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Understanding Kepler's Laws and Orbital Dynamics

Dive into Kepler's laws and orbital dynamics through an exploration of the Lagrangian, energy conservation, eccentricity, conic sections, apsidal positions, angular momentum, and effective potential. Discover the intricacies of orbital motion and their mathematical foundations in this comprehensive guide.

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Understanding Kepler's Laws and Orbital Dynamics

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  1. Kepler

  2. Force can be derived from a potential. k < 0 for attractive force Choose constant of integration so V() = 0. Inverse Square Force F2int m2 r = r1 – r2 m1 R r2 F1int r1

  3. The Lagrangian can be expressed in polar coordinates. L is independent of time. The total energy is a constant of the motion. Orbit is symmetrical about an apse. Kepler Lagrangian

  4. The right side of the orbit equation is constant. Equation is integrable Integration constants: e, q0 e related to initial energy Phase angle corresponds to orientation. The substitution can be reversed to get polar or Cartesian coordinates. Kepler Orbits

  5. The orbit equation describes a conic section. q0 init orientation (set to 0) s is the directrix. The constant e is the eccentricity. sets the shape e < 1 ellipse e =1 parabola e >1 hyperbola Conic Sections r q s focus

  6. Elliptical orbits have stable apses. Kepler’s first law Minimum and maximum values of r Other orbits only have a minimum The energy is related to e: Set r = r2, no velocity Apsidal Position r r1 q r2 s

  7. The change in area between orbit and focus is dA/dt Related to angular velocity The change is constant due to constant angular momentum. This is Kepler’s 2nd law Angular Momentum dr r

  8. The area for the whole ellipse relates to the period. semimajor axis: a=(r1+r2)/2. This is Kepler’s 3rd law. Relation holds for all orbits Constant depends on m, k Period and Ellipse r r1 q r2 s

  9. Effective Potential • Treat problem as a one dimension only. • Just radial r term. • Minimum in potential implies bounded orbits. • For k > 0, no minimum • For E > 0, unbounded Veff Veff r r 0 0 possibly bounded unbounded

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