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Next: CONCLUSIONS Up: Results Previous: Capacity Dimensions and Correlation

Comparing Correlation Dimensions and Capacity Dimensions

If we now compare the capacity dimensions for elliptical galaxies with their correlation dimensions, for the entire range of intensities (Figure 9), not much difference is seen, but spiral galaxies show a noticeable amount of difference between both dimensions. Figure 11 shows histograms for the number of spiral galaxies and the number of elliptical galaxies versus the difference between the average correlation dimension and the average capacity dimension for the entire intensity range. Figure 12 shows similar histograms for 20% of the entire intensity range around the center. For elliptical galaxies, one can see a tendency towards negative values. This inspires one to reexamine the methods of calculating the dimensions.


 
Figure 11: Difference between corr. and cap. dimensions for spirals and ellipticals
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Figure 12: Difference between dimensions for selected intensity range
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When we look at the methods described in chapter 3, we see that the correlation dimension takes the distances between two pairs of points into account for the calculation of the correlation summation C(r). If more points are distributed at greater distances, C(r)would have higher values resulting in higher values of correlation dimensions, Dcorr.

Figure 13 shows intensity fall-offs for an elliptical galaxy and a spiral galaxy as we move away from the center. The horizontal solid lines show the intensity range from the mean sky value plus 4 times the standard deviation around the mean sky value to the point were the number of points on the contours around the galaxy are less than 80. The dashed lines show 20% of the intensity range around the center of the above mentioned range. It can be seen that the intensity curve for the elliptical galaxy in Figure 13 has a smoother fall-off than that of the spiral galaxy. The increase in the intensity for the spiral galaxy for greater distances from the center is because of its spiral arms. For such increases in the intensity, the contours around a range of intensities will break into smaller contours. For such ranges, one could expect more pairs of points at greater distances in their images. Figure 14 is good example for such an effect. When the contours are drawn around NGC 2403 for various values of magnitudes, they break into many small contours creating a distribution of points at greater distances. When we examine the fractal dimensions for NGC 2403, we can clearly see the correlation dimension curve above the capacity dimension curve. This is also true for the spiral galaxy NGC 4303, one of whose contours and fractal dimensions are shown in Figure 7 and Figure 8 respectively. When we look at the elliptical galaxy NGC 4374 also described by Figure 7 and Figure 8, we notice that the correlation dimension curve is below the capacity dimension curve for the most part.


 
Figure 13: Intensity fall off from the center of an elliptical and a spiral galaxy
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Figure 14: Some of the contours for NGC 2403 and its fractal dimensions
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Figure 15: Some of the contours for NGC 5813 and its fractal dimensions
\scalebox{0.60}{\rotatebox{270}{\includegraphics{ngc5813dat.ps}}}

NGC 5813 is an elliptical galaxy. From our discussion, we would expect the above mentioned difference to be negative but it turns out to be a positive number, making the average correlation dimension higher than the average capacity dimension. When we examine the contours around different intensity counts, we see that its image (Figure 15) contains a star that did not get removed by the reduction process. Because of this, we can expect the correlation summation C(r)to increase as we move to higher intensity levels. The fractal dimension plot of NGC 5813, also shown in Figure 15, confirms this explanation. It was found that the point where the correlation dimensions curve drops below the capacity dimension curve is the point where the star in the lower left corner disappears. Now, if we remove the star from the image (Figure 16), we find that (on average) the correlation dimension curve is below the capacity dimension curve.


 
Figure 16: Fractal dimensions of NGC 5813 after removing the star
\scalebox{0.60}{\rotatebox{270}{\includegraphics{ngc5813rnostardat.ps}}}

NGC 3031 is classified as a spiral galaxy. From our discussion, we would expect the difference between the averages of the correlation dimension and the capacity dimension to be positive but the difference turns out to be a negative number. When we look at its contours around various intensity levels (Figure 17), we find that the most of its contours are smooth and very much unlike the average spiral galaxies and do not break in to smaller contours.

All the final results for the entire intensity range are tabulated in Table 4 to Table 7 and the results for the selected intensity range are tabulated in Table 8 to Table 11.


