We have r e v i s i t ed a t o p i c e x p l o r e d b y A . C o n s t a n t i n ( 2 0 0 1 ) -- m e a s u r i n g t h e e f f e c t i v e t e m p e r a t u r e o f the c e n t r a l s t a r o f a p l a n e t a r y n e b u l a e ( P N ) u s i n g t h e Z a n s t r a m e t h o d . W e i n v e s t i g a t ed t h e Z a n s t r a m e t h o d t h r o u g h Hα p h o t o m e t r y o f t w o P N e , s p e c i f i c a l l y A b e l l 3 6 a n d N G C 4 3 6 1 . We were only able to obtain results from observations of NGC 4361, giving us a Zanstra HI temperature of approximately 30,000 K.
There are several competing methods for calculating the effective temperature of the central star of a PN. Of these well known methods, the most reliable is the Zanstra method (Zanstra, 1931). T h e Z a n s t r a m e t h o d c a n b e a p p l i e d o n l y a f t e r a c q u i r i n g f i r s t t h e H α flux of the PN's central star a n d s e c o n d l y t h e a m o u n t o f i o n i z i n g p h o t o n s . I n t h i s c a s e t h e n u m b e r o f i o n i z i n g p h o t o n s have b een c a l c u l a t e d f r o m t h e t o t a l n e b u l a r f l u x a t t h e H α r e c o m b i n a t i o n l i n e . In this work we will explore the physical and observational limits of using the Zanstra Method to calculate the effective temperature of the central stars of two carefully chosen PNe.
To resolve both a planetary nebula and its central star in narrowband H α using the GOT 0.25m telescope has proven difficult in the past (Constantin, 2001). Limitations and hindrances have included but are not limited to angular resolution, and poor choice of candidate PNe. Anticipating similar difficulties we have attempted to increase our chances of success by thoroughly examining and choosing two PNe to observe-- paying close attention to H α magnitudes, relative magnitudes of the central stars as compared to the nebulas which contain them, and angular sizes. Our primary source, A b e l l 3 6, i s k n o w n t o h a v e a w e l l s e p a r a t e d ( i n d i s t a n c e ) c o r e a n d h a l o , as well as a central star that is bright in relation to its nebular emission. N G C 4 3 6 1 i s c h o s e n a s our s e c o n d a r y s o u r c e b e c a u s e i t h a s a much stronger overall m a g n i t u d e , while also having a favorable star-nebula separation and relative magnitudes.
| Object | RA | Dec | Epoch | Filter | Exposure Time | # Exposures | Seeing | Conditions |
| NGC 4361 | 12:24:30.7 | -18:47:04 | 2000.00 | H α | 600 | 2 | 5" | light cirrus |
| Abell 36 | 13:40:41 | -19:52:56 | 2000.00 | H α | 30 | 12 | 5" | light cirrus, no guide star available |
| Table 1. Observations of NGC 4361 and Abell 36 taken 4 May, 2004 UT. |
Bias (Zero) frames were taken, as well as Darks, and Twilight and Dawn Flats. The number of exposures and exposure time are as follows: Twilight Flats, 1 x 30 s, 1 x 60 s, 1 x 90 s, and 1 x 150 s in H α. Zeros, 9 exposures for .11s at a CCD temperature of -20 degrees Celsius. Darks, 3 x 600 s, 3 x 30 s. Dawn Flats, 1 x 120 s, 1 x 90 s, 1 x 60 s in H α.
Zero frames are combined into a single frame, as are the dark frames. The resulting combined zero is then used to correct the combined dark for bias level. Next, the corrected combined dark is used along with the combined zero to correct the flat field exposures. Object frames are then corrected only with a combined dark without zero correction. Finally flats are then combined in Hα and then divided out of the object frames. We check the consistency of our reductions by visual inspection of flat frames and object frames.
