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ASTR2050 Spring 2005
Lecture 10am 8 March 2005
Please turn in your homework now!
In this class we will study “compact objects”:
• Degeneracy pressure
• White dwarfs: End state of “low mass” stars
• Neutron stars: End state of “high mass” stars
• Pulsars: Rotating neutron stars
• Next class: Black holes (“very high mass” stars)
1
Reminder: Evolution of a “few solar mass” star
White
dwarf
So what’s so special about
a “white dwarf ” star?
2
Review(?): The “Equation of State”
Consider N particles in a box of side L
A particle hits a wall giving a force:
!p
p
pv
F=
=
=
!t L/v
L
L
Therefore there is a pressure P on the
wall from all N particles that hit it, i.e.
F
N pv
P=N
= 3 = npv = nmv2
Area
L
where n is the number density of (nonrelativistic) particles
each with a mass m.
3
Simple example:The Ideal Gas Law
Recall: We used this to model main sequence stars
Particles move freely and independently.
Temperature is defined by the (average) kinetic energy
of the collection of particles:
1 2 1
mv = kT
2
2
This gives the “ideal gas equation of state”:
P = nkT
also known as
(We will use this form.)
4
PV = NkT
Degeneracy Pressure
A white dwarf is a very compact object.This leads to new effects.
For example, temperature is irrelevant for the equation of state!
...
For electrons packed closely together, the Pauli Exclusion
Principle forces them to fill up the available quantum states:
E5
E4
E3
E
E2
E1 kT
E0
See Kutner Fig.10.11
The average energy of the
electrons is determined by
quantum mechanics, and is
much greater than kT.
A “degeneracy pressure” is
created which holds off
gravitational collapse.
5
How can we calculate degeneracy pressure?
Use the Heisenberg Uncertainty Principle!
h
!p!x ≥
≡ h̄
2"
Average momentum is zero, so put p=Δp
Put Δx = space for one electron =
(1/ne)1/3
5/3
2 ne
⇒ p=
1/3
h̄ne
p
Therefore P = ne pv = ne p = h̄
me
me
We’ve made some crude approximations when dealing with the
quantum mechanics. A careful calculation gives about x2 larger.
6
Express the result in terms of
“matter”, instead of “electrons”
For atomic density nZ have ne=ZnZ
But mass density ρ = AmpnZ + mene ≈ AmpnZ
(since electrons are light)
Therefore ne=(Z/A)(ρ/mp) and
! "5/3
5/3
(!/m
)
p
Z
p
2
P = ne pv = ne p = 2h̄
me
A
me
Included x2 “correction” factor. See Kutner Eq.10.11.
7
White Dwarf Stars
Remember: We know a lot
about white dwarfs because
they’ve been seen as binaries!
(Recall 8 Feb 11am lecture.)
Example: Typical values are
M=M!
−2
R=10 R!
which give a density a million
times greater than the Sun, or
a million times that of water.
8
What controls?
Degeneracy pressure or thermal pressure?
(Kutner Example 10.3.)
6
Assumptions:
3
!=10 gm/cm
Z/A=1/2
6
T =10 × 10 K
You find that the degeneracy pressure (from our formula)
is about 100 times larger than the thermal pressure (from
the Ideal Gas Law). That is, a white dwarf behaves as
if it were a degenerate electron gas!
9
White Dwarf in Hydrostatic Equilibrium
M2
Recall (15 Feb) the central pressure of a star: PC = G
R4
Use degenerate electron gas equation of state to find
2
!
M
M
5/3
G 4 !" !
R
R3
"5/3
or
1
R ! 1/3
M
(See Kutner for details and proportionality constants.)
That is, add mass to a white dwarf and it shrinks!
Of course, it also gets more dense, too!
10
The Chandrasekhar Limit
This can’t go on forever. What gives?
We have ignored relativity, and have written v=p/m .
Consider instead the limiting equation of state with v=c :
! "4/3 ! "4/3
Z
!
P = ne pv = ne pc = 0.8h̄c
A
mp
! "4/3
2
M
M
Now, hydrostatic equilibrium gives G
!
4
R
R3
The radius drops out! Find M = 1.44M!
Largest possible mass for a white dwarf. After this...
11
Neutron Stars
12
Recall the pre-supernova phase of a high mass star:
Red supergiant star
−
e
+ p → n + !e
Core collapses, energy released, and
A burst of neutrinos, and a neutron star is born!
13
Neutron Stars in Hydrostatic Equilibrium
Treat the same way as
white dwarfs but with
me→mn (See Homework.)
However you find that
the density of a neutron
star with Chandrasekhar
mass is about the same
density as a neutron!
Complicated internal structure!
How much more dense can it be?
Neutron stars for undergraduates,
R. Silbar and S. Reddy, Am.J.Phys. 72(2004)892
14
See also
Kutner
Figure 11.7
Pulsars: Rotating Neutron Stars
Imagine the Sun collapsing to the size of Manhattan:
Angular momentum: Iω same before and after
2
2
9
2
I = MR so INS/I! = (RNS/R!) = 2 × 10
5
!! = 30 days, so Neutron star period = 1.3ms
Magnetic field: Expect (?) flux BR2 also conserved
So, also expect 2 billion times the Sun’s magnetic field.
Big!!
15
Kutner
Fig. 11.10
Model of
a Pulsar
The Crab Pulsar in
X-Ray (Chandra) and
Optical (Hubble)
16
Pulsar Signals
17
The Crab Pulsar (HST)
18