Eötvös effect

The Eötvös effect describes the vertical deflection of a body moving parallel to the latitude on the rotating earth. This movement is associated with a slight increase (in an east-west direction) or a decrease (in a west-east direction) in the force of gravity acting on the body. The Eötvös effect is a special case of the Coriolis effect . It is named after the Hungarian physicist Loránd Eötvös .

Basis of the effect

Division of the earth's angular velocity into horizontal and vertical components based on geographical latitude ${\displaystyle \varphi }$

When a body moves on the earth, which in turn rotates with angular velocity , this generally leads, from the point of view of the co-rotating earth observer, to a deflection from the rectilinear direction of movement by the Coriolis force . This deflection, physically understood as acceleration , takes place perpendicular to the vector of the angular velocity, which runs parallel to the polar axis of the earth. While this deflection near the pole is still oriented horizontally to the earth's surface, the same effect appears to the observer at the equator as a vertical deflection; in this case one speaks of the Eötvös effect. ${\displaystyle \omega }$

For movement in the geographical latitudes lying in between, the angular velocity can be broken down into a vertical and a horizontal component; only the latter causes a deflection of the movement in the vertical direction.

Formally, the description of the Eötvös effect is based on the definition of the Coriolis acceleration; since the velocity vector is perpendicular to that of the angular velocity , and : ${\displaystyle v}$ ${\displaystyle \omega }$ ${\displaystyle \theta =90^{\circ }}$ ${\displaystyle \sin \theta =1}$

${\displaystyle {\vec {a}}_{C}=-2\,{\vec {\omega }}\times {\vec {v}}}$

and as an amount :

${\displaystyle a_{C}=2\,\omega \,v\sin \theta =2\,\omega \,v}$

Numerical Example : At a speed of and the angular velocity of the Earth from is on the equator ${\displaystyle v=30\mathrm {\,m\,s^{-1}} (=108\,\mathrm {km\,h^{-1}} )}$ ${\displaystyle \omega =7{,}3\,\cdot 10^{-5}\,\mathrm {s} ^{-1}}$

${\displaystyle a_{C}=2\,\omega \,v=2{,}19\,\cdot 10^{-3}\,\mathrm {m\,s} ^{-1}}$

and thus only a small fraction of the gravitational acceleration acting in the same direction of . ${\displaystyle g_{0^{\circ }}=9{,}78\,\mathrm {m\,s} ^{-1}}$

For movements parallel to the latitude in any geographic latitude, the following applies: ${\displaystyle \varphi }$

${\displaystyle {\vec {\omega }}_{\mathrm {hor} }={\vec {\omega }}\,\cos \varphi }$

${\displaystyle a_{C}=2\,\omega \,\cos \varphi \,v}$

whereby the amount becomes even smaller: At 60 ° N or S it is only half as large as at the equator. ${\displaystyle \cos 60^{\circ }=0{,}5}$

Dynamic explanation

Gravitation and centrifugal force on the earth's surface ${\displaystyle {\vec {G}}}$ ${\displaystyle {\vec {F}}}$

Component decomposition of centrifugal force at the earth's surface

The Eötvös effect is a result of the earth's rotation , which has slightly deformed the earth into a shape that roughly corresponds to an ellipsoid of revolution. Every body on its surface is subject to two forces, gravitation and centrifugal force (as an inertial force due to its rotation). At the equator, the centrifugal force reduces the gravity slightly, the resulting gravity is a small fraction smaller than the gravity. ${\displaystyle {\vec {G}}}$ ${\displaystyle {\vec {F}}}$

Away from the equator, gravity is the result of gravitation and centrifugal force in the force parallelogram . Only at the poles, where the centrifugal force is zero, are gravitation and gravity identical.

