Transcript Space Time and Gravity - Florida State University

Relativity, Quantum Theory
and
White Dwarfs
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Topics




The Sun
Hydrostatic Equilibrium
The Sun’s Central Pressure
White Dwarfs
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The Sun
Power Output of Sun
 3.826 x 1026
Watts
Unit of Power
 1 Watt = 1
Joule/second
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The Sun is a Fusion Reactor
1 p+p collision in 1022
leads to fusion
+ 1H → 2H + e+ + n
e+ + e- → g + g
2H + 1H → 3He + g
1H
3He
+ 3He → 4He + 1H + 1H
4H → 4He
Deuterium
creation
3He
creation
4He
creation
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4 1H → 4He
Mass of Proton
1.007276 amu
(i.e., 1H nucleus)
Mass of a-particle
(i.e., 4He nucleus)
4.001506 amu
where 1 Atomic Mass Unit (amu) is
10-3 kg of Carbon (126C) / NA = 1.66 x 10-27 kg
NA = 6.022 x 1023 is Avogadro’s Number
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4 1H → 4He
Mass Deficit
4 x 1.007276 amu
=
=
4.029104 amu 4 1H
4.001506 amu 4He
0.0276
amu
That is, 0.0276 / 4.0291 = 0.007 of the mass of
a proton is converted to photons and
neutrinos.
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How Long will the Sun Shine?
Available Hydrogen Fuel


0.1 of the Sun’s mass is hot enough to fuse
hydrogen to helium
0.007 of that mass is converted
fuel
= (0.1) x (0.007) x (2 x 1030) kg
= 1.4 x 1027 kg
that is, about 233 times the Earth’s mass
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How Long will the Sun Shine?
Available Energy
E = mc2
= (1.4 x 1027 kg) x (3 x 108 m/s)2
= 1.26 x 1044 Joules
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How Long will the Sun Shine?
Estimated Lifetime
1.26 x 1044 J / (4 x 1026 J/s x 3.15 x 107/s/yr)
10 billion years
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The Sun’ Old Age
In about 5 billion years, the Sun will, after
passing through a red giant phase, become a
hot Earth-sized star called a white dwarf
A white dwarf star is a super-dense object
whose existence is direct consequence of
relativity and the quantum nature of the
world.
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Helix
Nebula
450 ly
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Hydrostatic Equilibrium
Pressure exterior to shell
Pressure due to gravity
A star is in
hydrostatic
equilibrium if at any
radius r the net
outward force, due
to the pressure,
balances the inward
force of gravity
Pressure interior to shell
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Hydrostatic Equilibrium - II
Pressure exterior to shell
Consider a thin shell of
gas of thickness Dr.
Its mass is
Dm = (A Dr) r(r),
where A is the
shell’s area and r(r)
is the shell’s density
Pressure due to gravity
Pressure interior to shell
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Hydrostatic Equilibrium – III
The gravitational
force on the shell is
Dm m(r )
F  G
r2
where m(r) is the
mass up to radius r
r
m(r )  4  z 2 r ( z ) dz
0
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Hydrostatic Equilibrium – IV
The pressure difference
Dp between the interior
and exterior of the shell
must equal the pressure
due to gravity F / A
ADrr (r )m(r ) 

Dp   G
/
A
2

r


In the limit of infinitely
thin shells we have
dp
r ( r ) m( r )
 G
2
dr
r
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Pressure at Sun’s Core
Equation of Hydrostatic Equilibrium
dp
r ( r ) m( r )
 G
dr
r2
r
m(r )  4  z 2 r ( z ) dz
0
The pressure p(0) in the
p(0)  G 
dr core is the sum of the
2
r
R
pressure differences
Problem: Compute core between all shells, from
r = R at the surface to
pressure assuming a
r = 0 at the center
constant density
0
r (r ) m(r )
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Pressure at Sun’s Core - II
For a constant density r(r)  d, the central
pressure is
3 GM
p(0) 
8 R 4
2
Using M = 2 x 1030 kg, R = 7 x 108 m for the Sun
we find
p(0) = 1.3 x 1014 Pa (1 Pascal = 1 N/m2)
Note: pressure at Earth’s surface ~ 105 Pa
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The Standard Solar Model
Central density
r(0) = 1.5 x 105 kg /
m3
Central Pressure
p(0) = 2.3 x 1016 Pa
Approximate Profile
r
r r   1.755 10 exp( 7.434 )
R
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Bahcall – Pinsonneault
http://www.sns.ias.edu/~jnb
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Challenge Problem
Compute, by numerical integration, the pressure
in the Sun’s core assuming the density profile
r
r r   1.755 10 exp( 7.434 )
R
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Compare your result with the value
p(0) = 2.3 x 1016 Pa
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Pressure at Core of White Dwarf
A white dwarf star is so dense that the
gravitational compression forces within it
must be enormous.
But since these stars are stable these forces
must be balanced by an equally enormous
outward pressure.
What provides this pressure?
The electrons, which form a relativistic
quantum gas within the core!
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White Dwarf – II
According to the uncertainty principle,
Dp Dx = h
Dx
the momentum of an object, confined to a
region of linear dimension Dx, will fluctuate by
an amount Dp = h / Dx in that dimension
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White Dwarf – II
We assume that each electron occupies, on
average, a volume Dx3 and
they are as tightly packed
as possible.
Dx
In this case, the electron density, that is, the
number of electrons per unit volume is
n = 1 / Dx3
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White Dwarf – III
The momentum fluctuations Dp can therefore
be written as
Dp = h / Dx
= h n1/3
Dx
The electron pressure P = F / Dx2, where F, the
force, is the rate of change of momentum:
F = Dp / Dt
Therefore, the pressure is P = Dp / (Dt Dx2)
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White Dwarf – IV
The pressure is P = Dp / (Dt Dx2). We now
assume that the electrons
are moving at
the speed of light, c
Dx
We can then write
P = Dp c / (c Dt Dx2)
= Dp c / Dx3
= h c n4/3 (using, n = 1 / Dx3, Dp = h n1/3)
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White Dwarf – V
Since the star is electrically neutral, the
number of electrons equals the number of
protons
N = M / mp where mp is the proton mass
and M is the mass of the Sun.
Therefore, the electron density is roughly
n = N / (4/3  R3)
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White Dwarf – V
We assume that the electron pressure hcn 4/3 is
sufficient to balance the pressure due to
gravity.
Since the pressure needed, at the core, is
roughly GM2 / R4, we can write the equilibrium
condition as
hc n4/3 = GM2 / R4
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White Dwarf – VI
The equilibrium condition yields the remarkable
prediction that a relativistic electron gas can
support a mass no greater than about
M ~ N3 mp
Problem:
where
Complete
N = MPlanck / mp
calculation
= 1.3 x 1019
and
MPlanck = √(hc / G)
(The Planck mass)
= 2.18 x 10-8 kg
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White Dwarf – VII
We have found that the pressure of a
relativistic electron gas can support a
maximum mass of about 1.8 x Sun’s mass!
A more accurate calculation gives 1.4 x Sun’s
mass, a result first worked out in the 1920s
by the Indian physicist Chandrasekhar and
now known as the Chandrasekhar limit
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Summary


The Uncertainty Principle
 Because of the uncertainty principle, the
smaller the region to which an object is
confined the greater the fluctuations in its
momentum
Chandrasekhar Limit
 The fluctuating momentum leads to a
pressure, which, for electrons, is sufficient
to prevent a mass of up to 1.4 x Sun’s mass
from collapsing under its own gravity.
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