21 Photometric calibration

The purpose of the photometric calibration functions in CURSA is to convert a list of instrumental magnitudes, typically measured for a set of objects in a series of CCD frames, into calibrated magnitudes in some standard photometric system. To fix ideas, think of a group of programme objects for which instrumental magnitudes have been determined from a set of CCD frames using an aperture photometry package such as PHOTOM (see SUN/45[14]). These instrumental magnitudes are to be calibrated into standard $R$ magnitudes in the Johnson-Morgan UBVRI system.

Astronomical photometry is a diverse subject. There are many different standard photometric systems, many ways of making photometric observations and many ways of reducing them. CURSA provides only some simple and basic facilities. Though they will be useful and give reasonably accurate results in many circumstances they are certainly not appropriate in all circumstances. In particular, they are not suitable for high precision photometry. Whether they are suitable for you will depend on the details of your programme.

This section is not an introduction to how to calibrate photometric observations. Rather, it describes the principles behind the CURSA photometric calibration functions so that you can decide whether they are suitable for your purposes and describes how to use them. For a more general introduction to calibrating photometric observations see SC/6: The CCD Photometric Calibration Cookbook[22]. SC/6 also includes a tutorial example (a ‘recipe’ in the jargon of cookbooks) of using the CURSA photometric calibration functions.

21.1 Description

The CURSA photometric calibration functions, in common with most photometric calibration methods, use standard stars. In essence, as well as observing instrumental magnitudes for the programme objects that you are studying you also observe instrumental magnitudes for selected standard stars. These standard stars have a known brightness in your target photometric system. Numerous catalogues of photometric standard stars are available (see SC/6[22] for a brief discussion). You then define the transformation between the instrumental and standard system for the standard stars and apply this transformation to calibrate the instrumental magnitudes of the programme objects into the standard system.

In addition the observed brightness of a star varies throughout a night because of atmospheric extinction or the dimming of starlight by the terrestrial atmosphere. The longer the path length the starlight traverses through the atmosphere the more that it is dimmed. Thus, a star close to the horizon will be dimmed more than one close to the zenith. The path length through the atmosphere is known as the air mass. The air mass can be calculated from the zenith distance. In order to calibrate photometry air masses must be available for both the programme and standard stars.

Thus, a basic set of photometric data consists of:

• a set of standard stars with measured instrumental magnitudes, known ‘catalogue’ magnitudes (so-called because they are obtained from catalogues of photometric standards) in the target photometric system and air masses,
• a set of programme objects with measured instrumental magnitudes and air masses.

The standards are invariably stars; the programme objects can be any sort of astronomical object. Photometric calibration is a two-stage process:

(1)

define the transformation between the instrumental and catalogue magnitudes for the standard stars, typically by some sort of least squares fitting,

(2)

apply the transformation to convert the instrumental magnitudes into calibrated magnitudes for the programme objects.

In CURSA the relation between instrumental and catalogue magnitudes is assumed to be of the form:

where:

$m_{catalogue}$

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 the photometric zero point between the standard and instrumental systems,

$\kappa$

is the atmospheric extinction correction,

$X$

is the air mass.

See SC/6 for further discussion of the arbitrary constant $A$. This equation is a particularly simple form for the relation between instrumental and catalogue magnitudes. In particular, it omits any ‘colour corrections’ caused by the instrumental and standard systems being sensitive to different wavelengths. Thus, the CURSA photometric calibration functions should only be used when the instrumental photometric system is well-matched to the target photometric system. Though this may seem a serious limitation, in practice with modern instrumentation the instrumental system is often a good match to the standard system. For the same reason the CURSA applications are not suitable for very high precision work, where even small discrepancies between the instrumental and standard systems must be allowed for.

The basic reason why colour corrections are ignored is because by doing so the functions are much more general. They do not impose constraints on the photometric system that you are using (other than that the instrumental and standard systems should be well-matched) and they do not require you to make observations in any given colours.

