New information about the distribution of the galactic dark matter

The newest discovery relating to the galactic dark matter is described in the popular article Milky Way Has a "Squashed Beachball"-Shaped Dark Matter Halo. In more formal terms the title states that the orbit of the dwarf galaxy Sagittarius around Milky Way can be understood if the cold dark matter halo is not spherical but ellipsoid with different half axes in each three orthogonal directions. The dark matter distribution allowing the best fit is nearly orthogonal to the galactic plane and looks like a flattened sphere with height equal to one half of the diameter (see the illustration of the article).

The result is surprising since the most natural expectation is a complete spherical symmetry or ellipsoid with a rotational symmetry around the axes orthogonal to the galactic plane. The complete breaking of the rotational symmetry raises the question whether something might be wrong with the usual dark matter models. The following text is strongly updated version of the original one, which contained several errors and was badly organized.

Observations

Consider first in some detail what has been observed. Since the life span of the astronomers is not astronomical, they are not able to measure the orbit of the dwarf galaxy directly. The orbit of the dwarf galaxy can be however deduced from the stream of stars which Milky Way has ripped out from the dwarf galaxy.

Sagittarius is one of the 14 dwarf galaxies forming a gravitational bound state with Milky Way. It is an elliptic dwarf with a diameter of 10⁴ light years (about size as the core of Milky Way). It has rotated about 1 My around Milky Way and already made about 10 full rotations. Now (in astronomical sense) Sagittarius is about to traverse the plane of Milky Way. During its motion Sagittarius experiences enormous tidal forces ripping out stars from it. The resulting stream of ripped out stars marks the orbit of Sagittarius. Obviously Sagittarius loses its mass to Milky Way and has already lost a considerable fraction. The ability of Sagittarius to maintain its coherence has been explained in terms of unusually high dark matter content.

The article states that the study of the paths for the parts of Sagittarius gives different parameters for the dark matter distribution. Maybe the "parts" refer to the four globular clusters of stars belonging to Sagittarius. In any case, a highly refined study of the structure of the star stream left behind by Sagittarius is carried out and one goal has been to find a gravitational potential allowing to fit the paths of the parts deduced from the star debris left behind by Sagittarius. The fact that Sagittarius has made several rotations around Milky Way explains why the "leading star debris" is present in the illustration. The Sagittarius flyaround movie gives an artistic simulation about the situation. It seems that an illustration of the actual track from different angles in the galactic plane must be in question.

The basic observation is that the track is in a good approximation in plane. What one can conclude from this depends on what happens in the ripping out process. The star becomes part of Milky Way in some sense. The ripped out star experiences a free fall in the gravitational field of the Milky way. The question concerns what happens to the velocity of the star as it is ripped out.

1. The most natural guess is that the initial velocity is in a good approximation parallel to the velocity at the moment of ripping out.
2. A much stronger assumption is that the star eventually rotates with the same velocity as the distant stars of Milky Way around its center after the ripping out. If the dark matter is also rotating as it should be and forms a halo the gravitational interactions with it could force the hydrodynamic behavior. If one believe that dark matter in astrophysical length scales can have gigantic value of Planck constant, then hydrodynamics behavior looks natural.

Two models of dark matter

TGD allows to consider two alternative models for the dark matter. Contrary to the first guess both models are consistent if the ripping out process is interpreted in the first manner and need not therefore be hydrodynamic. Both models are consistent with the assumption that dark matter corresponds to particles at magnetic flux tubes, which are dark in the sense that they reside at different pages of the book like structure defined by the generalized imbedding space with pages labeled by differed values of Planck constant. Magnetic flux tubes can be regarded as outcomes of cosmic expansion thickening the extremely thing cosmic strings and weakening the extremely strong magnetic fields inside them.

Classically dark matter corresponds to the magnetic energy of cosmic string. This interpretation is not locally consistent with the General Relativistic form of the Equivalence Principle if one considers a model for the string like object itself. Einstein's equations however make sense when one considers only the long range gravitational fields created by cosmic strings.

The two models are following.

1. The first model is very similar to the standard models of dark matter. If the galactic dark matter consists of decay products of a closed non-circular cosmic string approximately vertical to the galactic plane, a non-spherically symmetric distribution of dark matter is expected and there is qualitative consistency with the observed squeezed sphere character. If the ripping out leads rapidly to a hydrodynamic behavior the stream of the particles should rotate around Milky Way destroy the planarity of the debris stream. This would be like rocket in straight path through a rotating liquid: the used fuel would start to rotate with fluid.

