Re: Mathematica code for Kepler's radial velocity equation?

*To*: mathgroup at smc.vnet.net*Subject*: [mg128436] Re: Mathematica code for Kepler's radial velocity equation?*From*: Roland Franzius <roland.franzius at uos.de>*Date*: Fri, 19 Oct 2012 02:43:04 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <k5o8gd$ic3$1@smc.vnet.net>

Am 18.10.2012 08:44, schrieb Virgil Stokes: > I would appreciate a look at Mathematica code for Kepler's radial velocity > equation (Nonlinear with 6 parameters). > > For the radial equation of motion there are only four, energy, angularMomentum, time t0 and radius r[t0]==r0 of start. NDSolve[ {r'[t]==Sqrt[2 energy-angularMomentum^2/r[t]^2 +SolarMass* GravitationalConstant/r[t], r[t0]==r0},{t, t0, tmax}) This works only in an open intervall between zeros of r'[t] with motion direction outward r'[t]> 0. For the other case of moving downwards to the gravitational center at r=0, you have to take the negative sqrt. Caution: At r'[t] = 0 the equation is not Lipschitz in r. Newtons equation of second order for the radius does not suffer from signs and zeros of r'[t] (remember: centrifugal force r omega^2 -> (r omega)^2/r^3 -> angularMomentum^2/r^3) NDsolve[ {r''[t]==angularmomentum^2/r[t]^3 -SolarMass*GravitationalConstant/r[t]2, r[t0]==R, r'[t0]==v0}, {t,t0,tmax}] Here the energy parameter is replaced by r'[t0] by replacement energy -> r'[t0]^2/2 + angularMomentum^2/r[t0]^2/2 -- Roland Franzius