Below you will find instructions on how you can complete your analysis of photometric data. We will guide you through the process of converting your instrumental magnitudes into calibrated magnitudes while accounting for the presence of the atmospheric extinction. We assume that you have completed the Beginner tutorial and therefore already know how to compute several of the required quantities. After reduced, your data will be well-suited for submission to the AAVSO for long-term storage.
What you will need:
A list of instrumental magnitudes for your variable star
A list of instrumental magnitudes for at least three comparison stars (six preferred)
An Internet connection (to lookup catalog values)
The date, time, and location at which your observations were conducted
Excel or OpenOffice Calc
Computing Calibrated Magnitudes
Unlike the beginner tutorial, this document is suited for a wide range of calibrator-to-variable distances and zenith angles < 85 degrees. This tutorial will introduce the concept of Air Mass and guide you through the process of correcting for its presence in your data. This tutorial is designed specifically to be used with the Intermediate-level sample data, although it can be used with the Beginner-level data as well as the data you collect using your camera. We have attempted to keep the look and feel of this tutorial similar to the Beginner tutorial in order to ease the transition to the more complicated procedure. Steps 1 and 2 provide files and background information. It is likely that these steps will only need to be read once. Steps 3 and onward are things you must do every night to ensure proper photometric correction.
If you do not have it, download the Intermediate-level reduction spreadsheet (Excel, OpenOffice) and open.
In cells B12 and B13, enter the latitude and longitude where you took observations. (The sample data contains the latitude and longitude in the Readme file). To determine your latitude and longitude you can use a GPS device or Google Maps (center your observation site in the map and click the link button. The latitude and longitude are encoded in the URL beginning with “ll=”. The Mt. Evans observatory has ll=39.586963,-105.641012 so the latitude is 39.586963, and the longitude is -105.641012)
If you have used the beginner-level reduction spreadsheet, there are a few differences. For instance some of the colors of the cells have changed. In this spreadsheet, blue cells have data which you need to enter from your observations. Green cells should contain data you look up in a catalog. Yellow cells are ones that you should check during reduction, and orange cells are the results of the reduction process that could be reported to AAVSO if the results make sense.
In the Beginner Tutorial you were introduced to the concept of transformation coefficients and zero point offsets. Here we consider how the atmosphere effects your data by including air mass calculations.
Wikipedia has a very nice description of what air mass is, what it does, and how to compute it so we will not elaborate much further. The basic principle is that as light goes through air, a little bit of it gets scattered away. The more air in between you and the star, the more light gets scattered. Fortunately several atmospheric models exist to account for this form of extinction, we just need to modify our equations to account for it's effect.
The mathematics used to compute the V-magnitude when the atmosphere is present are very similar to the equation presented in the beginner tutorial, except it has one additional term and lots of subscripts:
(V - v)i = -k' Xi - e * (B-V)i - Z
Here the subscript "i" indicates these values are measured with respect to the ith star in the field of view. Just as before this equation describes how the catalog V-magnitude (capital V) changes in response to several parameters. From left to right, they are:
V = (upper-case V) the catalog value for the V magnitude of the star
v = (lower-case v) the instrumental magnitude of the star
e = the color transformation coefficient
(B - V) = the mean catalog "color" of the star
Z = your camera's zero-point offset from the standard V filter.
-k' = atmospheric extinction coefficient (more on this later)
X = the air mass for the star
For our calibration stars, the upper-case variables (V and B) are known and with a measurement their instrumental magnitude, v, is known too. The air mass, X, may be computed (more on this later) from your location on Earth, the date and time of the observation, and the coordinates of the target star. This leaves three constants that haven't been determined, e, Z, and -k'. The atmospheric extinction coefficient, -k', describes how much light (in magntiudes) is lost per unit air mass (therefore it has units of magnitudes per air mass).
For those of you who have studied geometry, the above equation looks a lot like the equation of a plane in 3D:
z = Ax + By + C
where the x and y positions and constants (A, B, and C) determine the height of the point. In our case the constants are -k' (A), e (B), and Z (C). Because we have three unknowns in the equation, linear algebra requires that we have at least three calibration stars in order to solve for the coefficients, but what happens if one of the calibration stars is mis-measured? We get the wrong coefficients!
It would be nice if we could measure several stars simultaneously and derive the coefficients from that. Fortunately there is a method called least-square fitting that permits us to do so. The mathematics behind this method will be described in the Advanced Reduction tutorial (and in a corresponding JAAVSO article from the DSLR Documentation and Reduction Team), so we will just mention that this method minimizes the chance that a outlying data point can significantly alter the coefficients.
Calibrating the data is very similar to the beginner tutorial, except air mass must now be computed.
Step 3 (once for observing campaign):
Fill in the star names and their corresponding green cells for the calibration standards page. When you are done your sheet should look similar to what is shown in the image below:
The Intermediate-Level Reduction File already has a proposed selection of calibration stars that are reasonably close to Epsilon Aurigae. But you can edit in your own favorite stars, but make sure to insert all required data correctly (color index, magnitude, sky-coordinates). You can look up other well-calibrated stars in the Calibration Standards page.
