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CCD Observing Manual4.0 Photometry4.1 Introduction to PhotometryPhotometry is the process of taking brightness estimates of stars in the sky. Some call it an art and some others even call it black magic because it can be quite challenging and confusing. But don't let that scare you, it is also very fun. Variable stars are addictive. When you have been following the same star for a long amount of time it can almost seem like part of the family. You expect certain activity, and when it acts up, you get concerned! Photometry is definitely a challenge, but it is a worthy one and a unique area of science where amateur help is not only needed but required. There are many different methods to photometry. Each has different requirements and results. In general, the more work you put in the more accurate your data will be. So what needs to be done is a comparison of the methods with the level of accuracy you need for your observing program. 4.2 Differential
 Aperture Photometry
 Some random notes:
4.3 All Sky PhotometryAll sky photometry uses comp stars that are more than a field of view away from the variable star. Because of this distance you must take into account extinction caused by the airmass you are looking into and many systematic variables. It is a complicated procedure but one worthy of mastering by those who want to get the absolutely most accurate photometry possible for fields without comparison stars. Also, all-sky photometry is very important for making new charts. If you feel you have mastered it then please contact our chart making team for a list of fields we need to be observed. These are the equations we need to start All Sky BVRI photometry:
B-V = (b-v)Tbv + Kbv * X + Zbv
V-R = (v-r)Tvr + Kvr * X + Zvr
R-I = (r-i)Tri + Kri * X + Zri
V-I = (v-i)Tvi + Kvi * X + Zvi
R = r Tr + Kr * X + Zr
V = v Tv + Kv * X + Zv
Upper case BVRI implies Standard magnitudes,
Lower case bvri implies instrumental magnitudes.
T - are color transform coefficients
K - are atmospheric extinction coefficients
X - are the Air Mass terms.
Z - are zero point corrections
To avoid mistakes being introduced from exposures of different lengths, correct all instrumental magnitudes to the same exposure time. If one chooses 10 seconds as the standard exposure for the night, one would correct all magnitudes by the following formula for even faint stars:
m(10) = m(exp) + 2.50 LOG(exp/10)
An exposure (exp) of 20 seconds will produce instrumental magnitudes that appear 0.753 mag too bright relative to an exposure of 10 seconds.
m(10) = m(20) + 2.50 LOG(20/10) = m(20) + 0.753
The known quantities are the instrumental magnitudes, Standard magnitudes and Air Mass, X. The unknown quantities are the Transformation coefficients T, Extinction coefficients K, and Zero point corrections Z . One method of determining the three unknown coefficients in each equation would be to observe a number of stars at different altitudes and with a range of colors. Then do a Least Squares solution for the coefficients. Since some of the coefficients vary from night to night while others are fairly stable, it is better to determine the stable ones apart from the nightly ones. 4.4 Transformation CoefficientsFirst determine the T coefficents from a cluster such as M67. M67 is an ancient open cluster with a number of well determined secondary standards with a range of color. It is compact enough so that one can consider all the stars to be at the same Air Mass. This allows us to rewrite the original equations as :
B-V = (b-v)Tbv + Qbv
V-R = (v-r)Tvr + Qvr
R-I = (r-i)Tri + Qri
V-I = (v-i)Tvi + Qvi
R = r + (r-i)Tri + Kr * X + Zr
V = v + (v-i)Tvi + Kv * X + Zv
In principle, the Y term should be the more uncertain and X term the better known quantity. In practice it makes little difference whether one plots Y vs X or X vs Y if ones photometry is done with some care to get the signal significantly above the noise.
