Keplerian Orbital Motion PDF

1 MOTION IN A CENTRAL FIELD. CONIC SECTIONS 1.1 POLAR COORDINATES We can determine the instantaneous position of an object (i.e., our planet) in the x − y plane in terms of standard Cartesian coordinates, (x, y), or polar coordinates, (r, θ), as illustrated in Figure (1.1). Here, r = x...

1 MOTION IN A CENTRAL FIELD. CONIC SECTIONS 1.1 POLAR COORDINATES We can determine the instantaneous position of an object (i.e., our planet) in the x − y plane in terms of standard Cartesian coordinates, (x, y), or polar coordinates, (r, θ), as illustrated in Figure (1.1). Here, r = x + y and θ = tan (y/x). p 2 2 −1 It is helpful to dene two unit vectors, e ≡ r/r and e = e × e , at the instantaneous position of the planet. The rst always points radially away from the origin, whereas the second is normal to the rst, r θ z r e ⊥e , in the direction of increasing θ. As is easily demonstrated, the Cartesian components of e and e are, respectively, θ r r θ e = (cosθ, sinθ) (1.1) e = (−sinθ, cosθ) (1.2) r θ Figure 1.1: Polar coordinates We can write the position vector of our planet as r = rer (1.3) 1 Thus, the planet's velocity becomes dr = ṙe + rė v= dt r r (1.4) where the dot is shorthand for d/dt. Note that e has a non-zero time-derivative (unlike a Cartesian unit vec- tor) because its direction changes as the planet moves around. As is easily demonstrated, from dierentiating r Equation (1.1) with respect to time, ė = θ̇ (−sinθ, cosθ) = θ̇e r θ (1.5) Thus, v = ṙe + rθ̇e r θ (1.6) Now, the planet's acceleration is written dv = r̈e + ṙė + ṙθ̇ + rθ̈ e + rθ̇ė (1.7) a= dt r r θ θ Again, e has a non-zero time-derivative because its direction changes as the planet moves around. Dieren- tiation of Equation (1.2) with respect to time yields θ θė = θ̇ (−cosθ, −sinθ) = −θ̇e r (1.8) Hence, (1.9)     2 a = r̈ − rθ̇ e + rθ̈ + 2ṙθ̇ e r θ It follows that the equation of motion of our planet, d r = − GM r, 2 (1.10) dt 2 r 3 can be written as GM (1.11)     2 a = r̈ − rθ̇ e + rθ̈ + 2ṙθ̇ e = − r eθ 2 r r Since e and e are mutually orthogonal, we can separately equate the coecients of both, in the above equation, to give a radial equation of motion, r θ r̈ − rθ̇ = − 2 GM r , 2 (1.12) and a tangential equation of motion, rθ̈ + 2ṙθ̇ = 0, (1.13) Multiplying our planet's tangential equation of motion, (1.13), by r, we obtain 2 r θ̈ + 2rṙθ̇ = 0. (1.14) However, the above equation can be also written d r θ̇   (1.15) 2 dt = 0, 2  Figure 1.2: Conic sections which implies that r θ̇ is constant in time. On the other hand, the angular momentum of our planet, is given 2 by h = r × v, (1.16) which means that 2 h = r θ̇ (1.17) is the magnitude of the vector h dened in Equation (1.16). Thus, the fact that h is constant in time is equivalent to the statement that the angular momentum of our planet is a constant of its motion. This is the case because gravity is a central force. 1.2 CONIC SECTIONS The ellipse, the parabola, and the hyperbola are collectively known as conic sections, since these three types of curve can be obtained by taking various dierent plane sections of a right cone. An ellipse, centered on the origin, of major radius a and minor radius b, which are aligned along the x− and y−axes, respectively (see Figure 1.3), satises the following well- known equation: (1.18) 2 2 x y 2 + 2 = 1. a b Likewise, a parabola which is aligned along the +x-axis, and passes through the origin (see Figure 1.4a), satises: 2 y − bx = 0, where b > 0. (1.19) 3  Figure 1.3: An ellipse (a) A parabola (b) A hyperbola Figure 1.4 Finally, a hyperbola which is aligned along the +x-axis, and whose asymptotes intersect at the origin (see Figure 1.4b), satises: (1.20) 2 2 x y − 2 = 1. 