Cosmology · The evidence motivating dark matter

Galactic motion and inferred mass profiles

Infer a chosen spherical mass profile from circular motion, then test whether one change in an ordinary-matter normalization reproduces its radial shape.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Explain how circular acceleration constrains a gravitational potential.
Compare fitting one radius with fitting an entire supplied profile.
Keep a conditional mass inference separate from particle identification.

Motion supplies a gravitational test

A star or gas parcel responds to gravity through its motion. For pressure-free circular motion in a spherical Newtonian potential, the inward acceleration is v_c²/r = GM(<r)/r². The implied enclosed mass is M(<r) = rv_c²/G.

This statement names its assumptions. A disk is not spherical, and a gas parcel can also feel pressure forces. An observed line-of-sight speed needs a viewing geometry before it becomes a circular speed.

Historical rotation measurements in selected spiral galaxies helped motivate extended mass distributions. Their interpretation also requires attention to noncircular motions, inclination and distance; one historical sample does not establish the same rotation-curve shape in every galaxy.

Use a spherical mathematical example to isolate a narrower question: if the radial mass shape is supplied, can a single adjustment of its normalization reproduce a different chosen gravitational profile? No observed galaxy or measurement uncertainty is supplied below.

Specify mass and length before comparing speeds

Choose positive mass M_* and length b, and write x = r/b. Supply two Plummer components: one labelled ordinary matter with total mass M_* and scale b, and an additional component with total mass 2M_* and scale 3b.

Their enclosed masses in units M_* are B(x) = x³/(1+x²)³ᐟ² and E(x) = 2x³/(9+x²)³ᐟ². The total is B+E. The labels assign the supplied mass accounts; the calculation does not identify the additional component’s physical nature.

For a Plummer component of mass M_s and scale b_s, Φ_s = −GM_s/√(r²+b_s²) is gravitational potential energy per unit test mass. Its density is ρ_s = 3M_sb_s²/[4π(r²+b_s²)⁵ᐟ²], which is positive.

Assume weak gravity, GM_*/(bc²) ≪ 1, and circular passive tracers. Set v_* = √(GM_*/b). The dimensionless total speed obeys (v_c/v_*)² = (B+E)/x. The reference speed v_* is a chosen unit, not a measured galactic speed.

The required normalization changes with radius

A restricted ordinary-matter-only comparison retains the shape B(x) and changes its normalization by a positive dimensionless factor Υ. Its enclosed mass is ΥB M_*.

Matching the total gravitational mass at a radius requires Υ_required(x) = 1+E(x)/B(x). At x = 1 this is about 1.178885. At x = 4 it is about 2.121485.

One constant Υ therefore cannot match these two chosen profiles everywhere. Fitting at x = 1 fixes the comparison speed there, but leaves its speed about 0.177143 v_* below the target at x = 4.

The factor Υ is a normalization of this prescribed mass shape. It is not an observed stellar mass-to-light ratio, and its failure does not test every possible ordinary-matter distribution.

Chosen positive spherical profiles; the comparison uses the same Υ = 1.178885 fitted at x = 1
x = r/b	Total M(<r)/M_*	Target v_c/v_*	Base ordinary-matter speed/v_*	Fitted comparison speed/v_*	Required Υ
0.5	0.0983292	0.443462	0.422949	0.459223	1.099353
1	0.416799	0.645600	0.594604	0.645600	1.178885
2	1.056896	0.726944	0.598140	0.649439	1.477057
4	1.937075	0.695894	0.477775	0.518751	2.121485
6	2.390818	0.631245	0.399945	0.434246	2.491124

