Download Today in Astronomy 142

Today in Astronomy 142
Dead and stillborn stars:
! Degeneracy pressure of
electrons and neutrons
! White dwarfs
! Neutron stars
! Brown dwarfs and giant
planets
Figure: HST picture of the faint main sequence star Gliese 229A on the left -- heavily overexposed -and its brown-dwarf companion Gliese 229B on the right (Nakajima et al./Caltech and NASA).
Astronomy 142
1
How stars can support themselves against gravity
! Thermodynamics: gas and radiation pressure….
….support stars in which thermonuclear energy
generation occurs (also in some settings
)
! Quantum mechanics: degeneracy pressure sets in under
extreme states of compression and/or low temperatures.
This is the means of support for objects with no fusion:
dead stars (white dwarfs, neutron stars), stillborn stars
(brown dwarfs), and the cores of giant planets (Jupiter,
Saturn).
Astronomy 142
2
Degeneracy pressure
Degeneracy pressure is due to:
the Pauli exclusion principle: no two identical fermions
(spin-1/2 particles) can be in the same state simultaneously.
! Other things equal, this means that a larger density of
identical fermions (e.g. electrons) involves confinement of
each particle to a smaller space by mutual exclusion.
x p ~
the Heisenberg uncertainty principle:
(Δ=statistical uncertainty in..., p = momentum, x = position)
! Other things equal, this means that an electron confined to
a smaller box (smaller Δx) could have a larger momentum
component along the box’s side (larger Δpx).
Handwaving, 1-D derivation of the equation of state (P vs. n
relationship) for degeneracy pressure follows….
Astronomy 142
3
Pressure
numbers of
particles
passing
through a wall
with area A
per unit time
n numbers of particles per
unit volume
v velocity of particles
Astronomy 142
⇠ nvA
4
Degeneracy pressure, continued
Consider identical fermions in a stack of 1-D boxes bounded
by a pair of walls with area A. If particles have finite
momentum they hit the walls and exert pressure:
1
F
1 dp
⇠ nvA ⇥ p
P ⇠
⇠
A
A
A dt
Number of particles that hit one
wall per unit time
Typical momentum per particle
⇠ nvp
Length of boxes determined by Pauli principle from space not
taken up by other particles past the walls.
n = density (particles per unit volume)
1/n = volume per particle
l = n 1/3
= length of box
Each particle must be somewhere within its box.
Astronomy 142
5
Degeneracy pressure, continued
Typical momentum per particle:
p⇠
p
⇠ ~/ x
⇠ ~n
by uncertainty principle
1
3
v ⇠ p/m
Nonrelativistic motion:
5
( p)2
~2 n 3
Thus
P ⇠n v p⇠n
⇠
m
h
~=
2⇡
m
Done with the Fermi-Dirac probability distribution for p, v,
etc. and you get the leading factor :
Very different from an
ideal gas! (P=nkT)
Astronomy 142
6
Degeneracy pressure in the laboratory
Clusters of ultracold atoms
in a magneto-optical atom
trap: 7Li (spin 0, bosonic,
left), and 6Li (spin ½,
fermionic, right).
! Note that the 6Li cluster
doesn’t get smaller at
temperatures below
~300 nK: degeneracy
pressure prevents
further condensation.
Truscott & Hulet 2010
Astronomy 142
7
Electron degeneracy pressure
Consider a gas of electrons, produced by ionization from
atoms with nuclear charge Ze and baryon number A
(number of neutrons and protons in nucleus). The electron
and nucleus densities are related by
mass from
protons+neutrons
and the mass density is
degeneracy
pressure from
electrons
so
, and
✓ ◆5/3 5/3
h
Z
⇢
Pe = 0.0485
5/3
me A
mp
2
Astronomy 142
Equation of State
Electron
degeneracy
pressure
8
Degenerate electrons in stars
Use the new equation of state, instead of the ideal gas law, to
balance gravity. (Recall that previously we found scaling
relationships for M, R, T, PC, etc.)
Our former results from the ideal gas law and gravity:
Better calculations for
and gravity give:
Suppose this pressure from weight is balanced by electron
degeneracy pressure…..
Astronomy 142
9
Degenerate electrons in stars, continued
central pressure = degeneracy pressure
✓ ◆5/3 5/3
Z
⇢
5/3
A
mp
✓ ◆5/3 ✓
◆5/3
GM 2
h2 Z
1.43M
1
0.77 4 = 0.0485
R
me A
mp
R5
GM 2
h2
0.77 4 = 0.0485
R
me
R = 0.114
h
2
5/3
Gme mp
✓ ◆5/3
Z
M
A
1/3
! No temperature dependence! Simpler than a normal star.
! Much smaller than a normal star of the same mass.
Astronomy 142
10
White dwarf stars
Numerical example:
Mass of a star, size of a planet: white dwarf.
