Construction of Doppler Tomogram for KU Cygni

Construction of Doppler Tomogram for KU Cygni

The FORTRAN code FFBP was obtained from Allen W. Shafter (1997) and written following Horne (1991). The author of this thesis revised the original code to include a Gaussian cut-off in the application of the Fourier filter to the spectral data. Also, to later facilitate the creation of the Doppler tomogram, a FITS header was included at the top of the output data file. Appendix A contains a listing of the FFBP code.

The user of the FFBP code defines the following variables: (1) Beginning wavelength of flux data, (2) rest wavelength of spectral feature used in tomogram, (3) dimension of velocity map, (4) velocity extent of map, and (5) wavelength dispersion of data points. Prior to the actual application of back-projection mathematics to the flux data, the flux information is filtered by a Fourier transform. The Fourier-transform techniques are extensively described in Numerical Recipes in FORTRAN (Press 1992). The Fourier-filtering techniques utilized the fast Fourier Transform (FFT) (forward and inverse), as applied to real data sets. The FFT techniques require that the number of data points in the sample equal some integer power of two. This condition constrained the dimension of the Doppler image, created by the FFBP code. For this reason, the author created a Doppler image with the dimension of $(65\times65)$. Note that the number of data points filtered through the Fourier transform was sixty-four (64), an integer value of two. The Doppler image required one extra value in each dimension to accommodate the zero velocity axes. With the chosen dimension, as stated above, the velocity dispersion in the Doppler map ( $\Delta v=15.625$ km/sec) corresponds closely to the velocity dispersion of the data ( $\Delta v \approx 18.272$ km/sec) as determined by the wavelength dispersion of the spectra ($\approx0.4$Å/pixel).

The back-projection routine simply iterates through each velocity coordinate in the Doppler map and calculates the observed radial velocity of this coordinate for a particular orbital phase, using Eqn. 5.13. The program sums the flux contributions of this velocity coordinate in every spectrum as determined by the orbital phase of observation. Then, the relative intensity is averaged over all the orbital phases. Once completed for every velocity coordinate, the program outputs the averaged intensity contribution of each velocity coordinate in the user-defined dimensions of the Doppler image. The RTEXTIMAGE task of IRAF was utilized to transform the average intensity values into IRAF image format.

The Doppler map of KU Cygni was created by utilizing all available spectroscopic observations (3+ years of data), excluding the disk and primary eclipse phases. From the original sixty-six (66) spectra, forty (40) spectra contributed to the creation of the Doppler maps. Table 5.1 lists the spectroscopic observations used in the creation of the Doppler tomograms for KU Cygni. The Doppler tomograms extend to $\pm500$ km/sec. This maximum velocity is greater than the maximum Keplerian velocity at the surface of the primary star of KU Cygni ( $v_{surface}\approx437$ km/sec). Thus, the velocity extent of these Doppler tomograms are adequately large enough to display the (assumed) maximum velocity components of gaseous material within the accretion disk regions. The spectra of KU Cygni were originally normalized to a continuum level of unity. Therefore, the average intensity contribution of a particular velocity coordinate on the Doppler map is scaled above this normalized continuum flux.

Phase Listing of Observations Used in Doppler Tomogram

Table 5.1:

Orbital Phase	Image Number	Cycle Number
0.09173 *	6423	379
0.10842 *	10264	400
0.11455	6448	379
0.12467 *	4471	371
0.13739	6472	379
0.14676 *	1587	360
0.16615	6528	379
0.17360 *	10028	399
0.19101	6565	379
0.19759 *	10076	399
0.21813	6609	379
0.22524 *	10096	399
0.24835 *	10124	399
0.27487 *	10156	399
0.29035 *	1061	359
0.39593	2139	361
0.39765 *	3327	368
0.41212	3642	369
0.42242 *	2198	361
0.42684	6661	379
0.45445 *	6693	379
0.46314	3714	369
0.50134 *	3418	368
0.51671	3812	369
0.52735	3459	368
0.54080 *	3876	369
0.67637 *	10292	400
0.70190 *	10312	400
0.72999 *	10336	400
0.75371 *	10368	400
0.77931 *	10376	400
0.80534 *	10428	400
0.83123 *	10448	400
0.85728 *	10496	400
0.87935 *	5929	378
0.88324	10520	400
0.88426	5961	378
0.88968	1299	359
0.90495 *	6005	378
0.91032	10540	400

Three Doppler tomograms of KU Cygni are displayed in Figures 5.4-5.6. In all three tomograms, there is a clear indication for the existence of an accretion disk in KU Cygni. In Fig. 5.4 the back-projection (BP) techniques without the Fourier-filtering routines were used to construct the Doppler map. Streaks (or spokes) are visible in this tomogram, which resulted from the unequal phase spacing between observations. For example, only one spectrum of KU Cygni was obtained between phases 0.60-0.69. The streak angled 125$^o$(counterclockwise) from the x-axis on the Doppler tomogram is a result of inadequate observations within the phase interval of 0.6-0.69. The second tomogram displayed in Fig. 5.5 employed the Fourier filter without the Gaussian cut-off term. As one can see, the high-pass ramp filter, which does not include the Gaussian cut-off, amplified the high frequency noise noticeably in this tomogram, to the extent that the intensity of the emitting regions are ``washed-out'' by the background noise. Figure 5.6 displays the Doppler tomogram of KU Cygni, which was created with Gaussian cut-off in the Fourier-filtering techniques. This tomogram appears essentially similar to the back-projection ``only'' tomogram, although the radial spokes appear to be less evident. Also, in this map (Fig. 5.6), the resolution of features appear sharper than in Fig. 5.4, where the features appeared to be smoothed together. To reduce the artifacts produced by unequal phase sampling, another Doppler tomogram was produced with only 25 observations. The spectroscopic observations used to create this Doppler map are marked with an asterisk in Table 5.1. Figure 5.7 displays the velocity map of the H$\alpha$ emission regions. Notice the difference from Fig. 5.6. The streaks are less apparent in Fig. 5.7. The Doppler tomogram displayed in Fig. 5.7 will be analyzed in the final chapter of this thesis, in order to understand the accretion disk in KU Cygni.

Figure 5.4: Doppler tomogram of KU Cygni in H$\alpha$, BP only. The steaks apparent in this tomogram results from unequal phase sampling of spectroscopic data.

Figure 5.5: Doppler tomogram of KU Cygni in H$\alpha$, FFBP without Gaussian cut-off. The high-frequency noise inherent in the spectroscopic data was highly amplified in this tomogram, as a result of the application of a high-pass ramp filter.

Figure 5.6: Doppler tomogram of KU Cygni in H$\alpha$, FFBP with Gaussian cut-off. Notice the difference in applying a Gaussian cut-off term to the high-pass ramp filter, in order to suppress the high frequency noise. Also, the image appears sharper than the image created from pure back-projection.

Figure 5.7: Doppler tomogram of KU Cygni in H$\alpha$. Twenty-five (25) spectroscopic observations were used to create this tomogram. This sample of observations is more equally spaced in orbital phase than that used to create the previous tomograms.