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Astrophysics/Virial Theorem & Total stellar energy

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Expert: - 4/9/2007


Question
I've read that the virial theorem can be used to show the total energy of a star is negative and equal to one half the gravitational potential energy (or the negative of the gas kinetic energy. How can this be shown?

Answer
Hello,

The virial theorem is very important in stellar astrophysics. According to the virial theorem:

2K  +  W = 0

for any spherical system in hydrostatic equilibrium, where K is the gas kinetic energy:

K = 3/2[y - 1] U

y = the ratio of specific heats (c_p/c_v)

and U is the internal energy

W is the gravitational potential energy

From this one can obtain the binding (or total energy) of a star as:

E(S) = K +  W

Combining the above with:

2K  +  W = 0

we obtain:  

2K = -W   or K = -W/ 2

E(S) = -W/2 +  W  =  W/2

But W = -2 K

so E(S) =  W/2 = (-2 K)/ 2 =  -K

Thus, the total energy of a star E(S) is equal to half the gravitational potential energy (e.g. W/2) or to the negative of the gas kinetic energy:

E(S) = - K = - 3/2[y - 1] U

Hope this sheds light!

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Volunteer

Philip A. Stahl

Expertise

I specialize in stellar and solar astrophysics. Can answer any questions pertaining to these areas, the spectroscopic analysis of stars – as well as the magneto-hydrodynamics of sunspots and solar flares. Sorry – No homework problems done or research projects! I will provide hints on solutions.

Experience

Have published papers on the relationship between sunspot morphology and solar flares; discovery of SID flares related to this, constructed computerized stellar models; MHD research.

Organizations
American Astronomical Society (Solar physics and Dynamical astronomy divisions), American Geophysical Union, American Mathematical Society, Intertel.

Publications
Solar Physics, Journal of the Royal Astronomical Society of Canada, Journal of the Barbados Astronomical Society, Meudon Solar Flare Proceedings (Meudon, France). Books: 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level'.

Education/Credentials
B.A. degree in Astronomy; M.Phil. degree in Physics - specializing in solar physics.

Awards and Honors
Postgraduate research award- Barbados government; Studentship Award in Solar Physics - American Astronomical Society

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