Elliptical Orbits Follow from the Equation of Motion
The planet is launched from the point closest to the Sun, perpendicular to the line joining the Sun and the planet. The diagram shows roughly 1 orbit under the Sun's gravity alone.
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次の近日点に達するまで、太陽のまわりを何度回るか？
1/r³の引力を足すと、次の近日点までに何度回るか？
The equations explain the phenomenon
1Acceleration is expressed in terms of distance and anglear=r − r θ²，aθ=(1r) d(r² θ)dt……
2Newton's law of universal gravitation gives the equations of motion in the 2 directionsm (r − r θ²)=−G M mr²，(mr) d(r² θ)dt=0……
3Conservation of angular momentum follows from the angular equation of motionr² θ=h (constant)……
4The substitution u=1/r converts time derivatives into derivatives with respect to θ.u=1r，r=−h² u² d²udθ²……
5Substitution into the equation of motion gives an equation for u.d²udθ²+u=G Mh²……
6The solution for simple harmonic motion gives the angle at which u reaches its maximum again.u=(G Mh²)(1 + e cosθ)……
7Converting back to r reveals the shape of the orbit.r=ℓ1 + e cosθ……
8Perihelion and aphelion link the orbital shape to energy.e²=1+2 E h²(G M)²……
9The rate at which area is swept out gives the period of the elliptical orbit.T²=(4π²G M) a³……
10An additional 1/r³ attraction changes the equation for u.d²udθ²+(1 − κh²) u=G Mh²……
11The angle at which u reaches its maximum again determines the perihelion advance.Θ=360°√(1 − κ/h²)……
What this chapter has shown
For an elliptical orbit under a 1/r² attraction, u=1/r oscillates with a period of 360° in the angle θ swept out. When θ increases by 360°, the distance r from the Sun returns to its original value in every direction. The orbit therefore closes after 1 revolution.
Adding the 1/r³ attraction with the strength used here advances the next perihelion direction by 7.42°.
Mercury's actual orbitMercury's additional perihelion advance of 43 arcseconds per 100 years was explained by general relativity.
Detailed summary
Using conservation of angular momentum to eliminate time and substituting u = 1/r reduced the planet's equation of motion to an equation for simple harmonic motion with respect to angle. For a 1/r² force, the period is exactly 360°, so the orbit is a closed conic section described by r = ℓ/(1 + e cosθ), with its eccentricity determined by the energy. When the force departs from the 1/r² dependence, the period of oscillation of u differs from 360°, and the perihelion advances with each revolution.