Copernicus
Copernicus is a program written to simulate the luminosity curve of a variable star, based on the principle of vector addition of velocities.
First, an elliptical orbit is computed.
Then it is rotated in longitude of periastron and inclination to the celestial plane. Up to 1,000,000 points around the ellipse are computed at the discretion of the user. (More is possible, but takes a excessive time to compute)
Next, light is sent on its way to the telescope, adding the velocity of light, c, to the velocity of the star, v, along the line of sight. In the diagram to the left the value of v has been greatly exaggerated to demonstrate what happens.
As the light arrives, it is accumulated into columns, and the curve is then computed. A filter is available to smooth the curve of an orbit with few points.
Below are animations of the output of the program with changing parameters.
Effect of changing pitch only:
The "spikes" when the orbit in seen face-on are a consequence of automatic scaling (extreme magnification) of the vertical axis by the program when all the values are near zero.
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Effect of changing yaw only, with pitch preset to 45 degrees:
The chaotic dance of the curve is real, as can seen by the colour bands progressing smoothly.
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Effect of changing distance only, with pitch zero, yaw zero:
Here the number of lines representing the slope (speed) from source at bottom to observer at top has been reduced to show how the slope changes. Automatic scaling has changed the distance between source and observer, or the graph would be miles high.
Effect of changing eccentricity only, pitch at 45 degrees:
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Effect of reducing the period only, all other parameters constant:
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This curve is reproduced here with the permission of the British Astronomical Association.
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Changing b-Persei to d-Cephei and back again without changing distance. Pitch, eccentricity, Major Axis, Period and Yaw all change:
A light curve will not change it's characteristic shape as a function of distance if the orbital inclination compensates to reduce v.cos f where f is the angle of inclination.
In other words, the same shaped curved is obtained at 60 parsecs with the orbit tilted 60 degrees as an identical orbit at 30 parsecs seen edge-on (cos 60o = 0.5).
Further, a light curve will not change its characteristic shape if the ratio of magnification of Distance, Period and Major Axis is the same for all three parameters, as with similar triangles having the same angle at the vertices.
Thus if the period is known and major axis is known, the distance to a cepheid is also known. Kepler's third law relates major axis to period:
The squares of the periods of the planets are proportional to the cubes of their semimajor axes.
The fly in the ointment is the angle of inclination to the line of site of the observer. Further research is necessary to determine stellar distances.
When fast light passes slow light on it's way to us, there is a change in the spectrum (velocity curve) here exaggerated.
The program which produced the diagrams can be downloaded here: