ISIMA lectures on celestial mechanics. 1 Scott Tremaine, Institute for Advanced Study July 2014 The roots of solar system dynamics can be traced to two fundamental discoveries by Isaac Newton: first, that the acceleration of a body of mass m subjected to a force F is d2r F = ; (1) dt2 m and second, that the gravitational force exerted by a point mass m2 at position r2 on a point mass m1 at r1 is G m1m2(r2 − r1) F = 3 : (2) jr2 − r1j Newton's laws have now been superseded by the equations of general relativity, but remain accurate enough to describe all planetary system phenomena when they are supple- mented by small relativistic corrections. For general background on this material look at any mechanics textbook. The standard advanced reference is Murray and Dermott, Solar System Dynamics. 1. The two-body problem The simplest significant problem in celestial mechanics, posed and solved by Newton but often known as the two-body problem (or Kepler's problem), is to determine the orbits of two particles of masses m1 and m2 under the influence of their mutual gravitational attraction. From (1) and (2) the equations of motion are 2 2 d r1 G m2(r2 − r1) d r2 G m1(r1 − r2) 2 = 3 ; 2 = 3 : (3) dt jr2 − r1j dt jr1 − r2j We change variables from r1 and r2 to m1r1 + m2r2 R ≡ ; r ≡ r2 − r1; (4) m1 + m2 R is the center of mass of the two particles and r is the relative position. These equations can be solved for r1 and r2 to yield m2 m1 r1 = R − r; r2 = R + r: (5) m1 + m2 m1 + m2 { 2 { Taking the second time derivative of the first of equations (4) and using equations (3), we obtain d2R = 0; (6) dt2 which implies that the center of mass travels at uniform velocity, a result that is a consequence of the absence of any forces from external sources. Taking the second derivative of the second of equations (4) yields d2r G M = − r; (7) dt2 r3 where r = jrj and we have introduced M ≡ m1 + m2 to denote the total mass of the two particles. Equation (7) equally well describes the motion of a particle of negligible mass (a test particle) subject to the gravitational attraction of a fixed point mass M. Equation (7) can also be written as d2r = −rΦ ; (8) dt2 K where ΦK (r) = −G M=r is the Kepler potential. The solution of equation (7) is known as the Kepler orbit. The angular momentum per unit mass is dr L = r × : (9) dt Then dL dr dr d2r G M = × + r × = − r × ^r = 0: (10) dt dt dt dt2 r2 Thus the angular momentum is conserved; moreover, since the angular momentum vector is normal to the plane containing the test particle's position and velocity vectors, the test particle must be confined to a plane (the orbital plane). 1.1. The shape of the Kepler orbit We let (r; ) denote polar coordinates in the orbital plane, with increasing in the direction of motion of the orbit; writing r = r^r and using the relations d^r=dt = _ ^ and d ^ =dt = − _^r, the equation of motion (7) becomes dΦ (r) r¨ − r _ 2 = − K ; 2r _ _ + r ¨ = 0: (11) dr { 3 { The second equation may be multiplied by r and integrated to yield r2 _ = constant = L; (12) where L = jLj. This is just a restatement of the conservation of angular momentum. We may use equation (12) to eliminate _ from the first of equations (11), L2 dΦ r¨ − = − K : (13) r3 dr Multiplying byr _ and integrating yields L2 1 r_2 + + Φ (r) = E; (14) 2 2r2 K where E is a constant that is equal to the energy per unit mass of the test particle. We can also use equation (12) to write d d L d = _ = ; (15) dt d r2 d applying this relation in equation (14) yields L2 dr 2 L2 + + Φ (r) = E: (16) 2r4 d 2r2 K Equation (14) implies that 2G M L2 r_2 = 2E + − : (17) r r2 As r ! 0, the right side approaches −L2=r2 which is negative, while the left side is positive. Thus there must be a point of closest approach of the test particle to the central body, which is known as the periapsis or pericenter1. The periapsis distance q is given implicitly by the conditionr _ = 0, which yields the quadratic equation 2G M L2 2E + − = 0: (18) q q2 In the opposite limit, r ! 