A Bayesian Approach to Locating the Red Giant Branch Tip Magnitude (Part I)

Abstract

We present a new approach for identifying the Tip of the Red Giant Branch
(TRGB) which, as we show, works robustly even on sparsely populated targets.
Moreover, the approach is highly adaptable to the available data for the
stellar population under study, with prior information readily incorporable
into the algorithm. The uncertainty in the derived distances is also made
tangible and easily calculable from posterior probability distributions. We
provide an outline of the development of the algorithm and present the results
of tests designed to characterize its capabilities and limitations. We then
apply the new algorithm to three M31 satellites: Andromeda I, Andromeda II and
the fainter Andromeda XXIII, using data from the Pan-Andromeda Archaeological
Survey (PAndAS), and derive their distances as $731^{(+ 5) + 18}_{(- 4) - 17}$
kpc, $634^{(+ 2) + 15}_{(- 2) - 14}$ kpc and $733^{(+ 13)+ 23}_{(- 11) - 22}$
kpc respectively, where the errors appearing in parentheses are the components
intrinsic to the method, while the larger values give the errors after
accounting for additional sources of error. These results agree well with the
best distance determinations in the literature and provide the smallest
uncertainties to date. This paper is an introduction to the workings and
capabilities of our new approach in its basic form, while a follow-up paper
shall make full use of the method's ability to incorporate priors and use the
resulting algorithm to systematically obtain distances to all of M31's
satellites identifiable in the PAndAS survey area.

Full text

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arXiv:1107.3206v1 [astro-ph.CO] 16 Jul 2011
submitted to Astrophysical Journal
Preprint typeset using LATEX style emulateapj v. 11/10/09
A BAYESIAN APPROACH TO LOCATING THE RED GIANT BRANCH TIP MAGNITUDE (PART I)
A. R. Conn1, 3, G. F. Lewis2, R. A. Ibata3, Q. A. Parker1, 4, D. B. Zucker1, 4, A. W. McConnachie5, N. F. Martin6, M. J. Irwin7, N. Tanvir8, M.
A. Fardal9, and A. M. N. Ferguson10
submitted to Astrophysical Journal
ABSTRACT
We present a new approach for identifying the Tip of the Red Giant Branch (TRGB) which, as we show,
works robustly even on sparsely populated targets. Moreover, the approach is highly adaptable to the avail-
able data for the stellar population under study, with prior information readily incorporable into the algorithm.
The uncertainty in the derived distances is also made tangible and easily calculable from posterior probabil-
ity distributions. We provide an outline of the development of the algorithm and present the results of tests
designed to characterize its capabilities and limitations. We then apply the new algorithm to three M31 satel-
lites: Andromeda I, Andromeda II and the fainter Andromeda XXIII, using data from the Pan-Andromeda
Archaeological Survey (PAndAS), and derive their distances as 731(+5)+18
kpc respectively, where the errors appearing in parentheses are the components intrinsic to the method, while
the larger values give the errors after accounting for additional sources of error. These results agree well with
the best distance determinations in the literature and provide the smallest uncertainties to date. This paper is
an introduction to the workings and capabilities of our new approach in its basic form, while a follow-up paper
shall makefulluse ofthemethod’sabilityto incorporatepriorsanduse the resultingalgorithmto systematically
obtain distances to all of M31’s satellites identifiable in the PAndAS survey area.
Subject headings: galaxies: general — Local Group — galaxies: stellar content
(−4)−17kpc, 634(+2)+15
(−2)−14kpc and 733(+13)+23
(−11)−22
1. INTRODUCTION
The Tip of the Red Giant Branch is a very useful standard
candle for gauging distances to extended, metal-poor struc-
tures. The tip corresponds to the very brightest members of
theFirst AscentRedGiantBranch(RGB),atwhichpointstars
areonthebrinkoffusingheliumintocarbonintheircoresand
hence contracting and dimming to become Horizontal Branch
stars. The result is a truncationto the Red Giant Branch when
the Color-Magnitude Diagram (CMD) for an old stellar pop-
ulation is generated, beyond which lie only the comparatively
rare Asymptotic Giant Branch Stars and sources external to
the system of interest. The (highly variable) contamination
from such objects provides the principal obstacle to simply
‘readingoff’ the tip position fromthe RGB’s luminosityfunc-
tion and the truncation of the AGB can even masquerade as
the TRGB in certain instances. The I-band is the traditionally
favoredregionofthespectrumforTRGBmeasurements,min-
imizing the interstellar reddening that plagues shorter wave-
1Department of Physics & Astronomy, Macquarie University, Sydney
2109, Australia
2Institute of Astronomy, School of Physics, A29, University of Sydney,
Sydney NSW 2006, Australia.
