<\/p>\n\n\n\n
The rotation curves of disc galaxies are\nflat and dark matter is speculated as explanation. Alternatively, the gravity\nof material disk could explain the flat curves. Using the gravitational force\nthat a disk exerts on a body in the disk, we have computed the the rotation\ncurves of disc galaxies and the curve of their mass densities. The numerical\nresult fits the flat curves and the observed mass densities of galaxies. This\ntheory gives a new way to measure the masses of galaxies using their rotation\nvelocities and shape.<\/p>\n\n\n\n
In a disc galaxy stars orbit the center of\nthe galaxy at velocities that depend on the radial distance from the centre.\nThe measured orbital velocities of typical galaxies are plotted versus radial\ndistance in Figure 1, which are the rotation curves of these disk galaxies.\nFigure 1 shows that the observed rotation curves are flat for large radial\ndistance, which means that these velocities are roughly constant with respect\nto radial distance [1].<\/p>\n\n\n\n
The flat aspect of the rotation curves is\npuzzling because the Newtonian theory of gravity predicts that, like the\nvelocities of the planets in the solar system, the velocity of an orbiting\nobject decreases as the distance from the attracting body increases. Since the\ncenters of galaxies are thought to contain most of their masses, the orbital\nvelocities at large radial distance should be smaller than those near the\ncenter. But the observed the rotation curves clearly show the contrary and the\nobserved orbital velocities are bigger than expected.<\/p>\n\n\n\n
The masses of galaxies estimated using the\nluminosity of visible stars are too low to maintain the stars to move at such\nhigh speed. So, large amount of matter is needed to explain the observed\nvelocity, but we do not see this matter. The missing matter should act\ngravitational force because it should hold the flying stars, but should not\nradiate light because invisible. So, it is dubbed as dark matter. However, dark\nmatter has not been observed directly despite the numerous actively undertaken\nexperiments to detect it. Several alternatives to dark matter exist to explain\nthe rotation curve.<\/p>\n\n\n\n
We propose to model galaxy as regular\nmatter disk. Because of its shape, the resultant gravitational force of\nmaterial disk within the plane of a galaxy is different from that of the galaxy\ntaken as concentrated masses. But, is this resultant force bigger?<\/p>\n\n\n\n
Galaxies are made of stars that move in\ncircular orbits. Let us take out one circular orbit with all its stars and put\nthem in free space without an attracting body at the center. These stars form a\ncircle and attract each other gravitationally, which would make them to fall\ntoward the center if they were stationary, see Figure 2. <\/p>\n\n\n\n
Figure 2 shows a simplified image of stars\nin a circle with F_A being the combined gravitational force that the other\nstars act on the star A. For the star A not to fall out of the circle, it must\nrotate at a nonzero velocity which is labeled as v_A. By symmetry, all the\nstars feel the same gravitational force and must rotate at the same velocity to\nstay in the circle. So, this circle of stars should rotate although no\nattracting body is at the center. It is the proper mass of all the stars of the\ncircle that maintains the stars rotating.<\/p>\n\n\n\n
In the explanation, we will often use the\ncase of a single object in a circular orbit around an attracting body. For\nreferring to this case, we give it the following name. <\/p>\n\n\n\n
Definition: The Newtonian orbital\nacceleration and velocity of a single object in a circular orbit around an\nattracting body are named Single-Orbit acceleration and velocity.<\/p>\n\n\n\n
If a star orbits in circle around a\ncentral mass M, the gravitational force on it is from M only and is labeled as\nF_M. The star would move at the Single-Orbit velocity which is v_s, see\nequation (1).<\/p>\n\n\n\n
Now, let us add the mass M at the center\nof the circle of stars, see Figure 3. The gravitational force acted on the star\nA by the other stars of the circle is still F_A. But in addition it feels the\nforce F_M from the central mass M. So, the total force on the star A is F_A+F_M\nwhich is bigger than F_M. In consequence, to stay in the circle the star A\nshould orbit at a velocity bigger than v_s, say at v_s+\u2206v_A with \u2206v_A being the\ncontribution of the force F_A. So, if a circle of stars orbit an attracting\nbody at the center, their orbital velocity should be bigger than the Single-Orbit\nvelocity of one star around the same body, which is kind of like the case of\nthe bigger than expected orbital velocity in galaxies.<\/p>\n\n\n\n
Now, let us smear the stars of the circle\ninto a disk of dust around the central mass M which is not held by cohesion but\nby gravitational force, see Figure 4. Like the stars of the circle, the\ngravitational force on a dust is from the mass M and the proper mass of the\ndisk. So, the orbital velocity of the dust, v_d in Figure 4, will be bigger\nthan the Single-Orbit velocity around the mass M like the stars in the disk of\na galaxy.<\/p>\n\n\n\n
This is the working principle of our model\nthat explains the faster than expected orbital velocity in galaxies. Now, let\nus see if the gravity of material disk could make the orbital velocity constant\nfor large radial distance.<\/p>\n\n\n\n
For computing the gravitational force that\nthe entire disk exerts on a chunk of material of mass m_1, we use the Newtonian\nlaw of gravitation which expresses the gravitational force that an elementary\nmass dm exerts on the chunk of material, see equation (2) where dF_d is the\ngravitational force, G the universal gravitational constant, R_3 the distance\nbetween m_1 and dm, e_3 the unit vector pointing from m_1 to dm, see Figure 5.<\/p>\n\n\n\n
\u2026<\/p>\n\n\n\n
Figures and equations are in the article\nbelow:<\/p>\n\n\n\n
How galaxies make<\/a> their rotation curves flat and what about dark matter<\/a>?<\/p>\n\n\n\n https:\/\/www.academia.edu\/46903516\/How_galaxies_make_their_rotation_curves_flat_and_what_about_dark_matter<\/a><\/p>\n\n\n\n https:\/\/pengkuanonphysics.blogspot.com\/2021\/04\/how-galaxies-make-their-rotation-curves.html<\/a><\/p>\n\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" 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, … Continue reading