Can the Euler Method Accurately Solve Orbital Gravity Equations?

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Discussion Overview

The discussion centers on the use of the Euler Method for solving gravitational differential equations related to orbital mechanics. Participants explore the reliability of this method compared to more advanced techniques, particularly in the context of simulating orbital trajectories.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether the Euler Method can provide accurate solutions for gravity differential equations, considering the potential for divergence in approximations.
  • One participant suggests that while the Euler Method can be made precise with small time steps, it may not be practical for orbital calculations due to the required step size.
  • Another participant recommends using higher-order methods like Runge-Kutta for better accuracy in orbital simulations.
  • One user shares their experience with a simulation that incorrectly models the Earth's orbit, attributing the issue to the method used and the time step size.
  • Participants discuss the importance of using appropriate equations and coordinate systems, with some emphasizing the need for polar coordinates in orbital mechanics.
  • There is a mention of the Euler-Cromer method as a variation that updates velocities before positions, which may yield better results in certain scenarios.
  • Some participants express uncertainty about the implications of using large time steps in simulations, with one noting that a time step of 12 hours may be too large for stability.

Areas of Agreement / Disagreement

Participants generally do not reach a consensus on the effectiveness of the Euler Method for orbital simulations, with multiple competing views on its reliability and the necessity of using more advanced methods like Runge-Kutta. The discussion remains unresolved regarding the best approach for accurate orbital calculations.

Contextual Notes

Limitations include the dependence on time step size and the choice of coordinate systems, which may affect the accuracy of simulations. Some participants note that the assumptions regarding angular momentum may only hold in specific scenarios, such as the two-body problem.

  • #31
I've been tooling around with this, and found that the method known in some quarters as "Leapfrog" gives pretty good results even with modest computing power. The method is the same as described in Feynman Lectures volume 1 chapter 9. It seems to be the same as what some people call Verlet. There's an excellent discussion here:
http://young.physics.ucsc.edu/115/leapfrog.pdf

For a free iPad app to show the results of the method see here:
https://itunes.apple.com/nz/app/orbit-simulator/id1048345753?mt=8

The main problem I see is that highly elliptical orbits do start to precess, depending on the time step you use. Also when the speed gets very high the orbits develop sharp corners. To get round this you's need smaller steps or a method with adaptive step size, but Leapfrog is well good enough just to play around with.
 
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