Einstein ring and Einstein Cross

[hongseok]

TL;DR Summary

Why does an Einstein ring sometimes appear and sometimes an Einstein Cross appear?

Some say it is due to the distribution of the mass of the galaxy in front, while others say it is due to alignment. What is right? Are both correct?

If it is due to the galaxy's mass distribution, I wonder if it can be affected by the type of galaxy and the angle between the galaxy's rotation axis and the Earth. I also wonder if the distribution of dark matter could also affect this.

Why does an Einstein ring sometimes appear and sometimes an Einstein Cross appear?

Some say it is due to the distribution of the mass of the galaxy in front, while others say it is due to alignment. What is right? Are both correct?

If it is due to the galaxy's mass distribution, I wonder if it can be affected by the type of galaxy (like an Elliptical Galaxy, Spiral galaxy) and the angle between the galaxy's rotation axis and the Earth. It's the same principle as how the shape of the dot behind the glass looks different depending on the glass model.

I also wonder if the distribution of dark matter could also affect this.

 

[Ibix]

You get a ring from a circularly symmetric mass distribution lensing light from an object more or less directly behind it. You get a cross (or at least you can) from an elliptical mass distribution like a galaxy seen edge on. So yes, if you have two identical disc-shaped galaxies lensing light from two identical sources, you could get a cross from one and a ring from the other depending on the orientations of the lensing galaxies.

Dark matter affects light paths the same as normal matter.

 

[hongseok]

You get a ring from a circularly symmetric mass distribution lensing light from an object more or less directly behind it. You get a cross (or at least you can) from an elliptical mass distribution like a galaxy seen edge on. So yes, if you have two identical disc-shaped galaxies lensing light from two identical sources, you could get a cross from one and a ring from the other depending on the orientations of the lensing galaxies.

Dark matter affects light paths the same as normal matter.

So, is it wrong to say that it is due to alignment?
And can I get a source for your claim?

 

[Ibix]

So, is it wrong to say that it is due to alignment?

The exact shape will depend on alignment, yes. You'll only get a perfect ring around a perfectly symmetric source perfectly aligned with a perfectly symmetric lens, but you'll get things like the ring illustration in your OP from nearly symmetric situations. I don't think alignment can turn a ring into a cross, although you can certainly have crosses from nearly symmetric situations that can be very ring-like.

And can I get a source for your claim?

Which one?

 

[Ibix]

And can I get a source for your claim?

If it's just the point about alignment, all orbits around spherically symmetric sources lie in a plane. A source, a lens and an observer that are not colinear define a plane, and only light emitted in this plane can reach the observer. Light emitted in any other direction has a path that lies in a plane that doesn't include the observer. So you will get at most two images, one each side of the lens, that smear into a ring if the source, lens and observer become colinear. Thus alignment can't create a cross from a spherical lens.

Elliptical sources are harder to reason about, but there's detail here: https://lweb.cfa.harvard.edu/~dfabricant/huchra/ay202/lectures/lecture12.pdf. If you can program in python (or anything else where someone's written a numerical integrator) you can get lensing patterns from the weak field metric for arbitrary lens mass distributions if you really want.