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Recently, as a generalization of general relativity, a gravity theory has
been proposed in which gravitational field equations are described by the
Cotton tensor. That theory allows an additional contribution to the
gravitational potential of a point mass that rises linearly with radius as
$\Phi = -GM/r + \gamma r/2$, where $G$ is the Newton constant. The coefficients
$M$ and $\gamma$ are the constants of integration and should be determined
individually for each physical system. When applied to galaxies, the
coefficient $\gamma$, which has the dimension of acceleration, should be
determined for each galaxy. This is the same as having to determine the mass
$M$ for each galaxy. If $\gamma$ is small enough, the linear potential term is
negligible at short distances, but can become significant at large distances.
In fact, it may contribute to the extragalactic systems. In this paper, we
derive the effective field equation for Cotton gravity applicable to
extragalactic systems. We then use the effective field equation to numerically
compute the gravitational potential of a sample of 84 rotating galaxies. The 84
galaxies span a wide range, from stellar disk-dominated spirals to
gas-dominated dwarf galaxies. We do not assume the radial density profile of
the stellar disk, bulge, or gas; we use only the observed data. We find that
the rotation curves of 84 galaxies can be explained by the observed
distribution of baryons. This is due to the flexibility of Cotton gravity to
allow the integration constant $\gamma$ for each galaxy. In the context of
Cotton gravity, “dark matter” is in some sense automatically included as a
curvature of spacetime. Consequently, even galaxies that have been assumed to
be dominated by dark matter do not need dark matter.

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