Hill's differential equation (three-body problem)

The Hill differential equation was developed by George William Hill (1838-1914) for orbital calculations in astronomy . It is mainly used for solutions to the three-body problem , but also for the movement of particles in synchrotrons .

Hill's equation represents a preliminary stage to the canonical equations and works with dimensionless coordinates and masses, which makes the method easier and more versatile.

In the case of multi-body problem of celestial mechanics - in which generally two masses , highly dominant - is for. B. equated the unit of mass with the product of ( ) times the gravitational constant . What in Hill's time had the goal of becoming independent of the numerical quantities of the solar system (e.g. the astronomical unit ), which were not exactly known at the time, also turned out to be a theoretical advantage. ${\displaystyle M_{1}}$ ${\displaystyle M_{2}}$ ${\displaystyle M_{1}+M_{2}}$

The mass of the second largest body (i. A. a planet ) is in the equations as a ratio to total mass sun introduced + Planet, ie . ${\displaystyle m=M_{2}/(M_{1}+M_{2})}$

Furthermore, the coordinate system is placed in the barycentre (the center of gravity of M ₁ and M ₂ ) and the x -axis in their connecting line. In the simplified 2D notation (related to the path plane , i.e. z = 0) the equations are:

${\displaystyle x''-2y'=(3-m/d^{3})x=dU/dx}$ and

${\displaystyle y''+2x'=-(m/d^{3})y=dU/dy}$,

where and (primed quantities are the single or double derivatives with respect to time). Accordingly, the right-hand sides of the equations can be represented as partial derivatives of the same function U , which is a significant advantage of the Hill equations. ${\displaystyle U=3x^{2}/2+m/d}$ ${\displaystyle d^{2}=x^{2}+y^{2}}$

literature

• C. Murray, F. Dermott: Solar System Dynamics. Cambridge University Press, 1999, pp. 60-116.
• Manfred Schneider : Himmelsmechanik Volume IV. Spectrum, Heidelberg, Berlin 1999, pp. 405-440.