Celestial Mechanics I :Celestial Mechanics I


Today’s Lecture :Today’s Lecture Central orbits How to obtain the orbital shape from an (almost) arbitrary force function, and vice versa. Relativistic two-body problem Newtonian mechanics is only approximately correct, which is revealed by the orbit of Mercury


Why is this important? :Why is this important? Uniqueness test for Newtonian gravitational law Orbital geometries for other force functions Celestial mechanics important for understanding of fundamental physics


Conservative central forces :Conservative central forces Conservative: Work done by force independent of path Central: Force directed towards a fixed point Can be described by a potential V (here, over mass) Energy conservation Angular momentum conservation (K II always fulfilled)


Clairaut’s equation :Clairaut’s equation Based on energy 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)


(Generalized) Binet’s equation :(Generalized) Binet’s equation Given the Law of Force and the orbital angular momentum, Binet’s equation yields the geometry of the orbit, r()


Summary :Summary


Shape from force: Kepler I (again) :Shape from force: Kepler I (again) Newtonian r--2 law inserted into Binet’s formula yields conic section solution, i.e., leads to Kepler I


Shape from force: Cotes’ spirals :Shape from force: Cotes’ spirals B=0 (h=k) Hyperbolic spirals B>0 (h>k) Epi spirals B<0 (h

Shape from force: Cotes’ spirals :Shape from force: Cotes’ spirals B=1/3 B=1/10 |B|=2/3


Force from shape :Force from shape If we require lim rV(r)0 But h2/p=μ Of all conservative central forces, only r--2 results in Kepler I


The orbit of Mercury :The orbit of Mercury Planetary perturbations makes ω time-dependent. Apsis line moves 1000” per century (0.3º) Perturbations can only explain 97% of the observed shift! There is an unexplainable residual of 43” per century. Urbain Jean Joseph Leverrier (1811-1877) Bild: http://archive.ncsa.uiuc.edu/Cyberia/NumRel/Images/mercury.jpeg


The General theory of Relativity :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 :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 c2 c2


Relativistic orbital motion :Relativistic orbital motion Light and bodies move on straight lines in the spacetime The 3D “shadow” of a straight line in 4D is the orbit We need an equation for straight lines in spacetime! Euler-Lagrange equation The metric tensor g in the Lagrangian L describes spacetime curvature


Solving the Euler-Lagrange equation :Solving the Euler-Lagrange equation Relativistic version of Binet’s equation Newtonian version 2


Approximate solution to relativistic version of Binet’s equation I :Approximate solution to relativistic version of Binet’s equation I


Approximate solution to relativistic version of Binet’s equation II :Approximate solution to relativistic version of Binet’s equation II


Periapsis shift of Mercury :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 :Relativistic effects and distance