Rotational Curve Models

1This discussion is more of a general overview of some of the alternative approaches to resolving the galactic rotational curve. While an attempt will be made to outline some of the issues raised by the Rouke model in the subsequent pages, it is possibly more useful to first step back in order to see the bigger picture of uncertainty that appears to surround the astrophysics of galaxies. Today, the basic structure of a galaxy might be broken down into regions, as illustrated in the basic diagram opposite, although the actual processes within this description might still be open to much debate. However, if it is the case that we do not really understand all the processes going on in our own local galaxy, even within the present era, then we might also have to reflect on the need to continually question our understanding of the complexity  of a much, much bigger universe, which also extends  literally into the dim past and the far future. Therefore, on this basis, we shall table a number of hypotheses forwarded as an explanation of the observed galactic rotation:

  1. Accepted model based on dark matter
  2. Modified Newtonian dynamics (MOND) with no dark matter
  3. Mass  distribution model with no dark matter
  4. Rouke’s frame dragging model with no dark matter
  5. Plasma models with no dark matter

Today, the mainstream consensus appears to be weighted towards a model that requires dark matter and while there are other arguments for the existence of dark matter, which extend beyond the explanation of galactic rotation, it is still a hypothesis that has no accepted foundation within the current particle model. However, in the list above, we see 4 other alternatives, none of which appear to specifically require dark matter, but which are forwarded as an alternative explanation of the observed galactic rotation curves. At this point, the only thing we might conclude with any certainty is that they cannot all be right, which might also tell us something about the general state of verification in this field of study. However, it is possible that the list above simply maps the scope of complexity taking place within all galaxies, e.g.

  1. Unknown physics
  2. Extension to accepted physics
  3. Physics based on incorrect data models
  4. Relativistic spacetime
  5. Non-gravitational models

Clearly, the scope of the list above is probably enough to occupy theoretical speculation for many years to come, especially in the absence of any conclusive verification, and therefore a full comparative review of all these ideas has not been attempted. However, given that the topic of plasma cosmology will be discussed in a subsequent section and the accepted model is now well documented, we might focus the following discussions on the ideas encapsulates in list items 2, 3 and 4. However, before discussing these items in the following sub-pages, it might be worth establishing some of the basic observational processes associated with the astrophysics of galaxies. Originally, it was assumed that the rotation of the stars within a galaxy might comply with the same laws as those guiding the planets within our local solar system, such that classical physics might provide some sort of 1st order approximation of the rotational velocity [v] and the mass [M] within a galaxy , i.e.

[1]      1

Encapsulated in [1] are the classical ideas of planetary motion as formulated by Kepler, first published in 1609, and Newton’s subsequent formulation of the more fundamental gravitational laws, first published in 1687. Surprisingly, many sources still quote [1] as a method for calculating the mass of a galaxy, although the following outline suggests that the modern process is a little more involved. One approach considers the light from the stars, as measured at different radii within the galaxy, which we might visualised as follows:

2

Then, at each radii [r], the 'Doppler' shift of the starlight is measured, such that the relative velocity of the stars can be determined, both towards and away from us. These velocity measurements then have to be corrected for the relative velocity of the galaxy, as a whole, as measured with respect to the galactic centre. There is also a need to adjust for the relative tilt of the galaxy from our observational perspective, such that the rotational velocity of the galaxy can be estimated as a function of the radius [r]. So based on the general principles encapsulated in [1], the original expectation was that the rotational velocity should look something like the following diagram:

3

However, when the observational data was processed, the result, as illustrated below, appeared to suggest that the rotation of galaxies must be subject to some additional process, i.e. missing mass or some revision of the gravitational laws:

4

Subsequently, as the observational data improved, it was realised that while most of the spectral light was coming from the central region of the galaxy, i.e. the bulge, much of the mass had to be distributed throughout the galaxy, such that its mass distribution would not align to a model of a solar system. Today, the generally accepted dark matter model assumes that most of the gravitational energy-mass exists in an outer region called the halo; although as the following discussions will outline, there are other possibilities. However, before delving into these alternative hypotheses, it might be worth highlighting why the rotation curve initially increases in a linearly fashion near the  centre of the galaxy, as illustrated in the following diagram, which might also be linked to the previous cross-reference to Newton shells.

5

The shaded circular area in the diagram represents a spherical volume of homogeneous density [ρ] of finite radius [a], which has a total mass [M=pV] defined by its density [ρ]  and volume [V]. However, any mass [m] at a radius [r] smaller than [a], i.e. inside the homogeneous density sphere is subject to a gravitational force [F] defined by its internal radius [r], not the total radius [a], i.e.

[2]      2

As a result, the inner gravitational force is a linear function of the internal positional radius [r]. In contrast, when the radius [r>a], the mass [M] remains constant and the gravitational force takes the normal form of an inverse square function. However, the inner and outer functions depend on the assumption of a spherical volume defining the mass [M]. In the case of our solar system, the sun can be said to have a spherical radius [a], while all the planets obviously orbit at a radius [r>a] far away from the sun. As such, all the rotational velocities of the planets are defined by the outer inverse square function. However, it is clear that stars orbiting within the energy-mass density of a galaxy, which is not a homogeneous function of radius and is often disk-shaped rather than spherical, will not comply to the inverse square law. However, this statement does not exclude an explanation of the galactic rotation curve that is Newtonian in form; although equally, it does not imply that it should. So, on this basis, it would appear there is still scope for debate, as outlined in the following sub-pages.