 
Figure 17: Some of the contours for NGC 3031 and its fractal dimensions
\scalebox{0.60}{\rotatebox{270}{\includegraphics{ngc3031dat.ps}}}


 
Table 4: NGC 2403 to NGC 3953 (Data for the entire intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.
NGC 2403 S 8.93 1.02 1.27 0.25
NGC 2541 S 12.26 0.90 1.26 0.36
NGC 2683 S 10.64 1.19 1.23 0.04
NGC 2715 S 11.79 1.17 1.24 0.07
NGC 2768 E 10.84 1.13 1.18 0.05
NGC 2775 S 11.03 1.26 1.20 -0.06
NGC 2903 S 9.68 1.21 1.28 0.08
NGC 2976 S 10.82 1.23 1.30 0.07
NGC 2985 S 11.18 1.27 1.28 0.01
NGC 3031 S 7.89 1.16 1.15 -0.01
NGC 3079 S 11.54 0.97 1.13 0.16
NGC 3147 S 11.43 1.20 1.22 0.02
NGC 3166 S 11.32 1.18 1.17 -0.01
NGC 3184 S 10.36 1.17 1.35 0.18
NGC 3198 S 10.87 1.11 1.30 0.19
NGC 3319 S 11.48 0.92 1.23 0.31
NGC 3344 S 10.45 1.22 1.34 0.12
NGC 3351 S 10.53 1.16 1.22 0.06
NGC 3368 S 10.11 1.18 1.17 -0.01
NGC 3377 E 11.24 1.27 1.29 0.02
NGC 3379 E 10.24 1.32 1.26 -0.05
NGC 3486 S 11.05 1.13 1.18 0.05
NGC 3556 S 10.69 1.13 1.19 0.06
NGC 3596 S 11.95 1.22 1.21 -0.01
NGC 3623 S 10.25 1.13 1.14 0.01
NGC 3631 S 11.01 1.18 1.27 0.10
NGC 3672 S 12.09 1.20 1.20 -0.01
NGC 3675 S 11.00 1.28 1.25 -0.03
NGC 3726 S 10.91 1.24 1.37 0.14
NGC 3810 S 11.35 1.13 1.18 0.05
NGC 3877 S 11.79 1.16 1.19 0.02
NGC 3893 S 11.16 1.15 1.20 0.05
NGC 3938 S 10.90 1.15 1.27 0.12
NGC 3953 S 10.84 1.19 1.28 0.09


 
Table 5: NGC 4013 to NGC 4527 (Data for the entire intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.
NGC 4013 S 12.19 1.30 1.43 0.13
NGC 4030 S 11.42 1.20 1.24 0.04
NGC 4088 S 11.15 1.15 1.26 0.11
NGC 4123 S 11.98 1.25 1.37 0.13
NGC 4125 E 10.65 1.13 1.22 0.10
NGC 4136 S 11.69 1.18 1.28 0.10
NGC 4144 S 12.05 1.20 1.28 0.08
NGC 4157 S 12.66 1.20 1.27 0.07
NGC 4178 S 11.90 1.11 1.22 0.11
NGC 4189 S 12.51 1.28 1.34 0.07
NGC 4192 S 10.95 1.23 1.23 0.01
NGC 4216 S 10.99 1.17 1.21 0.04
NGC 4242 S 11.37 1.31 1.41 0.09
NGC 4254 S 10.44 1.04 1.20 0.17
NGC 4258 S 9.10 1.16 1.29 0.13
NGC 4303 S 10.18 1.12 1.27 0.15
NGC 4321 S 10.05 1.01 1.22 0.21
NGC 4365 E 10.52 1.24 1.20 -0.04
NGC 4374 E 10.09 1.23 1.15 -0.09
NGC 4394 S 11.73 1.15 1.23 0.07
NGC 4406 E 9.83 1.26 1.20 -0.05
NGC 4414 S 10.96 1.17 1.20 0.03
NGC 4450 S 10.90 1.16 1.22 0.06
NGC 4472 E 9.37 1.21 1.12 -0.09
NGC 4486 E 9.59 1.21 1.10 -0.10
NGC 4487 S 11.63 1.17 1.30 0.12
NGC 4498 S 12.79 1.22 1.26 0.04
NGC 4501 S 10.36 1.23 1.31 0.08
NGC 4527 S 11.38 1.20 1.27 0.07


