3. Data reduction and analysis

3.1. Deriving the apparent magnitudes

Originally it was hoped to reduce the data using well calibrated standard stars, as would be the case with near IR data sets. However, it quickly became apparent that in general there is a lack of 10 and 20 m data with known pass bands and a known zero points, particularly for the narrow band filters. For a summary of the current situation in 10 and 20m photometry see Hanner & Tokunaga (1991). It was therefore decided to set up new zero points for the IRTF filter set. The zero point that was adopted was to define that the photospheric Vega (HR 7001) is 0.00 magnitudes in all filters. The magnitudes of the stars measured were then tied to this zero point using the Kurucz 1993 model atmosphere grid.

The method used compares the predicted in-band flux density for each star in each filter with the measured signal. The in-band flux density (E ) is

where is the spectral energy distribution (SED) of the star is the total transmission function for the entire system, i.e. the multiplication of the filter profile, detector response and the terrestrial atmospheric transmission function.

3.1.1. Initial flux densities

The programme RED (developed at ESO Chile by P. Bouchet) was used to remove the differential atmospheric extinction to provide a table whose values are proportional to the flux density of the star in each filter. The programme also calculates an error for each measurement based on the fluctuations in the measured signals and the accuracy of the airmass calibration.

3.1.2. The total system transmission

The total system transmission is the multiplication of the detector response, the filter profiles and the atmospheric transmission above Mauna Kea. The detector response was ignored as the response of a bolometer is fairly flat over the wavelength range of interest. The filter profiles come from cold scans of an identical set of filters provided by OCLI to UKIRT, the IRTF and the University of Minnesota (Cohen 1995b), whilst the atmospheric transmission is that used in Cohen et al. (1992). Figure 1 (click here) shows the adopted transmission function for the 5 filters.

  
Figure 1: The total transmission curves for the 5 filters

3.1.3. The absolute photospheric Vega model

Tokunaga (1984) shows that there is no evidence of an IR excess in Vega in the N-Q colour. However, part of the aim of this project is to assess the possibility of predicting IR flux densities to beyond 30m using the model SEDs where the IR excess of Vega (Aumann et al. 1985) would be significant. Hence the true Vega flux density cannot be used as the zero point for the longest wavelengths. We have therefore chosen to use a photospheric Vega model as the zero point.

There remains some controversy over what is the absolute Vega flux density in the infrared, a discussion of which is provided in Mégessier (1995) or Cohen et al. (1992). However, the black body approximation to Vega suggested by Mégessier is unacceptable in this case as it is not clear that the approximation is applicable at 25m. Further the zero points are to be set up using the Kurucz model grids to tie the measured stars to Vega; therefore it is preferable to use a Vega model also derived from the same source.

The photospheric Vega model that we have chosen to adopt is that presented in Cohen (1992), which is a customised Kurucz model that incorporates the metal poor character of Vega. This has a nominal error of 1.45% and in general is within a few percent of other determinations in the near IR. Over the IRTF filter range the flux densities agree with the values given in Hanner & Tokunaga (1991) to within the errors. For the purpose of this paper we shall ignore the error in the Vega flux density and take the model as being a definition. When better determinations become available it is simple to update the given flux densities.

3.1.4. The spectral energy distributions for the stars and predicted magnitudes

A spectral energy distribution was derived for each star measured in this observing programme. It was decided to base the SEDs on the Kurucz 1993 model grids because they cover a wide range of effective temperatures (), metallicities and surface gravities and have a sufficiently high spectral resolution in the near IR that the majority of the major stellar features (e.g. CO) are clearly visible. This grid is derived using the ATLAS 9 code Kurucz (1991).

The stars already have accurate near IR and visible magnitudes Selby et al. (1988); Hammersley et al. (1997), Van de Bliek et al. (1996). The effective temperatures are based on Blackwell et al. (1991). For the stars in common with Blackwell et al. (1991) the was used directly, whilst for the remainder the was derived from the V-K versus relationship, such as that presented in Fig. 4 (click here) of Blackwell et al. (1990). The accuracy of such a determination of is comparable to that of the IRFM and the dominant errors will be systematic. These effective temperatures were then used with the surface gravity and metallicity (Cayrel de Strobel et al. 1992) to extract a spectral shape from the Kurucz model grid. When a log g was not available it was assumed from the spectral type and when a metallicity was not available it was assumed to be solar.

