Transcript Second

PHYS377:
A six week marathon
through the firmament
Week 1.5, April 26-29, 2010
by
Orsola De Marco
[email protected]
Office: E7A 316
Phone: 9850 4241
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
1. Stars are held together by gravitation.
2. Collapse is resisted by internal thermal pressure.
3. These two forces play a key role in stellar structure –
for the star to be stable they must be in balance.
4. Stars radiate
into space. For stability they
need to also
. It follows that sometime
stars run out of equilibrium and change, or evolve.
5. To describe stars (to make a model) we need to know
how energy is produced and how it is transported to
the surface.
Inspired by S. Smartt lectures – Queens University, Belfast
A stellar model
• Determine the variables that define a star, e.g., L, P(r), 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
Stellar Equilibrium
Net gravity force
is “inward”:
g = GM/r2
Pressure
gradient
“outward”
1. Equations of hydrostatic equilibrium
Balance between gravity and internal pressure
Mass of element
m  r(r)sr
where r(r)=density at r.
Forces acting in radial direction:
1. Outward force: pressure exerted by stellar material
on the lower face:
P(r)s
2. Inward force: pressure exerted by stellar material
on the upper face, and gravitational attraction of all
stellar material lying within r

GM(r)
m
2
r
GM(r)
 P(r  r)s 
r(r)sr
2
r
P(r  r)s 
In hydrostatic equilibrium:
GM(r)
P(r)s  P(r  r)s 
r(r)sr
2
r
GM(r)
 P(r  r)  P(r)  
r(r)r
2
r
If we consider an infinitesimal element, we write
P(r  r)  P(r) dP(r)

r
dr
Hence rearranging above we get
dP(r)
GM(r)r(r)

dr
r2
The equation of hydrostatic support
for
r 0
The central pressure in the Sun
• Just using
hydrostatic
equilibrium and
some
approximations
we can determine 
the pressure at
the centre of the
Sun.
8
Pc   Gr2 R
3
2

Dynamical Timescale
(board proof)
tdyn = √ ( 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
Consider a thin shell inside the star with
radius r and outer radius r+ r
V  4r 2r
 M  Vr(r)  4 r 2rr(r)
In the limit where
r
0
dM(r)

 4 r 2 r (r)
dr
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(r) the local mass density at r.
Two equations in three unknowns
dP(r)
GM(r)r(r)

dr
r2


dM(r)
 4r 2 r(r)
dr
3. Equation of State
P(r) 
r(r)kT(r)
m
p

Where , the mean molecular weight, is a function of
composition and ionization, and we can assume it to be
constant in a stellar atmosphere (≈0.6 for the Sun).
Three equations in four unknowns
dP(r)
GM(r)r(r)

dr
r2


dM(r)
 4r 2 r(r)
dr
r(r)kT(r)
P(r) 
m
p
Radiation transport
Energy transport by
Dominant in opaque solids.
Energy transport by matter
bulk motion. Dominant in opaque liquids
and gasses.
• Radiation: Energy transport by photons.
Dominant in transparent media.
.
Equation of radiative energy transport
dT(r)
3r(r) (r)L(r)

dr
64 r 2T(r) 3

The Solar Luminosity
T
Tc 3r (r)L(r)


r
R 64r 2T(r) 3


Tc 64 (Tc /2) 3 (RSun /2) 2
LSun  
RSun
3 Sun rSun
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 r will
change. P will equalise with the environment.
• If rp < rsurr. 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
dL
 4r 2 r(r)(r)
dr
