Question
The Laplace–Runge–Lenz vector is conserved for motion along these paths. For 10 points each:
[10e] Name these paths described by Kepler’s laws, which planets trace out around the Sun.
ANSWER: planetary orbits [accept ellipses]
[10m] For Keplerian orbits, the LRL vector is proportional to a vector form of this dimensionless non-negative quantity. This quantity is less than one for bound orbits and is greater than or equal to one for escape orbits.
ANSWER: orbital eccentricity [accept eccentricity vector; prompt on e]
[10h] Description acceptable. The conservation of the LRL vector results from this property of ideal Keplerian orbits, deviations from which lead to apsidal precession. Bertrand’s theorem characterizes the bound orbits of simple harmonic oscillators and systems with this property.
ANSWER: inverse-square central force [accept answers describing that the magnitude of the central force, which is gravity, obeys an inverse-square law or that the central force is proportional to the negative second power of distance; accept answers describing that the potential energy is proportional to inverse distance or potential energy is proportional to the reciprocal of distance or potential energy is proportional to the negative first power of distance; accept radius or r or d in place of “distance”; prompt on gravity or gravitational force or gravitational potential with “What feature of the gravitational force?”]
<Joseph Krol, Science - Physics> ~18161~ <Editor: David Bass>
Summary
| 2023 PACE NSC | 06/10/2023 | Y | 1 | 20.00 | 100% | 100% | 0% |
Data
| Barrington A | Thomas Jefferson A | 10 | 10 | 0 | 20 |