Projected linear separation between companion stars

[guv]

TL;DR Summary

projected linear separation between companion stars

A quasar with a bolometric flux of approximately 10−12 erg s−1 cm−2 is observed at
a redshift of 1.5, i.e. its comoving radial distance is about 4.4 Gpc.
Assume that the quasar in the previous question is observed to have a
companion galaxy which is 5 arcseconds apart. What is the projected linear separation of the
companion galaxy from the quasar?

The answer key given divides the comoving distance multiplied by angle by (1 + z), z = 1.5 such that
$d = \frac{4.4 Gpc * 5"}{1 + 1.5}$

This answer doesn't make sense because the 5" should stay as a constant as both the distance between the companion and the distance from companion to the Earth should grow at the same rate (Hubble's law). Therefore the correct answer I believe should just be $5.5 Gpc * 5"$ Can anyone more knowledgeable explain if the above reasoning is correct and which solution is correct (even if the question may be silly)?

Thanks,

 

[Bandersnatch]

You're right that the angular separation between objects comoving with the Hubble flow remains constant. But the second galaxy being a companion suggests - to me at least - not merely a visual association, but a bound object. I.e., the question seems to want you to treat the two galaxies as decoupled from the Hubble flow w/r to one another.

So, while the radial distance today is the comoving distance (or, equivalently, proper distance at reception), the proper distance between the two today is the same as the proper distance at emission. If they were X parsecs distant then, they remained X parsecs distant now. Unlike with the radial distance, which has since grown by the factor of z+1 to the present comoving distance.

That's why you need to divide the distance you get from the small angle approximation by the stretch (z+1) - since this distance has >not< grown together with the expansion of space. I.e. it is not true that the observer-galaxy-companion triangle drawn today has the same angles as it had at emission.
It would be true for triangles drawn between unbound objects.

This is the same idea as with the angular diameter of distant galaxies (i.e. bound objects that do not swell with the expansion of space) appearing larger beyond certain distance than similar galaxies nearby. The high-z galaxies seem to loom large in the sky (relatively speaking), as their angular size is given by an image of the light they sent when they were close by.
Cf. 'angular diameter distance'.

 

[guv]

Thanks, how to reconcile with your explanation with the solution of another question for this problem?
What is the bolometric luminosity of the quasar?

The answer given was
$L = I 4 \pi (d_c ( 1 + z))^2 = I 4 \pi ( 4.4 Gpc * ( 1 + 1.5))^2$

I had thought this made sense but now I am more perplexed.

Note that these problems and questions were not written by professionals, so there may be caveats other than being somewhat silly at times. I would just like to make sure it's not a misunderstanding on my part.

 

[Bandersnatch]

For the second question, if there were no expansion, you could just plug in the distance to an inverse square law and get the right answer. And you can still do it for very close galaxies (z=0). But expansion causes photons to be received over a longer interval (by a factor of 1+z), as well as shifting them towards lower energy (by the same factor, again). The two combine to nett the $(1+z)^2$ that you see in there.

That is, if a galaxy is observed at redshift z>0, it appears dimmer for reasons other than just the inverse square of the actual (comoving) distance. Plugging in the comoving distance would nett you too low a number for its bolometric luminosity. You need to correct for the extra $(1+z)^2$ factor to get the right luminosity.

This is briefly discussed in these notes, on pages 16-18 (section 'Flux, diameter...'):
https://www.astronomy.ohio-state.edu/weinberg.21/A873/notes3.pdf

You may want to review the various distance measures in cosmology. Luminosity distance, angular diameter distance, those things. That's what these questions seem to be about.

BTW, combining the effects from the two questions lets you test for whether the universe is expanding, as in such universes the surface brightness of distant galaxies should be lower by a factor of $(1+z)^4$. Two (1+z)'s come from what's discussed in the second question, i.e. the time delay and the redshift. The other two (1+z)'s come from the topic of the first question, i.e. the enlarged angular diameter (squared because it concerns the area). In other words, it seems to have lower overall luminosity, and a larger area, than it otherwise should.
This is called the Tolman surface brightness test, should you want to look it up.

 

[guv]

So you are saying the two given solutions do not contradict each other where in the first part, you divide $d_c$ by $1 + z$; while in the second part, you multiply $d_c$ by $1+z$. I will need to read your posts more carefully. I am relatively familiar with these concepts but the difference in the solution had struck me as odd.

 

[guv]

Interesting read of page 17 https://www.astronomy.ohio-state.edu/weinberg.21/A873/notes3.pdf

The formula after simplification seems to agree with the given solution. While the formula still seem counter-intuitive which I am still struggling to coneptualize, the key relationship is the luminous distance, angular distance, and comovming distance $D_l = (1 + z)^2 D_A = (1 + z) D_c$. This relationship depends on a conclusion drawn from the paper by Hogg 1999 equation 21.

Unfortunately the note has a dimensional error: $D_A$ is dimensionless but $D_L$ is measured in parsecs. There is something contradicting going on using Hogg's formula in the note. Otherwise the note is of very high quality.

 

[George Jones]

Unfortunately the note has a dimensional error: #D_A# is dimensionless but #D_L# is measured in parsecs.

If you put in the steps of the derivation of the note's expression for $D_A$, you can identify the mistake fairly quickly. I could put in the details, but it might be more useful for you to find the mistake.

