Simulations were also carried out to quantify the
range in parameter space which could be filled by stars varying in accordance
with the spot hypothesis. Spotted stars were modeled in a two step process.
First, an unspotted star was produced. To do this, colors of a given spectral
type (taken from Bessell and Brett 1992) were converted to fluxes for a
normalized distance. The flux was summed over 100 grid locations to
represent a 100% filling factor. In this model, each portion of the
star's surface is weighted evenly and limb darkening is not accounted
for. No correction
was applied for spectral class IV relative to spectral class V since the goal
of the simulation was to gain a qualitative
understanding of the expected result.
As a test, summation over the 100 grid locations returns the original colors.
To simulate spots, between five and 40 of the grid locations were
refilled with
values corresponding to temperatures between 100K and 1000K cooler than the
photosphere. Summation over this grid returns the colors of the spotted
star. The original and final colors are subtracted to produce
, 
and 
.
All spectral types between K0 and M5 were simulated. The change in the colors for each given band, the observable quantity, was calculated. For any given waveband, the flux change for spots with a spot filling factor of about 20% (assuming spots 500K cooler than the ambient photosphere) is about 10%. The largest change in any given color was about half a magnitude. This was for a spot 1500K cooler than the photosphere covering 40% of the observable surface. There have been some stars in the literature with reported spot modulation of over one magnitude. For example, DR Tau has had several reported periods ranging from 2.8 to 9 days (Bouvier et al. 1995). Its peak to peak variation ranges from 1.5 magnitudes at I to 3 magnitudes at U. It is not possible to produce such large and color sensitive variation using cool spots. Bouvier et al. conclude that, in this case, the modulation is caused by hot spots, presumably due to accretion. Herbst et al. (1994) suggest more generally that variations between 0.5 and three magnitudes are usually the result of hot spots, variable accretion or variable circumstellar obscuration. Eaton et al. (1995) discuss large amplitude variables (LAVs) and found similar results.
In Figure 7, the results of the simulation
are displayed in terms of the change observed in R and I relative to change
in V. One can see that these values are all 
1. Values close to unity
indicate stars with very cool spots relative to their photosphere.
It is interesting to note that spotted stars occupy a fairly small triangle
of parameter space. The right--hand side of the triangle is defined by the M4
stars, the left--hand side by the K0 stars, and the bottom by stars with spots
just slightly cooler than their photosphere and large spot filling factors.
Another result is the apparent uniqueness of any given point in the parameter
space. Each point on the plot corresponds to
exactly one combination of spectral type, spot
temperature and filling factor (the uniqueness breaks down as
, 
and 
approach unity).
While this uniqueness quality provides hope for the future determination of spot
properties based solely on photometric data, it should be pointed out that
this would require signal to noise in the modulation of about 100, which is a
factor of 20 better than were obtained. Also, the
model uses only a single spot temperature which is a vast over--simplification.
The color behavior is an important criterion for period determination.
The change in the bluer color must be greater than or equal to the
change in the redder color. The details are far too subtle any useful
study to be made with the data here due to insufficient signal to noise.
