The ‘Matter’ of Galactic Rotation

The ‘Matter’ of  Galactic Rotation – observable but ‘invisible’

Scientists have probed the motions of interstellar gas, dust as well as the motions of stars to determine rotation rates around the centre of spiral galaxies.  The question arises because of the rotation curve we observe:

If galaxies have mass distributions similar to the observed distributions of stars and gas then, the orbital speed would always decline at increasing distances in the same way as do other systems with most of their mass in the centre, such as the Solar System or the moons of Jupiter around Jupiter. With declining speed at the outskirts, the outer material  at the ends of the spiral arm would be disrupted as it would lag due to the slower speed.

If the rotation of the material around the centre of our spiral Galaxy (the Milky Way) followed the same Solar System dynamics , then  wouldn’t we  see the arms ‘ wind up’?

So, lets have a look at our Solar System dynamics. From the table below we can see that as the distance from the Sun increases, the time it takes to make one orbit (the orbital period) increases even more so that the orbital velocities slow down significantly – courtesy The Orbital Velocities of the Planets and Kepler’s Law

      Planet        Velocity (relative to Earth's Orbit)
Mercury       1.607
Venus         1.174
Earth         1.0
Mars          0.802
Jupiter       0.434
Saturn        0.323
Uranus        0.228
Neptune       0.182

Another way of understanding this is that the more distant planets rotate much more slowly in their orbit around the Sun.

planets
Animation courtesy: http://jrscience.wcp.muohio.edu/movies/planets.GIF

 

This is what was observed and formulated by Johannes  Kepler in the 16th century:

Kepler’s 3rd law of planetary motion:

The square of the orbital Period T is a proportional to the cube of the semi-major axis(R) of its orbit keplarrdLaw

Rotational velocity in terms of T:
nonConstantMotion
So that:
tSubstition
Substituting for velocity v for a given distance R, we find that the rotational velocity is:
finalVKeplerbecause:
substition1
kDeriv
So that the speed of the arm should decrease as the radial distance increases. 
The angular speed for each radius slows down noticeably as we slow down by roots of 'R'.
finalVKepler
omegaKepler

Observed Galactic Rotation Curve – not completely Keplerian!

Alas, (or perhaps thankfully) this is not what we observe… At large radial distances, we must be encountering ‘dark matter’ which is not detectable in telescopes, but is likely responsible for flattening out  the  galactic rotation curve. The rotation of the arms in this region of the ‘dark’ halo  looks more constant rather than decreasing in velocity. That is to say:

vLargeR

For a rotating  disk of a galaxy, we start to observe uniform circular motion  The cube of the  period T is no longer proportional to the  square of the radial distance R. For this uniform circular motion, the period T is simply the time for one revolution, the rotational motion is constant, and the angular motion is:

angularCircular

animation – courtesy P. Browne – using Geogebra

(Click on the image to start the animation)

The model shows:

The tighter winding Keplerian (blue) orbital velocity varies with:

oneOverSqrtRThe non-Keplarian (green) orbital velocity (constant rather than decreasing at the “arms length”  varies with:

oneOverRThis animation shows both the Keplerian angular rotation and the more gradual non-Keplerian model for galactic rotation . Notice that the non-Keplerian (green) arm shows less of a tendency to wind-up over time . The not-observable dark matter is distributed out at the periphery of the spiral arm. There is evidence that the distribution of matter satisfies a rotational model that is non-Keplerian with non-detectable mass, dark matter at the periphery.

Less ‘windup’ in the arms = More Matter at the Periphery:

“Because gravity becomes weaker with larger (increasing) distance, the stars on the outskirts of the visible galaxy should be moving slower than those closer to where most of the visible mass resides. Instead, they are moving at about the same speed, even though they are farther from most of the visible matter! There must be extra mass in the Galaxy we cannot see to create the extra amount of gravity force.” – courtesy Nick Strobel’s Astronomy Notes – Deriving the Galactic Mass from the Rotation Notes

rotcurv-darkhalo

 

Image courtesy : http://en.m.wikipedia.org/wiki/Galaxy_rotation_curve
Image courtesy :
http://en.m.wikipedia.org/wiki/Galaxy_rotation_curve

Image of another similar, Spiral galaxy M33 in Triangulum.

Another  theory that attempts to account for the behaviour of  spiral galaxies with sections of the arms having higher mass density is called the theory of Density Wave Theory  http://en.wikipedia.org/wiki/Density_wave_theory
Since the 1960s, there have been models for the spiral structures of galaxies: Star formation caused by density waves in the galactic disk. See Density Wave animation: 
Spiral arms of a galaxy formed by simplified model of density waves. In this model the arms remain fixed despite the rotation of the galaxy. Stars are moving in and out of the Spiral arms at a slower rate at the periphery.

Final word and seasonal greeting: … Back to Keplerian orbits:

Since the earth is in an elliptical orbit, following Kepler’s laws there is a point in our orbit  closest to the Sun. ,  Early January is this time – the time to celebrate Perihelion: peri (close ) helios( sun). Mary Lou Whitehorne, Canadian astronomy educator, has this to say: “It’s winter in the Northern Hemisphere and we’re at our closest point to the Sun. Closest? Yes, you read that right. Closest! For northerners, the winter solstice has just passed.”

But the truth is, on January 4, 2015, Earth reaches perihelion, its closest point to the Sun in its yearly orbit around our star. This is  thanks to Kepler’s Law #1 The orbit of a planet is an ellipse. See Mary Lou Whitehorne’s   Happy Perihelion!