Now we would like to combine all the observations of each object to reach our optimal and intended signal to noise ratio. Registering and co-adding are attempted for both A36 and NGC 4361. Upon inspection, re-processing and further inspection, observations of A36 are shown to have no useful data. Unfortunately, observations had to be performed without the use of a guide star and so could only be done in 30-second increments. The short exposure time of these frames does not yield a high enough signal to noise ratio to show the nebula. Also, since we had no guide star to reference during observation, the telescope may have not been positioned correctly. Therefore, no stars appear in the frame, and so co-adding cannot be performed. More fortunately, observations of NGC 4361 yield useful frames, and they are registered and co-added using four visible stars in the frame. Multiple trials of registering and co-adding ensures that the frames are shifted correctly and that the end result was properly co-added. From now on our reduction is only performed on NGC 4361 data.
Our method of calculation requires that we obtain the observed fluxes of the central star and the nebula of NGC 4361. Referencing Figure 1, we can see that the flux of the central star and the nebula mix and overlap in our final co-added frame. We must reduce our data so as to separate these two flux amounts, which proves to be tricky.
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| Figure 1. Image of NGC 4361, which has been bias corrected, registered and co-added from 2 frames. The central star can be seen in the nebula only slightly separated by eye. The frame is cropped around the nebula and so the set of stars used to create our PSF is not visible. |
To create our PSF we must first identify proto-type stars in our field that are preferably bright and isolated, so that we can begin to see how our optics treats point sources. To find the stars in our field we use a tool called daofind. Daofind takes a set background sky level, an input fitting radius, and a prototype FWHM and looks for candidate stars, at a chosen level above the sky, which fit these characteristics. We are able to locate the coordinates of all the stars in the image using this tool. Running the task for several chosen values above the background we are able to filter out noise that may masquerade as stars, and to include stars that may be elusive.
After settling on a set of strong, isolated stars, we allow daofind to create a coordinate file pinpointing their locations. This file is then sent through another script called phot, which will read in the coordinates and calculate, within a certain fitting radius, the magnitudes of the corresponding stars. The file produced by phot will give us the last piece of information we need to create a prototype Point Spread Function (PSF) which eventually be used to model the central star of the nebula.
Fortunately IRAF includes a script named psf, which takes the coordinate file from doafind and the magnitude file from phot and combines the information to create the prototype PSF characteristic of our optics. Obviously the PSF will vary depending on how many stars are used to create the fit, and the quality of the stars (brightness, isolation, etc.) may come into play as well. Throughout our frame there are a maximum of six usable stars. We run many iterations of psf, using different numbers of stars and selecting particularly strong stars. Using IRAF functions called nstar and allstar we can subtract stars from our field using the model PSF to see how well the function fits the stars in our frame. It takes many attempts before star subtraction is acceptable across the field. In the end we have chosen a set of four stars to create the ideal PSF.
Now that we have our prototype PSF, it should represent a typical star in our frame. Therefore if we subtract this PSF from the area centered on the coordinate of the central star in our nebula, we should be able to subtract the star and isolate it from the nebula. Taking our original co-added image, we subtract the psf to fit the central star of the nebula using coordinates of the central star given by daofind at a level of σ=14 above the background. The resulting image, Figure 2, gives us a final frame with only nebula contribution throughout the radius of the nebula.
| Figure 2. Final image of NGC 4361, which has been bias corrected, registered and co-added from 2 frames. The central star has now been removed by the fitting and subtracting of a PSF. Using phot over the radius of the nebula will now result in only nebular flux. |
| Figure 3. Radial profile of NGC 4361 showing intensity versus radius in pixels. The green section is the intense, constant flux output of the center of the nebula. The blue section shows the drop-off of nebular emission as we go to larger radius. The red section allows us to see at which radius nebular emission disapears and only background counts remain. |
To ensure our results are as accurate as possible, we must keep track of error in all of these processes and apply proper error propagation as we move along in reduction. Some sources of error we account for and try to minimize are: systematic error in the PSF caused by choice of the star set used to create the proto-type, error in flux measurements caused by the use of apertures of different radii, and error in the calculation of the effective radius of the nebula via the radial plot.
