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Ellipse
Rectangular coordinates
The equations √(x−e)2+y2+√(x+e)2+y2=2a and (xa)2+(yb)2=1 are equivalent, they define the same ellipse in rectangular coordinates.
- e is the focal distance or linear eccentricity,
- a is the semi-major axis,
- b2:=a2−e2 is the semi-minor axis and
- ε:=ea=√1−b2a2 is the eccentricity.
Polar coordinates
Important angles to describe the ellipse include- the eccentric anomaly E and
- the true anomaly ν.
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)=b√1−ε2cos2φc 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(cosE−ε√1−ε2sinE)=r(cosνsinν). It follows from (3) that tanν=√1−ε2sinEcosE−ε, or equivalently, tanν2=√1+ε1−εtanE2=√1+εsinE2√1−εcosE2. The radius r=a(1−εcosE) and the polar form of the ellipse (with respect to the focus) r(ν)=a1−ε21+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=(¨r−r˙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=r′′H2r4−2H2r5r′2. With this, the remaining equation ¨r−r˙ν2+GMr2=0 rewrites as the ordinary differential equation H2r4(r′′−2r′2r−r)+GMr2=0. To solve this equation define the new function s(ν):=1r(ν) so that r′=−s′s2 and r′′=2ss′2−s2s′′s4. The latter equation rewrites as H2(s′′+s)=GM with general solution s(ν)=GMH2(1+εcos(ν−ν0)) or r(ν)=H2GM11+εcos(ν−ν0).