12 February 2013 Astronomy 142, Spring 2013 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 nearby A1 star Sirius A (center, heavily overexposed) and its white-dwarf companion Sirius B (lower left) (NASA).

12 February 2013 Astronomy 142, Spring 2013 2

How stars can support themselves against gravity

Thermodynamics: gas and radiation pressure, for which the equations of state (EOSs) are support stars in which thermonuclear energy generation occurs.

Quantum mechanics: degeneracy pressure sets in under extreme states of compression and/or low temperatures. As we will see this implies quite different EOSs. This is the means of support for objects with no internal energy generation: dead stars (white dwarfs, neutron stars), stillborn stars (brown dwarfs), and the cores of giant planets (e.g. Jupiter, Saturn).

4

gas radiation4, ,

3kT TP nkT P

cρ σµ

= = =

12 February 2013 Astronomy 142, Spring 2013 3

Degeneracy pressure

Degeneracy pressure (Fowler 1926) 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: (∆=statistical uncertainty in..., p = momentum, x = position) Other things equal, this means that an electron confined to

a smaller box (smaller ∆x) could have a larger momentum component along the box’s side (larger ∆px).

Handwaving, 1-D derivation of the equation of state (P vs. ρ relationship) for degeneracy pressure follows….

xx p h∆ ∆ ≥

12 February 2013 Astronomy 142, Spring 2013 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: 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.

1 1 /2

.2

x

x x

dpF nA xP pA A dt A tnv p

δδ

≅ = ≅

=

Number of particles that hit one wall Typical momentum per particle Time interval in which they hit

( )1/31 /n=x∆ =

12 February 2013 Astronomy 142, Spring 2013 5

Degeneracy pressure, continued

Typical momentum per particle: Nonrelativistic motion: Thus Done with the Fermi-Dirac probability distribution for p, v, etc. (AST 241 or 453-style) and you get the leading factor right:

1/3

not exactly, but within a factor ~1by the uncertainty principlethink of a cube bounding each particle

x xp ph x

hn+

≈ ∆≈ ∆

=

x xv p m=2 2 5/31 1

2 2 2x x xnv p np h nP

m m= = =

2 5/30.0485 h nP

m=

The EOS is polytropic (P α ργ ), one of the few forms for which the equations of stellar structure yield simple solutions.

Degeneracy pressure in the laboratory

12 February 2013 Astronomy 142, Spring 2013 6

Clusters of ultracold atoms in a magnetooptical 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

5/32 5/3

5/30.0485ee p

h ZPm A m

ρ =

12 February 2013 Astronomy 142, Spring 2013 7

Electron degeneracy pressure

Consider a gas of electrons, produced by ionization from atoms with nuclear charge Ze and baryon number A. Then the electron and nucleus densities are related by and the mass density is so , and

en Zn+=

( )

( )e e p p n

p p e

m n Am n m m

Am n m m

ρ +

+

= + ≅

≅ >>

e pn Z Amρ=

EOS for electron degeneracy pressure

12 February 2013 Astronomy 142, Spring 2013 8

Degenerate electrons in stars

Let’s use the new EOS, instead of the ideal gas law, to balance gravity and support a star. (Recall that the last time we did this we got useful scaling relationships for M, R, T, PC, etc.) Our former results from the ideal gas law and gravity: Precise calculations for and gravity turn out to give Suppose this pressure from weight is balanced by electron degeneracy pressure…..

2

4 3C CGM MP

R Rρ≈ ≈

2

4 30.77 1.43C CGM MP

R Rρ= =

5/3P ρ∝

12 February 2013 Astronomy 142, Spring 2013 9

Degenerate electrons in stars, continued

No temperature dependence! Much simpler than a normal star.

Much smaller than a normal star of the same mass.

