A disk that is intrinsically circular will appear elliptical in projection on the sky. The viewing angles of the disk (see Figure 6) are then easily determined: the inclination is i = arccos(1 - ), where is the apparent ellipticity on the sky, and the line-of-nodes position angle is equal to the major axis position angle PAmaj of the projected body. The viewing angles of the LMC have often been estimated under this assumption, using the projected contours for many different types of tracers (de Vaucouleurs & Freeman 1973; Bothun & Thompson 1988; Schmidt-Kaler & Gochermann 1992; Weinberg & Nikolaev 2001; Lynga & Westerlund 1963; Kontizas et al. 1990; Feitzinger et al. 1977; Kim et al. 1998; Alvarez et al. 1987). However, it now appears that this was incorrect. The kinematics of carbon stars imply = 129.9° ± 6.0° (see Section 5.2), whereas the near-IR morphology of the LMC implies PAmaj = 189.3° ± 1.4° (see Section 2). The result that PAmaj implies that the LMC cannot be intrinsically circular. The value of PAmaj is quite robust; studies of other tracers have yielded very similar results, although often with larger error bars. The result that PAmaj therefore hinges primarily on our confidence in the inferred value of . There have been other kinematical studies of the line of nodes, in addition to that described in Section 5.2. These have generally yielded values of that are both larger and twisting with radius (e.g., Kim et al. 1998; Alves & Nelson 2000). However, the accuracy of these results is suspect because of the important simplifying assumptions that were made in the analyses (see Section 5.3). No allowance was made for a potential solid-body rotation component in the velocity field due to precession and nutation of the LMC disk, which is both predicted theoretically (Weinberg 2000) and implied observationally by the carbon star data (see Section 5.2).
Arguably the most robust way to determine the LMC viewing angles is to use geometrical considerations, rather than kinematical ones. For an inclined disk, one side will be closer to us than the other. Tracers on that one side will appear brighter than similar tracers on the other side. This method does not rely on absolute distances or magnitudes, which are notoriously difficult to estimate, but only on relative distances or magnitudes. To lowest order, the difference in magnitude between a tracer at the galaxy center and a similar tracer at a position (, ) in the disk (as defined in Section 5.1) is
where the angular distance is expressed in degrees. The constant in the equation is (5) / (180 ln 10) = 0.038 magnitudes. Hence, when following a circle on the sky around the galaxy center one expects a sinusoidal variation in the magnitudes of tracers. The amplitude and phase of the variation yield estimates of the viewing angles (i, ).
Van der Marel & Cioni (2001) used a polar grid on the sky to divide the LMC area into several rings, each consisting of a number of azimuthal segments. The data from the DENIS and 2MASS surveys were used for each segment to construct near-IR CMDs similar to that shown in Figure 1. For each segment the modal magnitude (magnitude where the luminosity function peaks) was determined for carbon-rich AGB stars selected by color, as had been suggested by Weinberg & Nikolaev (2001). Figure 9 shows the inferred variation in magnitude as function of position angle for the radial range 2.5° 6.7°. The expected sinusoidal variations are confidently detected. The top panel shows the results for stars selected from the DENIS survey with the color selection criterion 1.5 J - K 2.0. The bottom panel shows the results from the 2MASS survey with the same color selection. The same sinusoidal variations are seen, indicating that there are no relative calibration problems between the surveys. Also, the same variations are seen in the I, J, H and Ks bands, which implies that the results are not influenced significantly by dust absorption. The middle panel shows the variations in the TRGB magnitudes as a function of position angle, from the DENIS data. RGB stars show the same variations as the AGB stars, suggesting that the results are not influenced significantly by potential peculiarities associated with either of these stellar populations. The observed variations can therefore be confidently interpreted as a purely geometrical effect. The implied viewing angles are i = 34.7° ± 6.2° and = 122.5° ± 8.3°. The value thus inferred geometrically is entirely consistent with the value inferred kinematically (see Section 5.2). Moreover, there is an observed drift in the center of the LMC isophotes at large radii which is consistent with both estimates, when interpreted as a result of viewing perspective (van der Marel 2001).
