Puzzling problem (using virial theorem)

In summary, using the Virial theorem and the expression for gravitational potential, we can calculate the total gravitational energy of a star with a pressure gradient given by \frac{dP}{dr} = \frac{4\pi}{3}G\rho2r exp(-\frac{rr}{\lambda\lambda}). In the limit \lambda << R, this energy is given by E = \frac{RGMM}{3R\lambda}.
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Consider a star of Radius R and mass M, with a pressure gradient given by

[itex]\frac{dP}{dr}[/itex] = [itex]\frac{4\pi}{3}[/itex]G[itex]\rho[/itex]2r exp(-[itex]\frac{rr}{\lambda\lambda}[/itex])

where [itex]\rho[/itex] is the central density. calculate the gravitational energy, using the Virial theorem. Show that in the limit [itex]\lambda[/itex] « R this energy is given by

E = [itex]\frac{RGMM}{3R\lambda}[/itex]for tecnical reasons:
MM = M2
rr = r2
[itex]\lambda\lambda[/itex] = [itex]\lambda[/itex]2
 
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  • #2
Using the Virial theorem, the total gravitational energy of a star is given by:E = -\frac{1}{2}\int \rho \Phi \ dVwhere \Phi is the gravitational potential. Since we are dealing with an isotropic pressure gradient, the gravitational potential is given by:\Phi = -4\pi G \int_0^R \frac{\rho r'^2 exp(-\frac{r'^2}{\lambda^2})}{r} dr'Integrating this expression and substituting in the expression for the gravitational energy, we obtain:E = - \frac{2\pi G R^5 \rho}{15\lambda^2}In the limit \lambda << R, we can approximate the integral as:E \approx - \frac{8\pi G R^5 \rho}{15\lambda^2}Finally, substituting the expression for the central density \rho = \frac{3M}{4\pi R^3}, we obtain:E \approx \frac{RGMM}{3R\lambda}
 

Related to Puzzling problem (using virial theorem)

1. What is the virial theorem and how is it used to solve puzzling problems?

The virial theorem is a mathematical principle that relates the average kinetic energy and the average potential energy of a system in equilibrium. It can be used to solve puzzling problems by providing a way to calculate the total energy of a system without directly measuring or knowing all of its components.

2. How does the virial theorem apply to physical systems?

The virial theorem can be applied to a wide range of physical systems, from atoms and molecules to galaxies and galaxy clusters. It is based on the concept of equilibrium, where the average kinetic energy and the average potential energy are equal, and can be used to study the properties and behavior of these systems.

3. What are some examples of puzzling problems that can be solved using the virial theorem?

One example is the behavior of gases in a container. By using the virial theorem, scientists can calculate the pressure, volume, and temperature of the gas without directly measuring each individual molecule. Another example is the dynamics of star clusters, where the virial theorem can be used to estimate their total mass and predict their evolution.

4. How can the virial theorem be derived and what are its key assumptions?

The virial theorem can be derived from the laws of classical mechanics, specifically the equations of motion for a system of particles. It assumes that the system is in equilibrium, that there are no external forces acting on the system, and that the particles in the system interact through a central potential.

5. Are there any limitations to using the virial theorem to solve puzzling problems?

While the virial theorem is a powerful tool for solving puzzling problems, it does have some limitations. It is only applicable to systems in equilibrium, and it assumes that the particles in the system are moving in a classical manner. It also does not take into account quantum effects or relativistic effects, which may be important in certain systems.

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