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Given the different Sun incidence in different positions in the orbit, it is necessary to define a standard point of the orbit of the planet, to define the planet position in the orbit at each moment of the year w.r.t such point; this point is called with several names: vernal equinox, spring equinox, March equinox, all equivalent, and named considering northern hemisphere seasons.
For each rotation R of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes A and B each of which is invariant and whose direct sum A ⊕ B is all of 4-space. Hence R operating on either of these planes produces an ordinary rotation of that plane.
Rotation period with respect to distant stars, the sidereal rotation period (compared to Earth's mean Solar days) Synodic rotation period (mean Solar day) Apparent rotational period viewed from Earth Sun [i] 25.379995 days (Carrington rotation) 35 days (high latitude) 25 d 9 h 7 m 11.6 s 35 d ~28 days (equatorial) [2] Mercury: 58.6462 days [3 ...
Astronomy of the Earth's Motion in Space, high-school level educational web site by David P. Stern Newtonian Dynamics Undergraduate level course by Richard Fitzpatrick. This includes Lagrangian and Hamiltonian Dynamics and applications to celestial mechanics, gravitational potential theory, the 3-body problem and Lunar motion (an example of the ...
This is due to the solar day being shorter than the sidereal day for retrograde rotation, as the rotation of the planet would be against the direction of orbital motion. If a planet rotates prograde, and the sidereal day exactly equals the orbital period, then the formula above gives an infinitely long solar day (division by zero).
A general rotation in four dimensions has only one fixed point, the centre of rotation, and no axis of rotation; see rotations in 4-dimensional Euclidean space for details. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes.
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Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space. With this convention, a vector is the same as an element of the vector space R 3 . {\displaystyle \mathbb {R} ^{3}.} [ b ]