Download Sections F and G

Stars
M. R. W. Masheder
Room 4.15
[email protected]
Level 1 – 2006-07
Tranche 5 (28th February 2007)
Sections F and G
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(Section F) Stellar Activity
By stellar activity, we mean transient phenomena on the surfaces of stars. These are usually
associated with magnetic fields and the build up and release of magnetic energy. We see these
phenomena in more detail on the Sun since it is so much closer than any other star.
The Structure of the [Figs 65, 65a]
Starting from the outermost, least dense regions, we have first the corona. This is composed of
tenuous gas at T~106K and is visible optically during a solar eclipse. The temperature decreases
as we move in, reaching a minimum at the chromosphere. [Fig 67] Just below this is the
photosphere, the edge of the visible disc of the Sun, at T ≈5800K. Under the visible surface, there
is a turbulent layer where energy transport is by convection, and under the convective layer is the
radiative layer, where energy transport is via radiation. Finally at the centre is the nuclear burning
core.
We can image different layers of the Sun using radiation at different wavelengths since these are
emitted or absorbed differently.
Recall from Astrophysical Concepts, the idea of ‘optical depth’. If radiation of intensity I0 is
incident on one side of a slab of material of thickness x then the emergent intensity is I0 = I exp(τ) where τ is the optical depth (= κx if κ is the absorption coefficient). However, the material in
the slab will also be emitting (black bodies are both perfect absorbers and perfect emitters). Thus,
roughly speaking, we will see radiation from the layer one optical depth down (i.e. where τ ≈1):
nearly all the radiation from lower down is absorbed before it can escape while there is little
radiation emitted further up
If absorption is strong (as in absorption lines) then we reach τ = 1 for a thinner layer. Thus at the
wavelength of lines like Hα we see higher layers in the Sun.
Optical images show the photosphere because the corona is almost transparent to visible light
(i.e. optically thin) while the photosphere is optically thick to visible radiation. Note though that
in Hα we see the slightly higher chromosphere. [Fig 67 again] The photosphere appears patchy
due to uneven heating by convection cells in the layer below.
[Fig 68: Solar flare, but showing photosphere is UV as well ]
Radio images show the corona because it is opaque to radio waves because of plasma
oscillations. No transmission is possible at frequencies below the plasma frequency
νplasma = 9ne1/2 Hz , where ne is the plasma electron density (in m−3).
X-ray images also show the corona (although it is transparent to X-rays) since the regions lower
down are too cool to produce X-rays. [Fig. 69: Corona in X-rays]
Ionized Gas in Magnetic Fields
Charged particles move in helical orbits about magnetic field lines because of the Lorentz, B×v
force. Thus ionized gas can move freely along field lines but only diffuses slowly across them. A
magnetic field B has an energy density B2/20, where 0 is the permeability of free space, while
the moving plasma will have energy density ρv2/2. If the magnetic field density is larger,
B2 > 0ρv2 the field will contain the plasma, otherwise the plasma will distort the field.
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The Magnetic Field of the Sun
The Sun has a dipole field with average strength ~10−4T. It is probably generated by the dynamo
process, by electric currents in the solar interior. It is distorted by the differential rotation of the
Sun and by convective cells. The field varies in strength with a 22 year period, reversing
direction every 11 years (at ‘Solar minimum’).
Sunspots [Figs. 70, 71: Sunspots] [Fig. 72: B-field in sunspots]
Sunspots are cooler regions of the photosphere ( ~3800K) which therefore appear as dark regions.
They contain strong magnetic fields ( ~0.3T) and generally occur in pairs with the field emerging
from one (N pole) and entering the other (S pole). The polarity of the leading spot (in terms of the
rotation direction) is opposite in the two hemispheres and reverses with the field reversal. Spots
