Transcript Stellar Structure - Astronomy Centre

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)
A
δr
r
P(r), ρ(r) – pressure, density
at r; M(r) – mass within r
Hence (see blackboard):
Horizontal pressure forces cancel.
Balance net outwards pressure force
against inward gravitational force.
Spherical symmetry: Newton’s theorem
allows replacement of mass distribution
by equivalent mass at centre (just mass
within r).
P
GM
 2
r
r
(2.1)
Mass conservation
Dynamical timescale
• Definition of M(r) yields (see blackboard)
M
 4r 2 
r
(2.2)
• What happens if forces not in balance?
• Find (see blackboard) departures from equilibrium on a very
short timescale – the dynamical timescale, tD:
tD 
1
G
, where   mean density.
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):
GM s2
Pc 
8Rs4
• (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:
 3
4 3
Rs
3
 PdV
0
• (for proof, see blackboard, and Handout).
(2.13)