How to record the potential energy of gravity

Planetary system

If you shoot a body from a mountain on earth that protrudes above the atmosphere with the velocity \ (v \) parallel to the earth's surface, the following trajectories result:

1) For small v a "trajectory parabola" which is actually part of an ellipse, but ends when it hits the earth. (Curve 1)

2) If the speed is high enough, the result is a circle with its center at the center of the earth (curve 2).
Therefore the gravitational force corresponds to the centripetal force. So it follows: \ ({F_G} = {F_Z} \ Rightarrow \ frac {{m \ cdot {v ^ 2}}} {r} = \ frac {{G \ cdot m \ cdot M}} {{{r ^ 2}}} \)

The body has a kinetic energy \ ({E_ {kin}} = \ frac {1} {2} m \ cdot {v ^ 2} = \ frac {1} {2} \ cdot \ frac {{G \ cdot m \ cdot M}} {r} \)
(That's half of the negative potential energy)

This results in a speed of \ (v = \ sqrt {\ frac {{G \ cdot M}} {r}} \), i.e. \ (v = \ frac {{{v _ {\ rm {Escape}}}} } {{\ sqrt 2}} \).

The total energy of the circling body is dependent on \ (r \) (energy in infinity is assumed to be zero): \ [{E_0} = {E_ {kin}} + {E_ {pot}} = \ frac {1} { 2} \ cdot \ frac {{G \ cdot {\ rm {m}} \ cdot {\ rm {M}}}} {r} - \ frac {{G \ cdot {\ rm {m}} \ cdot { \ rm {M}}}} {r} = - \ frac {1} {2} \ cdot \ frac {{G \ cdot {\ rm {m}} \ cdot {\ rm {M}}}} {r } \]

This energy relation is valid for all circular orbits and also for elliptical orbits if one replaces r by a.

3) If the speed increases, the result is elliptical orbits, with the launch point in the perigee, as long as \ (v

For \ (v = v _ {\ rm {Escape}} \) there would be a parabola. The body does not return and has the total energy 0 (outside of the earth's area of ​​attraction it has no more kinetic energy). The escape speed of the earth is \ (v _ {\ rm {Escape}} = 11 {,} 2 \, \ rm {\ frac {km} {s}} \)

4) For \ (v> v _ {\ rm {Escape}} \) there is a hyperbolic trajectory (curve 4). The total energy of the body is greater than 0.

Derivation for the total energy of an elliptical orbit

The total energy \ (E_0 = E _ {\ rm {kin}} + E _ {\ rm {pot}} \) is the same everywhere on the ellipse. At a great distance from the central body, \ (E _ {\ rm {pot}} \) is larger and \ (E _ {\ rm {kin}} \) smaller, in the vicinity it is the other way round.

For perihelion, \ ({E_0} = \ frac {1} {2} m \ cdot {v_ {Perihel}} ^ 2 - \ frac {{G \ cdot {\ rm {m}} \ cdot {\ rm {M }}}} {{{r_ {Perihel}}}} \); for aphelion, \ ({E_0} = \ frac {1} {2} m \ cdot {v_ {aphelion}} ^ 2 - \ frac {{G \ cdot {\ rm {m}} \ cdot {\ rm {M }}}} {{{r_ {Aphelion}}}} \)
Multiplying the equations by \ (r_ {Perihel} ^ 2 \) or \ (r _ {{Aphel}} ^ 2 \) leads to the equations \ [{E_0} \ cdot {r_ {Perihel}} ^ 2 = \ frac {1} {2} m \ cdot {v_ {Perihel}} ^ 2 \ cdot {r_ {Perihel}} ^ 2 - G \ cdot {\ rm {m}} \ cdot {\ rm {M}} \ cdot { r_ {perihelion}} \ tag {1} \] \ [{E_0} \ cdot {r_ {aphelion}} ^ 2 = \ frac {1} {2} m \ cdot {v_ {aphelion}} ^ 2 \ cdot { r_ {Aphelion}} ^ 2 - G \ cdot {\ rm {m}} \ cdot {\ rm {M}} \ cdot {r_ {Aphel}} \ tag {2} \] In addition, the following applies: \ ({v_ { Perihel}} \ cdot {r_ {Perihel}} = {v_ {Aphelion}} \ cdot {r_ {Aphelion}} \ Rightarrow \ frac {1} {2} m \ cdot {v_ {Perihel}} ^ 2 \ cdot { r_ {perihelion}} ^ 2 = \ frac {1} {2} m \ cdot {v_ {aphelion}} ^ 2 \ cdot {r_ {aphelion}} ^ 2 \).

If one subtracts equation (1) from equation (2), the result is:

\ [E_0 \ cdot r _ {{aphelion}} ^ 2-E_0 \ cdot r _ {{perihel}} ^ 2 = -G \ cdot m \ cdot M \ cdot r _ {{aphelion}} + G \ cdot m \ cdot M \ cdot r _ {{Perihel}} \] \ [\ Rightarrow E_0 \ cdot \ left (r_ {Aphelion} ^ 2-r _ {{Perihel}} ^ 2 \ right) = - G \ cdot m \ cdot M \ cdot \ left (r_ {Aphelion} -r _ {{Perihel}} \ right) \] \ [\ Rightarrow {E_0} = - G \ cdot m \ cdot M \ cdot \ frac {{{r_ {Aphel}} - {r_ { Perihel}}}} {{{r_ {Aphelion}} ^ 2 - {r_ {Perihel}} ^ 2}} \] \ [\ bbox [aquamarine, 10pt, border: 2px solid black] {{E_0} = - \ frac {{G \ cdot m \ cdot M}} {{{r_ {Aphelion}} + {r_ {Perihel}}}} = - \ frac {{G \ cdot m \ cdot M}} {{2a}}} \]

For \ (a = r \) the total energy of a circular path follows from this, as already shown above.

Track speed

For the orbital velocity of a body on the ellipse at a distance \ (r \) \ [\ frac {1} {2} m {v ^ 2} - \ frac {{G \ cdot m \ cdot M}} {r} = - \ frac {{G \ cdot m \ cdot M}} {{2a}} \] Thus the orbital velocity \ (v \) results from \ [\ bbox [aquamarine, 10pt, border: 2px solid black] {v = \ sqrt {2GM \ cdot \ left ({\ frac {1} {r} - \ frac {1} {{2a}}} \ right)}} \]