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Determining a Spiral Galaxy Rotation

Dark matter assumes we fully understand the motions of the stars, dust clouds, gas clouds, in a spiral galaxy.

There are 2 big problems for this certainty.
1) We started off with our solar system as the model for a galaxy. A galaxy has many more objects with no single Sun for the orbits. Our model is probably inadequate for a galaxy.

2) we do not have an extensive  data base to know how these components are moving now. We have limitations.

Hopefully this post is not too long. Details are important when addressing dark matter.

Details for the two problems:

1) our model

from Wikipedia:

'
The pioneer of studies of the rotation of the Galaxy and the formation of the spiral arms was Bertil Lindblad in 1925. He realized that the idea of stars arranged permanently in a spiral shape was untenable. Since the angular speed of rotation of the galactic disk varies with distance from the centre of the galaxy (via a standard solar system type of gravitational model), a radial arm (like a spoke) would quickly become curved as the galaxy rotates. The arm would, after a few galactic rotations, become increasingly curved and wind around the galaxy ever tighter. This is called the winding problem. Measurements in the late 1960s showed that the orbital velocity of stars in spiral galaxies with respect to their distance from the galactic center is indeed higher than expected from Newtonian dynamics but still cannot explain the stability of the spiral structure.
'

Among those attempts to explain the velocities and stability is dark matter.

Our Sun is observed (by observing other stars) to rotate in a circular orbit (taking 225 million years), not like the planets which all move in an ellipse.

The Sun is a large mass affecting the planets in their orbits.

From online:

'
Although it is convenient to think of the Sun as the stationary anchor of our solar
system, it actually moves as the planets tug on it, causing it to orbit the solar
system's barycenter. The Sun never strays too far from the solar system
barycenter. The barycenter is often outside the photosphere of the Sun, but
never outside the Sun's corona.
'

A spiral galaxy will have billions of stars with no single  huge mass
to mimic the Sun for a behavior like a barycenter. A spiral galaxy has its mass distributed over distances of many light years.

Several recent studies have concluded the galactic magnetic field and magnetic fields in the spiral arms affect the spiral arm formation and motion.

From a Caltech.edu document:

'
Paradoxically, the rotation curve of the nearest galaxy [Milky Way] remains poorly known. Extinction is too large to observe the stars and too small to observe the gas. It is preferable to observe the gas, either at 21 cm or at 2.7 mm, because it extends at much greater radii. Thus we must rely on the corotation of both the stellar and the gaseous systems, an assumption that is not always justified, as mentioned previously.

There is a crucial date (1965) prior to which, as reviewed by Schmidt (1965), it was thought that the outer rotation curve was Keplerian and the estimated mass of the Milky Way was about 2 × 10^11 solar masses. After this year, various authors began to realize that the outer curve was more or less flat, and the conclusion that our Milky Way may contain large amounts of dark matter became more and more widely accepted.
'





Before concluding dark matter is required first we must demonstrate our understanding of spiral galaxies.


2) Limits on our understanding

Here is the technique for getting the gas rotation in the galaxy:

'
Neutral Hydrogen atoms have an emission line at a frequency of 1420.4 MHz, or a wavelength 21 cm. This emission line has proven to be an extremely useful probe for mapping the interstellar medium of the Milky Way Galaxy (or of any galaxy, for that matter). The interstellar gas of the Milky Way Galaxy is about 90% Hydrogen, by number of atoms, and most of the interstellar gas is relatively cold (about 100 K) so the Hydrogen atoms are mostly in the ground state.  The 21-cm line is due to a spin flip of the electron in Hydrogen atoms in the ground state orbital.  This transition is forbidden and so the photons are not easily absorbed by another Hydrogen atom. Additionally, at a wavelength of 21 cm, these photons are also not absorbed by interstellar dust, which blocks our view at optical wavelengths. Because there are so many Hydrogen atoms in the Galaxy, the strength of the total emission along any line of sight in the plane of the Galaxy is significant. And, because this radiation is not absorbed easily, we can detect the emission from Hydrogen atoms far across the galaxy.
'

'
Measurements of the HI region of the Galaxy can be used in various calculations. For example, observations of the 21-cm line can be used to create the rotation curve for our Milky Way Galaxy. If hydrogen atoms are distributed uniformly throughout the Galaxy, a 21-cm line will be detected from all points along the line of sight of our telescope. The only difference will be that all of these spectra will have different Doppler shifts. Once the rotation curve for the Galaxy is known, it can be used to find the distances to various objects. By knowing the Doppler shift of a body, its angular velocity can be calculated. Combining this angular velocity and the plot of the rotation curve, the distance to a certain object can be inferred. Using measurements of the HI region, the mass of the Galaxy can also be determined.

My observation:

We are using the analysis of the relative motion of hydrogen atoms to both a) create the rotation curve, and b) calculate the mass of the galaxy based on that curve.

