If spiral galaxies did not contain dark matter, how would their rotation curves behave?

In the absence of dark matter, the rotation curves of spiral galaxies would exhibit a behavior where the orbital speeds fall off sharply with increasing distance from the galactic center. This is based on the understanding of gravitational dynamics and how mass is distributed within a galaxy.

In Newtonian dynamics, if only the visible matter (stars, gas, and dust) were present, the mass that influences the orbital speed of stars would follow the distribution of this visible matter. As one moves further away from the dense center of the galaxy, the amount of mass that is gravitationally influencing the stars decreases, leading to a decrease in orbital speed. Mathematically, this relationship can be described using Kepler's laws, where the circular orbital speed decreases with the square root of the distance ( v \propto \sqrt{M/r} ), where ( M ) is the mass within the orbit and ( r ) is the radius.

However, observations of spiral galaxies show that their rotation curves tend to remain flat or even rise slightly at larger distances, which indicates that there is additional unseen mass, attributed to dark matter, exerting additional gravitational influence at these outer radii. Therefore, in a scenario devoid of dark matter, one would expect to see a