ASTR 3730: Fall 2003

White dwarfsEnd state of stars with masses M < 8 Msun is a white dwarf:stellar remnant in which the pressure is provided by degenerateelectrons and there is no significant nuclear burning.

Very well described as polytropes. Basic equations:

†

dmdr

= 4pr2r

dPdr

= -Gmr2 r

P = Krg

conservation of mass

hydrostatic equilibrium

polytropic equation of state (dependsonly upon density)

For non-relativistic electrons, g = 5/3. This corresponds to apolytropic index n = 3/2, where (by definition):

†

g =1+1n

ASTR 3730: Fall 2003

Step 1: combine the two relevant stellar structure equations

†

dPdr

= -Gmr2 r

r2

rdPdr

= -Gm

ddr

r2

rdPdr

Ê

Ë Á

ˆ

¯ ˜ = -G dm

dr

start with hydrostatic equilibrium

Substitute for right-hand-side using the mass equation:

†

ddr

r2

rdPdr

Ê

Ë Á

ˆ

¯ ˜ = -G ¥ 4pr2r

1r2

ddr

r2

rdPdr

Ê

Ë Á

ˆ

¯ ˜ = -4pGr

ASTR 3730: Fall 2003

Step 2: make use of the polytropic equation of state toeliminate pressure from the equation

†

P = Krg = Kr1+

1n

dPdr

= K n +1n

Ê

Ë Á

ˆ

¯ ˜ r1 n dr

dr

Substitute this into the combined equation to get:

†

n +1( )K4pGn

1r2

ddr

r2

rn-1n

drdr

Ê

Ë

Á Á

ˆ

¯

˜ ˜ = -r

Looks messy, but note that this equation only involves the unknown function r(r) - everything else is a knownconstant.

ASTR 3730: Fall 2003

Step 3: introduce a dimensionless variable for the density

Define a new variable q(r) that is related to the densityvia the definition:

†

r = rcqn

…where rc is the central density of the star. q is adimensionless quantity, and varies between q = 1 (at r = 0) and q = 0 (at r = R, the stellar radius)

This substitution further simplifies the equation:

†

n +1( )K

4pGrc

n-1n

È

Î

Í Í Í

˘

˚

˙ ˙ ˙

1r2

ddr

r2 dqdr

Ê

Ë Á

ˆ

¯ ˜ = -q n

constants

ASTR 3730: Fall 2003

At this point, possible to reduce the whole equation to adimensionless form by replacing r with a dimensionlessvariable (see Section 5.3 in the textbook). Can derive themass - radius relation without doing this:

Stellar mass:

†

M = 4pr2rdr0

R

ÚHaving made the previous substitutions, this integral is nowtrivial:

†

n +1( )K

4pGrc

n-1n

È

Î

Í Í Í

˘

˚

˙ ˙ ˙

1r2

ddr

r2 dqdr

Ê

Ë Á

ˆ

¯ ˜ = -q n = -

rrc

†

M = 4p ¥ -rcn +1( )K

4pGrc

n-1n

È

Î

Í Í Í

˘

˚

˙ ˙ ˙

ddr

r2 dqdr

Ê

Ë Á

ˆ

¯ ˜ dr

0

R

Ú

ASTR 3730: Fall 2003

†

M = -n +1( )Krc

1 n

GÈ

Î Í

˘

˚ ˙

ddr

r2 dqdr

Ê

Ë Á

ˆ

¯ ˜ dr

0

R

Ú

= -n +1( )Krc

1 n

Gr2 dq

drÈ

Î Í ˘

˚ ˙ 0

R

= -n +1( )Krc

1 n

G¥ R2 dq

dr r= R

Since q is dimensionless, the value of dq / dr at r = R is proportional to R-1 for a fixed value of n.

†

M µ rc1 nR

Substitute for the central density:

†

rc µMR3

Mass - radius relationfor polytropes

†

R µ Mn-1n-3

ASTR 3730: Fall 2003

Application to white dwarfs

Non-relativistic degeneracy pressure: n = 3/2

†

R µ M-1 3More massive white dwarfs have smallerradii. Central density increases with mass,toward the point where the most energeticelectrons have relativistic velocities:

-2 2 6 10

4

6

8

log (T / K)

log (r / gcm-3)

Radiationpressure

Idealgas

Degenerate,non-relativistic

Degenerate,relativistic

ASTR 3730: Fall 2003

Relativistic degeneracy pressure: n = 3

†

M µ Rn-3n-1 = constant

The mass of an n = 3 polytrope is not a free parameter, it is uniquely specified by the constant K in the equationof state.K depends upon the composition. For a white dwarf made ofcarbon and oxygen the mass is:

†

MCh =1.46Msun

Chandrasekhar mass - the maximum mass possible for awhite dwarf star.Essentially the same maximum mass applies to the degeneratecore of a star whatever the composition: MCh depends upon the number of nucleons per electron which is ~2 for all elements heavier than hydrogen.

ASTR 3730: Fall 2003

Endpoints of stellar evolution1) Stars with M < 8 Msun:

These stars never develop a degenerate core more massivethan the Chandrasekhar limit (for the more massive stars,this requires a lot of mass loss).Endpoint is a white dwarf with a mass smaller than MCh, inwhich the pressure is provided by non-relativistic degenerateelectrons.An isolated white dwarf simply cools off and becomes dimmerand dimmer for all time.

2) Stars with M > 8 Msun:Nuclear reactions in these stars cease once an iron corehas developed. Core is too massive to be supported byelectron degeneracy, leading to core collapse.

ASTR 3730: Fall 2003

Exceeding the Chandrasekhar limiting massSuppose we add mass to a white dwarf, for example in a mass transfer binary system, to bring it up to the Chandrasekharlimit. What happens?

Possibility 1Once MCh is reached, thepressure of degenerate electrons can no longerhold the star up:

collapse

If this accretion-induced collapse occurs, the end state wouldbe a neutron star. The collapse would produce very little inthe way of observable phenomenon.

ASTR 3730: Fall 2003

Possibility 2As M approaches MCh, the temperature and density in thecore ignite fresh nuclear reactions.Unlike in the case of ordinary stellar nuclear reactions, thisis devastating to the star. Recall:

†

P = Kr5 3 …with no temperature dependenceHence, large energy release from nuclear reactions heats the material up without changing the pressure or density.

Reactions runaway, eventually lifting the degeneracy butnot before all the star has been burned:

Supernova explosion, production of ~1 Msunof radioactive nickel.

ASTR 3730: Fall 2003

Observe several classes of supernova, which are classifiedaccording to the spectral lines seen in their spectra:

Type I: no lines of hydrogen in the spectrumType II: lines of hydrogen seen in spectrum

Type Is are further divided into subclasses (Ia, Ib and Ic)again based on their spectral properties.Type Ia supernovae are believed to result from the explosionof Chandrasekar mass white dwarfs. Other types are thoughtto result from the collapse of massive stars.

Note: classification predates any physical understanding, andso is potentially confusing!

Spectra of different classes of supernovae

Type IIClear lines ofhydrogen inspectrum

Type Ia

Type Ia supernovae

Identification of Type Ia supernovae with exploding whitedwarfs is circumstantial but strong. Main clues are:

• No H lines but presence of Si lines in absorption

• Observed in elliptical galaxies as well as spirals

• Remarkably homogenous properties

• Lightcurve fit by radioactive decay of about a Solarmass of 56Ni

At most ~ 0.1 Solar masses of H in vicinityNuclear burning all the way to Si must occur

Old stellar population - not massive stars

`Same object’ exploding in each case

Does not mean that accretion-induced collapse does notoccur in some circumstances as well…