Can we see the south Pole of the Moon standing on North Pole of the Earth

Question

As the Moon's orbit is inclined to Earth's Equator and radius of earth is only 6350Km but Moon is 385000Km away, we should be able to see the south pole of the Moon from below- i.e standing on North Pole of the Earth

Answer 1

No, people at the North pole (of Earth) cannot see the South Pole (of Moon).

We can actually find this by some basic trigonometric relations and considering the orbits of Earth and Moon: ( taken from Orbit of the Moon)

Since, the south pole of the moon can be viewed only at an angle that is tangential to the moon like:

Here, the south pole of the moon can only be viewed below the red line: that is the yellow region on the Earth. Note that this image is not scaled and hence the scaled image might be quite different. But if we solve it, we can find the yellow region mathematically just like below:

We have:

1. $D_m$ = Distance between Earth and Moon
2. $R_e$ = Radius of Earth
3. $r_m$ = Radius of Moon
4. $\theta = 5.14^{\circ}$
5. $\phi = 23.44^{\circ}$
6. $\alpha = \angle ACB$

In triangle CDF: we have $sin \theta = \frac{r_m+a}{D_m}$. Using this we can find the value of a.

In triangle AGC: we have $sin \alpha = \frac{R_e}{a}$. Using this we can find the value of $\alpha$.

Solving these equations, we get $a$~$34000$ Km and $\alpha$ ~ $10.775^{\circ}$.

Hence, anything above $\phi + \alpha = 34.215^{\circ}$ above the Earth's equatorial plane cannot see the south pole of the Moon directly.

Answer 2

The moon rotates at exactly one revolution per orbit. This keeps the same side of the moon facing the Earth. The moon is indeed inclined to the Equator (in fact it orbits close to the plane of the ecliptic) But this fact is not very significant to the answer to the question. The moon could orbit in any plane in such a way that the same face was always pointed towards the Earth.

However, as the moon does orbit in the ecliptic, it is visible from the North Pole.

Now there is a 6.7-degree axial tilt between the moon's orbit and its equator. This means that at some points in its orbit the South pole of the moon is tilted by 6.7 degrees towards Earth. It is visible from the North Pole as well as everywhere else on Earth. At the north pole, the Earth's radius reduces this to about 5.7 degrees From the South pole it would increase to 7.7 degrees. But 7.7, 6.7 or 5.7 is still a very difficult observing angle. You can't see easily what the polar regions are like, even at times of maximum tilt. You can't see the South pole of the moon "from above" from anywhere on Earth.

I think your misconception is that you think that the North pole of the moon points in the same direction as the North pole of the Earth. It doesn't. It is tilted so that it is nearly perpendicular to the moon's orbit in the plane of the ecliptic.