Galactic Rotation Curve

Introduction

A galactic rotation curve is a graph of the orbital speeds of visible stars or gas in a galaxy versus their radial distance from that galaxy’s center. It is crucial in the study of galaxy formation and evolution and has significant implications for the understanding of dark matter.

Theoretical Framework

Keplerian Rotation

If the mass of a galaxy were concentrated in its center, stars farther out would move slower in accordance with Kepler’s third law. The expected velocity of a star a distance from the center of a galaxy is given by:

where is the gravitational constant and is the mass of the galaxy within the orbit of the star. This equation predicts decreasing velocities (a “Keplerian decline”) as at large radii, leading to a rotation curve that drops off sharply.

Observational Data

However, observations of spiral galaxies do not show this Keplerian decline. Instead, the rotation curve remains flat or even rises slightly with increasing radius, indicating that the mass of a galaxy is not concentrated in its center but is distributed more uniformly throughout its disk.

Dark Matter

The Dark Matter Hypothesis

The disparity between the observed rotation curves and the ones predicted by Kepler’s laws implies that there is more mass in the outer regions of galaxies than can be accounted for by visible matter. This unseen matter, which does not emit or reflect enough electromagnetic radiation to be detected directly, is termed “dark matter.”

Density Distribution

The mass density ρ of dark matter as a function of radius is often modeled with a Navarro-Frenk-White (NFW) profile or an Einasto profile. The NFW profile, for example, is given by

where is the density of the universe at the scale radius .

Limitations and Alternatives

While the dark matter hypothesis is the most widely accepted explanation for the discrepancy in rotation curves, alternative theories exist. Modified Newtonian Dynamics (MOND) modifies Newton’s second law at low accelerations to match the observed rotation curves without invoking dark matter. However, MOND struggles to account for observations on larger cosmological scales.

Conclusion

Understanding the galactic rotation curve is key to modern astrophysics and cosmology. The flatness of these curves has been one of the main pieces of evidence leading to the theory of dark matter, a major unsolved mystery in physics. The study of galactic rotation curves remains an active area of research in the quest to better understand the universe’s composition and structure.

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