When can Moon occult Venus?

I'm trying to solve this task:

At what age of the Moon (in days after the new moon) is it possible for the Moon to occult Venus?

How I was solving it:

Angular velocity of the Moon is 13.2 degrees per day, which means that covering of Venus is possible during approximately 3,5 days after one new moon and 3.5 days before next new moon because the Moon should be not further than 47 degrees from the Sun. $$\frac{47}{13.2} = 3.56.$$

But: My teacher told me that 13.2 is wrong. I found some dates of when Moon covers Venus and it seems like they happen almost exactly 3 days before and after new Moon. Examples: new Moon - 6. October, covering - 9. October; new Moon - 5. November, covering - 8. November; covering - 28. March, new Moon - 1. April.

So why is my solution wrong?

But: My teacher told me that 13.2 is wrong. I found some dates of when Moon covers Venus and it seems like they happen almost exactly 3 days before and after new Moon. Examples: new Moon - 6. October, covering - 9. October; new Moon - 5. November, covering - 8. November; covering - 28. March, new Moon - 1. April.

So why is my solution wrong?

tl;dr: 13.2°/day angular velocity is not wrong, the number is correct. It's just that it's not the correct number to use in this case. There's one more step you need to add.

I can't emphasize enough how much I disagree with this answer telling you that you should not have asked your question in Stack Exchange.

It's possible that you were not given the explanation why you were wrong as a way to encourage self-research. One of the greatest and most rewarding things is to build the skills for finding things out on our own, either through seeking out alternative sources or (in some cases) experimenting.

We can't experiment on the Moon, so turning to Stack Exchange was a good move. Unfortunately rather than blurt out the answer then tell you you should not have asked, had I been here first I would have left a comment suggesting you sketch the Earth orbiting the Sun and the Moon orbiting the Earth (ignore Venus for the moment) and then draw their positions ~27.3 days later.

The Moon is in the same position relative to the Earth, but now where is the Earth? A comment or two later and you'd have solved it yourself, and then you could have posted an answer to your own question.

Instead you got "Your wrong, here's what's right, and you should not have asked here in the first place. You did something wrong by not insisting your teacher tell you the answer, after all, they're paid for that."

No. A great teacher will inspire you to explore. Welcome to Stack Exchange!

But I digress!

Now, suppose you have your 27.3 day lunar orbital period giving the 13.2°/day angular velocity, and can't go look up anything else, but want to derive a better answer. What can you do?

There are two ways. You can calculate the lunar month from the synodic month (the 27.3 day orbital period) or you can simply subtract the angular velocities!

The Moon does indeed rotate around the Earth at 13.2°/day. No question about it. But the Earth rotates around the Sun at 360/365.2564 = 0.986°/day in the same direction. The difference is 12.2°/day or a period of 29.5 days, which is a lunar month, the average period between new Moons or full Moons.

If all you know is 27.3 and 365 days, you can just use:

$$\frac{1}{T} = \frac{1}{27.3} - \frac{1}{365}$$

To learn more about the difference between synodic and sidereal periods (we have the same distinction for days as well!) see the following:

Because some folks pay so much attention to how the Moon moves in the heavens, there are a surprisingly large number of kinds of months for the Moon! Here's a table from this answer to Is there a timekeeping word for the orbit of a moon?. The first two are the ones discussed above.

  kind            frame - or - criteria      Earth's moon (days)

sidereal          inertial wrt stars              27.321661

synodic           rotating with Sun-Earth         29.530588

draconitic        orbital plane (precesses)       27.212220
(or draconic)     time between ascending nodes
(or nodal)

anomalistic       time between periapses          27.554551

tropical          time between alignments of      27.321582
Moon's axis with the
planet-moon line

• Wow! That is a really great explanation, thank you! Commented Sep 29, 2021 at 15:28

Your teacher told you that 13,2 is wrong. Why do you think that is? Where did you get this number from? What does it mean to you?

That number looks like 360/27.3, the angular distance that the moon moves around the Earth each day. But the sun is also moving. Relative to the sun, the moon takes 29.5 days to go from new moon to new moon. Since Venus is moving with the sun, perhaps you should use 360/29.5 = 12.2 degrees per day, relative to the sun.

47/12.2 = 3.85

But really you would do better to ask your teacher. They are paid to answer this kind of question.