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Escape velocity
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== Calculation == Escape speed at a distance ''d'' from the center of a spherically symmetric primary body (such as a star or a planet) with mass ''M'' is given by the formula<ref>{{cite book |title=New Understanding Physics for Advanced Level |author1=Jim Breithaupt |edition=illustrated |publisher=Nelson Thornes |year=2000 |isbn=978-0-7487-4314-8 |page=231 |url=https://books.google.com/books?id=r8I1gyNNKnoC}} [https://books.google.com/books?id=r8I1gyNNKnoC&pg=PA231 Extract of page 231]</ref><ref>{{cite book |title=Black Holes: A Very Short Introduction |author1=Katherine Blundell |edition=illustrated |publisher=Oxford University Press |year=2015 |isbn=978-0-19-960266-7 |page=4 |url=https://books.google.com/books?id=13PNCgAAQBAJ}} [https://books.google.com/books?id=13PNCgAAQBAJ&pg=PA4 Extract of page 4]</ref> : <math>v_\text{e} = \sqrt{\frac{2GM}{d}} = \sqrt{2gd\ }</math> where: * ''G'' is the [[gravitational constant|universal gravitational constant]] ({{nowrap|''G'' ≈ {{physconst|G|round=2}}}}) * ''g'' = ''GM''/''d''<sup>2</sup> is the local gravitational acceleration (or the [[surface gravity]], when {{nowrap|1=''d'' = ''r''}}). The value ''GM'' is called the [[standard gravitational parameter]], or ''μ'', and is often known more accurately than either ''G'' or ''M'' separately. When given an initial speed ''V'' greater than the escape speed ''v''{{sub|e}}, the object will asymptotically approach the ''[[Hyperbolic trajectory|hyperbolic excess speed]]'' ''v''{{sub|∞}}, satisfying the equation:<ref>{{Cite book |last1=Bate |first1=Roger R. |url=https://books.google.com/books?id=UtJK8cetqGkC&pg=PA39 |title=Fundamentals of Astrodynamics |last2=Mueller |first2=Donald D. |last3=White |first3=Jerry E. |publisher=[[Courier Corporation]] |year=1971 |isbn=978-0-486-60061-1 |edition=illustrated |page=39}}</ref> : <math>{v_\infty}^2 = V^2 - {v_\text{e}}^2 .</math> For example, with the definitional value for [[standard gravity]] of {{convert|9.80665|m/s2|abbr=on}},<ref>{{cite book | last = Bureau International des Poids et Mesures | author-link = International Bureau of Weights and Measures | date = 1901 | title = Comptes Rendus des Séances de la Troisième Conférence· Générale des Poids et Mesures | section = Déclaration relative à l'unité de masse et à la définition du poids; valeur conventionnelle de {{math|''g''{{sub|n}}}} | url = https://www.bipm.org/fr/committees/cg/cgpm/3-1901/resolution-2 | location = Paris | publisher = Gauthier-Villars | page = 68 | language = FR | quote = Le nombre adopté dans le Service international des Poids et Mesures pour la valeur de l'accélération normale de la pesanteur est 980,665 cm/sec², nombre sanctionné déjà par quelques législations. Déclaration relative à l'unité de masse et à la définition du poids; valeur conventionnelle de {{math|''g''<sub>n</sub>}}.}}</ref> the escape velocity from Earth is {{convert|11.186|km/s|km/h mph ft/s|abbr=on}}.<ref>{{Cite book |last=Lai |first=Shu T. |url=https://books.google.com/books?id=JjrdCG5BFwUC&pg=PA240 |title=Fundamentals of Spacecraft Charging: Spacecraft Interactions with Space Plasmas |publisher=[[Princeton University Press]] |year=2011 |isbn=978-1-4008-3909-4 |page=240}}</ref> === Energy required === For an object of mass <math>m</math> the energy required to escape the Earth's gravitational field is ''GMm'' / ''r'', a function of the object's mass (where ''r'' is [[Earth radius|radius of the Earth]], nominally 6,371 kilometres (3,959 mi), ''G'' is the [[gravitational constant]], and ''M'' is the mass of the [[Earth]], {{nowrap|''M'' {{=}} 5.9736 × 10<sup>24</sup> kg}}). A related quantity is the [[specific orbital energy]] which is essentially the sum of the kinetic and potential energy divided by the mass. An object has reached escape velocity when the specific orbital energy is greater than or equal to zero. === Conservation of energy === [[Image:RIAN archive 510848 Interplanetary station Luna 1 - blacked.jpg|thumb|[[Luna 1]], launched in 1959, was the first artificial object to attain escape velocity from Earth.