The SEDs needed to calibrate the IRTF bolometry are only required for G to K giants between about 8 and 25 m. Even for the more general case of calibrating ISO the required SEDs are for A0 to M0 dwarfs and giants at wavelengths longer than 2.5 m. In both cases the stars are close to the Rayleigh-Jeans part of their flux curves in the wavelength range of interest and the opacities are dominated by H^- free-free. Therefore, extracting the SEDs was done in such a way that the error in the wavelength range of interest would be minimised.

The final error in a SED is made up of a number of factors.

Error in the initial photometry.

The spectral shapes are turned into absolute flux densities by normalising to filter photometry. This photometry will have an error, typically of order 1%, or better, for high quality VRIJHK data but 2 or 3% at L and M.

The error in .

There is an error in the derived of 1.5 to 3% (Blackwell et al. 1994) which will lead to a wavelength dependent error in the shape of the SED. The dominant source of this error is a systematic difference in derived depending on which model grid is adopted (Mégessier 1994)

In effect the method used here predicts a colour from the shape of the SED. Hence, the magnitude in one filter is obtained using the measured magnitude in the other filter and the predicted colour. However, as the shape of the SED has a temperature dependent error, the predicted colour will also have an error which will depend on the and the filters chosen. Figure 2 (click here) shows the difference in 5 colours when changes by 2% at 4100 K, 5000 K, 6000 K and 9000 K. The longest wavelength is in all cases L, so that real measurements could be used. At these effective temperatures, L is nearly on the Rayleigh-Jeans part of the stellar spectra. Therefore the L-N colour will be within a few percent of zero and the values in Fig. 2 (click here) would be almost identical if a longer wavelength than L were used. The JHKL data were taken from Van der Bliek et al. (1996) and the BV data from the Hipparcos Input Catalogue (Turon et al. 1992). Curves for each colour against were calculated using a polynomial fit. The colour at each , and then 2% different, were then determined from the fits. As can be seen when both filters are close to the Rayleigh-Jeans part of the spectrum (K-L for all four temperatures and JHK-L at 9000 K) the colour is insensitive to small changes in temperature. However, if one, or indeed both, filters are far from the Rayleigh-Jeans part of the spectrum then the change in will produces a significant change in the derived colour. Hence, an error in of 2% would give an error in predicted J-L of around for the stars presented here.

  
Figure 2: The change in colour when the is changes by 2%

In order to minimise the error in the SED longwards of 2.5 m, the filters used to normalise the spectra must lie close to the Rayleigh-Jeans part of the star's spectrum. For the data set presented here this means that only filters with a wavelength longer than H should be used. If K is taken as the basis filter, then an error of 3% in at 4100 K (typical for the used stars here) would produce an error of 0.004 mags in the predicted Q magnitude.

The error in metallicity and Log .

Close to the Rayleigh-Jeans part of the spectra the shape of the SEDs are, in general, insensitive to small changes in metallicity. The stars used will be almost exclusively disc stars, as opposed to low metallicity halo stars, so for those stars without a metallicity the error will be of the order dex which will cause a negligible error in the SED. In the same manner the SEDs are insensitive to small changes in log g .

Errors in the models.

The models can only be considered to be approximations to the true SED of the stars, for example note the controversy over the Vega flux density. However, there are specific regions in the spectrum which could be particularly difficult to model. One example could be in the H window where the spectra change from being dominated by the H^- bound-free opacity to the H^- free-free opacity. This causes a knee in the SED of cooler stars at 1.6 m. A full discussion of this is beyond the scope of the paper. However, the error in the predicted colour is likely to be less if two wavelengths are dominated by the same opacity. As we are interested in wavelengths between 8 and 25 m then the wavelengths used to normalise the SED should be longer then 1.6m implying that K should be the shortest filter to be used.

From the above discussion it is clear that if a weighted mean, using the near IR filters, is used to normalise the SED of K giants, then the K filter will be by far the most important. The J filter would have too large an error because of the uncertainty in the and the L filter has a significantly larger error in the photometry. In order to simplify matters it was decided to use only the K filter to normalise the SEDs. As up to 26 stars are used to set the zero point, using just the K filter will not significantly influence the final error.

Once the SED has been normalised, the predicted magnitude for a star in each of the IRTF filters is then

where pm_s,i is the predicted magnitude of star s in filter i.

3.2. Calculation of the system constant

In principle the system constant (SC ) for each filter is then

where pm_s is the predicted flux density for star s .

is the measured signal after being corrected for the atmospheric extinction. n is the number of stars.

The main calibration stars are bright, well measured at all wavelengths and were measured a number of times. Therefore, it was decided to work with the individual measurements to weight the result in favour of the most measured stars.

