**Gravitation**

**Law of universal gravitation: F=GMm/r**, where M, m represents the mass of the two bodies (M is usually larger), r represents the distance, and G is the gravitation constant,

^{2}**g=6.67*10**with a unit of m

^{-11}^{3}kg

^{-1}s

^{-2}.

Gravitational field strength is the gravitation force due to an object per unit mass. Mathematically

**g = F/m = GM/r**, with the unit of Nkg^{2}^{-1}, the formula suggests that gravitational field strength is independent of the mass of object is feeling the gravitational force.Gravitational field strength on Earth's surface is g

_{0 }= 9.81Nkg^{-1}(or 10), and the gravitational field strength above Earth's surface is given by**g = g**, where r_{0}r_{0}^{2}/R^{2}_{0}is Earth's radius and R is the distance from the object to the center of Earth.Gravitational field strength is often written in the form of g = GM/r

^{2 }= (GM/r_{0})(1+R/r_{0})^{-2 }=**g**, where this R is the distance from Earth's surface to the object. When R<<r_{0}(1+R/r_{0})^{-2}_{0}(maybe a difference of thousands times), we estimate the field strength as**g = g**._{0}(1-2R/r_{0})It is suggested that outside the planet the field strength is proportional to r

^{-2}, but inside the planet the field strength is proportional to r (under the assumption of the perfectly mass-distributed and sphere planet) since M is proportional to V = kr^{3}, then g = GM/r^{2}= Gkr.**Kepler's Laws on Planetary Motion**

1. The orbit of every planet is an ellipse with the Sun at one of the two foci.

2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit in the same solar system. i.e. T

^{2}is proportional to r^{3}.The third law can be derived from circular motion, and it's suggested that

**T**.^{2 }= 4π^{2}r^{3}/GMT

^{2}= (2π/ω)^{2}= 4π^{2}/ω^{2}= 4π^{2}r/g = 4π^{2}r^{3}/GM.
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