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Kepler Orbit Hoodie

FAST WHEN CLOSE, EXACTLY | Pixelated Physics

\[\frac{dA}{dt} = \frac{L}{2m} = \text{const}\]
$54.99 USD
$54.99 USD
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A planet on an ellipse speeds up near the Sun and slows down far away. Johannes Kepler found the exact rule in 1609. An e = 0.6 orbit cut into 12 wedges swept in equal times, from Kepler’s equation M = E − e sin E. Equal areas, so fast when close. 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

A planet sweeps out equal areas in equal times, which is angular momentum conservation, so it moves fastest when it is closest to the Sun. Kepler's equation, M = E − e sin E, converts the elapsed time into the planet's position on its ellipse.

A planet on an ellipse speeds up near the Sun and slows down far away. Johannes Kepler found the exact rule in 1609.

The line from the Sun to the planet sweeps out equal areas in equal times. Newton later showed that this is simply conservation of angular momentum, and holds for any central force.

The art splits one orbit with eccentricity 0.6 into 12 wedges, each swept in one twelfth of the period. The boundaries come from Kepler’s equation, M = E − e sin E, solved numerically. The wedges near the Sun are short and wide; the far ones are long and thin. Their areas match.

The Sun sits at one focus, not at the centre. The small dot at the other focus marks an empty point.

Fast when close, exactly.

Equations

\[\frac{dA}{dt} = \frac{L}{2m} = \text{const}\]
\[M = E - e\sin E,\qquad M = \frac{2\pi t}{P}\]
\[\tan\frac{\nu}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}\]
SymbolMeaningUnit
\(A\) area swept by the line from the Sun to the planet \(\mathrm{m^2}\)
\(L\) orbital angular momentum \(\mathrm{kg·m^2/s}\)
\(m\) planet mass \(\mathrm{kg}\)
\(M\) mean anomaly \(\mathrm{rad}\)
\(E\) eccentric anomaly \(\mathrm{rad}\)
\(e\) eccentricity (here 0.6) \(\mathrm{1}\)
\(P\) orbital period \(\mathrm{s}\)
\(\nu\) true anomaly, the angle seen from the Sun \(\mathrm{rad}\)

Sources

  1. Kepler (1609) Astronomia Nova, Prague
  2. Newton (1687) Philosophiæ Naturalis Principia Mathematica, Book I, Prop. 1
  3. Murray & Dermott, Solar System Dynamics (1999), Ch. 2

Source Kepler, J. (1609). Astronomia Nova. Heidelberg: G. Voegelin.

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