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Sandbox Physics
Galaxy rotation curves and mass models

Galaxy rotation curves · L1

Why a disk and a sphere pull differently

Sandbox Physics ·

Circular speed directly gives the required radial acceleration. Spherical symmetry supplies the extra step that turns it into enclosed mass. A stellar disk needs its own gravitational calculation.

Choose the direction before writing the force

Let RR be cylindrical radius in the disk plane, increasing outwards. Write the gravitational potential as Φ\Phi. The outward acceleration is −∂Φ/∂R-\partial\Phi/\partial R. It is convenient to define the signed inward acceleration gin=∂Φ/∂Rg_{\rm in}=\partial\Phi/\partial R; a positive value means a pull towards the centre.

A circular tracer has constant speed and a continually changing velocity direction. Newton’s second law equates its inward force to its mass times the circular acceleration. The tracer’s mass cancels. The remaining relation works for a disk, a sphere or their combined field when a circular orbit in an axisymmetric potential is a suitable approximation.

mvc2R=mgin,vc2=Rginm\frac{v_c^2}{R}=m g_{\rm in},\qquad v_c^2=R g_{\rm in}
mm is the tracer mass. Potentials and accelerations below are per unit test mass. Speed squared has units km2 s−2\mathrm{km^2\,s^{-2}}.

Reading: Independent saved-observation rotation-curve calculation

Spherical shells reduce the problem to interior mass

Divide a spherically symmetric body into concentric thin shells. A complete shell outside the test point exerts zero net force there. Every interior shell acts, at that point, as though its mass were concentrated at the centre. Adding those interior shells gives a force determined by enclosed mass.

For volume density ρ(r)\rho(r), a shell at radius rr with thickness drdr has volume approximately 4πr2dr4\pi r^2dr. Multiplying by density gives the shell mass. The integral adds these shells from the centre to RR. This is why the upper integration limit and the spherical geometry enter together.

M(<R)=4π∫0Rρ(r)r2 dr,gin(R)=GM(<R)R2,M(<R)=Rvc2GM(<R)=4\pi\int_0^R\rho(r)r^2\,dr,\qquad g_{\rm in}(R)=\frac{G M(<R)}{R^2},\qquad M(<R)=\frac{R v_c^2}{G}
GG is the gravitational constant. The final mass estimator inherits the spherical-symmetry assumption.
ρ(r)=ρ0⟹M(<R)=4π3ρ0R3,vc=R4πGρ03\rho(r)=\rho_0 \Longrightarrow M(<R)=\frac{4\pi}{3}\rho_0R^3,\qquad v_c=R\sqrt{\frac{4\pi G\rho_0}{3}}
Inside a uniform-density sphere, circular speed grows linearly with radius. Moving outward includes more mass in the region that supplies the force.

Reading: Spherical shell theorem foundation Lab · Independent saved-observation rotation-curve calculation

Worked example: weigh a spherical flat-speed interval

Suppose the circular speed is 200 km s−1200\,\mathrm{km\,s^{-1}} from 10 kpc10\,\mathrm{kpc} to 20 kpc20\,\mathrm{kpc}, and explicitly assume a spherical gravitating source. Use G=4.3009173×10−6 kpc (km s−1)2 M⊙−1G=4.3009173\times10^{-6}\,\mathrm{kpc}\,(\mathrm{km\,s^{-1}})^2\,\mathrm{M}_\odot^{-1}.

Squaring the speed gives 4.00×104 (km s−1)24.00\times10^4\,(\mathrm{km\,s^{-1}})^2. At the inner radius, Rvc2=4.00×105 kpc (km s−1)2Rv_c^2=4.00\times10^5\,\mathrm{kpc}\,(\mathrm{km\,s^{-1}})^2. Divide by GG to obtain 9.30×1010 M⊙9.30\times10^{10}\,\mathrm{M}_\odot. Doubling the radius at unchanged speed doubles the enclosed mass to 1.86×1011 M⊙1.86\times10^{11}\,\mathrm{M}_\odot.

This inference covers the specified finite interval. It does not make a measurement of an infinite halo or determine the galaxy’s total mass. Extending the result beyond the observed radii requires further observations or an additional model assumption.

For an ideal continuously flat spherical curve, differentiate M(<R)=Rvc2/GM(<R)=Rv_c^2/G and divide by the shell-area factor. This gives ρ(R)=vc2/(4πGR2)\rho(R)=v_c^2/(4\pi G R^2). An inverse-square density profile adds the same mass in each equal-width radial shell, explaining the linear growth of enclosed mass.

