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Data: A Galactic Rotation Curve

by Luke Woolfenden

Conclusions

The Rotation Theory of the Milky Way

The Milky Way is a typical twin-armed spiral galaxy approximately 50 kpc in diameter with the sun located 10 kpc from the galactic centre. Observations have shown that spiral galaxies typically obey the exponential surface brightness profile:

Formula

where

I(r) = the intensity per unit surface area at a radius r from the galactic centre.

I0 = the intensity per unit surface area at the centre of the galaxy.

r = radius.

a = characteristic scale length, the radius at which the I(r) has decreased to 1/e x I0.

Because the profile is exponential the luminous matter can be thought of as residing almost entirely at the centre. If it is assumed that all matter within a galaxy is luminous Newton?s simple law of gravitation can be used to predict a rotational model.

Newton's law of gravitation states:

Formula

For equilibrium and with no other forces this must be equal to the radial circular motion force:

Formula

Equating these, cancelling through and rearranging for v, gives:

Formula

M = the mass of the galaxy within the radius r.

However we are constructing a model where all matter is luminous so this can be approximated to the total mass of the galaxy, a constant.

v = the rotational velocity of the test mass m at a radius r.

The rotation curve for luminous matter

Graph showing the rotation curve for luminous matter

From this model we would expect to see v drop off approximately with r^(-1/2), as in the image.

This is the central mass approximation model. Notice how the curve is initially linear, this is due to high dynamic drag causing the inner region of the Galaxy to rotate like a solid body.