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Is it possible to explain the difference between synodic and sidereal month by degrees - in the following way?

Just recently I learnt about these two types of months, and I want to see if I can understand the difference by degrees method. Assuming the orbit's moon is 360 degrees, the sidereal month starts from the point where 1st degree to (include) 1st degree, while the synodic month starts from 1st degree to the 0.5 degree (after full orbit's moon, because the angular size of the moon is 0.5 degree). Is it possible to understand why it that's way?

enter image description here

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A better diagram is something like this:

Moon's sidereal and synodic period

  1. Day 0. The Moon is aligned with the Sun (New Moon) and some star X behind the Sun (of course :-), as shown by the arrow at 1. (Neither the Sun nor the star X are shown in the figure.)
  2. Day 27.32. One sidereal month after figure 1. The Moon has travelled through 360 degrees orbiting around the Earth. (During that time, an arrow from the Earth through the Moon is pointing at different stars.) After one sidereal month, the Moon is again aligned with star X, as shown by the solid arrow in figure 2. (The arrow in 1 and 2 are parallel because the star is very far away.) Because the Earth has moved in its orbit around the Sun, the position of the Sun (shown by the dashed arrow in figure 2) has changed relative to figure 1. The Moon is still a few days prior to New Moon.
  3. Day 29.53. One synodic month after figure 1. The Moon is again aligned with the Sun (New Moon), as shown by the solid arrow in figure 3. The direction to star X is shown by the dashed arrow which is parallel to the solid arrows in figures 1 and 2.

Note how the arrows pointing to the Sun all point to the center of the Earth's orbit while the arrows pointing to star X are parallel.

You are probably interested in the angle between the two arrows in position 3; that is, how far has the Moon moved in its orbit from the position of the sidereal month to the position of the synodic month. The average angle = (360 degrees)/(27.32 days) * (29.53-27.32 days) = 29.12 degrees.

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  • $\begingroup$ Thank you for the answer. Just want confirm some things due to my ignorance. 1. When you say star X, does it mean it can be any satr that I choose by which I determine as a starting point. Right? 2. You wrote "Because the Earth has moved in its orbit around the Sun, the Moon is still a few days prior to New Moon". This is exactly the point I have a difficulty to understand. I don't understand why this this difference (2.2 days) appear. $\endgroup$ – Reckless Glacier Apr 15 at 18:25
  • $\begingroup$ 3. I said that the angle size of the moon is 1/2 degree relying on this reference: lco.global/spacebook/sky/… $\endgroup$ – Reckless Glacier Apr 15 at 18:25
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    $\begingroup$ 1. To keep things simple, you can choose any star as long as it is behind the Sun. 2. I changed the diagram and text to help clarify the difference. It might also help if you draw the full orbit of the Earth going around the Sun on a large piece of paper and repeat the close-up figures I made on the full orbit. 3. The angular size of the Moon has nothing to do with this discussion. $\endgroup$ – JohnHoltz Apr 15 at 19:31
  • $\begingroup$ Thank you very much for drawing the diagram, it's really was helpful to understand it better. Now I understand that the Sun moves faster than the rate of the moon, because of my ignorance I thought the that in the heliocentric system, everything orbit the Sun but the Sun is 'fixed' in one place (even-though I knew it rotates, I didn't know it's also moving further beside its rotation, as the Earth and Moon). Now I understand that everything moves all the time. Right? $\endgroup$ – Reckless Glacier Apr 15 at 21:56
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    $\begingroup$ @UbiquitousStudent - Imagine the Earth, Moon, Sun, and a very distant quasar exactly line up at some point in time. One sidereal month later, the Earth, Moon, and quasar will once again more or less line up. But the Earth, Moon, and Sun will not line up because the Earth-Moon system has moved in its orbit about the Sun. For the Earth, Moon, and Sun to more line up once more, the Moon will have to go about 1/12th of its orbit about the Earth further, 1/12 because that's roughly the number of months in a year. A twelfth of 360° is 30°. $\endgroup$ – David Hammen Apr 16 at 9:35
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The new moon is when the center of the Moon and the center of the Sun are at the same ecliptic longitude. The angular diameters of the Sun and Moon are not considered.

Angular speeds give us another way to look at the situation. The Earth moves 360° around the Sun per sidereal year, or an average of 0.9856°/day. The Moon moves 360° around the Earth per 27.32-day sidereal month, or an average of 13.18°/day.

In a 29.53-day synodic month, the Earth moves 29.1° around the Sun, and the Moon moves 389.1° = 360° + 29.1° around the Earth. The 2.21-day difference between synodic and sidereal months is the time the Moon takes to cover the additional 29.1° at 13.18°/day.

Also consider that in one year there are 13.37 sidereal months or 12.37 synodic months. In that time the Moon moves 13.37 × 360° = 12.37 × 389.1° = 4813° around the Earth.

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