Download The Virial Theorem

The Virial Theorem •  A fundamental equa-on necessary for proper understanding of stellar forma-on is the virial theorem. •  The virial theorem gives the rela-on between the poten-al and kine-c energies of a system of par-cles at equilibrium. •  For a star (or an interstellar cloud) in hydrosta-c equilibrium, the virial theorem gives the rela-on between its gravita-onal energy (Ω) and its thermal energy (U). •  For a spherically symmetric mass distribu-on in the hydrosta-c equilibrium equa-on •  The gravita-onal energy of a two-­‐mass system is the work necessary to bring one of these masses (m1 or m2) from infinity to distance r from other mass r •  • 
−Gm
m
/r
1 2
m
m
2 1 •  The work needed to bring a spherical shell of mass dM to a spherical mass M of radius r is €
dr /r
−GMdM
€
dM r •  M •  then the poten-al energy of a star of mass M* €
where dM is the mass of a shell between r and r+dr and M(r) is the mass within the radius r. •  This integral measures the poten-al energy when progressively assembling the mass M* shell by shell. •  For spherically symmetrical stars, the mass inside a shell between r and r+dr is dr dM r •  M •  When each side of the hydrosta-c equilibrium equa-on is mul-plied by the volume inside radius r, namely V(r)=4πr3/3, •  since the mass inside a spherical shell between r and r+dr is where 4πr2dr is the volume of the shell •  If we integrate this equa-on over the en-re volume of the star •  where V* is the volume of the star. Since the pressure at the surface and the volume at the centre are zero, the first term on the right -­‐ hand side of this equa-on is zero. The two equa-ons above lead to where M* is the mass of the star. •  By defini-on, the poten-al gravita-onal energy of the star, which has a nega-ve value, is equal to •  and hence •  the average kine-c energy of the par-cles in a gas at temperature T is equal to 3kT/2. •  If N is the total number of par-cles in a given volume V small enough inside the star so that the temperature is constant within it and equal to T, the thermal energy density ε in units of erg/cm3 is •  For an ideal gas which leads to following equa-on •  The total thermal energy of the star U is •  The rela-on between the thermal energy and the gravita-onal energy or the so-­‐called “virial theorem” is finally found •  During the contrac-on –Ω increases, and so does U. •  The virial theorem shows that half of the gravita2onal energy is used to heat the gas (i.e. ΔU=−ΔΩ/2 ). •  The other half is radiated into the interstellar space in the form of electromagne-c radia-on •  The virial theorem only applies to a system of par1cles in equilibrium. Therefore can not apply to collapsing interstellar clouds! •  The virial theorem can be used to state approximately half of the gravita1onal energy is transformed to thermal energy during collapse of a cloud! •  During such a collapse, the internal temperatures increase. If the mass of the cloud is sufficient (or the gravita-onal energy is enough) then the central temperatures become enough to commence the hydrogen fusion (star is born). •  The contrac-on of the star persists un-l the energy generated by the nuclear reac-ons equals the luminosity of the star. •  Example: Calculate the poten-al gravita-onal energy of a star of mass M* and radius R* assuming it possesses a constant density. •  Example: Make use of the virial theorem to es-mate the average temperature inside a star with mass M* and radius R*. Use this equa-on to derive the average temperature of the Sun. Compare it with its surface temperature. Discuss the result.