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Lagrange Points Hoodie

FIVE PLACES TO PARK | Pixelated Physics

\[U = -\frac{1-\mu}{r_1} - \frac{\mu}{r_2} - \frac{x^2+y^2}{2}\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Two bodies orbit each other. In the frame that turns with them, there are five places where a third, small body can sit still. Contours of the rotating-frame potential of two masses with μ = 0.03, through all five Lagrange points. L4 and L5 are stable here. 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

In a frame rotating with two orbiting bodies, gravity and the centrifugal effect combine into one effective potential. It has five flat points where a third body can stay in place, and the triangular ones are stable only if the mass ratio is below a critical value.

Two bodies orbit each other. In the frame that turns with them, there are five places where a third, small body can sit still.

They are where gravity and the centrifugal effect balance, the zero-gradient points of the effective potential U on the back. Euler found the three on the line in 1767; Lagrange found the two triangle points in 1772.

The art uses a mass ratio of 0.03 and draws contours of U through each point’s level. L1, L2 and L3 are saddles, and anything parked there drifts away unless it is steered.

L4 and L5 sit on hilltops of U, but the Coriolis force keeps nearby orbits close, as long as the mass ratio is below about 0.0385. Jupiter’s Trojan asteroids live there.

Five places to park.

Equations

\[U = -\frac{1-\mu}{r_1} - \frac{\mu}{r_2} - \frac{x^2+y^2}{2}\]
\[\nabla U = 0\;\text{at } L_1\ldots L_5\]
\[\mu < \mu_c = \tfrac12\left(1-\sqrt{\tfrac{23}{27}}\right) \approx 0.0385\;\text{(}L_4, L_5\text{ stable)}\]
SymbolMeaningUnit
\(U\) effective potential in the co-rotating frame (scaled) \(\mathrm{1}\)
\(\mu\) mass ratio m_2/(m_1+m_2), here 0.03 \(\mathrm{1}\)
\(r_1, r_2\) distances to the two masses \(\mathrm{1}\)
\(x, y\) position in the rotating frame, separation = 1 \(\mathrm{1}\)
\(L_1 \ldots L_5\) Lagrange points \(\mathrm{1}\)
\(\mu_c\) Routh critical mass ratio \(\mathrm{1}\)

Sources

  1. Lagrange (1772) Essai sur le problème des trois corps, Prix de l’Académie Royale des Sciences de Paris 9
  2. Euler (1767) De motu rectilineo trium corporum se mutuo attrahentium, Novi Comm. Acad. Sci. Petrop. 11, 144
  3. Routh (1875) On Laplace’s three particles, Proc. London Math. Soc. 6, 86
  4. Murray & Dermott, Solar System Dynamics (1999), Ch. 3

Source Lagrange, J.-L. (1772). Essai sur le problème des trois corps. Prix de l’Académie Royale des Sciences de Paris, tome IX.

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Field notes

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