### 11 Calibrating Instrumental Magnitudes

The purpose of calibrating instrumental magnitudes is to convert them into magnitudes in the target standard photometric system. To fix ideas, think of the target system as being the Johnson-Morgan UBV system. You can think of the calibrated magnitudes as the magnitudes which would be recorded by a detector which perfectly matched the standard UBV system and was operating above the terrestrial atmosphere. The magnitudes and colours recorded by such a detector still do not correspond to the intrinsic colours of the object observed because of the effects of interstellar material in-between the object and the detector. Interstellar material reddens and dims light which passes through it. However, correcting the effects of interstellar reddening and extinction is normally considered to be part of the astrophysical interpretation and analysis of the observations rather than routine data reduction. Interstellar extinction and reddening are briefly described in Appendix A, but not otherwise considered in this cookbook.

Thus, calibrating instrumental magnitudes consists of correcting two effects:

• discrepancies between the instrumental and target standard systems,
• atmospheric extinction.

Any given instrumental magnitude is, of course, simultaneously affected by both these effects. Two distinct cases can be considered for performing the calibration:

(1)
the instrumental system is well-matched to the target standard photometric system,
(2)
the instrumental system is less well or poorly matched to the standard system.

In the first case the detectors and filters used have been chosen carefully to match the responses of the target standard system as closely as possible. Thus, for example, the transmission profiles of an instrumental UBVRI system would be similar to those for the standard system shown in Figure 2. Often the instrumental system will closely match the corresponding standard one (and considerable effort and attention will have been expended at the observatory providing the instrumentation to ensure that this is the case). Staff at the observatory should be able to advise on how well matched the systems are. Other useful sources of information are handbooks, World Wide Web pages, newsletters and instrument manuals issued by the observatory. SG/10: Preparing to Observe[63] includes a list of URLs for the Web pages of the observatories usually used by British astronomers.

The two cases of whether the instrumental and target standard system are well or less well matched really correspond to whether it is necessary to make a colour correction when calibrating the instrumental magnitudes. The judgement of whether or not the target and instrumental systems are well matched is not absolute, but rather will depend on the precision which you wish to achieve in your photometry, which in turn will depend on the astronomical aims of your programme. Sets of observations made with the same instrumentation for different programmes may well be reduced with or without colour corrections, depending on the accuracy required and the aims of the programmes. In particular, if observations are only made in a single band then clearly colour corrections cannot be made. The following section discusses the simpler case of calibration without a colour correction and the subsequent one calibration with colour corrections.

Finally, there are types of programmes where it is not necessary to calibrate instrumental magnitudes into standard magnitudes. For example, if you are only interested in determining the periods of a variable star then these periodicities can be extracted from a time series of instrumental magnitudes as easily as from one of calibrated, standard magnitudes. (However, in this particular case it is still, of course, necessary to correct for atmospheric extinction).

#### 11.1 Calibration without a colour correction

Calibration without a colour correction is appropriate when the instrumental system is well matched to the target standard system. The calibrated magnitude is computed solely from the corresponding instrumental magnitude. Because magnitudes are logarithmic quantities and the standard and instrumental systems are being assumed to be well matched the principal difference between them is a zero-point correction. In this case the relation between instrumental and calibrated magnitudes is of the form:

 ${m}_{calib}={m}_{inst}-A+Z+\kappa X$ (15)

where:

${m}_{calib}$
is the calibrated magnitude,
${m}_{inst}$
is the instrumental magnitude,
$A$
is an arbitrary constant which is often added to the instrumental constants,
$Z$
is a photometric zero point between the standard and instrumental systems,
$\kappa$
is the atmospheric-extinction coefficient,
$X$
is the air mass.

For programme objects $A$ is arbitrarily chosen by you, ${m}_{inst}$ is measured and $X$ is known (remember that the air mass depends solely on the zenith distance which, in turn, can be calculated from the celestial coordinates of the object, the location of the observatory and the time of observation; see Section 8 and Appendix B). $Z$ and $\kappa$ are constants which are initially unknown. Once they have been determined Equation 15 can be used to calculate the calibrated magnitudes.

There are various methods of determining $Z$ and $\kappa$. For example, if a single standard star9 is repeatedly observed throughout the night then the instrumental magnitude can be plotted against the air mass. Such a plot should be a straight line with a slope of $\kappa$. Figure 4 shows a schematic example of such a plot.

However, the most common method of determining the constants is to intersperse observations of your programme objects with observations of standard stars. Suitable standard stars will typically have been selected from one of the catalogues of standard stars (see Sections 7.3 and 9). For each of the observations of standard stars ${m}_{calib}$ is known in addition to ${m}_{inst}$, $A$ and $X$ and it is possible to simply solve for $Z$ and $\kappa$ using least squares or some similar technique.

Once $Z$ and $\kappa$ have been determined Equation 15 can be used to simply calculate the calibrated magnitudes for the programme objects.

