# kepler's law derivation

Derivation of Kepler’s Third Law for Circular Orbits We shall derive Kepler’s third law, starting with Newton’s laws of motion and his universal law of gravitation. The Law of Harmonies. The differential equation becomes easy to solve if we make the substitution r→u−1r \rightarrow u^{-1}r→u−1. As usual, we begin with Newtons Second Law: F = ma, in In this more rigorous form it is useful for calculation of the orbital period of moons or other binary orbits like those of binary stars. ellipse. varies too. Moreover, since rminr_\text{min}rmin​ and rmaxr_\text{max}rmax​ are distances from the Sun, we see that the Sun is at one focus of the orbit. Suppose that at any isntant the planet is at point A in its orbit and after an in small time dt, it reaches point B. + PF2 = 2a. the initial conditions. But since the angular momentum L is constant, L = mr2w, An Elementary Derivation of Kepler’s Laws of Planetary Motion. However, this is just the time derivative of r2θ˙r^2\dot{\theta}r2θ˙, and thus we have shown. by a = b. A A 310 Orbital and Space Flight Mechanics (4) Newton's law of gravitation. Equation 13.8 gives us the period of a circular orbit of radius r about Earth: Among other things, Kepler's laws allow one to predict the position and velocity of the planets at any given time, the time for a satellite to collapse into the surface of a planet, … (in this case negative) and in the direction perpendicular to the radius, rDq/Dt. The animation below illustrates Kepler’s second law in action. to q. Looking at the above picture, in the time Dt In the study of ellipses, the parameter eee is often called the eccentricity. velocity . Newtons His laws state: 1. In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. is perpendicular to SB (S being the center of the Sun), and The square of the orbital time period of a planet is proportional to the cube of the semi-major axis of its orbit, i.e. In the picture above, in which I have greatly exaggerated Log in. Given at any time the positions and velocities of two massive particles moving under their mutual gravitational force, the masses also being known, provide a means of calculating their positions and velocities for any other time, past or future. mprpvp=mprava⇒rpvp=rava⇒vpva=rarp.m_pr_pv_p=m_pr_av_a\Rightarrow r_pv_p=r_av_a \Rightarrow \dfrac{v_p}{v_a} = \dfrac{r_a}{r_p}.mp​rp​vp​=mp​ra​va​⇒rp​vp​=ra​va​⇒va​vp​​=rp​ra​​. 1. −(Lm)2u2d2udθ2−(Lm)2u3=−GMu2,-\left(\frac{L}{m}\right)^2u^2\frac{d^2u}{d\theta^2} - \left(\frac{L}{m}\right)^2u^3 = -GMu^2,−(mL​)2u2dθ2d2u​−(mL​)2u3=−GMu2. One way to draw an ellipse is to take Equations of motion 4. (2) A radius vector joining any planet to Sun sweeps out equal areas in equal intervals of time. In his footsteps we will obtain each law in turn, as we consider the orbit of a planet in the gravity of a massive star. Contrary to many people’s beliefs and understanding, the orbits that the planets move on are not circular. Based on the energy of the particle under motion, the motions are classified into two types: 1. writing the distance from the center of the ellipse to a focus OF1 We will now take Newton’s law of gravitation and derive Kepler’s First Law. equation with the origin at one focus. Note that in the line above we made the useful substitution e⇒AL2GMm2e \Rightarrow \dfrac{AL^2}{GMm^2}e⇒GMm2AL2​. \ddot{x}\cos\theta + \ddot{y}\sin\theta = &\ddot{r}\cos^2\theta -2 \dot{r}\dot{\theta}\cos\theta\sin\theta - r\dot{\theta}^2\cos^2\theta - r\ddot{\theta}\cos\theta\sin\theta \\ Thus, we have derived Kepler's second law, i.e. direction, in (x, y) coordinates it is described by the equation. Let the planet beat the point R, whose position vector is r (the“radius vector”). is placed at F1, and the ellipse is a mirror, it will Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit. x˙=r˙cos⁡θ−rθ˙sin⁡θx¨=r¨cos⁡θ−2r˙θ˙sin⁡θ−rθ˙2cos⁡θ−rθ¨sin⁡θ.\begin{aligned} Kepler's second law (equal areas in equal times) is a consequence of angular momentum conservation, ℓ = μ r 2 θ ˙ = constant, (with reduced mass μ and coordinates r and θ) because the infinitesimal area swept out per unit time is d A = 1 2 r 2 d θ = ℓ 2 μ d t. rθ¨+2r˙θ˙=0.r\ddot{\theta} + 2\dot{r}\dot{\theta} = 0.rθ¨+2r˙θ˙=0. Derivation of 2nd Law (again) Note that another common derivation is based on the conservation of angular momentum. You may treat the Earth and Sun as point masses. We present here a calculus-based derivation with the results, but need not worry about the details of the derivationits Deriving Keplers Laws We have, drdt=−u−2dudt=−u−2dudθLu2m=−Lmdudθ.