Monday, 4 July 2011

Mekanika Orbit: Teknik-Teknik Matematika

"Teknik-Teknik Mekanika dalam IPTEK Antariksa Sangat Perlu Dikembangkan" 
*Arip Nurahman*

Persamaan Kepler

M=E-\varepsilon\cdot\sin E.
where M is the mean anomaly, E is the eccentric anomaly, and \varepsilon is eccentricty.
  1. Compute the eccentric anomaly E from true anomaly θ
Finding the angle at a given time is harder. Kepler's equation is transcendental in E, meaning it cannot be solved for E analytically, and so numerical approaches must be used. In effect, one must guess a value of E and solve for time-of-flight; then adjust E as necessary to bring the computed time-of-flight closer to the desired value until the required precision is achieved. Usually, Newton's method is used to achieve relatively fast convergence.
The main difficulty with this approach is that it can take prohibitively long to converge for the extreme elliptical orbits. For near-parabolic orbits, eccentricity e is nearly 1, and plugging e = 1 into the formula for mean anomaly, E − sinE, we find ourselves subtracting two nearly-equal values, and so accuracy suffers. For near-circular orbits, it is hard to find the periapsis in the first place (and truly circular orbits have no periapsis at all). Furthermore, the equation was derived on the assumption of an elliptical orbit, and so it does not hold for parabolic or hyperbolic orbits at all. These difficulties are what led to the development of the universal variable formulation, described below.

Perturbation theory

Main article: Perturbation theory
One can deal with perturbations just by summing the forces and integrating, but that is not always best. Historically, variation of parameters has been used which is easier to mathematically apply with when perturbations are small.

Conic orbits


The patched conic approximation

The transfer orbit alone is a poor approximation for interplanetary trajectories because it neglects the planets' own gravity. Planetary gravity dominates the behaviour of the spacecraft in the vicinity of a planet. It severely underestimates delta-v, and produces highly inaccurate prescriptions for burn timings.



The universal variable formulation

To address the shortcomings of the traditional approaches, the universal variable formulation was developed. It works equally well on circular, elliptical, parabolic, and hyperbolic orbits; and also works well with perturbation theory. The differential equations converge nicely when integrated for any orbit.





Non-ideal orbits

  • Equatorial bulges cause precession of the node and the perigee
  • Tesseral harmonics of the gravity field introduce additional perturbations
  • lunar and solar gravity perturbations alter the orbits




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