(B-V)-(b-v)Tbv = Kbv * X + Zbv
(V-R)- (v-r)Tvr = Kvr * X + Zvr
(R-I)-(r-i)Tri = Kri * X + Zri
(V-I)-(v-i)Tvi = Kvi * X + Zvi
R - r*Tr = Kr * X + Zr
V - v*Tv = Kv * X + Zv
Again these equations have the form y = ax + b One can observe one standard star as it rises or sets through a range of airmass and plot the Y, left hand side, vs the Air Mass, X. The slope will be the extinction coefficient K and the Y intercept at X=0 will be the Zero Point Z. X can be given as the secant of the zenith distance angle, zd. This model implies a flat earth! At an altitude of 30 degrees the secant model will have an airmass in error of 0.005. A power series will give a better airmass as follows: X= sec(zd)-0.0018167*Del-0.002875*Del^2 -0.0008083*DEL^3 where Del= sec(zd)-1.0 A quicker method is to observe a number of stars at high and low altitude, being sure to include a range of colors too. These observations can be plotted as before and the extinction and zero point determined. Excess sky signal is introduced here; but since most of the standard stars are brighter objects, this is not too objectionable. Once the Transformation and Extinction Coefficients have been determined, one can insert the air mass and instrumental magnitudes from the unknown images into the first equations and tie them into the Standard BVRI magnitude system. Transformation: If you apply transformation coefficients that are less than a year old then put: "Transform: Yes" in the Comments Explained portion of your observation (or click the appropriate box in WebObs & PCObs). Zero Points Zero points are one of the most difficult concepts to understand in photometry. Ron Zissell has written a discussion about zero points for JAAVSO Vol. 26 No. 2. It is available below: 4.5 Pitfalls
4.6 Calculating ErrorIntroduction Symantic Note of the Day: In everyday English, observers tend to use the term error to mean uncertainty. So here we define photometric error as measurement uncertainty. "Uncertainty" includes the stochastic component (related to precision, which can be calculated using statistics) and the calibration component (related to systematic errors, that can only be estimated). At present, this manual explains how to include an "error" that is based only on the stochastic component. At a future date we hope to have a tutorial on how to determine the calibration component. Reporting your photometric error/uncertainty to the AAVSO is a very important part of the observing process. When professionals are looking at your data they are much more likely to respect the data if it includes an error term. Why? Consider as an example a professional who has a Target of Opportunity (TOO) observing project on a satellite. They need to observe U Gem when it goes into outburst (usually > V=13.5). We receive an observation from one observer at 13.2 and another at 13.7. What does the astronomer do? If the observers submitted error then the astronomer would know which observation is more likely to be real.
Note: For bright stars the error you calculate may seem pretty small. That is because most of the error with bright star measurements is of systemic nature and not statistical. That is okay, we'll get to measuring systemic error soon enough. Signal to Noise Ratio (SNR)The signal-to-noise ratio (SNR) is a term that represents how much information you have from the measured object compared to how much information comes from other sources. It is a way to measure the confidence of your observation. SNR is proportional to information; it is the ratio of signal to the uncertainty of that signal value. The signal comes from the contribution of the star to the reading from the signal circle that encloses the star. Noise comes from uncertainty in both the signal circle and sky annulus. The uncertainty of the total counts within the signal circle is proportional to the square-root of the number of pixels inside the signal circle, whereas the uncertainty of the average level for the sky annulus (used to establish a reference for excess counts in the signal circle) is proportional to the inverse of the square-root of the number of pixels enclosed by the sky annulus. A star's "intensity" is the number of "excess counts" within the signal circle, which is simply the total counts within the signal circle minus the number that would be expected using the average level for the sky annulus. Hence, SNR is maximum when the signal circle is small (but not smaller than the FWHM) and when the sky annulus is large. What that means: For a perfect system SNR is maximum when the radius of the signal circle is 3/4 FHWM. For a system with imperfect star shapes and imperfect tracking it is a good practice to use a radius twice FWHM. The sky annulus should be large for high SNR, but not so large as to include other stars or be affected by background gradients. Most photometric software packages will calculate SNR for you. However, if they don't there are a few ways to do it manually:
...where Cnet is the net number of electrons (counts multiplied by gain) from the star once the sky background has been subtracted. This is good for stars much brighter than background. Roughly speaking, 10,000 electrons gives a SNR of 100 which is approximately equivalent to a observational error of 0.01 magnitude. This equation is an over simplification of the formula for SNR. A more detailed discussion regarding calculation of the SNR can be found in the articles by Howell 1989 and Newberry 1991. For Bright Stars