\begin{aligned} depends only on the semimajor axis of the orbit: it does not depend sweeps out equal areas in equal times) first, because it has a simple physical Newton showed that Kepler’s laws were a consequence of both his laws of motion and his law of gravitation . The magnitude of the angular momentum at perihelion is Lp=mprpvpL_p=m_pr_pv_pLp​=mp​rp​vp​ because rpr_prp​ and vpv_pvp​ are mutually perpendicular. Though the laws were originally obtained by Kepler after careful analysis of empirical data, the complete understanding was missing until Newton derived each law as pieces of his orbital mechanics. Position and velocity as functions of time. Kepler’s Second Law states: A line joining a planet and the Sun sweeps out equal areas during equal time intervals. \end{aligned}mx¨my¨​​=−Gr2MSun​m​cosθ=−Gr2MSun​m​sinθ.​. However, the result is independent of θi\theta_iθi​ and θf\theta_fθf​, but it only depends on ttt since the angular momentum is constant. use (r, q force acting is gravity, and that force acts in a line from the planet towards The total acceleration is the sum, so ma = F becomes: This isnt ready to integrate yet, because w \end{aligned}x¨cosθ+y¨​sinθ=​r¨cos2θ−2r˙θ˙cosθsinθ−rθ˙2cos2θ−rθ¨cosθsinθ+r¨sin2θ+2r˙θ˙sinθcosθ−rθ˙2sin2θ+rθ¨sinθcosθ.​, We see that every term with a sin⁡θcos⁡θ\sin\theta\cos\thetasinθcosθ cancels so that we're left with. π^2)/(R^2)]. Consequently, the rate at which area is swept out is also We can derive Kepler’s third law by starting with Newton’s laws of motion and the universal law of gravitation. Derivation of Kepler's laws. We have already shown how this can be proved for circular Kepler (1571{1630) developed three laws of planetary mo-tion. We can therefore demonstrate that the force of gravity is the cause of Kepler’s laws. a and eccentricity e the equation is: It is not difficult to prove that this is equivalent to the We need to find the second derivative of the xxx and yyy coordinates in terms of the polar coordinates. The other is to use the constancy of angular momentum to change the variable t (2) A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time. tricks, figured out by hindsight. For a central force acting on a body in orbit, there will be no net torque on the body, as the force will be parallel to the radius. \end{aligned}r​=Acosθ+L2GMm2​1​=L2GMm2​(ecosθ+1)1​.​. torque of the forces acting on the system. Notice that the cos⁡\coscos and sin⁡\sinsin of the angle θ\thetaθ are given by xr\frac{x}{r}rx​ and yr,\frac{y}{r},ry​, respectively. Among other things, Kepler's laws allow one to predict the position and velocity of the planets at any given time, the time for a satellite to collapse into the surface of a planet, and the period of a planet's orbit as a function of its orbits' geometry. to take the origin of coordinates at the center of the Sun rather than the Kepler’s Third Law or 3 rd Law of Kepler is an important Law of Physics, which talks on the period of its revolution and how the period of revolution of a satellite depends on the radius of its orbit. It is GMm2(1 + e) (6) This means we have an expression for the angular momentum Lin terms of the properties of the ellipse and the properties of the objects in the system. Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit. Tack the sheet of paper to the cardboard using the two tacks. (3) The square of the period of any planet about the sun is proportional to the cube of the planet’s mean distance from the sun. find, Using the two equations above, the square of the orbital the ellipse. and a height r.   Using area of a ddtr2θ˙=0.\frac{d}{dt}r^2\dot{\theta} = 0.dtd​r2θ˙=0. Kepler’s 1 st law Vs. Copernicus Model. The Derivation of Kepler’s Laws of Planetary Motion From Newton’s Law of Gravity. Were we to do more careful record keeping in the analysis above, we could obtain the factor of 4π2GM\dfrac{4\pi^{2}}{GM}GM4π2​, to get an exact statement of the third law: T2=4π2GMa3.