Figure 3 (click here) shows the system constant (i.e. ) against temperature for every measurement in the 5 filters. In general the scatter is small, however there are a few problems.

  
Figure 3: The system constants in the 5 filters

3.2.1. The SiO bands

The Kurucz model grids, which were used for constructing the SEDs, do not contain the SiO absorption bands. These affect the region between about 7.8 and 11m in cool stars, such as many of those observed in this data set. The result is that the system constant calculated for these stars will be too low over this wavelength range. The effect of the SiO can be clearly seen in the N, 8.7m and, to a lesser extent, the 9.8m filters for temperatures cooler than about 4400 K. From spectroscopy is clear that there is still a small amount of SiO in stars as hot as 4900 K, however, Fig. 1 (click here) suggests that the SiO apparently disappears in these filters for stars hotter than about 4500 K. Hence in order to determine the system constant in the N, 8.7 m and 9.8m filters only those stars hotter than 4500 K have been used. The residual SiO could lead to a small error, but probably well under 0.01 magnitudes. The effect of the SiO is not significant in the other filters so all of the measurements have been used.

3.2.2. Bad points

There are a few obvious bad points which were deleted. However, there are a couple of anomalies.

In the Q filter there are a group of points with a system constant near 2.55, well away from the rest of the group. It turns out these are all of the points that were taken on the 26th Feb., except one. The exception was a HR 5340 point which was used to calibrate the day. It was decided that HR 5340 measurement was in error, so it was deleted, and the remaining Q values on that day had 0.195 mags subtracted.

In the 12.5 m filter there are 7 points which are near 3.07. Two of these stars were re-measured and the values were the expected ones. The other five were only measured once. The points come from different days and there is no clear indication of a cause. As none of these points belong to HR 5340 or HR 1457 which were the main calibration sources, and as the magnitudes for these stars in the other filters are as expected, these points have been deleted from the 12.5 m filter set.

3.2.3. Calculated system constants

The after taking acount of the above points, the calculated system constants are









The quoted error is the standard error of the distribution.

3.3. The final magnitudes

The magnitude (m ) for each observation is calculated in the normal way for each filter, i.e.:

 

