Determination of The Kepler Orbit That Corresponds To A Given Initial State
This is the "initial value problem" for the differential equation (1) which is a first order equation for the 6-dimensional "state vector" when written as
|
(48) |
|
(49) |
For any values for the initial "state vector" the Kepler orbit corresponding to the solution of this initial value problem can be found with the following algorithm:
Define the orthogonal unit vectors through
|
(50) |
|
(51) |
with and
From (13), (18) and (19) follows that by setting
|
(52) |
and by defining and such that
|
(53) |
|
(54) |
where
|
(55) |
one gets a Kepler orbit that for true anomaly has the same r, and values as those defined by (50) and (51).
If this Kepler orbit then also has the same vectors for this true anomaly as the ones defined by (50) and (51) the state vector of the Kepler orbit takes the desired values for true anomaly .
The standard inertially fixed coordinate system in the orbital plane (with directed from the centre of the homogeneous sphere to the pericentre) defining the orientation of the conical section (ellipse, parabola or hyperbola) can then be determined with the relation
|
(56) |
|
(57) |
Note that the relations (53) and (54) has a singularity when and
i.e.
|
(58) |
which is the case that it is a circular orbit that is fitting the initial state
Read more about this topic: Kepler Orbit
Famous quotes containing the words determination of, orbit, corresponds, initial and/or state:
“No man is quite sane; each has a vein of folly in his composition, a slight determination of blood to the head, to make sure of holding him hard to some one point which nature has taken to heart.”
—Ralph Waldo Emerson (1803–1882)
“The human spirit is itself the most wonderful fairy tale that can possibly be. What a magnificent world lies enclosed within our bosoms! No solar orbit hems it in, the inexhaustible wealth of the total visible creation is outweighed by its riches!”
—E.T.A.W. (Ernst Theodor Amadeus Wilhelm)
“the full analysis of the notions of saying something and understanding what one said inevitably involves a concept which, as I will show in detail, essentially corresponds to the Cartesian idea of thought.”
—Zeno Vendler (b. 1921)
“For those parents from lower-class and minority communities ... [who] have had minimal experience in negotiating dominant, external institutions or have had negative and hostile contact with social service agencies, their initial approaches to the school are often overwhelming and difficult. Not only does the school feel like an alien environment with incomprehensible norms and structures, but the families often do not feel entitled to make demands or force disagreements.”
—Sara Lawrence Lightfoot (20th century)
“He was discontented and wasted his life into the bargain; and yet he rated it as a gain in coming to America, that here you could get tea, and coffee, and meat every day. But the only true America is that country where you are at liberty to pursue such a mode of life as may enable you to do without these, and where the state does not endeavor to compel you to sustain slavery and war and other superfluous expenses which directly or indirectly result from the use of such things.”
—Henry David Thoreau (1817–1862)