In contrast to gravity, gravity depends not only on the mass and location of a body, but also on its movement. The centrifugal force acts on a resting body with the mass on the surface of the earth , whereby the centrifugal acceleration is at rest; for this, the horizontal distance from the earth's axis applies : ${\displaystyle m}$ ${\displaystyle {\vec {F}}=m\,a_{z}}$ ${\displaystyle a_{z}}$ ${\displaystyle R}$

${\displaystyle a_{z}=\omega ^{2}\cdot R}$

If a body moves with the relative speed on the earth parallel to the latitude, an additional component for the angular velocity and an additional component for the centrifugal acceleration follow , so that the total centrifugal acceleration results: ${\displaystyle v=\omega _{+}\,R}$ ${\displaystyle \omega _{+}}$ ${\displaystyle a_{+}}$

${\displaystyle a_{g}=a_{z}+a_{+}}$

For the additional component it follows: ${\displaystyle a_{+}}$

${\displaystyle a_{+}=a_{g}+a_{z}=(\omega +\omega _{+})^{2}\,R-\omega ^{2}\,R=\omega ^{2}\,R+2\,\omega \,\omega _{+}\,R+\omega _{+}^{2}\,R-\omega ^{2}\,R=2\,\omega \,\omega _{+}\,R+\omega _{+}^{2}\,R}$

${\displaystyle =2\,\omega \,v+\omega _{+}^{2}\,R}$

The first term describes the Coriolis acceleration, the second the centrifugal acceleration resulting from the relative movement of the body. Depending on whether the relative movement of the body takes place in or against the direction of rotation of the earth, it increases or decreases the centrifugal force and as a result decreases or increases the acting gravity. ${\displaystyle {\vec {F}}}$

If the centrifugal force is broken down into a horizontal and a vertical component , both also increase as they grow. The result is not only an equatorial deflection ( Coriolis force , to the right on the northern hemisphere, to the left on the southern hemisphere), but also a perpendicular force that reduces gravity: ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {F}}_{h}}$ ${\displaystyle {\vec {F}}_{v}}$ ${\displaystyle {\vec {F}}}$

"... that a well-fed man weighing 100 kg is 2 g lighter when he walks slowly at 1 m speed per second on a normal surface of the earth [...] when he moves east than when he then returns to the west."

Similarly, in an east-west movement, the centrifugal force is reduced and the force of gravity increases. While the direction of the horizontal Coriolis effect is reversed on transition to the other half of the earth, the direction of the perpendicular Eötvös effect is the same for both halves of the earth.

Meaning of the effect

The importance of the Eötvös effect for climatology lies in the fact that it tends to stabilize the trade winds flowing in westerly direction through the gain in gravity. On the other hand, the rise of humid air masses is promoted in the tropical monsoon winds blowing with an easterly component , especially the Indian south-west monsoons .

In rocket technology , the Eötvös effect is used when rockets with an eastern component are launched in order to take advantage of the reduced force of gravity.

A patent has been granted for a compact motor based on the Eötvös effect .

Research history

In 1859, Jacques Babinet and Charles Eugène Delaunay pointed in theoretical considerations to the diversion of west-east directed movements. In the first decade of the 20th century, gravitational measurements were carried out on several research expeditions in the three oceans under the direction of Oskar Hecker . In the results, Eötvös noted that the measurements tended to provide smaller values ​​when the ships were moving in an easterly direction than when they were moving in the opposite direction. This was confirmed in 1908 during a measurement expedition in the Black Sea.

literature

• Roland Eötvös: Experimental proof of the change in gravity that a body moving in an easterly or westerly direction on a normally shaped earth's surface suffers as a result of this movement. In: Annalen der Physik Volume 59, 1919, pp. 743-752.
• Anders Persson: The Coriolis Effect - a conflict between common sense and mathematics. The Swedish Meteorological and Hydrological Institute: 8th 2005.

Web links

Commons : Eötvös experiment  - collection of images, videos and audio files

Individual evidence

1. ↑ Roland Eötvös: Experimental proof of the change in gravity which a body moving in an easterly or westerly direction on a normally shaped earth's surface suffers as a result of this movement. In: Annalen der Physik Volume 59, 1919, pp. 744-745
2. ↑ Compact motor for energetic use of the Eötvös effect. Patent DE19750235A1 May 27, 1999.
3. ↑ Anders O. Persson: The Coriolis Effect: Four centuries of conflict between common sense and mathematics. In: History of Meteorology. Volume 2, 2005, p. 10
4. ↑ Anders Persson: The Coriolis Effect - a conflict between common sense and mathematics. The Swedish Meteorological and Hydrological Institute: 8. 2005, pp. 7-8.