Light, mass and circular motion

Galaxy Rotation Curves and Disk–Halo Degeneracy

Does a flat rotation curve uniquely reveal a galaxy’s dark halo? Circular motion constrains radial gravity. A heavier stellar disk leaves less force for the halo, while distance and inclination shift the inferred mass budget. A flat curve alone does not uniquely determine the halo or identify a dark-matter particle.

Change a stellar disk’s mass-to-light ratio and a spherical halo. Compare their forces, circular speed and residuals against the same saved curve.

Circular speed at probe $115\,\mathrm{km}\,\mathrm{s}^{-1}$

Spherical halo mass inside probe $7.5\times10^{9}\,\mathrm{M}_\odot$

Stellar share of inward force $0.342$

Working residual score per sample $0.956$

The same curve, different mass budgets

Move mass between disk and halo

Disk mass-to-light ratio

0.55

\Upsilon_d/(\mathrm{M}_\odot\,\mathrm{L}_\odot^{-1})

Halo density scale

1.4\times10^7

\rho_h/(\mathrm{M}_\odot\,\mathrm{kpc}^{-3})

Halo scale radius

9\,\mathrm{kpc}

r_h/\mathrm{kpc}

Halo centre

Drag the gold probe to select a saved orbit radius. Changing the mass model prepares another set of test orbits.

Distance, inclination and source files

Distance relative to the saved source

1

Global inclination

63^\circ

These settings move every sample together. The native radial shape is held; this is not a new photometric or warp reconstruction.

Download the independent fixed-source recipe

Run the Python recipe on your own computer, then select NGC2403_rotmod.dat and sparc-table1-metadata.body together. The two exact identities are checked locally; nothing is uploaded.

Read the two named local files

Reset restores the own synthetic record. No optimizer is included; the real object’s simple conditional fits remain poor.

Start from the spherical shell theorem

Model assumptions

The default observations are our own synthetic record, not a measured galaxy. The synthetic record uses analytic thin exponential disk bases. Local SPARC imports instead use native finite-thickness gas and stellar force inputs. A static axisymmetric model adds these signed terms to a spherical NFW or pseudo-isothermal halo. Native stellar columns use unit mass-to-light ratio; gas already includes helium. Circular test tracers show the resulting speed field with explicit time compression, not a galaxy evolution or a recovered matter image. Halo contours and disk mass samples are schematic. The probe selects a saved radial sample; plot segments connect samples without reconstructing the native finite-thickness field. Distance changes follow a frozen-shape homology and inclination is a single global nuisance, not a new warp or photometric deprojection. The full working Gaussian likelihood uses a fixed reference-projected coordinate; its score is not a calibrated probability. Independent finite corners, joint cases and holdouts do not certify every continuous setting. There is no optimizer in this Lab. A separate local-only import accepts the two exact named SPARC files; no original observations are hosted or uploaded. Simple conditional fits of that real object remain poor. A visually plausible curve does not identify a particle or establish a theory winner.

The same saved curve, force budget and residuals

Own synthetic observations. This is a conditional mass model, not a measured galaxy image.

Saved samplesTotal modelStellar diskGasSpherical halo

Circular speed

(\mathrm{km}\,\mathrm{s}^{-1})

Horizontal distance $R/\mathrm{kpc}$ . Component lines show signed equivalent speeds; forces are added before the square root.

Residual in the fixed observation coordinate

$(\mu_j-y_j)/\sigma_{y,j}$ ; changes in distance or inclination affect all radii together.

Probe selects saved sample 30 of 48. $3.93\,\mathrm{kpc}$

Physical assumptions, likelihood and data export

The disk-plane view shows conditional mass samples and circular test tracers. Velocity arrows share the displayed scale within each accepted record; that scale can change between mass models. Halo density is a model choice; the shading is schematic, not luminosity. The default synthetic disk basis is analytic and thin; imported native SPARC forces instead retain their finite-thickness basis.

The probe snaps to a saved radius. Plot segments connect samples; no force is inferred outside the saved radius window. Distance uses fixed-shape homology and inclination is a single global correction.

The full working Gaussian log likelihood includes its normalizer in a fixed reference-projected coordinate. Original errors do not supply a full covariance or a calibrated Gaussian noise law. This Lab does not optimize parameters or select a particle/theory.

\log L=

-105

Own presets, compound corners and withheld cases have independent Python comparisons. They do not certify every continuous request. Unsupported parameters or local files retain the last complete record.

Start with the essentials

Focus question: Does a flat rotation curve uniquely reveal a galaxy’s dark halo?
One-sentence intuition: Component forces add; their velocity-curve heights do not.

Core mathematical model

Add radial force terms

v_c^2=V_g|V_g|+\Upsilon_d V_d|V_d|+\frac{GM_h(<R)}R

A negative native gas term points outward. Add signed contributions before taking the total square root.

Only the halo is spherical here

M_h(<R)=4\pi\int_0^R\rho_h(r)r^2\,dr

The integral counts spherical halo mass, not the stellar disk’s mass.

One global distance change

R^\prime=dR,\qquad A_b^\prime=dA_b,\quad A_b=Rg_b

Here $d$ is the distance ratio and $g_b$ the signed inward baryonic acceleration. Fixed-shape homology gives inferred mass proportional to distance squared.

A fixed error coordinate

y_j=V_{0,j}\sin i_0,\quad \mu_j=v_{c,j}\sin i,\quad \log L=-\frac12\sum_j\left[\frac{(y_j-\mu_j)^2}{\sigma_{y,j}^2}+\log(2\pi\sigma_{y,j}^2)\right]

The saved speed error is multiplied by the same reference sine. Inclination is measured from the disk normal to the line of sight. The full working likelihood retains its normalization; this App has no optimizer or calibrated posterior.

Common difficulties

A disk is not a sphere

Typical misconceptionMultiplying the total speed squared by radius gives the true enclosed mass of any galaxy.

Better mental modelThat enclosed-mass relation applies to a spherical source. The disk also feels material outside the chosen radius; the displayed halo enclosed mass is spherical halo only.

A plausible curve is not a theory verdict

Typical misconceptionThe smaller residual score proves a particular dark-matter particle.

Better mental modelThe score is conditional on the mass profile, geometry and a working error model. Original SPARC systematic/correlated errors and the poor simple fits remain relevant.

Run the experiment

01
Select a circular orbit
Drag the gold probe and pause the tracers. Compare its speed with the marked point in the curve.
What to observe: The probe uses a saved radial sample; it never asks an interpolated disk to provide an unverified force.
02
Make the disk heavier
Increase only the stellar mass-to-light ratio. Watch the disk force and total speed change.
What to observe: Stellar speed scales with the square root of the mass-to-light ratio; component speeds are not summed linearly.
03
Change the centre of the halo
Compare the extended-cusp and flat-centre presets, then move the halo scale.
What to observe: The same saved observations can be compared with distinct conditional mass profiles. A different visual centre is a model choice, not a recovered image.
04
Move a whole uncertainty together
Open distance and inclination, change one at a time and inspect the residuals.
What to observe: Each change affects every radius coherently; it cannot be treated as independent noise at each point.

Galaxy Rotation Curves and Disk–Halo Degeneracy

The same saved curve, force budget and residuals

The force behind a rotation curve

Core mathematical model

Add radial force terms

Only the halo is spherical here

One global distance change

A fixed error coordinate

Common difficulties

A disk is not a sphere

A plausible curve is not a theory verdict

Run the experiment

Select a circular orbit

Make the disk heavier

Change the centre of the halo

Move a whole uncertainty together

Related experiments and articles

From circular motion to conditional mass models