Explaining Mira

Keplerian Orbit plot created with Google Sketchup using the Microsoft Excel spreadsheet: Orbit.xls

Here we have a star and a planet in orbit about a common barycentre.

The same, but with the image hurtling away from the source at the speed of light ­ which means it approaches you, the observer, with the speed of light plus the speed of the star. If a star moving at 1 m/s emits light at 299 792 458 m/s then the light approaches you at 299 792 459 m/s from one side of the orbit and 299 792 457 m/s from the other. Eventually the faster light emitted later must pass the slower light emitted earlier.

The image races through space, and if the distance to the observer is great enough the image from the slow side of the star's orbit about the barycentre is caught up by the fast image from the other side, at which point (together with all images in between, not shown here) the two overlapping images are seen at the same moment, increasing the apparent brightness. However, the star is so distant that not even the most powerful telescopes in existence can see more than just a point of light -- until this picture of Mira, a giant star, was taken by the Hubble Space Telescope.

Credit: M. Karovska (Harvard-Smithsonian CfA) et al., FOC, ESA, NASA

Ref: http://antwrp.gsfc.nasa.gov/apod/ap010121.html

When we take the star's spin into account, the blurred image makes sense. There is no actual "tail", Mira is spherical like any other star; the "tail" is an earlier image of the star's equatorial region travelling at a different speed to the poles. There is a corresponding flattening opposite the "tail" and some light bleeding between the pixels of the CCD.

Three views of the same orbit, without barycentre showing.

Cepheid                                                  "Eclipsing" variable                                         Recurrent Nova      

Multiple planets in the orbit are shown to highlight the velocity change. It can clearly be seen that the change in brightness (shape of the luminosity curve) is related to the attitude of the orbit as presented to the observer.

The actual light curves are not from reflected light as shown above, of course,  but are related to the change in the velocity of light as shown in this distance/time plot taken from Copernicus.exe. Light leaves the bottom of the plot at regular time intervals and gradually catches up or falls further behind so that upon arrival (top of the plot) more and then less photons arrive per second, thereby varying the apparent luminosity. In this plot a peak is produced: 

Seen from the opposite side of the orbit, the characteristic luminosity curve of Algol is reproduced.