! Remarkable feature of R-M
relation: R decreases with
increasing M. Reason:
larger mass requires
larger supporting pressure, which in turn
requires larger electron
momenta, which in
turn requires that each
electron be confined to
a smaller box.
Astronomy 142
11
Massive white dwarfs: relativity and
Chandrasekhar’s WD mass limit
! To support higher mass (smaller) white dwarfs, larger
electron momenta (and speeds) are required.
! Speeds cannot exceed the speed of light! And when they
get close to c, p isn’t simply given by mv any more.
! Electron degeneracy pressure in extreme relativistic limit
(v approaching c) is
The relativistic and nonrelativistic expressions for electron
degeneracy pressure are equal at
, about
that of the core of a 0.3 M⊙ white dwarf.
Astronomy 142
12
Massive white dwarfs: relativity and
Chandrasekhar’s WD mass limit (continued)
For
and gravity, the central pressure and density
Balance that with relativistic degeneracy pressure, and even
radius disappears from the equation:
Maximum mass of
a white dwarf
(corresponds to zero
radius)
Astronomy 142
13
Theory vs. observations: high-precision R and M
measurements for white dwarfs
White dwarfs are hard to detect in binaries because they are
so much fainter than main-sequence stars. Best to worst:
! Four visual binaries with good orbits, three with Hipparcos
parallaxes (M from orbits, R from distance, L and Te).
! Known eclipsing binaries: 49 WD-MS, 4 WD-WD (M from
orbits, R from ingress time). Some WDs with MS companions have H atmospheres, which affect R measurements.
! 15 with cluster or proper-motion companions which have
Hipparcos parallaxes, for which gravitational redshift has
been measured (R as before, M from R, GR).
! 11 with Hipparcos parallaxes of their own, and surface
gravitational acceleration g determined from shapes of
spectral lines (R as before, M from g = GM/R2).
Astronomy 142
14
Nearby white dwarfs
✚  Stars within 25
parsecs of the Sun
(Gliese and Jahreiss
1991)
✚  Nearest and
Brightest stars (Allen
1973)
✖  Pleiades X-ray
sources (Stauffer et
al. 1994)
✚  Binaries with
measured
temperature and
luminosity (Malkov
1993)
White
dwarfs
Astronomy 142
15
Chandrasekhar mass
Chandrasekhar’s
relativistic white
dwarf theory
Data on white
dwarfs in visual
binary systems (all
four of them) from
the Hipparcos
satellite, by
Provencal et al.
1998, Astrophys. J.
494, 759.
Astronomy 142
16
Earth compared to a 1M⊙ white dwarf (Sirius B)
~105 times more
massive than Earth
Apollo 17/NASA
Astronomy 142
17
White dwarf cooling, masses and ages
1.25 M⊙
Main
sequence
0.2 M⊙
0.5 M⊙
White
dwarfs
Cooling curve for a
0.5 M⊙ carbon
white dwarf, with
time from zero to
1010 years marked
in 109 year intervals
(circles) and
compared to the
white dwarfs in the
third Gliese
catalogue (crosses).
The starting central
temperature was
108 K.
Effective Temperature K
Astronomy 142
18
Theory vs. observations for WDs
Nondegenerate hydrogen
atmospheres: R is that of
the atmosphere, not the
degenerate star.
Radius 109cm
Z/A taken to be 0.5.
WDs in binaries with
precisely-known
orbits (!): Provencal
et al. 1998, 2002;
Parsons et al. 2010,
2011, 2012a, 2012b;
Pryzas et al. 2012;
Hermes et al. 2012.
Those in visual
binaries are labeled
with their names.
40 Eridani B
Procyon B
Stein 2051 B
(no parallax)
Astronomy 142
Sirius B
19
WDs detected in
WD-MS systems by
the Sloan Digital Sky
Survey
Rabassa-Mansergas
et al. 2012 and
eclipsing
Parsons et al. 2013
Radius 109cm
Theory vs. observations for WDs (cont’d)
40 Eridani B
Procyon B
Stein 2051 B
Astronomy 142
Sirius B
20
WDs with companion
parallax and gravitational redshift (○):
Provencal et al. 1998,
Casewell et al. 2009.
Radius 109cm
Theory vs. observations for WDs (cont’d)
Nearby WDs with
parallax and surface
gravity measurements
(o): Provencal et al. 1998.
Astronomy 142
21
Exceeding degeneracy pressure
What does it mean to exceed degeneracy pressure
Pressure estimated from hydrostatic equilibrium is higher
than that possible from degeneracy pressure
" System cannot be in hydrostatic equilibrium
" System collapses
" When system collapses nuclear processes can take place
Astronomy 142
22
Beyond the Chandrasekhar mass: neutron stars
A dead star more massive than 1.4 M⊙ simply cannot be
supported by electron degeneracy pressure; add a little too
much mass and it will collapse gravitationally.
! During the collapse, the extra energy liberated from
gravity, and the high density, can help drive some
endothermic nuclear reactions, e.g.,
! But neutrons are fermions, and neutron degeneracy
pressure can balance gravity: a neutron star is formed.