5/32 2 5/3

4 5/3

5/35/32

5

0.77 0.0485

1.43 10.0485

e p

e p

GM h Zm AR m

h Z Mm A m R

ρ =

=

5/321/3

5/30.114e p

h ZR MAGm m

− ⇒ =

12 February 2013 Astronomy 142, Spring 2013 10

White dwarfs

Numerical example: Mass of a star, size of a planet: white dwarf. Remarkable feature of R-M

relation: R decreases with increasing M (Stoner 1929).

Reason: larger mass ⇒ larger supporting pres- sure ⇒ larger electron momenta ⇒ each electron confined to a smaller box ⇒ whole object smaller

Mi1 1 0 3 0 . 0 1 0 . 1 1

0

5 1 09

1 1 01 0

81 , / 0.5 9 10 cm.M M Z A R= = ⇒ = ×10 8(compare 6.96 10 cm, 6.4 10 cm)R R⊕= × = ×

( )M M

R (c

m)

Earth compared to a 1M

white dwarf (Sirius B)

12 February 2013 Astronomy 142, Spring 2013 11

Apollo 17/NASA

12 February 2013 Astronomy 142, Spring 2013 12

Massive white dwarfs: special relativity and the white-dwarf mass limit

To support higher mass (smaller) white dwarfs, larger electron momenta (and speeds) are required.

Speeds cannot exceed the speed of light! Thus WD masses have a maximum (Anderson 1929, Stoner 1930, Chandrasekhar 1931, …).

And when v approaches c, p isn’t simply given by mv. In this extreme relativistic limit (compare page 5), The special-relativistic and nonrelativistic expressions for electron degeneracy pressure are equal at , about that of the core of a 0.3 M

white dwarf (Stoner 1929).

4/30.123e eP hcn=

30 -310 cmen =

12 February 2013 Astronomy 142, Spring 2013 13

Massive white dwarfs: special relativity and the white-dwarf mass limit (continued)

For and gravity, the central pressure and density turn out to be Balance that with relativistic degeneracy pressure, and even radius disappears from the equation (see Homework #3):

4/3P ρ∝

2

4 311 12.9C CGM MP

R Rρ= =

3/22

SAC 20.2

1.44 for / 0.5.

pp

Z hcM mA Gm

M Z A

= = =

Stoner-Anderson-Chandrasekhar mass

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 meas.

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).

12 February 2013 Astronomy 142, Spring 2013 14

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5

Radi

us (1

09cm

)

Mass (M

)

Theory vs. observations for WDs

You’ll learn how to do the calculation ( ) in AST 241. 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 labelled with their names. 12 February 2013 Astronomy 142, Spring 2013 15

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)

1.44 M

Theory vs. observations for WDs (cont’d)

“Best” WDs joined with others detected in WD-MS systems by the Sloan Digital Sky Survey (Rabassa-Mansergas et al. 2012) and shown to be eclipsing (Parsons et al. 2013).

12 February 2013 Astronomy 142, Spring 2013 16

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5

Radi

us (1

09cm

)

Mass (M

)

Sirius B

40 Eridani B Procyon B

Stein 2051 B (no parallax)

1.44 M

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 (): Provencal et al. 1998.

12 February 2013 Astronomy 142, Spring 2013 17

1.44 M

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5

Radi

us (1

09cm

)

Mass (M

)

12 February 2013 Astronomy 142, Spring 2013 18

White dwarfs and the history of physics

The theory of degeneracy pressure and white dwarfs represented the first correct combination of quantum mechanics and special relativity, and a radical result in the study of stellar structure.

For both of these reasons the development of the theory involved a great deal of intrigue and torment for several of the theorists. The intrigue is illustrated in AST 102 (lectures of 6 and 8 October 2009); the torment, here.

Chandrasekhar, by then the only survivor of these theorists, shared the 1983 Nobel Prize in Physics, having been nominated mostly on the strength of this work. • And several others who played even greater roles,

particularly Edmund Stoner, have largely been forgotten.

