### A Bayesian Approach to Locating the Red Giant Branch Tip Magnitude (Part II); Distances to the Satellites of M31

It sometimes comes as a surprise to non-astronomers that one of the hardest things to do when you look at the Universe is to measure distances to objects out there. People have head about the almost 100 year battle to measure Hubble's Constant, but what people fail to realise is that all of the uncertainty was in measuring distances; how far is it to that galaxy, or that supernova, or that star.

Books have been written about the titanic struggle of measuring and calibrating distances in the Universe, so I am not going to cover that here again. But let's talk about my (and my collaborators) effort in the field.

I've written before about some work I've been doing with PhD student, Anthony Conn, using the tip of the Red Giant Branch to measure the distances to the dwarf galaxies orbiting our nearest neighbours, the Andromeda (M31) and Triangulum (M33) galaxies.

It's easy to understand the method, basically it says that things are fainter when they are further away. If you know how bright things truly are, you can calculate their distance using the inverse square law.
The problem is know how bright something is really is. This is where the tip comes in.

Here's some colour-magnitude diagrams for a globular cluster. The stars are not all over the place, but lie in particular places.

The main sequence is where stars are burning hydrogen into helium in their cores. This is where th Sun finds itself. One day, however, the hydrogen fuel in the core is used up, so what happens then?

Stars are simple objects. Basically gravity squeezing inwards is balanced by energy (in terms of pressure) pushing outwards. So, when the energy flow form the core is used up, the star starts to collapse in on itself. The squeezing rises, the temperature sizes, until a shell of hydrogen starts to burn into helium just outside the core.

However, this burning changes the properties of the star, with the flow of energy into the outer parts of the star, causing it to swell up. As it swells, the atmosphere cools and becomes red, but because the star is getting larger, it actually emits more radiation into space. The star has become a Red Giant (this is the future for the Sun!).

The swelling stars are the line of stars up the right hand side of the picture. The star continues to swell, and get brighter and brighter. Due to continual squeezing though, the core gets hotter and hotter, until it BOOOM, the core ignites again, burning helium into heavier elements. This is called the helium flash.

The outer layers of the become less luminous and the star drops back down the giant branch. The cool thing is that the point that this happens is the same for all stars (there is an effect of the chemical composition of the star, but that's a smallish effect). So, the tip of the red giant branch, the point in the colour-magnitude diagram where the stars stop getting brighter and fall back down, is a standard candle, something we can use to measure distances. And this is what Anthony did.

Now, that might make it sound easy, but the data we are working with is not as clean as the picture up there, there are a mess of contamination from stars in our galaxy, to faint galaxies at the limit of detection. Here's an example of what we are working with;
The top is the colour-magnitude diagram, with the box being the area of the red giant branch we are interested in. The inset box shows a dwarf galaxy orbiting Andromeda, where we have used colour-coding to note how far the star is away from the centre of the dwarf; this allows us to more robustly measure the tip.

The bottom right box is the luminosity function, with the bright being on the left, and faint on the right (I know, I know, astronomers are stupid for using the barse-ackward magnitude system). Above the tip, no stars, then we have a sharp jump at the tip and then more and more stars below.

The bottom left is our measurement of the location of the tip in this case. Notice that we don't have a single number, we have a probability distribution function; the peak of this distribution might be the bestest value for the location of the top, but the width of the distribution is also very important, show how accurately we have made the measurement. I will stress again, you don't get the Nobel prize for measuring a number, you get it for measuring a number and its uncertainty.

To cut to the chase, we now know the three dimensional distribution of dwarf galaxies about Andromeda. What does it look like, here's the picture from the paper;
Now, the question is, is this distribution of dwarfs just a random scattering of galaxies, or does it agree with our computer simulations of galaxy formation and evolution, or does it look like something else? That's a story for another day, hopefully a day not to far in the future. For now, well done Anthony!