 
Table 6: NGC 4535 to NGC 5746 (Data for the entire intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.
NGC 4535 S 10.59 1.25 1.39 0.14
NGC 4548 S 10.96 1.19 1.31 0.11
NGC 4559 S 10.46 1.18 1.36 0.18
NGC 4564 E 12.05 1.20 1.26 0.06
NGC 4569 S 10.26 1.20 1.26 0.06
NGC 4571 S 11.82 1.32 1.39 0.07
NGC 4579 S 10.48 1.14 1.14 0.01
NGC 4593 S 11.67 1.12 1.20 0.08
NGC 4594 S 8.98 1.21 1.14 -0.07
NGC 4621 E 10.57 1.20 1.23 0.03
NGC 4636 E 10.43 1.23 1.14 -0.09
NGC 4651 S 11.39 1.08 1.16 0.08
NGC 4654 S 11.10 1.24 1.33 0.09
NGC 4689 S 11.60 1.33 1.39 0.06
NGC 4725 S 10.11 1.12 1.22 0.10
NGC 4731 S 11.90 1.13 1.25 0.13
NGC 4826 S 9.36 1.30 1.32 0.02
NGC 4861 S 12.90 1.20 1.35 0.16
NGC 5005 S 10.61 1.21 1.27 0.06
NGC 5033 S 10.75 1.21 1.24 0.03
NGC 5055 S 9.31 1.19 1.29 0.10
NGC 5204 S 11.73 1.21 1.26 0.05
NGC 5248 S 10.97 1.20 1.26 0.06
NGC 5322 E 11.14 1.20 1.27 0.06
NGC 5334 S 11.99 1.41 1.50 0.09
NGC 5364 S 11.17 1.13 1.22 0.10
NGC 5371 S 11.32 1.22 1.35 0.13
NGC 5377 S 12.24 1.19 1.26 0.08
NGC 5585 S 11.20 1.10 1.22 0.11
NGC 5669 S 12.03 1.20 1.25 0.05
NGC 5701 S 11.76 1.07 1.13 0.06
NGC 5746 S 11.29 1.18 1.18 0.00

         


 
Table 7: NGC 5792 to NGC 6503 (Data for the entire intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.
NGC 5792 S 12.08 1.21 1.35 0.14
NGC 5813 E 11.45 1.20 1.31 0.12
NGC 5850 S 11.54 1.21 1.27 0.06
NGC 5985 S 11.87 1.25 1.31 0.06
NGC 6015 S 11.69 1.14 1.23 0.09
NGC 6118 S 12.42 1.23 1.30 0.07
NGC 6384 S 11.14 1.17 1.24 0.07
NGC 6503 S 10.91 1.11 1.15 0.04


 
Table 8: NGC 2403 to NGC 3953 (Data for a selected intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.
NGC 2403 S 8.93 1.11 1.36 0.25
NGC 2541 S 12.26 0.90 1.31 0.41
NGC 2683 S 10.64 1.12 1.13 0.00
NGC 2715 S 11.79 1.04 1.21 0.16
NGC 2768 E 10.84 1.03 1.07 0.05
NGC 2775 S 11.03 1.24 1.15 -0.09
NGC 2903 S 9.68 1.28 1.35 0.07
NGC 2976 S 10.82 1.25 1.33 0.07
NGC 2985 S 11.18 1.22 1.17 -0.05
NGC 3031 S 7.89 1.21 1.30 0.09
NGC 3079 S 11.54 1.08 1.26 0.18
NGC 3147 S 11.43 1.26 1.26 0.00
NGC 3166 S 11.32 1.13 1.07 -0.05
NGC 3184 S 10.36 1.14 1.37 0.23
NGC 3198 S 10.87 1.18 1.39 0.21
NGC 3319 S 11.48 0.87 1.10 0.22
NGC 3344 S 10.45 1.16 1.29 0.13
NGC 3351 S 10.53 1.32 1.40 0.08
NGC 3368 S 10.11 1.24 1.22 -0.02
NGC 3377 E 11.24 1.37 1.33 -0.04
NGC 3379 E 10.24 1.33 1.23 -0.11
NGC 3486 S 11.05 1.07 1.14 0.07
NGC 3556 S 10.69 1.17 1.24 0.07
NGC 3596 S 11.95 1.37 1.34 -0.03
NGC 3623 S 10.25 1.20 1.24 0.04
NGC 3631 S 11.01 1.23 1.35 0.12
NGC 3672 S 12.09 1.13 1.18 0.05
NGC 3675 S 11.00 1.28 1.24 -0.03
NGC 3726 S 10.91 1.25 1.35 0.10
NGC 3810 S 11.35 1.06 1.16 0.10
NGC 3877 S 11.79 1.18 1.15 -0.02
NGC 3893 S 11.16 1.24 1.29 0.05
NGC 3938 S 10.90 1.19 1.25 0.06
NGC 3953 S 10.84 1.15 1.24 0.09


