The Energy-Angular-Momentum Diagram is proposed in this paper to clarify the orbit transfer problem of a space vehicle. With the use of this diagram, some properties of parameter variations in the orbit transfer can be represented in a more intuitive manner.
On this diagram, the orbit is shown by the point (orbit point)
x=2c/μ, y=h2/μ where
c=energy,
h=angular momentum, μ=gravitational parameter (3.98604×10
5km
3/s
2 for the earth).
The Energy-Angular-Momentum Diagram has properties as follows:
1. The orbit having the eccentricity
e is shown by the point on the hyperbola 1+
xy=
e22. No orbit point appears in the domain 1+
xy<0. Let the boundary hyperbola (1+
xy=0) be
H.
3. The ordinates of two points of contact on
H where the tangential lines from the orbit point touch the hyperbola
H, are the apoapsis radius
rap and the periapsis radius
rpg, respectively.
4. We consider two parallel straight lines on the diagram. The tangential line of the hyperbola
H that passes the point (-2/
r, 0) is designated as
T. And the line
L is the line that is parallel to
T and that passes the orbit point (
x, y). We assume that this line
L crosses
X-axis at the point (
xl, 0). Then the axial velocity component
va and the transversal velocity component
vt of the space vehicle are related to this diagram by:
xl+(2/r)=va2/μ, x-xl=vt2/μ where
r=radius from the focus to the space vehicle.
5. The orbit transfer of the space vehicle due to some velocity change can be explained by the nonlinear transformation.
That is, velocity components which are expressed in orthogonal coordinate are transformed to coordinates of the orbit point expressed in the skew coordinate whose axes are the
x-axis and the straight line
T and whose scale is proportional to the squared value of velocity, in the Energy-Angular-Momentum Diagram.
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