The Tully-Fisher relation and its implications for the halo density profile and self-interacting dark matter
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We show that the Tully-Fisher relation observed for spiral galaxies can be explained in the current scenario of galaxy formation without invoking subtle assumptions, provided that galactic-sized dark haloes have shallow, core-like central profiles, with a core radius proportional to halo circular velocity. In such a system, both disk and halo contribute significantly to the maximum rotation of the disk, and the interaction between the disk and halo components acts to reduce the scatter in the Tully-Fisher relation. With model parameters chosen in plausible ranges, the model can well accommodate the zero-point, slope, scatter of the observed Tully-Fisher relation, as well as the large range of disk sizes. The model predicts that LSB disks obey a Tully-Fisher relation similar to that of normal disks, if disk mass-to-light ratio is properly taken into account. The halo profile required by the Tully-Fisher relation is as shallow as that required by the rotation curves of faint disks, but much shallower than that predicted by conventional CDM models. Our results cannot be explained by some of the recent proposals for resolving the conflict between conventional CDM models and the rotation-curve shapes of faint galaxies. If dark matter self-interaction (either scattering or annihilation) is responsible for the shallow profile, the observed Tully-Fisher relation requires the interaction cross section \sigma_X to satisfy <\sigma_{X}|v|>/m_{X}~10^{-16} cm^3/s/GeV.Keywords:
Tully–Fisher relation
Fundamentally for the extended disc region of a spiral galaxy, an alternative solution to Laplace equation has been presented for a potential that is radially symmetric on the disc plane. This potential, unlike newtonian one, is shown to be logarithmic in distance from the centre, which allows for the rotation velocity to be constant along the disc radius. It is also shown that this potential easily manifests into a relationship between inner mass of the galaxy and terminal rotation velocity, which has been empirically observed and known as Baryonic Tully-Fisher relations.
Tully–Fisher relation
Logarithmic spiral
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This paper investigates the statistical properties of the Tully-Fisher (TF) relations for a volume-limited complete sample of spiral galaxies in the nearby Ursa Major Cluster. The merits of B, R, I, and K' surface photometry and the availability of detailed kinematic information from H I synthesis imaging have been exploited. In addition to the corrected H I global profile widths W, the available H I rotation curves allow direct measurements of the observed maximum rotational velocities Vmax and the amplitudes Vflat of the outer flat parts. The dynamical state of the gas disks could also be determined in detail from the radio observations. The four luminosity and three kinematic measures allowed the construction of 12 correlations for various subsamples. For large galaxy samples, the M-log W correlation in conjunction with strict selection criteria is preferred for distance determinations with a 7% accuracy. Galaxies with rotation curves that are still rising at the last measured point lie systematically on the low-velocity side of the TF relation. Galaxies with a partly declining rotation curve (Vmax > Vflat) tend to lie systematically on the high-velocity side of the relation when using W or Vmax. However, systematic offsets are eliminated when Vflat is used. Residuals of the M-log(2Vflat) relation correlate consistently with global galaxy properties along the Hubble sequence like morphological type, color, surface brightness, and gas mass fraction. These correlations are absent for the near-infrared M-log(2Vflat) residuals. The tightest correlation (χ = 1.1) is found for the M-log(2Vflat) relation, which has a slope of -11.3 ± 0.5 and a total observed scatter of 0.26 mag with a most likely intrinsic scatter of zero. The tightness of the near-infrared correlation is preserved when converting it into a baryonic TF relation that has a slope of -10.0 in the case (gas/L) = 1.6 while a zero intrinsic scatter remains most likely. Based on the tightness of the near-infrared and baryonic correlations, it is concluded that the TF relation reflects a fundamental correlation between the mass of the dark matter halo, measured through its induced maximum rotational velocity Vflat, and the total baryonic mass bar of a galaxy where bar ∝ V. Although the actual distribution of the baryonic matter inside halos of similar mass can vary significantly, it does not affect this relation.
