Index相対性理論と宇宙Hoodie

Rotation Curve Hoodie

THE EDGES SPIN TOO FAST | Pixelated Physics

\[v^2(r) = \frac{G\,M(<r)}{r}\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Stars orbit the centre of a galaxy, and their speed tells you how much mass lies inside their orbit. A spiral galaxy with its rotation curve: measured speeds stay flat while stars alone predict a fall. The gap points to dark matter. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

How fast stars orbit at each radius reveals how much mass lies inside that radius. The visible disk alone predicts speeds that fall off at large radius, but measured curves stay flat, which requires an extra dark-matter halo.

Stars orbit the centre of a galaxy, and their speed tells you how much mass lies inside their orbit.

If the stars and gas were all the mass, speeds would peak and then fall with distance. The dashed amber curve is that prediction for an exponential stellar disk, using Ken Freeman’s 1970 solution.

What is measured stays flat, the cream curve. Vera Rubin and Kent Ford made this clear in the 1970s. Flat speeds mean the enclosed mass keeps growing in proportion to radius, far beyond the light. Here the extra comes from an NFW dark-matter halo.

The galaxy is drawn tilted, with the curve laid along its long axis: one side moves toward you, the other away.

The edges spin too fast.

Equations

\[v^2(r) = \frac{G\,M(<r)}{r}\]
\[v^2 = v_{\mathrm{disk}}^2 + v_{\mathrm{halo}}^2\]
\[v_{\mathrm{disk}}^2 = 4\pi G\Sigma_0R_d\,y^2\left[I_0K_0 - I_1K_1\right](y),\quad y = \frac{r}{2R_d}\]
SymbolMeaningUnit
\(v\) circular orbital speed \(\mathrm{m/s}\)
\(r\) radius from the galactic centre \(\mathrm{m}\)
\(G\) gravitational constant \(\mathrm{m^3·kg^{-1}·s^{-2}}\)
\(M(<r)\) mass enclosed within r \(\mathrm{kg}\)
\(\Sigma_0\) central surface density of the stellar disk \(\mathrm{kg/m^2}\)
\(R_d\) disk scale length \(\mathrm{m}\)
\(I_n, K_n\) modified Bessel functions \(\mathrm{1}\)

Sources

  1. Rubin & Ford (1970) Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions, ApJ 159, 379
  2. Freeman (1970) On the Disks of Spiral and S0 Galaxies, ApJ 160, 811
  3. Navarro, Frenk & White (1996) The Structure of Cold Dark Matter Halos, ApJ 462, 563

Source Freeman, K. C. (1970). On the Disks of Spiral and S0 Galaxies. The Astrophysical Journal 160, 811.doi.org/10.1086/150474

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