Consider a star to be "bright" when its intensity is three times the average level of the sky annulus. For other stars use the formula below. The higher the SNR, the better your photometry will be. The easiest way to increase SNR is to stack many images. Collect images in groups of 3 to 5, and perform a median combine in order to remove cosmic ray defects that could give false photometry readings. Then average all your median combined images. Stacking images preferentially increases the signal over the noise. The signal is summed but the noise increases only by the square root of the number of images you are adding. So the more images you stack, the better your SNR. Stacking 10 images increases signal by 10X while increasing noise by only ~3.2X! For other ways to increase SNR see Maximizing your SNR. Going further... For even better uncertainty, add the 1/SNR value to the standard deviation of your comparison/check stars in quadrature. err = sqrt( (1/SNR)2 + comp_stdev2) Online SNR Tool: Click here for Michael Richmond's online SNR Calculator. A More Detailed Look at Error/UncertaintyIn this section we will look at the the basic definition that relates to errors, common sources of errors, and ways to reduce the errors in CCD photometry. By the end of this section you should have a good idea on how to quantify the photometric errors in your system. Before discussing errors in CCD photometry we must understand the definitions of the mean, standard deviation, and the signal-to-noise ratio (SNR). If you measure the magnitude of a star five separate times you will get five different measurements. The average or mean will give you a more accurate estimate of the star's magnitude than any one single measurement. The average is calculated by summing all the measurements and dividing by the total number of them. In mathematical terms: ...where xi are the values of the individual measurements and N is the total number of measurements. The summation sign is a mathematical symbol that simply means "the sum of x values from i=1 to N." The next definition we need to be familiar with is the standard deviation. The standard deviation tells us the amount of dispersion (or spread or scatter) we should expect from a single measurement relative to the average. The mathematical formula for the standard deviation is:
...where the Greek symbol sigma represents the standard deviation. In some texts, the standard deviation will be represented by s.d. Note that the standard deviation is always positive and has the same units as xi. The average and standard deviation calculations are standard features in most spread sheet programs such as Microsoft Excel. Two good references for a general discussion about the measurement of errors are:
Henden and Kaitchuck 1982, and Lyons 1991.
Maximizing SNR
SNR can be improved by choosing the right sizes for the "signal circle" and
"sky annulus."
Larger sky annulus areas are always better, except when it
includes interfering stars or extends
across an area where an imperfect flat field can have an effect.
There's an optimum size for the signal circle. Theoretically, the signal
circle radius should be ~0.8 * FWHM. This assumes that a lot of things are
text-book perfect. In the real world, a better choice would be:
"Signal Circle Radius = 1.5 * FWHM".
The reason there's an optimum signal circle radius is that as the radius
increases starting from zero the counts from the star increase in
proportion to the number of pixels included in the circle whereas the noise
from the signal circle total counts increases as the square-root of the
number of pixels. For large signal circle radii the star's contribution to
intensity does not increase with increasing radius whereas the the
uncertainty of the total counts within the circle keep increasing in
proportion to the square-root of the number of pixels. The radius where
these two competing effects balance is the radius of 80 % of FWHM.
In choosing a signal circle radius consideration should be given to changes
in FWHM throughout an observing session; the worst FWHM should be
used: Signal Circle Radius = 1.5 * FWHM(worst).
Sources of Error
There are several sources of error that creep into CCD measurements, such as, lack of
color correction, not using a V band filter, and a poor signal-to-noise ratio. Even with a
V band filter CCD measurements will suffer from errors resulting from color differences
between the comparison and variable star. This error can be removed by determination
the instruments color transformation coefficients and then transforming to the standard
Johnson V band photometric system. This topic is covered in section 4.4 of this manual.
Unfiltered photometric observations introduce anther source of error similar to the lack of
color correction. Since no CCD camera has a response that exactly replicates the
standard V (or any other filter band) band, the wide band response of the camera
measures flux from outside the desired wavelength region. For example, Mira variables
are very red (B-V > 1), when using a red sensitive CCD camera they will appear to be
much brighter than their actual V magnitude. The red sensitive camera and the star's
intrinsic brightness at long wavelengths give a large error in V magnitude. This error can
be minimized by using a V band filter.
The last source of error results from a low SNR. Figure 4.6.1 shows how the error
increases as the magnitude increases. As the stars become fainter there are fewer photons
for the camera to detect which results in a low signal for the camera to record. For a
given exposure, the noise remains constant regardless if the star is bright or dim.
Therefore, fainter stars have a lower SNR which leads to a larger difference in each
individual measurement.
