Tully-Fisher关系，重子物质的暗物质伴侣的关键
Tully-Fisher relation, key to dark matter companion of baryonic matter
摘要：
旋涡星系的自转曲线\emph{i}）比开普勒曲线下降得要平缓得多，而\emph{ii}）的渐近速度几乎与星系质量的四次方根成正比，即塔利-费希尔关系。仅凭这些特征就足以以明确的方式将暗伴星分配给星系。在球形系统之外的区域，我们设计了一个球对称时空来适应刚才提到的特殊性。引力的出现超过了可观测物质所能产生的引力。我们把多余的引力归因于星系的一个假设的、黑暗的、完美的流体伴星，并诉诸塔利-费舍尔关系来推导它的密度和压力。暗密度与星系质量的平方根成正比，并以$r^{-(2+α)}, α\ll 1$的形式下降。暗状态方程是正压的。对于位形的内部，我们要求系统边界处的总力场连续。这使我们能够根据可观测物质的结构来确定内部暗密度和压力的大小和分布。这种形式是非定域的和非线性的，在任何时空点上的暗物质的密度和压力都以非线性的方式依赖于重子物质在系统的全部或部分上的某些积分。
英文摘要：
Rotation curves of spiral galaxies \emph{i}) fall off much less steeply than the Keplerian curves do, and \emph{ii}) have asymptotic speeds almost proportional to the fourth root of the mass of the galaxy, the Tully-Fisher relation. These features alone are sufficient for assigning a dark companion to the galaxy in an unambiguous way. In regions outside a spherical system, we design a spherically symmetric spacetime to accommodate the peculiarities just mentioned. Gravitation emerges in excess of what the observable matter can produce. We attribute the excess gravitation to a hypothetical, dark, perfect fluid companion to the galaxy and resort to the Tully-Fisher relation to deduce its density and pressure. The dark density turns out to be proportional to the square root of the mass of the galaxy and to fall off as $r^{-(2+α)}, α\ll 1$. The dark equation of state is barrotropic. For the interior of the configuration, we require the continuity of the total force field at the boundary of the system. This enables us to determine the size and the distribution of the interior dark density and pressure in terms of the structure of the observable matter. The formalism is nonlocal and nonlinear, and the density and pressure of the dark matter at any spacetime point turn out to depend on certain integrals of the baryonic matter over all or parts of the system in a nonlinear manner.
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