Actually, I'd advise against looking for that, without first steeling oneself against crackpottery. The most popular one I know of is 1. somewhat inaccurate, 2. loaded with nonsense annotations (paraphrasing: see Earth does not circle around the sun, but rather it's a 'vortex'! therefore scientific cabal; ivory tower; we're being lied to; etc.).
As for the question, it's all a matter of choosing a reference frame (i.e. with respect to what you want to describe the motion) and adding all component motions together.
For the reference frame, on the largest scale, since cosmic expansion was mentioned, I'd choose the CMBR-rest frame. It's a family of frames in which the radiation coming at us from the earliest universe looks roughly the same in every direction. It's not a bad way to think of it as 'how is the Earth moving w/r to the average distribution of matter in the universe'.
1. So you start with Earth going around the Sun in somewhat wobbly, nearly circular ellipses. It does so at 30km/s.
2. Add to that the motion around the galactic centre. Here, we need a few steps:
- include the roughly circular orbit due to the bulk rotation of the galaxy. This changes the circular path into a helical one, where the distance between the 'bends' in the helix are approx. 40 times the its radius (so it's a pretty elongated one). But the helix is angled backwards - like a slinky toy leaning to one side. This is due to the plane of the solar system being angled at approx 60° w/r to the direction of its motion.
Kinda like this:
View attachment 193714
For added complexity, this alignment does not change as the Sun orbits the galaxy - the helix gets slowly squashed sideways, then relaxed, then squashed again.
- if we looked at this elongated, bent helix from afar, it'd be just a line. This line has to bend around the galactic centre to form a circle. The radius of curvature is so large, however, that it would not be noticeable at the scale where you can resolve the helix.
- additionally, it needs to follow a sinusoidal path perpendicular to the plane of rotation (plane of the galaxy). This is due to peculiar motion in the plane-normal direction taking it slightly above the disc, where mass in the disc pulls it back, and the Sun overshoots in the opposite direction, moving below the disc. And so on. I don't remember the estimated frequency of those oscillations, but it's probably on the order of a dozen per orbit. Again, only really noticeable when you zoom out from the helix.
The two above look like this:
View attachment 193715
(the picture has the +z axis pointing towards the galactic south)
- finally, the remaining motion due to interactions with local stellar neighbourhood should be added (i.e., the remaining 'peculiar motion'). This further changes the circular base orbit into a somewhat elliptical one.
The peculiar motions are subject to local interactions, and are likely to chaotically change over time.
3. Adding to that motion of the Milky Way w/r to the Local Group of galaxies - this means mostly just the hurtling towards the Andromeda galaxy.
This changes the wobbly, elliptical orbit around the galactic centre into another leaning, helical one. This one looks very squashed, since MW moves towards Andromeda almost edge-on, with inclination of only 20°
View attachment 113456
The distance between bends of this helix is approx 3 times its radius. It's much more tightly wound than the previous one.
4. Finally, add the motion of the Local Group w/r to the CMBR-rest frame. This is pretty much a straight line in the direction of the Pump (Antlia) constellation.
This changes direction of the last helix like so (rather crudely eyeballed and drawn, I know, but mostly preserves proportions):
View attachment 113465
You need to imagine the already twisted helix from 3 as going towards Andromeda changing direction towards the 'net motion' one. Think of it, again, as a slinky toy with one end at the origin, and the other being dragged from the one arrow to the other. The trick is to preserve the angle of the bends as you do so.
So this step makes the helix from 3 more elongated and twisted in yet another way.
It is worth keeping in mind that these are the motions that can be thought of as representing reality only in this particular moment in time or in rough terms. As millions of years go by, and the stars and galaxies dance around each other in their gravitational ballet, there will come deviations and largely unpredictable changes.Now, can you imagine all these component motions happening at the same time? If so, good for you. :)
Personally, I tend to stick to the geocentric frame most of the time, else I get dizzy every time the Sun rises over the horizon.
