Astronomy 142 1

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).

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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).

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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.

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….

�x�p � ~

Pressure

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numbers of particles passing through a wall with area A per unit time

⇠ nvAn numbers of particles per unit volume v velocity of particles

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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:

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

= length of box Each particle must be somewhere within its box.

Number of particles that hit one wall per unit time Typical momentum per particle

l = n�1/3

P ⇠ F

A⇠ 1

A

dp

dt⇠ 1

AnvA⇥ p

⇠ nvp

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Degeneracy pressure, continued

Typical momentum per particle:

Nonrelativistic motion: Thus

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)

p ⇠ �p

⇠ ~/�x

⇠ ~n 13

by uncertainty principle

P ⇠ n�v�p ⇠ n(�p)2

m

�v ⇠ �p/m

⇠ ~2n 53

m

~ =h

2⇡

Degeneracy pressure in the laboratory

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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

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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

and the mass density is

so , and Equation of State Electron degeneracy pressure

Pe = 0.0485h2

me

✓Z

A

◆5/3 ⇢5/3

m5/3p

mass from protons+neutrons degeneracy pressure from electrons

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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…..

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Degenerate electrons in stars, continued

! No temperature dependence! Simpler than a normal star. ! Much smaller than a normal star of the same mass.

0.77GM2

R4= 0.0485

h2

me

✓Z

A

◆5/3 ⇢5/3

m5/3p

0.77GM2

R4= 0.0485

h2

me

✓Z

A

◆5/3 ✓1.43M

mp

◆5/3 1

R5

R = 0.114h2

Gmem5/3p

✓Z

A

◆5/3

M�1/3

central pressure = degeneracy pressure

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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 pres- sure, which in turn requires larger electron momenta, which in turn requires that each electron be confined to a smaller box.

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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.

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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)

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 compan-ions 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).

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✚  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)

Nearby white dwarfs

White dwarfs

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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.

Chandrasekhar mass

Earth compared to a 1M⊙ white dwarf (Sirius B)

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Apollo 17/NASA ~105 times more massive than Earth

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White dwarf cooling, masses and ages

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.

1.25 M⊙

0.5 M⊙

0.2 M⊙ Main sequence

White dwarfs

Effective Temperature K

Theory vs. observations for WDs

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.

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Sirius B

Nondegenerate hydrogen atmospheres: R is that of the atmosphere, not the degenerate star.

40 Eridani B Procyon B

Stein 2051 B (no parallax) R

adiu

s 10

9 cm

Theory vs. observations for WDs (cont’d)

WDs detected in WD-MS systems by the Sloan Digital Sky Survey Rabassa-Mansergas et al. 2012 and eclipsing Parsons et al. 2013

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Sirius B

40 Eridani B Procyon B

Stein 2051 B R

adiu

s 10

9 cm

Theory vs. observations for WDs (cont’d)

WDs with companion parallax and gravita-tional redshift (○): Provencal et al. 1998, Casewell et al. 2009.

Nearby WDs with parallax and surface gravity measurements (o): Provencal et al. 1998.

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Rad

ius

109 c

m

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

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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

✓M�M

◆1/3

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Figu

re: a

1.4

M⊙

neu

tron

sta

r an

d N

ew Y

ork

City

, sho

wn

at th

e sa

me

scal

e. Fr

om C

hais

son

and

McM

illan

, Ast

rono

my

Tod

ay.

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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

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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).

E

N

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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

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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.

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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”).

Neutron stars (continued)

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This theoretical calcu-lation ( ; 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.)

Oppenheim

er-Volkov lim

it

Degenerate stars

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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.

Degenerate stars near their size limits

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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.

rochester

Below these gray lines, the interior of a stable star would have:

event horizon

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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⊙ .

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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.

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Planets, Brown Dwarfs and Planets

Burrows et al. 97

H-burning D-burning stars

brown dwarfs

planets

log10 age (year)

Lum

inos

ity

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New spectral classes – cooler than M stars L-dwarfs and T-dwarfs Discovered by the 2MASS.

Kirkpatrick et al 1999

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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