Download 5 log5 − = − d . N

10. Observational Evidence for White Dwarfs
Do white dwarfs actually exist? Yes – we can see them:
Although very faint, white dwarfs can be imaged directly with
telescopes.
Measuring the amount of light we receive from a white dwarf
gives us its apparent magnitude, N.
If we can determine the
distance, d, to the star (by parallax, or using adjacent Cepheid
variables, etc.), we can then work out its absolute magnitude, .,
using the distance modulus formula:
N − . = 5 log10 d − 5
The absolute magnitude then gives us the star’s luminosity, L, from
Pogson’s equation:
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Lecture 2.6
⎛ L ⎞
⎛ F ⎞
. − .~ = −2.5 log10 ⎜⎜ ⎟⎟ = −2.5 log10 ⎜⎜ ⎟⎟
⎝ L~ ⎠
⎝ F~ ⎠
If we examine the spectrum of the light we receive from the star
(and assume it emits like a blackbody), we can determine the
temperature of the white dwarf using Wien’s Displacement Law:
λmaxT = constant
If we have L and T, we can then obtain the radius of the star, R,
using the Stefan-Boltzmann law:
L = 4πR 2σT 4
If the star is in a binary system, we can also work out its mass from
observations of its orbital motion (see SP I).
We can then determine the mean stellar density, and whether the
white dwarf stars we observe follow our derived R ∝ M -1/3
behaviour.
So what do the observations tell us?
Take as an example perhaps the most famous white dwarf,
Sirius B, the binary companion of Sirius A, the ‘Dog Star’.
Observed properties:
• N = 5.67 (bolometric magnitude)
• d = 2.63 pc (determined by parallax)
• T = 25000 K (from Wien’s Displacement Law)
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Lecture 2.6
• orbital period is 49.94 years, and system is a visual binary, so
orbital parameters can be observed directly: semi-major axis
of system a = 20.04 AU; ratio of the distances of each star
from the centre-of-mass r1/r2 = 0.466
Using these values, we can calculate:
• . = 8.57 (and .~ = 4.83), so L = 0.03 L~
• R = 0.009 R~ = 6.42 × 106 m
• M = 1.03 M~ = 2.05 × 1030 kg
• ρ = 1.85 × 109 kg m-3
So Sirius B is about the size (~ 106 m), density (~ 109 kg m-3), and
temperature (> 10000 K) we expect for a white dwarf.
Another example: 40 Eridani B (the first white dwarf to be
observed, by William Herschel in 1783)
•
•
•
•
L = 0.013 L~
R = 0.014 R~
M = 0.5 M~
T = 16500 K~
Taking mass and radius
ratios with Sirius B, we
find
R1 ⎛ M 1 ⎞
⎟
=⎜
R2 ⎜⎝ M 2 ⎟⎠
−0.61
Provencal et al. 1998, Astrophysical Journal
Theoretically, the power should be -0.33 – the agreement is not too
bad (and the sign is correct). Observations of other white dwarfs
broadly confirm the relationship, but observational uncertainty
remains high.
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11. Accretion
Stellar remnants in close binary systems may gravitationally attract
material away from their companion stars. This material flows
onto the surface of the remnant in a process called accretion.
Consider the gravitational
potential in a binary system.
The region of space around a
star in which material is
gravitationally bound to that
star is called its Roche Lobe.
In a binary system, these touch
at the system’s Lagrange point
where the gravitational effect of each star exactly balances.
(Roche Lobes are gravitational equipotential surfaces.)
Suppose that one member of
the binary system is a compact
object (e.g. a white dwarf),
and that the other member is
large enough (e.g. a giant star)
or close enough that it
completely fills its own Roche
Lobe.
Matter will ‘spill
across’ from the companion star’s Roche Lobe at the Lagrange
point and stream into the white dwarf’s Roche Lobe, falling down
towards the white dwarf.
Since the whole
system is rotating
(as the stars orbit
around each other),
the flowing stream
of matter possesses
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Lecture 2.6
angular momentum. Therefore, it doesn’t fall directly down onto
the white dwarf, but misses the surface and swings round the white
dwarf, forming a spiral of in-falling matter.
Particles feed down into the orbital plane from above and below,
so collisions between particles cancel out the component of
momentum perpendicular to the orbital plane. However, the
particles are streaming parallel to the orbital plane (flowing from
one star to the other), so the component of momentum parallel to
the orbital plane is conserved. The net effect is that the spiraling
flow flattens itself into a thin accretion disc in the plane of the
orbit.
orbital
plane
Collisions between particles
in the accretion disc cause
friction, which heats up the
gas and makes it glow. The
glowing accretion disc can
be observed telescopically.
Mira system in X-rays, and artist’s impression
Image credit: NASA/CXC/SAO/ M. Karovska et al., M. Weiss
What is the source of the energy for this emission? Release of
gravitational potential energy during the in-fall.
By a similar argument to the one we used to calculate energy
release in a supernova (see Lecture 2.1), we can see that the
gravitational potential energy released by a mass m falling onto a
body of mass M, radius R, from a height à R is
E≈
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GMm
R
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[6.1]
Lecture 2.6
The accretion luminosity is then given by
LAcc =
dE GM dm GM
=
=
m&
R dt
R
dt
[6.2]
where m& is the mass accretion rate (kg s-1).
If we assume the accretion disc radiates like a blackbody, we can
relate the disc accretion luminosity to the disc temperature Td
using the Stefan-Boltzmann law:
LAcc = AσTd4
[6.3]
Since the emitting area A is both sides of a flat disc, rather than the
usual sphere, the emitting area for an accretion disc of radius Rd is
A = 2πRd2
Equating [6.2] and [6.3] we find
LAcc
GMm&
=
= 2πRd2σTd4
R
[6.4]
Accretion discs are usually observed in X-rays, giving Td ≈ 106 K.
Orbital measurements of the binary system can give M, and thus R
from the mass-radius relationship. If the disc can be resolved, Rd
can be determined, thus allowing us to calculate m& from
measurements of LAcc.
Novae: Accretion transfers H on to the white dwarf. This mass
‘piles up’ on the surface, and the weight of this overlying material
causes the local density to increase. Eventually the density (and
therefore pressure and temperature) becomes sufficiently high for
H fusion to begin. This fusion releases a large amount of energy,
causing a sudden bright outburst of emission – a nova. This
thermonuclear explosion temporarily sweeps away the accretion
disc: when it builds up again sufficiently, another nova occurs.
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Lecture 2.6
If an accreting carbon white dwarf’s mass approaches MCh, the
density within the star becomes sufficiently high that rapid carbon
fusion can begin throughout the entire star – a carbon detonation
supernova (Type Ia). This is probably sufficiently violent that it
disrupts the entire star, gravitationally unbinding it and blowing it
apart: a stable neutron star probably never gets a chance to form.
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