Celestial Mechanics IV Central orbits Force from shape

Celestial Mechanics IV Central orbits Force from shape, shape from force General relativity correction

Conservative central force field • Conservative: Work done by force independent of path • Central: Force directed towards a fixed point • Can be described by a potential V(r) – i. e. , potential energy per unit mass Force per unit mass: Equation of motion:

Integrals of motion • Scalar multiplication of the equation of motion by the velocity and vector multiplication by the radius vector leads to integrals of energy and angular momentum

Clairaut’s equation • Based on energy and angular momentum conservation • Given an orbital shape r( ), energy E and angular momentum h, Clairaut’s equation yields the potential V(r) and hence, force Fc(r)

Binet’s equation We make the variable substitution: Look for the dependence of u on the angular variable :

Binet’s equation, ctd Acceleration vector in polar coordinates: The equation of motion then yields: The term Binet’s equation in principle refers to this equation for the particular Keplerian case but can be used more generally

Summary Finding the force field from the shape of the orbit (Clairaut) Finding the shape of the orbit from the force field (Binet)

Orbits in central force fields • Due to Clairaut’s equation, we must have: • For negative energy (bound orbits) the general shape is a rosette orbit oscillating between rmin and rmax at a frequency noncommensurable with the circulation • The angular momentum prevents the object from reaching r 0, unless with n > 2

Orbits in central force fields, ctd Central point mass Stellar cluster

Special cases I: Keplerian orbits Central point mass with GM= : Analytic solution: Inserting this into Binet’s equation gives:

Keplerian orbits, ctd With: we get: i. e. , an ellipse (or parabola, or hyperbola) with the Sun at a focal point, for the case of planetary motion. Proof of Kepler I

Special cases II: Cotes spirals This is the limiting case with V(r) proportional to r-2 Inserting this into Binet’s equation gives: The solutions have the shape of spiral curves that depend on the parameter:

Cotes spirals, ctd B=0 B=1/9 B=1/100 B=– 9/4

Force from shape, Keplerian case Insert into Clairaut’s equation:

Force from shape, ctd By requiring that V(r) 0 as r , we get: By putting h 2/p = : Only the inverse-square force law is consistent with orbits in the shape of conic sections!

The orbit of Mercury • Planetary perturbations make ω time-dependent. • Apsis line moves 1000” per century (0. 3º) • Perturbations can only explain 97% of the observed shift! • There is an inexplicable residual of 43” per century. Urbain Jean Joseph Leverrier (1811 -1877)

The General Theory of Relativity • Introduces a novel treatment of space, time, and gravity • Relativistic version of Binet’s formula predicts a new orbital geometry (ellipse with periapsis shift) • Predicted rate of periapsis shift identical to residual between observations and Newtonian theory

Schwarzschild metric Line element in Euclidean 3 D space Line element in Euclidean 4 D space, empty “spacetime”: Line element in 4 D spacetime, containing one massive body in origo: Schwarzschild metric

Relativistic orbital motion • Light and bodies move on geodetics in the spacetime Euler-Lagrange equation • The 3 D “shadow” of a geodetic in 4 D is the orbit • We need an equation for “straight lines” in The metric tensor g describes spacetime! curvature, contains the Schwarzschild metric coefficients

Solving the Euler-Lagrange equation Relativistic version of Binet’s equation Newtonian version

Approximate derivation of the orbit Put:

Derivation of the orbit, ctd Rearrangements, manipulations… Final result: Slowly precessing ellipse!

Periapsis shift of Mercury • • a=0. 39 AU e=0. 206 =7. 9293· 10 -8 2 =4. 9821· 10 -7 rad shift each revolution • 410. 6 revolutions in 100 years • ω=42. 2” per century ( ω=43. 03” with more accurate calculation) • Observations: ω=43. 11” 0. 45”

Relativistic effects and distance OBJECT Mercury PREDICTED PERIAPSIS SHIFT [“ /100 yr] 43. 03 OBSERVED PERIAPSIS SHIFT [“/100 yr] 43. 11 0. 45 Venus 8. 6 8. 4 4. 8 Earth 3. 8 5. 0 1. 2