3Observatoire Astronomique, Universite de Strasbourg, CNRS, 67000
Stras- bourg , France.
4Australian Astronomical Observatory, PO Box 296, Epping, NSW
2121, Australia
5NRC Herzberg Institute of Astrophysics, 5071 West Saanich Road,
Victoria, British Columbia, Canada V9E 2E7
6Max-Planck-Institut f¨ ur Astronomie, K¨ onigstuhl 17, D-69117 Heidel-
berg, Germany
7Institute of Astronomy, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK
8Department of Physics and Astronomy, University of Leicester, Le-
icester LE1 7RH, UK
9University of Massachusetts, Department of Astronomy, LGRT 619-
E, 710 N. Pleasant Street, Amherst, Massachusetts 01003-9305, USA
10Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
lengths, while keeping dependance on metallicity lower than
it would be at longer IR wavelengths. It should also be re-
membered that stars approachingthe TRGB generally exhibit
peak emission in this regime. Iben & Renzini (1983) deter-
mined that low mass (< 1.6M⊙for Pop. I, < 1M⊙for Pop.
II), metal-poor ([Fe/H] < -0.7 dex) stars older than 2 Gyr
produce a TRGB magnitude that varies by only 0.1 magni-
tudes. More recently, Bellazzini, Ferraro, & Pancino (2001)
determined the tip magnitude to lie at an I-band magnitude of
MTRGB= −4.04 ± 0.12. This low variation can be attributed
to the fact that all such stars have a degenerate core at the
onset of helium ignition and so their cores have similar prop-
erties regardless of the global properties of the stars. The re-
sult is a standard candle that is widely applicable to the old,
metal-poor structures that occupy the halos of major galax-
ies. Distances derived from the TRGB, unlike those from the
Cepheid variable or RR Lyrae star for example, can be deter-
minedfroma single epochof observation,makingit very use-
ful for wide-area survey data. Furthermore, Salaris & Cassisi
(1997) confirmed agreement between Cepheid and RR Lyrae
distances and TRGB distances to within ∼5 %.
Up until Lee, Freedman, & Madore (1993) published their
edge-findingalgorithm,the tip had always been found by eye,
but clearly if the wide-reaching applications of the TRGB
standardcandle were to be realized, a moreconsistent, repeat-
able approach was in order. The aforementionedpaper shows
that, if a binned luminosity function (LF) for the desired field
is convolved with a zero sum Sobel kernel [-2, 0, +2], a max-
imum is produced at the magnitude bin corresponding to the
greatest discontinuity in star counts, which they attribute to
thetip. Usingthis method,theywereable toobtainaccuracies
of better than 0.2 magnitudes. Sakai, Madore, & Freedman
(1996) set out to improve on this approach by replacing the
binned LF and kernel with their smoothed equivalents. To do
this, they equate each star with a Gaussian probability distri-
bution whose FWHM is determined by the photometric error

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2 Conn et al.
at the magnitude actually recorded for the star. Then rather
than each star having to fall in a single bin, it contributes to
allbins, butmoststronglytothebinat themagnituderecorded
for it and less so the further a bin is from that recorded mag-
nitude, weighted by the photometric error. This is illustrated
in equation 1:
Φ(m) =
N
?
i=1
1
?
2πσ2
i
exp
−(mi− m)2
2σ2
i

(1)
where m is the magnitude of the bin in question and miand
σ2
gaussian probability distribution for the ith star. This method
halved the error associated with the non-smoothed version of
the algorithm and an identical smoothing is hence justly in-
corporated into the model LF for our Bayesian approach.
In a more recent variation on the Edge Detection methods,
Madore, Mager, & Freedman(2009) once again applied a So-
bel kernel, but fit to a luminosity function built from compos-
ite stellar magnitudes T ≡ I − β[(V − I)0− 1.50] where β is
the slope of the TRGB as a function of color. This they ar-
gue results in a sharper output response from the filter, and
allows all stars, regardless of color, to contribute equally to
the derived tip position. Rizzi et al. (2007) derived a value of
0.22± 0.02 for β after a study of 5 nearby galaxies, and show
that it is quite consistent from one galaxy to another.