12 February 2013 Astronomy 142, Spring 2013 19

Neutron stars

A dead star more massive than MSAC simply cannot be supported by electron degeneracy pressure; add a little too much mass and it will collapse gravitationally or explode. During the collapse, the extra energy liberated from

gravity, and the high density, can help drive some endothermic nuclear reactions, notably

But neutrons are fermions, and neutron degeneracy pressure can balance gravity: a neutron star is formed (Tolman 1939; Oppenheimer & Volkov 1939). Nonrelativistic R-M relation:

The mass of a star, the size of a city.

.eE e p n ν∆ + + → +

21/3

8/30.0685p

hR MGm

−=

Rochester compared to a 1M

neutron star

12 February 2013 Astronomy 142, Spring 2013 20

12 February 2013 Astronomy 142, Spring 2013 21

Pulsars

Most of the neutron star we know appear to us as pulsars, or as X-ray flaring members of low-mass binary systems (LMXBs). This is the pulsar at the center of the Crab Nebula (M1), the remnant of the supernova visible in the year 1054. The visible-light images were taken 0.015 second apart.

On

Off E

N

12 February 2013 Astronomy 142, Spring 2013 22

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

12 February 2013 Astronomy 142, Spring 2013 23

Neutron stars (continued)

The maximum mass is not as easy to calculate as in the white dwarf case; it involves general relativity and a strong-interaction EOS.

The maximum mass turns out to be about 2.2 M

(e.g. Lattimer 2012); it could not possibly be > 3 M

, else the EOS would imply sound speed in excess of c.

A handful of neutron stars are observed in LMXB binary systems, and their masses and radii have been measured (see Steiner et al. 2010). • R comes in these cases from luminosity and

temperature determined from spectra taken in between flares (“quiescent LMXB”).

• NSs are so small that eclipses enabling the determination of R from timing are extremely rare.

Neutron stars (continued)

12 February 2013 Astronomy 142, Spring 2013 24

65

67

69

71

73

75

77

79

0.0 1.0 2.0

Circ

umfe

renc

e (k

m)

Mass (M

)

50

60

70

80

90

100

110

120

130

140

150

0.0 0.5 1.0 1.5 2.0 2.5

Circ

umfe

renc

e (k

m)

Mass (M

)

This theoretical calcu-lation ( ; Lattimer & Prakash 2007) , unfortunately beyond the scope of all UR undergraduate classes, is currently the closest match to the data on neutron stars in bina-ries (; Steiner et al. 2010). (Note: circumference plotted, instead of radius.) 2.24M

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10

1.E+11

0.0 0.1 1.0 10.0

Circ

umfe

renc

e (c

m)

Mass (M

)

Degenerate stars

12 February 2013 Astronomy 142, Spring 2013 25

Z/A = 0.5 white dwarfs

Neutron stars

Earth

Rochester

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.

25.8C GM cπ=

2 ,P cρ∂ ∂ >

0.E+00

5.E+06

1.E+07

1.0 1.5 2.0 2.5 3.0

Circ

umfe

renc

e (c

m)

Mass (M

)

Degenerate stars near their size limits

12 February 2013 Astronomy 142, Spring 2013 26

WDs

NSs sv c>

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

an event horizon (it would be a black hole). Proximity to the last one is how you can tell we need general relativity to describe degenerate stars in this regime.

P > ∞

12 February 2013 Astronomy 142, Spring 2013 27

Way smaller than MSAC: 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 turns out to be about 0.08 M

(HW #3).

2 36

4119 15.7 10 K

150C C C

CC

RP GM R MTk k M M RRµ µ

ρ

= ≅ = ×

12 February 2013 Astronomy 142, Spring 2013 28

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 thousands are known from deep near-infrared surveys, and from Spitzer Space Telescope observations.

But these objects could be very numerous in our galaxy and comprise a significant – and largely invisible – component of the galaxy. We will touch on this again, under the rubric of dark matter or missing mass.