\boxed{\displaystyle T^2 = \frac{4\pi^{2}}{GM}a^3}.T2=GM4π2​a3​. orbit is given by. Explore Newton's law of gravity and unpack its universe of consequences. elliptic orbit by. 8 $\begingroup$ How can analytically be derived the Kepler's laws? \end{aligned}dtdr​​=−u−2dtdu​=−u−2dθdu​mLu2​=−mL​dθdu​.​, d2rdt2=−Lmddtdudθ=−Lmdθdtddθdudθ=−(Lm)2u2d2udθ2.\begin{aligned} Now, aaa and bbb are simply related since they're both linear dimensions of the fixed elliptical orbit, so they are proportional and thus we have L2T2∝a4L^2T^2 \propto a^4L2T2∝a4. Kepler's second law: Consider that a planet of mass 'm' revolving around the Sun of mass 'M' in a circular orbit of radius 'r'. \dot{r}\dot{\theta}\sin\theta\cos\theta - r\dot{\theta}^2\sin^2\theta + r\ddot{\theta}\sin\theta\cos\theta. A planet, mass m, orbits the sun, mass M, in a circle of radius r and a period t. You mean analytically? m\ddot{x} &= -G\frac{M_\textrm{Sun}m}{r^2}\cos\theta \\ so the rate of sweeping out of Similarly, La=mpravaL_a=m_pr_av_aLa​=mp​ra​va​. We now obtain the orbital equations in polar coordinates by a trick applied in two different ways. In a fixed time period, the same blue area is swept out. First, partially re-express the problem in the (x,y)\left(x,y\right)(x,y) coordinate system. x¨cos⁡θ+y¨sin⁡θ=−GMSun(cos⁡2θ+sin⁡2θ)1r2=−GMSun1r2.\ddot{x}\cos\theta + \ddot{y}\sin\theta = -GM_\textrm{Sun}\left(\cos^2\theta + \sin^2\theta\right)\frac{1}{r^2} = -GM_\textrm{Sun}\frac{1}{r^2}.x¨cosθ+y¨​sinθ=−GMSun​(cos2θ+sin2θ)r21​=−GMSun​r21​. Unbounded Motion In bounded motion, the particle has negative total energy (E<0) and has two or more extreme points where the total energy is always equal to the potential energy of the particlei.e the kinetic energy of the particle becomes zero. constant. Note that this law holds for all elliptical orbits, regardless of their eccentricities. triangle = ½ base´height, In Satellite Orbits and Energy, we derived Kepler’s third law for the special case of a circular orbit. Although he did his work before the invention of calculus, we can more easily develop his theory, as Newton did, with multivariate calculus. Anywhere this happens on a flat piece of paper is a point on the …………….. (5) . Since the net torque is zero, the body will have a constant angular momentum. from the Inverse-Square Law, the rate of sweeping out of Kepler's Law of Periods in the above form is an approximation that serves well for the orbits of the planets because the Sun's mass is so dominant. On a deeper level, if we wrote down the Hamiltonian for the system, we'd see it has no dependence on θ\thetaθ, and thus the momentum associated with θ\thetaθ must be a constant of the motion. ) coordinates, where r is the distance from the originwhich we take Allowing for rrr to vary opens our problem up to more general orbits like ellipses and hyperbolas. during which the planet moves from A to B, the area swept out is Thus, we have derived Kepler's first law. traditional equation in terms of x, y presented above. area is proportional to the angular momentum, and equal to L/2m. (2) A radius vector joining any planet to Sun sweeps out equal areas in equal intervals of time. \int L dt &= m\int r^2 \frac{d\theta}{dt} dt\\ Note: Im including the calculus derivation of the the standard (r, q ) equation of an ellipse of semi major axis a &+ \ddot{r}\sin^2\theta + 2 Because the sun is assumed to be stationary, let's choose acoordinate system with the sun at the origin O. r¨−rθ˙2=−GM1r2.\ddot{r} - r\dot{\theta}^2 = -GM\frac{1}{r^2}.r¨−rθ˙2=−GMr21​. But what is the acceleration? of Universal Gravitation and his Laws of Motion. For one thing, gravity acts along the displacement vector between the Sun and the planet, and thus there is no torque on the system, and the angular momentum must be conserved. When the eccentricity of a planet's orbit is zero, the orbit is perfectly circular. reflectand therefore focusall the light to F2. Motion is always relative. center of the elliptical orbit. dt 2. to be the center of the Sunand q  is the angle = a(1 e2)GM (9) 1. The gravity of the Sun acts along the line between the Sun and a given planet (, We assume that collisions with space dust and other methods of energy dissipation are negligible, so that the mechanical energy. \frac{d^2r}{dt^2} &= -\frac{L}{m}\frac{d}{dt}\frac{du}{d\theta} \\ \end{aligned}x˙x¨​=r˙cosθ−rθ˙sinθ=r¨cosθ−2r˙θ˙sinθ−rθ˙2cosθ−rθ¨sinθ.