Figure 4.6.1 Graph showing the error in photometric measurements as it depends on the magnitude of the star. (graph provided courtesy of Arne Henden). Once you are familiar with your camera and telescope the next step is to develop a scatter plot similar to figure 4.6.1. You can either make it error verse magnitude or more preferably, error verse SNR. To take the observations necessary for this plot refer to one of the standard calibration star fields in Martinez and Klotz 1997, Montgomery et.al. 1993, Romeo 1989 et.al., (http://binaries.boulder.swri.edu/binaries/fields/ngc7790.html. Once you have acquired a few images of a standard field, next reduce the magnitudes of the stars in the field. Choose one of the stars, preferably with 0 < (B-V) < 1 and calculate the magnitude and SNR of as many stars in the field as possible. Your SNR values should range from greater than 100 to about 10. Now find the difference between the standard star magnitudes you calculated and the values listed in reference for that field. Next remove the minus sign from any of the error that were negative. At this point you should have a table that looks approximately like the following: Table 4.6.1 The absolute error as a function of SNR
Table 4.6.1 can be used to quantify your errors when reporting observations to the AAVSO. For example, you wouldn't report a measurement to an accuracy of 0.01 if your SNR was only 20. According to the table, your measurements shouldn't be reported to 0.01 magnitude accuracy unless your SNR is at least 90. Example: Average and Standard Deviation To determine the precision of the CCD camera, filter, and telescope combination, 13 V band differenential photometric measurements were taken of the star SAO 69879 using SAO 69888 as a comparison star. The results are shown below.
Using the equations above, the average is 8.577 and the standard deviation is 0.022 magnitude. What this tells us is that the precision of a single measurement with SNR > 100 is approximately 0.02 magnitude. SAO 69888 and SAO 69879 are not standard stars with magnitudes known to 0.005 and transformation to the standard system was not performed. These measurements only provide the information about the precision of the measurement and not on the accuracy. A Derititave of the SNR error equation By Bruce Gary Given: SNR Definition of magnitude difference: dM = -2.5 * log (Y/Yo), where log is logarithm to the base 10, Y and Yo are two brightness values (think of Yo as average magnitude solution) Find: "magnitude SE" (standard error in magnitude units) Solution: SNR = Yo/SE Where Yo = average intensity solution and SE is stochastic uncertainty (SE) of Yo solution. Adding and subtracting SE to and from Yo yields upper and lower Y-values encompassing ~68% of the probability of finding the true solution (assumes Gaussian statistics). These two Y-values, Y+ and Y-, are simply Y+ = Yo + SE and Y- = Yo - SE Another way of writing these two Y-values is Y+ = Yo + Yo/SNR and Y- = Yo - Yo/SNR This can be re-written Y+ = Yo * (1 + 1/SNR) and Y- = Yo * (1 - 1/SNR) Consider the ratios of these two Y-values to the average value:
Y- / Yo = (1 - 1/SNR) These ratios can be written as magnitude differences:
dM- = -2.5 * log (Y-/Yo) Substituting the equations for Y+/Yo and Y-/Yo yields:
dM- = -2.5 * log (1 - 1/SNR) "magnitude SE" = average of dM+ and dM- QED Note 1: The + and - magnitude errors are different, the more so for smaller SNR. This is the way things are, and when SNR is large the two dM values can be averaged to give one SE. Hopefully AAVSO observers will always have large SNR, so that a simple average of the two dM values can be used. Whenever SNR <= 15 then the inequality of dM errors will exceed 22%. The value of dM+ will always exceed the absolute value of dM-. Note 2: These errors, dM+ and dM-, are standard errors, or SE - also referred to as 1-sigma errors. This means that the truth has a 68% probability of existing within the range Yo - SE and Yo + SE (or Mavg - dM- and Mavg + dM+). Of course, this assumes that calibration uncertainties are small compared to stochastic uncertainties. Note 3: When using results obtained with the same instrument, the same "seeing," the same quantity of measurements, and the same analysis procedure (same annuli set for the photometry, same reference stars), whenever these same conditions apply, it is acceptable to ignore calibration uncertainties in order to monitor CHANGES in brightness. Clearly, each observer has different calibration offsets, and this is the reason it's difficult to combine observations from different observers to construct a light curve. It is much easier to construct a light curve from a single observer (provided that observer has sufficient data and adheres to the same analysis procedure). When observations are pooled from many observers either empirical offsets must be derived for each observer or each observer must present an estimated calibration uncertainty. The estimated calibration uncertainty should then be orthogonally added (square-root of the sum of squares) to the stochastic uncertainty, described above as dM- and dM+ before combining with other observations. 