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| In the [[gravitational two-body problem]], the '''specific [[orbit]]al energy''' <math>\epsilon\,\!</math> (or '''vis-viva energy''') of two [[orbiting body|orbiting bodies]] is the constant sum of their mutual [[potential energy]] (<math>\epsilon_p\,\!</math>) and their total [[kinetic energy]] (<math>\epsilon_k\,\!</math>), divided by the [[reduced mass]]. According to the [[Vis-viva equation|orbital energy conservation equation]] (also referred to as '''vis-viva equation)''', it does not vary with time:
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| :<math> \epsilon = \epsilon_k+\epsilon_p \!</math>
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| :<math> \epsilon = {v^2\over{2}}-{\mu\over{r}} = -{1\over{2}}{ \mu^2\over{h^2}}\left(1-e^2\right) = -\frac{\mu }{2a} </math>
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| where
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| *<math>v\,\!</math> is the relative [[orbital speed]];
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| *<math>r\,\!</math> is the [[orbital position vector|orbital distance]] between the bodies;
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| *<math>\mu = {G}(m_1 + m_2)\,\!</math> is the sum of the [[standard gravitational parameter]]s of the bodies;
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| *<math>h\,\!</math> is the [[specific relative angular momentum]] in the sense of [[relative angular momentum]] divided by the reduced mass;
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| *<math>e\,\!</math> is the [[eccentricity (orbit)|orbital eccentricity]];
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| *<math>a\,\!</math> is the [[semi-major axis]].
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| It is expressed in J/kg = m<sup>2</sup>[[second|s]]<sup>−2</sup> or
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| MJ/kg = km<sup>2</sup>s<sup>−2</sup>. For an [[elliptical orbit]] the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to [[escape velocity]] ([[parabolic orbit]]). For a [[hyperbolic orbit]], it is equal to the excess energy compared to that of a parabolic orbit. In this case the specific orbital energy is also referred to as [[characteristic energy]].
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| ==Equation forms for different orbits==
| | Our world is driven by existing plus demand. My husband and i shall examine the Greek-Roman model. Consuming special care to highlight the task of clash of clans hack tool no survey within the vast system which usually this materials.<br><br>Replacing from band blueprint to positively a besprinkle blueprint gives some sort of supplied authentic picture. The accumbent time arbor will be actually scaled evenly. But it's adamantine to be able to acquaint what's activity now inside bottom-left bend now. An ethics are so bunched up you simply can't [http://Www.Bing.com/search?q=acquaint&form=MSNNWS&mkt=en-us&pq=acquaint acquaint] them afar yeast infection.<br><br>Pay attention to a game's evaluation when purchasing a present. This evaluation will allow you to learn what age level clash of clans hack tool is right for and will reveal to you when the sport can violent. It can help you figure out whether you have to buy the sport.<br><br>A marvelous method to please children with a gaming circle and ensure they [http://Statigr.am/tag/continue continue] to exist fit is to purchase a Wii. This system needs real task to play. Your children won't be put for hours on end playing clash of clans hack. They are moving around as easy methods to play the games on this particular system.<br><br>We each can use this approach to acquisition the discount of any time anywhere between 1hr and one special day. For archetype to acquisition the majority of dispatch up 4 a too long time, acting x = 15, 400 abnormal or you receive y = 51 gems.<br><br>Purchase are playing a wearing game, and you don't have any experience with it, set the difficulty even to rookie. This is considered help you pick rising on the unique showcases of the game learn your way close by the field. If in case you set it large than that, you commonly tend to get frustrated plus not have any awesome.<br><br>Bide time until game of the month versions of major titles. These often come out a year or maybe more right after the original title, but comprise lots of the downloadable and extra content which had been released in stages had been first title. If you loved this posting and you would like to receive far more details regarding [http://prometeu.net clash of clans hack tool download no survey] kindly pay a visit to our web site. Involving games offer a much bang for the cent. |
| For an [[elliptical orbit]], the specific orbital energy equation, when combined with [[Specific relative angular momentum|conservation of specific angular momentum]] at one of the orbit's [[apsis|apsides]], simplifies to:<ref name="Bong Wie SVDC">{{cite book|last=Wie|first=Bong|title=Space Vehicle Dynamics and Control|publisher=[[American Institute of Aeronautics and Astronautics]]|location=[[Reston, Virginia]]|year=1998|series=AIAA Education Series|chapter=Orbital Dynamics|isbn=1-56347-261-9|accessdate=2008-11-09|page=220}}</ref>
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| :<math>\epsilon = -{\mu \over{2a}}\,\!</math>
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| where
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| *<math>\mu={G}(m_1 + m_2)\,\!</math> is the [[standard gravitational parameter]];
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| *<math>{a}\,\!</math> is [[semi-major axis]] of the orbit.