2 a b Here, a is the distance of closest approach to the origin. The asymptotes subtend an angle ϕ = tan (b/a) −1 with the x-axis. It is not obvious, from the above formulae, what the ellipse, the parabola, and the hyperbola have in common. It turns out, in fact, that these three curves can all be represented as the locus of a movable point whose distance from a xed point is in a constant ratio to its perpendicular distance to some xed straight-line. Let the xed pointwhich is termed the focus of the ellipse/parabola/hyperbolalie at the origin, and let the xed linewhich is termed the directrixcorrespond to x = −d (with d > 0). Thus, the distance of a general point (x, y) (which lies to the right of the directrix) from the focus is 1 p r = x +y ,2 2 (1.21) 4 whereas the perpendicular distance of the point from the directrix is r2 = x + d (1.22) see Figure (1.5). In polar coordinates, we have r1 = r, cosθ + d. r2 = r (1.23) Hence, the locus of a point, for which r and r are in a xed ratio, satises the following equation: 1 2 Figure 1.5: Conic sections in polar coordinates (1.24) p r1 x2 + y 2 r r2 = x+d = r cosθ + d = e, where e ≥ 0 is a constant called eccentricity. When expressed in terms of polar coordinates, the above equation can be rearranged to give r= r 1 − e cosθ , c (1.25) where r = e d. In the case of an ellipse, each of the two lines parallel to the minor axis, and at a distance of d from it, is called a directrix of the ellipse, with c d= a2 c=. a e (1.26) For an arbitrary point P of the ellipse, the quotient of the distance to one focus and to the corresponding directrix (see Fig. (1.7) ) is equal to the eccentricity: |P F | 1 |P l | 1 = |P F | |P l | 2 =e=. 2 c a (1.27) 5 Figure 1.6: Ellipse parameters: a: semi-major axis, b: semi-minor axis, c: linear eccentricity, p: semi-latus rectum (usually ℓ). On the other hand, when written in terms of Cartesian coordinates, (1.24) can be rearranged to give, for e 1, (1.34) 2 2 (x − x )c y 2 − = 1, 2 a b 6  Figure 1.7: Ellipse: directrix property with a= rc e2 −1 , (1.35) b= √ rc e2 − 1 p = a e2 − 1, (1.36) e rc xc = 2 e −1 = e a. (1.37) Equation (1.34) can be recognized as the equation of a hyperbola whose asymptotes intersect at (x , 0), and which is aligned along the +x−direction. The asymptotes subtend an angle with the x−axis [cf., Equation c (1.20)] such that ϕ = tan = tan (1.38)   −1 b p −1 2  e −1. a In conclusion, Equation ((1.25)) is the polar equation of a general conic section which is confocal with the origin (i.e., the origin lies at the focus). For e < 1, the conic section is an ellipse. For e = 1, the conic section is a parabola. Finally, for e > 1, the conic section is a hyperbola. 1.3 MOTION IN A GENERAL CENTRAL FORCE-FIELD This section examines the motion of a celestial body in a general central potential (i.e., a potential that is a function of r, but does not necessarily vary as 1/r). Consider the motion of an object in a general (attractive) central force-eld characterized by the potential function V (r). Since the force-eld is central, it still remains true that h = r θ̇ 2 (1.39) is a constant of the motion. Our planet's radial equation of motion, (1.12), can be combined with Equation (1.39) to give (1.40) 2 h GM r̈ − 3 =−. 2 r r 7 Suppose that r = u. It follows that −1 u̇ u θ u d d d (1.41) ṙ = − u2 = −r2 θ t = −h. θ d d d Likewise, d u θ̇ = −u h d u. 2 2 (1.42) dθ dθ 2 2 r̈ = −h 2 2 Hence, Equation (1.40) can be written in the linear form, where u = r. −1 d u + u = GM. 2 (1.43) dθ 2h 2 The general solution to the above equation is u(θ) = GM h2 [1 − e cos(θ − θ0 )] , (1.44) where e and θ are arbitrary constants. Without loss of generality, we can set θ = 0 by rotating our coordinate system about the z−axis. Thus, we obtain 0 0 r(θ) = r 1 − e cos θ , c (1.45) where (1.46) 2 h r = c. GM We immediately recognize Equation (1.45) as the equation of a conic section that is con- focal with the origin (i.e., with the Sun). Specically, for e < 1, (1.45) is the equation of an ellipse. For e = 1, (1.45) is the equation of a parabola. Finally, for e > 1, (1.45) is the equation of a hyperbola. However, a planet cannot have a parabolic or a hyperbolic orbit, since such orbits are only appropriate to objects which are ultimately able to escape from the Sun's gravitational eld. Thus, the orbit of our planet around the Sun is a confocal ellipsethis is Kepler's rst law of planetary motion. On the other hand, gravity is