10/25/2026

Galactic Rotation Curves

The central question

Why do stars far from the centers of many spiral galaxies orbit faster than we would expect from the visible distribution of matter?

Starting from Newtonian mechanics

For a circular orbit around a spherically symmetric mass distribution, we have:

mv2r=GM(r)mr2\frac{mv^2}{r} = \frac{GM(r)m}{r^2}

Therefore,

v(r)=GM(r)rv(r) = \sqrt{\frac{GM(r)}{r}}

If the enclosed mass approaches a constant,

v(r)∝r−1/2v(r) \propto r^{-1/2}

However, observed rotation curves of many spiral galaxies remain approximately flat at large radii.

Questions to investigate

  1. What assumptions went into the equation above?
  2. What mass distribution could produce a flat curve?
  3. What observations support the dark matter hypothesis?
  4. Could modifications of gravity explain the data?

Further exploration

To be developed.