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| Proof:
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| ::For an elliptical orbit with [[specific angular momentum]] ''h'' given by
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| ::<math>{h^2} = \mu p = \mu a (1-e^2)\,\!</math> | |
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| ::we use the general form of the specific orbital energy equation,
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| ::<math>\epsilon={v^2\over{2}}-{\mu\over{r}}</math>
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| ::with the relation that the relative velocity at [[periapsis]] is
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| ::<math>{v_p^2} = {h^2 \over{{r_p}}^2} = {h^2 \over{{a^2(1-e)}}^2} = {\mu a (1-e^2) \over{{a^2(1-e)}}^2} = {\mu (1-e^2) \over{{a(1-e)}}^2}\,\!</math>
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| ::Thus our specific orbital energy equation becomes
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| ::<math>\epsilon = {\mu \over{a}} {\left [ {(1-e^2) \over{{2(1-e)}}^2} - {1 \over{(1-e)}} \right ] } = {\mu \over{a}} {\left [ {{(1-e)(1+e)} \over{{2(1-e)}}^2} - {1 \over{(1-e)}} \right ] } = {\mu \over{a}} {\left [ {(1+e) \over{{2(1-e)}}} - {2 \over{2(1-e)}} \right ] } = {\mu \over{a}} {\left [ {{e-1} \over{2(1-e)}} \right ] }\,\!</math>
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| ::and finally with the last simplification we obtain:
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| ::<math>\epsilon = -{\mu \over{2a}}\,\!</math>
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| For a [[parabolic orbit]] this equation simplifies to
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| :<math>\epsilon=0\,\!.</math>
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| For a [[hyperbolic trajectory]] this specific orbital energy is either given by
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| :<math>\epsilon = {\mu \over{2a}}\,\!.</math>
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| or the same as for an ellipse, depending on the convention for the sign of ''a''.
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| In this case the specific orbital energy is also referred to as [[characteristic energy]] (or <math>C_3\,\!</math>) and is equal to the excess specific energy compared to that for a parabolic orbit.
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| It is related to the hyperbolic excess velocity <math>v_{\infty} \,\!</math> (the [[Kinetic energy|orbital velocity]] at infinity) by | |
| :<math>2\epsilon=C_3=v_{\infty}^2\,\!.</math>
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| It is relevant for interplanetary missions.
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| Thus, if [[orbital position vector]] (<math>\mathbf{r}\,\!</math>) and [[orbital velocity vector]] (<math>\mathbf{v}\,\!</math>) are known at one position, and <math>\mu\,\!</math> is known, then the energy can be computed and from that, for any other position, the orbital speed.
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| ==Rate of change==
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| For an elliptical orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is
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| :<math>\frac{\mu}{2a^2}\,\!</math>
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| where
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| *<math> \mu={G}(m_1 + m_2)\,\!</math> is the [[standard gravitational parameter]];
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| *<math>a\,\!</math> is [[semi-major axis]] of the orbit.
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| In the case of circular orbits, this rate is one half of the gravity at the orbit. This corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy.
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| ==Additional energy==
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| If the central body has radius ''R'', then the additional energy of an elliptic orbit compared to being stationary at the surface is
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| <math>\ -\frac{\mu}{2a}+\frac{\mu}{R} = \frac{\mu (2a-R)}{2aR}.</math>
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| *For the Earth and ''a'' just little more than <math>R/2</math> this is <math>(2a-R)g</math> ; the quantity <math>2a-R</math> is the height the ellipse extends above the surface, plus the [[periapsis distance]] (the distance the ellipse extends beyond the center of the Earth); the latter times ''g'' is the kinetic energy of the horizontal component of the velocity.
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| ==Examples==<!-- This section is linked from [[Low Earth orbit]] -->
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| ===ISS===
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| The [[International Space Station]] has an orbital period of 91.74 minutes (5.5 ks), hence the semi-major axis is 6,738 km.
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| The energy is −29.6 MJ/kg: the potential energy is −59.2 MJ/kg, and the kinetic energy 29.6 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 3.4 MJ/kg, the total extra energy is 33.0 MJ/kg. The average speed is 7.7 km/s, the net [[delta-v]] to reach this orbit is 8.1 km/s (the actual delta-v is typically 1.5–2 km/s more for [[atmospheric drag]] and [[gravity drag]]).
