Flattening

The title of this article is ambiguous. For shaping in wood, furniture and interior work, see table .

Haumea with moons, artist impression.
The dwarf planet is very flattened due to the strong rotation .

Under the flattening refers to the deformation of a planet or other celestial body by its rotation . ${\displaystyle f}$

The geometric flattening of an ellipsoidal body is the relative difference between the radii  a at the equator and  b at the poles :

${\displaystyle f={\frac {a-b}{a}}=1-{\frac {b}{a}}<1\quad \mathrm {mit} \quad b<a}$

The flattening is usually specified in the form "1: x" with a value x> 1, which is also referred to as the number of flattening; a strong flattening corresponds to a comparatively small value of x.

The dynamic flattening is less than the geometric.

One speaks of flattening also with a more irregular body, if z. B. its meridional section is not an ellipse , but a spheroid . This results in clear anomalies of the gravitational field , which also cause orbital disruptions of satellites or moons .

causes

Gravitation alone forms spherical bodies, the flattening is caused by the centrifugal force that results from the rotation: ${\displaystyle F_{\text{Zf}}}$

${\displaystyle F_{\text{Zf}}=m\,\omega ^{2}\,r}$

With

• the mass of a mass element ${\displaystyle m}$
• its angular velocity ${\displaystyle \omega }$
• its distance from the axis of rotation . ${\displaystyle r}$

The flattening is countered by the deformation of two celestial bodies circling each other into elongated ellipsoids , which is caused by the tidal friction and is directed in the direction of the mutual gravitational forces. Usually, however, the flattening of the body caused by the centrifugal force predominates , so that the shape of the elongated ellipsoid can hardly be observed.

Connections

If one considers rigid bodies made of - to simplify - constantly dense material, then both the gravitational force and the centrifugal force (each on a test mass) increase linearly with the radius of the body. Therefore, the flattening of such bodies, regardless of their radius, is only determined by the rotational frequency: If a body rotates faster (= smaller period of rotation or higher angular speed), it will flatten more than another with the same structure.

If currents are possible, differences in density cause concentric stratification - the lightest (on earth: air and water) on top, the dense in the core in the center of the sphere. Such bodies with different density layers on the inside behave the same on the surface, i.e. flatten to the same extent, provided that the average densities of the respective complete bodies match.

A lower average density, however, results in a lower gravity on the surface and thus greater flattening at the same rotational frequency (example gas planets ).

A core of higher density will therefore flatten less than the entire body with lighter, higher layers.

Considerations of gaseous stars with zones of different rotation frequencies as they occur on the sun ( differential rotation ) are more complex .

When a body contracts, expands or changes its density locally, its angular velocity changes with the moment of inertia and thus also the flattening, while the angular momentum remains constant. This is particularly relevant when stars or galaxies change significantly in diameter. See also Coriolis force .

Similar to how regional density irregularities cause anomalies of gravity , the flattening of an entire body or its core causes a flattening of gravity , i.e. H. the gravitational force on the surface of the body is not the same everywhere, but depends on the location.

values

The sun is almost spherical because of its long period of rotation (just under 1 month).

Already at the earth (rotation time 24 hours) the diameter from pole to pole is 42 km smaller than at the equator (1: 298, earth flattening ), but this cannot be seen from space with the naked eye .

With Jupiter and Saturn the flattening is clearly visible in the small telescope due to the rapid rotation of about 10 hours (1:15 and 1:10 respectively).

Larger or very rapidly rotating stars ( red giants , pulsars, etc.), on the other hand, are strongly flattened. The object that has the strongest flattening known to date is Achernar , which is flattened to the limit of what is theoretically possible due to its rapid rotation (1: 3 or even more). See also Wega .

In our galaxy ( Milky Way ), a similar effect has resulted in a lens shape of around 1: 4 since its formation , even though it takes around 250  million years to rotate.

Web links

Wiktionary: Flattening  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ↑ The Guardian: Sun is the most perfect sphere ever observed in nature (August 16, 2012)
2. ↑ ESO Press Release 31/03: Biggest Star in Our Galaxy Sits within a Rugby-Ball Shaped Cocoon (November 18, 2003)
3. ↑ ESO Press Release 14/03: Flattest Star Ever Seen (June 11, 2003)