Fitting the instrumental and standard magnitudes for the standard stars is usually an ‘iterative’, interactive process. Typically, you will start by fitting all the standard stars, examine the residuals, reject the stars with large residuals, fit the remaining stars and continue until you have a satisfactory solution. (Aberrant results for individual stars can be caused by various effects, including passing clouds.)

For completeness, the subroutine used by the CURSA photometric calibration applications to fit the instrumental and catalogue magnitudes for the standard stars is PDA_DBOLS. This subroutine is described in SUN/194[19].

21.2 Assembling the input catalogues

You need to prepare two catalogues: one containing the observations of the standard stars, the other the observations of the programme objects. Neither catalogue is likely to contain more than, at most, a few score entries. The most convenient way to create these catalogues is to use the STL format (see Appendices D and E) and type them in using an editor. Note that separate sets of catalogues should usually be prepared for each night that observations were made; observations from different nights should not normally be combined prior to calibration.

The instrumental magnitudes will be assembled from the output of other programs, such as PHOTOM. The standard or catalogue magnitudes will ultimately come from the catalogues of standards which you used when selecting the standard stars to observe. The air mass (or zenith distance) will often be included in either your observing logs or the header information of your CCD frames. If the air mass is not available then the CURSA applications can automatically calculate it from the zenith distance. Note that it is the observed zenith distance, that is as affected by atmospheric refraction, which is required. If the zenith distance is not available either then you will have to calculate it from whatever information you have about the celestial coordinates and times of your observations. Most standard textbooks on spherical astronomy give the requisite formulæ (see, for example, Spherical Astronomy by R.M. Green[15]).

The catalogues of standard stars and programme objects are discussed separately below.

21.2.1 Standard star catalogue

Figure 14 shows an example catalogue of standard stars. The observations used in this example were kindly provided by John Lucey. The example is available as file:

The catalogue must contain columns containing the instrumental magnitude, the catalogue magnitude and the air mass (or alternatively the observed zenith distance). It may optionally contain a column containing a name for each of the standard stars and a column of ‘include in the fit’ flags. All five columns are included in the example. If supplied, the star name is listed in the table of residuals produced when the fit is made. Often being able to identify each standard star will be useful to you. The ‘include in the fit’ flag column is of data type LOGICAL and determines whether each star is included in the fit or not. To include or exclude a given star in the fit you simply edit the STL format catalogue and toggle the value of the flag for the star to ‘T’ (or ‘TRUE’) or ‘F’ (or ‘FALSE’) to include or exclude it as appropriate. This procedure is much less troublesome and error-prone than deleting and reinserting stars from the catalogue. Initially set the flags for all the stars to ‘T’ (or ‘TRUE’) so that they are all included in the fit. In the example all the stars are included in the fit except 99Z367 (the penultimate one in the list). This star is excluded as an illustration. When preparing your own catalogues you will usually initially include all the stars.

The zenith distance is an angle and if it is used it must ultimately be presented to the CURSA applications in radians. If you wish you can simply type the values into the STL catalogue in radians. Alternatively, if it is more convenient, you can define the zenith distance column as containing a sexagesimal angle, usually in degrees, and type in the values as sexagesimal degrees. The example catalogue of programme objects in Figure 15 includes a column of zenith distances in this form.

Though both the columns of star names and ‘include in the fit’ flags are optional I recommend that you use them.

The columns do not have to have the names shown in the example. However, if you use these names you will be able to accept the defaults from the prompts in the CURSA applications.

Obviously the catalogue can contain additional columns, though these are not used. For example, if you are calibrating multi-colour photometry you could prepare a single catalogue containing the instrumental and catalogue magnitudes in all the colours observed. Obviously the columns for magnitudes in different colours would have to have different names. If you did not observe all the stars in all the colours simply use the STL mechanism for indicating null values (see Section C.3.2) to represent the missing measurements.

21.2.2 Programme object catalogue

Figure 15 shows an example catalogue of programme objects. This example is available as file:

As an illustration this catalogue contains columns of both the air mass and the observed zenith distance. It does not need to contain both, but must contain one or the other. Here the zenith distance has been entered as sexagesimal degrees and minutes.