A Bayesian Approach to Locating the Red Giant Branch Tip Magnitude (Part II); Distances to the Satellites of M31

Anthony R. Conn, Rodrigo A. Ibata, Geraint F. Lewis, Quentin A. Parker, Daniel B. Zucker, Nicolas F. Martin, Alan W. McConnachie, Mike J. Irwin, Nial Tanvir, Mark A. Fardal, Annette M. N. Ferguson, Scott C. Chapman, David Valls-Gabaud
In `A Bayesian Approach to Locating the Red Giant Branch Tip Magnitude (PART I),' a new technique was introduced for obtaining distances using the TRGB standard candle. Here we describe a useful complement to the technique with the potential to further reduce the uncertainty in our distance measurements by incorporating a matched-filter weighting scheme into the model likelihood calculations. In this scheme, stars are weighted according to their probability of being true object members. We then re-test our modified algorithm using random-realization artificial data to verify the validity of the generated posterior probability distributions (PPDs) and proceed to apply the algorithm to the satellite system of M31, culminating in a 3D view of the system. Further to the distributions thus obtained, we apply a satellite-specific prior on the satellite distances to weight the resulting distance posterior distributions, based on the halo density profile. Thus in a single publication, using a single method, a comprehensive coverage of the distances to the companion galaxies of M31 is presented, encompassing the dwarf spheroidals Andromedas I - III, V, IX-XXVII and XXX along with NGC147, NGC 185, M33 and M31 itself. Of these, the distances to Andromeda XXIV - XXVII and Andromeda XXX have never before been derived using the TRGB. Object distances are determined from high-resolution tip magnitude posterior distributions generated using the Markov Chain Monte Carlo (MCMC) technique and associated sampling of these distributions to take into account uncertainties in foreground extinction and the absolute magnitude of the TRGB as well as photometric errors. The distance PPDs obtained for each object both with, and without the aforementioned prior are made available to the reader in tabular form...

1. Questions:

1. If we could follow a main sequence star as it evolves towards a red giant, what "route" would we observe it taking across the H-R diagram? Would it follow the main sequence upwards and then head off along the red giant branch; or would it proceed as the crow flies from its passive main sequence position to its new red giant position?

2. Would a helium flash be a detectable event in itself or would it only be detected by the star's change of direction on the H-R diagram?

3. Is it possible to estimate when a helium flash is imminent by measuring how close a star is getting to the RGBT?

4. Is any work being done to test the RGBT standard candle theory with deeper objects?

2. 1. The Sun would spend most of its time sitting on the main sequence, with the motion up and down the RGB being relatively short compared to the life-time of the stars. You can see the path here

http://www.uni.edu/morgans/ajjar/Astrophysics/sunevolve.html

The helium burning phase is relatively stable and the stars linger there in what is known as the red clump.

2. I don't think so -I think the time scale for the turn around is relatively slow on human scales (but an expert on stellar evolution can correct me?)

3. Again, I don't think so as it still relatively long time scales on human scales.

4. Yes, the tip can be used out to significant distances, but the key point is that you need to resolve individual stars. As you get too far, the star light smushes into extended light and you can make colour-magnitude diagrams anymore. Then you need to use other methods.

3. Thanks, Geraint, for another great post. That makes it very clear now. Stellar evolution really is fascinating.

Just two follow up questions, if I may (I could ask a lot more but I will go and research them if I can):

(a) on the linked "Solar Evolution on the H-R Diagram" there are two pointy bits as the Sun evolves, one acutely at about 3440 Solar Luminosity; and another more obtuse at about 130,000. I assume it is the second (brighter) one that is the RGBT?

(b) Your first image, the coloured H-R diagram, shows a lot of cool bright stars seemingly adrift of the main sequence, (inside the boomerang). Is it right to assume these are the stars that are leaving the main sequence and evolving towards their red giant stage?

Regards

4. I'm happy to answer questions :)

1) The tip of the red giant branch is the very sharp tip at 3440 Solar Luminosities. The second trip up and the more gentle turn over at extreme brightness is the asymptotic giant branch. This is where the core runs out of helium fuel and then a shell of helium starts burning outside the core. When it becomes really bright, the star starts to pulse and the outer layers are thrown off and you have a planetary nebula.

2) That's the horizontal branch, where stars settle down after they have been on the red giant branch. It is basically the equivalent of the main sequence for stars burning helium in their core.