<ref>{{Cite web |url=https://nssdc.gsfc.nasa.gov/nmc/spacecraft/display.action?id=1959-012A |title=NASA – NSSDC – Spacecraft – Details<!-- Bot generated title --> |access-date=21 August 2019 |archive-date=2 June 2019 |archive-url=https://web.archive.org/web/20190602031816/https://nssdc.gsfc.nasa.gov/nmc/spacecraft/display.action?id=1959-012A |url-status=live }}</ref> (See [[List of Solar System probes]] for more.)]] The existence of escape velocity can be thought of as a consequence of [[conservation of energy]] and an energy field of finite depth. For an object with a given total energy, which is moving subject to [[conservative force]]s (such as a static gravity field) it is only possible for the object to reach combinations of locations and speeds which have that total energy; places which have a higher potential energy than this cannot be reached at all. Adding speed (kinetic energy) to an object expands the region of locations it can reach, until, with enough energy, everywhere to infinity becomes accessible. The formula for escape velocity can be derived from the principle of conservation of energy. For the sake of simplicity, unless stated otherwise, we assume that an object will escape the gravitational field of a uniform spherical planet by moving away from it and that the only significant force acting on the moving object is the planet's gravity. Imagine that a spaceship of mass ''m'' is initially at a distance ''r'' from the center of mass of the planet, whose mass is ''M'', and its initial speed is equal to its escape velocity, ''v''{{sub|e}}. At its final state, it will be an infinite distance away from the planet, and its speed will be negligibly small. [[Kinetic energy]] ''K'' and [[gravitational potential energy]] ''U''<sub>g</sub> are the only types of energy that we will deal with (we will ignore the drag of the atmosphere), so by the conservation of energy, : <math>(K + U_\text{g})_\text{initial} = (K + U_\text{g})_\text{final}</math> We can set ''K''<sub>final</sub> = 0 because final velocity is arbitrarily small, and {{nowrap|''U''<sub>g</sub>{{hsp}}<sub>final</sub>}} = 0 because final gravitational potential energy is defined to be zero a long distance away from a planet, so : <math>\begin{align} \Rightarrow {} &\frac{1}{2}m{v_\text{e}}^2 + \frac{-GMm}{r} = 0 + 0 \\[3pt] \Rightarrow {} &v_\text{e} = \sqrt{\frac{2GM}{r}} \end{align}</math> === Relativistic === The same result is obtained by a [[Theory of relativity|relativistic]] calculation, in which case the variable ''r'' represents the ''radial coordinate'' or ''reduced circumference'' of the [[Schwarzschild metric]].<ref>{{Cite book |last1=Taylor |first1=Edwin F. |url=https://books.google.com/books?id=y_waLQAACAAJ |title=Exploring Black Holes: Introduction to General Relativity |last2=Wheeler |first2=John Archibald |last3=Bertschinger |first3=Edmund |publisher=Addison-Wesley |year=2010 |isbn=978-0-321-51286-4 |edition=2nd revised |pages=2–22}} [http://www.eftaylor.com/pub/chapter2.pdf Sample chapter, page 2-22] {{Webarchive|url=https://web.archive.org/web/20170721192815/http://www.eftaylor.com/pub/chapter2.pdf |date=21 July 2017 }}</ref><ref>{{Cite book |last=Choquet-Bruhat |first=Yvonne |url=https://books.google.com/books?id=rOYwBQAAQBAJ&pg=PA116 |title=Introduction to General Relativity, Black Holes, and Cosmology |publisher=[[Oxford University Press]] |year=2015 |isbn=978-0-19-966646-1 |edition=illustrated |pages=116–117}}</ref>
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