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Stars M. R. W. Masheder Room 4.15 [email protected] Level 1 – 2006-07 Tranche 5 (28th February 2007) Sections F and G Page 1 in tranche 5 of 6 pages. There were 35 previous pages (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 B2/20, 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. Page 2 in tranche 5 of 6 pages. There were 35 previous pages 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. Page 3 in tranche 5 of 6 pages. There were 35 previous pages 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 Page 4 in tranche 5 of 6 pages. There were 35 previous pages 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 4R12T14 R22 T 1 2 2 2 4 2 F2 R1 L2 R1 4R2 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 =2a/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. Page 5 in tranche 5 of 6 pages. There were 35 previous pages 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. Page 6 in tranche 5 of 6 pages. There were 35 previous pages