From an MIT undergrad document online:

'
Measurement of Galactic Rotation Curve


Objective:

The 21cm line produced by neutral hydrogen in interstellar space provides radio astronomers with a very useful probe for studying the differential rotation of spiral galaxies. By observing hydrogen lines at different points along the Galactic plane one can show that the angular velocity increases as you look at points closer to the Galactic center. The purpose of this experiment is to create a rotational curve for the Milky Way Galaxy using 21-cm spectral lines observed with a small radio telescope. The sample observations for this experiment will be made using the small radio telescope located at the Haystack Observatory. The rotational curve will be created by plotting the maximum velocity observed along each line of sight versus the distance of this point from the Galactic center.

Introduction:

The 21-cm line of neutral hydrogen

Hydrogen is the most abundant element in the cosmos; it makes up 80% of the universe's mass.  Therefore, it is no surprise that one of the most significant spectral lines in radio astronomy is the 21-cm hydrogen line. In interstellar space, gas is extremely cold. Therefore, hydrogen atoms in the interstellar medium are at such low temperatures (~100 K) that they are in the ground electronic state. This means that the electron is as close to the nucleus as it can get, and it has the lowest allowed energy. Radio spectral lines arise from changes between one energy level to another.

A neutral hydrogen atom consists of one proton and one electron, in orbit around the nucleus. Both the proton and the electron spin about their individual axes, but they do not spin in just one direction. They can spin in the same direction (parallel) or in opposite directions (anti-parallel). The energy carried by the atom in the parallel spin is greater than the energy it has in the anti-parallel spin. Therefore, when the spin state flips from parallel to anti parallel, energy (in the form of a low energy photon) is emitted at a radio wavelength of 21-cm. This 21-cm radio spectral line corresponds to a frequency of 1.420 GHz.

Optical observations of the Galaxy are limited due to the interstellar dust, which does not allow the penetration of light waves. However, this problem does not arise when making radio measurements of the HI region. Radiation from this region can be detected anywhere in our Galaxy.

Measurements of the HI region of the Galaxy can be used in various calculations. For example, observations of the 21-cm line can be used to create the rotation curve for our Milky Way Galaxy. If hydrogen atoms are distributed uniformly throughout the Galaxy, a 21-cm line will be detected from all points along the line of sight of our telescope. The only difference will be that all of these spectra will have different Doppler shifts. Once the rotation curve for the Galaxy is known, it can be used to find the distances to various objects. By knowing the Doppler shift of a body, its angular velocity can be calculated. Combining this angular velocity and the plot of the rotation curve, the distance to a certain object can be inferred. Using measurements of the HI region, the mass of the Galaxy can also be determined.
'


Model of the Galactic rotation

We model the Galactic motion as circular motion with monotonically decreasing angular rate with distance from the center. 
 The rotational velocity doesn't depend strongly on the distance from the center at least for R > 3 kpc.  [Measurements close to the Galactic center are difficult and the integration needs to be sufficient to reduce the noise so that the maximum velocity edge of the emission can be determined.]  If the motion was entirely Keplerian (i.e. classical circular orbits around a central mass) the rotational velocity would decrease with a R^-½ dependence.  The lack of decline with distance suggests that not all the mass is in the center of the Galaxy.
'

Everything important with a spiral galaxy. like its mass and rotation, depends on hydrogen gas atoms. The data suggests not all of the mass is at the center which should not be a surprise with billions of stars, plus separate globular clusters and satellite galaxies.


2a) No alternative
It is essentially impossible to use stars to measure a rotation curve.

A visual analysis of stellar motion requires a multitude of images taken over a very long time to determine the relative motions among subsets of the billions of stars. Over time changes in positions could derive an initial trajectory and velocity for each individual star in that subset. Further observations will adjust that analysis. Many observations of our Sun's relative motion concluded its orbit takes 225 million years. At least that much time is required to confirm the understanding of the orbit is correct by confirming its position after one orbit.

I do not know what size of a sample of individual stars could be available within our nearest spiral galaxies < 15 Mly.
These are similar spiral galaxies: Milky Way = SBbc, M31= Sb, M33=Sc,  IC342 = Sc, M81= Sab, M98 = SAB(s)ab.
The time required for an accurate analysis makes this approach impractical.

My conclusion:

We base our understanding of a spiral galaxy rotation on the relative motions of individual hydrogen gas atoms.
I expect relative motions of stars would be a better indicator. Maybe they behave similarly but which data set justifies that assumption for using atoms?


I suspect the use of individual atoms is not an indicator reliable enough to justify dark matter in a galaxy of billions of stars.

I added this comment:

M64 has an interesting anomaly with its gas clouds:
'
While the interstellar gas in the galaxy's outer regions rotates in the opposite direction from the gas and stars in the inner disk, all the stars in M64 are rotating in the same direction.
'

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