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Canonical Equations of Celestial Mechanics
Published in G.A. Gurzadyan, Theory of Interplanetary Flights, 2020
where α and e are the major semiaxis and the eccentricity of the orbit of the second component of the binary system, respectively, i is the inclination of the orbit’s plane, n(t – T) is the mean anomaly, ω is the angular distance of perihelion from the node, Ω is the longitude of the ascending node, and M=m+m* is the total mass of the binary system. Under canonical transformations the volume element of phase space is conserved or, in other words, the Jacobian of transformation equals unity: dxdydzdpxdpydpz=dLdGdHdldgdh.
Satellite Optical Imagery
Published in Victor Raizer, Optical Remote Sensing of Ocean Hydrodynamics, 2019
The ideal elliptical orbit, with the earth at one node, is described by six Keplerian elements: α true anomaly (instantaneous angle from satellite to perigee)ω argument of perigee (twist)Ω longitude of the ascending node (pin)a semi-major axis of the elliptical orbit (size)e eccentricity of the orbital ellipse (shape)i inclination of the orbital plane (tilt).
Satellite positioning
Published in W. Schofield, M. Breach, Engineering Surveying, 2007
It is now necessary to rotate the orbital plane about the X0-axis, through the inclination i, to make the orbital plane coincide with the equatorial plane and the Z0-axis coincide with the Z-axis of the Earth fixed system (IRP). Thus: XE=X0YE=Y0cosiZE=Y0siniwhere i=i0+i˙ts−tp+Ciccos2(ω+f)+Cissin2(ω+f)and i0 is the inclination of the orbit plane at reference time tpi˙ is the linear rate of change of inclinationCic, Cis are the amplitude of the cosine and sine correction terms of the inclination of the orbital plane. Finally, although the ZE-axis is now correct, the XE-axis aligns with the Ascending Node and requires a rotation about Z towards the Zero Meridian (IRM) usually referred to as the Greenwich Meridian. The required angle of rotation is the Right Ascension of the Ascending Node minus the Greenwich Apparent Sidereal Time (GAST) and is in effect the longitude of the ascending node of the orbital plane (λ0) at the time of observation ts.
Fully distributed finite-time adaptive robust time-varying formation-containment control for satellite formation
Published in International Journal of Control, 2023
Pingli Lu, Qing Jiang, Ye Tian, Haikuo Liu, Changkun Du
Consider a satellite formation system consisting of four leader satellites labelled as 1, 2, 3, 4, five follower satellites labelled as , respectively. The reference signal is labelled as 0 and only available for the 1th leader satellite. The relative motion dynamics of the ith satellite can be described by (2). Assume that the reference satellite moves along a near-circular orbit around the Earth with the following initial orbit elements, where a is the semi-major axis (in meters) of the orbit, e is the eccentricity, i is the inclination, Ω is the longitude of the ascending node, ϖ is the argument of perigee, and θ is the true anomaly.
Numerical Solving of Radiation Geometrical Inverse Problem
Published in Heat Transfer Engineering, 2023
Aleksey V. Nenarokomov, Evgeniy V. Chebakov, Dmitry L. Reviznikov, Irina V. Krainova
In the general case the orientation of an arbitrary surface element of a spacecraft can be determined by the following nine angles: Three angles determine the relative position of the equatorial XYZ and orbital nrb coordinate systems (Figure 1a): where Ω is the longitude of the ascending node, i is the inclination of orbit, u is the argument of latitude. The planetocentric equatorial coordinate system can be considered as an inertial coordinate system for most engineering problems. We assume that the corresponding angles are always known from the predetermined spaceflight control program.