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  Dark Matter Margin of Error

When scientists make calculations or predictions a value often includes a margin of error to identify the extent of uncertainty.
With no stated margin of error one could conclude the stated value is absolutely certain.

Counting the number of stars in a galaxy is very difficult. In the case of the Milky Way we know a substantial extent of the galaxy cannot be observed because of intervening dust clouds and so many stars making it difficult to resolve all the individual stars. With the Milky Way, instead of a margin of error the number is specified as a range, from 200 to 400 billion stars. This is appropriate because 300 +/- 100 implies 300 is our best guess but that +/- usage is misleading with this large uncertainty.
There are estimated counts of globular clusters and dwarf satellite galaxies because these can be difficult to resolve a distance from the galactic disk.
For an unknown reason, Andromeda Galaxy, M31 is specified differently.
Its number of stars is consistently specified as 1 trillion stars. This value must have a margin of error (never stated) because we view the galaxy at a slight angle so we cannot see the entire galaxy. I certainly doubt we can resolve all 1 trillion stars from our distance of about 2.5 million light years, even counting those hidden by the disk. This number is certainly an estimate.

This uncertainty is important for dark matter.

Many astronomers have observed the rotation curve of M31. Those stars closer to the core are usually found to be moving as expected, but those stars further away are usually not as expected.

Vera Rubin, Kent Ford, and Ken Freeman's work in the 1960s and 1970s, provided further strong evidence, also using galaxy rotation curves. Rubin and Ford worked with a new spectrograph to measure the velocity curve of edge-on spiral galaxies with greater accuracy. This result was confirmed in 1978. An influential paper presented Rubin and Ford's results in 1980.They showed most galaxies must contain about six times as much dark as visible mass; thus, by around 1980 the apparent need for dark matter was widely recognized as a major unsolved problem in astronomy.

From a 2001 Caltech paper by V. Rubin and Y. Sofue:

Rotation curves are tools for several purposes: for studying the kinematics of galaxies; for inferring the evolutionary histories and the role that interactions have played; for relating departures from the expected rotation curve Keplerian form to the amount and distribution of dark matter; for observing evolution by comparing rotation curves in distant galaxies with galaxies nearby. Rotation curves derived from emission lines such as Halpha, HI and CO lines are particularly useful to derive the mass distribution in disk galaxies, because they manifest the motion of interstellar gases of population I, which have much smaller velocity dispersion, of the order of 5 - 10 km/s, compared to the rotation velocities. This allows us to neglect the pressure term in the equation of motion for calculating the mass distribution in a sufficiently accurate approximation.
There are three immediate alarms.
1) 'Keplerian form' - but our Sun is known to move not in an ellipse. There is no justification to expect stars in other spiral galaxies will move in an ellipse.
The largest deviations in velocities are for stars furthest from the core so this anomaly could be expected when the star is not actually moving in a Keplerian ellipse.

2) 'sufficiently accurate approximation' - so the astronomers know critical details are estimates. Since the total number of stars is not certain nor their locations; their mass distribution needs an approximation.

3) 'useful to derive the mass distribution ' - so the curve is assumed to be driven by the mass distribution but when the curve is not as expected then the conclusion is the mass distribution is now wrong due to the presence of dark matter. This is not logical.

Cosmology should use the scientific method for dark matter: a) hypothesis, b) make a prediction based on the hypothesis, c) test the prediction. d) If the test fails then the hypothesis must be fixed so it makes valid predictions.

This simple approach of basic science is not followed with dark matter.

With the claim of dark matter, the model's mass distribution must be adjusted to include the claimed distribution of dark matter - like X kg at each specific XYZ location (using which coordinate system?) among those 1 trillion stars.

If the updated model does not create the observed rotation curve then the mass distribution of dark matter must be fixed until the rotation curve matches exactly that observed. That is the hypothesis being tested, to be confirmed or denied.

In my opinion this exercise is not the correct approach.
Eventually rotation curves for other spiral galaxies must be explained, like M33, M51, M63, M74, M81, M101, and many more.

Each galaxy will have estimates for their star count and mass distribution. Each prediction amounts to a probability and a probability can never be absolutely certain. I expect a model for a trillion stars must use probabilities or generalities.

this iterative process of just fixing numbers does not improve our understanding, represented by the galaxy model.
The simple alternative to this iterative process is: the rotation curve needs a margin of error.

After the model is accurate for 95% of the stars (a margin of error of 5%) that should be close enough given the known uncertainties with spiral galaxies, especially those galaxies with estimates of billions of stars. Our own Sun has a disturbed orbit. Since its orbit is estimated at 226 million years no one bothers to make a prediction.

The model will improve over time. Perhaps imaging M31 in multiple wavelengths will help understand more details.

Claiming dark matter as the solution implies the rotation curve is solved.
That is not how the scientific method should be applied for rotation curves.
After the model itself, with no dark matter, eventually achieves 99% accuracy (or whatever margin of error is defined by cosmologists) then dark matter will not be claimed to be in M31.

Dark matter is the correction when failing to achieve perfection.

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