HistoryRadial VelocityHistory of Classical MechanicsKeplers LawsNewton's ContributionsThe Center of MassAn Equation to Explain the Motion of the BodiesThe Energy of the SystemAngular MomentumAn Equation for Angular MomentumAngular Momentum and the Total Energy of the systemEnergy, Angular Momentum, and the Center of MassPutting it All TogetherDetermining the CharacteristicsThe Distance between the Planet and the StarThe Planets VelocityThe Planet's MassDetermining Exo-Planetary Orbits using RadialVelocity MeasurementsBrian Reid and Geoff ZelderMay 15, 2009HistoryRadial VelocityHistory of Classical MechanicsKeplers LawsNewton’s ContributionsThe Center of MassAn Equation to Explain the Motion of the BodiesThe Energy of the SystemAngular MomentumAn Equation for Angular MomentumAngular Momentum and the Total Energy of the systemEnergy, Angular Momentum, and the Center of MassPutting it All TogetherDetermining the CharacteristicsThe Distance between the Planet and the StarThe Planets VelocityThe Planet’s MassHistory of Radial Velocity MeasurementsIn 1952 Otto Struve proposed a method of using high poweredtelescopes to observe the slight displacement of a star created by amassive planet, such as Jupiter, orbiting it. Struve predicted thatthere would be a slight Doppler shift created by the star as theplanet orbits it. Although at the time there were no instruments ortechniques that would be able to detect such slight shifts in theelectromagnetic spectrum. In the 1980’s and early 1990’s advancesin spectral measuring devices enabled humans to observe slightvariations in a stars Doppler signature. This technique is calledRadial Velocity Measurement using Doppler Spectroscopy. InOctober 1995, 51 Pegasi b was the first exoplanet to be recordedand cataloged. Since then over 300 planets have been cataloguedwhere many have been found using Doppler Spectroscopy.What is a Doppler ShiftDoppler shifts work on the fact that as an object that is movingtoward you has a higher recieved frequency as opposed to anobject that is moving away from you.Radial VelocityUsing this equation we can determine the radial velocity of anobject.dλdt= λ0(1 + v/c)We solve for v here to find the radial velocity of the star, wheredλ/dt is the change in the doppler spectrum and λ0is the initialdoppler value, and c is the speed of light.51 Pegasi graphIf we plot time of one orbit versus the radial velocity we obtain agraph similar to this oneKepler’s Three Laws of Orbital MotionIn the mid 1600’s Johannes Kepler devised these laws of planetarymotion1. planets travel in elliptical orbits with the parent star at onefocus2. planets sweep out equal areas in equal times3. the ratio of the cube of the mean distance to the square of itsorbital period is a constant.Later in the 1700’s Sir Issac Newton contributed these laws1. a body in motion will remain in motion unless acted upon byan outside force, basically if a mass experiences a force it willexperience an acceleration such thatF = ma2. where F is the force,m is the mass, and a is the acceleration,lastly for every action there is an equal and opposite reaction.~F12= −~F21Newton also contributed his universal law of gravitation whichis~F =Gm1m2r2ˆrHere G is the universal gravitation constant, m1and m2are themasses of the two objects, r is the distance between the twomasses, and ˆr is a unit vector that points in the direction of m2with its origin at m1. By combining these equations we obtainm~a = −Gm1m2r2ˆrmd2~rdt2= −Gm1m2r2ˆrThis leads us to this view of the systemConverting to a Center of Mass frame of referenceSince the star’s mass is creating a force on the planet, so to is theplanet creating a force on the star. In turn both objects areactually orbiting a common center of mass. To find the effectivemass of both objects at the center we obtain1Me=1m1+1m2Me=m1m2m1+ m2Me=m1m2MAlso the ratio between the masses ism2(1 + m2/m1)keep these equations in mind as we will use them later.Viewing the body systemNow that we have set up a different view of the system with acenter of mass introduced both bodies move about it with thierown associated direction vectors r1and r2.Energy in Orbital MotionTo simplify the calculations for now we will assume the moremassive object to be fixed in the center. We will use kinetic energydenoted KE, and potential energy denoted PEand note that bothof these added together will give us the total energy E .1. KE= 1/2m2v222. PE= Gm1m2/r, which is gravitational potential energyPutting all of these facts together we obtainE = KE+ PEE =12m2v22−Gm1m2rE =12m2dxdt+dydt2−Gm1m2rA new cordiante system is nessasarySince we now have dx’s, dy’s and r ’s we can simplify this byconverting over to polar coordinates since both objects lie in aplane. We know that1. x = r cos θ and2. y = r sin θIt follows that1.dxdt=drdtcos θ − r sin θdθdt2.dydt=drdtsin θ + r cos θdθdtsubstituting back into our original equation for dx/dt and dy/dtwe getE =12m2drdtcos θ − r sin θdθdt+drdtsin θ + r cos θdθdt2−Gm1m2rE =12m2 drdt+ r2dθdt2!−Gm1m2rWe will label this one and come back to it in a little bitDeriving the Angular Momentum of the systemWe will now look at the angular momentum of the system by thedefinition which is~L =~r ×~p. Graphically this looks something like thisAngular Momentum in Polar CoordinatesIn the case of orbiting bodies the angular momentum is theposition vector crossed with the velocity vector. But we want thisto be in polar form to match our previous equation of the totalenergy.~L =~r ×~pL = rp sin θL = xdydt− ydxdtL = r2dθdtThis result is very familiar from the last equation so lets substituteit in to try to simplify our result for the total energy.Substituting with Angular MomentumOur last equation of the energy in the system wasE =12m2 drdt2+ r2dθdt2!−Gm1m2rIf we now substitute in L for r2dθ/dt we getE =12m2 drdt2+L2r2!−Gm1m2rE =m2~L22r2+m22drdt2−Gm1m2rNow we can put this equation back into the reference frame withthe center of mass and obtainE =m2L22r2+m22drdt2−Gm1m2rE =12m21 +m2m1+drdt+12m2(1 + m2/m1) L21 + m22/m21r2−Gm1m2(1 + m2/m1) rSo now we have found a differential equation that relates the twoorbiting bodies to a common center of mass and describes theorbital motion of both bodies. We are now able to correlate this tothe radial velocity measurements taken by observing a stars motionover a certain interval of time. If