The Rigel Telescope Installation and Calibration Tests:
Relationship between Signal to Noise Ratio and Apparent Magnitude
When making observations with the Rigel Telescope it is extremely useful to know the signal to noise ratio (SNR) that will be attained for a star at a given apparent magnitude at a given exposure time in a given filter. This is useful because it gives the observer an idea of what exposure time and filter to use if they’re observing objects that they hope to get a certain SNR for. An SNR is a measure of the degree that the object of interest stands out from the background, or noise. It is found by measuring the peak of a Gaussian fit that is fit to the star and then dividing by the standard deviation of an area of the image which contains no stars. The standard deviation is the "noise" of the image. So, the SNR is a measure of to what degree one can measure a change in flux. For example, if a measurement of an occultation by a planet were to be made (approximately a change in flux of 1/100 the total flux), then a signal to noise ratio of greater than 100:1 would be needed.
To right, the +2907 is the peak of the Gaussian fit of the object of interest (the signal). To left, an exaggerated diagram of a Gaussian fit which illustrates the signal and the noise.
This relationship was determined at the North Liberty, IA observatory during primary testing of the telescope, but it was repeated in Sonoita, AZ for two reasons. First, the skies in AZ are generally darker with less atmospheric disturbance, which results in a smaller full width half maximum (FWHM) for an average star in a given field. Since the FWHM’s were smaller this meant that the same amount of light was being collected, but in a smaller area, effectively concentrating the light more effectively on a smaller portion of the CCD sensor. It is not hard to imagine that this would result in an increased SNR because the Gaussian fit would be steeper and have a higher maximum. It turns out that the relationship between the FWHM and the SNR scales as one over the square of the FWHM. This makes sense because as the FWHM decreases, the SNR increases. The FWHM is squared because it is effectively a measure of the area of the star on the CCD sensor, which is pr2. Second, the primary mirror had been replaced due to a mistake that made the original primary mirror unusable. Later it will be determined if the new mirror was more or less effective by comparing the data from North Liberty with the data from Sonoita.
To find a relationship between the SNR and the apparent magnitude it was first necessary to find a reference field with very well determined reference magnitudes with which to compare the observed signal to noise ratios. While in North Liberty, the reference was Montgomery from the Astronomical Journal in 1993, for the field of M67. Reference magnitudes were supplied for B, V, and I filters. The reference stars were matched to images taken at North Liberty using a finding chart that was included in the reference materials. Data was collected from North Liberty and SNR’s were measured for the three filters. The log of the SNR’s were plotted against the reference apparent magnitudes. This was done because the SNR is directly proportional to the flux, which is logarithmically related to the apparent magnitude. This data showed that the V filter gave better SNR’s than B or I by a factor of about 10, which is quite significant. This data was taken for 40 second exposures with average FWHM’s of about 3.0".
Once in Sonoita, a similar plot was created using M57 as the reference source. The reference apparent magnitude values were taken from Skiff using the Lowell observatory in 1996. Reference magnitudes were included for B, V, and R filters. This data was taken for 30 second exposures with average FWHM’s of 2.7". Click here for plots of both of the above.
The SNR’s from Sonoita are much better than those in North Liberty. This can be seen by first correcting for the difference in exposure times and then plotting the two sets of data together. The SNR scales as the square root of the exposure time. Therefore, the SNR’s corresponding to a 40 second exposure time (to make them comparable to the ones taken in North Liberty) would be the original SNR’s time the square root of 4/3. This plot effectively shows the difference in SNR that can be attributed to the difference in the FWHM. This plot showed that the SNR’s in Sonoita were a factor of ten better for a given magnitude in the V filter. The SNR’s were increased even more for a given magnitude in the B filter. Only V and B filter were compared because those were the only common filters given in the different references. Next it was thought instructive to examine the difference in the plots once they were corrected for the difference in FWHM’s. This was done by multiplying the data from Sonoita by the square of 2.7/3.0, effectively giving the data from Sonoita an FWHM of 3.0" so that it would be comparable to the North Liberty data. This plot would show what effect the change in the primary mirror had had on the SNR’s. This plot demonstrated that the new mirror could be attributed to increasingthe SNR’s by a factor of two for a given magnitude in the V filter. Click here for the above plots.
Finally it was thought to be interesting to determine the maximum apparent magnitude of Rigel by using differential photometry on an image with a long exposure time and attempting to determine the magnitude of the faintest object in the image that has an SNR of at least 5:1. The field used was M57 and the image was taken in a clear filter at a 100 second exposure. The reference star was again taken from Skiff. The maximum measured magnitude was 18.71 in the 100 second exposure. The maximum magnitude was 18.2 in the 30 second exposure. Note that the first measured star in the 100 second exposure and the last measured star in the 30 second exposure are in the same position. In fact they are the same star, it is instructive to note that in this case the SNR does in fact scale as the square root of the exposure time as expected. 11.8 times the square root of 3/10 is equal to 6.5.
Data for determining the max. apparent magnitude.
|
Exposure |
R. A. |
Dec. |
Mag. |
SNR |
|
100 |
18:53:37.74 |
+32:59:30.0 |
15.27 (reference) |
|
|
100 |
18:54:05.09 |
+33:07:19.3 |
17.82 |
11.8:1 |
|
100 |
18:53:44.25 |
+33:01:44.1 |
18.01 |
10.4:1 |
|
100 |
18:53:50.53 |
+33:01:44.1 |
18.71 |
5.6:1 |
|
30 |
18:53:37.71 |
+33:59:30.0 |
15.27 (reference) |
|
|
30 |
18:54:09.20 |
+33:05:07.5 |
17.29 |
12.4:1 |
|
30 |
18:54:09.04 |
+33:06:16.4 |
18.03 |
8.0:1 |
|
30 |
18:54:05.09 |
+33:07:19.5 |
18.17 |
6.5:1 |
