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Discussion in comments under the question Does seeing a gravitationally lensed/magnified galaxy imply that they could also see us as well? have got me wondering about the geometry behind detection of distant galaxies that were made possible by gravitational lensing-induced magnification (in size but more importantly in brightness) like the scenarios discussed in What exactly is it that is being magnified 50 times in this gravitational lensing observation? and its answers.

Acknowledging that these are not simple lenses with well-defined focal lengths, we perhaps can no longer relate the magnification to the ratio of the distances from the lens to the conjugate focal planes. So I'd like to ask.

Question: For detection scenarios of distant galaxies that were made possible by gravitational lensing-induced magnification, roughly what are the distance ratios from the lensing systems to the observing and detecting galaxies?

note: In this question the "detecting galaxy" is always our galaxy

Is it like 1:10? Or 1:1000. Or larger? Or smaller?

Related questions and answers, some may provide helpful examples for this question:

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    $\begingroup$ Relevant: en.wikipedia.org/wiki/Einstein_ring which has the equation $$\theta_1 = \sqrt{\frac{4GM}{c^2}\;\frac{D_{LS}}{D_S D_L}}$$ $\endgroup$
    – PM 2Ring
    Commented Feb 14 at 1:07
  • $\begingroup$ Observed angles of deflection are typically several arcmin. If they get much larger, the image would be too faint. So you could play around with different sets of $D_\mathrm{L}$ and $D_\mathrm{S}$ in @PM2Ring's equation to see what would give you such values of $\theta$. However, those distances are angular diameter distances, so you would need to convert to physical (comoving) distances to get your ratio. $\endgroup$
    – pela
    Commented Feb 14 at 22:09

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For $z < 1.5$ the diameter distance grows linearly so we have,

$$\sqrt{\frac{D_{LS}}{D_S D_L}} = \sqrt{\frac{1 - D_L/D_S}{D_L}}$$ If the source galaxy is twice as far as the lens galaxy, then the deflections are at about 70% of the maximum at infinity. But, if the source is only 1% farther, then the effect is only 10% of maximum.

Therefore, the answer is that a source to lens anglar distance ratio smaller than 10:1 still has detectable lensing. A ratio of 2:1 gives nearly maximal magnification. A ratio of 1.01:1 would affect the photometry, but the source images will probably be within the lensing galaxy and may be hard to distinguish from galaxy structure. Whether lensing can be detected at such low ratios would depends on the light distribution of the lens.

For high z or for higher precision one should use the more precise formula for $D_{LS}$, and the above formula becomes: $$\sqrt{\frac{D_{LS}}{D_S D_L}} = \sqrt{\frac{1 - \frac{(1+Z_L)D_L}{(1+Z_S)D_S}}{D_L}}$$

which compensates for the angular diameter distance D_S slowly shrinking at high z. It is now saying that if the redshift ratio is 2:1 the lensing deflections can be reasonably strong.

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    $\begingroup$ The equation you give implies that $D_\mathrm{LS} = D_\mathrm{S}-D_\mathrm{L}$, but this is not true. Remember that it is the angular diameter distance, so the expression for $D_\mathrm{LS}$ is not so trivial. $\endgroup$
    – pela
    Commented Feb 14 at 21:42
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    $\begingroup$ (not my downvote though) $\endgroup$
    – pela
    Commented Feb 14 at 21:49
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    $\begingroup$ @pela - This is an order of magnitude type question. If D_S/D_L is of order 2, then one will have reasonable lensing, maybe not exactly 71% of maximum. And it is not as I see it focused on lenses z above 2 where things become nonlinear. But, I will put in a warning about high z. $\endgroup$
    – eshaya
    Commented Feb 15 at 0:45
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    $\begingroup$ Right, okay. I guess I'm mostly into the high-z Universe, where using the first formula gives imaginary numbers. $\endgroup$
    – pela
    Commented Feb 15 at 10:27

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