Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.

Calibrating magnitudes


The problem: sharing measurements of a star

Astronomers, like all scientists, don't work all alone; instead, they share their results and ideas and problems and solutions with a community of other astronomers. So, for example, if you make some measurements of a particular star, and find that it varies in a periodic way, you might want to compare your measurements to those acquired at some other observatories, to make sure that those other observers found a similar variation. Checking results from independent efforts is a VERY good way to avoid mistakes!

The problem is ... the measurement of the brightness of one particular star, on one particular night, with one particular instrument, depends on a lot of factors, such as

The goal of calibration is to remove (as much as possible) all these effects, and to provide a value for stellar brightness that may be compared fairly to those determined by others.


Step 1: measure intensities

We'll assume that we have acquired some images of a field of stars, among which is the target of our study. As an example, let's adopt the binary star V445 Cas. The chart below is based on an image taken at the RIT Observatory.

Using techniques we will learn later in the semester, we can measure the intensity of each star in one of these images. Below are examples from one particular image:



Star         intensity
----------------------------
  A          298,064
  B          177,047
  
  C           96,166
  D           25,530

  E           19,612
  P        1,128,567

  V445        86,340
----------------------------


Step 2: choose a reference star

We can, and will, try to use the measurements of all these stars, but in order to achieve the best results, we ought to pick one star to serve as the photometric reference. In one of the final steps of the calibration procedure, we will look up the magnitude of this star in a good catalog and use it to shift our measurements of all the other stars to a standard magnitude system. That shift to a standard system will allow us to compare our measurements of the target star to measurements made by others.

So, what are some criteria for choosing a good reference star?

  1. constant over time
  2. roughly the same brightness as the target
  3. close to the target in the image
  4. but not TOO close to the target, or any other stars
  5. roughly the same color as the target
  6. has precise measurements in a good photometric catalog

Let's look again at our choices in the field of V445 Cas.



  Q:  Which of these stars might be a good reference star?

          Why?









As you are thinking things over, please take a look at one of the original, RAW images of this field taken by our telescope. Notice what happens at the sides of the field, and in the corners.


Step 3: compute instrumental magnitudes

Okay, now that we've chosen our reference star, we can compute the instrumental magnitude for our target, and for each of the other comparison stars. The general formula we will use is

Please fill in the following table.



Star         intensity      instr_mag
---------------------------------------------
  A          298,064
  B          177,047
  
  C           96,166
  D           25,530

  E           19,612
  P        1,128,567

  V445        86,340
----------------------------------------------


Step 4: verify that the other stars don't vary

If we've chosen the reference star properly, its brightness (and so magnitude) ought to be constant all night long. Most stars in the sky don't vary perceptibly, so it OUGHT to be the case that most, if not all, of the other comparison stars also should be constant all night long.

That means that the instrumental magnitude of those stars ought to be constant all night long, too. So, if we have many images of a field over some period of time, and we perform these calculations for the stars in each image, we ought to create a long list of instrumental magnitudes that are -- mostly -- the same. There might be some noise and random scatter from one image to the next, but if we chose a good reference star, we ought to see a bunch of nearly horizontal light curves.

If most of the comparison stars show some significant, and similar, variation, then there is something rotten in the state of Denmark. Perhaps we picked a reference star that happens to be variable itself (unlucky), or perhaps there was some big problem with the equipment or weather (also unlucky).


Step 5: locate comparison stars in a catalog

Once we've verified that our choice of the reference star was a good one, it's time to look up its properties in a catalog. I recommend using the catalog called APASS 9 , which contains good photometric measurements of millions of stars similar in brightness to the ones we will study.



  Q:  Find the entry for our reference star in the APASS 9 catalog.

         What is its B-band magnitude?

         What is its V-band magnitude?






You should also look up the magnitudes of the other comparison stars in the field.



  Q:  Find the entry for star "C" in the APASS 9 catalog.

         What is its B-band magnitude?

         What is its V-band magnitude?







Step 6: choose the appropriate catalog value

We are about to make a momentous decision: choosing the real, calibrated magnitude of the reference star in our image. In order to do so, we need to know WHICH of the many magnitudes in the catalog is the right one.

The answer is pretty simple: choose the magnitude corresponding to the filter through which you acquired your images. In the example we've been using, of V445 Cas, the image was taken through a V-band filter.



  Q:  What is the V-band magnitude of our reference star?

          What is the estimated uncertainty in the catalog's magnitude?






Step 7: shift instrumental magnitudes to calibrated magnitudes

Okay, it's time for the big fix. We are about to convert our instrumental magnitudes into calibrated magnitudes. The basic idea is to apply a constant shift to all the instrumental magnitudes, so that the magnitude of the reference star changes from its instrumental value to its catalog value. If we apply the SAME shift to all the stars, then their instrumental magnitudes ought to turn into calibrated magnitudes, too.



   Q:  What is the instrumental magnitude of our reference star?

   Q:  What is the calibrated, catalog magnitude of our reference star?

   Q:  What constant value should we add to the reference star's
               instrumental value?


  

   Once you know this constant, add it to ALL the instrumental magnitudes,
        for all the stars.





Star         intensity      instr_mag       calib_mag
-------------------------------------------------------
  A          298,064
  B          177,047
  
  C           96,166
  D           25,530

  E           19,612
  P        1,128,567

  V445        86,340
--------------------------------------------------------

Now, at this point, it's time for a sanity check. If we did everything properly, and if these stars are all ordinary, run-of-the-mill stars, then we should find that the shifted magnitude values of all our comparison stars are reasonably close to the catalog magnitudes for those stars.

If so, then we can trust our measurements of the target star. If not, we may have made a mistake somewhere along the way; or, perhaps, our choice of reference star was a poor one, and we should choose a different one and repeat the analysis.


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Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.