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 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)
“In America there are two classes of travel—first class, and with children. Travelling with children corresponds roughly to travelling third-class in Bulgaria. They tell me there is nothing lower in the world than third-class Bulgarian travel.”
—Robert Benchley (1889–1945)
“Capital is a result of labor, and is used by labor to assist it in further production. Labor is the active and initial force, and labor is therefore the employer of capital.”
—Henry George (1839–1897)
“Indiana was really, I suppose, a Democratic State. It has always been put down in the book as a state that might be carried by a close and careful and perfect organization and a great deal of—[from audience: “soap”Ma reference to purchased votes, the word being followed by laughter].
I see reporters here, and therefore I will simply say that everybody showed a great deal of interest in the occasion, and distributed tracts and political documents all through the country.”
—Chester A. Arthur (1829–1886)