HistoryClasical Planetary MotionAngular MomentumBack to the Center of MassRadial Velocity and Doppler ShiftsThe Distance from the Center of Mass to the PlanetConclusionHistoryRadial Velocity and . . .ConclusionHome PageTitle PageJJ IIJ IPage 1 of 15Go BackFull ScreenCloseQuitDetermining Exo-Planetary Orbits using RadialVelocity MeasurementsBrian Reid and Geoff ZelderMath 55- College of the Redwoods11 May 2009AbstractThe detection of extra solar planetary systems has become an active field ofstudy through the use of viewing the displacement of a star due to a planet orbitingit, also known as radial velocity measurement. Using classical planetary motionand differential equations we can find out vital information about the planet suchas mass and distance from the parent star. If the mass and the distance of theplanet seem close to that of Earth we can then analyze it using other techniquesto determine its atmosphere and the possibility for life on the planet.HistoryRadial Velocity and . . .ConclusionHome PageTitle PageJJ IIJ IPage 2 of 15Go BackFull ScreenCloseQuit1. History1.1. Clasical Planetary MotionThe study of planetary motion owes much of its creation to Johannes Kepler and SirIssac Newton. Kepler contributed his three laws based on observational data which are;1. planets travel in elliptical orbits with the parent star at one focus2. planets sweep out equal areas in equal times3. the ratio of the cube of the mean distance to the square of its orbital period is aconstant.Newton contributed his own three laws which are;1. a body in motion will remain in motion unless acted upon by an outside force,basically if a mass experiences a force it will experience an acceleration such thatF = ma (1)2. where F is the force,m is the mass, and a is the acceleration, lastly for everyaction there is an equal and opposite reaction.~F12= −~F21(2)Newton also contributed his universal law of gravitation which is~F =Gm1m2r2ˆr. (3)Here G is the universal gravitation constant, m1and m2are the masses of the twoobjects, r is the distance between the two masses, and ˆr is a unit vector that points inHistoryRadial Velocity and . . .ConclusionHome PageTitle PageJJ IIJ IPage 3 of 15Go BackFull ScreenCloseQuitFigure 1: Where one objects center is the center of mass.the direction of m2with its origin at m1. Combining Equation (1), Equation (2), andEquation (3) and noting that a can also be seen as the second derivative of the positionvector with respect to time.m~a = −Gm1m2r2ˆrmd2~rdt2= −Gm1m2r2ˆr(4)But Newton’s second law states that for every action there is an equal and oppositereaction, which means that just as the star pulls on the planet the planet pulls onthe star. So Figure 1 is not exactly correct because both objects will actually orbitaround an unseen point in space due to their mutual gravitational attraction much likeFigure 2.Since both objects are in motion now this complicates our equations quite a bit. Thefirst thing we must do is define what the center of mass is. We can denote M as thecombined mass, so M = m1+ m2, we will also use the effective mass Mewhich isHistoryRadial Velocity and . . .ConclusionHome PageTitle PageJJ IIJ IPage 4 of 15Go BackFull ScreenCloseQuitFigure 2: Where both objects are orbiting a common center of mass.defined as1Me=1m1+1m2Me=m1m2m1+ m2Me=m1m2M(5)We will keep Equation (5) in mind and return to this thought later. To obtain anequation for the orbital motion of both bodies we will begin by assuming the moremassive object to be fixed and look at the total energy of the system which we willdenote as E. We will use the fact that E = KE+ PEwhere KEis the kinetic energyof the system and PEis the potential energy of the system, which in this case is thegravitational potential energy. If we set KE= 1/2m2v22and PE= Gm1m2/r then weHistoryRadial Velocity and . . .ConclusionHome PageTitle PageJJ IIJ IPage 5 of 15Go BackFull ScreenCloseQuitarrive at the equation for E.E =12m2v22−Gm1m2rE =12m2dxdt2+dydt2−Gm1m2r(6)Here we can begin to see the trouble in that we have r, dx/dt, and dy/dt so we willremember Equation (6) and convert to polar coordinates, where everything will be interms of r and θ.x =r cos θdxdt=drdtcos θ − r sin θdθdtand for the y-coordinate we obtainy =r sin θdydt=drdtsin θ + r cos θdθdtInputting these equivalencies back into Equation (6) we obtainE =12m2 drdtcos θ − r sin θdθdt2+drdtsin θ + r cos θdθdt2!−Gm1m2rE =12m2 2drdt2+ r2dθdt2!2−Gm1m2r(7)HistoryRadial Velocity and . . .ConclusionHome PageTitle PageJJ IIJ IPage 6 of 15Go BackFull ScreenCloseQuitFigure 3: Angular Momentum.1.2. Angular MomentumTo obtain a true picture of the orbital pattern of the two objects in is imperative thatwe introduce the concept of angular momentum. Angular momentum states that theeffect on one object will be translated to the other object if no other external forcesare involved. Simply stated, if one object speeds up the other will slow down, andif one slows down the other speeds up. This is the reason why in elliptic two bodysystems both objects will have the same eccentricities and when one is at shortest pointfrom the focal point the other will be at its furthest. For now we are only lookingat the orbiting body so we can simple write the angular momentum of the body L as~L = ~r × ~p. Since the angular momentum is a cross product it is a vector perpendicularto the displacement vector ~r and the momentum vector ~p, but~L is a constant becauseif ~r becomes shorter the velocity vector will become larger and vice-versa.HistoryRadial Velocity and . . .ConclusionHome PageTitle PageJJ IIJ IPage 7 of 15Go BackFull ScreenCloseQuit~L =~r × ~pL =rp sin θL =xdydt− ydxdtL =r2dθdtThis result seems familar because the value r2dθ/dt appeared in Equation (7) so we cannow substitute this value back into Equation (7) and notice that it is a scaling factorfor the kinetic energy.E =12m2 2drdt2+ r2dθdt2!−Gm1m2rE =12m2 2drdt2+L2r2!−Gm1m2rE =m2L22r2+ m2drdt2−Gm1m2r(8)From Equation (8) we can see the orbital motion of an body if the center of mass isfixed, and a body with a lot more mass than the orbiting body. Equation (8) is acoupled first order differential equation because both θ and ~r are allowed to vary overa certain time interval. Also the total energy~E and the angular momentum~L areinput parameters that not only define the orbit of the body but also the velocity at anyinstant in time