We now look at one of the most basic problems in terms of the "virial theorem" and how it 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.
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
This is the putative basis for how a collapsing gas cloud eventually emerges as a star with a total energy equal to the negative of the gas kinetic energy.
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
This is the putative basis for how a collapsing gas cloud eventually emerges as a star with a total energy equal to the negative of the gas kinetic energy.
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