FYI: if you put a second # at the beginning and end of in-line mathematical expressions, then the LaTeX wll be rendered.

 

[guv]

I found a good stackexchange thread on this https://physics.stackexchange.com/q...vable?newreg=ee254fcb84bb473e9e577b216d1bded0

Is there a formal textbook that I can read to look up this information further. Thanks,

 

[guv]

If you put in the steps of the derivation of the note's expression for $D_A$, you can identify the mistake fairly quickly. I could put in the details, but it might be more useful for you to find the mistake.

FYI: if you put a second # at the beginning and end of in-line mathematical expressions, then the LaTeX wll be rendered.

oops, it looks like I can't edit the posts any more. Thanks for the reminder though.

 

[Bandersnatch]

Is there a formal textbook that I can read to look up this information further. Thanks,

I'm pretty sure virtually every cosmology textbook will cover this one way or another as it's rather fundamental.
A. Liddle's introductory text, 2nd ed. has it in the Advanced Topic 2 section. B. Ryden covers it in chapter 7.
But you seem to have already found the Hogg paper, and that pretty much covers the same material on a similar level. If you want something more advanced, maybe try some of the references from that paper.

 

[berkeman]

oops, it looks like I can't edit the posts any more.

(Fixed the double-# delimiters for you)

 

[guv]

Okay in reviewing the textbooks and hogg's paper which is great, I have arrived at a conclusion, that comoving distance between two objects is greater than their proper distance. $d_C = (1 + z) d_P$ I found my conclusion contradicting to my understanding between the two distances in the past. I had thought $d_P = (1 + z) d_C$ from reading introductory material on this topic. First I want to check if my latest conclusion is correct with folks here.

Then I searched for 'is comoving distance greater than proper distance' and found the following enlightening thread :-)

https://www.physicsforums.com/threa...comoving-distance-and-proper-distance.787000/

I found a great explanation by non-other but Bandersnatch (an equation would have helped more :-)) and really enjoyed the story given by marcus, I was able to match my understanding with his story until the very last bit of definition he gave on comoving distance,

"Because we might want to talk about how much matter is currently in our observable region, and how has that increased over time, as light from more and more matter has come in and we "hear from" a wider and wider circle of stuff. this is not EXPANSION, this is the growing amount of matter of all forms that we have observational data for.
How do we describe how much we expect that to grow in the future? We need a measure of radius which is not affected by expansion and then we can say "that much" matter: out to such and such comoving radius."

I found the last sentence contradicting, how much we expect that to grow in the future, it grows because light from remote objects has enough time in the future to arrive at Earth. If the radius of the visible universe is a comoving radius, it is changing over time. But the defintion of comoving distance (from wikipedia) says:

"Comoving distance factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space (though this may change due to other, local factors, such as the motion of a galaxy within a cluster)."

By the way, equations are welcome, I found a combination of concepts and equations can make things most clear. Thanks!

 

[guv]

So, while the radial distance today is the comoving distance (or, equivalently, proper distance at reception),

I am re-reading this post and I am confused what's 'today' versus 'at reception'. Is it proper distance at emission?

 

[Bandersnatch]

I am re-reading this post and I am confused what's 'today' versus 'at reception'. Is it proper distance at emission?

If it's confusing, it's only because of unclear wording. There's just two distances under consideration here: at emission of a signal, and at reception of the same signal. I.e., how far something we see was then vs today.About the comoving distance, and I hope I'm not just repeating myself too much here, the basic idea is that the comoving distance between any two set points that move solely because of expansion does not change. Like us, and a particular galaxy. Or some two specific galaxies elsewhere.

You take any proper distance at any time in the history of the universe and factor out the expansion by dividing it by the scale factor $a$ at that time, where $a=(z+1)^{-1}$. So that indeed $d_c=(z+1)d_p$ (and not the other way around).
If you've just received a signal that was stretched twice (z+1=2), it means all the distances in the universe back when it was emitted were half the current size (a=1/2). So if you want to factor out the expansion you need to multiply the distance then (i.e. at emission) by 2.

This means, that its numerical value will be always the same as the proper distance between those points today. It's just how it's defined. It's like a grid we paint on the universe, where each galaxy is stationary, but we understand that the proper distances between the nodes on this grid do change as the universe expands. We've painted the numbers on the grid to reflect the distances as they are today.

If, now, we wanted talk about distances on that grid that are changing for some other reason than expansion, like the distance covered by light travelling between galaxies (or the size of the observable universe) we could have the comoving distance vary. Not because we're changing the definition of that grid we had prepared earlier, but because we're traversing it. The comoving size of the observable universe grows, because we're not talking about the same two points, the same two 'stationary galaxies' any more. This is also what the last sentence of the Wiki quote says.

BTW, I'd love to show you more equations here, but it's really just dividing or multiplying by z+1. I think the difficulty tends to be conceptual.

Something else you might find helpful, is comparing the light cone graphs drawn in proper vs. comoving coordinates, like the ones below:

(from Lineweaver & Davis 2003)
The dotted lines are example 'galaxies' observed today at redshifts marked at the top of the graphs. Think of them as nodes on the comoving grid.
Compare the shape of the drawn light cone when you factor out the expansion. Its base is the size of the observable universe.

 

[guv]

I get it now. Thanks!