# Finding variables

The Find variables tool is based on the dependency of noise level to mean brightness. The fainter an objects is, the smaller is the signal-to-noise ratio and the more noisy is the measurement of the object. It can be shown that two objects of similar brightness which do not vary in brightness during the observation also have similar noise levels, which is computed as the sample standard deviation. If one of the objects changes its brightness due to reasons other than measurement noise, for example it is an eclipsing binary, its sample standard error is greater than a constant object of the same mean brightness. Using this idea, we can make a graph of sample standard deviation vs. mean brightness of all objects detected on a frame set. Supposing the majority of objects are constant, the graph will reveal a curve, which grows towards lower brightness. Any outlying point located above the curve denotes an object that has brightness variations other than measurement noise and thus it might be a variable star.

The Find variables tool does not find variable stars automatically. It presents the graph described above and provides a means of selecting any object on the graph and showing its light curve. The decision whether an object is a variable or not is left to the user.

Making the mag-dev graph

A list of source frames is analyzed and raw instrumental magnitudes stored in memory. Each detected object must be uniquely identified across all the frames. One star is chosen as the comparison star and we assume that this star was constant during the observation.

However, an object might not be measured correctly on all frames in a frame set. If the comparison star has an invalid measurement on one or more frames, we cannot derive a differential magnitude for any object on those frames, so we have to reject these frames as whole. For the rest of the frames, it may happen that an object does not have a valid measurement on one or more frames. In this case we cannot derive a differential magnitude of that particular object on those frames.

Let us denote N as the number of source frames and N_C as the number of frames where a comparison star has valid measurements. For each object i we can derive a number of frames N_i where both the object and the comparison star has valid measurements. If N_i is less than a minimum number of frames N_{min}, we will rule out the object from the output. The minimum number of frames N_{min} is calculated from N_C and configurable threshold t in units of percent.

(1)N_{min} = \lfloor{\frac{t}{100}\,N_C \rfloor}

For each object i that has a valid measurement on frame j, we compute the differential magnitude m_{i,j}:

(2)m_{i,j} = mag_{i,j} - mag_{C,j}

where mag_{i,j} is instrumental magnitude of the object i and mag_{C,j} is instrumental magnitude of the comparison star C on frame j. Then, the mean value \overline{m_i} is computed by means of the robust mean algorithm and the sample standard variance s_i^2 is also determined. Please note, that although the robust mean algorithm is used to compute mean brightness of an object, all valid measurements should be taken into account in the computation of the sample variance. Otherwise for a variable star, some of its valid measurements would be unjustly rejected as outliers, resulting in lower variance. The term \frac{N_i}{N_i-1} is known as Bessel’s correction.

(3)s_i^2 = \frac{N_i}{N_i-1}\,\frac{1}{N_i}\,\sum_{j=1}^{N_i}(m_{i,j} - \overline{m_i})^2 = \frac{1}{N_i-1}\,\sum_{j=1}^{N_i}(m_{i,j} - \overline{m_i})^2

Figure 1 shows a scatter graph of stars that were measured in the field of eclipsing binary CR Cassiopeiae showing the relationship between the stars’ mean magnitude x_i versus their sample standard deviations s_i.

Mag-dev graph of stars in the field of eclipsing binary CR Cas. The majority of stars are constant forming a typical “background” curve. The variable star CR Cas, labeled var, has greater standard deviation than stars of similar mean magnitude.

Choosing a comparison star

If the user does not pick a comparison star, the program uses the following algorithm to choose one and uses it to make a mag-dev curve. There are several criteria that the program can use to determine the suitability of an object to be a comparison star. A good comparison star should be 1) on the greatest number of frames to avoid more data rejection than necessary, 2) it should not vary in brightness and 3) the noise of its measurements should be low to reduce a noise of differential magnitudes and the resulting data.

Let us take all objects that were found on a reference frame. We assign a number of frames F_i which each object $i$ has been successfully identified on. The maximum of values F is determined and all objects that have the maximum value of F are kept on the list, the others are removed.

For each \frac{N(N-1)}{2} pairs of objects that were left on the list, a differential light curve is derived and a robust mean and a sample standard deviation is computed for the light curve data. Finally, the star that has the lowest sum of standard deviations is picked as the comparison star.

Please note, that although the robust mean algorithm is used to compute mean brightness, all valid measurements should be taken into account in the computation of the sample standard deviation. If there were a variable star on the list of candidates, some of its valid measurements would be unjustly rejected as outliers, resulting in lower deviation.