EPER S AWS OF PANETARY MOTION 1. Intoduction We ae now in a position to apply what we have leaned about the coss poduct and vecto valued functions to deive eple s aws of planetay motion. These laws wee empiically detemined in the ealy 1600 s by the Geman mathematician and astonome Johannes eple, based on lage amounts of expeimental data that had been collected in the pevious centuy by the Danish astonome Tycho Bahe. The thee laws ae as follows: (1) A planet evolves aound the sun in an elliptical obit with the sun at one focus. () The line joining the sun to a planet sweeps out equal aeas in equal times. (3) The squae of the peiod of evolution of a planet is popotional to the cube of the length of the semi-majo axis of its obit. In 1687 Si Isaac Newton showed that these laws follow fom his second law of motion, F = m a, and his aw of Univesal Gavitation: The gavitational foce between two point paticles is attactive, and popotional to the poduct of thei masses and invesely popotional to the squae of the distance between them. If we put the sun at the oigin of ou coodinate system, and denote by the vecto fom the sun to the planet in question, then the gavitational foce F on the planet is: (1.1) F GMm = ˆ whee = and ˆ = is the unit vecto in the diection. The masses of the 11 m3 two bodies ae M and m, and G = 6.67 10 kg s is the univeal constant of gavitation. The minus sign indicates that the foce is attactive. We will follow Newton (with moden notation) and deive eple s thee laws fom F = m a and equation 1.1.. The Geomety of the Ellipse Thee ae two useful ways to descibe an ellipse in R. It can be descibed as the set of points (x,y) which satisfy: x (.1) a + y b = 1. This ellipse is cented at the oigin and passes though the points (±a,0) and (0, ±b). The length of the semi-majo axis is a and that of the semi-mino axis is b. (We can assume that a > b without loss of geneality.) See Figue. The scala c = a b is called the focal length of the ellipse. This gives the two focal points of the ellipse as (±c,0). You may have also seen the definition of an ellipse as the set of points in the plane fo which the sum of the distances to the two focal points is constant. To see that this is equivalent, and to help deive 1
EPER S AWS OF PANETARY MOTION y Ellipse x y + = 1 a b (c,b) a b ( a,0) c (c,0) (a + c,0) x c = a b a = = 1 e cos( ) b = Figue 1. An Ellipse c = e eple s aws, we need to descibe an ellipse in pola coodinates. Ou claim is that the pola equation (.) = 1 ecos(θ) descibes an ellipse cented on the x-axis but with the oigin at one focus. Hee is a positive constant, and 0 e < 1, whee e is the eccenticity of the ellipse. If e = 0, then = and the equation descibes a cicle of adius cented at the oigin. The maximum value of occus when θ = 0: max = 1 e > and the minimum value coesponds to θ = π: min = 1+e <. In the case of planetay motion about the sun, the point in the obit of minimum distance (closest appoach) is called peihelion and the point of maximum distance is called aphelion. Twice the length of the semi-majo axis is the sum of these two lengths: This gives us a = min + max = (.3) a =
EPER S AWS OF PANETARY MOTION 3 as the length of the semi-majo axis, and this gets vey lage when e appoaches 1. The cente of the ellipse is at (x 0,0) whee (.4) x 0 = a min = e = ea When we ae at the top vetex of the ellipse, cos(θ) = x0. We can eaange equation. to = + ex 0 = 1 e using equation.4. Now the Pythagoean theoem tells us that the length of the semi-mino axis, b, is given by: (.5) b = x 0 = 1 e You can check that this desciption is indeed an ellipse cented at (x 0,0) by algebaically tansfoming equation. to: ( ) x e (1 e ) y ( ) + ( ) = 1 (1 e ) 1 e and compaing with equation.1. It emains to pove the focal popety of the ellipse: One focus is at (0,0) and the othe is at (x 0,0) and we need to show that the sum of the distances fom any point on the ellipse to these two focal points is constant. et P be a point on the ellipse, and let be the distance to the oigin and s the distance to the second focus (x 0,0). Then the law of cosines gives: s = + 4x 0 4x 0 cos(θ) Fom equation., we get cos(θ) = e, and using also equation.4, some manipulation yields: s = ( a) Since < a we need to take the negative squae oot: s = a. This gives + s = a = constant as claimed. 3. Some Useful Notation in Pola Coodinates With ou pola coodinate desciption of ellipses in hand, we still need some moe notation befoe we can tackle eple s aws. We denote by the vecto fom the oigin (which will be the sun) to the planet. This has pola coodinates (,θ), so = and the unit vecto ˆ in the diection is: (3.1) ˆ = = cos(θ)î + sin(θ)ĵ Fom this we can constuct anothe unit vecto ˆθ, pependicula to ˆ and in the diection detemined by the equiement that ˆ ˆθ = ˆk: (3.) ˆθ = sin(θ)î + cos(θ)ĵ As the planet moves aound the sun, its pola coodinates ((t),θ(t)) will be functions of t. If we let a dot denote diffeentiation with espect to t, then fom equations 3.1 and 3., we obtain (emembeing θ is a function of t): (3.3) ˆ = sin(θ) θî + cos(θ) θĵ = θˆθ
