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Ellipse


Rectangular coordinates


The equations (xe)2+y2+(x+e)2+y2=2a and (xa)2+(yb)2=1 are equivalent, they define the same ellipse in rectangular coordinates.

Polar coordinates

Important angles to describe the ellipse include

Polar form with respect to the center

Employing polar coordinates (rc,φc) with respect to the center the points of the ellipsoid are (xy)=(acosEbsinE)=rc(cosφcsinφc). The relation tanφc=batanE derives from (2) and the polar form rc(φc)=b1ε2cos2φc from (1).


Polar form with respect to the focus

The position, expressed in polar coordinates (r,ν) as seen from the focus (e,0), is r=a(cosEε1ε2sinE)=r(cosνsinν). It follows from (3) that tanν=1ε2sinEcosEε, or equivalently, tanν2=1+ε1εtanE2=1+εsinE21εcosE2. The radius r=a(1εcosE) and the polar form of the ellipse (with respect to the focus) r(ν)=a1ε21+εcosν are immediate from (3).


Kepler orbits



From Newton's 2nd law of motion and Newton's universal law of gravitatioin it follows that m¨r=GmM|r|2r|r|. The angular momentum H:=r×˙r satisfies ddt(r×˙r)=˙r×˙r+rרr=0 by (5), so that the motion is in a plane and we can assume that r(t)=r(t)(cosν(t)sinν(t)). Note that H:=|r×˙r|=|r(cosνsinν)×(˙r(cosνsinν)+r˙ν(sinνcosν))|=r2˙ν is constant. It follows from (5) that ¨r=(¨rr˙ν2)(cosνsinν)+(2˙r˙ν+r¨ν)(sinνcosν)=GMr2(cosνsinν). Now 2˙r˙ν+r¨ν=0 integrates to r2˙ν=H. By changing the dependence of r so that r(t)=r(ν(t)) we find ¨ν=ddtHr2=2Hr3r˙ν=2H2r5r and consequently ˙r=r˙ν and ¨r=r˙ν2+r¨ν=rH2r42H2r5r2. With this, the remaining equation ¨rr˙ν2+GMr2=0 rewrites as the ordinary differential equation H2r4(r2r2rr)+GMr2=0. To solve this equation define the new function s(ν):=1r(ν) so that r=ss2 and r=2ss2s2ss4. The latter equation rewrites as H2(s+s)=GM with general solution s(ν)=GMH2(1+εcos(νν0)) or r(ν)=H2GM11+εcos(νν0).

Kepler's first law

Compare the equations (6) and (4) to see that the orbit is an ellipse with H=GMab.


Kepler's 2nd law

The area is A(t,t0)=νν012r(ν)r(ν)dν=tt012r(ν(u))2˙ν(u)du. Now recall that r2˙ν=H so that A(t,t0)=H2(tt0) grows linearly with time.


Kepler's third law

Using r=a(cosEε1ε2sinE) and ˙r=a˙E(sinE1ε2cosE) we find that H=|r×˙r|=ab(1εcosE)˙E, which integrates to Ht=ab(EεsinE) or EεsinE=GMa3t=2πtP. This is Kepler's equation, where the period P satisfies P2a3=4π2GM, Kepler's third law. Note as well the relation ˙E=GMa311εcosE.


The velocity vector

We finally notice that ˙r=GMa11εcosE(sinE1ε2cosE)=aGMr(sinE1ε2cosE), an explicit expression of the velocity vector. With this and Kepler's equation (7), Newton's equation (5) can be verified directly ex post.