedings you take part involving game for most months, you might finally obtain the nirvana of five-star barbarians.<br><br>Obviously individuals who produced this important Crack Clash of Loved ones are true fans for the sport themselves, and as well this is exactly what ensures the potency created by our alternative, because we will needed to do this task ourselves.

	==''~~Vis viva~~'' ~~equation==~~
	For any [[Kepler orbit]] ([[elliptic orbit\|elliptic]], [[parabolic trajectory\|parabolic]], [[hyperbolic trajectory\|hyperbolic]] or [[Radial trajectory\|radial]]), the ''vis viva'' equation<ref>Tom Logsdon, [http://~~books~~.~~google~~.de/~~books~~?id=~~C70gQI5ayEAC Orbital Mechanics: theory and applications~~]~~, John Wiley & Sons, 1998</ref> is as follows:~~

	~~:<math>v^2 = GM \left({{ 2 \over{r}} - {1 \over{a}}}\right)</math>~~
	~~where:~~

	*<math>v\,\!</math> is the relative speed of ~~the two bodies~~
	*<math>r\,\!</math> is ~~the distance between the two bodies~~
	*<math>a\,~~\!</math>~~ is ~~the [[semi-major axis]] (a>0 for [[ellipse]]s,~~ <~~math~~>~~a=\infty~~<~~/math~~> ~~or <math>\tfrac{1}{a}</math>=0 for [[parabola]]s,~~ and ~~a<0 for~~ [~~[hyperbola]]s)~~
	*<math>G\,\!</~~math> is the [[gravitational constant]]~~
	*<math>M</~~math> is the mass~~ of ~~the central body~~

	~~== Derivation ==~~

	~~In the vis-viva equation the mass m~~ of ~~the orbiting body (e.g.,~~ a ~~spacecraft) is taken to~~ be ~~negligible~~ in ~~comparison to~~ the ~~mass M~~ of the ~~central body (e~~.~~g., the Earth). In the specific cases of an elliptical or circular orbit, the vis~~-~~viva equation may be readily derived from conservation of energy~~ and ~~momentum~~.

	~~Specific total energy is constant throughout~~ the ~~orbit~~. ~~Thus, using the subscripts a and p to denote apoapsis (apogee) and periapsis (perigee), respectively,~~
	:<~~math~~> ~~\epsilon = \frac{v_a^2}{2} - \frac{GM}{r_a} = \frac{v_p^2}{2} - \frac{GM}{r_p}~~ <~~/math~~>

	~~Rearranging,~~

	~~:<math> \frac{v_a^2}{2} - \frac{v_p^2}{2} = \frac{GM}{r_a} - \frac{GM}{r_p} </math>~~

	~~Recalling that for an elliptical orbit (and hence also~~ a ~~circular orbit) the velocity and radius vectors are perpendicular at apoapsis and periapsis, conservation~~ of ~~angular momentum requires <math> h = r_pv_p = r_av_a = const</math>~~, ~~thus <math>v_p = \frac{r_a}{r_p}v_a</math>:~~

	~~:<math> \frac{1}{2} \left( 1-\frac{r_a^2}{r_p^2} \right) v_a^2 = \frac{GM}{r_a} - \frac{GM}{r_p} </math>~~

	~~:<math> \frac{1}{2} \left( \frac{r_p^2 - r_a^2}{r_p^2} \right) v_a^2 = \frac{GM}{r_a} - \frac{GM}{r_p} </math>~~

	~~Isolating the kinetic energy at apoapsis and simplifying~~,

	~~:<math> \frac{1}{2}v_a^2 = \left( \frac{GM}{r_a} - \frac{GM}{r_p}\right) \left( \frac{r_p^2}{r_p^2-r_a^2} \right) </math>~~

	~~:<math> \frac{1}{2}v_a^2 = GM \left( \frac{r_p - r_a}{r_ar_p} \right) \left( \frac{r_p^2}{r_p^2-r_a^2} \right) </math>~~

	~~:<math> \frac{1}{2}v_a^2 = GM \left( \frac{r_p}{r_a(r_p+r_a)} \right) </math>~~

	~~From the geometry of an ellipse, <math>2a=r_p+r_a</math> where a~~ is ~~the length~~ of ~~the semimajor axis~~. ~~Thus,~~

	:<~~math~~> ~~\frac{1}{2}v_a^2 = GM \left( \frac{2a-r_a}{r_a(2a)} \right)~~ <~~/math~~>

	~~Substituting this into our original expression for specific orbital energy,~~

	~~:<math> \epsilon = \frac{v_a^2}{2} - \frac{GM}{r_a} =GM \left( \frac{2a-r_a}{2ar_a} \right) - \frac{GM}{r_a} </math>~~

	~~:<math> \epsilon = GM \left( \frac{2a-r_a}{2ar_a} - \frac{1}{r_a} \right) = - \frac{GM}{2a} </math>~~

	~~Thus, <math> \epsilon = - \frac{GM}{2a} </math> and~~ the ~~vis-viva equation may be written~~

	~~:<math> \frac{v^2}{2} - \frac{GM}{r} = - \frac{GM}{2a} </math>~~

	or

	~~:<math> v^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right) </math>.~~

	~~==Practical applications==~~
	~~Given~~ the ~~total mass~~ and the ~~scalars ''r''~~ and ~~''v'' at~~ a ~~single point~~ of ~~the orbit, one can compute ''r''~~ and ~~''v'' at any other point in~~ the ~~orbit~~.~~<ref>For~~ the ~~[[three-body problem]] there~~ is ~~hardly a comparable vis-viva equation: conservation of energy reduces~~ the ~~larger number of [[Degrees of freedom (physics~~ and ~~chemistry)\|degrees of freedom]] by only one~~.<~~/ref~~>

	~~Given the total mass and the scalars ''r''~~ and ~~''v'' at~~ a ~~single point~~ of ~~the orbit~~, ~~one can compute~~ the ~~[[specific orbital energy]]~~ <~~math~~>~~\epsilon\,\!~~<~~/math~~>~~, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being "[[suborbital]]" (a ballistic missile,~~ for ~~example), having enough energy to be "orbital", but without~~ the ~~possibility to complete a full orbit anyway because it eventually collides with the other body~~, ~~or having enough energy to come from~~ and~~/or go to infinity (~~as ~~a meteor~~, ~~for example)~~.

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Vis-Viva Equation}}~~
	~~[[Category:Orbits]]~~
	~~[[Category:Conservation equations]]~~

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Stellar structure - Revision history

en>Tom.Reding: +{{Solar mass}}

en>Klenze197: /* Energy transport */

en>BattyBot: fixed CS1 errors: dates & General fixes using AWB (9832)

en>StringTheory11: fixed dashes using a script