 
Table 9: NGC 4013 to NGC 4527 (Data for a selected intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.
NGC 4013 S 12.19 1.35 1.52 0.17
NGC 4030 S 11.42 1.21 1.27 0.06
NGC 4088 S 11.15 1.16 1.23 0.07
NGC 4123 S 11.98 1.16 1.31 0.15
NGC 4125 E 10.65 1.10 1.16 0.06
NGC 4136 S 11.69 1.11 1.17 0.06
NGC 4144 S 12.05 1.14 1.19 0.05
NGC 4157 S 12.66 1.27 1.29 0.02
NGC 4178 S 11.90 1.12 1.19 0.07
NGC 4189 S 12.51 1.36 1.41 0.05
NGC 4192 S 10.95 1.27 1.31 0.05
NGC 4216 S 10.99 1.20 1.27 0.07
NGC 4242 S 11.37 1.31 1.39 0.08
NGC 4254 S 10.44 1.18 1.35 0.18
NGC 4258 S 9.10 1.31 1.34 0.04
NGC 4303 S 10.18 1.24 1.36 0.12
NGC 4321 S 10.05 1.06 1.30 0.25
NGC 4365 E 10.52 1.22 1.14 -0.08
NGC 4374 E 10.09 1.20 1.09 -0.11
NGC 4394 S 11.73 1.14 1.19 0.05
NGC 4406 E 9.83 1.19 1.13 -0.07
NGC 4414 S 10.96 1.22 1.27 0.05
NGC 4450 S 10.90 1.06 1.13 0.08
NGC 4472 E 9.37 1.22 1.13 -0.09
NGC 4486 E 9.59 1.23 1.11 -0.12
NGC 4487 S 11.63 1.17 1.28 0.10
NGC 4498 S 12.79 1.21 1.32 0.11
NGC 4501 S 10.36 1.24 1.31 0.07
NGC 4527 S 11.38 1.21 1.32 0.11


 
Table 10: NGC 4535 to NGC 5746 (Data for a selected intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.
NGC 4535 S 10.59 1.23 1.32 0.09
NGC 4548 S 10.96 1.17 1.18 0.01
NGC 4559 S 10.46 1.15 1.24 0.09
NGC 4564 E 12.05 1.18 1.22 0.05
NGC 4569 S 10.26 1.28 1.30 0.02
NGC 4571 S 11.82 1.26 1.43 0.16
NGC 4579 S 10.48 1.15 1.14 -0.01
NGC 4593 S 11.67 1.06 1.10 0.04
NGC 4594 S 8.98 1.33 1.31 -0.02
NGC 4621 E 10.57 1.14 1.13 -0.01
NGC 4636 E 10.43 1.23 1.12 -0.11
NGC 4651 S 11.39 1.02 1.10 0.08
NGC 4654 S 11.10 1.28 1.34 0.06
NGC 4689 S 11.60 1.19 1.32 0.13
NGC 4725 S 10.11 1.06 1.21 0.15
NGC 4731 S 11.90 1.14 1.18 0.04
NGC 4826 S 9.36 1.25 1.29 0.04
NGC 4861 S 12.90 1.14 1.32 0.17
NGC 5005 S 10.61 1.18 1.25 0.07
NGC 5033 S 10.75 1.26 1.19 -0.08
NGC 5055 S 9.31 1.21 1.36 0.14
NGC 5204 S 11.73 1.21 1.24 0.03
NGC 5248 S 10.97 1.31 1.38 0.07
NGC 5322 E 11.14 1.16 1.24 0.08
NGC 5334 S 11.99 1.49 1.60 0.11
NGC 5364 S 11.17 0.93 1.06 0.13
NGC 5371 S 11.32 1.31 1.41 0.10
NGC 5377 S 12.24 1.15 1.20 0.05
NGC 5585 S 11.20 1.03 1.18 0.14
NGC 5669 S 12.03 1.19 1.19 0.00
NGC 5701 S 11.76 1.09 1.11 0.02
NGC 5746 S 11.29 1.21 1.21 0.01


 
Table 11: NGC 5792 to NGC 6503 (Data for a selected intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.
NGC 5792 S 12.08 1.34 1.51 0.17
NGC 5813 E 11.45 1.09 1.21 0.12
NGC 5850 S 11.54 1.24 1.33 0.09
NGC 5985 S 11.87 1.18 1.25 0.07
NGC 6015 S 11.69 1.18 1.23 0.05
NGC 6118 S 12.42 1.22 1.30 0.08
NGC 6384 S 11.14 1.05 1.14 0.09
NGC 6503 S 10.91 1.03 1.08 0.04

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Next: CONCLUSIONS Up: Results Previous: Capacity Dimensions and Correlation
Sandip Thanki
1999-07-29