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Celestial Mechanics IV Central orbits Force from shape, shape from force General relativity correction
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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:
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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
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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 F c (r)
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Binet’s equation We make the variable substitution: Look for the dependence of u on the angular variable :
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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
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Summary Finding the force field from the shape of the orbit (Clairaut) Finding the shape of the orbit from the force field (Binet)
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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 r min and r max at a frequency non- commensurable with the circulation The angular momentum prevents the object from reaching r 0, unless with n > 2
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Orbits in central force fields, ctd Central point massStellar cluster
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Special cases I: Keplerian orbits Central point mass with GM= : Inserting this into Binet’s equation gives: Analytic solution:
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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
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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:
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Cotes spirals, ctd B=0B=1/9 B=1/100B=–9/4
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Force from shape, Keplerian case Insert into Clairaut’s equation:
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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!
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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)
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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
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Schwarzschild metric Line element in Euclidean 3D space Line element in Euclidean 4D space, empty “spacetime”: Line element in 4D spacetime, containing one massive body in origo: Schwarzschild metric
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Relativistic orbital motion Light and bodies move on geodetics in the spacetime The 3D “shadow” of a geodetic in 4D is the orbit We need an equation for “straight lines” in spacetime! Euler-Lagrange equation The metric tensor g describes spacetime curvature, contains the Schwarzschild metric coefficients
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Solving the Euler-Lagrange equation Relativistic version of Binet’s equation Newtonian version
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Approximate derivation of the orbit Put:
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Derivation of the orbit, ctd Rearrangements, manipulations… Final result:Slowly precessing ellipse!
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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”
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Relativistic effects and distance OBJECTPREDICTED PERIAPSIS SHIFT [“ /100yr] OBSERVED PERIAPSIS SHIFT [“/100 yr] Mercury43.03 43.11 0.45 Venus8.6 8.4 4.8 Earth3.8 5.0 1.2
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