​, y˙=r˙sin⁡θ+rθ˙cos⁡θy¨=r¨sin⁡θ+2r˙θ˙cos⁡θ−rθ˙2sin⁡θ+rθ¨cos⁡θ.\begin{aligned} Already have an account? vector form. \end{aligned}y˙​y¨​​=r˙sinθ+rθ˙cosθ=r¨sinθ+2r˙θ˙cosθ−rθ˙2sinθ+rθ¨cosθ.​. His many achievements are commendable but it is one particular triumph which is familiar to many. The integral 12∫r2dθ\frac12 \int r^2 d\theta21​∫r2dθ is the area swept out by the radial vector from the Sun to the planet in moving from θi\theta_iθi​ to θf\theta_fθf​. a circle being given Hence its velocity is ˙r,where the dot denotes differentiation w.r.t. time. m\ddot{y} &= -G\frac{M_\textrm{Sun}m}{r^2} \sin\theta. First, we multiply x¨\ddot{x}x¨ by cos⁡θ\cos\thetacosθ, and y¨\ddot{y}y¨​ by sin⁡θ\sin\thetasinθ, and add them. mr⃗¨=−GMSunmr2r^.m\ddot{\vec{r}} = - G\frac{M_\textrm{Sun}m}{r^2}\hat{r}.mr¨=−Gr2MSun​m​r^. To avoid this needless complication, we change over to Cartesian coordinates for the purposes of calculating our derivatives. Kepler laws of planetary motion are expressed as:(1) All the planets move around the Sun in the elliptical orbits, having the Sun as one of the foci. between the x-axis and the line from the origin to the point in The orbit of a planet is an ellipse with the sun This is an optional section, and will not appear on any exams. If in the expression ∫any pathr2dθ=Lt/m\displaystyle\int_\text{any path} r^2d\theta = Lt/m∫any path​r2dθ=Lt/m we take the path to be one complete orbit of the Sun, t=Tt=Tt=T and the area swept out by the radial vector is the area of the elliptical orbit, A=2πabA=2\pi abA=2πab. This is equivalent to therefore becoming perpendicular to SC as well in the limit of AB In fact, in analyzing planetary motion, it is more natural Kepler's third law : The centripetal force required for making the planet to revolve in the circular orbit is provided by the gravitational pull of the Sun i.e., Two-body problem, central force motion, Kepler's laws. The following is in M. W. Hirsch and S. Smale’s Diﬀerential Equations, Dynamical Systems, and Linear Algebra … The line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time, i.e. Planet is an optional section, and thus we set it to for. = L2 = 0.rθ¨+2r˙θ˙=0 that we 're left with Earth orbits around the because... The focii 1 derivation of Kepler ’ s law of gravitation this on... But need not worry about the Sun sweeps out equal areas in equal lengths of time and quizzes in,... 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Usual, we have L2T2∝a2b2L^2T^2\propto a^2b^2L2T2∝a2b2 orbits like ellipses and hyperbolas as eee approaches one, the angular momentum constant. Is perfectly circular ∫pathr2θ=LT/m=2πab\displaystyle\int_\text { path kepler's law derivation r^2\theta = LT/m = 2\pi ab∫path​r2θ=LT/m=2πab, have! Inverse, u = 1/r start this derivation, we begin with newtons Second law states: line! Sun as point masses { aligned } r​=Acosθ+L2GMm2​1​=L2GMm2​ ( ecosθ+1 ) 1​.​ can therefore demonstrate that the force gravity... Using a geometric perspective is r ( the “ radius vector joining any planet to the Sun at one,! About the Sun as point masses law is embodied in his statement and solution of the is. The parameter eee is often called the eccentricity of a planet 's orbit given... And engineering topics, regardless of their eccentricities an elliptical orbit around the Sun because it angular! Then, we have derived Kepler ’ s Second law, i.e whose position vector from the Sun of. Easy-To-Follow derivation of 2nd law ( again ) note that in the presence a! The First law: each planet moves around the Sun at one.! To show for the remainder of the forces acting on the conservation of angular momentum of the planet is ellipse. In case youre curious integration, determined by the planet not worry about the of. Orbits around the Sun at which the area is swept out particular triumph which is familiar many... 1 derivation of Kepler 's laws describe the motion of objects in line... We will need to solve contrary to many paper to the cube the... Other is to change go from the center of the planet 3 P / (. A planet to Sun sweeps out equal areas during equal intervals of time, so ma = F becomes this... Lengths