Rung 2: Main Sequence Fitting

The second rung of the distance ladder is main-sequence fitting. Most stars are burning hydrogen to form helium, and if you plot their luminosity against their colour (a so-called Hertsprung-Russell diagram), you find that they lie in a line called the “Main Sequence”. More massive stars are luminous and hot, and hence appear blue. Less massive stars are faint and red. Using parallax distances to the Hyades Cluster we can convert the measured fluxes into luminosities and hence fit the main sequence and thus determine the luminosity of a star of a main-sequence star of any particular colour.

We then go out to the nearest easily observable galaxy, the Large Magellanic Cloud (LMC). We measure the fluxes and colours of lots of stars in the LMC, and identify the main sequence. We then look at how much fainter stars of a given colour are compared to similar stars in the Hyades.

Consider two stars of the same luminosity L. Their observed fluxes are given by the inverse square law

f=L4πD2

Take the ratio of the fluxes of the two stars. Rearranging, you find that

D2D1=f1f2

Using this, you can estimate the distance to the LMC. Unfortunately, you cannot get distances to any more distant galaxies, and main-sequence stars are too faint and too crowded to measure individual brightness’s clearly even in the Andromeda galaxy.

One problem – the stars in the Hyades have a different chemical composition to those in the LMC. In particular, they have more “metals”. To an astronomer, a metal is any element heavier than hydrogen and helium! It could well be that this composition difference affects the luminosities and colours of main-sequence stars. In addition, the luminosity of a star on the main sequence depends somewhat on how old it is – the Hyades is much younger than most LMC stars. It is possible to correct for these effects, but it is unclear how accurate these corrections are.

Rung 3: Cepheids

To go beyond the LMC, we need something brighter than main sequence stars, so they can be seen out to greater distances. Cepheid variables are pulsing giant stars (and hence nice and luminous). They have a layer of doubly ionised Helium within them that traps heat inside. This causes the stars to expand as the heat builds up. As they expand, they cool down, until eventually the Helium ceases to be doubly ionised. At this point, the built-up heat can escape and the star shrinks. Eventually it gets hot enough for the helium to become ionised once more, and the whole process repeats.

Crucially, the luminosity of cepheid variables correlates with their pulsation period (the Leavitt law). This calibration is measured in the Large Magellanic Cloud (LMC). You can then look for pulsing stars in more distant galaxies and use the ratio of fluxes to work out the ratio of distances. With ground-based telescopes this method easily gets you out to the Andromeda galaxy, and perhaps ten times further. But that’s still only far enough to get a handful of galaxies (space-based measurements extend this considerably further).

Unfortunately, many Cepheid variables are partially obscured by dust clouds, which can make them appear abnormally faint. Infra-red observations can reduce this effect but are hard. It may also be that the Leavitt law depends on the chemical composition (amount of “metals”), which would be a problem as the law is calibrated in the LMC (low metals) but mostly applied to galaxies like Andromeda (high metals). The metal dependence is controversial.

Rung 4: Tully-Fisher

Using galaxies in clusters, it has been found that for certain types of spiral galaxy, their luminosity correlates with their rotation speed, in the sense that brighter galaxies spin faster. This tells us that the amount of dark matter in a galaxy (which determines the rotation speed) correlates with the number of stars (which determines the luminosity).

We can use the doppler effect to measure the rotation speed, and hence the luminosity. The flux then gives us the distance, via the inverse square law. Unfortunately, there are very few spiral galaxies close enough to measure their distances using Cepheid variables, so the Tully-Fisher relation is not well calibrated. It is also not very accurate – there is an 18% scatter in distances measured using it.

Rung 5: Type Ia Supernovae

Type Ia supernovae are carbon-oxygen white dwarf stars that become dense enough to ignite fusion and hence explode. They are all roughly the same luminosity. It turns out that their exact peak luminosity correlates with how long they stay bright and using this relationship they can be used to produce quite accurate distances. And they are so bright that they can be seen out to enormous distances.

Unfortunately, few have occurred close enough that we have some other way to measure their distance, so while we know that they all have the same peak luminosity, we have trouble knowing what this peak luminosity is.

In Conclusion, measuring distances is hard. We have five rungs in our distance ladder, and each of them has problems. The problems all multiply together as you move up the ladder, so that the end result is extremely uncertain. For decades, cosmologists fought over this, often getting widely inconsistent answers.

I enjoyed re visiting this subject in my research for this article but am now considering that I will probably need to shorten my answer for our tour.  Otherwise we could be there from sunset until sunrise!  But I hope for those of you that managed to reach the end of it, you found some answers to your questions.

Until next time this is Stargazing Tenerife team signing off and wishing you all a very Happy & Prosperous New Year.