Step 4 (this and following steps: for every observation night):
Fill in the date and time of your observation in UTC/GMT near the top of the sheet in the cells B8-G8. Make sure to account for daylight saving time if applicable when converting your local date and time to GMT. Note: For an accurate correction of extinction effects, the observation time should be accurate to at least a few minutes. If you are using the timestamps embedded in your DSLR's images, make sure the clock of your DSLR is set accurately.
Step 3 :
Now fill in the Right Ascention (RA), Declination (DEC), Catalog V-value and Catalog B-V value from the calibration standards page for your calibration stars. Do this in Cells C19-F24. Repeat this process for the check star (cells C28-F28), and the target object (cells C33-E33).
Now that the catalog values have been filled in, we need the instrumental magnitudes for the calibration stars. Enter these values into cells B19-B24. When you are done, the spreadsheet will have automatically computed a least-squares fit to the data. This least-squares fit is reported in the tables entitled “Planar Fit Table w/ Extinction” and “Lin Regr. w/o Extinction” located between rows 40 and 44. The mathematics behind these tutorials will be discussed in the Advanced Tutorial when it is written as well as in an upcoming article in JAAVSO. You are, of course, welcome to ask us about it in the Forums to.
After -k', e and, Z are computed, you may now insert the instrumental magnitud values for the check star and target object in the Target and check star Calibrated Magnitudes Table (the air mass values for check star and target star are calculated automatically for you by the spreadsheet). Here we use the values of -k', e, and Z to compute the calibrated V-magnitude for the stars using a rearrangement of the above equation:
V = vi + -k' Xi - e * (B-V)cat - Z
Now compute the average and standard deviation of the check star's "Check V Mag" column and "V Mag" column using the same procedure as discussed in the Beginner Tutorial.
Lastly we need to make sure our answer makes sense. Compare the average and standard deviation of the check star to its catalog value. If the catalog value falls within the average value plus or minus a standard deviation, you have a very good result and the calibrated magnitude for the target object can be considered valid. If not please post in the Photometry section of the forum indicating that you had a problem with the tutorial and we'll try to help you.
Congratulations you are done calibrating your data. There are a few more useful bits of information below that can help you determine if your data is of good quality. Once you have read those, proceed on to the Imaging and lastly, how to submit your data tutorials.
Below we discuss some of the features we added to the spreadsheet in order to make it easier to identify erroneous data.
We have integrated three plots that can show if the spreadsheet is functioning normally:
Residual Mag vs. Color Correction
Residual Mag vs. Air Mass
Residual Mag vs. Full Correction
All three of these plots show the redisuals, what is left over, after the given fit. The most important quantity to look at is the R-squared term in the linear fit and the slope of the line.
If the R-squared is small (i.e. less than 0.7) then it is likely that there isn't any underlying effect left in the data. If you find an R-squared bigger than 0.7 your data might be showing second-order effects. For instance, in the image above you can see that there appears to be something left over in the air mass correction. This is anticipated because air mass correction should have both a first-order (X1) and a second-order (X2) correction coefficient, but we have only included a first-order coefficient in our spreadsheet (we needed to leave something for the Advanced Tutorial!). Also, pay attention to the slope of the line. If the slope is near zero (i.e. a mostly horizontal line) it is likely that the second-order corrections would have a minimal effect.
This spreadsheet also has some built-in indicators that try to help you determine the quality of your data. If you are using more than three calibration stars, they will not all fit perfectly to the computed calibration coefficients. To see if one star is causing a bad fit, check the stars lited in the Data Quality Check table. This will show you the V-magnitude for the calibration star as computed by the fit, as well as the error from the catalog value. Through some experimentation we have found good DSLR data will have maximum errors in the 0.01-0.05 mag level. If your errors are significantly bigger, please contact us in the forums and we'll help you determine what could be causing the problem..
Disabling the extinction correction
Although we have tried to make the spreadsheet as error-free as possible, the least-squares fit can sometimes do funny thing is low air-mass scenarios. If you know you are observing near your zenith (i.e. at angles less than 30 degrees from the zenith), try turning off the extinction correction by entering a zero into cell B47. This make the spreadsheet behave like the Beginner-level spreadsheet by applying no air mass corrections. Be sure to double-check that the errors go down and make sense.
The spreadsheet allows you to apply the calculated calibration results (color transformation coefficient, extinction coefficient and zero point) to several measurement of instrumental magnitude. So during an observation night, you can do one set of measurements for calibration (measuring the calibration stars) and then re-use the results for many additional measurements which need only include the check star and target star. However, atmospheric conditions can change even during a long observation night, so you should not re-use the calibration results from one night for measurements of a different night. You need to do the calibration for extinction effects on every observation night to get good results!