a conservative force, which means that it can be derived from a potential energy, U(r), f ≡ −∇U(r), (1.47) where U(r), in the Sun's gravitational eld, takes the form U(r) = − GMm r. (1.48) Thus, Eq. (1.43) generalizes to d u + u = − 1 dV. 2 (1.49) dθ 2 h du 2 where V (r) is the gravitational potential at the location (r). In other words, it is the gravitational potential energy (U ) at that location per unit mass, V (r) = − GM r. (1.50) 8 1.3.1 SPIRAL ORBIT Suppose, for instance, that we wish to nd the potential V (r) that causes an object to execute the spiral orbit r=r θ. 0 2 (1.51) Substitution of u = (r θ ) into Equation (1.49) yields 0 2 −1 dV = −h 6r u + u
. (1.52) du 2 2 0 Integrating, we obtain (1.53) 2   2 u 3 V (u) = −h 2r u + 0 , or 2 (1.54)   2 2r 1 0 V (r) = −h + 3. 2 r 2r In other words, the spiral pattern (1.51) is obtained from a mixture of an inverse-square and inverse-cube potential. 2 Conservation Laws As we have already seen, gravity is a conservative force. Hence, the gravitational force f =− GMm r3 r, (2.1) can be written f = −∇U(r), (2.2) where the potential energy, U(r), of our planet in the Sun's gravitational eld takes the form U(r) = − GMm r. (2.3) It follows that the total energy of our planet is a conserved quantity. In other words, E= v2 2 − GM r , (2.4) is constant in time. Here, E is actually the planet's total energy per unit mass, and v = dr/dt. Gravity is also a central force. Hence, the angular momentum of our planet is a con- served quantity. In other words, h = r × v, (2.5) which is actually the planet's angular momentum per unit mass, is constant in time. Taking the scalar product of the above equation with r, we obtain h · r = 0. (2.6) This is the equation of a plane which passes through the origin, and whose normal is parallel to h. Since h is a constant vector, it always points in the same direction. We, therefore, conclude that the motion of our planet is two-dimensional in nature: i.e., it is conned to some xed plane which passes through the origin. Without loss of generality, we can let this plane coincide with the x − y plane. 9 3 Orbital Energies Let us now generalize our analysis to take into account the orbits of asteroids and comets about the Sun. Such orbits satisfy Eq. (1.45), r= r 1 − e cosθ , c (3.1) but can be parabolic (e = 1), or even hyperbolic (e > 1), as well as elliptical (e < 1). For an elliptical orbit the closest distance to the Sunthe so-called perihelion distanceis [see Equations (1.29) and (3.1)] r = p r 1+e c = a(1 − e). (3.2) Thus, from Eqs. (1.46) and (3.2), we can write (3.3) q h = GMr (1 + e). p According to Equations (2.4) and (1.6), the total energy per unit mass of an object in a general orbit around the Sun is given by (3.4) 2 2 2 ṙ + r θ̇ GM E= −. 2 r It follows from Equations (1.17), (1.41), and (1.46) that (3.5) " 2 # h2 du 2 E= + u − 2uuc , 2 dθ where u = r , and u −1 c = rc−1. However, according to Equation (3.1), u(θ) = uc (1 − e cos θ). (3.6) The previous two equations can be combined with Equations (1.46) and (3.2) to give E= u2c h2 2 2
GM e −1 = 2rp (e − 1). (3.7) We conclude that elliptical orbits (e < 1) have negative total energies, whereas parabolic orbits (e = 1) have zero total energies, and hyperbolic orbits (e > 1) have positive total energies. This makes sense, since in a conservative system in which the potential energy at innity is set to zero, we expect bounded orbits to have negative total energies, and unbounded orbits to have positive total energies Thus, elliptical orbits, which are clearly bounded, should indeed have negative total energies, whereas hyperbolic orbits, which are clearly unbounded, should indeed have positive total energies. Parabolic orbits are marginally bounded (i.e., an object executing a parabolic orbit only just escapes from the Sun's gravitational eld), and thus have zero total energy. For the special case of an elliptical orbit, whose major radius ”a” is nite, we can write E =− GM 2a. (3.8) It follows that the energy of such an orbit is completely determined by the orbital major radius. 10