MOdified Gravity (MOG) is a covariant modification of Einstein's general relativity. This theory is one of the current alternatives to dark matter models. We describe dynamics of collisionless self-gravitating systems in the context of MOG. By studying the weak field approximation of this theory, we derive the equations governing the dynamics of the self-gravitating systems. More specifically, we consider the Jeans instability for self-gravitating fluid and stellar systems, and derive new Jeans mass limit $\tilde{M}_J$ and wave-number $\tilde{k}_J$. Furthermore, considering the gravitational instability in star forming regions, we show that MOG has not a significant difference with general relativity on this astrophysical scale. However, at larger scales such as intergalactic space MOG may lead to different galaxy and structure formation processes

The observation of the hot gas surrounding Sgr A* and a few other nearby galactic nuclei imply that electron and proton mean free paths are comparable to the gas capture radius. So, the hot accretion flows are likely to proceed under week‐collision conditions. Hence, thermal conduction has been suggested as a possible mechanism by which the sufficient extra heating is provided in hot advection‐dominated accretion flow (ADAF) accretion discs. We consider the effects of thermal conduction in the presence of a toroidal magnetic field in an ADAF around a compact object. For a steady‐state structure of such accretion flows, a set of self‐similar solutions are presented. We find two types of solutions which represent high and slow accretion rate. They have different behaviours with saturated thermal conduction parameter, φ.

We find the dispersion relation for tightly wound spiral density waves in the surface of rotating, self-gravitating disks in the framework of Modified Gravity (MOG). Also, the Toomre-like stability criterion for differentially rotating disks has been derived for both fluid and stellar disks. More specifically, the stability criterion can be expressed in terms of a matter density threshold over which the instability occurs. In other words the local stability criterion can be written as Σ 0 < Σ crit (v s , κ, α, µ 0 ), where Σ crit is a function of v s (sound speed), κ (epicycle frequency) and α and µ 0 are the free parameters of the theory. In the case of a stellar disk the radial velocity dispersion σ r appears in Σ crit instead of v s . We find the exact form of the function Σ crit for both stellar and fluid self-gravitating disks. Also, we use a sub-sample of THINGS catalog of spiral galaxies in order to compare the local stability criteria. In this perspective, we have compared MOG with Newtonian gravity and investigated the possible and detectable differences between these theories.