# Artificial comparison star

The artificial comparison star is used to enhance a quality of a light curve by utilizing multiple stars that are used to compute differential magnitudes of a variable star.

A new virtual comparison star is created from a set of stars and its brightness in magnitudes and error estimation are derived for each frame. Then, a normal course of light curve construction is followed. The brightness of an artificial star is computed by averaging the intensities (not magnitudes) of stars from a set. The same set of stars must be incorporated in the average on each frame, otherwise the results were incorrect. Therefore, if there is missing measurement for one of stars, it is not possible to compute the brightness of a comparison star for this frame.

Deriving brightness of the artificial comparison star

First, we use inverse Pogson’s law to convert brightness in magnitudes m_i into intensity I_i for each star i from the set.

(1)\frac{I_i}{I_0} = 10^{-0.4\,m_i}

Next, intensities are combined by means of arithmetic mean. Please note, that the factor 1/N is used to keep the brightness in the same range as the source data.

(2)\frac{I}{I_0} = \frac{1}{N}\,\sum{\frac{I_i}{I_0}}

Then, the intensity is converted back to magnitudes:

(3)m = -2.5 \log_{10}\left(\frac{I}{I_0}\right)

Estimating the measurement error

The inverse equation (17) can be used to transform the error estimation from magnitudes to intensity unit. Supposing that noise sources are independent, we can compute the variance of I as:

(4)\sigma_{I}^2 = (\frac{1}{N})^2\,\sum{\sigma_{I_i}^2}

The equation (17) is used to convert the resulting variance back to magnitudes. Putting all three formulas together we get the following formula for the error estimation of the resulting brightness of a artificial comparison star in magnitudes:

(5)\sigma_{m} = \frac{\sqrt{\sum{(\frac{I_i}{I_0}\sigma_{m_i})^2}}}{\sum{\frac{I_i}{I_0}}}