Thus, in essence, photometric calibration consists of making a least squares (or similar) fit to a series of observations of standard stars to determine the photometric zero point and the atmospheric extinction coefficient. However, such a fit should not be made blindly. (At least) the following caveats should be borne in mind.

(1)
$Z$ depends on the details of the instrumentation (CCD detector, filter, telescope etc.) and should remain fairly constant throughout an observing run. However, atmospheric extinction definitely varies from night to night10. Hence:

observations of standard stars should only be used to calibrate observations of programme objects made on the same night.

That is, observations made on different nights should be calibrated separately.

(2)
When a fit is made to the standard-star observations, the individual residuals should be examined, any observations with large residuals discarded and the remaining observations re-fitted. Passing clouds and other transient phenomena can cause temporary variations leading to aberrant and invalid observations.
(3)
The residuals and/or the coefficients themselves should be plotted as a function of time of observation throughout the night. Systematic variations can occur during a single night and it may be necessary to discard the observations for a portion of the night or make separate fits for different parts of the night.

Section 16 gives a simple recipe for calibrating photometric observations without a colour correction.

#### 11.2 Calibration with colour corrections

Calibration with colour corrections is usually appropriate in two cases:

• where the instrumental system is not well matched to the target standard system,
• where very high precision photometry is being carried out and even small discrepancies between the instrumental and target standard systems must be corrected for.

Calibration with colour corrections is similar to calibration without a colour correction. The calibrated magnitude is still computed from the instrumental magnitude in the corresponding band in a manner similar to Equation 15. However, an additional term is added corresponding to a colour index determined from an adjacent band. This term compensates for the mismatch between the instrumental and standard systems. For example, for the Johnson-Morgan UBV system the calibration formulæ are: $\begin{array}{rcll}U& =& {U}_{inst}-{A}_{u}+{Z}_{u}+{C}_{u}\left(U-B\right)+{\kappa }_{u}X& \text{}\\ B& =& {B}_{inst}-{A}_{b}+{Z}_{b}+{C}_{b}\left(B-V\right)+{\kappa }_{b}X& \text{(16)}\text{}\text{}\\ V& =& {V}_{inst}-{A}_{v}+{Z}_{v}+{C}_{v}\left(B-V\right)+{\kappa }_{v}X& \text{}\end{array}$

where:

$U$, $B$ and $V$
are the calibrated magnitudes in the three bands,
${U}_{inst}$, ${B}_{inst}$ and ${V}_{inst}$
are the instrumental magnitudes in the three bands,
${A}_{x}$
is an arbitrary constant which is often added to the instrumental constants,
${C}_{x}$
is the colour-correction term,
${Z}_{x}$
is the photometric zero point between the standard and instrumental systems,
${\kappa }_{x}$
is the atmospheric extinction coefficient,
$X$
is the air mass, and
subscripts $x=u,b,v$
refer to the individual bands.

The operational procedure is similar to that for calibration without a colour correction. For a set of observations of standard stars ${Z}_{x}$, ${C}_{x}$ and ${\kappa }_{x}$ (where $x=u,b,v$) are unknowns which can be solved for by least squares fitting of Equations 16. Once the coefficients have been determined, they can be used to compute the calibrated magnitudes of the programme objects. For really accurate work more-complex equations including higher-order terms may be introduced. For example:

 $V={V}_{inst}-{A}_{v}+{Z}_{v}+{C}_{1}\left(B-V\right)+{\kappa }_{v}X+{C}_{2}X\left(B-V\right)+\cdots$ (17)

Sometimes the atmospheric-extinction coefficient is not constant, but includes a colour term. That is:

 $\kappa ={\kappa }^{\prime }+{\kappa }^{″}\left(colour\phantom{\rule{1em}{0ex}}index\right)$ (18)

where ${\kappa }^{\prime }$ and ${\kappa }^{″}$ are constants. Often ${\kappa }^{″}$ is sufficiently small that $\kappa$ can be assumed to be constant. If the colour term is significant then the lines in Figure 4 will appear curved.

The three cautionary caveats given in the preceding section for calibrating without colour corrections are equally, if not more, applicable when colour corrections are included. Briefly: programme objects should only be calibrated with observations of standards made on the same night, when standards are fitted the residuals should be examined individually and aberrant observations discarded and the residuals should be checked for systematic trends.

Often bespoke software is used for reducing photometric observations with colour corrections, partly because the colour correction terms used will depend on the bands that were observed. There is no recipe for calibration with colour corrections in this cookbook. Further discussions are given by Massey et al.[55], Da Costa[15], Harris et al.[36] and Stetson and Harris[69].

# Part IIThe Recipes

9Strictly speaking it is not necessary to use a standard star; any star can be used as long as it is not variable on the time-scale of a night’s observing.

10Obviously the atmospheric extinction depends on the prevailing atmospheric conditions. An unusual example of variation caused by atmospheric conditions is the disruption following the eruption of Mount Pinatubo, as described by Forbes et al.[27].