Cataclysmic Variable
 Note that to compare these two graphs you have to compare identical phases. The bottom graph spans phases 0.5 to 1.5, whereas the top graph spans phases 0.9 to 1.9. To understand these graphs imagine a long line of identical graphs joined end-to-end, each link in this chain of graphs represents a single orbit. A phase from any value to that same value + 1 also spans a complete orbit! In both cases we have used the convention of setting the eclipse of the white dwarf to phase 1.0 (and hence to 0, 2, 3, 4, ... also). Thus, the point at phase 1.8 on the top graph, which is the orbital hump is equivalent to phase 0.8 on the bottom diagram. We can fix the horizontal scale wherever we like, and not everyone uses the same start and end phase for this axis! I did those just to get you used to such graphs, however, for convenience I have rescaled the axes on the graph of our model to match the actual data for Z-Cha, this rescaled graph is shown below. Looking at it you can see the similarities more clearly. So you see, it is possible to tell a lot even without a clear photographic image! There are additional techniques that may be used and we consider these next.
 The inclination of a binary is the angle it makes with the observer, and is equal to 90 degrees if the binary 9disc) is edge-on and zero degrees if it is face-on (like looking down from above). If the disc is observed edge-on, then the donor star will eclipse the various structures - hot spot, accretion stream, white dwarf companion (in the centre of the disc) and the disc itself as it completes its orbit. The way the light-curve varies reveals the structure of the system. For illustration, we can study the average intensity of the pixels in each animation frame (as we did for the simpler case of the contact binary). The result is shown below, in which apparent brightness 9in arbitrary units) is plotted against orbital phase. The orbital phase is the number of whole or fractional orbits completed, from the point we begin recording, which conventionally is chosen as the point at which the larger star eclipses the smaller star and the light-curve is at a minimum. The phase keeps counting indefinitely, thus the smaller star gets eclipsed at phase 0, 1, 2, 3, ... etc. At these phases the larger (donor) star is pointing straight at us, and the white dwarf is eclipsed.
 Compare this to the sketch of actual publsihed data obtained for the system Z Cha. The similarities are striking. The bars in the picture above indicate the phases at which the following structures are eclipsed: WD, white dwarf; HS, hot spot; and disc by the donor star, and the donor star being partially eclipsed itself by the accretion disc (note that the disc may be more-or-less opaque depending on the system and the rate of mass transfer). The bright spot (hot spot, HS) causes the orbital hump (phase 1.8 in the above diagram, phase 0.8 in the below diagram, which are equivalent phases!) as it is on the near-side of the disc. Hot spots may account for up to 30% of the total light emitted by the system. (Note, our spot varies in shape as the stars rotate to add more realistic variation). The minimum of the light-curve occurs when either the white dwarf (WD) or hot spot (HS) is eclipsed, or both if this is observed. There is a shallower, but more prolonged dip due to the accretion disc being eclipsed (how broad this dip is will depend on the relative sizes of the donor star and the disc). In reality, we must do this modelling process in reverse - start with the light-curve and deduce the structure. In the example below, the various components making up the total light-curve have been teased apart to show the contributions from eclipsing the white dwarf, hot spot and disc.
