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Introduction:

Because ingress and egress are essentially linear, a technique using Linear Regression can determine the best fit of a straight line for a set of data points. This best fit can be expressed as an equation so any Y value (magnitude) can be used to calculate the corresponding X date (RJD). This method is sensitive to what the out-of-eclipse average magnitudes are and with the out-of-eclipse variability that makes it difficult. However, a fair average can be determined from December 2003 to January 2009.

The following data are all from the Hopkins Phoenix Observatory observations. Other observers are encouraged to use their data to determine contact points.

Second Contact is very sensitive to the Totality Magnitude and for the most accurate determination will need to wait past third cognac and determine average magnitudes with the mid-eclipse brightening disagreed.

Linear Regression

The equation for a straight in is in the form of:

Y = mX + b

where Y is the vertical axis value, X is the horizontal axis value, m is the slope of the line (DY/DX) and b is the Y intercept (where the line incepts the Y axis). Each point on the line has a X and Y value.

By taking a series of data points that have Y (magnitude) and X (RJD) values, a technique known as linear regression can be used to calculate the slope and Y intercept. This then determines an equation for the set of values.

From that one can determine the point the line intercepts different Y values (magnitudes) and the corresponding X (RJD) values by re arranging the equation as:

X (Contact date) = (Y * m)/b

Where X is the Contact Date and Y is the magnitude at that point.

Using this technique it is possible to determine eclipse contact points. The following is an image of a FileMaker Pro sample program designed to run a linear regression on a series of photometric data points to determine the slope (m) and Y intercept (b). Then by adding the magnitude at which point the RDJ is desired that date is calculated.

 

 

2009 - 2011 Eclipse

First Contact Estimations
as of 22 October 2009

JD Dates are RJD plus 2,400,000

Out-of-Eclipse Averages are assumed to be:
V = 3.0360
B = 3.6050
U = 3.7264

From the Linear Regression Calculation:

First Contact
V Band = 5,5065.890 ** 22 August 2009
B Band = 5,5067.772 ** 24 August 2009
U Band = 5,5073.143 ** 30 August 2009

2009 -2010 Eclipse

Second Contact Estimations
as of 22 October 2009

JD Dates are RJD plus 2,400,000

Totality Averages are assumed to be:
V = 3.746
B = 4.305
U = 4.516

From the Linear Regression Calculation:

Second Contact
V Band = 5,5183.368 ** 18 December 2009
B Band = 5,5174.797 ** 09 December 2009
U Band = 5,5170.836 ** 05 December 2009