M´ endez et al. (2002) made a departure from the simple
‘edge-finding’algorithmsabovebyadaptingamaximumlike-
lihood model fitting procedure into their technique. They
point out that the luminosity function faint-ward of the tip is
well modeled as a power law:
iare the central magnitude and variance respectively of the
L(m ≥ mTRGB) = 10a(m−mTRGB)
(2)
where m ≥ mTRGBand a is fixed at 0.3. They then ascribe
the location of the tip to the magnitude at which this power
law truncates - i.e. m = mTRGB. Bright-ward of the tip they
assume a functional form:
L(m < mTRGB) = 10b(m−mTRGB)−c
(3)
where b is the slope of the power law bright-ward of the tip
and c is the magnitude of the step at the RGB tip.
Sucha model,thoughsimplistic, is robustagainstthestrong
Poisson noise that is inevitable in more sparsely populated
LFs, making it a significant improvement over the previous,
purely ‘edge-finding’methods.
Makarov et al. (2006) follow in a similar vein, demon-
strating the proven advantages of a Maximum Likelihood
approach over simple Edge Detection techniques, despite a
model dependance. Unlike M´ endez et al. (2002) however,
they allow a as a free parameter, arguing its notable variance
from 0.3 and importantly,they smooth their model LF using a
photometric error function deduced from artificial star exper-
iments. One shortcoming of both of these methods however,
is that the most likely parameter values alone are obtained,
withouttheir respectivedistributionsorrepresentationof their
dependance on the other parameters. Also, with regard to the
background contamination, the RGB LF in fact sits on top
of non-system stars in the field and so rather than model the
background exclusively bright-ward of the TRGB, the trun-
cated power law of Eq. 2 can be added onto some predefined
function of the contamination.
Arguably the most successful method developed so far has
been that devised by McConnachie et al. (2004). It has been
used to ascertain accurate distances to 17 members of the Lo-
cal Group (McConnachie et al. 2005). It combines aspects of
both ‘edge-finding’ and model fitting to zero in more accu-
rately on the tip. They argue that as the precise shape of the
luminosity function at the location of the tip is not known,
a simple Sobel Kernel approach that assumes a sharp edge
to the RGB, does not necessarily produce a maximum at the
right location. They instead use a least-squares model fitting
technique that fits to the LF in small windows searching for
the portion best modeled by a simple slope function. This,
they reason, marks the location of the steepest decline in star
counts which is attributable to the tip location. This methodis
capable of finding the tip location accurate to better than 0.05
magnitudes, although is still susceptible to be thrown off by
noise spikes in a poorly populated LF.
Despite the merits of previous methods such as these, none
of them work particularly well when confrontedwith the high
levels of Poisson noise that abound in the more poorly popu-
lated structures of galaxy halos. Furthermore, in such condi-
tions as these where the offset between detected and true tip
position will likely be at its greatest, it is of great use to have
a full picture of likelihood space, as opposed to merely the
determined, most probable value. This has lead us to develop
a new, Bayesian approach to locating the TRGB, specifically,
one that incorporates a Markov Chain Monte Carlo (MCMC)
algorithm. As shall become apparent in the next section, such
a method is very robust against noise spikes in the luminosity
function and allows all prior knowledge about the system to
be incorporated into the tip-finding process - something lack-
ing in the previous approaches. Further to this, the MCMC
provides for a remarkably simple, yet highly accurate error
analysis. It also makes it possible to marginalize over param-
eters to provide posterior probability distributions (PPDs) of
each parameter, or to obtain plots of the dependance of each
parameter on every other. In §2, a detailed explanation of
our approach and its limitations is given. §2.1 introduces the
method by applying the algorithm to one of M31’s brightest
dwarf spheroidals, Andromeda I. §2.2 discusses the nature of
systematic errors that apply to the method. §2.3 investigates
the accuracy that the basic method (before addition of priors)
is capable of given the number of stars populating the LF for
the field and the strength of the non-RGB background while
§2.4 deals with its performance when faced with a compos-
ite luminosity function. §3 then applies our new approach to
two additional M31 dwarf satellite galaxies and §4 summa-
rizes the advantages of the method and outlines the expected
applicability of the method in the immediate future.