 Pov-Ray model of a cataclysmic variable: above, inclination = 57 degrees; below, inclination = 90 degrees
 The velocities of disc material can be estimated by assuming that the material follows orbits described by Kepler's laws of orbital motion. These are Keplerian orbits. In a Keplerian orbit, such as a planet in stable orbit around the Sun, the wider the orbit (the further the planet or disc material is from the central star) the more slowly it orbits. We explained this by the fact that as matter falls into the central star, it speeds up as gravitational potential energy is converted into kinetic energy. The inner disc rotates more quickly, sliding past the outer disc (though the velocity change from one region to an adjacent region is small). However, the velocities of the outermost disc region are 10-30% lower than those predicted by Keplerian orbits, which suggests that the donor star may be perturbing the outer disc and dragging upon it. Note the high velocities of material in the disc, these velocities are supersonic. S-Waves - If we take the spectrum, that is a graph of wavelength (horizontal axis) against light intensity (vertical axis) at regular time-intervals over one complete orbit and then stack these plots on top of one-another (down the page) then we see something interesting: the double-peak snakes across in a sinusoidal fashion, shifting back and forth in wavelength during a complete orbit. This is because, although the double peak results from rotation of the disc about the central white dwarf, the white dwarf and its accompanying disc rotate as a whole about the centre of mass of the binary system, causing the wavelengths of the double-peak to Doppler shift during the course of an orbit. Considering the bright spot, this also forms a sinusoidal curve or s-wave in this manner, except it is single-peaked, since the bright spot is either coming towards us or away from us, not both together as it only occurs on one side of the disc. Other features also produce s-waves, such as lines from the white dwarf and donor star, though these are generally fainter - the disc and bright-spot dominate the emission. The result is illustrated below:
 Above: Top: a map of a CV as seen from above, in plane view (inclination = 0); bottom: a double-peaked spectral line, the height of which is the light intensity and the horizontal axis shows velocity of the material emitting the corresponding part of the line. We could plot wavelength instead on the horizontal axis, as the shift in wavelength is proportional to velocity. The shaded regions on the spectral line correspond to the shaded regions of the disc. That is, we have mapped the velocities onto the disc. The higher velocities are seen in the innermost regions of the disc, which equate to the wings of the spectral line, when that part of the disc is moving along our line of sight. Negative velocities indicate motion away from the observer, positive velocities towards. Although the inner parts of the disc are hotter and brighter, the outer regions have a much larger surface area and so contribute more to the brightness of the whole image. Thus, the faster moving inner regions emits lower total light intensities and so the intensity drops off at the wings of the line.
 Above: the observation of s-waves. Notice the break at phase = 1 where the disc and bright spot are eclipsed by the donor star: the edge of the disc approaching us is eclipsed first. The use of S-waves. Thus, we can obtain an s-wave for every observable feature in the disc, though the bright-spot s-wave is the most obvious. The amplitude of the s-wave gives us the velocity of that component - the faster it moves the larger the height (width in our view) of the wave from peak to trough. For example, this has shown that the speed of the bright-spot is neither equal to the lower speed of the outer disc, nor the higher speed of the mass-transfer stream, but somewhere in-between the two. This shows that as the stream moves from the donor star to the accretion disc, it impacts the disc and slows dramatically, and this sudden conversion of kinetic energy into heat causes the bright-spot to shine so brightly. From these speeds, and given the fact that the material is moving in circular Keplerian orbits, we can obtain two velocity components - the component of velocity along our line-of-sight (y-velocity) and that at right angles to it (x-velocity). This allows us to construct a sort of 2D picture of the disc by plotting y-velocity against x-velocity (what we call velocity-space). This is Doppler tomography and it gives us a tomogram. We can not go the whole way and produce an actual 2D image of the disc in space, since that would mean going from the 1D information in the spectral lines, to a 2D image, and again many uncertainties exist due to the missing information. However, tomograms do assist in visualising discs and may be combined with other methods to build-up a more complete picture. Some example sketches of tomograms are shown below:
 It was analysis of the tomogram, obtained by observations of the spectrum that lead to the discovery that the accretion disc of IP Pegasi contains a spiral-shock pattern. This is caused by perturbation of the disc by the donor star's gravity and this pattern rotates more-or-less in-sync with the binary. This shock pattern corresponds to uneven temperatures of the disc. We make no assumptions as to the effect on the disc shape (such as warping the disc or causing it to bulge). (Note that although this pattern adds a certain patchiness to the disc, it is no more patchy than it needs to be to fit the data). To better visualise the relationship between velocity coordinates on a tomogram and space coordinates on the actual disc, consider the diagram below:
Top left: a tomogram of a CV showing the
velocity components of the disc and hot spot.
The topmost X marks the centre of the donor
(secondary) star; the middle X marks the centre
of mass of the system (about which the
components revolve); the bottom-most X marks
the centre of the white dwarf. The system and
disc in a tomogram is inside-out (the inner disc
rotates faster than the outer). Bottom left:
another tomogram showing a spiral shock
pattern. The accretion stream is also indicated.
Bottom right: the model of the spirally
shocked-disc viewed in normal spatial
coordinates, this model will produce the
tomogram on the bottom left.