Applying the Tremaine-Weinberg Method

DATA ANALYSIS

3.3 Applying the Tremaine-Weinberg Method

that the dust ring would significantly affect the measurements, the kinematic profiles show that only a few points deviate significantly from the average around that region.

To aid the interpretation of the results, we also plot the light profile of the galaxy along each slit. For this, the continuum image (Figure 2.11) was simply collapsed in the dispersion direction and left uncalibrated. Both the original and folded values are plotted to clearly show the location of the dust lane.

As a test for template mismatch, each of the template stars was used to extract the stellar kinematics (two stars of type K1III and one star of type GO). When compared with one another, the results are consistent, so that template mismatch does not seem to be a problem. Having more diverse stellar types would however provide a more stringent test.

The resulting kinematic profiles along each slit are shown in Figure 4.1-4.4 and discussed in Chapter 4.

all offset slits of a given PA (but independent of them).

For the velocities, this is ensured by the fact that all data show the same wavelength (and thus velocity) calibration, and that all were processed with the same template star. However, when collapsing the data spatially, one must make sure to either symmetrically exclude or interpolate over foreground stars, which can bias the velocity measurements if bright enough.

For the positions, it is much harder to ensure consistent zero-points between different slits. This is because the galaxy may have been moved along the slit for different offsets and because, for triaxial systems, the maximum of the light profile along the slit may not follow the so-called Y-axis of the TW method. In other words, the photometric minor-axis need not be, and generally is not, perpendicular to the line- of-nodes and the PA adopted (this is most obvious for barred spiral galaxies). The only way forward is thus to determine < X > with respect to the maximum of the light profile along the slit, and to i) calculate the offset between the latter and a fixed axis perpendicular to the line-of-nodes (the same for all offsets) using an image or ii) to assume that the latter is at a fixed angle with respect to the PA adopted and calculate the offset from there (note: method i) was chosen for this project). Again, however, one must either symmetrically exclude or interpolate over foreground stars when determining the mean position.

To determine the mean velocity < v >, we used the non-normalized two-dimensional blue spectra, which are collapsed in the spatial direction for each PA and offset, yielding GT(I^), the integrated spectrum. This is then normalized as for the original

along the slit. The luminosity-weighted mean velocity < v > is then determined in the standard manner:

E VjMvj)

<v>=~ , (3-13) E FT(VJ)

j=0

where m is the number of pixels along the LOSVD in the velocity direction.

To find the mean position < X > , we again used the non-normalized two-dimensional blue spectra. For each PA and offset, the spectra are collapsed in the dispersion direction, yielding S ( X ) , the luminosity profile along the slit. Foreground stars are corrected for when necessary and the luminosity-weighted mean position is again determined in the standard manner:

E XiT,(Xi)

<X>=^ , (3.14)

i=0

where n is the number of pixels along the slit in the spatial direction. For both < X >

and < v > measurements, it is necessary in practice to limit the summations above to only part of the E(X) and FT(V) profiles. Figures 3.2 - 3.12 show both the S(X) and Fr(v) profiles used in the calculations above. To obtain accurate results, one must use only the central portion of these profiles (assuming a monotonic decline), especially for FT(V). The other bumps and wiggles on either side of the main peak are spurious features and may bias the results.

The resulting plots of < v > vs. < X > for the different PAs are shown in Figures 4.5 - 4.8 and discussed in Chapter 4.

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Fig. 3.2.— Profile of E(X) (left) and FT(v) (right) for NGC 5266 at PA = 259°, offset = 0". Note that the x-axis of the S(X) plots in Fig. 3.3 - 3.13 are incorrect.

They should read "Position (pixels)". Also note that for plots of FT(V) in Fig. 3.3 - 3.13, the x-axis has been incorrectly labeled by IRAF as "Wavelength (angstroms)".

It should read "D ( k m s^{- 1}) " . Flux scales are linear but arbitrary.

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Fig. 3.3.— Profile of £ ( X ) (left) and F_T(v) (right) for NGC 5266 at PA = 259°, offset = +11".

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Fig. 3.4.— Profile of E(X) (left) and FT(v) (right) for NGC 5266 at PA = 259°, offset = - 1 1 " .

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Fig. 3.5.— Profile of E(X) (left) and FT{v) (right) for NGC 5266 at PA = 274°, offset = 0".

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Fig. 3.6.— Profile of E(X) (left) and F_T(v) (right) for NGC 5266 at PA = 274°, offset = +11".

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Fig. 3.7.— Profile of S ( X ) (left) and FT(v) (right) for NGC 5266 at PA = 274°, offset - - 1 1 " .

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Fig. 3.8.— Profile of S ( X ) (left) and FT(v) (right) for NGC 5266 at PA = 304°, offset = 0".

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Fig. 3.9.— Profile of S ( X ) (left) and FT{v) (right) for NGC 5266 at PA = 304°, offset = +11".

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Fig. 3.10.— Profile of E(X) (left) and Fr(u) (right) for NGC 5266 at PA = 304^c offset = - 1 1 " .

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Fig. 3.11.— Profile of E(X) (left) and F_T(v) (right) for NGC 5266 at PA - 319°, offset = 0".

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Fig. 3.12.— Profile of E(X) (left) and FT(v) (right) for NGC 5266 at PA = 319°, offset = +11".

Table 3.1. One-dimensional NGC 5266 spectra: position angle, offset, and number of spectra produced for each signal-to-noise ratio threshold.

PA (degrees) 259 259 259 274 274 274 304 304 304

Offset (arcsecs) 0 + 11

-11 0 + 11

-11

Number of Spectra Produced S/N=10

49 23 41 51 59 45 59 43 59

S/N=13 39 17 25 39 43 29 43 27 37

S/N=15 25 13 19 31 31 23 31 23 31

S/N=20 19

7 13 19 19 13 21 13 21 319 0 53 43 31 21 319 + 1 1 25 17 13 9

Chapter 4

Dalam dokumen Kinematics and dynamics of the elliptical galaxy NGC 5266. (Halaman 61-70)

Applying the Tremaine-Weinberg Method

DATA ANALYSIS

3.3 Applying the Tremaine-Weinberg Method

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Chapter 4

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