5/12/99 Reduction Procedures for Data obtained with the HB Filter Set ------------------------------------------------------------- ************************************************************************ Important Note: Although the information in this file is still relevant, it has not been updated to include recent additions to the data reduction procedures (e.g. OH extinction). For the final procedures, refer to the postscript version of the paper that we submitted to Icarus (specifically Appendices A and D). ************************************************************************ The reduction process for the HB filters is more complicated than for the IHW filters, primarily because there is an additional continuum filter that must be accounted for. Since few observers use all 11 filters, the reduction process must account for continuum measurements at any combination of the four wavelengths, and still produce results that can be compared to observers using a different combination of filters. We feel that the procedures outlined here provide the best calibration of comet data for the variety of situations that may be encountered, and give results which can be directly intercompared between observers. ======================================================================== The basic procedure for reducing photometric data is outlined here, with a brief discussion of each step giving any rationale or explanation that is needed, and then any equations or step-by-step instructions that are required. To make these instructions easier to follow, the first section outlines a nominal reduction procedure, which assumes that, given specific emission band measurements, all of the nominal continuum filter measurements are available. After that is a section describing how to deal with any missing continuum filters that are needed. (Once the missing filters are accounted for, the nominal procedures are then used.) The simpler equations are given here in ascii format (terminology is given in Appendix A), however, some of the equations get complicated, and ascii format would be confusing, so a list of equations will be faxed or mailed to anyone interested. A flow chart showing the logic in accounting for missing filters, etc. will also be provided to help in programming. Other components that you should have to perform these reductions: 1) A list of equations. A list of eight groups of equations referred to in these instructions. Each equation has a unique number in the form group-#. 2) A list of the constants and coefficients that are used in the equations. These values are not embedded in the equations because they are subject to revisions during the calibration process, and because embedding them would make it less obvious what is being done in the equations. 3) A list of the standard star magnitudes. A magnitude is given for each star at each filter wavelength. The table lists the magnitudes for Flux Standards and Solar Analogs, and then gives the solar color for each filter (relative to the BC filter). The flux star magnitudes are utilized in section 2 of these instructions. The solar analog magnitudes are given for completeness only. They were measured for use in determining the color of the sun so that it could be removed from the underlying continuum colors. Note that it is not necessary for the solar analogs to be observed in a standard comet observing program. Finally, the solar color is listed for each of the filters. These colors were determined from the average of the colors of the starred solar analogs. 4) A flow chart that illustrates the logic involved in steps 4-9 below. The nominal procedures discussed below are shown as the boldfaced path, and deviations from this path are points at which missing filters are filled in. =========================================================================== NOMINAL PROCEDURES =========================================================================== 0. Starting point. The starting point for these instructions requires the photometric data for each filter be in the form of an instrumental magnitude. Steps 1-3 are not specific to the HB system, but they give a brief review of general photometric reductions and allow a lead-in to the sections that _are_ HB specific. ---------------------------------------------------------------------- 1. Calculate the extinction coefficients using the standard star measurements. The extinction for each filter is determined from a standard star, measured at different airmasses as it rises or sets. Plotting the measured magnitudes against the airmass at which they were observed should result in a linear relation, with the slope of the line defining the extinction coefficient (in magnitudes per airmass). The Y intercept of this relation gives the star's reduced (above the Earth's atmosphere) instrumental magnitude. The extinction coefficients from more than one star can be averaged or used individually for objects in the same region of the sky. Alternatively, a global solution for the extinction coefficients can be done using measurements of several stars at various airmass. Once the extinction coefficients have been determined, they are used to determine the reduced instrumental magnitude of other objects. Note: the extinction varies across the OH filter, so a linear fit will not work well over a significant range of airmass. Details for determining the OH extinction are still in progress and will be given in a later update. ---------------------------------------------------------------------- 2. Convert the standard star reduced instrumental magnitudes to HB Magnitudes and determine the instrumental correction. The reduced instrumental magnitude differs from the magnitude in the HB system by a shift known as the instrumental correction. For each filter, this correction is the same for every object observed on a given night. Taking the difference between a standard star's instrumental magnitude and its magnitude listed in the HB filter system gives a measure of the instrumental correction. Averaging the values from several stars can improve the result still further. Note: Ideally, the instrumental correction should be determined using the flux standards, but if no flux standards are available, solar analogs can be used. Once the extinction coefficients and the instrumental corrections are determined from the standard stars, they are used to transform observations of other objects (e.g. comets) to the HB system so that an absolute