are cooler than their surroundings because strong magnetic field inhibits convection in the layer
below the spot
[Fig. 72: Sunspot numbers] [Fig. 73 : ‘Butterfly diagram’]
Typical spots last 2 - 4 weeks. The rotation of the Sun (period ~27 days near the equator) carries
the spots across the face of the Sun. The number of sunspots visible at any one time varies in the
same 11 year cycle as the dipole field. The preferred latitude of occurrence also varies, changing
from around ±30° just after minimum to ±15° at maximum and ±8° just before the next
minimum. This leads to the ‘butterfly diagram’ showing the spots positions as a function of time.
The Solar Corona and Solar Wind
The very hot (106K) coronal gas is probably heated by one or both of magnetohydrodynamic
waves or magnetic reconnection events, where tangled magnetic fields reconnect into a lower
energy configuration. The corona is continuously expanding (i.e. it is not stable) and as particles
escape from the Sun they become the Solar Wind. When charged particles in the Solar wind
reach the Earth’s magnetosphere, they can be channelled along the field line to the North and
South magnetic poles. Their interaction with the atmosphere produces Aurora [Fig 74]
X-ray images of the Sun show the corona to be very non-uniform, consisting of bright regions
where the material is trapped by magnetic loops and darker regions (coronal holes) where the
corona is escaping along open field lines. [Fig 69 again: Corona in X-rays]
Solar Flares
Flares are intense bursts of radiation at all wavelengths from a small region of the Sun. They
occur in the lower corona and are probably caused by sudden release of magnetic energy by
reconnection above a coronal loop. This results in large currents heating the plasma to 107K and
the acceleration of electrons and nuclei to high energies. [Fig 68 again: Solar flare]
Some of these particles escape (e.g. to reach the Earth), travelling along field lines. Others travel
downwards to ‘foot points’, losing energy via collisions with gas at the surface of the Sun,
producing X-rays and rays. Interaction of the high energy (relativistic) electrons with the
magnetic field produces radio emission through the synchrotron process (as in particle
accelerators).
Activity on Other Stars
Several types of star are observed to have activity. For instance, low mass M dwarf stars show
intense optical flares coincident with radio bursts.
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RS CVn stars (named after their prototype) are tidally locked binaries, probably also
magnetically linked. They show very strong radio flares and optical variations consistent with
rotation of very large sunspot regions across the face of the star.
High resolution imaging of nearby red giants (e.g. Betelgeuse) show direct evidence for
brightness variations across the surface of the star. [Fig. 75: Betelgeuse in Infra-red ]
(Section G) Binary Stars
As we saw earlier, about 60% of stars are in binary systems. Their binary nature may be revealed
in several ways.
[Figs 13, 14 again: Binary star orbit; Orbit of Sirius B],
Firstly, for visual binaries, we can directly observe both components and we can relate their
separations and orbital periods to the masses of the two stars. Unless observing from space, this
requires their angular separation to be at least 1″ (because of atmospheric blurring, or ‘seeing’).
Since most stars are at least 10pc away, this requires separations of 10AU or more, so for typical
masses of order a solar mass, our standard equation
(M1+M2)T2 = (r1+r2)3
implies T2 > 500 or T> 22 yr. For many binary stars the periods are greater than 100 years.
(Example: Sirius)
Spectroscopic Binaries [Figs 15, 15a again ] [See also simulation]
At smaller separations the binary nature can be revealed by the periodic Doppler shift of the
spectral lines from the stars. Consider a star in a circular orbit which is edge on to the observer
(i.e. inclination angle i = 90o). If the star orbits at speed v1, the Doppler shift will vary over the
orbit as
z1 
 v1 cos( ) where θ is the angle between the line of sight and the velocity vector of


