Lecture 20Lecture 20

White dwarfs

Sirius BSirius B

•Wobble in the position of Sirius A led to the prediction of an unseen companion.

Sirius B

Sirius A

M/MSun 1.053 2.3

L/LSun 0.03 23.5

Sirius B: detection and spectroscopySirius B: detection and spectroscopy

•The temperature of Sirius B is 27,000 K, almost three times larger than Sirius A. Surprisingly hot!

Given its low luminosity, it must be very small

•Thus it has the mass of the Sun in a volume smaller than Earth.

An enormous density and force of gravity.

Estimate the central temperature and pressure

Clearly the low luminosity does not arise from hydrogen fusion.

Are white dwarfs white?Are white dwarfs white?

•White dwarfs have a range of temperatures (i.e. colours)

CompositionComposition

•Heavy nuclei are pulled below the surface, while hydrogen rises to the top, layered above the helium

Degenerate matterDegenerate matter

Pauli exclusion principle: at most one fermion can occupy any given quantum state.

The Fermi energy is the energy that divides occupied and unoccupied states at 0K.

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22

32

HeF mA

Z

m

Degenerate matterDegenerate matter

•At non-zero temperature, the degeneracy is not complete

We call a gas degenerate if its average kinetic energy is less than the Fermi energy

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22

32

HeF mA

Z

m

Degenerate matterDegenerate matter

The electron degeneracy pressure is derived from the Pauli exclusion principle and the Heisenberg uncertainty principle:

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5

3

He mA

Z

mP

2

px

(non-relativistic matter)

Mass-Volume RelationMass-Volume Relation

Calculate the relationship between mass and volume for a completely degenerate star of constant density.

constantMVMore massive stars are smaller.

Electrons must be more closely packed in more massive stars, for degeneracy to provide sufficient pressure.

Clearly a problem here because if you keep piling mass on it’s volume must go to zero. The derivation ignored relativity, and at high enough densities the velocities of the electrons approach the speed of light.

Chandrasekhar limitChandrasekhar limit

The velocities of the electrons are actually smaller than predicted by ignoring relativity. Thus they contribute less pressure: the volume will be even smaller than predicted earlier.

In fact, volume goes to zero for a finite mass. There is a maximum mass that a white dwarf can have.

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4

3

HmA

ZcP

The relativistic expression for pressure is:

Chandrasekhar limitChandrasekhar limit

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4

3

HmA

ZcP

The relativistic expression for pressure is:

This leads to the Chandrasekhar mass limit:22/3

1

8

23

HCh mA

Z

G

cM

(contains elements of quantum mechanics, relativity, and gravity!)

A more careful calculation shows:SunCh MM 44.1

BreakBreak

White dwarfs: coolingWhite dwarfs: cooling

Electron conduction is very efficient, so the interior of a white dwarf is nearly isothermal. The luminosity, mass, and interior temperature are related by:

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c

SunSun

T

M

M

L

L

The cooling time can then be calculated from the thermal energy and the luminosity.

White dwarfs: coolingWhite dwarfs: cooling

•As the white dwarf cools, the carbon (and oxygen) crystallize, leaving something like a huge diamond in the sky.

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5.2

70

0 1010122.11

yr

t

K

TA

T

T 5/7

9

5.2

70

0 1010122.11

yr

t

K

TA

L

L

White dwarfs: star formation historyWhite dwarfs: star formation history

• Observations of the number of white dwarfs as a function of their luminosity, compared with theoretical models with different epochs of initial star formation.

Accretion disksAccretion disks

•Orbital motion of the stars means mass transfer will form an accretion disk.

NovaeNovae

•Accretion of fresh hydrogen builds up until a shell of hydrogen fusion (CNO cycle) is created.

The sudden change in luminosity is known as a nova.

Type 1a supernovaeType 1a supernovae

•Type 1a supernovae arise from an accreting white dwarf in a close binary system.

When the mass exceeds the Chandrasekhar limit, the core collapses

•These are important because they all appear to have the same peak luminosity (MB=-19.6±0.2).

Since they are so bright, they are excellent distance indicators for the Universe.

Example: Type 1a supernovaeExample: Type 1a supernovae

How far away can a Type 1a supernova be seen, using large telescopes sensitive to apparent magnitudes mB~25 ?

pcd

Mm

pc

d

9103.8

92.915

6.19251

5log

At this distance, the light we are seeing was emitted when the Universe was only a third of its present age.

The most distant supernova ever seen, at a distance of 12.7 Gpc the light was emitted when The Universe was only 3.8 billion years old

The geometry of the UniverseThe geometry of the Universe

• It has been known since the 1930s that the Universe is expanding: more distant galaxies are moving away from us more quickly.

• By comparing the distance of the supernova to their redshift (recession velocity) we can measure not only the velocity of this expansion, but how it has changed over time (i.e. acceleration of deceleration).

•But the observations of the most distant supernova indicate that the expansion has actually been accelerating!