2. METHOD
2.1. The MCMC Method
TheMarkovChainMonteCarlo methodis an iterativetech-
nique that, given some model and its associated parameters,
rebuilds the model again and again with different values as-
signed to each parameter, in order that a model be found that
is the best fit to the data at hand. It does this by comparing
the likelihood of one model, built from newly proposed pa-
rameter values, being correct for the data, as opposed to the
likelihood for the model built from the previously accepted
set of model parameters. The MCMC then accepts or rejects
the newly proposed parameter values weighted by the rela-
tive likelihoods of the current and proposed model parameter

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A Bayesian Approach to locating the TRGB3
values. At every iteration of the MCMC, the currently ac-
cepted value of each parameter is stored so that the number
of instances of each value occurring can be used to build a
likelihood distribution histogram - which can be interpreted
as a PPD - for each model parameter. Hence, the MCMC is a
way of exploring the likelihood space of complicated models
with many free parameters or possible priors imposed, where
a pure maximum likelihood method would be quickly over-
whelmed. With the PPD generated, the parameter values that
produce the best fit model to the data can simply be read off
from the peak of the PPD for each parameter. Similarly, the
associated error can be ascertained from the specific shape of
the distribution. A detailed description of the MCMC with
worked examples can be found in Gregory (2005).
To illustrate the precise workings of our MCMC tip-finding
algorithm, its application to a well-populated dwarf galaxy in
the M31 halo is described. Andromeda I was discovered by
van den Bergh (1971) and at a projected distance of ∼ 45 kpc
from M31 (Da Costa et al. 1996), it is one of its closest satel-
lites. Da Costa et al. (1996) ascribed to it an age of ∼ 10 Gyr
and a relatively low metallicity of ?Fe/H? = −1.45 ± 0.2 dex
which is clearly exemplified in the color-magnitude diagram
for Andromeda I presented in Fig 2. Here the RGB of An-
dromeda I lies well to the blue side of that of the Giant Stellar
StreamwhichliesbehindAndromedaIbutinthesamefieldof
view. Mould & Kristian (1990) provide the first TRGB based
distance measurement to Andromeda I, which they deduce as
790 ± 60 kpc, based solely on a visual study of the red giant
branch. McConnachie et al. (2004) improve on this signifi-
cantly, producing a distance determination of 735 ± 23 kpc,
based on a tip magnitude of 20.40+0.03
Andromeda I’s position with respect to M31 and the Gi-
ant Stellar Stream is presented in Fig. 1, where the red circle
indicates the precise field area fed to our MCMC algorithm.
An Object-to-Background Ratio (OBR) of 11.0 was recorded
for this field with the color-cut applied, based on comparisons
of the signal field stellar density with that of an appropriate
background field. The data presented in this figure, as with
all other data discussed in this paper, was obtained as part
of the Pan-AndromedaArchaeologicalSurvey(McConnachie
2009), undertaken by the 3.6 m Canada-France-Hawaii Tele-
scope (CFHT) on Mauna Kea equipped with the MegaCam
imager.CFHT utilizes its own unique photometric band
passes i and g based on the AB system. We work directly
with the extinction-corrected CFHT i and g magnitudes and
it is these that appear in all relevant subsequent figures. The
extinction-correction data applied to each star has been in-
terpolated using the data from Schlegel, Finkbeiner, & Davis
(1998).
At the heart of our tip-finding algorithm is the model lumi-
nosity function that the MCMC builds from the newly chosen
parameters at every iteration. The luminosity function is a
continuous function which we subsequently convolve with a
Gaussian kernel to account for the photometric error at each
magnitude. This is achieved by discretising both functions on
a scale of 0.01 magnitudes. Like M´ endez et al. (2002), we as-
sume the LF faint-ward of the tip to follow a simple power
law, of the form given in Eq. 2; however, we set a as a free
parameter. The bin height at each magnitude is then calcu-
lated by integrating along this function setting the bin edges
as the limits of integration. The value for the bin which is set
tocontaintheRGB tip forthecurrentiterationis calculatedby
integrating along the function from the precise tip location to
the faint edge of the bin. All other bins are then set at 0. A bin
−0.02in I band.
Fig. 1.— The position of Andromeda I relative to the M31 disk. The satu-
rated disk dominates the North West corner of the field whilst Andromeda I
itself appears as an over-density within the Giant Stellar Stream (GSS). The
GSS in actuality lies well behind Andromeda I, as is evidenced by the CMD
in Fig. 2. A strict color-cut was imposed on the data to highlight the location
of the satellite and the extent of the stream with greatest contrast.