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We also note that the nebular luminosity also depends on recombination throughout its entire volume:
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The so-called Zanstra discrepancy between Teff(H) and Teff(He II) describes the usual behavior that the Zanstra temperature measured from the He II recombination line is greater than that of the temperature measured from the H I recombination line (i.e. Teff(H) < Teff(He II)). This discrepancy has been a matter of debate for a long time. The main causes of this discrepancy are: The possible excess of photons able to ionize He+ with energies greater than 54.4eV, the condition of optical thinness to the H continuum, and differential dust absorption in the nebula. Having available previously published values of the He II Zanstra temperature of NGC 4361, it may prove interesting upon completion of our results to compare the two temperature values superficially.
We begin our calculations by defining constants that will be used throughout. Referencing Eq. 8:
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Having identified our constants, we now focus on the bulk of the calculation--the Planck Function. Firstly, the integral on the right hand side of Eq. 8 may be tricky in its original form. Therefore the form of the Planck Function
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The Mathematica notebook detailing our calculations can be downloaded here.
NGC 4361 turned out to be a much more fruitful observation. With our exposures we were able to resolve and separate the flux of the central star from the flux of the nebula itself. Using a PSF created from field stars we were able to cleanly subtract the central star from the nebula (Fig. 2).
Using aperture photometry in IRAF we measured an Hα flux of 1009.24±195.32 counts from the central star itself. From the nebula after subtraction we measured a flux of 623943.10±2873.46 counts. This gave us a flux ratio of 0.0016±0.0003. Using Mathematica routines and first principles of the Zanstra method as described in the two previous sections, we plotted flux ratio vs. effective central star temperature and thus were able to determine the temperature of NGC 4361's central star. Our measurements give us a Teff of approximately 30,000 K, with an upper limit of 31,100K and a lower limit of 29,700K.
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To have a lower temperature than known values means that our measured flux ratio is much higher than those of previous observations. This means that either the flux of the central star was measured high, or the flux of the nebula was found to be low. Each of these situations can arise as problems caused by the reduction process. If our constructed PSF was too strong and subtracted away much more flux than actually belong to the central star, this can account for a high stellar flux. On the other hand if our photometric aperture used to measure the flux of the nebula was taken to be smaller than the actual radius of the nebular contribution, then we will have a low nebular flux. Fortunately these two problems can be reinvestigated with reprocessing of the data through the software.
Also, we know the central stars belonging to PNe are in the end of their lives and are cooling off. This begs the question--can the temperature discrepancy seen be attributed to cooling of the star? This question is easily cast aside however, with a few short calculations. The radii of the largest PNe are approximately 2 light years. Diving this size by the typical expansion rate of 20 km/s we get a rough estimate of the lifetime of a PN, somewhere around 30,000 years. Thus, noting that our observations differ only 14 years at most from known measurements, we see that the star has most likely not cooled by 10,000 K in this short span of time.
Though hardly unexpected, the results of our comparison with published He II temperatures lead us into deeper contemplation of the physics of PNe and the suitability of the Zanstra method in determining their stellar temperatures. It seems that there are important deviations between the fluxes from stars and the Planck function, especially in regions where there are large changes in frequency opacity. Further study of these processes would make for interesting project extensions.
In conclusion it seems that the accuracy of the measurement of the Zanstra temperature strongly depends on the quality of both the observations and the theoretical model used to convert flux ratios into effective temperature. However, it is more likely that statistical and systematic errors inherent in the reduction process are the most likely culprit of our deviated results. Higher resolution data (not from the GOT) would definitely improve results, as you would be more able to separate stellar flux from nebular flux. However a more in depth study into reduction methods and the errors they introduce would be a more worthwhile continuance of this project.