**Introduction**

The standard equation relating the observed apparent magnitude of star observed to the true (zero air mass) magnitude can be written

where V0 is the true apparent magnitude [i.e., at zero air mass], Vobs is the observed magnitude, Z is the air mass, and B-V is the color index of the star. Note that the magnitudes and coefficients are filter-dependent. We determined these coefficients for the B, V, R, and I filters using observations of field stars near the `little dipper’ asterism in the open cluster M67 on April 28, 2012. We used apparent magnitudes for these stars reported by Brian Skiff at Lowell Observatory (2002).

**Equipment**

The observations were made using the Rigel robotic telescope, a 37cm f/14 classical Cassegrain system on an equatorial fork mount. The imaging camera is a Finger Lakes Instruments (FLI) Proline CCD camera with a Kodak 16803 front-illuminated sensor cooled to -35 C. The sensor has 4096 x 4096 (16M) 9-micron pixels. The field of view is 25’ x 25’, with a plate scale 0.73" per pixel using 2x2 pixel binning. The telescope is located at the Winer Observatory near Sonoita, Arizona, and is operated remotely from Iowa City, Iowa.

The filter system consists of a FLI ‘Centerline’ ten-position filter wheel with 50 mm x 50 mm glass filters, each 3 mm thick. The bandpass response of the B,V, R, and I filters [from the FLI website] are shown below. Note that although the center wavelength and bandwidth of these filters are similar to the older Johnson-Cousins [J-C] system, they are not identical, so detailed comparisons to J-C photometric measurement may require small correction terms (probably of order 1%).

Figure 1a. B filter spectral response.

Figure 1b. V filter spectral response.

Figure 1c. R filter spectral response.

Figure 1d. I filter spectral response.

**Observation and data reduction**

First, a field was chosen containing several stars with a range of accurately measured photometric colors. In this paper, the field selected is Messier 67 (NGC 2682), an old open cluster located in Cancer. This field was selected for its compact size, permitting it to be observed in one field of view, and the presence of several blue straggler stars amongst the largely uniform cluster population.

The target field was observed on April 28, 2012 using blocks of three 20-sec exposures with B, V, and R filters. This was repeated every 15 minutes over 10 hours centered near transit. This allows sampling sequence of air mass range 1.0 < Z < 3.5. We determined color indices using precisely measured apparent magnitudes reported for this field by Skiff (2002). A finding chart of the target stars is shown below.

** **

Figure 2. M67 finding chart.

Star | Skiff name | RA (J2000) | Dec (J2000) | B | V | R | I |

1 | F81 | 8:51:11.7 | +11:45:23 | 9.941 | 10.027 | 10.059 | 10.095 |

2 | I-12 | 8:51:14.8 | +11:47:24 | 13.377 | 12.821 | 12.492 | 12.170 |

3 | F95 | 8:51:15.4 | +11:47:31 | 13.175 | 12.668 | 12.375 | 12.071 |

4 | F108 | 8:51:17.4 | +11:45:24 | 11.074 | 9.701 | 8.986 | 8.350 |

5 | F111 | 8:51:18.0 | +11:45:54 | 13.294 | 12.730 | 12.402 | 12.076 |

6 | F117 | 8:51:18.7 | +11:47:03 | 13.417 | 12.630 | 12.163 | 11.729 |

7 | F124 | 8:51:19.9 | +11:47:00 | 12.577 | 12.118 | 11.838 | 11.558 |

8 | F127 | 8:51:20.1 | +11:46:42 | 13.328 | 12.769 | 12.439 | 12.118 |

9 | I-198 | 8:51:20.4 | +11:45:53 | 13.726 | 13.152 | 12.819 | 12.524 |

10 | I-199 | 8:51:20.6 | +11:46:05 | 13.729 | 13.152 | 12.819 | 12.451 |

11 | F130 | 8:51:20.6 | +11:46:16 | 13.323 | 12.869 | 12.580 | 12.289 |

12 | F132 | 8:51:20.8 | +11:45:02 | 13.700 | 13.091 | 12.741 | 12.386 |

13 | F134 | 8:51:21.2 | +11:45:53 | 12.840 | 12.256 | 11.919 | 11.587 |

14 | F135 | 8:51:21.6 | +11:46:06 | 12.502 | 11.436 | 10.880 | 10.383 |

15 | F170 | 8:51:29.9 | +11:47:17 | 11.020 | 9.663 | 8.199 | 7.497 |

**Extinction coefficients **

A series of images were taken of the cluster using three filters: blue, green, and red, all over a varying range of air masses. Once the images are calibrated, differential photometry was performed upon the images using a Unix script named photom that compared each star in each image to a selected reference star and calculated the differential magnitude, generating a table containing these differences.

To calculate the extinction coefficient, k’, the brightest star, F81, was used as a reference and differential photometry was performed on the series of images from least air mass to greatest air mass. The output file thus contained the differential apparent magnitude of this star. Next, a script was written to perform a linear fit of V-Z for each filter and plot data and fitted line. This was done for each filter, as shown in figure 2.

Figure 3. Magnitude vs. airmass.