Nonrelativistic formula:
R = 15 km
.
Astronomy 142
✓
M
M
◆1/3
23
Astronomy 142
24
Figure: a 1.4 M⊙ neutron star and New York City, shown at the same scale.
From Chaisson and McMillan, Astronomy Today.
A neutron star
observed directly
The neutron star at the
center of the Crab
Nebula, the remnant
of the supernova
visible in the year
1054. It is seen as a
pulsar in these visible
images taken 0.03
second apart.
On
Off
Astronomy 142
25
Pulsars (continued)
Same again (i.e. images taken 0.015 sec apart) , but this time
in X-rays, with the Chandra X-ray Observatory (CXO).
N
E
Astronomy 142
26
Neutron stars observed-- Pulsars
Objects pulsing in radio and
X-rays.
Radio objects discovered by
Jocelyn Bell with periods of
order ms to 1000s.
Radio signals highly dispersed
in interstellar medium, so
advanced searches dechirp to
discover them
Magnetic dipole light house
model for radio/X-ray
emission
Astronomy 142
27
The Vela Pulsar, a neutron
star corpse left from a
supernova explosion. A jet is
emitted from one of the
neutron star's rotational
poles. Now a counter jet in
front of the neutron star has
also been imaged by the
Chandra X-ray observatory.
The Vela Pulsar as a bright
white spot in the middle of
the picture. The counter jet
can be seen wiggling from
the hot gas in the upper
right.
Astronomy 142
28
Beyond the Chandrasekhar mass: neutron stars
! The maximum mass calculation involves general relativity
and an equation of state that includes the strong
interaction. The maximum mass is about 2.2 M⊙; it could
not possibly be > 3 M⊙.
! Neutron stars generally have very large magnetic fields
(conservation of flux) and rotate rapidly (conservation of
angular momentum), and are observed as pulsars:
apparently pulsed radio (or visible/X-ray) emission from
high energy electrons moving along poloidal field lines.
! A handful of pulsars are observed in binary systems and
their masses have been measured. Curiously, they all are
about 1.4 M⊙. (see Steiner et al. 2010).
•  R comes estimated from luminosity and temperature
determined from spectra taken in between flares
(“quiescent LMXB”).
Astronomy 142
29
Neutron stars (continued)
OppenheimerVolkov limit
This theoretical calculation ( ;
Lattimer & Prakash
2007) is currently the
closest match to the
data on neutron stars
in binaries (!;
Steiner et al. 2010).
(Note: circumference
plotted, instead of
radius.)
Astronomy 142
30
Degenerate
stars
Causality cutoff
A star smaller than
turns out to require
an EOS with
which in turn
implies speed of
sound vs > c :
violates order of
cause and effect.
Astronomy 142
31
Degenerate stars near their size limits
Below these gray
lines, the interior of a
stable star would
have:
rochester
on
z
i
r
t ho
even
an event horizon (it
would be a black
hole).
Proximity to the black
line implies we need
general relativity to
describe degenerate
stars in this regime.
Astronomy 142
32
Smaller than the Chandrasekhar mass: brown
dwarfs and giant planets
When stars form, they contract until they are hot enough in
the center (about 3×106 K) to ignite the pp-chain fusion
reactions. Recall that
for solar-type stars, if gravity is supported by gas pressure.
! For small masses this involves gas pressures that become
smaller than electron degeneracy pressure – so that
degeneracy pressure can stop the contraction and prevent
the object from reaching fusion temperatures. This
imposes a lower mass limit on what can become a star.
The limit is about 0.08 M⊙ .
Astronomy 142
33
Brown dwarfs and giant planets (continued)
Depending upon how they are formed and what their mass
is, such objects are called brown dwarfs or giant planets.
! Because they cannot replace the energy that leaks away in
the form of light, they simply remain at the size
determined by degeneracy pressure, and cool off forever.
! Thus if they are very old, they are very faint. This
prevented their detection until just a few years ago. Now
many are known from the 2MASS, DENIS and SDSS
surveys.
! These objects are not so numerous in our galaxy as to be a
significant – and largely invisible – fraction of dark matter
component of the galaxy.
Astronomy 142
34
Planets, Brown Dwarfs and Planets
H-burning
D-burning
Luminosity
stars
brown
dwarfs
Burrows et al. 97
planets
log10 age (year)
Astronomy 142
35
Astronomy 142
36
New spectral
classes – cooler than
M stars
L-dwarfs
and T-dwarfs
Discovered by the
2MASS.
Kirkpatrick et al
1999
Astronomy 142
37
Summary
! Degeneracy pressure and how to estimate it from the
exclusion principle
! Radii of white dwarfs
! Classes of objects supported by degeneracy pressure: white
dwarfs, brown dwarfs, planet cores, neutron stars.
! Chandrasekhar mass limit for white dwarfs
! Oppenheimer-Volkov limit for neutron stars
Astronomy 142
38