Speaker
Ms
Virginia Raposo Pulido
(SDG-UPM)
Description
The Kepler equation for the elliptical motion, *y − e* sin *y* − *x* = $0$, involves a nonlinear
function depending on three parameters: the eccentric anomaly *y* = E, the eccentricity e and
the mean anomaly *x* = M. For given e and x values the numerical solution of the Kepler equation
becomes one of the goals of orbit propagation to provide the position of the object orbiting around
a body for some specific time (see references [1–6]). In this paper, a new approach for solving Kepler
equation for elliptical and hyperbolic orbits is developed. This new approach takes advantage
of the very good behavior of the Laguerre method [7] when the initial seed is close to the looked
for solution and also of the existence of symbolic manipulators which facilitates the obtention of
polinomial approximations. The central idea is to provide an initial seed as good as we can to the
modified Newton-Raphson method, because when the initial guess is close to the solution, the algorithm
is fast, reliable and very stable. To determine a good initial seed the domain of the equation is
discretized in several intervals and for each one of these intervals a fifth degree interpolating polynomial
is introduced. The six coefficients of the polynomial are obtained by requiring six conditions
at both ends of the corresponding interval. Thus the real function and the polynomial have equal
values at both ends of the interval. Similarly relations are imposed for the two first derivatives. Consequently,
given e and *x* = M, selecting the interval [$x_i$, $x_{i+1}$] in such a way that M $\epsilon$ [$x_i$, $x_{i+1}$]
and using the corresponding polynomial p$_i$(*x*), we determine the starter value y$_o$ = E$_o$. However,
the Kepler equation has a singular behavior when M is small and e close to unity (singular corner).
In this case, the exact solution of the equation has to be described in a different way to guarantee the
enough accuracy to be part of the seed used to start the numerical method. In order to do that, an
asymptotic expansion in power of the small parameter $\varepsilon$ = 1−*e* is developed. In most of the cases,
the seed generated by the Space Dynamics Group at UPM(SDG-code) leads to reach machine error
accuracy with the modified Newton-Raphson methods with no iterations or just one iteration. The
final algorithm is very stable and reliable. This approach improves the computational time compared
with other methods currently in use. The advantage of our approach is its applicability to
other problems as for example the Lambert problem for low thrust trajectories.
[1] RA Serafin. Bounds on the solution to kepler’s equation. Celestial mechanics, 38(2):111–121, 1986.
[2] F Landis Markley. Kepler equation solver. Celestial Mechanics and Dynamical Astronomy, 63(1):101–
111, 1995.
[3] Toshio Fukushima. A method solving kepler’s equation without transcendental function evaluations. Celestial Mechanics and Dynamical Astronomy, 66(3):309–319, 1996.
[4] Toshio Fukushima. A fast procedure solving kepler’s equation for elliptic case. The Astronomical Journal, 112:2858, 1996.
[5] Daniele Mortari and Antonio Elipe. Solving kepler’s equation using implicit functions. Celestial Mechanics and Dynamical Astronomy, 118(1):1–11, 2014.
[6] RH Gooding and AW Odell. The hyperbolic kepler equation (and the elliptic equation revisited). Celestial mechanics, 44(3):267–282, 1988.
[7] Bruce A Conway. An improved algorithm due to laguerre for the solution of kepler’s equation. Celestial mechanics, 39(2):199–211, 1986.
| Applicant type | First author |
|---|
Primary authors
Dr
Jesús Peláez
(SDG-UPM)
Ms
Virginia Raposo Pulido
(SDG-UPM)
