IX. Data Processing

A. Remote processing of the observer data

B. Data Transmittal

Each observer transmits their separation data sheet(s) (filled out Attachment A) to Mizar Consulting by private e-mail, via the Quad-A group mail, or by postal service.

When four to five sets of separations are obtained, it may be beneficial to have a preliminary review of the data performed. The spreadsheet used has some capacity to predict Jovian moon positions, a capability that assists in moon identification in later observations.

C. Effects of Changes in the Distance to Jupiter

As the Earth and Jupiter move in their respective orbits (the orbits are indicated by circles) the apparent size of Jupiter changes with time. When Jupiter is at Opposition, its apparent size is larger than when it is at Conjunction. See the diagram on the next page for a visual representation.

For other than the measurements using the JD method, the EXCEL spreadsheet compensates for the changes in the apparent angular size of Jupiter (See Attachment G). Thus, the observed separations are normalized to a common Earth-Jupiter distance.

D. Effect of Planet Positions

During the observations, both Jupiter and Earth have orbital motion. The orbital motion not only changes the distance between the planets, but also the angle of viewing. As the angle between Earth and Jupiter changes the observer sees a different orientation of the orbit of the moon of Jupiter. At the beginning of the observations, for a moon at its maximum Eastward extent, let the position of the moon be 0°. The maximum Westward extent would then be 180°.

As the observations continue the angle between Earth and Jupiter changes by some amount, say 16°. Thus at the end of the observations the maximum extents are now 16° and 196° (each extent progressed by 16°).

If an orbit period was equal to the observation time, then the moon would have begun at 0° and ended at 196° (more than 180°). If the orbit period were half the observation period, then the moon would have begun at 0° and ended at 188°. Thus while each orbit appears to have 180° between maximum extents, the true travel angle slightly different. The observed data is modified to account for this effect by the EXCEL spreadsheet.

E. Goodness of Fit

For Project Jupiter the best estimate of the orbital period, and other parameters, is determined mathematically from the observation data. Mathematically there are several measures that may be used to judge when the observed data is best reproduced by the computer. Prior tests have shown that the true elliptical orbit variation from an assumed circular orbit need not be considered for Project Jupiter.

Four measures of the Goodness of Fit used in the computer model are:

1. Standard Deviation

One measure of the fidelity of the data ( how well it is trending) is the statistical measure called the Standard Deviation. The larger the Standard Deviation, the larger is the variation in the data being considered.

When the orbital period is being estimated, the observer’s moon positions are subtracted from the position computed by the EXCEL program. As the EXCEL program inputs begin to match the observer’s data the Standard Deviation value begins to drop.

The value does not get to zero because of observational biases, errors, transcription mistakes, difficulty in estimating small separations, etc. The EXCEL program computes the Standard Deviation, so the Quad-A observer need not be concerned with the mathematics.

The concept here is that the statistical measure called the Standard Deviation is used as an indicator of when the orbital inputs best match the observed data. When a good match is found the Standard Deviation value reaches a minimum.

2. Correlation Coefficient

The Correlation Coefficient is a statistical comparison of two sets of data. The Correlation Coefficient ranges from -1 to +1, with ±1 representing the strongest (best) similarity measure ("correlation").

The observer’s Project Jupiter separation data is compared with EXCEL generated data for a sinusoidal curve. When a computed "correlation coefficient" approaches 1.0 its is an indication that the computed separations are strongly matching the observed separations.

One of the EXCEL inputs for the sinusoidal formula is the orbital period, so a high correlation coefficient is one indicator that the orbital period input is the best estimate possible. As the Correlation Coefficient value approaches 1.0, it is also confirmation that the orbit is not strongly elliptical, validating an earlier assumption.

3. Least Squares

When two data sets are compared, one measure of their similarity is to examine the differences in the individual data values. By taking each value and multiplying it by itself ("squaring it") negative values do not cancel positive values, and the check becomes a sensitive check of the similarity of data sets. When the sum of the squared values is a minimum, the best agreement between the data sets has been determined.

The EXCEL generated data is automatically subtracted from the observer’s data, squared, and then summed. The program automatically displays the result so that the user can know when the minium value has been found. This provides added assurance that the best determination of the orbital period, based on the transmitted Quad-A observer’s data, has been found.

4. Residuals

Residuals are the differences between what was observed and the mathematical representation of the process observed. Usually the residuals themselves do not have a pattern to then, they are randomly distributed. The Project Jupiter analysis makes reasonable graphical efforts to ensure that no residual biases impact the determined orbital period.

F. Processed Data

The observer’s data is be processed into a graphical format, similar to the below image. The EXCEL program will be used to determine the orbit period rather than subjectively reading from the graph.