Fig. 2.— Color-Magnitude Diagram for a circular field of radius 0.2◦cen-
tered on Andromeda I. Two red giant branches are clearly visible, that of
Andromeda I (within the red rectangle color-cut) and that of the Giant Stellar
Stream which lies behind Andromeda I in the same line of sight.
width of 0.01 magnitudes for our model was found to provide
a good balance between magnitude resolution, which is lim-
ited by the photometric error in the MegaCam data (∼ 0.01
mag at m = 20.5), and the computational cost for a higher
number of bins. We stress here however, that each star’s like-
lihood is calculated from the model independently,so that the

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4 Conn et al.
actual data LF is ‘fed’ to the MCMC in an un-binned state.
A faint edge to the model LF was imposed at m = 23.5 to
remove any significant effects from data incompletion and in-
creasing photometric error.
Further to this, we add a background function to this trun-
catedpowerlaw. Whilethescalingofthebackgroundstrength
relativeto theRGB signalstrengthcouldbeset as anotherfree
parameter, and indeed was initially, it makes better use of our
prior information to instead determine the fraction of back-
ground stars or ‘background height’ (f) manually. This is
achieved simply by calculating the average density of stars in
the background field DBGand in the ‘signal’ field DSIGwith
f then being the ratio of the two, i.e. f = DBG/DSIG. Note
that this is not directlythe inverseof theobject-to-background
star ratio, OBR = (DSIG− DBG)/DBG, as f represents the per-
centage of all stars lying inside the signal field that can be ex-
pected to be external to the object of interest. Hence when we
normalize the area under the model LF so that it may be used
by the MCMC as a probability distribution, the background
component will have area f while the RGB component will
have area 1 − f. Now, with f known, what we then have is a
simplified2 parametermodel,allowingforfaster convergence
of the MCMC algorithm.
We have thus devised our model so that the MCMC is
tasked with the problem of finding just two parameters,
namely the slope of the RGB luminosity function (a) and
of course the location of the RGB tip magnitude (mTRGB).
For simplicity in this first paper, we impose uniform priors
on each of these parameters, where 19.5 ≤ mTRGB ≤ 23.5
and 0 ≤ a ≤ 2. We also do not account for the color de-
pendence of the tip magnitude which is only slight in I band
(see Rizzi et al. (2007)) and for the metal poor targets exam-
ined here, but these effects will be dealt with in future pub-
lications. While it is true that two parameters are tractable
analytically, we apply the numerical MCMC in order to set
the framework for computationally more challenging models
with non-uniform priors that will become necessary for the
more sparsely populated structures presented in future contri-
butions. There are however, several more complexities to the
model that are yet to be discussed. Firstly, the choice of back-
ground function is not arbitrary. It has been found that the
best way to model the background is to fit it directly by tak-
ing the luminosity function of an appropriate ‘background’
field. The best choice of background field is arguably one
that is at similar galactic latitude to the structure of interest,
as field contamination is often largely Galactic in origin, and
hence closely dependent on angular distance from the Galac-
tic plane. Furthermore, the field should be chosen so that the
presence of any substructure is minimal, so as to prevent the
signatureofanotherhalo objectinterferingwith theLF forthe
structure of interest.
Inadditiontotheseconstraints,owingtothelowstellarden-
sity of the uncontaminated halo, it is preferable that the back-
ground field be as large as possible to keep down the Pois-
son noise and hence it will of necessity be much larger than
that of the field of interest. As a result, the main error in the
background fit will arise from background mismatching and
is not random. In addition, the large background field size
may inevitably contain some substructure, requiring removal.
Thismaybedonebyphysicallysubtractingcontaminatedpor-
tions of the background area, but this is often unnecessary as
the CMD color-cut imposed on the signal field must also be
applied to the background field, usually ridding the sample
of any substantial substructure that may be present. In the
case of our Andromeda I background field however, we have
removed a large 2.4◦portion crossing numerous streams (as
shown in Fig. 3) as these streams do trespass into the chosen
AndromedaI color-cut. Nevertheless,this is just a precaution,
because for well populated systems such as Andromeda I and
Andromeda II, the algorithm is impervious to small discrep-
ancies in the functional form of the background.