calibration is possible. ---------------------------------------------------------------------- 3. Convert the comet measurements to HB magnitudes. Apply the extinction coefficients to the comet observations, to convert them to instrumental magnitudes above the atmosphere. These magnitudes are then converted to HB magnitudes by applying the instrumental correction for each filter. ---------------------------------------------------------------------- 4. Convert comet magnitudes to linear values. After the HB magnitudes have been determined for the comet observations, they are converted to a linear values for use in the following procedures. The linear format is simply f_Xj = 10^(-0.4 m_Xj) where the Xj subscript denotes a particular HB filter. These linear values will be converted to absolute fluxes in a later step. ---------------------------------------------------------------------- 5. Fill in any missing continuum filter measurements. If any required continuum filters are missing, this is the point at which they are accounted for. See the "ACCOUNTING FOR MISSING CONTINUUM FILTERS" section below to determine what filters are required and how to fill in for them if they are not available. If gas filter measurements are missing, then they are simply left out of the reduction process. ---------------------------------------------------------------------- 6. Remove gas contamination from the continuum measurements. The design and manufacturing of the filters has minimized the amount of contamination from undesired species. There is some C2 contamination in the GC filter and some C3 contamination in the UC filter, but otherwise the continuum bands are assumed to be clean. (In actuality, there is likely to be some contamination from NH2 or other gases, but this is at a very low level (typically about 1%), and without measurements of these species, the contamination cannot be correctly removed anyway.) The gas contamination of the continuum is removed using the appropriate gas filter measurement. The amount of contamination is a fixed fraction of the measured gas flux, and removing this flux from the continuum is, in principle, a simple matter. The catch is that, before the gas is removed from the continuum, the continuum must be removed from the gas measurement. To deal with this problem, the gas contamination must be removed with an iterative process that reduces the mutual contamination with each step. Tests performed with dusty, gassy, and average comets show that, at most, three iterations are sufficient for convergence (to 1 part in 10^4) in all cases. The problem with using the gas measurements to decontaminate the continuum is that, if C2 was not measured, the GC filter cannot be decontaminated. If this is the case, it must be assumed that all of the measured GC flux comes from the continuum. Similarly, if C3 was not measured, then the UC filter cannot be decontaminated, and all of the UC flux must be assumed to come from the continuum. Specific Notes: BC: Since the blue continuum filter is uncontaminated, equation (2-1) is used to note that the "decontaminated" measurement (f'_BC) is the same as the "contaminated" value. GC: The green continuum measurement is decontaminated using equation (2-2) along with the BC and C2 measurements. An iteration is required, so initially f'_GC is set to f_GC, and three iterations are performed. (If the C2 measurement is not available, then f'_GC = f_GC) UC: Similarly, the UV continuum filter is decontaminated using equation (2-4) with the BC and C3 measurements. An iteration is again required, so initially f'_UC is set to f_UC, and three iterations are performed. (If the C3 measurement is not available, then f'_UC = f_UC) RC: The red continuum filter is also uncontaminated, so equation (2-5) indicates that the "decontaminated" measurement is the same as the "contaminated" value. ---------------------------------------------------------------------- 7. Determine the continuum colors and interpolate to get the continuum underlying the gas measurements. After all of the necessary continuum filter measurements exist and have been decontaminated, they are used to estimate the amount of dust underlying the various gas species. The shape of the continuum, after the solar colors have been removed, is assumed to be linear between the continuum filters, so the dust underlying each gas filter is found by interpolation (or extrapolation). The interpolation is done using the continuum color and the solar color to step from one of the continuum wavelengths to the gas wavelength of interest. The result is a linearized magnitude for the underlying dust at each of the gas wavelengths. The color of the continuum at each of the three intervals is determined from equations (3-1), (3-2) and (3-3) (BC to GC, UC to BC, and GC to RC respectively. These equations give the continuum color after removal of the solar color. After these colors have been obtained, the continuum at any other wavelength can be obtained. Equations (4-1) through (4-7) are used to calculate the linearized continuum magnitude at each of the HB filter wavelengths. Note that equation (4-7) has a different format, since the color is measured off the GC filter rather than the BC filter. ---------------------------------------------------------------------- 8. Convert all linearized magnitudes to absolute fluxes. Convert the linearized magnitudes to absolute fluxes by multiplying by the absolute flux of a 0 magnitude star at that particular wavelength. All of the pertinent linearized magnitudes should be converted at this stage, so that the following calculations are done with absolute values. Absolute fluxes are obtained using: equations (5-1) through (5-4) for the continuum filter measurements, equations (6-1) through (6-7) for the interpolated continuum under the gas species, and equations (7-1) through (7-7) for the measurements with the gas filters themselves. ---------------------------------------------------------------------- 9. Correct gas measurement for continuum, contamination, and fraction of the emission band actually measured. The next step in obtaining flux values for the gas species is to remove the continuum and any undesired gas contamination from each measurement. The underlying continuum, which affects all of the gas measurements, was determined in step 7. The contamination from other gases is a fixed fraction of the flux measured by that gas filter, and only affects a few filters. Fortunately, there is no