The columns do not have to have the names shown in the example. However, if you use these names you will be able to accept the defaults from the prompts in the CURSA applications.

The catalogue can contain additional columns; indeed a programme catalogue will often contain celestial coordinates and/or object names. Also, if you are calibrating multi-colour photometry you could prepare a single catalogue containing the instrumental magnitudes in all the colours observed. Obviously the columns for magnitudes in different colours would have to have different names. If you did not observe all the objects in all the colours simply use the STL mechanism for indicating null values (see Section C.3.2) to represent the missing measurements.

21.3 Applications for photometric calibration

CURSA contains three applications for photometric calibration:

catphotomfit

define the transformation coefficients from the standard stars,

catphotomtrn

apply the transformation coefficients to determine the calibrated magnitudes for the programme objects,

catphotomlst

list the contents of a transformation coefficients file.

The usual sequence of using these applications is:

(1)

run catphotomfit to determine the transformation coefficients. Examine the residuals, exclude aberrant standard stars and re-run. Repeat this process until you get a satisfactory fit,

(2)

run catphotomtrn to apply the transformation coefficients to the programme objects and determine calibrated magnitudes for them.

The transformation coefficients are passed from catphotomfit to catphotomtrn via a file, the so-called ‘transformation coefficients file’. Normally you do not need to inspect this file. However, if you wish to do so then catphotomlst is available for this purpose.

The details of running the individual applications are described below.

21.4 Running catphotomfit

To perform a simple fit to a set of standard stars type:

Your catalogue of standard stars should contain an air mass for each star. catphotomfit will determine the transformation coefficients, display them together with the residuals and write the coefficients to a file. If your catalogue contains a column of observed zenith distances rather than air masses then type:

See Section 21.7 for details of how the air mass is calculated from the zenith distance. If some of the transformation coefficients are fixed (that is, you know them beforehand) type:

You will be prompted for details of which coefficients are fixed and their values. If all the coefficients are fixed then obviously no fit is made. However, the residuals are still computed and listed and a file of transformation coefficients is written. To suppress the listing of residuals type:

These options can be combined. Thus, to read a catalogue containing zenith distances rather than air masses and fix some of the transformation coefficients type:

You then answer a series of prompts. All the possible prompts are listed below, identified by the corresponding ADAM parameter name. All the prompts will not appear in a given run. For example, none of the prompts FZEROP, ZEROP, FATMOS or ATMOS appear if none of the transformation coefficients are fixed.

FZEROP

Specify whether the zero point is fixed. The possible replies are:

TRUE

the zero point is fixed,

FALSE

the zero point is not fixed.

ZEROP

Enter the value of the fixed zero point.

FATMOS

Specify whether the atmospheric extinction is fixed. The possible replies are:

TRUE

the atmospheric extinction is fixed,

FALSE

the atmospheric extinction is not fixed.

ATMOS

Enter the value of the fixed atmospheric extinction.

INSCON

Enter the arbitrary constant previously added to the instrumental magnitudes.

CATALOGUE

Enter the name of the catalogue containing the standard and catalogue magnitudes.

NAME

Enter the name of the column containing names of the standard stars. The special value ‘NONE’ indicates that a column of star names is not required.

INCLUDE

Enter the name of the column of ‘include in the fit’ flags for the standard stars. The special value ‘ALL’ indicates that all the stars are to be included in the fit.

CATMAG

Enter the name of the column or expression holding the standard or catalogue magnitudes.

INSMAG

Enter the name of the column or expression holding the instrumental magnitudes.

AIRMASS

Enter the name of the column or expression holding the air masses.

ZENDST

Enter the name of the column or expression holding the observed zenith distances.

FILNME

Enter the name of the file which is to contain the transformation coefficients.

