Solving Kepler's 2nd Law: Determine a & e from E, l, m, M, and G

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SUMMARY

This discussion focuses on deriving the semimajor axis (a) and eccentricity (e) of a satellite's orbit using its total energy (E), angular momentum (l), mass (m), parent body mass (M), and Newton's constant (G). The relationship among energy, angular momentum, and the orbit radius at the apside is crucial for solving the problem. Participants emphasize the importance of expressing apsidal radii (r*) in terms of the semimajor axis and eccentricity, specifically using the equation r* = a(1 ± e). The discussion encourages sharing initial thoughts or attempts to tackle the problem for effective assistance.

PREREQUISITES
  • Understanding of Keplerian orbits and their properties
  • Familiarity with Newton's laws of motion and gravitation
  • Knowledge of energy and angular momentum in orbital mechanics
  • Ability to manipulate algebraic equations and solve for variables
NEXT STEPS
  • Study the derivation of orbital parameters from energy and angular momentum
  • Learn about the mathematical relationships in orbital mechanics
  • Explore the implications of eccentricity on satellite orbits
  • Investigate the role of gravitational constant (G) in orbital calculations
USEFUL FOR

Astronomy students, astrophysicists, and aerospace engineers interested in orbital mechanics and satellite dynamics will benefit from this discussion.

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A satellite of mass m is in Keplerian orbit around a parent body of mass M >>m. The satellite has total energy E and angular momentum l, where angular momentum is measured about the parent body and the zero of potential energy is at infinity. Determine the semimajor axis a and the eccentricity e of the orbit, in therms of the above quantities and Newton'ss constant G (use the symbol, not the numerical value). HINT: Find the relation among the energy, angular momentum and orbit raduis at the apside of the orbit. Solve this relation for the apsidal radii r*, and use the fact that the apsidal radii are r*= a(1+-e).
 
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