Reading: Independent saved-observation rotation-curve calculation

A disk also feels material beyond the test radius

Flatten the mass into a disk. An outer ring has a nearer side and a farther side, and their attractions no longer cancel as a spherical shell’s do. An exterior ring can supply an outward radial component. The disk calculation must add rings on both sides of the chosen radius.

A surface density Σ(R)\Sigma(R) describes mass per disk area; it differs from volume density and can be measured in M⊙ pc−2\mathrm{M}_\odot\,\mathrm{pc}^{-2}. An exponential disk has Σ(R)=Σ0e−R/Rd\Sigma(R)=\Sigma_0e^{-R/R_d}, where RdR_d is the radial scale over which surface density falls by a natural exponential factor.

Integrate the Newtonian potential of those rings and then differentiate with respect to the test radius to obtain the disk-plane force. The next chapter supplies a convenient special-function expression. We can already test the geometrical shortcut by applying it to exactly the same disk.

The actual thin exponential disk and the wrong spherical mass-in-radius speed-squared calculation
Own analytic exponential mass distribution. One curve keeps disk geometry; the other inserts interior disk mass into the spherical formula. Their difference isolates the geometry assumption. Native SPARC finite-thickness components are a separate input. Open the full figure

Reading: Independent saved-observation rotation-curve calculation · Lelli, McGaugh & Schombert (2016), SPARC, AJ152,157

A numerical counterexample with one fixed disk

Let the disk have total mass MdM_d and evaluate it at R=2.2RdR=2.2R_d. The actual razor-thin disk gives vd2/(GMd/Rd)=0.386912v_d^2/(GM_d/R_d)=0.386912. Inserting only the interior disk mass into the spherical formula gives 0.2933770.293377. Its speed-squared estimate is about 24.2 percent too low.

The two calculations use the same exponential mass distribution. Their difference comes from gravitational geometry. The radius label alone cannot decide which enclosed-mass relation is valid.

This test concerns an ideal thin disk. Imported SPARC stellar components retain their native finite-thickness disk calculation. Accurately evaluating the thin-disk benchmark checks the mathematics; it does not reconstruct a native finite-thickness force field.

Reading: Independent saved-observation rotation-curve calculation

Exercises and explained answers

  1. Add mass to a complete spherical shell outside a test point. How does the circular speed at the point change? What if the added mass instead forms a thin exterior ring?

    Read the explanation

    The complete exterior shell still gives zero net force at the point, so the circular speed is unchanged. An exterior ring lacks that shell cancellation and can add an outward radial force. Knowing that the added material is outside the test radius is therefore insufficient to determine its effect.

  2. Inside a uniform-density spherical core, what happens to enclosed mass and circular speed when radius doubles?

    Read the explanation

    Mass grows as R3R^3, so it increases eightfold. Since vc2=GM/Rv_c^2=GM/R, speed squared grows as R2R^2 and speed doubles. The density law and geometry determine the speed scaling.

  3. Can a disk’s measured speed be inserted into the spherical mass estimator and called the disk’s true interior mass?

    Read the explanation

    No. The step that identifies radial force with interior mass used the shell theorem. Exterior disk material also contributes to the force. Retain the disk geometry, or explicitly call the inferred quantity a spherical-equivalent mass.

Try the prediction

Compare the stellar disk and conditional halo

  1. Move the probe and compare circular speed with the spherical halo mass inside that radius.
  2. Make the stellar disk heavier. Watch total speed change and check that the halo enclosed-mass readout still follows only the halo and selected radius.
  3. Explain why that halo mass is not the combined interior mass of disk and halo.

Sources and calculation scope

  1. Lelli, McGaugh & Schombert (2016), SPARC, AJ152,157

    Official fixed Table1 and Rotmod_LTG.zip; named NGC2403 member only. The master paper sections3.1–3.3/5.1–5.3 and original Table2 notes establish signed gas, unit stellar mass-to-light, finite thickness and error meaning. Fresh source acquisition is separate from cached replay. Original rows and measured-derived plots stay local.

  2. Spherical shell theorem foundation Lab

    Foundation reuse: spherical-shell cancellation and exterior central force. The new rotation-curve route adds saved component data, geometry and conditional fitting.

  3. Independent saved-observation rotation-curve calculation

    Own MIT Python produces synthetic observations, actual six-start fits, profiles, commonD/i scenarios, positive-density/ring-potential/Jacobian references and SVGs. Optional exact named-source acquisition and73-row conversion/fits were actually run. Default is offline; ZIP contains no App Engine, original SPARC rows, paper or measured-derived graph.

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