Difference between revisions of "User:Tohline/H Book"
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# <font color="darkblue">'''[http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations Spherically Symmetric Configurations]:'''</font> | # <font color="darkblue">'''[http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations Spherically Symmetric Configurations]:'''</font> | ||
If the self-gravitating configuration that we wish to construct is spherically symmetric, then the coupled set of multidimensional, partial differential equations that serve as our [http://www.vistrails.org/index.php/User:Tohline/PGE principal governing equations] can be simplified to a coupled set of one-dimensional, ordinary differential equations. This is accomplished by expressing each of the multidimensional spatial operators — gradient (<math>\nabla</math>), divergence (<math>\nabla\cdot</math>), and Laplacian (<math>\nabla^2</math>) — in spherical coordinates (<math>r, \theta, \varphi</math>) (see, for example, the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates]) then setting to zero all derivatives that are taken with respect to the angular coordinates <math>\theta</math> and <math>\varphi</math>. After making this simplification, our governing equations become, | |||
<div align="center"> | |||
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br /> | |||
<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr} \biggr] = 0 </math><br /> | |||
<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br /> | |||
<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{d\Phi}{dr} </math><br /> | |||
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br /> | |||
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br /> | |||
{{User:Tohline/Math/EQ_FirstLaw02}} | |||
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br /> | |||
<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math><br /> | |||
</div> | |||
==Appendices== | ==Appendices== | ||
{{LSU_HBook_footer}} | {{LSU_HBook_footer}} | ||
Revision as of 20:22, 1 February 2010
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Preface from the original version of this HyperText Book (H_Book):
November 18, 1994
Much of our present, basic understanding of the structure, stability, and dynamical evolution of individual stars, short-period binary star systems, and the gaseous disks that are associated with numerous types of stellar systems (including galaxies) is derived from an examination of the behavior of a specific set of coupled, partial differential equations. These equations — most of which also are heavily utilized in studies of continuum flows in terrestrial environments — are thought to govern the underlying physics of all macroscopic "fluid" systems in astronomy. Although relatively simple in form, they prove to be very rich in nature... <more>
Context
- Virial Equations
Applications
If the self-gravitating configuration that we wish to construct is spherically symmetric, then the coupled set of multidimensional, partial differential equations that serve as our principal governing equations can be simplified to a coupled set of one-dimensional, ordinary differential equations. This is accomplished by expressing each of the multidimensional spatial operators — gradient (<math>\nabla</math>), divergence (<math>\nabla\cdot</math>), and Laplacian (<math>\nabla^2</math>) — in spherical coordinates (<math>r, \theta, \varphi</math>) (see, for example, the Wikipedia discussion of integration and differentiation in spherical coordinates) then setting to zero all derivatives that are taken with respect to the angular coordinates <math>\theta</math> and <math>\varphi</math>. After making this simplification, our governing equations become,
Equation of Continuity
<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr} \biggr] = 0 </math>
Euler Equation
<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{d\Phi}{dr} </math>
Adiabatic Form of the
First Law of Thermodynamics
<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math>
Poisson Equation
<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math>
Appendices
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© 2014 - 2021 by Joel E. Tohline |