The following table was generated by a slightly different way, in that a standard Landolt field is imaged through four filters, B,V,R, and I, over a large range of air masses. The images are analyzed through another script, photcal, that returns a table of calculated k’ values for the field.

Filter | k prime |

Blue | 0.267 +/- 0.005 |

Visual | 0.153 +/- 0.004 |

Red | 0.114 +/- 0.003 |

Infrared | 0.088 +/- 0.006 |

Therefore the apparent magnitude of a star is given by

where V is the true apparent magnitude, Vobs is the observed magnitude, Z is the air mass at the time of the specific image, and k’ is the extinction coefficient for the specific filter. Note that, as a larger magnitude is a dimmer star, that higher air masses will increase the magnitude but decrease the star’s brightness.

**Color index coefficients**

For the color index coefficient, only one image was chosen for each filter, the image with the lowest air mass value being used in all three cases. However, photom was run on multiple stars, with F134 being the reference star due to its B-V value being in the middle of the range. Then, to cancel the k’ term of (1) as well as the V0 term, leaving only the k’’ term, (1) for the reference star was subtracted from (1) for another star, giving

However, the second term of (3) is defined to be zero by the calculations that photom performs, meaning that the desired expression for k’’ is

Another script was written to plot the difference in magnitude of the check star against the differential color index, as well as the best-fit lines for each filter.

Fig 4. Color index vs. differential magnitude. Note that fit for V and R filters are not shown: They are consistent with zero slope

For the visible, red, and infrared filters, the calculated color index coefficients are within one sigma of zero; thus the blue filter is the only filter exhibiting a significant color index correction. This correction was determined to be a k'' value of 0.26 +/- 0.03.

Once the filter response coefficients (k’’) have been determined, future photometric observations can be obtained by simply determining the extinction coefficients for that night by means of Equation (1) to a set of standard stars observed over a range of air masses. Even this step is not necessary for observation at low air masses, as the differential extinction in B is less than 0.2 magnitude at air masses below approximately 1.75, and correspondingly less in other wavelength filters.

**Signal to noise ratio**

To determine the relation between magnitude and signal to noise ratios for each filter, images were taken of the cluster through each filter for exposure times of 3, 10, 30, and 100 seconds. Then, the signal strengths of each star’s image was measured and divided by the signal of an area of dark background to determine the signal to noise ratio. Then, these values were plotted against the star’s magnitude, with the SNR axis displayed on a logarithmic scale.

If S is the signal to noise ratio for a star and V its magnitude through the specific filter, then the relationship between two stars of any magnitude is given by

Thus, a graph of SNR values against magnitude, with SNR values plotted logarithmically, will have a liner form with the slope being the log of 2.5, or approximately -0.4. To calculate the expected SNR for a star of any magnitude using this equation, the magnitude of a star with an SNR of 1, V0, is desired, so as to use the simpler equation

Figure 5. Signal-to-noise ratio for V-filter as a function of exposure time.

Exp (sec) | B | V | R | I |

3 | 17.00 +/- 0.11 | 17.21 +/- 0.04 | 17.02 +/- 0.06 | 17.69 +/- 0.04 |

10 | 18.87 +/- 0.04 | 18.73 +/- 0.01 | 18.28 +/- 0.03 | 19.05 +/- 0.01 |

30 | 20.20 +/- 0.05 | 19.96 +/- 0.16 | 19.46 +/- 0.07 | 20.47 +/- 0.01 |

100 | Not available | 21.19 +/- 0.04 | 20.28 +/- 0.13 | 21.48 +/- 0.01 |

**Example Calculation**

Assume a star is
observed to have a B-magnitude B_{obs }= 11.23 at airmass Z = 1.78. The
difference between the observed magnitude of 11.23 and the true (zero airmass)
magnitude is given by the following expression.

where V and Vobs are the true and observed magnitudes respectively. The photometric constant k’ for the blue filter is k’B = 0.267 +/- 0.005, and thus this gives a true magnitude of B = 10.75.

As for the color correction, the only significant effect is in the blue filter, so let us use the same example of a star of apparent blue magnitude 11.23. Using

let the reference star for instance have a B-V value of -0.123 (a quite blue star), and our star have a value of 0.890. Taking the calculated blue color correction of 0.26 and substituting it for k’’, we get a true magnitude of 10.97.

Thus, if the effects of both the extinction and color coefficients for this star are combined, the observed magnitude of 11.23 leads to a true magnitude of 10.49.

Now what would be the expected signal to noise ratio of this star through a 10 second exposure? The relationship between magnitude and signal to noise ratio is given by

where V0 for a 10 second blue filter exposure is 18.87. Substituting in 11.23 for V, the log of the SNR is 3.04, and thus the expected signal to noise ratio is about 1097:1.

This research made use of the SIMBAD database, operated at CDS, Strasbourg, France.

**References **

Cargile, P.A., and D.J. James. “Employing a New, BVIc Photometric Survey of IC 4665 to Investigate the Age of This Young Cluster.” The Astronomical Journal 140 (2010): 677. Table 3 enhanced HTML, Accessed 11 June 2012.

Skiff, Brian. “Table of Magnitude Values for Stars in M67.” Private communication. Accessed 21 May 2012