Why is Earth rotating along $x$(day and night) and $y$(tilt, causing seasons) axis, but not along the $z$ axis (taking $z$ as the axis along orbital)?

http://en.wikipedia.org/wiki/Axial_tilt

It is obvious that the rotation of any planet is a combination of rotation along three axis x,y,z. Lets take the z axis as the one being the one as the Earth moves along the orbital around the sun. It is obvious that Earth is then rotating along the x axis (let's take that one for days-nights) and there is a tilt (which we can then take as the rotation along the y axis, for seasons). Question:

1. But I found nothing on why Earth is not(or anything about it's rate) rotating along the z axis?
• Can anybody please tell me why I got a downvote on this? – Árpád Szendrei Dec 2 '16 at 18:27

This is ultimately down to Euler's rotation theorem; every rotation has a line of fixed points and hence an axis which it rotates around.

What is slightly more interesting physics-wise is why these axes line up with, or fail to line up with, the orbits of the planets around the sun, which are rotations of a different sort. Ultimately you get a cosmological picture of the early solar system as a big accretion disk with various calamities befalling the tilted planets since then.

It is obvious that the rotation of any planet is a combination of rotation along three axis x,y,z.

You happened to choose a frame of reference, the ecliptic frame, in which the Earth's rotation axis isn't oriented along one of the reference frame's principal axes. You wouldn't see this as a problem had you picked a frame in which the fundamental plane is the Earth's equator rather than the Earth's orbital plane.

Rotation in two dimensional space is tame, quite weird in four dimensional space, and weirder yet in even higher dimensional spaces. Rotation in three dimensional space (e.g., our universe) lies somewhere between tame and weird. One consequence of rotation in three dimensional space is that a rotating object can be described as having a single instantaneous axis of rotation. The is one of the many consequences of Euler's rotation theorem (referenced in CR Drost's answer).

From the perspective of an equatorial-based frame, the Earth's rotation is almost purely about the z axis. There are a number of factors that make this "almost purely" rather than "purely". A rotating rigid body with non-coinciding angular velocity and angular momentum vectors will undergo a torque-free precession. While the Earth isn't a rigid body, this torque-free precession does exist, with non-rigid complications. This is the Chandler wobble.

Another factor is that the Earth comprises multiple rotating elements: The mantle+crust, the outer core, inner core, atmosphere, and oceans. There are seasonal variations in the Earth's rotation rate and its rotation axis due to different behaviors of the atmosphere in northern hemisphere winter versus summer, transfer of water (as snow) from equatorial regions to Siberia during the winter and transfer of water (as melted snow) back to equatorial regions during summer, and transfer of water due to complex ocean currents. Exchanges between the mantle and core result in decadal variations in both the rotation rate and the direction of the rotation axis.

Even longer term, torques from the Moon and the Sun make the Earth undergo a slow but large precession of roughly 26,000 years. This axial precession is one of the key causes of ice ages. Even longer term, tidal forces from the Moon and the Sun are making the Earth's rotation rate slow down, and most likely are subtly changing the direction of the Earth's rotation axis.

The reason that any of the rotation axes are somewhat constant is due to conservation of angular momentum, which is generally true for any system, about any axis. We do have to take into account the whole system though, meaning not just the Earth, but also the sun, the moon, etc. In this complicated system of many bodies, the Earth might slowly transfer some of its angular momentum (through gravity) to other celestial bodies, tilting both of their rotation axes in opposite directions, but always keeping the total angular momentum constant. You might imagine defining a combined rotation axis for the entire solar system, which would then stay constant (that is, until you wait long enough and you have to take the other starts into account also...).

We thus see that your different rotation axes are unstable at different time scales. This time scale then depend on how quickly bodies can transfer angular momentum back and forth, compared to the size of the angular momentum in question. In the case of your $z$ axis, we are talking about the angular momentum due to the Earth's orbit around the sun, which is much bigger than the angular momentum due to the Earth's spin. We would then really also expect the $z$ axis to change, but on a bigger time scale.

• Thank you. "In the case of your z axis, we are talking about the angular momentum due to the Earth's orbit around the sun, which is much bigger than the angular momentum due to the Earth's spin." Does this mean that this is the reason why there is no rotation around the z-axis, because the orbital momentum is much bigger and cancels it out? – Árpád Szendrei Dec 1 '16 at 22:47
• Yes, but be careful with the language: There is rotation around the $z$ axis (this is just the Earth orbiting the sun). You are referring to the wobble/tilting of this axis, which as you state is zero (or at least very small). This is indeed because this orbital angular momentum is huge, and there is not much around (e.g. heavy, nearby planets) which can alter it within a reasonable time scale. – jmd_dk Dec 1 '16 at 22:52
• hmm. I think I do not understand. I am taking the z axis as being along the movement of orbit. so that z axis is adjusted as the earth orbits(from viewpoint of the sun) , so the z axis itself is adjusted and 'moving' around the sun (we do not feel that because to us it seems like the earth is moving straight). Now what I mean by earth – Árpád Szendrei Dec 2 '16 at 18:32
• 's rotation around the z axis is if the earth would be spinning around the z axis too and that would add to the combined rotation in 3D. But it is not. So to me it seems like earth's rotation in 3D is only a combination of spinning around the x(gives days and nights) and y(this gives the tilt) axis. If there would be a spin around the z axis, then Earth would also rotate somewhat from the north pole to the south pole (so the earth would also be tilted on the axis z, so along the orbital movement, which is the z axis for me). But the earth is only tilted on the x and y axis. – Árpád Szendrei Dec 2 '16 at 18:40
• The tilt does not involve the z axis. So the combined rotation does not make the z axis tilt not even a little bit (from the sun's point of view), right? – Árpád Szendrei Dec 2 '16 at 18:43

See this Wikipedia article. On shorter time scales, the Earth rotates around one single, fixed axis, which is what it should do in the absence of external torque to satisfy angular momentum conservation (assuming perfect rotational symmetry of the planet about its axis of rotation; this is not quite correct, see the comment by David Hammen below). Notice that this is the generic behavior of any body that is not subject to a torque acting on it. In other words, in the absence of torque any solid body with rotational symmetry about its axis of rotation will revolve around a single, unique axis of rotation, and the question for other axes of rotation has no meaning.

However, due to the fact that the Earth is not a perfect sphere, the presence of other heavy bodies (moon and sun, in particular) indeed causes a small amount of torque, which causes the slow precession of the Earth's axis of rotation.

• Re On shorter time scales, the Earth rotates around one single, fixed axis, which is what it should do in the absence of external torque to satisfy angular momentum conservation. This is incorrect. Even on shorter timescales, while the Earth's angular momentum is more or less constant, the Earth's angular velocity is not. An object with a non-spherical mass distribution that is not subject to external torques but is not rotating about a principal axis will undergo torque-free oscillations. In the case of the Earth, this is the Chandler wobble. – David Hammen Dec 1 '16 at 23:01
• Hmm, yes, you are correct. I had naively assumed rotational symmetry of our planet relative to its axis of rotation, which is not quite correct. I'll simply point to your comment in my answer above. – Pirx Dec 1 '16 at 23:13