Theoretical Description of Spacecraft Flybys by Variation of Parameters

Perturbations of a Sepacecraft Orbit during a Hyperbolic Flyby

“…More recently, a very simple analytical theory has been developed for flybys of small bodies (Anderson and Giampieri 1999), which also uses the method of variation of parameters, but which adopts unperturbed orbital elements based on the Born approximation for a massless central body. In AG's theory, all gravity terms are treated as perturbations, including the monopole coefficient µ, the product of the gravitational constant by the total mass of the central body.…”

“…For covariance analyses, these forces are not important since they have no effect on the observable. For trajectory calculations, the rotation perturbations derived by Anderson and Giampieri (1999) can be used to introduce the rotation perturbations into the body-fixed principal axes.…”

Stardust dynamic science at comet 81P/Wild 2

“…The observable is the range rate , or LOS (line‐of‐sight) Doppler expressed in terms of fractional frequency shift y = Δν/ν, which for purposes of error analysis can be approximated by the first‐order two‐way Doppler formula y = 2/ c , with c the speed of light. The LOS Doppler is obtained by projecting the spacecraft velocity vector along the Earth‐spacecraft direction as follows [ Anderson and Giampieri , 1999]: All quantities in are referred to the closest approach time t = 0, with α and β the direction cosines for the Earth‐spacecraft direction projected, respectively, along the radius and velocity vectors at closest approach. The modulus r of the time‐varying unperturbed radius vector is approximated by .…”

“…The mass parameter GM is designated by μ. The identification of α with the projection along the radius vector and β along the velocity vector is demonstrated by Anderson and Giampieri [1999].…”

Mass and density determinations of 140 Siwa and 4979 Otawara as expected from the Rosetta flybys

“…The vector can be resolved into two components along and across the trajectory direction. In the existing literature (Anderson 1971;Anderson & Giampieri 1999;Rappaport et al 2000) this is accomplished by integration over time with initial conditions at closest approach. Here, we choose to integrate with initial conditions at −∞, which makes a small difference, equal to one-half of the total deflection angle (2), in the direction of the two components.…”