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For such two- or restricted three-body problems as its simplest examples—e.g., one more massive primary astrophysical body, mass of m1, and a less massive secondary body, mass of m2—the concept of a Hill radius or sphere is of the approximate limit to the secondary mass's "gravitational dominance", [6] a limit defined by "the extent" of its ...
For astronomical bodies other than Earth, and for short distances of fall at other than "ground" level, g in the above equations may be replaced by (+) where G is the gravitational constant, M is the mass of the astronomical body, m is the mass of the falling body, and r is the radius from the falling object to the center of the astronomical body.
The most common base models to calculate the sphere of influence is the Hill sphere and the Laplace sphere, but updated and particularly more dynamic ones have been described. [ 2 ] [ 3 ] The general equation describing the radius of the sphere r SOI {\displaystyle r_{\text{SOI}}} of a planet: [ 4 ] r SOI ≈ a ( m M ) 2 / 5 {\displaystyle r ...
rad/s is the diurnal angular speed of the Earth axis, and km the radius of the reference sphere, and the distance of the point on the Earth crust to the Earth axis. [ 3 ] For the mass attraction effect by itself, the gravitational acceleration at the equator is about 0.18% less than that at the poles due to being located farther from the ...
^ Surface gravity derived from the mass m, the gravitational constant G and the radius r: Gm/r 2. ^ Escape velocity derived from the mass m , the gravitational constant G and the radius r : √ (2 Gm )/ r .
+ + () + = where = + is the mass fraction of M 2 and = is the normalised distance. If the mass of the smaller object (M 2) is much smaller than the mass of the larger object (M 1) then L 1 and L 2 are at approximately equal distances r from the smaller object, equal to the radius of the Hill sphere, given by:
The mass of any of the discs is the mass of the sphere multiplied by the ratio of the volume of an infinitely thin disc divided by the volume of a sphere (with constant radius ). The volume of an infinitely thin disc is π R 2 d x {\displaystyle \pi R^{2}\,dx} , or π ( a 2 − x 2 ) d x {\textstyle \pi \left(a^{2}-x^{2}\right)dx} .
The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance r 0 from the center of the mass distribution: [13] The portion of the mass that is located at radii r < r 0 causes the same force at the radius r 0 as if all of the mass enclosed within a sphere of radius r 0 ...