4 EPER S AWS OF PANETARY MOTION and the elation dˆθ (3.4) = cos(θ)î sin(θ)ĵ = ˆ dθ Finally, we have = ˆ and upon diffeentiation and using the poduct ule and equation 3.3, (3.5) v = = ṙˆ + ˆ = ṙˆ + θˆθ and fom ˆ ˆθ = 0 and v = v = v v, we have: (3.6) v = ṙ + θ We ae now eady to deive eple s aws. 4. Consevation of Angula Momentum Choose a coodinate system so that the sun is at the oigin, with mass M, and the planet we ae studying has mass m. Since the sun contains 99% of the mass of the sola system, it is a easonable assumption to ignoe the gavitational effects of othe planets. The planet has position vecto, and we define the angula momentum vecto to be: (4.1) = m v If we diffeentiate this equation using the poduct ule we get: d (4.) dt = m v + m v = 0 whee we have used = v and a = v is a multiple of fom equation 1.1. This says that is a constant vecto, o the angula momentum is conseved. Since by its definition, is always pependicula to, the motion of the planet is confined to a plane pependicula to. We can choose ou coodinate system so that = ˆk, whee =. Now using equation 3.5, we have: ( ) (4.3) ˆk = m v = m ṙˆ ˆ + θˆ ˆθ = m θˆk and so we have the useful elation = m θ. 5. Consevation of Enegy The potential enegy U of a planet in the gavitational field of the sun is: U = GMm and the kinetic enegy is 1 m v. This gives us the total enegy E: (5.1) E = 1 mv GMm Diffeentiating this equation with espect to t, de dt = m + GMm d dt But =, so d (5.) dt =. This gives ( de dt = m + GMm ˆ ) = 0
EPER S AWS OF PANETARY MOTION 5 because the tem in paentheses is zeo by equation 1.1 and F = m. So the total enegy E is conseved. Since the kinetic enegy is always non-negative, we have the inequality GMm E Fom this equation we see that if the total enegy E < 0, we can eaange this inequality to GMm E which says that the obit is bounded. This is the case fo planets, which have elliptical obits. It is also possible fo an object to have E = 0 o E > 0, in which case the obits ae paabolas o hypebolas, espectively. The object is not in a closed obit, it just passes aound the sun and leaves the sola system pemanently. Hypebolic obits ae typical if the incoming object has a high enough velocity to escape the sun s gavitational pull. Fom equation 3.6 and = m θ, we have the following expession fo the enegy E: E = 1 ( m ṙ + m GMm ) We can use methods of single vaiable calculus to find, as a function of, the minimum value fo the tems in paentheses. And since the fist tem in the expession fo E is always non-negative, it can be shown that: E G M m 3 This can be eaanged to the inequality (5.3) e = 1 + E G M m 3 0 which seves to define the nonnegative dimensionless scala e. Finally, in tems of e, the enegy becomes: E = e 1 G M m 3 So bounded obits coespond to negative enegy which coespond to e < 1. 6. eple s aws The aea A(t) swept out by the line fom the sun to the planet, fom the angle θ = θ 0 to θ = θ(t) is: θ(t) 1 (6.1) A(t) = θ 0 dθ This follows fom appoximating a vey small ac of length θ by an isosceles tiangle of side lengths (θ), which has aea 1 θ. Now diffeentiating and using the fundamental theoem of calculus and equation 4.3, we have: d da dθ (6.) A(t) = dt dθ dt = 1 θ = m Thus the aea is swept out at a constant ate, which is eple s second law.
6 EPER S AWS OF PANETARY MOTION (6.3) Fom the chain ule, a = v = d v dθ θ we obtain, using equations 1.1 and 4.3: GM d v dθ = ˆ m Now by compaing with equation 3.4, we see: = GMm ˆ (6.4) v = GMm ˆθ + C fo some constant vecto C. Befoe continuing, we notice that since ˆθ is a unit vecto, we have v(t) C = GMm fo all time t. This is sometimes called Hamilton s theoem. It says that as the planet moves aound in its obit, its velocity vecto taces out a cicle of adius R = GMm and cente C if we think of v(t) as having its tail at the oigin. At this point, we have chosen ou oigin and ou ˆk diection, but we ae still fee to oient the î and ĵ axes. et us choose these axes, and the scala e, so that C = egmm ĵ The scala e is detemined by the velocity v 0 of the planet at the point in its obit coesponding to θ = 0: (1 e)gmm v 0 = ĵ Note that fo the planet to evolve aound the sun counteclockwise, e 1 and e = 1 coesponds to v 0 = 0, which is the limiting case when the planet is infinitely fa away fom the sun at θ = 0 and has zeo speed. This is a paabolic obit. The bounded obits occu fo e < 1, and a moe caeful analysis can be done to show that this is exactly the same paamete as in equation 5.3, which we call the eccenticity of the obit. Now substituting equation 6.4 into equation 4.1, we have: (6.5) ˆk = GMm ( ) ˆ (ˆθ eĵ) = GMm (1 ecos(θ))ˆk which can be eaanged to yield (6.6) = GMm 1 ecos(θ) = 1 ecos(θ) which is an ellipse with constant = GMm, since 0 e < 1. This is eple s fist law. Now fom the second law, the aea is swept out at a constant ate by the obit, so the total aea of the ellipse is equal to the ate multiplied by the peiod T, which fom equation 6. is: ( ) Aea = T = πab m since the aea of an ellipse with semi-majo axis a and semi-mino axis b is πab. Squaing this equation, T = 4m π a b
EPER S AWS OF PANETARY MOTION 7 But fom equations.3 and.5, the semi-mino axis b is equal to Thus we have: b = = a (6.7) T = 4m π a b = 4m π a GMm a = 4π GM a3 Thus we have finally aived at eple s thid law: T = 4π GM a3. The squae of the peiod of the planet s obit is popotional to the cube of the semi-majo axis, whee the constant of popotionality is independent of the mass m of the planet: it depends only on the mass M of the sun and on the gavitational constant G.