4.7 Bright Star PhotometryA good discussion of short exposure photometry occured on the AAVSO_Photometry Discussion Group in July. It can be read via the archives here: http://mira.aavso.org/pipermail/aavso-photometry/2004-July/subject.html As previously announced, the AAVSO is in the process of creating a program to detect exoplanet transits. The project will involve the monitoring of many bright stars with known planets but of unknown inclination (thus we're not sure if they transit or not). With all this emphasis on faint objects recently we wanted to summarize the results of this discussion of bright objects. There are two main problems with high-precision short-exposure photometry (SEP from here on out). First, bright sources tend to get overexposed, especially if any nearby comparison star is fainter than the target. Antiblooming gate (ABG) CCDs typically saturate at about 50% of their full well depth, and many non-ABG chips saturate before the limit of their Analog/Digital Converters (ADCs). Second, the atmosphere itself conspires to degrade the photometry through a phenomenon called scintillation, where the turbulent bubbles of gas act like lenses, increasing or decreasing the amount of flux entering the front of your telescope. The twinkling of naked eye stars is caused by scintillation. To counteract saturation, first test your CCD camera to find out the limits of its linearity and where full well occurs in the dynamic range of your ADC. Then keep the signal level within the linear range and below the full well. For very bright sources, you may reach a limit where you cannot take a short enough exposure to prevent saturation. Techniques to go even brighter include:
Scintillation can be tricky. It is stronger nearer the horizon where you are looking through more atmosphere. It is also stronger for smaller telescope apertures, where each blob of atmospheric gas is closer to the telescope aperture size (bigger telescopes average many blobs). It is also a function of wavelength (redder is better) and exposure time (longer is better). Tips to avoid scintillation:
Here is a formula posted By Radu Corlan (CXR) for estimating scintillation without taking altitude into consideration: Where A is the airmass, D is the aperture in cm and t is the integration time in seconds. It is taken from a series of papers by Dravins et al. that begin with "Atmospheric Intensity Scintillation of Stars, I. Statistical Distributions and Temporal Properties" 1997, PASP. 109, 173. References Henden, A.A., and Kaitchuck, R.H., 1982, Astronomical Photometry, Van Nostrand Reinhold company. The book is available from Willmann Bell Publishing at http://www.willbell.com/photo/photo4.htm. Martinez, P., and Klotz, A., 1997, A Practical Guide to CCD Astronomy, Cambridge University Press. Montgomery, K.A., Marschall, L.A., and Janes, K.A., "CCD Photometry of the Old Open Cluster M67", The Astronomical Journal, Volume 106, Number 1, 1993. Romeo, G., Bonifazi, A., and Fusi Pecci, F, "CCD photometry of galactic open clusters - I. NGC 7790", Monthly Notices of the Royal Astronomical Society, 1989, 240, pages 459-485. Howell, S.B., "Two-dimensional Aperture Photometry: Signal-to-Noise Ratio of Point- Source Observations and Optimal Data-Extraction Techniques", Publications of the Astronomical Society of the Pacific, 101, June 1989, pages 616-622. Newberry, M.V., "Signal-to-Noise Considerations for Sky-Subtracted CCD Data", Publications of the Astronomical Society of the Pacific", 103, January 1991, pages 122- 130. Lyons, L., 1991, A Practical Guide to Data Analysis for Physical Science Students, Cambridge University Press |
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In differential photometry, apertures
(a.k.a. annuli) are selected around the
variable, comp, and
check stars. In each aperture are 2 or 3 circles. The inner circle's size is adjusted to fit around the star
itself. The outer circles are adjusted to include background light but not light from any neighboring stars. The
software then calculates the average sky value for the outer circles and for the inner circle. The outer (sky)
value is subtracted from the inner (star) value.
This also happens for the comp and the check star. The resulting star values are
compared to get the final
values for the image.
