The rotation curves of disc galaxies are
flat and dark matter is speculated as explanation. Alternatively, the gravity
of material disk could explain the flat curves. Using the gravitational force
that a disk exerts on a body in the disk, we have computed the the rotation
curves of disc galaxies and the curve of their mass densities. The numerical
result fits the flat curves and the observed mass densities of galaxies. This
theory gives a new way to measure the masses of galaxies using their rotation
velocities and shape.
- Rotation curve
In a disc galaxy stars orbit the center of
the galaxy at velocities that depend on the radial distance from the centre.
The measured orbital velocities of typical galaxies are plotted versus radial
distance in Figure 1, which are the rotation curves of these disk galaxies.
Figure 1 shows that the observed rotation curves are flat for large radial
distance, which means that these velocities are roughly constant with respect
to radial distance .
The flat aspect of the rotation curves is
puzzling because the Newtonian theory of gravity predicts that, like the
velocities of the planets in the solar system, the velocity of an orbiting
object decreases as the distance from the attracting body increases. Since the
centers of galaxies are thought to contain most of their masses, the orbital
velocities at large radial distance should be smaller than those near the
center. But the observed the rotation curves clearly show the contrary and the
observed orbital velocities are bigger than expected.
The masses of galaxies estimated using the
luminosity of visible stars are too low to maintain the stars to move at such
high speed. So, large amount of matter is needed to explain the observed
velocity, but we do not see this matter. The missing matter should act
gravitational force because it should hold the flying stars, but should not
radiate light because invisible. So, it is dubbed as dark matter. However, dark
matter has not been observed directly despite the numerous actively undertaken
experiments to detect it. Several alternatives to dark matter exist to explain
the rotation curve.
We propose to model galaxy as regular
matter disk. Because of its shape, the resultant gravitational force of
material disk within the plane of a galaxy is different from that of the galaxy
taken as concentrated masses. But, is this resultant force bigger?
Galaxies are made of stars that move in
circular orbits. Let us take out one circular orbit with all its stars and put
them in free space without an attracting body at the center. These stars form a
circle and attract each other gravitationally, which would make them to fall
toward the center if they were stationary, see Figure 2.
Figure 2 shows a simplified image of stars
in a circle with F_A being the combined gravitational force that the other
stars act on the star A. For the star A not to fall out of the circle, it must
rotate at a nonzero velocity which is labeled as v_A. By symmetry, all the
stars feel the same gravitational force and must rotate at the same velocity to
stay in the circle. So, this circle of stars should rotate although no
attracting body is at the center. It is the proper mass of all the stars of the
circle that maintains the stars rotating.
In the explanation, we will often use the
case of a single object in a circular orbit around an attracting body. For
referring to this case, we give it the following name.
Definition: The Newtonian orbital
acceleration and velocity of a single object in a circular orbit around an
attracting body are named Single-Orbit acceleration and velocity.
If a star orbits in circle around a
central mass M, the gravitational force on it is from M only and is labeled as
F_M. The star would move at the Single-Orbit velocity which is v_s, see
Now, let us add the mass M at the center
of the circle of stars, see Figure 3. The gravitational force acted on the star
A by the other stars of the circle is still F_A. But in addition it feels the
force F_M from the central mass M. So, the total force on the star A is F_A+F_M
which is bigger than F_M. In consequence, to stay in the circle the star A
should orbit at a velocity bigger than v_s, say at v_s+∆v_A with ∆v_A being the
contribution of the force F_A. So, if a circle of stars orbit an attracting
body at the center, their orbital velocity should be bigger than the Single-Orbit
velocity of one star around the same body, which is kind of like the case of
the bigger than expected orbital velocity in galaxies.
Now, let us smear the stars of the circle
into a disk of dust around the central mass M which is not held by cohesion but
by gravitational force, see Figure 4. Like the stars of the circle, the
gravitational force on a dust is from the mass M and the proper mass of the
disk. So, the orbital velocity of the dust, v_d in Figure 4, will be bigger
than the Single-Orbit velocity around the mass M like the stars in the disk of
This is the working principle of our model
that explains the faster than expected orbital velocity in galaxies. Now, let
us see if the gravity of material disk could make the orbital velocity constant
for large radial distance.
For computing the gravitational force that
the entire disk exerts on a chunk of material of mass m_1, we use the Newtonian
law of gravitation which expresses the gravitational force that an elementary
mass dm exerts on the chunk of material, see equation (2) where dF_d is the
gravitational force, G the universal gravitational constant, R_3 the distance
between m_1 and dm, e_3 the unit vector pointing from m_1 to dm, see Figure 5.
Figures and equations are in the article
How galaxies make their rotation curves flat and what about dark matter?