Deriving the Galactic Mass from the Rotation Curve

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Recall from the Gravity Applications
chapter that if you know the size of an object's orbit and how quickly it is
moving in its orbit, you can find the mass of the central object, or the mass of
the two object's orbiting around a common point. The same principle can be used
to calculate the mass of the Galaxy! (Isn't gravity wonderful?) In this case,
the two masses involved are the mass of the star and the mass of the part of the
Galaxy that is inside the star's orbit. For most orbits the Galaxy mass
inside the orbit is MUCH larger than the star's mass, so it is safe to ignore
the star's mass.

The gravity from the part of the Galaxy inside the star's orbit accelerates
the star so it moves in an orbit. The greater the gravity pulling on a star, the
faster the star will move. Applying the mass and orbit speed relations of the gravity chapter, the
enclosed mass inside the star's orbit = (orbital speed)2 × (star's
distance to the galactic center)/G, where G is the gravity
constant. The Galaxy's mass is found from measuring the orbital speed of stars
(or gas clouds) and their distance from the center.

From measurements of stars and hydrogen clouds in the galactic disk,
astronomers construct a rotation curve for the Galaxy. This is a plot of
the orbital speed of stars (or gas clouds) vs. their distance from the center of
the Galaxy. The ``rotation'' part of the term refers to the motion of the disk
as a whole. As a whole the disk seems to be rotating (spinning) even though it
is really the individual stars and clouds that are orbiting around the center.
The rotation curve tells you the the amount of mass inside a given distance from
the center. You find how the mass is distributed in the Galaxy by looking at how
the amount of enclosed mass changes with the size of the orbits.

A rigid body rotation is seen close to the center (as if the part close to
the center was rotating as one single object even though it is made of stars and
gas), then the curve drops off, then rises, and eventually flattens out as far
out as can be seen. 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. The Galaxy is spinning
too quickly---the visible matter does not have enough mass to keep the Galaxy
together!

The observed rotation curve can be decomposed into the individual parts
contributed by each component of the Galaxy: the disk, bulge + stellar halo, and
the rest of total mass is what is called dark matter---material that does
not emit any light (at least it is too faint to be detected yet) but has a
significant amount of gravitational influence. The highest solid line in the
left plot below is for all of the galactic components combined. The other curves
(dashed, dotted, and solid) are the contributions of the individual galactic
components (the bulge + stellar halo = ``bulge'', disk, and the dark matter
halo) to the rotational velocity. In the figure below, the dark matter halo is
called the ``corona''. Select the image to show a magnified view in another
window.

The plot on the right shows the enclosed mass inside a given
distance. If you reach the point where the enclosed mass does NOT increase with
distance, then you will have found the edge of the Galaxy's mass.
Beyond the edge of the Galaxy, the rotational velocity will decrease as
the distance from the center increases. Both plots show that the end of the
Galaxy's mass has not yet been found! The stars, dust, and gas in the disk and
stellar halo do not explain all the mass. The part of the Galaxy that fills up
the remainder of mass is the dark matter halo (corona). Ninety percent of the
Galaxy's mass is in the form of this dark matter! Unfortunately, because gravity
depends only on the distance and mass and not the composition, astronomers are
not certain what the dark matter is composed of. It could be in the form of
planets, brown dwarfs (stars too small to shine from nuclear fusion), white
dwarfs, black holes, neutrinos with mass, or other exotic particles that have
not been discovered in the laboratory yet.

You may be a bit skeptical about the existence of this dark matter. Are not
astronomers going ``too far'' with their speculations? Scientists, by nature,
are skeptical themselves of any radical change in our understanding of the
universe. However, nature and careful observations are the sole judge of
scientific truth and even the smartest scientist must yield to what nature is
telling us. The conclusion that dark matter exists has come from using the
standard scientific method of observation, interpretation, and testing of the
explanation. Looking back in history, the detection of dark matter is just
another application of Newton's law of gravity. Until Neptune was directly
observed, its presence was known from its gravitational effect on Uranus
(Neptune was ``dark matter''). Astronomers have used the orbits and wobbling of
stars to detect the presence of as yet, unseen companions around them (i.e.,
planets and black holes). The extrasolar planets and black holes are
technically, dark matter too. They may be faint or invisible themselves, but
they produce definite gravitational effects on nearby visible objects. At the
other end of the scale, no one has ever directly seen an electron, but you know
such a particle exists and is essential to our society of electricity and
electronics.

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last updated: 28 May 2001

Is this page a copy of Strobel's Astronomy Notes?

Author of original content: Nick Strobel