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Fritz Zwicky Sticker

Too Fast To Be Lit | Physics

\[\sigma^2 \sim \frac{GM}{R}\]
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What holds a galaxy cluster together? In 1933 Fritz Zwicky found galaxies in the Coma cluster moving far too fast for the visible matter to hold. By the virial theorem, σ² ~ GM/R, so fast motions need much more mass than shines. He called the missing mass dunkle Materie and refined the case in 1937. A 3" pixel-art physicist sticker; equation, symbols and sources in the physics panel below.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

In a gravitationally bound cluster, the galaxies' speeds are set by the total mass. Zwicky used their velocity spread to estimate the Coma cluster's mass and found it far larger than the light suggested.

In 1933 Fritz Zwicky timed the Coma cluster and found the galaxies moving too fast for the glowing mass alone.

The virial estimate is σ² ~ GM/R. Invert it: M_vir ≈ σ² R / G. If the observed dispersion exceeds what luminous mass can bind, you need dark mass.

On the sticker Coma appears as dots with velocity arrows too long for the lit matter, stamped MISSING MASS.

Zwicky called it dunkle Materie. Decades later the name stuck, and the mass budget of the universe still does not add up without it.

Too fast to be lit.

Equations

\[\sigma^2 \sim \frac{GM}{R}\]
\[M_{\mathrm{vir}} \approx \frac{\sigma^2 R}{G}\]
SymbolMeaningUnit
\(\sigma\) line-of-sight velocity dispersion of galaxies \(\mathrm{km/s}\)
\(G\) gravitational constant \(\mathrm{pc M_\odot^{-1} (km/s)^2}\)
\(M\) gravitating mass within radius R \(\mathrm{M_\odot}\)
\(R\) characteristic cluster radius \(\mathrm{pc}\)
\(M_{vir}\) virial mass implied by \sigma and R \(\mathrm{M_\odot}\)

Sources

  1. Zwicky (1933) Die Rotverschiebung von extragalaktischen Nebeln, Helvetica Physica Acta 6, 110
  2. Binney & Tremaine, Galactic Dynamics, Ch. on clusters and the virial theorem
  3. Wikipedia, Dark matter / Fritz Zwicky

Source Zwicky, F. (1933). Die Rotverschiebung von extragalaktischen Nebeln. Helvetica Physica Acta, 6, 110–127.

Field notes

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