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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