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If we assume the emission is derived from the synchrotron mechanism then we can evaluate the spectral index of the electrons and estimate the energies of the electrons which generate the photons in the various wavebands. We have the spectral index. (B in Tesla, E e in eV). and.

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  1. If we assume the emission is derived from the synchrotron mechanism then we can evaluate the spectral index of the electrons and estimate the energies of the electrons which generate the photons in the various wavebands. We have the spectral index (B in Tesla, Ee in eV) and If we take the value B ~ 5 10-8 the we obtain the following values There must be some very energetic electrons in the Crab. 13.2 POLARISATION OF THE EMISSIONS Rough contour of the field lines As far as technology will allow us to go, all wavelength emissions from the Crab appear to be polarised. The polarisation vector gives the direction of the magnetic lines inside the nebula, since the electrons are accelerated at right angles with respect to the magnetic field.. PHYS3010 - STELLAR EVOLUTION

  2. R t0 t 13.3 ACCELERATION OF THE FILAMENTARY STRUCTURE Whereas the amorphous mass (right) is related to the synchrotron emission, the filamentary structure seen left is derived from discrete optical emission line of the interstellar material (O, N, S, H, He…) swept up by the expanding envelope. Individual filaments are seen to be expanding outwards, and appear to expand from a single centre - presumably the site of the original explosion. The adjacent picture shows two views of the Crab, taken about 20 years apart, one printed in negative, one positive, the expansion is clearly seen. When the current rate of expansion is extrapolated back to a single point, a problem arises : they meet the axis at about 700 years ago, rather than the ~950 years. Thus instead of decelerating, or even expanding at constant velocity, this can only mean that acceleration of the filaments has taken place. PHYS3010 - STELLAR EVOLUTION

  3. dm/dt acceleration AN ESTIMATE OF THE LEVEL OF ACCELERATION In order to cause an acceleration an input of energy is required. We can readily evaluate this energy input. Since the conservation of energy does not apply we will have Power input giving Measurements givea = 7 10-6 m s-2, m = 2 1030 kg, R = 4 1016 m, v = 106 m s-1. and we obtain If nH ~ 106 H m-3 13.4 LIFETIMES OF ELECTRONS IN THE NEBULA The lifetime of energetic electrons in a magnetic field is related to the rate of energy loss by the synchrotron mechanism PHYS3010 - STELLAR EVOLUTION

  4. The half-life of the electrons with energy E may be approximated as so that E in electron volts we may substitute for E in terms of n Since and obtain B in Tesla, nin Hz THE VALUE OF THE MAGNETIC FIELD In The broad-band continuum emission from the Crab shows that there is a break in the synchrotron spectrum in the region between radio and optical frequencies atn ~ nb. We may use this to make an estimate of the magnetic field strength. nb We assume that n i) For n < nb the spectrum of the electrons has not been changed by synchrotron losses ii) For n > nb the loss has occurred and resulted in a steeper spectrum iii) For n ~ nb the electrons have a lifetime which is the age of the nebula PHYS3010 - STELLAR EVOLUTION

  5. Substitution into the t1/2 equation for break point in the spectrum of 1012 Hz photons, gives a value for the magnetic field of B ~ 2 10-8 Tesla The lifetimes of the electrons corresponding to the emissions in the various wavebands gives tRadio ~ 104 years tOptical ~ 102 years tX-ray ~ 1 year Since the nebula is ~ 1000 years old and still emits high energy photons, then there must be replenishment of the energetic electrons in real time. There must be a source of electrons within the nebula which is probably also responsible for providing the acceleration of the filaments. Where do these electrons originate? - perhaps a study of the size of the nebula at different wavelengths will help. 13.5 THE SIZE OF THE CRAB VS FREQUENCY uv Crab Optical Crab with schematic radio contours superposed NOTE this is an illustrative figure, images are not to precise scale X-ray Crab PHYS3010 - STELLAR EVOLUTION

  6. 13.6 A CENTRAL SOURCE OF ELECTRONS IN THE NEBULA A compilation of images of the Crab nebula taken over all wavebands clearly shows that the size of the object decreases with increasing photon energy. Furthermore the centroid of the expansion is the point to which the object appears to condense with increasing photon frequencies. This should be the point from which the electrons originate. If so we can explain why R = R(n). B Charged particles spiral in a magnetic field such that For the electrons which generate the synchrotron radiation the value of R will be very much less than the diameter of the nebula (1016 m). It automatically follows that the electrons must diffuse through the nebula. After a time t an injected electron will be found a distance R away from the source, given by The maximum distance an electron may be found away from the source is given when t = t the lifetime i.e. Substituting for t we get Since we have Giving if B is constant throughout the nebula PHYS3010 - STELLAR EVOLUTION

  7. The Central Region There is much optical evidence which indicates that continual violent activity exists in the central region. The ‘wisps’ (fast moving gas clouds) were noticed long ago to move outwards like bits of paper picked up in a strong wind. The velocities are typically 0.1 c. The left-hand picture is a ground-based image of the entire Crab nebula showing the filamentary structure and the blue inner glow of the synchrotron radiation emitted by electrons which spiral through the Crab’s magnetic field. The picture on the right show a Hubble Space telescope image of the inner parts of the Crab. This is one of a sequence of Hubble images taken over the course of several months. The sequence confirms the dynamic nature of this inner region. The complex of sharp knots and wisp-like features is clearly visible. The Crab pulsar is visible as the left of the pair of stars near the centre of the frame. The wisp motions stream away from the pulsar with inner velocities approaching half the speed of light. It is highly likely that the pulsar is the driving force behind these motions, we need to investigate further. 13.7 WISPS - THE CENTRAL REGION PHYS3010 - STELLAR EVOLUTION

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