Historical Data Analysis
Adapted from Kalée Tock and Ryan Caputo
After measuring your double star system and determining its physical properties, the next crucial step is to compare your results with historical observations. This analysis helps you understand how the system has changed over time and provides insights into its orbital behavior.
Note
For this assignment, you may choose to use this Python script to make your plots.
Working with Historical Data
Once you receive the historical data on your star system back from the US Naval Observatory, you should compare your measurement to the most recent measurement of the system and/or to the PA and sep measurements that apply to the star coordinates from Gaia. One way to do this is to visually plot the historical data of the system. In order to do this, you will need to get the PA and sep measurements out of the historical data file and into a spreadsheet. What you will find when you open the historical data file is shown in Figure 1 below:
[FIGURE 1: A sample double star historical data file from the US Naval Observatory.]
This file tells me that the first measurement, made in late 1824, had PA 205.0 degrees and sep 7.33 arcseconds. To plot this on the Cartesian plane, I would take the horizontal coordinate as:
x = Sep * sin(PA)
and the vertical coordinate as:
y = -Sep * cos(PA)
This translates from polar coordinates to Cartesian coordinates, where PA is the angle measured from Celestial North. But watch out: most spreadsheet programs assume that the position angle is in radians! Convert the PA from degrees to radians by multiplying by pi / 180.
Data Import Tip
Importing your data into a spreadsheet can be helpful to avoid copy-pasting individual points by hand. Most spreadsheet programs have functionality for importing this text file since the fields in the WDS historical data files have a fixed width. If you copy-paste the data columns into Google Sheets, you can use Data → split text to columns to get the numbers into the appropriate cells. Note that separation is the fourth column, not the third column, if you do it this way. (The third column is "position angle uncertainty.")
Creating Visualization Plots
If you change all of the historical measurements into Cartesian coordinates as demonstrated above, you can put them into a spreadsheet program such as Google sheets and make a scatterplot. Then, you can predict where the next measurement should fall. When you make your own measurement, you will plot it with the other points and compare its location to your prediction. Be sure to use a distinctive color or marking for your own measurements, as is done in Figure 2 below (note: the plots below are different systems from the sample data file labeled above).
[FIGURE 2: Historical data for two double star systems, with the team's measurement as a green square with an X. Darker points are more recent.]
Plot Formatting Guidelines
Also, when you make these plots, be sure to scale the axes so that both the vertical and horizontal axes have the same range. For example, if the horizontal axis goes from 3.5 to 6.5 arcseconds in RA, then the vertical axis might go from -7.5 to -10.5 arcseconds in Dec. Adjust the axes so that the data fills the maximum amount of screen real estate on the plot. For example, you would not want to use (0,0) as your origin for either of the plots above, because if you did, then the data would look like a teensy jumble in one small region of the plot.
Analyzing Trends Over Time
One important factor to consider with these plots is the element of time. Sometimes, scatter plots that appear to follow a trend actually do not, because the early and later measurements are all jumbled together. For example, compare the two plots of historical data shown in Figure 2. The points are color mapped according to time, with darker points being more recent. The plot on the left of Figure 2 shows a clear trend: the secondary is moving toward the right with respect to the primary as time passes, because you can see the points becoming darker from left to right. The plot on the left does not show a discernible trend in time, because the light and dark points are intermingled.
For a first look at your data, do not worry about color mapping your plots--the point of the exercise is to see how the parameter of time informs your interpretation of the plots that you construct. The question is: do the points follow a trend over time or not? If so, what sort of trend? Perhaps the simplest way of doing this is to look at a graph of your points and ask yourself: where is the first point in the series? Where is the last point? Use the X and Y coordinates of those points to see if you can locate them on the graph. Then for a few (maybe 4 or 5?) points in the middle, find the points in the graph in the same way. Do they fall where you expect, if the system is moving in a consistent way over the course of time? There are many ways to visualize this, but they all come down to staring at the numbers and thinking about what they mean.
Special Considerations
If you are imaging systems that have a small separation (less than 5 arcseconds), you should also correct the early measurements for the axial precession of the Earth since the time of the measurement. This is not a significant issue for the wider systems we are describing here.
Looking for Orbital Motion
Finally, if you are looking for evidence of curvature that might indicate an orbit, one technique is to fit the data with a linear versus 2nd order polynomial regression. If the 2nd order polynomial fits the data significantly better than a line, that increases the likelihood that the two stars have a gravitational relationship. However, before you do this, you will want to make sure that any curvature you notice curves around the 0,0 point of your plot. This is because the primary star is at the origin. If the position of the secondary as a function of time curves in such a way that it does not enclose the primary star, then any curvature you see cannot indicate that it might be bound to the primary.
When performing these fits, it is often useful to weight the historical measurements according to how accurate they are likely to be. For example, telescopes with larger apertures are generally more accurate, measurements made over multiple nights are more accurate, and measurements made with more modern methods are (usually) more accurate.
The effect of time, the correction for axial precession, and the implications of weighted linear vs. polynomial fits are all factors that should be considered for a thorough double star investigation. However, the technique described above is adequate to give you a rough sense of what your stars have been up to since they started being measured.
Submission Requirements
Please insert your plot into this template. You are welcome to use whatever plotting program you like. Here is a checklist that you should use for each of your plots:
Double Star Plots Checklist: - Plot is titled with the name of the system. - Axes are labeled (horizontal is arcseconds of RA, vertical is arcseconds of Dec). - Axes are scaled so that they have the same range (see above). - Plot is scaled to spread out the data points so that they cover as much screen real-estate as possible (see above). - Your measurement is labeled with some distinguishing color or marking. - Gaia point is labeled with some distinguishing color or marking.
Once you have made your plot, answer the questions on the template, and then save it as a pdf and turn it into Canvas as a single file.
Next Steps
After analyzing historical data and your own measurements, you'll integrate this information with your physical analysis to form a complete picture of your double star system's nature and behavior. This will become the foundation of your scientific paper.