The Size, Structure, and Variability of Late-Type Stars Measured ...

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second free parameter, A, to be the fraction of light coming from the stellar disk itself. In
this case, the visibity function for the range of resolutions reported is given by:
V (x, r, A) = 2AJ 1(2πrx)
2πrx
(2.1)
where x is the spatial frequency in rad −1 , J 1 is the Bessel function of order unity, and r
is the radius of the stellar disk in radians. The best-fit radius is given by the minimum of
χ 2 (r, A) where:
χ 2 (r, A) =
∑ N
i=1 (V 2 (x i , r, A) − V 2
i )2
σ 2 V 2
i
(2.2)
The square of the visibility was fit as opposed to the visibility because the former is the
directly measured quantity and its fluctuations are distributed normally. 4
It should also
be noted that any error in the calibration of the visibility data will be absorbed into the
parameter, A, when fit, and will not affect the diameter determination.
The error in the best-fit stellar disk radius, σ r , was estimated by considering the
region of the A-r plane in which the normalized χ 2 is increased by no more than unity
from its minimum. σ r was taken to be one-half the width of this region in r. This general
procedure is outlined in Bevington, 1969 [13]. The use of the normalized χ 2 causes the fit
of the stellar radius to have an error which depends only on the spread of the data about
the best fit and not on the magnitude of the σ V 2
i
.
An example of a uniform disk fit to a relatively good observing night’s data on
o Cet is shown in Figure 2.1. Figure 2.2 shows the same data averaged so to be clearer
visually. Often, as in this case, a single night of observations yields enough data onto which
we can reasonably fit a uniform disk. For weaker stars, or in worse seeing conditions, we
may combine several nights together before fitting. There is rarely a case in which we will
fit data spanning more than a week or so together.
The distinction between fitting a uniform disk to the data and implying that the
data represents a uniform disk should be noted.
The ISI employs effective baselines of
about 20 - 56 meters. This corresponds to angular resolutions between ∼20 and 60 mas.
The “sharp” edge of a uniform disk profile is too narrow a feature to be resolved with the
ISI. The correct interpretation of our data (as a certain sized disk) is conditional on the
4 At spatial frequencies near the null of the visibility function, half of the measured data points lie below
zero (because of random fluctuations in the fringe power). Their square roots would not be able to be fitted
to a uniform disk visibility curve.