These are supplied positive spherical Newtonian Plummer profiles, not an observed galaxy, disk or fitted rotation sample. Choose M_* > 0 and b > 0, x = r/b, v_* = √(GM_*/b), and weak gravity GM_*/(bc²) ≪ 1. The base ordinary-matter component has total M_* and scale b; the additional component has total 2M_* and scale 3b. Their normalized enclosed masses are B = x³/(1+x²)^(3/2) and E = 2x³/(9+x²)^(3/2). Both panels show only 0.5 ≤ x ≤ 6 with a shared linear radius axis. Panel A uses a zero-based linear speed axis from 0 to 0.8 in units v_*. Solid blue circles show the target √[(B+E)/x]. Dashed orange squares show the base √(B/x). Dotted violet diamonds show √[Υ_fit B/x], where the same constant Υ_fit = 1.1788854382 is fixed at x = 1 and retained throughout. Marker positions are exact synthetic model evaluations at x = 0.5, 1, 2, 4, 6, not measurements. The target and fitted markers at x = 1 genuinely coincide at about 0.645600, without artificial offsets. At x = 4 their speeds are about 0.695894 and 0.518751; the difference is about 0.177143 v_*. Panel B has its own zero-based dimensionless normalization axis from 0 to 3. Solid green circles show Υ_required = 1 + E/B; the dotted violet diamond line shows the one retained constant fit. Required values at x = 0.5, 1, 2, 4, 6 are about 1.099353, 1.178885, 1.477057, 2.121485 and 2.491124. A local normalization can match one radius, but a single value cannot reproduce these different supplied radial shapes. Υ is a chosen mass-profile normalization, not an observed stellar mass-to-light ratio. The construction assumes pressure-free circular passive tracers and known spherical geometry. It excludes disk forces, pressure support, projection, distance and inclination uncertainties and any measurement model. The profiles have finite total mass 3M_*; beyond the finite display their circular speed eventually declines as r^(−1/2), and no exactly flat universal curve is asserted. This restricted mismatch does not reject every ordinary-matter spatial distribution or identify a microscopic particle. Original Ark teaching model.

An unexplained acceleration does not name a particle

At x = 2, multiplying the supplied squared speed by x gives M(<r)/M_* ≈ 1.056896. This is a mass inference under the spherical circular-motion law.

The next step is a comparison with separately estimated ordinary matter, whose tracers and conversion assumptions need their own account. A faint component and a nonbaryonic component are different possibilities; darkness alone does not distinguish them.

The example rejects a particular fixed-shape, single-normalization comparison. A different spatial distribution, a different force law or additional pressure support would require a different forward prediction and further tests.

Nor does the added mass term identify a microscopic particle. Its mass, interactions and production history do not appear in the circular-speed equation once the smooth total density is supplied.

Use the whole profile and the observational response

The displayed curves are not exactly flat. Both components have finite total mass, so at sufficiently large radius their combined circular speed approaches the declining finite-mass behavior v_c ∝ r⁻¹ᐟ².

A real rotation analysis must connect a spatial mass model to projected velocities and account for distance, inclination, tracer motions and measurement errors. These can affect both normalization and radial shape.

Additional tracer populations can respond to the same potential through random motions rather than circular orbits. Their mass inference then needs an equilibrium and velocity-anisotropy model, rather than direct use of rv_c²/G.

The transferable lesson is to test a common physical prediction over the measured range. Agreement at one radius is useful calibration information; agreement over several radii or mechanisms imposes additional conditions.

Worked reasoning

Fit at b and test at 4b

Does the normalization fitted at x = 1 also reproduce the supplied target at x = 4?

At x = 1, B ≈ 0.353553 and E ≈ 0.0632456, so Υ = (B+E)/B ≈ 1.178885.
Retain that same Υ at x = 4, giving a comparison speed about 0.518751 v_*.
The target speed is about 0.695894 v_*, and the required local normalization is 2.121485.

No. The one-radius normalization misses the second radius because the supplied radial shapes differ.

Try the reasoning

At x = 2, what enclosed mass follows from x(v_c/v_*)² ≈ 2 × 0.52844786?
Show the answer and reasoning
About 1.056896 M_*.
The circular-motion relation converts the chosen speed into mass under the declared spherical assumptions.
Does the failed single-normalization comparison identify a dark-matter particle?
Show the answer and reasoning
No.
It tests a supplied radial mass comparison, while microscopic properties remain unspecified.
Has every possible ordinary-matter spatial distribution been excluded?
Show the answer and reasoning
No.
Only one prescribed shape with one adjustable normalization was compared.

What to carry forward

Circular motion constrains mass through a stated gravitational and geometric model.
A fit at one radius need not reproduce a radial profile.
Gravitational mass inference, ordinary-matter accounting and particle identity are distinct steps.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.