Name N Q

mag sd no mag sd no mag sd no mag sd no mag sd no

HR 1457 -3.000 0.004 10 -2.949 0.003 09 -3.026 0.004 09 -3.052 0.003 09 -3.071 0.004 09

HR 2077 1.329 0.009 01 1.334 0.013 01 1.323 0.015 01 1.328 0.038 01 1.389 0.068 01

HR 2335 2.296 0.008 01 2.329 0.008 01 2.278 0.015 01 2.289 0.049 01

HR 2443 1.699 0.010 01 1.697 0.006 01 1.707 0.010 01 1.694 0.053 01

HR 2459 1.396 0.009 01 1.434 0.008 01 1.385 0.010 01 1.455 0.043 01 1.344 0.044 01

HR 2533 2.181 0.008 01 2.228 0.005 01 2.149 0.010 01 2.217 0.018 01 2.128 0.020 01

HR 2970 1.528 0.008 01 1.526 0.005 01 1.524 0.009 01 1.541 0.043 01

HR 3145 1.305 0.009 01 1.285 0.010 01 1.265 0.039 01

HR 3304 2.232 0.008 01 2.270 0.008 01 2.214 0.011 01 2.230 0.028 01 2.137 0.036 01

HR 3738 1.931 0.006 01 1.973 0.005 01 1.907 0.008 01 1.893 0.024 01

HHR 3834 1.413 0.009 02 1.427 0.010 02 1.399 0.014 02 1.411 0.027 02 1.371 0.052 02

HR 3939 1.894 0.009 02 1.933 0.007 02 1.873 0.009 02 1.884 0.013 02 1.832 0.015 01

HHR 4094 0.255 0.012 01 0.291 0.009 01 0.222 0.007 01 0.168 0.012 01 0.220 0.048 01

HHR 4232 0.142 0.005 01 0.161 0.005 01 0.139 0.007 01 0.103 0.016 01 0.140 0.012 01

HR 4335 0.296 0.007 02 0.303 0.008 02 0.295 0.008 02 0.306 0.018 02 0.258 0.020 02

HHR 4402 0.837 0.013 02 0.888 0.010 02 0.810 0.011 02 0.791 0.026 02 0.768 0.086 02

HR 4701 1.948 0.009 01 1.982 0.011 01 1.935 0.019 01 1.939 0.043 01 1.899 0.089 01

HHR 4728 2.794 0.010 01 2.794 0.007 01 2.804 0.008 01 2.804 0.014 01 2.864 0.037 01

HR 4954 1.150 0.008 02 1.204 0.007 02 1.135 0.008 02 1.129 0.014 02 1.099 0.020 02

HR 5315 0.908 0.008 01 0.929 0.004 01 0.884 0.005 01 0.883 0.012 01 0.865 0.006 01

HR 5340 -3.154 0.003 12 -3.133 0.002 13 -3.168 0.003 13 -3.175 0.006 13 -3.182 0.003 12

HR 5616 1.501 0.007 02 1.518 0.007 02 1.488 0.011 02 1.471 0.018 01 1.498 0.028 02

HR 5622 1.338 0.008 01 1.379 0.007 01 1.317 0.012 01 1.271 0.025 01 1.260 0.028 01

HR 6108 1.610 0.010 02 1.642 0.004 02 1.573 0.007 02 1.577 0.013 02 1.551 0.019 02

HR 6159 1.094 0.009 01 1.138 0.007 02 1.070 0.007 02 1.054 0.012 02 1.009 0.013 02

HR 6705 -1.426 0.008 02 -1.394 0.009 02 -1.470 0.010 02 -1.472 0.023 02 -1.513 0.011 02

 

 

Name N Q

mag sd no mag sd no mag sd no mag sd no mag sd no

HR 2061 -5.079 0.011 03 -4.794 0.011 04 -5.262 0.006 04 -5.299 0.007 03 -5.700 0.010 04

HR 2990 -1.216 0.015 01 -1.214 0.015 01 -1.217 0.015 01 -1.234 0.013 01

HR 4069 -0.993 0.023 01 -0.925 0.005 02 -1.005 0.008 02 -1.086 0.025 02 -1.071 0.030 01

HR 4534 1.921 0.005 02 1.920 0.005 02 1.917 0.006 02 1.926 0.006 02 1.857 0.069 02

 

The final magnitudes are presented in Table 2 (click here). When there are multiple observations of a star these have been averaged using an inverse variance weighted mean and the error quoted is the standard error from the individual values. When there is only one measurement the error is calculated from the error in the original measurement or 0.01 which ever is the greater. The error given at this stage does not include the error in the zero points, which is discussed in more detail in the following section.

For completeness, Table 3 (click here) gives the measured magnitudes for the 4 IRTF standards which were are not in the ISO calibration set. Of these stars we note that HR 2061 and HR 4069 are listed in the Bright Star Catalogue as variables. Further HR 2061, although very bright, is a poor standard as it has a significant IR excess so the flux densities are difficult to predict.

3.4. Probable error in the zero points

The absence of well calibrated photometry for all of the IRTF filters makes a direct validation of the zero points difficult, so more indirect means have been used.

3.4.1. The 10m colours

In setting the zero points each filter was dealt with individually however, from the definition of the zero points, it is clear that any photospheric A0V star should have colours of 0.00 at all wavelengths. Hence for any of the stars measured, the colours using the IRTF filters should be also very close to zero as all are near the Rayleigh-Jeans part of their spectra. Unfortunately, the SiO bands means that at 8.7m and N, and to a lesser extent at 9.8m, the cooler stars in the sample will be significantly below the continuum flux density. However, there are sufficient hotter stars to determine the position of the zero colours. In Fig. 4 (click here) are plotted the colours using the 9.8m filter as the common wavelength. Although there is some SiO present, the accuracy of the photometry is better than in the 12.5m or Q filters. The error bars are calculated from Table 2 (click here). The solid line is the predicted colour from the models (i.e. no SiO). The solid square marks the position of HR 1457 and the solid triangle HR 5340, which were the main calibration stars.

  
Figure 4: The measured and predicted colours for the IRTF filters taking the 9.8 m filter as common. The predicted colours is the continuous line. The main calibration stars are marked as a solid square (HR 1457) and a solid triangle (HR 5340)

If there were a large relative error between the zero point of the 9.8m and that of another filter the values would be offset away from the predicted line, and in this case away from zero. As can be seen, once the SiO is allowed for, the bulk of the sources are within 1 of the predicted colour (all are within 2 ) with the average being very close to zero. The presence of the SiO makes an exact figure difficult to calculate but the figures suggest that the error in the zero points add no more than 1% to the error of the colours. Clearly this is just a relative error, but indicates that the zero points are internally consistent.

