Mass of the Milky Way Galaxy
The mass of an object is one of the most fundamental properties of the
object. As such, reliable ways to measure mass are exceedingly important.
- One way
is to simply measure the optical emission (luminosity of a galaxy) and
to (independently) determine the average mass of a star in the galaxy.
This will then lead to an estimate for the mass of the galaxy if all
of the luminosity is in fact due to the stars and all material radiates
in a similar manner. This method has problems because it is
sometimes difficult to know whether all mass produces radiation.
- Another way to determine masses is to use
our knowledge of physics (in particular,
gravity). Since all mass is expected to interact gravitationally whether
or not we can see it, we should be able to measure the mass of objects
even if they are invisible to our eyes by their effects on
nearby objects. This is the tact we now
Rotation Curves ===> Masses
The gist of the method is to note that all objects are in orbit around
the center of our Galaxy. To see how this allows us to infer the mass of
our Galaxy, consider orbital motion. First, consider orbits about the Earth.
- A cannon fires a cannoball horizontal to the surface of the Earth. The
Earth's gravity pulls the cannonball toward the center of the Earth.
- If the cannonball has a small horizontal velocity, it is pulled to the
ground before it has had much chance to move,
- If the cannon shoots the cannonball at a higher speed, then it will travel
further in the horizontal diretion before it is pulled to the ground,
Paths B anc C.
- Now, here is the key, because of the curvature of the Earth, the ground
falls away from the cannonball. If the cannonball is given the
correctvelocity, then the rate at which it falls and the rate at which
the ground falls aways from it can be made the same and the cannonball never
strkes the ground; it goes into orbit, Paths D, E, and
- The critical speed required to go into orbit is determined by how hard the
Earth pulls on the cannoball,
F = - G(MEmca)/R2
A similar situation arises for the Earth as it orbits the Sun.
- The gravitational attraction between the Earth and the Sun tries to pull
the two together.
- The Earth overcomes this by orbiting around the Sun with just the
right speed so that the
force just balances the attractive tug of
Gravity + Centrifugal Force = -GMSunmE/d2
+mEvorb2/d = 0
- The mass of the Sun and Earth and the distance between the Earth
and Sun determine the strength of the gravitational
attraction. The speed of the Earth's motion and the distance of the Earth
from the Sun determine the strength of the centrifugal force. For a given
size orbit, the mass of the Sun determines how fast the
Earth must move to stay in orbit. To turn this around, note that if we
can measure how fast the Earth moves in its orbit and the size of its orbit,
then we can infer the mass of the Sun! The simple expression for
the mass of the Sun is
Mass of the Sun = MSun = vorb2d/G
where G is the gravitational constant, vorb
is the orbital speed,
and d is approximately the distance from the Earth to the Sun.
This is the basis of the method used to determine the mass of
the Milky Way galaxy (and other disk
galaxies). However, before we move on, we have one further tweak to
Orbits Outside the Body
In the Solar System, the planets orbit about the Sun. None of the
planets orbit within the body of the Sun. The planets closer to
the Sun feel stronger gravitational tugs due to the Sun and thus must
travel faster to maintain their orbits. For example, the Earth travels
with a speed of ~30 km per second. Pluto travels with a speed of
~4.6 km per second. This is typical of a sytem where bodies orbit outside
of the attracting mass. This is shown by curve A.
Orbtis Inside the Body
Suppose we lived in a spherical system where the mass was spread out and we
orbited inside of the body. Newton showed that only the mass contained
within our radius contributed to the gravitational force on us.
Note that although there is mass behind us and in front of us, the effect
of gravity from these two sources cancel exactly!
So, as we approach the center of the object, the
gravitational force we feel weakens and then vanishes.
In such extended objects, orbital
speeds do not fall-off as quickly (as in our Solar System)
as one moves away from
the center of the object. In fact,
the orbital speed could stay the same and even possibly
increase. These possibilities are shown
by Curves B & C in the above figure.
For the Milky Way, we find that the orbital speed increases and then
remains roughly constant. This implies that we are still within the
body of the Milky Way, even at the largest distances from the center
of the Milky Way. We have not yet located the edge of the Milky
Way; the Milky Way extends much further than the edge of the disk
of stars (the visible disk)!
- Recasting the equation given above (for the determination of the mass
in a galaxy), we find
Mass = 9x1010(speed/220 km/s)2(orbit/25,000 ly)
- The mass contained in the Milky Way galaxy (in its visible disk)
is then 2x1011 Solar masses.
Note that if we count up the visible stars
to get the luminous mass, we find that we can only account for 30 % of the
mass; 70 % of the mass in the visible disk of our Galaxy
must be Dark Matter.
- the mass contained in the Milky Way galaxy (out to as far as we can
see HI gas) is 6x1011 Solar masses.
Oh my, the mass where we can see stars is only 1/3 of the mass of the
entire Galaxy of which only 30 % is normal luminous matter.
This suggests that a lot of mass in the Milky
Way is in some form which does not radiate large amounts of light
(Dark matter). This interesting result will keep popping up
throughout the rest of the course as we discuss the mass of the Universe.
Return to Lecture 2