Figure 16 shows the output displayed by catphotomfit. The transformation coefficients are self-explanatory. The minimum residual vector length is a measure of the goodness of the fit. The table of residuals is also mostly self-explanatory. The column of star names will be absent if parameter NAME was specified as ‘NONE’. A ‘Y’ in the ‘Fit’ column indicates that the star was included in the fit. The residuals are defined in the sense:

The calculated magnitudes and residuals are shown to three places of decimals. This format does not imply that the results are this accurate; the actual accuracy will depend on the data used. It is noteworthy, however, that in the example data the largest residual is only slightly larger than 0.01 magnitude, despite the method ignoring colour corrections.

The bar to the right of the residuals is a simple graphic representation of the absolute size of the residual; the length of the bar is scaled according to the absolute size of the residual for the star. The scaling is such that the largest absolute residual amongst the stars included in the fit is ten asterisks long. Stars which are included in the fit are shown as a row of asterisks (‘*’). Stars which are excluded from the fit are shown as a row of dashes (‘-’). Because excluded stars will often have larger residuals than the included stars, for excluded stars with residuals larger than the largest included residual a right chevron (‘>’) is shown beyond the last dash (thus forming an arrow).

For completeness, and to avoid any possible ambiguity, the formula used to compute the standard deviation, $s$, is:

where $n$ is either the number of stars included in the fit or the total number of stars, as appropriate.

21.5 Running catphotomtrn

To convert a catalogue of instrumental magnitudes into calibrated magnitudes for programme objects type:

A new catalogue will be written which contains the new calibrated magnitudes as well as all the columns in the original catalogues. Also the transformation coefficients are added as parameters to the output catalogue. If your original catalogue contains a column of zenith distances rather than air masses then type:

See Section 21.7 for details of how the air mass is calculated from the zenith distance. The amount of textual information written to the output catalogue is controlled using the command line mechanism described in Section 10.1.

You then answer a series of prompts. All the possible prompts are listed below, identified by the corresponding ADAM parameter name. In a given run either AIRMASS or ZENDST will appear, but not both.

FILNME

Enter the name of the file containing the transformation coefficients.

INSCON

Enter the arbitrary constant previously added to the instrumental magnitudes. The default will be the value read from the transformation coefficients file, which corresponds to the value added to the instrumental magnitudes for the standard stars. Usually it is good practice to add the same arbitrary value to the instrumental magnitudes for both the standard stars and programme objects.

CATIN

Enter the name of the input catalogue.

CATOUT

Enter the name of the output catalogue to contain the calibrated magnitudes.

INSMAG

Enter the name of the column or expression in the input catalogue holding the instrumental magnitudes.

AIRMASS

Enter the name of the column or expression in the input catalogue holding the air masses.

ZENDST

Enter the name of the column or expression in the input catalogue holding the observed zenith distances.

CALMAG

Enter the name of the column in the output catalogue to hold the calibrated magnitudes.

21.6 Running catphotomlst

To display the contents of a transformation coefficients file type:

By default the transformation coefficients are shown to six places of decimals. Usually this precision will be more than adequate given the accuracy of the photometry and the fitting technique. However, you can specify the number of decimal places used. For example, type:

to show the coefficients to eight places of decimals.

21.7 Calculating the air mass

catphotomfit and catphotomtrn can optionally calculate the air mass from the observed zenith distance. They use subroutine SLA_AIRMAS in the SLA subroutine library (see SUN/67[32]) for this task. This routine is more than sufficiently accurate for the present purposes. The following notes are based on the documentation for SLA_AIRMAS in SUN/67.

The air mass is calculated using Hardie’s[16] polynomial fit to Bemporad’s data for the relative air mass, $X$, in units of thickness at the zenith as tabulated by Schoenberg[25]. This method is adequate for all normal needs as it is accurate to better than 0.1% up to $X=6.8$ and better than 1% up to $X=10$. Bemporad’s tabulated values are unlikely to be trustworthy to such accuracy because of variations in density, pressure and other conditions in the atmosphere from those that he assumed. At zenith distances greater than about $87^{\circ }$ the air mass is held constant to avoid arithmetic overflows.