Figure 9. Variations in the magnitude of tracers as function of position angle from van der Marel & Cioni (2001). (a) Modal magnitude of AGB stars in the DENIS data with colors 1.5 J - Ks 2.0. (b) TRGB magnitude from DENIS data. (c) As (a), but using data from the 2MASS Point Source Catalog. All panels refer to an annulus of radius 2.5° 6.7° around the LMC center. Filled circles, open circles, four-pointed stars and open triangles refer to the I, J, H and Ks-band, respectively. Results in different bands are plotted with small horizontal offsets to avoid confusion. The dashed curve shows the predictions for an inclined disk with viewing angles i = 34.7° and = 122.5°.
The aforementioned analyses are sensitive primarily to the structure of the outer parts of the LMC. Several other studies of the viewing angles have focused mostly on the region of the bar, which samples only the central few degrees. Many of these studies have been based on Cepheids. Their period-luminosity relation allows calculation of the distance to each individual Cepheid from a light curve. The relative distances of the Cepheids in the sample can then be analyzed in similar fashion as discussed above to yield the LMC viewing angles. Cepheid studies in the 1980s didn't have many stars to work with. Caldwell & Coulson (1986) analyzed optical data for 73 Cepheids and obtained i = 29° ± 6° and = 142° ± 8°. Laney & Stobie (1986) obtained i = 45° ± 7° and = 145° ± 17° from 14 Cepheids, and Welch et al. (1987) obtained i = 37° ± 16° and = 167° ± 42° from 23 Cepheids, both using near-IR data. The early Cepheid studies have now all been superseded by the work of Nikolaev et al. (2004). They analyzed a sample of more than 2000 Cepheids with lightcurves from MACHO data. Through use of photometry in five different bands, including optical MACHO data and near-IR 2MASS data, each star could be individually corrected for dust extinction. From a planar fit to the data they obtained i = 30.7° ± 1.1° and = 151.0° ± 2.4°. Other recent work has used the magnitude of the Red Clump to analyze the relative distances of different parts of the LMC. Olsen & Salyk (2002) obtained i = 35.8° ± 2.4° and = 145° ± 4°, also from an analysis that was restricted mostly to the the inner parts of the LMC.
There is one caveat associated with all viewing angle results for the central few degrees of the LMC. Namely, it appears that the stars in this region are not distributed symmetrically around a single well-defined plane, as discussed in detail in Section 8.4. In the present context we are mainly concerned with the influence of this on the inferred viewing angles. Olsen & Salyk (2002) perform their viewing angle fit by ignoring fields south-west of the bar, which do not seem to agree with the planar solution implied by their remaining fields. By contrast, Nikolaev et al. (2004) fit all the stars in their sample, independent of whether or not they appear to be part of the main disk plane. Clearly, the (i,) results of Olsen & Salyk and Nikolaev et al. are the best-fitting parameters of well-posed problems. However, it is somewhat unclear whether they can be assumed to be unbiased estimates of the actual LMC viewing angles. For a proper understanding of this issue one would need to have both an empirical and a dynamical understanding of the nature of the extra-planar structures in the central region of the LMC. Only then is it possible to decide whether the concept of a single disk plane is at all meaningful in this region, and which data should be included or excluded in determining its parameters. This is probably not an issue for the outer parts of the LMC, given that the AGB star results of van der Marel & Cioni (2001) provide no evidence for extra-planar structures at radii 2.5°.
In summary, all studies agree that i is in the range 30°-40°. At large radii, appears to be in the range 115°-135°. By contrast, at small radii all studies indicate that is in the range 140°-155°. As mentioned, it is possible that the results at small radii are systematically in error due to the presence of out-of-plane structures. Alternatively, it is quite well possible that there are true radial variations in the LMC viewing angles due to warps and twists of the disk plane. Many authors have suggested this as a plausible interpretation of various features seen in LMC datasets (van der Marel & Cioni 2001; Olsen & Salyk 2002; Subramaniam 2003; Nikolaev et al. 2004). Moreover, numerical simulations have shown that Milky Way tidal effects can drive strong warps in the LMC disk plane (Mastropietro et al. 2004).