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| The increase per meter would be 4.4 J/kg; this rate corresponds to one half of the local gravity of 8.8 m/s².
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| For an altitude of 100 km (radius is 6471 km): | |
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| The energy is −30.8 MJ/kg: the potential energy is −61.6 MJ/kg, and the kinetic energy 30.8 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 1.0 MJ/kg, the total extra energy is 31.8 MJ/kg.
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| The increase per meter would be 4.8 J/kg; this rate corresponds to one half of the local gravity of 9.5 m/s². The speed is 7.8 km/s, the net delta-v to reach this orbit is 8.0 km/s.
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| Taking into account the rotation of the Earth, the delta-v is up to 0.46 km/s less (starting at the equator and going east) or more (if going west).
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| ===Voyager 1===
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| For [[Voyager 1]], with respect to the Sun:
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| *<math>\mu = GM\,\!</math> = 132,712,440,018 km<sup>3</sup>s<sup>−2</sup> is the [[standard gravitational parameter]] of the Sun
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| *r = 17 [[1000000000 (number)|billion]] kilometers
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| *v = 17.1 km/s
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| Hence:
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| * <math>\epsilon=\epsilon_k+\epsilon_p={v^2\over{2}}-{\mu\over{r}}
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| =</math> 146 km<sup>2</sup>s<sup>−2</sup> - 8 km<sup>2</sup>s<sup>−2</sup> = 138 km<sup>2</sup>s<sup>−2</sup>
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| Thus the hyperbolic excess velocity (the theoretical [[Kinetic energy|orbital velocity]] at infinity) is given by
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| :<math>v_{\infty}=\,\!</math> 16.6 km/s
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| However, Voyager 1 does not have enough velocity to leave the [[Milky Way]]. The computed speed applies far away from the Sun, but at such a position that the potential energy with respect to the Milky Way as a whole has changed negligibly, and only if there is no strong interaction with celestial bodies other than the Sun.
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| ==Applying thrust==
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| Assume:
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| *'''a''' is the acceleration due to [[thrust]] (the time-rate at which [[delta-v]] is spent)
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| *'''g''' is the gravitational field strength
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| *'''v''' is the velocity of the rocket
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| Then the time-rate of change of the specific energy of the rocket is <math> \mathbf{v} \cdot \mathbf{a}</math>: an amount <math>\mathbf{v} \cdot (\mathbf{a}-\mathbf{g})</math> for the kinetic energy and an amount <math>\mathbf{v} \cdot \mathbf{g}</math> for the potential energy.
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| The change of the specific energy of the rocket per unit change of delta-v is
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| :<math>\frac{\mathbf{v \cdot a}}{|\mathbf{a}|}</math>
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| which is |'''v'''| times the cosine of the angle between '''v''' and '''a'''.
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| Thus, when applying delta-v to increase specific orbital energy, this is done most efficiently if '''a''' is applied in the direction of '''v''', and when |'''v'''| is large. If the angle between '''v''' and '''g''' is obtuse, for example in a launch and in a transfer to a higher orbit, this means applying the delta-v as early as possible and at full capacity. See also [[gravity drag]]. When passing by a celestial body it means applying thrust when nearest to the body. When gradually making an elliptic orbit larger, it means applying thrust each time when near the periapsis.
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| When applying delta-v to ''decrease'' specific orbital energy, this is done most efficiently if '''a''' is applied in the direction opposite to that of '''v''', and again when |'''v'''| is large. If the angle between '''v''' and '''g''' is acute, for example in a landing (on a celestial body without atmosphere) and in a transfer to a circular orbit around a celestial body when arriving from outside, this means applying the delta-v as late as possible. When passing by a planet it means applying thrust when nearest to the planet. When gradually making an elliptic orbit smaller, it means applying thrust each time when near the periapsis.
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| If '''a''' is in the direction of '''v''':
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| :<math>\Delta \epsilon = \int v\, d (\Delta v) = \int v\, a dt</math>
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| {{earth orbits}}
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| ==See also==
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| *[[Tsiolkovsky rocket equation#Energy|Specific energy change of rockets]]
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| ==References==
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| {{reflist}}
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| {{orbits}}
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| {{Voyager program}}
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| [[Category:Astrodynamics]]
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| [[Category:Orbits]]
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| [[Category:Voyager program]]
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