Stellar Structure Section 2: Dynamical Structure Lecture 2 – Hydrostatic equilibrium Mass conservation Dynamical timescale Is a star solid, liquid or gas? Boundary conditions Limit on central pressure Gravitational potential energy

Force balance Hydrostatic equilibrium: balance between gravity and internal pressure (evidence: geological timescales) δrδr r A P(r), ρ(r) – pressure, density at r; M(r) – mass within r Horizontal pressure forces cancel. Balance net outwards pressure force against inward gravitational force. Spherical symmetry: Newtons theorem allows replacement of mass distribution by equivalent mass at centre (just mass within r). Hence (see blackboard): (2.1)

Mass conservation Dynamical timescale Definition of M(r) yields (see blackboard) (2.2) What happens if forces not in balance? Find (see blackboard) departures from equilibrium on a very short timescale – the dynamical timescale, t D :

Equation of state Two equations, 3 variables (P, ρ, M) Equation of state relates P, ρ – but introduces more variables (see blackboard) Is a star solid, liquid or gaseous? Mean density and surface temperature (see blackboard) suggest liquid. But actually a plasma – highly ionised gas, so that particle size ~ nuclear radius << typical separation (~ atomic radius). Hence stellar material behaves like an ideal gas (plus radiation pressure) – see blackboard

Limits on conditions inside stars: pressure Now have 3 equations, 5 variables (P, ρ, M, T, μ) – but can obtain some general results without more equations. Boundary conditions: take P = ρ = 0 at the surface. Can then find lower limit for the central pressure (Theorem I): (for proof, see blackboard, and Handout). For Sun, lower limit is ~450 million atmospheres. (2.11)

Gravitational potential energy Gravity is an attractive force – so the work done to bring matter from infinity to form a star is negative: positive work must be done to prevent the infall of material. This means that the gravitational potential energy, Ω, is negative (see blackboard). It is related to the internal pressure by Theorem II: (2.13) (for proof, see blackboard, and Handout).