PHYS377:A six week marathon

through the firmament

by

Orsola De Marcoorsola@science.mq.edu.au

Office: E7A 316Phone: 9850 4241

Week 1.5, April 26-29, 2010

Overview of the course

1. Where and what are the stars. How we perceive them, how we measure them.

2. (Almost) 8 things about stars: stellar structure equations.

3. The stellar furnace.

4. Stars that lose themselves and stars that lose it: stellar mass loss and explosions.

5. Stellar death: stellar remnants.

6. When it takes two to tango: binaries and binary interactions.

Things about stars

Inspired by S. Smartt lectures – Queens University, Belfast

A stellar model

• Determine the variables that define a star, e.g., L, P(r), r.• Using physics, establish an equal number of equations that

relate the variables. Using boundary conditions, these equations can be solved exactly and uniquely.

• Observe some of the boundary conditions, e.g. L, R…. and use the eqns to determine all other variables. You have the stellar structure.

• Over time, energy generation decreases, the star needs to readjust. You can determine the new, post-change configurations using the equations: you are evolving the star.

• Finally, determine the observable characteristics of the changed star and see if you can observe a star like it!

Equations of stellar structure

For a star that is static, spherical, and isolated there are several equations to fully describe it:

1. The Equation of Hydrostatic Equilibrium.2. The Equation of Mass Conservation. 3. The Equation of Energy Conservation. 4. The Equation of Energy Transport. 5. Equation of State. 6. Equation of Energy Generation.7. Opacity. 8. Gravitational Acceleration

Net gravity force is “inward”:g = GM/r2

Pressure gradient “outward”

Stellar Equilibrium

Mass of element

where (r)=density at r.

Forces acting in radial direction:

1. Outward force: pressure exerted by stellar material

on the lower face:

2. Inward force: pressure exerted by stellar material

on the upper face, and gravitational attraction of all

stellar material lying within r

1. Equations of hydrostatic equilibriumBalance between gravity and internal pressure

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δm = ρ(r)δsδr

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P(r)δs

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P(r + δr)δs +GM(r)

r2δm

= P(r + δr)δs +GM(r)

r2ρ(r)δsδr

In hydrostatic equilibrium:

If we consider an infinitesimal element, we write

for r0

€

P(r)δs = P(r + δr)δs +GM(r)

r2ρ (r)δsδr

⇒ P(r + δr) − P(r) = −GM(r)

r2ρ(r)δr

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P(r + δr) − P(r)

δr=

dP(r)

dr

Hence rearranging above we get

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dP(r)

dr= −

GM(r)ρ (r)

r2

The equation of hydrostatic support

The central pressure in the Sun

• Just using hydrostatic equilibrium and some approximations we can determine the pressure at the centre of the Sun.

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Pc ≈ −8π

3Gρ⊕

2 R ⊗2

Dynamical Timescale(board proof)

dyn = √ ( R3/GM )

It is the time it takes a star to react/readjust to changes from Hydrostatic equilibrium.It is also called the free-fall time.

2. Equation of mass conservation

This tells us that the total mass of a spherical star is the sum of the masses of infinitesimally small spherical shells. It also tells us the relation between M(r), the mass enclosed within radius r and (r) the local mass density at r.

In the limit where r 0

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δV = 4πr2δr

⇒ δM = δVρ(r) = 4πr2δrρ(r)

⇒dM(r)

dr= 4πr2ρ (r)

Consider a thin shell inside the star with radius r and outer radius r+r

Two equations in three unknowns

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dP(r)

dr= −

GM(r)ρ (r)

r2

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dM(r)

dr= 4πr2ρ (r)

3. Equation of State

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P(r) =ρ (r)kT(r)

μm p

Where , the mean molecular weight, is a function of composition and ionization, and we can assume it to beconstant in a stellar atmosphere (≈0.6 for the Sun).

Three equations in four unknowns

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dP(r)

dr= −

GM(r)ρ (r)

r2

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dM(r)

dr= 4πr2ρ (r)

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P(r) =ρ (r)kT(r)

μm p

Radiation transport

Equation of radiative energy transport

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dT(r)

dr= −

3ρ(r)κ (r)L(r)

64πσr2T(r)3

The Solar Luminosity

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ΔT

Δr= −

−Tc

R∗

=3ρ∗κ (r)L(r)

64πσr2T(r)3

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LSun = −−Tc

RSun

64πσ (Tc /2)3(RSun /2)2

3κ Sunρ Sun

Convection

• If rises, dT/dr needs to rise to, till it is very high. High gradients lead to instability. What happens then?

• Imagine a small parcel of gas rising fast (i.e. adiabatically – no heat change). Its P and will change. P will equalise with the environment.

• If p < surr. the parcel keeps rising.• So if the density gradient in the star is small

compared to that experienced by the (adiabatically) rising parcel, the star is stable against convection.

Giant star: convection simulation

Simulation by Matthias Steffen

Equation of Energy Conservation

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dL

dr= 4πr2ρ(r)ε(r)