How is a spacecraft's Sun speed different from its ground speed?

"It is found that the relative motion for a circular chief orbit corresponds to the motion of an epitrochoid."

"While the description of orbiting spacecraft relative motion is usually done in the rotating Hill frame due to analytical first-order solutions and the intuitive shape of the relative orbits, it is disadvantageous for mission design requirements that are fixed in the inertial frame. This includes distributed space telescopes aligned with inertial targets as well as formations and servicing operati"

"For spacecraft orbiting Earth, this imposes a keep-out or keep-in zone that is fixed in a quasi-inertial frame (because the Sun direction changes slowly)."

"Unlike the transition from the Hill frame \(\mathcal {H}\) to the perifocal frame \(\mathcal {P}\) , which results in a general change of the relative orbit shape, the mapping from the perifocal frame \(\mathcal {P}\) to the inertial frame \(\mathcal {N}\) is straightforward as it is a pure rotation."

"When they are instead moving parallel – so that one catches up with the other – the final velocity of the spacecraft relative to the star is 2u − u , which could be negative."

"Energy is conserved, so the magnitude of the asymptotic velocity of the spacecraft relative to the planet will be unchanged by the encounter, but the direction of travel will change."

"But in the star’s rest frame, because the planet is moving, the velocity of the spacecraft is changed."

"The principle of the gravity assist is very similar to this, with the spacecraft playing the role of the ball and the planet the role of the train."

"The Earth moves at about 30 km/sec in an orbit centered on the Sun."

"This is the heliocentric velocity of the spacecraft relative to the Sun."

"However, we need to compute the velocity relative to the Earth in order to calculate the launch energy required."

"The hyperbolic escape velocity, Vout, is the vector difference of the velocity of the Earth with respect to the Sun, VE, and the velocity required o"

"A spacecraft’s speed varies constantly — it accelerates falling toward the Sun and slows climbing away — so this ranking shifts over time."

"Relative to the Sun, it can leave faster or slower and pointing in a new direction, because it has borrowed a tiny share of the planet's orbital momen..."

"It is found that the relative motion for a circular chief orbit corresponds to the motion of an epitrochoid."

"For elliptic chief orbits, the inertial frame relative orbits are stretched and distorted compared to the epitrochoid curve for circular chief orbits, but similar relative orbit elements are defined as well."

"Note that while \(\varDelta \mathbf { r}={\varvec{\uprho } },\) we have that \(\varDelta {\dot{\mathbf{r}}}\ne {\dot{\varvec{\uprho }}}\) and \(\varDelta {\ddot{\mathbf{r}}}\ne {\ddot{\varvec{\uprho }}}\) since the time derivatives are taken by observers fixed with respect to different reference frames, the first of which is inertial, the other non-inertial."

"Thisnon-inertialreference frame,traditionallynamedLVLH(Local-Vertical-Local-Horizontal)ischosenasfollows"

"helps introduce a change of variable which transforms the relative orbital motion problemintotheclassicKeplerproblem. Thus,theproblemoftherelativeorbitalmotionis super-integrable."

"In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities as vectors"

"Note that the last equation holds for all Galilean transformations up to addition of a constant, and expresses the assumption of a universal time independent of the relative motion of different observers."