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| '''AQUAL''' is a [[theory of gravity]] based on [[Modified Newtonian Dynamics]] (MOND), but using a [[Lagrangian]]. It was developed by [[Jacob Bekenstein]] and [[Mordehai Milgrom]] in their 1984 paper, "Does the missing mass problem signal the breakdown of Newtonian gravity?". "AQUAL" stands for "A QUAdratic Lagrangian".
| | Disabilities Services Officer Rocky Paterno from Coldwater, has interests such as crafts (unspecified), [http://www.comoganhardinheiro101.com/inicio/ como ganhar dinheiro na internet] and walking. Likes to travel and had been motivated after visiting Yin Xu. |
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| The gravitational force law obtained from MOND,
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| :<math>m \mu(a/a_0) a = GMm/r^2</math>,
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| has a serious defect: it violates [[Newton's third law of motion]], and therefore fails to [[conservation law|conserve]] momentum and energy. To see this, consider two objects with <math>m \neq M</math>; then we have
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| :<math>\mu(a_m/a_0) m a_m = GmM/r^2 = GMm/r^2 = \mu(a_M/a_0) M a_M</math> | |
| but the third law gives <math>m a_m = M a_M</math>, so we would get <math>\mu(a_m/a_0) = \mu(a_M/a_0)</math> even though <math>a_m \neq a_M</math>, and <math>\mu</math> would therefore be constant, contrary to the MOND assumption that it is linear for small arguments.
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| This problem can be rectified by deriving the force law from a Lagrangian, at the cost of possibly modifying the general form of the force law. Then conservation laws could then be derived from the Lagrangian by the usual means.
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| The AQUAL Lagrangian is:
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| :<math>\rho \Phi + (8 \pi G)^{-1} a_0^2 F(|\nabla\Phi|^2/a_0^2)</math>;
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| this leads to a modified Poisson equation:
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| :<math>\nabla \cdot (\mu(|\nabla\Phi|/a_0) \nabla\Phi) = 4 \pi G \rho </math>.
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| Here, <math>\mu(x) = dF(x^2)/dx</math> and the predicted acceleration is <math>-\nabla\Phi = a</math>. These equations reduce to the MOND equations in the spherically symmetric case, although they differ somewhat in the disc case needed for modelling spiral or lenticular galaxies. However, the difference is only 10-15%, so does not seriously impact the results.
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| ==References==
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| *{{cite journal|authors=Jacob Bekenstein and M. Milgrom|title=Does the missing mass problem signal the breakdown of Newtonian gravity?|journal=Astrophys. J.|volume=286|pages=7–14|year=1984|bibcode = 1984ApJ...286....7B |doi = 10.1086/162570 }}
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| *{{cite journal|author=Milgrom, M|title=Solutions for the modified Newtonian dynamics field equation|journal=Astrophys. J.|volume=302||pages=617–625|year=1986|bibcode = 1986ApJ...302..617M |doi = 10.1086/164021 }}
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| *{{cite journal|journal=Nuclear Physics A|volume=827|pages=555c-560c|author=Bekenstein, Jacob D|title=Relativistic MOND as an alternative to the dark matter paradigm|arxiv=0901.1524|year=2009|doi= 10.1016/j.nuclphysa.2009.05.122|bibcode = 2009NuPhA.827..555B }}
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| [[Category:Astrophysics]]
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| [[Category:Theories of gravitation]]
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| [[Category:Lagrangian mechanics]]
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Disabilities Services Officer Rocky Paterno from Coldwater, has interests such as crafts (unspecified), como ganhar dinheiro na internet and walking. Likes to travel and had been motivated after visiting Yin Xu.