Tully–Fisher relation
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Strict criteria are applied to the sample of spiral galaxies with measured rotation curves in order to select those objects for which the observed rotation curve is an accurate tracer of the radial force law. The resulting sub-sample of 10 galaxies is then considered in view of two suggested explanations for the discrepancy between the luminous mass and the conventional dynamical mass of galaxies: dark haloes and the modified Newtonian dynamics (MOND). This is done by means of least-squares fits to the rotation curves. Three-parameter dark-halo models (M/L for the visible disc, the core radius and the asymptotic circular velocity of the halo) work well in reproducing the observed rotation curves, and it is found that, for the higher luminosity galaxies, the visible matter dominates the mass distribution within the optically bright disc. However, in the low-luminosity gas-rich dwarfs the dark component is everywhere dominant. MOND, with one free parameter, (M/L for the visible disc) generally works well in predicting the form of the rotation curves, in some cases better than multi-parameter dark-halo fits. If the distance to the galaxy is also taken as a free parameter, then the MOND fits are as good as three parameter dark-halo models and, with one exception, the implied distances are consistent with the adopted distances within the probable uncertainty in the distance estimates. Restricting the number of parameter in dark-halo models by making use of the disc-halo coupling does not produce satisfactory fits to the rotation curves. The overall conclusion is that MOND is currently the best phenomenological description of the systematics of the discrepancy in galaxies.
Modified Newtonian dynamics
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We show that the Tully-Fisher relation observed for spiral galaxies can be explained in the current scenario of galaxy formation without invoking subtle assumptions, provided that galactic-sized dark haloes have shallow, core-like central profiles, with a core radius proportional to halo circular velocity. In such a system, both disk and halo contribute significantly to the maximum rotation of the disk, and the interaction between the disk and halo components acts to reduce the scatter in the Tully-Fisher relation. With model parameters chosen in plausible ranges, the model can well accommodate the zero-point, slope, scatter of the observed Tully-Fisher relation, as well as the large range of disk sizes. The model predicts that LSB disks obey a Tully-Fisher relation similar to that of normal disks, if disk mass-to-light ratio is properly taken into account. The halo profile required by the Tully-Fisher relation is as shallow as that required by the rotation curves of faint disks, but much shallower than that predicted by conventional CDM models. Our results cannot be explained by some of the recent proposals for resolving the conflict between conventional CDM models and the rotation-curve shapes of faint galaxies. If dark matter self-interaction (either scattering or annihilation) is responsible for the shallow profile, the observed Tully-Fisher relation requires the interaction cross section \sigma_X to satisfy <\sigma_{X}|v|>/m_{X}~10^{-16} cm^3/s/GeV.
Tully–Fisher relation
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The observed stellar velocity dispersions of galactic discs show that the maximum rotation of a disc is on average 63% of the observed maximum rotation. This criterion can, however, not be applied to small or low surface brightness (LSB) galaxies because such systems show, in general, a continuously rising rotation curve until the outermost measured radial position. That is why a general relation has been derived, giving the maximum rotation for a disc depending on the luminosity, surface brightness, and colour of the disc. As a physical basis of this relation serves an adopted fixed mass-to-light ratio as a function of colour. That functionality is consistent with results from population synthesis models and its absolute value is determined from the observed stellar velocity dispersions. The derived maximum disc rotation is compared with a number of observed maximum rotations, clearly demonstrating the need for appreciable amounts of dark matter in the disc region and even more so for LSB galaxies. Matters have been illustrated for two examples; the galaxy NGC 6503 and LSB galaxy NGC 1560.