2. In the second model galactic dark matter as matter resides at long cosmic string perpendicular to galactic plane. The matter in galactic plane could be also partially dark and visible matter could have resulted as decay products of the cosmic string transformed to magnetic flux tube. Galactic strings would have been linked around the long strings like pearls in necklace and this would explain the observed long strings of galaxies.

Consider next in detail the latter model. The very heavy cosmic string like object along the axis perpendicular to the galactic plane creates (in the Newtonian approximation) 2-D logarithmic potential forcing everything to rotate with a constant velocity around it. Besides this there is a weaker nearly vertical acceleration orthogonal to the plane created by the matter in the galactic plane. If the density of the matter in the galactic plane is approximated with a constant density, the motion of the individual star is a superposition of a free fall in the perpendicular direction and scattering in a logarithmic potential of form Klog(ρ/ρ₀) in the approximation that the individual stars of the dwarf galaxy move completely independently. Second extreme would be a hydrodynamic flow.

Sagittarius rotates around the axis orthogonal to the plane of galaxy with the same velocity as the galactic matter identified as the velocity of the distant stars in the galactic plane (the constancy of this velocity led to the discovery of dark matter). Stating it differently, the motion of the stars of dwarf galaxy takes place in a a potential, which is sum of a potential V(ρ) depending on the radial coordinate of the plane and a potential V(z) depending on the vertical coordinate and created by the galactic matter.

The models differ from each other in several respects.

1. In the first model the simplest gravitational potential would be some function V(r) of the 3-D radial coordinate and in the first approximation logarithmic. The rotation around the axes of Milky way takes place with a smaller velocity as in case of Milky Way and dark matter. The ripping out process is not consistent with the hydrodynamic behavior. The necessity to modify the spherically symmetric distribution of matter might reflect the fact the behavior is actually hydrodynamic.

2. In the second model galactic matter and Sagittarius itself would rotate with approximately the same velocity around the cosmic string and the ripping out process could be rather smooth since the velocity component in the galactic plane would not be affected in the ideal case. This model is consistent with the hydrodynamic behavior. In the optimal situation only the vertical gravitational forces due to the matter in the galactic plane would tend to rip out stars. This might relate to the fact that Sagittarius has been able to maintain its coherence so long.

   Addition: The article Missing matter mystery of small galaxies in the latest New Scientist tells about mysterious missing dark matter. Roughly half of the dark matter predicted by theories is missing. The dark matter at the long cosmic strings would be the natural candidate for this missing dark matter if visible and dark matter in the plane of galaxy identifiable as decay products of galactic cosmic strings is responsible for the visible matter and already identified dark matter.

Some details related to the central string model

It is interesting to look in more detail the toy model based on cosmic string vertical to the galactic plane (also in this case matter in galactic plane could be decay remnants of a cosmic string). The energies for vertical and transverse motions are conserved separately as is also angular momentum component in vertical direction and one can solve the Newton's equations exactly. By Equivalence Principle one can speak about energy and angular momentum per unit mass: therefore notations e_z, e_T, l for energies and angular momentum are natural.

1. Energy conservation in the vertical direction gives

   v_z²+ 2g_G×z = 2e_z

   where g_G is the analog of gravitational acceleration at the Earth's surface and created by a constant density of the galactic matter in the galactic plane.

2. Angular momentum conservation gives

   ρ²ω=l

3. The conservation of energy in plane orthogonal to z-axis gives the third conservation law

   (dρ/dt)² +l²/ρ² +2Klog(ρ/ρ₀)= 2e_T.

These conditions allow to solve the equations of motions for e_z,e_T, and l for each star involved and the mass of the star does not matter at all. In hydrodynamical model correlations between velocities of stars are forced by idealization as continuous matter. In this case the flow lines correspond to classical orbits with gradient of pressure added as an additional force to gravitational force. Energy and angular momentum are conserved along flow lines also now. Situation becomes more complex (and realistic) when one takes into account the gravitational forces between stars.

For background see the chapter Cosmic Strings of "Physics in Many-Sheeted Space-Time".