Now that you have obtained instrumental magnitudes for the variable and comparison stars the next step is to perform the analysis to get calibrated magnitudes.
Below you will find instructions on how you can complete your analysis of photometric data. Below we will guide you through the process of converting your instrumental magnitudes into calibrated magnitudes that can be submitted 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 two comparison stars (six preferred)
An internet connection (to lookup catalogue values)
Excel or OpenOffice Calc
Computing Calibrated Magnitudes
We must first stress that this tutorial is suitable for calibrator-to-variable distances less than 3 degrees at zenith angles less than 34 degrees. Outside of this range, Air Mass must be taken into account to correctly calibrate your data. If your data does not meet these qualifications, please see the intermediate tutorial. The sample data provided by the DSLR Documentation and Reduction team is fine to use here.
Download the Reduction for Beginners file that the DSLR Documentation and Reduction team has created for you.
Fill in your name to the right of Observer, the Julian Date (a.k.a. JD) calculated using the USNO Julian Date Converter, and the time of your observation in UTC/GMT.
As you may have noticed, the spreadsheet contains cells filled in with yellow, green and orange. The green cells correspond to values looked-up in a catalogue, yellow cells are instrumental magnitudes that you provide, and orange cells are calibrated V magnitudes that *could* be reported to AAVSO.
As was mentioned in earlier portions of the tutorial, DSLR cameras record values in each of three colors: red, green, and blue. In the previous steps, you extracted the green portion of the Bayer array because it most closely corresponds to the photometric V-filter. The problem is that even though the filters are close, they are not exactly the same. Here we compute the necessary correction factors to convert your camera's Green filter into a V-filter.
|Read More: Transformation Equations
Astronomers have already created a transformation equation that accounts for this change:
V = v - e * (B-V) - z
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 catalogue value for the V magnitude of the star
v = (lower-case v) the instrumental magnitude of the star
e = the color transformation coefficient (more on this later)
(B - V) = the mean catalogue "color" of the star
z = your camera's zero-point offset from the standard V filter.
For our calibration stars, the upper-case variables (V and B) are known and with a measurement the instrumental magnitude, v, is known too. This leaves only unknowns in the equation (e and z). With at least two calibrators and a little algebra we can determine both the transformation coefficient and zero-point offset.
Rearranging the above equation, we have:
V - v = -e * (B-V) - z
Which, with a few more parentheses:
(V - v) = (-e) * (B-V) + (-z)
Looks a lot like the equation of a line:
y = m * x + b
In our case, y = (V - v), m = (-e), and b = (-z). If we plot a series of calibrators in which the V, B, and v values are known and fit a line to them, we can pick off the slope and intercept and solve for 'e' and 'z'! There are, of course, other methods to solving this problem, but doing it this way allows you to solve for 'e' and 'z' simultaneously without doing any algebra! (It also allows you to solve for uncertainties in the slope and intercept which will be used in the advanced tutorial that includes a little error analysis).
Before we can solve this problem though, we must know V, B, and v for the calibration stars. As you recall, v is simply the instrumental magnitude for the comparison star your measured so we only need to find V and B. Therefore, we'll fill in these values in the spreadsheet using the calibration tables provided. If you always observe the same stars with your camera, you will only need to do this step once and re-use the data during later calibration sessions
As noted earlier, we have already chosen calibration stars for you. Look up the V Cat and (B-V) Cat values for the stars in the Target Star Table, Check Star Table, and Comparison Star Table and enter those values into the appropriate green cell. When you are done, your sheet should look something like what is shown below:
Now that you have filled in the V and B-V values, we simply need to insert the instrumental magnitudes for the comparison stars that you found earlier in the tutorials. Type in the values you found earlier in the yellow cells in the Comparison Star Table. As you fill in these values, you will notice the Transformation Coefficient graph to the right changing.
A “best fit” line will appear on the graph along with the data points. The slope of this line is the Transformation Coefficient (TC). The y-intercept is the Zero Point.
|Read More: Slope and Intercept
|After all of the values are inserted the graph will stop changing. We have used two functions (SLOPE and INTERCEPT) to extract the slope and intercept from the graph to make analysis a little easier. You should notice that the line displayed in the graph has the same slope value as the cell to the right of 'm' in the transformation coefficient (TC) table. Furthermore, the zero-point intercept in the graph is to the left of the 'b' cell in the TC table
Now we may insert check star and variable star insturmental magnitudes in order to compute a check star V-magnitude and variable star V-magnitude.
Type the instrument magnitudes for your check stars in the yellow cells of the “Check I Mag” column and for your variable star in the yellow “I Mag” column cells.
Here we apply the equation listed earlier:
V = v - e * (B-V) - z
With with (B-V) as defined earlier and 'e' and 'z' as determined from the graph / TC output table. To do so, we fill out the Target and Check Star Calibrated Magnitudes table. Fill in the image numbers used in the "IMAGES" column, and the instrumental magnitudes (I Mag) for the check and comparison stars in the yellow columns. You may need to drag down the formulas in the "Check V Mag" and "V Mag" columns if you have several images
Now we compute the average and standard deviation of the check star's "Check V Mag" column and "V Mag" column. We will first do this on the "Check V Mag" Column:
In the cell immediately below the Target and check Star Calibrated Magnitudes table in column B type the word "Average". In the cell below that, type "StdDev" (for standard deviation). The table should now look like this:
Now in the cell immediately below the check V-mag column, type "=Average(" (without the quotation marks) and then click and drag down through the orange cells in the Check V Mag column and then type the closing parenthesis ")". In my sheet the input looks what is picture below. (Notice that the cells C39 to C42 are the orange cells and you could have typed in C39:C42 inside the parenthesis.) If your input looks similar to mine, press return and the average should have been calculated for you.
Now we are going to compute the standard deviation for the data. In the cell below the average we just computed, type "=Stdev(" then highlight the cells and close the parenthesis. Again your equation should look like what I have included below. If it is okay, press return. The standard deviation of your data (a measure of the spread around the average value) has now been calculated for you.
Repeat the last three points for the "V Mag" column (or, alternatively, copy cells from below the "Check V Mag" column and paste them below the "V Mag column".
Congratulations! You have now produced calibrated values for the variable star!
Now we check that our results make sense. Remember how we calculated the average and standard deviation for a check star? We can use this column to determine if our calculations were correct or in error. Compare the average V mag for the check star (which you just computed) to the catalog value for the check star (in cell B12). They should be in reasonable agreement, certainly within +/- 0.01 magnitude. 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 with this part of the tutorial. Please proceed on to the Intermediate-Level reduction tutorial to learn how to do air mass corrections. If you wish to skip it, you can go directly to the Imaging and how to submit your data tutorials.
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!