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Chaos Game

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Miscel-

laneousMotion of the Moon

Computing the true longitude of the Moon:

True longitude of Moon = mean longitude + major inequality + evection + variation + annual inequality + reduction to ecliptic + parallactic inequality + more terms

Select "Orbit" instead of "Diagram" to watch the geocentric motion of the Moon.

mean longitude of the Moon:

measured from the mean position of the perigee

L0 = 218.31617 + 481267.88088*T - 4.06*T*T/3600.0

the Moon's mean anomaly:

M = 134.96292 + 477198.86753*T + 33.25*T*T/3600.0

the Sun's mean anomaly:

MSun = 357.52543 + 35999.04944*T - 0.58*T*T/3600.0

the mean distance of the Moon from the ascending node

F = 93.27283 + 483202.01873*T - 11.56*T*T/3600.0

the difference between the mean longitudes of the Sun and the Moon

D = 297.85027 + 445267.11135*T - 5.15*T*T/3600.0

Time

T = (JD - 2451545)/36525Source: Montenbruck, Pfleger: Astronomy on the Personal Computer

True longitude of Moon = mean longitude + major inequality + evection + variation + annual inequality + reduction to ecliptic + parallactic inequality + more terms

L = L0 + ...

Period

(1) Major Inequality (equation of center)

= 22640"*sin(M) + 769*sin(2M)~27.5 d

(2) Evection

= -4586"*sin(M-2D)14.7 d

(3) Variation

= 2370"*sin(2D)

31.8

(4) Annual Inequality

= -668"*sin(MSun)1 y

(5) Reduction to the Ecliptic

= -412"*sin(2F)

~14 d

(6) Parallactic Inequality

= -125"*sin(D)

(7) More terms

term1

term2

term3

term4

term4

term5

term6

= -212"*sin(2*M-2D)

= -206"*sin(M+MSun-2D)

= +192"*sin(M+2D)

= -165"*sin(MSun-2D)

= +148"*sin(L-MSun)

= -110"*Math.sin(M+MSun)

= - 55"*Math.sin(2F-2D)

E

The eccentric anomaly (a parameterization of polar angle)

ν

The true anomaly specifies the position along the orbit.

M

The mean anomaly is the angle of the line joining the focus (Earth) to a hypothetical body that has the same orbital period but travels at a uniform angular speed:

M = n·(t - T)

Kepler's Equation for a body orbiting on an ellipse with eccentricity e:

M = E - e·sin(E)

Instructions:

Choose or hide the terms of interest.

The value of a term is red if active, otherwise black.

drawing the diagram:

The values for the annual inequality, the reduction to the ecliptic, the parallactic inequality, and the additional terms are

magnifiedby a factor of 10.You may use the keys

dormto increase the date, or month,

orShift keyanddormto decrease the date or month !

Click the applet first !Click the diagram to get the coordinates of a point.

Click a point and a second point holding down the Alt key to get a (horizontal) time intervall.

Click a point and a second point holding down the Shiftkey to get a (vertical) angular difference.

Select "Orbit" to see the orbit of the Moon for the month selected.

You may use the keys

d,morhto increase the date, month, or hour,

orShift keyandd,morhto decrease the date, month, or hour !

Click the applet first !

More applets:

More details:

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Last update: 2023, Oct 05