Solution of Lambert's Problem Assuming An Elliptic Transfer Orbit
First one separates the cases of having the orbital pole in the direction or in the direction . In the first case the transfer angle for the first passage through will be in the interval and in the second case it will be in the interval . Then will continue to pass through every orbital revolution.
In case is zero, i.e. and have opposite directions, all orbital planes containing corresponding line are equally adequate and the transfer angle for the first passage through will be .
For any with the triangle formed by, and are as in figure 1 with
and the semi-major axis (with sign!) of the hyperbola discussed above is
The eccentricity (with sign!) for the hyperbola is
and the semi-minor axis is
The coordinates of the point relative the canonical coordinate system for the hyperbola are (note that has the sign of )
where
Using the y-coordinate of the point on the other branch of the hyperbola as free parameter the x-coordinate of is (note that has the sign of )
The semi-major axis of the ellipse passing through the points and having the foci and is
The distance between the foci is
and the eccentricity is consequently
The true anomaly at point depends on the direction of motion, i.e. if is positive or negative. In both cases one has that
where
is the unit vector in the direction from to expressed in the canonical coordinates.
If is positive then
If is negative then
With
- semi-major axis
- eccentricity
- initial true anomaly
being known functions of the parameter y the time for the true anomaly to increase with the amount is also a known function of y. If is in the range that can be obtained with an elliptic Kepler orbit corresponding y value can then be found using an iterative algorithm.
In the special case that (or very close) and the hyperbola with two branches deteriorates into one single line orthogonal to the line between and with the equation
Equations (11) and (12) are then replaced with
(14) is replaced by
and (15) is replaced by
Read more about this topic: Lambert's Problem
Famous quotes containing the words solution of, solution, lambert, problem, assuming, transfer and/or orbit:
“I herewith commission you to carry out all preparations with regard to ... a total solution of the Jewish question in those territories of Europe which are under German influence.... I furthermore charge you to submit to me as soon as possible a draft showing the ... measures already taken for the execution of the intended final solution of the Jewish question.”
—Hermann Goering (1893–1946)
“I herewith commission you to carry out all preparations with regard to ... a total solution of the Jewish question in those territories of Europe which are under German influence.... I furthermore charge you to submit to me as soon as possible a draft showing the ... measures already taken for the execution of the intended final solution of the Jewish question.”
—Hermann Goering (1893–1946)
“Our duty now is to keep alive—to exist. What becomes of a nation if its citizens all die?”
—Dudley Nichols, U.S. screenwriter. Jean Renoir. George Lambert (George Sanders)
“A serious problem in America is the gap between academe and the mass media, which is our culture. Professors of humanities, with all their leftist fantasies, have little direct knowledge of American life and no impact whatever on public policy.”
—Camille Paglia (b. 1947)
“Quite generally, the familiar, just because it is familiar, is not cognitively understood. The commonest way in which we deceive either ourselves or others about understanding is by assuming something as familiar, and accepting it on that account; with all its pros and cons, such knowing never gets anywhere, and it knows not why.... The analysis of an idea, as it used to be carried out, was, in fact, nothing else than ridding it of the form in which it had become familiar.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“I have proceeded ... to prevent the lapse from ... the point of blending between wakefulness and sleep.... Not ... that I can render the point more than a point—but that I can startle myself ... into wakefulness—and thus transfer the point ... into the realm of Memory—convey its impressions,... to a situation where ... I can survey them with the eye of analysis.”
—Edgar Allan Poe (1809–1849)
“The Fitchburg Railroad touches the pond about a hundred rods south of where I dwell. I usually go to the village along its causeway, and am, as it were, related to society by this link. The men on the freight trains, who go over the whole length of the road, bow to me as to an old acquaintance, they pass me so often, and apparently they take me for an employee; and so I am. I too would fain be a track-repairer somewhere in the orbit of the earth.”
—Henry David Thoreau (1817–1862)