Talk:Equation of the center

storing this here for now - off-topic material

Related expansions may be used to express the true distance $r$ of the orbiting body from the central body as a fraction of the semi-major axis $a$ of the ellipse,

${\frac {r}{a}}=(1+e^{2}/2)-(e-{\frac {3}{8}}e^{3})\cos M-{\frac {1}{2}}e^{2}\cos 2M-{\frac {3}{8}}e^{3}\cos 3M-...$ ;

or the inverse of this distance $a/r$ has sometimes been used (e.g. it is proportional to the horizontal parallax of the orbiting body as seen from the central body):

${\frac {a}{r}}=1+(e-e^{3}/8)\cos M+e^{2}\cos 2M+{\frac {9}{8}}e^{3}\cos 3M+...$ . — Preceding unsigned comment added by Tfr000 (talk • contribs) 14:55, 30 January 2016 (UTC)

some misconceptions in this article

Ok, I think I finally found where this

In the case of the moon, its orbit around the earth has an eccentricity of approximately 0.0549. The term in \sin(M), known as the principal term of the equation of the center, has a coefficient of 22639.55",[2] approximately 0.1098 radians, or 6.289° (degrees).

came from. See [1], pg. 139. Two things — 1) Brown is not talking about the equation of the center. The "terms" he lists are for a completely different, much more complicated, theory of the Moon's motion. No "term in \sin(M)" is discussed in that theory. A search of the text for "equation of the center" yields no hits. It is possible that the sin(M) term of the equation of the center might have a similar numeric value, but Brown is not the source of it. 2) Nothing is "known as the principal term". Here is the text:

The addition of 1" to the adopted coefficient of the principal elliptic term in longitude (22639".550) requires an addition to the factors... blah blah blah etc.

Brown is not naming anything "the principal term". He is merely describing something to be done to the term with the largest numeric value, and "principal term" is easier to read than "term with the largest numeric value". Similar discussion found on pg. 80, and a few other pages. Tfr000 (talk) 04:18, 31 January 2016 (UTC)

also dropping this here for now - see comments above

Moon's equation of the center

In the case of the moon, its orbit around the earth has an eccentricity of approximately 0.0549. The term in $\sin(M)$, known as the principal term of the equation of the center, has a coefficient of 22639.55", approximately 0.1098 radians, or 6.289° (degrees).

The earliest known estimates of a parameter corresponding to the Moon's equation of the center are Hipparchus' estimates, based on a theory in which the Moon's orbit followed an epicycle or eccenter carried around a circular deferent. (The parameter in the Hipparchan theory corresponding to the equation of the center was the radius of the epicycle as a proportion of the radius of the main orbital circle.) Hipparchus' estimates, based on his data as corrected by Ptolemy yield a figure close to 5° (degrees).

Most of the discrepancy between the Hipparchan estimates and the modern value of the equation of the center arises because Hipparchus' data were taken from positions of the Moon at times of eclipses. He did not recognize the perturbation now called the evection. At new and full moons the evection opposes the equation of the center, to the extent of the coefficient of the evection, 4586.45". The Hipparchus parameter for the relative size of the Moon's epicycle corresponds quite closely to the difference between the two modern coefficients, of the equation of the center, and of the evection (difference 18053.1", about 5.01°).

Bibliography

• Brown, E.W. An Introductory Treatise on the Lunar Theory. Cambridge University Press, 1896 (republished by Dover, 1960).
• Brown, E.W. Tables of the Motion of the Moon. Yale University Press, New Haven CT, 1919.
• Neugebauer, O., A History of Ancient Mathematical Astronomy (Springer, 1975), vol. 1, pp. 315–319.

References