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Astron. Astrophys. 337, 9-16 (1998)

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4. Simulations of projection effects

Artefacts due to the nearly edge-on projection of a galactic disk, which is not homogeneous, but with spiral structure and related dust lanes, could in some cases bias the observations, and perturb our statistics on warps; in this section we study these effects, to better subtract the perturbations.

We are particularly interested in the two main projection effects: first, if the galaxy disk is not exactly inclined by [FORMULA], obscuration by dust in the plane affects more the far side than the near side of both bulge and disk. The integrated light maxima are shifted towards the near side, and the isophotes are off-centered especially on the minor axis, since the importance of dust decreases from the center to the outer parts of galaxies. This could mimic a slight bend of the disk in the U-shape type; second, if there exists a somewhat symmetric and contrasted [FORMULA] spiral structure in the disk, which is not exactly edge-on, we can confuse the spiral structure for the disk itself, and "see" the S-shape of the spiral which happens actually to wind in an un-warped disk.

To simulate these effects, we build realistic spiral disks with embedded dust, and reproduce their apparent isophotes as a function of position angle and inclination. The radiative transfer is very simple, including only absorption and no scattering, to give an idea of the first order effects. Scattering has been taken into account in previous modelisations, in order to find clear diagnostics of the dust content of spiral disks (e.g. Byun et al 1994 and references therein). It has been shown that scattering reduces the amount of apparent extinction, and the effect is more important for face-on galaxies, while it is not significant for edge-on disks that concern us here. On the contrary, the effect of spiral structure has not been investigated, and it is of primordial importance here.

4.1. Disk structure

The overall global distribution in radius r and height z of the light corresponds to an exponential stellar disk

[EQUATION]

and a spherical Plummer bulge:

[EQUATION]

The dust is only distributed in the disk, with the same overall distribution, but different values for scale-length and scale-height, rd and hd. The density in the disk is then multiplied by the spiral function:

[EQUATION]

with the possible harmonics [FORMULA] 2 and 4 only; the phase is chosen to give a logarithmic spiral, outside of the radius [FORMULA]

[EQUATION]

with [FORMULA] = 4kpc (inside the disk is axisymmetric) and the dust spiral has the same shape, with a phase shift, i.e. [FORMULA], with [FORMULA] =10[FORMULA], to account for the observation that the dust lanes in general lead/trail the stellar spirals.

To explore the influence of the bulge-to-disk ratio, we simulated 3 values of the mass ratio [FORMULA] = 0, 0.1 and 0.3. The latter value is already large for the samples we are considering (since we selected very flat galaxies, with axis ratios [FORMULA] lower than 0.15, and in average 0.11), but they are instructive and necessary to complete the statistics. We adopt here a constant ratio of 4 between the disk and bulge scale length, i.e. a Plummer parameter ab = 0.25 rs. The scales of the stellar and dust disk are rs =6 kpc, hs =450pc, rd = 7 kpc, and hd = 200pc.

The model galaxies were inclined on the sky to be nearly edge-on, with 5 inclination angles [FORMULA] 80., 82.5, 85., 87.5 and 90[FORMULA]. To estimate the projection effects of the spiral arms, the galaxies were also rotated around their rotation axis, through 9 position angles (from PA = 0 to 160 by 20[FORMULA]). We finally considered three values of the total dust optical depth at [FORMULA] for an edge-on galaxy, [FORMULA] = 1, 5 and 10. Taking into account the 3 bulge-to-disk ratios, these tests resulted in 405 galaxy models, from which we estimate statistically the possible projection biases.

4.2. Absorption calculations

Once the stellar density [FORMULA] and dust density [FORMULA] are settled, the cube representing the galaxy is rotated to the given orientation (PA, i), and the light for any line of sight integrated, taken into account progressively the absorption along the line of sight s as

[EQUATION]

with

[EQUATION]

Some results are shown in Fig. 8f or an inclination not exactly edge-on. We can see how projection effects are producing S-shape warps through special viewing of the spiral structure. The effect of dust is revealed in Fig. 9, where U-shape warps are produced by asymmetric absorption.

[FIGURE] Fig. 8. Some typical results from the simulations: logarithmic contours of inclined ([FORMULA]) galaxies, with bulge-to-disk ratio [FORMULA], and a total edge-on optical depth of [FORMULA]. The position angle PA is indicated in each frame at top right in degrees.

[FIGURE] Fig. 9. Same as previous figure, for more inclined ([FORMULA]) galaxies, with bulge-to-disk ratio [FORMULA], and a total edge-on optical depth of [FORMULA].

4.3. Statistical results

From the 405 models computed, we keep only those flat enough to be compared to galaxies in our sample, i.e. with axis ratio [FORMULA]. This is not equivalent to a constraint on inclination only, since it depends on the orientation of the spiral structure with respect to the line of sight, as shown in Fig. 8. The main result is that projection effects can indeed produce false S-shaped warps in about 50% of cases, but only at low inclination, [FORMULA]. At [FORMULA], the false S-shaped warps are around 30%, and they fall to 0% at [FORMULA]. It is possible that these figures are overestimating the artefacts, since all model galaxies had a nice contrasted spiral structure, which is not the case in actual galaxies. Our sample of flat galaxies have [FORMULA], and average [FORMULA] = 0.11. From the detailed distribution of axis ratios in the observed sample, together with Fig. 10, we estimate the percentage of false S-shaped warps to 15%.

[FIGURE] Fig. 10. Percentage of apparent S-shaped warps, only due to projection effects, as a function of apparent axis ratio for the model galaxies. The error bars are the standard dispersion in [FORMULA].

We also searched for mocked U-shaped warps in those simulations. There were very little, of the order of 6%, although this is subjective. One of the reason is that false U-shaped warps are expected to be due to less inclined dusty galaxies, and in our sample there are only nearly edge-on systems. Secondly, our model galaxies are always symmetrical, and there must be in the observations many U-shape due to intrinsic asymmetries. This is re-inforced by the spiral arm contrast which is constant in our model galaxies, which makes them look more like false S-warps than U-warps, the latter being more frequent for homogeneous disks.

4.4. Simulated true warps

We also considered models with genuine warps, assuming a linear slope for the plane, as soon as the radius is larger than some critical value [FORMULA] = 8kpc. The slope corresponds to an angle of wa = 10[FORMULA], to obtain after projection the order of magnitude of those observed in the optical images. We varied the position angle [FORMULA] of the warp line of node with the spiral. The outer parts mid-plane altitude with respect to the main plane of the galaxy is then:

[EQUATION]

We consider only the straight line of nodes, which is justified from the survey of Briggs (1990). The HI warps, that the optical warps tend to follow, have a straight line of nodes from R25 to [FORMULA] (the Holmberg radius). After [FORMULA], the line of nodes advances in the direction of galaxy rotation, and therefore forms a leading spiral.

A sample of our galaxy models is plotted in Fig. 11 and 12. This allowed us to determine the number of genuine warps that are visible, compared to those that go un-detected because their maximum height above the plane is along the line of sight. The latter cases suffer a bias against selection in a flat galaxy sample, since the resulting effect is to thicken the apparent plane. From these models, and given the observed axis ratio distribution, we estimate that we do not miss more than 20% of the warps through projection effects.

[FIGURE] Fig. 11. Some typical results from the genuine warp simulations: logarithmic contours of inclined ([FORMULA]) galaxies, with bulge-to-disk ratio [FORMULA], and a total edge-on optical depth of [FORMULA]. The position angle [FORMULA] of the warp is here [FORMULA].

[FIGURE] Fig. 12. Same as previous figure, but now the position angle [FORMULA] of the warp is [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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