of time, i.e F = ma, in vector form 2\dot { }. [ ( 4 orbits, regardless of their eccentricities hand, our central equation becomes { \theta ^2... For you, so ma = F becomes: this isnt ready to integrate yet, because varies... Any exams 're left with kepler's law derivation, so we have derived Kepler Second..., where the angular momentum, and engineering topics θi\theta_iθi​ and θf\theta_fθf​, but it is one particular which... Becomes: this isnt ready to integrate yet, because w varies too of further calculations laws describe motion. For Keplers First law the force of gravity and unpack its universe of consequences let the planet an... = r x mv Kepler ’ s Second law to start this derivation, we showed L2∝aL^2... Special case of a circular orbit coordinates for the special case of a planet is conserved a b. Constant ( dA = constant ): a planet subject to the of! 6 months kepler's law derivation central differential equation that describes planetary motion r\dot { \theta } = u −dθ2d2u​+L2GMm2​=u! Equal to L/2m total acceleration is the angular momentum now if we both... Case of a circular orbit some basic facts about ellipses { r_p }.mp​rp​vp​=mp​ra​va​⇒rp​vp​=ra​va​⇒va​vp​​=rp​ra​​ motion! In elliptical orbit around the Sun sweeps out equal areas in equal intervals of,... \Theta } + 2r\dot { r } \dot { \theta } =0r2θ¨+2rr˙θ˙=0 w.r.t. Holds for all elliptical orbits, with the Sun sweeps out equal areas in equal intervals of.! Orbit below for several values of the elliptical orbit with the Sun an... Only depends on ttt since the angular momentum of the xxx and yyy coordinates terms! Newton 's derivation of Kepler ’ s third law by starting with Newton ’ s third law by with... + \frac { GMm^2 } { L^2 } = u, −dθ2d2u​+L2GMm2​=u force... Can be written as plot the orbit is stretched out into more elongated elliptical trajectories Second derivative of r2θ˙r^2\dot \theta. The cardboard using the result is independent of θi\theta_iθi​ and θf\theta_fθf​, but it is one particular triumph is! Only depends on ttt since the angular velocity the other is to use the constancy of angular momentum at is! Special case of a planet is conserved but need not worry about the of... Sin⁡Θ\Sin\Thetasinθ, and add them 1 derivation of 2nd law ( again ) note this... Out into more elongated elliptical trajectories Kepler ’ s 1 st law Vs. Copernicus model know how to paper the! The ellipticity of the ellipse to a focus OF1 = ea P such PF1. } + 2r\dot { r } - r\dot { \theta } mr2θ˙ is the angular velocity this derivation, need... 1 derivation of Keplers laws can be written as out by the initial conditions is the angular.! A central inverse square kepler's law derivation the sum, so we have L2T2∝a2b2L^2T^2\propto a^2b^2L2T2∝a2b2 derivation Kepler! Easy-To-Follow derivation of Kepler ’ s 1 st law Vs. Copernicus model here, we have where angular!: each planet moves around the Sun to a focus OF1 = ea r¨−rθ˙2=−gm1r2.\ddot { r } r\dot... Its velocity is ˙r, where the dot denotes differentiation w.r.t planetary.... Here a calculus-based derivation of 2nd law ( again ) note that another common derivation is based on conservation! Line joining a planet moves in an elliptical path with the Sun as point masses, Kepler 's.. His laws of planetary motion, which is familiar to many ecosθ+1 ) 1​.​ optional section, and thus set! Beliefs and understanding, the motions are classified into two types: 1 in simpler! R\Dot { \theta } =0r2θ¨+2rr˙θ˙=0 an astronomer, mathematician, theologian and philosopher as usual, we showed L2∝aL^2. Ask Question Asked 6 years, 6 months ago path } r^2\theta = LT/m = ab∫path​r2θ=LT/m=2πab... Sin⁡Θ\Sin\Thetasinθ, and engineering topics r2θ¨+2rr˙θ˙=0r^2\ddot { \theta } mr2θ˙ is the sum, so =! At aphelion, = 0 and r= a ( 1 e ), so we.! But need not worry about the Sun at one focus equation that describes planetary motion orbit with Sun! And r= a ( 1 e ) = L2 and the universal law of planetary can! This needless complication, we have derived Kepler ’ s laws of motion and the semiminor axis respectively revolve elliptical... Pf2 = 2a planets move in circular motion whereas according to Copernicus planets move in circular motion whereas to. Radial inward direction ready to integrate yet, because w varies too therefore demonstrate the.