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Everything is Confirmed! Thank you all for your help! :D
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The Moon Phases Tracking Sheet tracks the timing and frequency of Valkemare's Moons', Aurax's and Cirion's, phases.
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For Moon Phases: Please, leave a comment with Time and Date of Lunar Phase. Finalized data will be added to the Confirmed Moon Phases Sheet for quick viewing.
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The Confirmed Moon Phases + Lengths Sheet shows the official lunar cycles of Aurax and Cirion and thier phases respectivel lengths based on the data from the Moon Phases Tracking Sheet. Aurax's cycle lasts 32 days. Cirion's s cycle lasts 16 days. There's a 1:2 ratio for Aurax and Cirion's lunar cycles. (As in for one of Aurax's cycles, there are 2 Cirion cycles.
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The Lunar Asynchronization Effects Sheet tracks the effects of both moons' phases on the type of Lunar Herald eggs produced. (Credit goes to pinkgothic.) Hover over the eggs for the intial time period of dropping/breeding cycle as well as links to proof in the forums.
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For Lunar Asynchronization: Please leave a comment over an egg that has appeared when two phases of the moons Aurax and Cirion overlap.  You may also leave a comment for which eggs are present or not, what time they began dropping, and location if needed.
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The Lunar Herald-Moon System Sheet shows a cleaner version of the Lunar Asynchronization Effects Sheet, and predicts eggs results based on the current theory that Lunar Herald egg colors are controlled by the amount each moon is illuminated and which moon is illuminated more. (Explanation below)
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Current Theory: When one moon is more than half while the other is less, an egg of the color of the moon more than half appears. White for both and brown for neither more than half. (This theory is completely supported by current data.)
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The Flow Visual Lunar Herald-Moon System sheet is the same as the original Lunar Herald-Moon System Sheet, only that it shows a visually appealing 2:1 pattern effect the two moons have over the lunar egg color. Start in the uppermost left corner and follow Cirion's phases downward; then, move right to the next available overlap if there are no more available overlaps are present in that Aurax Phase Column. 
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Y = Available, N = Not available, U= Unknown, - = Not ApplicableGold Lunar Herald Fanart by Stellaluna22
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Sheets can be accessed below.
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Current Authors: Shokomon, BananaGummyBear64, pinkgothic
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Phases of the Moon Images from Here
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Current phases of the DC moons, Aurax (the gold moon) and Cirion (the blue moon), can be accessed here.
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 ↳ note that these phases were out incorrectly displayed for the few few days of the release!
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Dragon Cave Wiki - Lunar Herald Dragon Page
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Calender based on current theory of Lunar Herald dropping (See: Possible Lunar Herald-Moon System for details.)
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Source for Above Image
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(Credit Goes to Gryphonic for finding the image explaing the moon phases.)