Compact Objects (Stellar Corpses)
Stellar evolution leads to one of 4 endpoints:
nothing (complete disintegration in a SN explosion)
white dwarfs
neutron stars
black holes
The last 3 are called compact objects.
Good ref. for further reading: Shapiro & Tseukolsky, “Black Holes, White Dwarfs
and Neutron Stars, The Physics of Compact Objects.”
Which stars end up as which remnant?
Main sequence mass
< 1 Msun
Outcome of evolution
Still on m.s., slow evolution
1 ≤ M/Msun ≤ 3-6
Mass loss, planetary nebula + white dwarf
(3-6) ≤ M/Msun ≤ (5-8)
a) Degenerate ignition of carbon
b) Mass loss, planetary nebula + white dwarf
(5-8) ≤ M/Msun ≤ 60
Core collapse –––> supernova –––>
neutron star or black hole.
M/Msun ≥ 60
Unstable from the outset.
What are white dwarfs, neutron stars, and black holes like?
Properties of Compact Objects
Object
M/Msun
R/Rsun
ρ(g/cm3)
Surface potential
(GM/Rc2)
Sun
1
1
1.4
10-6
W. D.
~0.65
~10-2
≤ 107
10-4
N. Star
1-3
~10-5
≤ 1015
~0.1
B. H.
Any
2GM/c2
M/R3
1
Now we’ll look in more detail at each type of object.
White Dwarfs (spelling correct!)
The first white dwarf ‘discovered’ was Sirius B, by W. S. Adams (1915). He
found:
Teff ≈ 8000K,
L ≈ Lsun/360,
M ≈ 1 Msun.
Using R2 = L/(4πσTeff4), he inferred a very small radius, and ρ ≈ 105 g/cm3.
He also measured the gravitational redshift, and found agreement with general
relativity.
1926 - publication of Fermi-Dirac statistics.
1926 + few months - R. H. Fowler deduced that white dwarfs are supported by
electron degeneracy pressure.
White dwarfs in (wide) binaries
for accurate mass determinations Object
M/Msun
R/Rsun
Sirius B
1.053 ± 0.28
0.0074 ± 0.0006
40 Eri
0.43 ± 0.02
0.0124 ± 0.0005
Stein 2051
0.50 ± 0.05
0.0115 ± 0.0012
or 0.72 ± 0.08
(Orbit traversed for only fraction of a period)
Masses determined from pulsation properties, e.g.,
PG 1159
0.59
White Dwarf Masses and Radii
- Masses cluster over a very narrow range.
- Mass-radius relation: in a degenerate gas pressure does not increase much
with increased M. Gravity is more effective, and R decreases as M increases.
More massive WDs are smaller!
- Chandrasekhar mass limit: If M ≥ 1.4 Msun, then electron degeneracy pressure
is overcome by gravity. Chandrasekhar (1931)
Landau (1932)
- The critical mass depends only on fundamental constants, not the details of
the formation process.
M max
3/2


hc
=
.
4 /3
 2πGM e 
Doesn’t depend on the WD composition: 4He, 12C, or 24Mg!
€
Because the degeneracy is due to tight packing of electrons.
White Dwarf Physics
WD gravity is 10,000 times stronger than the Sun’s.
A ping pong ball on a WD weighs as much as you on Earth!
“Normal” gas pressure can’t balance gravity.
S. Chandrasekhar (1930s): a new form of pressure supports WDs Electron Degeneracy Pressure
Pauli Exclusion Principle: electrons resist being squeezed together with
quantum mechanical forces. No more than one ‘Fermion’ per
quantum
level.
Electron degeneracy pressure supports WDs.
But,
If M ≥ Msun even e- degeneracy can’t stop collapse.
Landau’s energy argument
Begin with uncertainty principle from quantum mechanics ΔpΔx ≈ h in a degenerate gas (where h = h/2π)
Since Δx ≈ n-1/3 –––>
Δp ≈ p ≈ hn1/3.
The Fermi energy per particle (which plays the role of thermal energy here) in
the relativistic case is,
EF ≈ pc ≈ hn1/3c ≈ hN1/3c/R,
Where n = N/R3, and N is the total number of Fermions.
The gravitational energy per Fermion is,
EG ≈ -(GMmB)/R,
with M = NmB,
where mB is the mass of a baryon (most of the mass is in baryons). Then the total
energy is,
E = EF + EG = hN1/3c/R - (GMmB)/R.
Near the critical mass, we expect that EF ≈ |EG|, and E ≈ 0. Then we can solve for
N,
Nmax ≈ (hc/(GmB2))3/2 ≈ 2 x 1057,
Mmax ≈ NmaxmB ≈ 3 x 1029 kg ≈ 1.5 Msun.
Ultimate fate of WDs.
• they cool and fade into darkness
• they crystallize as T < 8000K, diamond stars?
• the timescale to cool to 5000K –––> age of the disk of the galaxy.
White Dwarf Compositions
Technical ref.: D’Antona and Mazzitelli (1990) Ann. Rev. Astr. Ap.
Core Composition:
2 possibilities:
- If M* ≈ Mup (upper limit for WD formation)
Rapid, but non-disruptive C-burning –––> O-Ne-Mg core.
This is thought to be the more common in close binaries.
- Else, M*<< Mup, He burning leaves C, O core.
Envelope Compositions
Spectral classes: DA - almost pure H atmosphere.
non-DA (e.g., DB, DO) - no H atm.
Seismological studies imply MH ≤ 10-8 Msun even in DAs. (Also statistical
evidence for the transition DA –––> non DA.
Envelope composition gradients determined by:
- “initial” composition
- gravitational separation
- diffusive counter-separation
- convective mixing
- ISM sweeping
Such gradients affect cooling times.