Stellar Structure Section 2: Dynamical Structure Lecture 3 – Limit on gravitational energy Limit on mean temperature inside stars Contribution of radiation pressure Virial theorem Properties of polytropes

Limits on conditions inside stars: gravitational potential energy The magnitude of the gravitational potential energy has a lower limit given by Theorem III: (2.14) For the Sun this is ~10 41 joules. (for proof, see blackboard, and Handout).

Limits on conditions inside stars: temperature For an ideal gas, with constant μ, and neglecting radiation pressure, the mean temperature satisfies Theorem IV: (2.15) (for proof, see blackboard, and Handout). For Sun, lower limit is ~ K.

Pressure in a star imply (see blackboard) that the material behaves like an ideal gas radiation pressure is much less than gas pressure. Neutral gas – particles overlapPlasma – separation >> size..

Limits on conditions inside stars: radiation pressure Radiation pressure can be shown (for proof see reference in Lecture Notes) to satisfy Theorem V: If the mean density at r does not increase outwards then, in a wholly gaseous configuration, the central value of the ratio of radiation pressure to total pressure satisfies 1 – β c 1 – β * (2.16) where β * satisfies the quartic equation (where μ c is the mean molecular weight at the centre of the star):. (2.17) Radiation pressure < 10% of total pressure for M < ~6 M sun (more detailed numbers in Table in Lecture Notes).

Virial Theorem The internal energy of a non-relativistic ideal gas is ½kT per degree of freedom per particle. Relating this to, the ratio of specific heats at constant pressure and constant volume, we can find (see blackboard) an integral expression for the total internal energy, U. Using Theorem II, if is constant throughout the star, we can then prove the Virial Theorem: (2.20) This can be used to show (see blackboard) that a self-gravitating gas has a negative specific heat. For = 5/3, half the energy released by a contracting star goes into heating up the star; the other half is radiated away.

Polytropes – simple models for stars For details see Lecture Notes The force balance and mass conservation equations were derived long before anything was known about the nature of stellar material, and are independent of that nature. To form a closed set of equations, early workers introduced polytropes – models that had a power-law P(ρ) relation, with an index n (see blackboard and Lecture Notes); n = 0 corresponds to a liquid star, so these are mathematical generalisations of that. Combining the three equations gives a single second-order differential equation, the Lane-Emden equation of index n. This produces model stars with a finite radius for 0 n < 5. For polytropes, explicit expressions can be obtained for the potential energy and the mean temperature (see blackboard).