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The cosmic distance ladder has a wide range of length scales, which are quite difficult to measure and to conceptualize. These distances are commonly quoted, particularly in less technical articles, in light years - the distance that light covers in one year. I would like to probe the precise relationship between astronomical distance measurements and the light year.

A brief look at the cosmic distance ladder shows that it has two distinct types of rung.

  • Most of the rungs depend on standard candles - objects whose intrinsic luminosity can be determined by independent means, such as the period of a Cepheid variable - and the fact that the luminosity $L(r)$ of an object at distance $r$ scales as $1/r^2$. Thus, if we have two objects of the same population (i.e. same intrinsic luminosity) and we have determined object 1 to be at distance $r_1$, then the distance $r_2$ to object 2 can be determined from the observed luminosities $L_1$ and $L_2$ as $$r_2=\sqrt{\frac{L_1}{L_2}}r_1.$$ In essence, then, we are using clever physics to peg the ratio $r_2/r_1$ for some standard object 1 whose distance we determine in some other way.

  • The very first rung, however, is different. From what I can gather, the distances to the closest useful standard candles (visual binaries) are determined via dynamical parallax.

    If I understand this rung correctly, we have a population A of visual binaries which are close enough for parallax to give meaningful distances. This enables us to accurately measure the stars' intrinsic luminosities (via their parallax distances) and their masses (via their orbital parameters), and this population is big enough to get a good handle on the standard mass-luminosity relation. This mass-luminosity relation then enables us to get the intrinsic luminosity of binaries which are too far for parallax but whose orbital parameters (and hence their masses) can still be identified.

    I'm pretty shaky on that first rung - please correct me if I'm wrong.

In the end, though it's the parsec that rules, and we are still in essence measuring astronomical distances in terms of astronomical units - the distance from the Earth to the Sun. We can then use earthbound methods to measure the AU in terms of earthbound distances, and radar methods to compare the AU to the light year, but in the end we're fundamentally measuring in parsecs. Is this all correct? Or did I miss some crucial subtlety?


Some background: I've been reading M.J. Duff's various papers on the meaning of units and fundamental constants, and trying to understand what "varying speed of light" theories, like the one reported here, actually mean. I agree with Veneziano's succinct statement that

the "time variation of a fundamental unit", like $c$, has no meaning, unless we specify what else, having the same units, is kept fixed

so I'm slightly annoyed that papers like this one are getting published without making that clear. In a VSL theory, does light cover more or less than one light year in one year? How could you possibly test that experimentally using yardsticks which are fundamentally in light years? Or, more operationally, in the practical test that Salzano et al. are proposing, what are the yardsticks in?

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    $\begingroup$ The very first rung is radar measurements of distances in the solar system, followed by parallax measurements to nearby stars. This is sufficient to get main sequence fitting distances to clusters, some of which contain Cepheids... All in parsecs. Gaia is important! $\endgroup$ – Rob Jeffries Apr 7 '15 at 22:20
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    $\begingroup$ The proper motion and Doppler shift of ballistically coupled clusters gives us a second, distinct rung for distances larger than the solar system but smaller than the galaxy. And I believe that that rung is independent of the parsec. It is possible that this is the core of a good answer, but I am now at the limit of what I recall on the matter. $\endgroup$ – dmckee Apr 7 '15 at 22:28
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    $\begingroup$ @RobJeffries So, are we going to get an answer from a professional astrophysicist? $\endgroup$ – WetSavannaAnimal Apr 7 '15 at 23:01
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    $\begingroup$ @Rob en.m.wikipedia.org/wiki/Moving_cluster_method Doesn't get a lot of press, and never been very accurate but independent of the AU and these day we can range some of those clusters by parallax. $\endgroup$ – dmckee Apr 8 '15 at 4:57
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    $\begingroup$ More on the moving cluster method. $\endgroup$ – Emilio Pisanty Apr 8 '15 at 18:40
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You are correct. If you worked through the same steps on the distance ladder in a universe with a different speed of light, you would find the answer in light years would be different.

Your question on whether light travels one light year in one year, though trivial sounding, opens the door to difficulties that are inherent in the nature of space-time. Your analogy with the yardstick is a common example used in GR texts to illustrate the difficulty with saying that an object is some distance d away from an observer. Fundamentally, there is no such distance, because, well, distance is relative! Aside from luminosity distance, other distances such as angular diameter distance may be used. Angular diameter distance, as an example, compares intuitively to seeing nearby objects as being larger than distant objects. However, the angular diameter distance differs from the luminosity distance for the same object.

In Salzano et al., it appears that angular diameter distance is the primary measure used. The reason for this is that when looking at the Cosmic Microwave Background you see perturbations on a map, so you measure their size by measuring how large they appear to be on the sky. There is no simple way to measure how far away these perturbations are, as it is impossible to extend a ruler into past. The important point to understand, though, is that as long as the same distance measure is always used, useful comparisons can be made.

As an interesting example of how luminosity distance and angular diameter distance differ, imagine looking at the same object at more and more distant locations (and thus further in the past). The further away the object is, the dimmer it will become. However, as you get closer to the big bang, the object becomes larger relative to the universe. Indeed, the entire observable universe would have fit inside a star at some point. If you look back from the earth at an object far enough away that it took up a large portion of the universe, suddenly that object takes up a large portion of the sky. Thus, objects eventually appear to grow in size as they get further away. This is why, in papers like Salzano et al., you find that more distant/earlier perturbations appear to be larger than closer/later perturbations.

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  • $\begingroup$ Hi Blake. This sounds like an erudite answer and highly plausible: do you work in the field? $\endgroup$ – WetSavannaAnimal Apr 7 '15 at 23:01
  • $\begingroup$ To be honest I'm quite confused by the angular diameter distance. Wikipedia defines it as "An object of size $x$ at redshift $z$ that appears to have angular size $\delta\theta$ has the angular diameter distance of $d_A(z)=x/\delta\theta$", but how do you measure $x$? What sort of objects is this usually applied to? $\endgroup$ – Emilio Pisanty Apr 7 '15 at 23:09
  • $\begingroup$ @ Rod Vance: Yes. @ Emilio Pisanty: Again, it's not something you can actually measure, but it's a quantity which is shared between all the distance measures. $x$ isn't any physical or measurable quantity, but it allows you to relate comoving distance, angular diameter distance, luminosity distance, etc. $\endgroup$ – Blake Hartley Apr 7 '15 at 23:09
  • $\begingroup$ So how do you measure angular diameter distance? $\endgroup$ – Emilio Pisanty Apr 7 '15 at 23:14
  • $\begingroup$ You don't actually measure it. It is incorporated in your model of the growth of perturbations in the universe. Indeed, looking at Salzano's paper, they only make reference to $D_A(z)$ and how it behaves in their model universes. $\endgroup$ – Blake Hartley Apr 7 '15 at 23:18

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