Abarian point

In physics, the abarian point is the point between two masses where their forces of attraction cancel each other out. This point is basically closer to the lighter mass. The exact calculation is made by equating the forces of attraction that the two masses exert on a test body of the mass at the abaric point. Since, for geometric reasons, the Abarian point always lies on the imaginary connecting line between the two masses, the problem is reduced to a one-dimensional one. This is shown here using the example of earth - moon : ${\displaystyle F}$ ${\displaystyle m_{\mathrm {K{\ddot {o}}rper} }}$

${\displaystyle {\begin{aligned}F_{\text{Erde}}&=F_{\text{Mond}}\\\Leftrightarrow \gamma \cdot {\frac {m_{\mathrm {K{\ddot {o}}rper} }\cdot M_{\text{Erde}}}{r_{1}^{2}}}&=\gamma \cdot {\frac {m_{\mathrm {K{\ddot {o}}rper} }\cdot M_{\text{Mond}}}{r_{2}^{2}}}\\\Leftrightarrow {\frac {M_{\text{Erde}}}{r_{1}^{2}}}&={\frac {M_{\text{Mond}}}{r_{2}^{2}}}\end{aligned}}}$

With

• the gravitation constant from Newton's law of gravitation ${\displaystyle \gamma }$
• the distance of the center of the earth (geocenter) from the abarian point ${\displaystyle r_{1}}$
• the distance of the center of the moon from the abaric point. ${\displaystyle r_{2}}$

Since the mass of the earth is approx. 81 times larger than that of the moon:

${\displaystyle M_{\text{Erde}}\approx 81\cdot M_{\text{Mond}}}$

applies:

${\displaystyle \Rightarrow {\frac {M_{\text{Erde}}}{M_{\text{Mond}}}}=\left({\frac {r_{1}}{r_{2}}}\right)^{2}=81\quad \Rightarrow \quad r_{1}=9\cdot r_{2}}$

The abarian point of the earth-moon system is about 1/10 of the total distance between the moon and the center of the earth. With the approximation that the distance between the two celestial bodies is approximately 60  earth radii , one obtains that the abaric point in the moon-earth system is 54 earth radii from the center of the earth and 6 earth radii from the moon center. The abarian point is located in the universe between the two celestial bodies, much closer to the moon than to the earth. ${\displaystyle r_{1}+r_{2}}$

Differentiation from similar excellent points

The abarian point must not be confused with the common center of gravity ( barycenter ) of the moon-earth system ( earth-moon center of gravity ), which lies in the interior of the earth .

In contrast to the Lagrange points, the centrifugal forces are not taken into account with the Abarian point .

See also

• Two-body problem