Stellar population
Tully–Fisher relation
Surface brightness fluctuation
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Using 22 hydrodynamical simulated galaxies in a LCDM cosmological context we recover not only the observed baryonic Tully-Fisher relation, but also the observed "mass discrepancy--acceleration" relation, which reflects the distribution of the main components of the galaxies throughout their disks. This implies that the simulations, which span the range 52 < V$_{\rm flat}$ < 222 km/s where V$_{\rm flat}$ is the circular velocity at the flat part of the rotation curve, and match galaxy scaling relations, are able to recover the observed relations between the distributions of stars, gas and dark matter over the radial range for which we have observational rotation curve data. Furthermore, we explicitly match the observed baryonic to halo mass relation for the first time with simulated galaxies. We discuss our results in the context of the baryon cycle that is inherent in these simulations, and with regards to the effect of baryonic processes on the distribution of dark matter.
Tully–Fisher relation
Mass distribution
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Fundamentally for the extended disc region of a spiral galaxy, an alternative solution to Laplace equation has been presented for a potential that is radially symmetric on the disc plane. This potential, unlike newtonian one, is shown to be logarithmic in distance from the centre, which allows for the rotation velocity to be constant along the disc radius.It is also shown that this potential easily manifests into a relationship between inner mass of the galaxy and terminal rotation velocity, which has been empirically observed and known as Baryonic Tully-Fisher relations.
Tully–Fisher relation
Modified Newtonian dynamics
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The rotation curves of 20 spiral galaxies are examined in the light of a toy model (Soares 1992) which has as the main feature the assignment of a high M/L ratio (=30; Ho=50 km/s/Mpc) to the visible matter. The observed rotation of all galaxies can be accommodated without the assumption of a dark halo.
Moreover, the suggestion is made that the fact that almost all available rotational velocity measurements are derived from emission lines emitted by galaxian gas (either neutral or ionized) makes them inappropriate as tracers of the galaxy gravitational potential. To account for that, the model introduces an effective potential meant to describe the hydrodynamics inside a gaseous disk. The general morphology of the curves (i.e., the presence of a plateau in V(r) X r) is interpreted in this framework as a consequence of the hydrodynamical characteristics of galaxian disks.
The Tully-Fisher relation is expressed in terms of model parameters and used as an additional constraint in the process of fitting the model to the observed rotation of the galaxies.
Tully–Fisher relation
Gravitational potential
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We use the concept of the spiral rotation curves universality to investigate the luminous and dark matter properties of the dwarf disc galaxies in the local volume (size $\sim11$ Mpc). Our sample includes 36 objects with rotation curves carefully selected from the literature. We find that, despite the large variations of our sample in luminosities ($\sim$ 2 of dex), the rotation curves in specifically normalized units, look all alike and lead to the lower-mass version of the universal rotation curve of spiral galaxies found in Persic et al. We mass model the double normalized universal rotation curve $V(R/R_{opt})/V_{opt}$ of dwarf disc galaxies: the results show that these systems are totally dominated by dark matter whose density shows a core size between 2 and 3 stellar disc scale lengths. Similar to galaxies of different Hubble types and luminosities, the core radius $r_0$ and the central density $\rho_0$ of the dark matter halo of these objects are related by $ \rho_0 r_0 \sim 100\hspace{0.1cm} \mathrm{M_\odot pc^{-2}}$. The structural properties of the dark and luminous matter emerge very well correlated. In addition, to describe these relations, we need to introduce a new parameter, measuring the compactness of light distribution of a (dwarf) disc galaxy. These structural properties also indicate that there is no evidence of abrupt decline at the faint end of the baryonic to halo mass relation. Finally, we find that the distributions of the stellar disc and its dark matter halo are closely related.
Dwarf galaxy problem
Cuspy halo problem
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In this paper, the effect of halo substructures on galaxy rotation curves is investigated using a simple model of dark matter clustering. A dark matter halo density profile is developed based only on the scale-free nature of clustering that leads to a statistically self-similar distribution of the substructures at the galactic scale. A semi-analytical method is used to derive rotation curves for such a clumpy dark matter density profile. It is found that the halo substructures significantly affect the galaxy velocity field. Based on the fractal geometry of the halo, this self-consistent model predicts a Navarro–Frenk–White-like rotation curve and a scale-free power spectrum of the rotation velocity fluctuations.
Cuspy halo problem
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