# How distorted will our galaxy be if we are viewing it from several thousand light years away?

I saw this answer and read this sentence regarding the distortion of our galaxy when viewed above the galactic plane:

Once you got to a point where the entire [sic] galaxy was within your vision, the entire galaxy would look distorted, since our galaxy is 100,000 light years [sic] across, the light from the edges of the Galaxy would be thousands of years old, and would not match up with how the spiral should appear.

I am thinking that this could potentially be true, but I am not sure why or how distorted it would be. Does this fact hold, and for what distances will it no longer be distinguishable?

Note: I have thought that it will be well-blended because the distance does not spike, it increases gradually, but I need verification.

## 2 Answers

The light travel time of 100,000 years is quite small compared to the time it takes the Milky Way's spiral arms to complete an appreciable fraction of one rotation. The arms have a pattern angular speed of $$\Omega_{\text{p}}=28.2\pm2.1\text{ km}\text{ s}^{-1}\text{ kpc}^{-1}$$ (Dias et al. 2019), so they should complete one full rotation on the order of $$\tau=2\pi/\Omega_{\text{p}}\approx218\text{ Myr}$$.

An observer a distance $$h$$ above the center of the galaxy would observe light coming from a radius $$R$$ to have to travel an additional time $$\Delta t=\frac{1}{c}(\sqrt{h^2+R^2}-h)=\frac{R}{c}\left(\sqrt{\left(\frac{h}{R}\right)^2+1}-\frac{h}{R}\right)$$ which is a decreasing function of $$h/R$$. Its maximum is at $$h=0$$, where it would be on the order of 100,000 years for stars on the outer edge of the disk. Clearly, $$\Delta t\ll\tau$$, so there would not be an appreciable distortion, per se, because of the additional distance. While it's true that the objects you'd see would indeed have moved some amount by the time the light reached you, it wouldn't be significant.

• Also, just take a look at any other galaxy. They don't look distorted. – pela May 9 at 19:12
• @pela "They don't look distorted"... considering we have but one view of any galaxy (changes to our position are negligible on that kind of scale) how is it possible to ascertain whether or not that view is distorted? – spender May 10 at 8:51
• @pela: That doesn't quite follow. For pretty much any galaxy other than the Milky Way when viewed from the Milky Way, the distance to the galaxy $h$ is much larger than its radius $R$. And since $\Delta t \to 0$ when $h \gg R$, we wouldn't expect significant distortion of such galaxies. – Michael Seifert May 10 at 15:41
• @pela that sounds like the allegory of the cave. – Eric Duminil May 10 at 17:02
• @MichaelSeifert Why would $\Delta t \rightarrow 0$ for $h>>R$? For any [image of a] galaxy, the near end is $R$ closer than the far end (modulo its inclination). It's true that $\Delta t/t \rightarrow 0$, but the appropriate number is $\Delta t$, not $\Delta t / t$ (if I understand correctly what the OP means by "distortion"). – pela May 10 at 20:07

Yes, there would be some distortion, but not enough to make a visible difference. Let's do a rough calculation.

Assume the galactic disc has a diameter of $$100,000$$ LY, and we are viewing it from a point $$50,000$$ LY directly above the galactic centre, so the galaxy has an angular diameter of $$90°$$. The distance from the edge is $$\sqrt2×50,000 \approx 70,710$$ LY, so the image of the edge is lagging the image of the centre by almost $$21,000$$ years.

However, that's not much time compared to the time it takes the galaxy to rotate. Galaxy rotation is rather complicated, but roughly speaking, the inner part of a spiral galaxy rotates much like a solid disc, with a constant angular speed, so the rotation speed at a given radius is approximately proportional to the radius. The outer parts tend to rotate at a roughly constant linear speed. Wikipedia has some details for the Milky Way, including this graph:

Galaxy rotation curve for the Milky Way – vertical axis is speed of rotation about the galactic center; horizontal axis is distance from the galactic center in kpcs; the sun is marked with a yellow ball; the observed curve of speed of rotation is blue; the predicted curve based upon stellar mass and gas in the Milky Way is red; scatter in observations roughly indicated by gray bars, the difference is due to dark matter

Our Solar System, at a radius of roughly $$27,000$$ LY from the galactic centre, takes around $$225-250$$ million years to orbit the galaxy, and the outer parts of the galaxy rotate at roughly the same speed. So we can expect stars at $$50,000$$ LY to have a rotation period around $$420-460$$ million years.

So that $$21,000$$ year time lag we calculated earlier corresponds to only about 1 arc-minute ($$\frac1{60}$$ of a degree) of rotation. I don't think that's very noticeable. ;)

• Aren't you only examining one aspect of the situation? Individual planets can collide,died, born, or etc and we won't see that for potentially 100,000's of years or millions. A blackholes and all sort of other things could be forming and changing and we won't see those changes for a long time. Individual planets in other solar systems orbit in periods much short than we are talking about here. – cybernard May 10 at 19:49
• @cybernard Fair point, but the OP is asking about visible distortion of the galaxy's spiral structure due to the time differential. At distances of 50-70 thousand lightyears, you aren't going to see individual stars or star systems with the naked eye, or even a small telescope. – PM 2Ring May 10 at 20:12