3.4.2. Comparison with other 10 micron data

Tokunaga (1984) has published magnitudes for a number of stars in the IRTF N and Q filters. These are calibrated against Vega, which is defined as having a magnitude of 0.00. A second source is Rieke et al. (1985). Rieke et al. provide a comparison between their system and that of Tokunaga and conclude at 10m the maximum difference for the stars in common is 0.02 magnitudes. However, Rieke et al. have defined their zero point to agree with that Tokunaga value.

 

N Q

This work Tokunaga Rieke This work Tokunaga Rieke

HR 1457 -3.000 -3.071

HR 2990 -1.216

HR 5340 -3.154 -3.182

HR 6705 -1.426 -1.513

 

Table 4 (click here) lists the magnitudes for the brightest, multiply measured stars in this work which are also in Tokunaga & Rieke et al. There are two other stars in common however HR 4069 is listed as being variable and HR 4534 is amongst the faintest presented here (even if these stars were included the comparison presented below would be very similar). The comparison magnitudes have the errors which include the errors in the zero points. Using weighted means the difference between the zero points are

Note the has a positive N-Q colour for HR 5340, which is unreasonable for a K giant. If the for HR 5340 is ignored then the , however, this is based solely on HR 1457.

Hence there appears to be a small offset between the N and Q zero points used here and those of Tokunaga/Rieke, however it is only, at most, at the 2 level. From the discussion on the colours we suggest that the simplest method to convert all of the magnitudes presented here to be compatible with the zero points as defined by Rieke et al. or Tokunaga is to subtract 0.024 from all the magnitudes given in Tables 2 (click here) and 3.

3.4.3. Comparison with spectral composite/templates

In Cohen et al. (1995a) a method is presented to construct complete 1.2 to 35 m spectra of K giants by splicing together measured spectral fragments. The spectrum that is produced is linked though Sirius to the absolute Vega model used in this work (Cohen et al. 1992). Therefore, apart from the last step of conversion to flux density, these spectral composites are independent of the Kurucz model grid and will include the SiO features. As the composites are complete spectra it is possible to generate magnitudes for any filter set in the wavelength range and so give, in effect, measured magnitudes.

The difference between the magnitudes presented here and those from the spectral composites (e.g. ) for the main calibration stars (i.e. HR 1457, HR 5340 and HR 6705) are.

In Cohen et al. (1996) a method of extending the spectral composites to fainter stars is developed. These are called spectral templates and assume that the spectral shape is defined from the spectral type and luminosity class and the flux density level set using near IR photometry. These are not as accurate as the composites but there are many more stars available for comparison. As templates are not as accurate as the composites and the stars are not as well measured the template values given below should not be used to convert the zero points to the Cohen system. Rather they give an indication of the internal consistency.

The differences between the magnitudes presented here and the magnitudes predicted from spectral templates (e.g. ) for 9 stars in the list for which templates exist are:

3.4.4. Assumed error in the zero points

From the above it would appear that the zero points are close to those defined at the outset, i.e. Vega having 0.00 mags in all filters. The maximum error when comparing with other determinations of the zero points is 0.028 mag, with the average error being less than this. It is very difficult to accurately determine the true error in the zero points as each of the above data sets used for reference has possible systematic errors which are difficult to evaluate. Taking the average difference as being the error in the zero points, gives a probable error in the zero points for the data set presented here of about 0.015 magnitudes (rms) in all filters.

3.5. Conversion to flux density

As the data presented here are based on broad band filter photometry, the flux density given are the isophotal flux .

i.e.

 

Filter Jy

1.924E-12 3.415E13 49.59

1.221E-12 3.045E13 39.64

4.841E-13 2.409E13 25.07

Q 7.182E-14 1.511E13 9.947

 

In Table 5 (click here) is presented the isophotal fluxes for Vega in the 5 IRTF filters. The fluxes are also given in terms of Jy which are calculated using the above equation but in frequency space. The wavelength at which the continuum spectrum of a star has is known as the isophotal wavelength, (). Table 5 (click here) lists the for Vega in the 5 filters. It also lists the which is the frequency at which the continuum spectrum has the stated flux density in Jy. It should be noted that is not the same as . For a full discussion of the significance of isophotal fluxes and wavelengths, refer to Golay (1974).

The isophotal flux for any star, s , is then

where m_s is the apparent magnitude of the star.

 

  Table 6: The fluxes Wmm^-1 and Jy for the 5 IRTF filters. The percentage error is also given and includes the error in the zero points

In Table 6 (click here) are presented the isophotal fluxes in Wm^-2m^-1 and Jy for the 26 stars. The error in the flux is also given, which is determined from the error in the measurement and the zero point. As stated previously, the error in the absolute Vega flux has been ignored.