Astronomy:Chandrasekhar potential energy tensor
In astrophysics, Chandrasekhar potential energy tensor provides the gravitational potential of a body due to its own gravity created by the distribution of matter across the body, named after the Indian American astrophysicist Subrahmanyan Chandrasekhar.[1][2][3] The Chandrasekhar tensor is a generalization of potential energy in other words, the trace of the Chandrasekhar tensor provides the potential energy of the body.
Definition
The Chandrasekhar potential energy tensor is defined as
- [math]\displaystyle{ W_{ij} = -\frac{1}{2} \int_V \rho \Phi_{ij} d\mathbf{x} =\int_V \rho x_i \frac{\partial \Phi}{\partial x_j} d\mathbf{x} }[/math]
where
- [math]\displaystyle{ \Phi_{ij}(\mathbf{x}) = G \int_V \rho(\mathbf{x'}) \frac{(x_i-x_i')(x_j-x_j')}{|\mathbf{x}-\mathbf{x'}|^3} d\mathbf{x'}, \quad \Rightarrow \quad \Phi_{ii} = \Phi = G \int_V \frac{\rho(\mathbf{x'})}{|\mathbf{x}-\mathbf{x'}|} d\mathbf{x'} }[/math]
where
- [math]\displaystyle{ G }[/math] is the Gravitational constant
- [math]\displaystyle{ \Phi(\mathbf{x}) }[/math] is the self-gravitating potential from Newton's law of gravity
- [math]\displaystyle{ \Phi_{ij} }[/math] is the generalized version of [math]\displaystyle{ \Phi }[/math]
- [math]\displaystyle{ \rho(\mathbf{x}) }[/math] is the matter density distribution
- [math]\displaystyle{ V }[/math] is the volume of the body
It is evident that [math]\displaystyle{ W_{ij} }[/math] is a symmetric tensor from its definition. The trace of the Chandrasekhar tensor [math]\displaystyle{ W_{ij} }[/math] is nothing but the potential energy [math]\displaystyle{ W }[/math].
- [math]\displaystyle{ W= W_{ii} = -\frac{1}{2} \int_V \rho \Phi d\mathbf{x} = \int_V \rho x_i \frac{\partial \Phi}{\partial x_i} d\mathbf{x} }[/math]
Hence Chandrasekhar tensor can be viewed as the generalization of potential energy.[4]
Chandrasekhar's Proof
Consider a matter of volume [math]\displaystyle{ V }[/math] with density [math]\displaystyle{ \rho(\mathbf{x}) }[/math]. Thus
- [math]\displaystyle{ \begin{align} W_{ij} &= -\frac{1}{2} \int_V \rho \Phi_{ij} d\mathbf{x} \\ &= - \frac{1}{2} G \int_V \int_V \rho(\mathbf{x})\rho(\mathbf{x'}) \frac{(x_i-x_i')(x_j-x_j')}{|\mathbf{x}-\mathbf{x'}|^3}d\mathbf{x'}d\mathbf{x} \\ &= -G \int_V \int_V \rho(\mathbf{x})\rho(\mathbf{x'}) \frac{x_i(x_j-x_j')}{|\mathbf{x}-\mathbf{x'}|^3}d\mathbf{x}d\mathbf{x'} \\ &= G \int_V d\mathbf{x}\rho(\mathbf{x})x_i \frac{\partial}{\partial x_j} \int_V d\mathbf{x'} \frac{\rho(\mathbf{x'})}{|\mathbf{x}-\mathbf{x'}|}\\ &= \int_V \rho x_i \frac{\partial \Phi}{\partial x_j} d\mathbf{x} \end{align} }[/math]
Chandrasekhar tensor in terms of scalar potential
The scalar potential is defined as
- [math]\displaystyle{ \chi(\mathbf{x}) = -G \int_V \rho(\mathbf{x'}) |\mathbf{x}-\mathbf{x'}|d\mathbf{x'} }[/math]
then Chandrasekhar[5] proves that
- [math]\displaystyle{ W_{ij} = \delta_{ij} W + \frac{\partial^2 \chi}{\partial x_i\partial x_j} }[/math]
Setting [math]\displaystyle{ i=j }[/math] we get [math]\displaystyle{ \nabla^2\chi = -2W }[/math], taking Laplacian again, we get [math]\displaystyle{ \nabla^4\chi = 8\pi G \rho }[/math].
See also
References
- ↑ Chandrasekhar, S; Lebovitz NR (1962). "The Potentials and the Superpotentials of Homogeneous Ellipsoids" (PDF). Ap. J. 136: 1037–1047. Bibcode: 1962ApJ...136.1037C. doi:10.1086/147456. Retrieved March 24, 2012.
- ↑ Chandrasekhar, S; Fermi E (1953). "Problems of Gravitational Stability in the Presence of a Magnetic Field" (PDF). Ap. J. 118: 116. Bibcode: 1953ApJ...118..116C. doi:10.1086/145732. Retrieved March 24, 2012.
- ↑ Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969.
- ↑ Binney, James; Tremaine, Scott (30 October 2011) (in en). Galactic Dynamics (Second ed.). Princeton University Press. pp. 59–60. ISBN 978-1400828722. https://books.google.com/books?id=6mF4CKxlbLsC.
- ↑ Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969.
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