Orbital elements of binary components in a hierachical system can be fit to either the visibility data directly or to the separation and position angle data. In the first case, the orbital motion during the observations of a single night in case of short orbital periods are handled properly. In our example, Mizar has a 20 d orbital period, so is not very fast and we will choose the second method.

Initial guess

Since fitting orbital elements is a non-linear problem, it has to be solved iteratively beginning with a good initial estimate.

Begin by changing the binary mode parameter to "1".


Then use Data|Astrometry to read a previously prepared file ("zetuma.psn") containing the positional data. Do this by selecting the (only) file (zetuma.psn) displayed, and then by clicking the "OK" button.

After that, type:

positions.emajor=positions.emajor/30 positions.eminor=positions.eminor/30

It so happens that the major and minor axes of the uncertainty ellipses are too big by a factor of 30 in the data file.

Use Plot|Astrometry to bring up the plot widget for positional data and click "Plot|Screen".

With Util|Ellipse we define roughly an apparent ellipse as shown below, and then use Fit|Astrometry|Ellipse (after selecting all parameters for the fit) to optimize the ellipse parameters.

Re-plot the data (make sure you select the "Ellipse" option) and see the result.

Center x = 1.9497254
Center y = 2.0485887
Semi-major axis = 8.4214471
Semi-minor axis = 4.7663976
Position angle = 101.95642

We can now use the Thiele-Innes method to estimate the orbital elements from this apparent ellipse and the data, as implemented with the Util|Orbit button.
Orbit is PROGRADE. Thiele-Innes estimates:
Semi-major axis = 9.8266080
Eccentricity = 0.53497164
Inclination = 60.554283
Periastron = 104.53198
Ascending node = 105.81683
Period = 20.528109
Epoch = 10203.557
Initial chisq= 2.5939877
Final chisq= 2.3497009
Iter= 8, Alamda= 1.00, Status=0

Finally, we use Fit|Astrometry|Orbit to improve these initial estimates through fitting, and the results should look like these. In order to get the orbit plotted (plot option "Orbit"), you have to transfer the orbit parameters to the general model parameters using the "Set Model" button in the Fit|Astrometry widget, and type calcmodel to compute the model data.

Semi-major axis = 9.8441526
Eccentricity = 0.53703750
Inclination = 60.629149
Periastron = 104.25061
Ascending node = 105.95661
Period = 20.538443
Epoch = 7636.0598

Alternatively, we could have used the "Fit" feature of the general model; as it happens, both were implemented over the course of time.