1, the right side of equation (17) approaches 2E. Since the left side is positive, when E < 0 there is a maximum distance Q that the particle can achieve, which is known as the apoapsis or apocenter, and is determined by solving equation (18) 1For specific central bodies other names are used, such as perihelion (Sun), perigee (Earth), perijove (Jupiter), periastron (a star), etc. \Periapse" is incorrect|an apse is not an apsis. { 4 { with Q replacing q. Orbits with E < 0 are referred to as bound orbits since there is an upper limit to their distance from the central body. Particles on unbound orbits, with E ≥ 0, have no apoapsis and eventually travel arbitrarily far from the central body. To solve equation (13) we introduce the variable u ≡ 1=r, and change the independent variable from t to using the relation (15). With these substitutions,r ¨ = −L2u2d2u=d 2, so equation (13) becomes d2u 1 dΦ + u = − K : (19) d 2 L2 du 2 Since ΦK = −G M=r = −G Mu, the right side is equal to a constant, G M=L , and the equation is easily solved to yield 1 G M u = = [1 + e cos( − $)]; (20) r L2 where e ≥ 0 and $ are constants of integration. We replace the angular momentum L by another constant of integration, a, defined by the relation2 L2 = G Ma(1 − e2); (21) so that the shape of the orbit is given by a(1 − e2) r = ; (22) 1 + e cos f where f = − $ is known as the true anomaly3. The closest approach of the two bodies occurs at azimuth = $ and hence $ is known as the azimuth of periapsis. The periapsis distance is q = a(1 − e): (23) Substituting this result and equation (21) into (18) reveals that the energy is simply related to the constant a: G M E = − : (24) 2a Bound orbits have E < 0, so that a > 0 by equation (24) and hence e < 1 by equation (21). The apoapsis distance is obtained from equation (22) with f = π and is given by Q = a(1 + e): (25) 2The combination a(1 − e2) is sometimes called the semilatus rectum. 3The term \anomaly" is used to refer to any angular variable that is zero at periapsis and increases by 2π as the particle travels from periapsis to apoapsis and back. { 5 { The azimuth of the apocenter is = $ + π; thus the periapsis and the apoapsis are joined by a straight line known as the line of apsides. Equation (22) describes an ellipse with one focus at the origin. Its major axis is the line of apsides and has length q + Q = 2a; thus the constant a is known as the semi-major axis. The semi-minor axis of the ellipse is easily determined as the maximum perpendicular distance of the orbit from the line of 2 2 1=2 apsides, b = maxf [a(1 − e ) sin f=(1 + e cos f)] = a(1 − e ) . The eccentricity of the ellipse, (1 − b2=a2)1=2, is therefore equal to the constant e. The radial velocity is found by differentiating equation (22) and using the relation f_ = L=r2 (eq. 15): (G M)1=2e sin f r_ = : (26) [a(1 − e2)]1=2 Unbound orbits have E > 0, a < 0 and e > 1. In this case equation (22) describes a hyperbola with focus at the origin and asymptotes at azimuth = $ ± cos−1(−1=e). The constants a and e are still commonly referred to as semi-major axis and eccentricity even though these terms have no direct geometric interpretation. In the special case E = 0, a is infinite and e = 1, so equation (22) is undefined; however, in this case equation (18) implies that the periapsis distance q = L2=(2G M) so that equation (20) implies 2q r = ; (27) 1 + cos f which describes a parabola. 1.2. Motion in the Kepler orbit The period P of a bound orbit is the time taken to travel from periapsis to apoapsis and back. Since d =dt = L=r2, we have R t2 dt = L−1 R 2 r2d ; the integral on the right side t1 1 is twice the area contained in the ellipse between azimuths 1 and 2, which leads to the simple result that P = 2A=L where A is the area of the ellipse. Combining the geometrical result that the area is A = πab = πa2(1 − e2)1=2 with equation (21) we obtain a3 1=2 P = 2π ; (28) G M the period, like the energy, depends only on the semi-major axis. The mean motion or mean rate of change of azimuth, usually written n and equal to 2π=P , thus satisfies n2a3 = G M; (29) { 6 { which is Kepler's third law. If the particle passes through periapsis at t = t0, the dimen- sionless variable t − t ` = 2π 0 = n(t − t ) (30) P 0 is referred to as the mean anomaly. The position and velocity of a particle in the orbital plane at a given time t are specified by four orbital elements: two specify the size and shape of the orbit, which we can take to be e and a (or e and n, q and Q, L and E, etc.); one specifies the orientation of the orbit ($); and one specifies the location or phase of the particle in its orbit (f, `, or t0).