mutual gas contamination of any two filters, so the contamination can be removed in a step-by-step manner, with no iteration. OH, C3, C2 and H2O+ filters are assumed to be uncontaminated NH, CN and CO+ filters are contaminated by C3 After the continuum and contamination have been removed, the remaining value represents the flux (from the desired species) that was transmitted through the filter. This flux is only a fraction of the total emission band flux, so it must be scaled up to account for the rest of the light that was blocked by the filter. The gamma factors are related to the fraction of the total flux that is passed through the filter. (In actuality, gamma is the fraction of flux passed through the filter, normalized by the equivalent width of the filter.) Thus, dividing the measured flux by gamma gives the total band flux. Equations (8-1) through (8-7) are used to perform these last steps, producing the total flux from the gas species of interest. =========================================================================== ACCOUNTING FOR MISSING CONTINUUM FILTERS =========================================================================== The procedures described above provide a systematic means of reducing photometric comet measurements, so that the results from different observers are comparable. However, few observers will be using all 11 of the filters and there are many combinations that can result when some of the filters are not used, so a plan must exist to allow filter subsets to be reduced in the same systematic manner as the full set. If a gas species was not measured, there is no way to constrain the value that should be used, and that missing filter measurement cannot be accounted for. On the other hand, the continuum at one wavelength can be related to other wavelengths through its color. Even if the continuum was only measured at one wavelength, solar color can be assumed to fill in the missing filters. This section describes how missing continuum filters should be accounted for if they are needed. When all of the necessary values have been filled in, then the data can then be reduced in the same manner as described above. --------------------------------------------------------------------------- Required Filters: The continuum filters necessary to reduce a given gas species are listed here. The required list indicates that a measurement in at least one of the listed filters is necessary for proper continuum determination. If only one continuum filter exists, then solar colors will be used to step to other wavelengths. The preferred list gives the best combination of continuum filters to properly determine the color for that wavelength interval. Gas Filter Required Cont. Preferred Cont. ------------ ---------------- --------------- OH NH BC or GC BC and UC CN C3 CO+ BC or GC BC and UC or BC and GC C2 BC or GC BC and GC H2O+ BC or GC or RC GC and RC Note that either the BC or GC filter can be used alone as a measure of the basic continuum level, from which solar colors are assumed to obtain the level at any other wavelength. The UC filter does not have enough signal and the RC filter is too far away to be used reliably in this manner. In general, a continuum value should exist for the continuum filters that bracket a gas species (except for OH and NH, in which the UC and BC are used to extrapolate). For this reason, some continuum filters may not be needed: if OH and NH were not measured, then filling in a missing UC value is not necessary, and if H2O+ was not measured, then the RC value is not needed. --------------------------------------------------------------------------- IF-THEN structure for filling in continuum filters IF both BC and GC measurements are missing and are needed THEN quit UC does not have enough S/N to be reliably used as basis for calculating BC and GC IF GC is missing and BC exists THEN calculate the GC value off BC, using solar colors set f'_BC = f_BC (equation 2-1) assume solar color (k_1 = 0) (equation 1-1) compute f'_GC using equation (1-2) NOTE: when returning to the nominal section, GC does not need to be decontaminated IF BC is missing and GC exists THEN decontaminate GC, and use the solar color to get BC assume solar color (k_1 = 0) (equation 1-3) if C2 measurement is missing, then set f'_GC = f_GC if C2 measurement exists then decontaminate GC Calculate f'_GC using equation (2-3) calculate f'_BC using equation (1-4) Now that BC and GC values both exist, UC and RC can also be determined IF UC is missing THEN use BC and the BC/GC color to get UC set k_2 = k_1 (equation 1-5) use equation (1-6) to compute f'_UC NOTE: when returning to the nominal section, UC does not need to be decontaminated IF RC is missing THEN use GC and BC/GC color to get RC set k_3 = k_1 (equation 1-7) use equation (1-8) to compute f'_RC At this point, a value exists for all of the continuum regions, and the procedures for the nominal reduction can be continued. =========================================================================== APPENDIX A -- TERMINOLOGY =========================================================================== Xj Reference to a particular filter (BC,C2, etc) m_Xj HB magnitude of an object in the Xj filter (reduced to above the atmosphere and adjusted for the zero point) f_Xj Linearized magnitude measured with the Xj filter (f_Xj = 10^[-0.4*m_Xj]) f'_Xj Decontaminated linearized magnitude from the Xj filter f'_Xjc Decontaminated linearized magnitude of the continuum underlying the Xj filter F0_Xj Absolute flux of a zero magnitude star at the Xj wavelength F_Xj Absolute flux measured with the Xj filter (before decontamination) F_Xj = F0_Xj * f_Xj F'_Xj Absolute flux measured with the Xj filter (after decontamination) F'_Xj = F0_Xj * f'_Xj k_1 Continuum color between BC and GC (mag/1000A) k_2 Continuum color between UC and BC (mag/1000A) k_3 Continuum color between GC and RC (mag/1000A) K_UCn Decontamination coefficients for UC and GC being K_GCn contaminated by C3 and C2, respectively =========================================================================== If there are any problems in following these instructions please contact me and I will try to help solve them. Also, if you find any errors, I would appreciate hearing about them so they can be fixed. Tony Farnham farnham@lowell.edu Lowell Observatory (520) 774-3358 x232 1400 W. Mars Hill Road FAX (520) 774-6296 Flagstaff, AZ 86001 ===========================================================================