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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.
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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 ~1041 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 ~4106 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 overlap Plasma – 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 – βc ≤ 1 – β* (2.16) • where β* satisfies the quartic equation (where μc is the mean molecular weight at the centre of the star): • . (2.17) • Radiation pressure < 10% of total pressure for M < ~6 Msun (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 starsFor 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).