c
the star at any given point of its orbit. Thus ()max = v1/c
but as v1 = 2πr1/T we have r  v1T  cT (  )
1
max
2
2
If spectral lines from both stars are visible then r1 and r2 can be determined and the system can be
analysed as for a visual binary. Note that it is not necessary to know the distance to the system.
Elliptical orbits distort the cosine curve but the system parameters can still be found if i = 90o.
For an orbit inclined at an unknown angle
r1 
cT (  ) max and we can determine only M1 sin3 i.
2 sin(i )
Eclipsing Binaries [Simulation]
When i = 90o the stars will periodically eclipse each other (from our viewpoint). Unless the two
stars happen to be equal in size, the smaller star will block out only part of the light from the
larger one (annular eclipse), but will itself be completely hidden when it is eclipsed. In general
we will see unequal depth eclipses in the overall light curve of the system. If the two stars have
individual fluxes at the Earth F1 and F2, then (assuming 2 to be the smaller star) we will see
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fluxes F = F1 + F2out of eclipse, E2 = F1 during the secondary eclipse and
 R 2  R 2  during the primary eclipse.
E1  F2  F1  1 2 2 
 R1

The ratio of eclipse depths is then
F  E1 ( F1  F2 )  F2  F1 ( R12  R22 ) / R22

F  E2
F2
Now, simplifying and putting in the luminosities at temperatures T, this ratio is:
F1R22
L1R22
4R12T14 R22
T 


  1 
2
2
2
4 2
F2 R1 L2 R1 4R2 T2 R1  T2 
4
For either eclipse, label the start of the eclipse (first contact) by tp, the time when the secondary is
completely behind or in front of the primary as tq and the corresponding events at the end of the
eclipse by tr and ts. The relative velocity of the two stars across the observers line of sight v = v1
+ v2 = 2(r1 + r2)/T =2a/T, where a is the separation between the two stars.
Hence ts - tp = 2(R1 + R2)/v and
tr - tq = 2(R1 - R2)/v,
allowing us to solve for R1 and R2 in terms of a. Most eclipsing binaries are also spectroscopic
binaries, so we know v and hence can determine the radii absolutely.
Close Binaries
These are systems in which one (or both) stars influences the structure and evolution of the other.
Often there is a transfer of mass between the two. Particles in the system move in the joint
gravitational potential of the two stars. Surfaces on which this potential has a given value are
called equipotential surfaces.
For a single non-rotating star, equipotentials are spherical. For a single rotating star, centripetal
acceleration causes the equipotentials to be ellipsoids (e.g. the shape of the Sun). For two stars in
a binary system, there is also the gravitational attraction due to the other star (or equivalently the
centripetal acceleration due to the orbital motion).
Near each star, the equipotentials are pear shaped, pointing towards the other star. For some value
of the potential these two surfaces just meet, at what is called the ‘inner Lagrangian point’,
defining the ‘Roche lobes’ of the system. If one star fills its Roche lobe, material from the star
can flow through the Lagrange point to the other star. This material will have angular momentum,
from its orbital motion, so will not ‘fall’ directly onto the companion star. Instead it will form a
rotating ‘accretion disc’ around it.
Accretion Discs
Gas in the accretion disc will move in approximately Keplerian orbits with
mv 2 GMm
 2 or v 2  GM
r
r
r
Since v varies with r, viscous effects between the gas at different r cause the gas to be heated.
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The energy released will be proportional to M/R , where R is the size of the central star (which
determines how small the accretion disc can get). Thus accretion discs around dense, compact
objects are hotter than those around larger, less dense objects. Accretion discs around white
dwarfs lead to optical emission (novae), while those around neutron stars (or black holes) result
in X-ray emission (X-ray binaries).
Novae
A nova (short for stella nova, or new star) is a star which undergoes a large increase in brightness
(by factors up to 107) on a short time scale.
In many cases outbursts recur. In dwarf novae, the interval between outbursts is only a few
weeks, but for more dramatic outbursts the interval can be 100s or 1000s of years. Novae are
close binary systems with a normal star filling its Roche lobe and overflowing onto a white dwarf
via an accretion disc.
The luminosity emitted by the accretion disc has its origin in the potential energy lost by the gas
as it falls into the potential well of the white dwarf.
X-ray Sources
Most stars, such as the Sun, are only weak X-ray sources. But there are several hundred strong Xray emitting sources in the Galaxy. These are binaries which consist of a normal star filling its
Roche lobe and a neutron star plus hot accretion disc.
Near the neutron star, the strong magnetic field (~108T) funnels the gas down onto the magnetic
poles of the star, forming two X-ray emitting hot spots. These are carried around the star by
rotation (if the magnetic axis does not coincide with the rotation axis) leading to variability in the
observed X-ray flux.
In some cases the deduced masses are much more than 3M (e.g. Cygnus X-1 at about 16M).
These are almost certainly too massive to be neutron stars so are presumably black holes
surrounded by accretion discs.
In General Relativity, around any mass M there is an ‘event horizon’ from within which no
matter or radiation can escape. This has the Schwarzschild radius
Rs= (2GM)/c2
(In a simple Newtonian analogue, this would be the radius at which the escape velocity becomes
the speed of light).
Any body which contracts to within its own event horizon becomes a black hole. No known
pressure mechanism can hold up a degenerate star of more than about 3M, so any such star must
collapse to a black hole, hence the conclusion that the most massive X-ray binaries must harbour
black holes.
Much larger black holes with masses of millions of solar masses or more are believed to be
present at the centres of galaxies. Accretion onto these fuels active galactic nuclei (AGN), the
most powerful of which are quasars.
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