Once an appropriate backgroundfield has been selected, its
LF can be fitted by a high-orderpolynomial. This polynomial
then becomes the function added to our model and scaled by
f as describedearlier. Our choice of backgroundfield for An-
dromeda I (along with Andromeda II and Andromeda XXIII)
and the polynomial fit to its LF are presented in figures 3 and
4 respectively.
Fig. 3.— A map of the entire PAndAS survey area, with color-cut chosen to
favor the low metallicities exhibited by many of M31’s satellite galaxies. The
three dwarf spheroidal companions of M31 studied in this paper are labeled,
along with the signal fields (small circles of radius 0.2◦) and their respective
background fields fed to our algorithm. Note that the background fields are
chosen to be as narrow as possible in Galactic latitude while retaining as large
an area as possible. In each case, the signal field areas are subtracted from
their respective background fields to prevent contamination.
The other major consideration that is yet to be addressed
is the effect of photometric error on the luminosity function.
This is dealt with by convolvingthe initial binned model with
a normalized Gaussian whose width is adjusted as a function
of magnitude in accordance with the error analysis conducted
on the PAndAS data. This is equivalent to the method of
Sakai, Madore, & Freedman (1996) described in Eq. 1. As
described earlier, this procedure has the added advantage of
making the model independent of binning. It is also impor-
tant in this stage, as it is at every stage, that the model and all
constituentparts are normalizedso that the model can be used
as a probability distribution.
With these issues addressed, the MCMC algorithm can be
set in motion. The i-band magnitudes and (g − i)0data for
the desired field is read into data arrays, spurious sources
are rejected and a color-cut imposed to remove as many non-
members of the structure’s RGB as possible. The same con-
straints are of course applied to the background field as well.

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A Bayesian Approach to locating the TRGB5
Fig. 4.— Top) CMD for the Andromeda I background field (see Fig. 3).
The same color-cut is applied as in the CMD for the signal field (Fig. 2).
Bottom) The binned luminosity function for the background with the fitted
polynomial superimposed. A polynomial of degree 7 was found adequate to
represent the luminosity function.
The MCMC then applies pre-set starting values of a and
mTRGBand builds the corresponding model for the first iter-
ation. Within this iteration, the MCMC proposes new values
foreachparameter,displacedbysome randomGaussian devi-
ate from the currently set values and re-constructs the appro-
priate model. The step size, or width of the Gaussian deviate
is chosen so as to be large enough for the MCMC to explore
the entire span of probability space, while small enough to
provide a high resolution coverage of whatever features are
present. The ratio of the likelihoods of the two models is then
calculated(theMetropolisRatio r) anda swap ofacceptedpa-
rameter values made if a new, uniform random deviate drawn
from the interval [0,1], is less than or equal to r. The calcu-
lation of the Metropolis Ratio for our model is exemplified in
equations 4 and 5:
r =Lproposed
Lcurrent
(4)
with the value for each of the likelihoods L being calculated
thus:
L =
ndata
?
n=1
M(mTRGB,a,mn)(5)
with
M(mn≥ mTRGB) = RGB(mn) + BG(mn)
M(mn< mTRGB) = BG(mn)
where RGB(mn) = 10a(mn−mTRGB)
?m=23.5
m=mTRGB
?m=23.5
m=19.5
where mTRGBand a are the parameters currently chosen for
the model by the MCMC, ndata is the numberof stars and mn
is the i-band magnitude of the nth star. BG represents the fit-
ted backgroundfunction (see Fig. 4). The MCMC then stores
the new choice for the current parameter values and cycles to
the next iteration. In order to ascertain a reasonable number
of iterations, the chains for each parameter were inspected to
insure that they were well mixed, resulting in posterior distri-
butions that appeared smooth (by eye).
When the MCMC has finished running, the posterior prob-
ability distribution (PPD) for each parameter is generated. By
binningupthenumberofoccurrencesofeachparametervalue
over the course of the MCMC’s iterations, the probability of
each value is directly determined and the most probable value
can be adopted as the correct model value for the data. If one
assumes a Gaussian probability distribution, then the 1-sigma
errors associated with each parameter value can be obtained
simply by finding the value range centred on the best-fit value
that contains 68.2 % of the data points. As our PPDs are not
always Gaussian, our quoted 1-sigma errors in the tip magni-
tude represent more strictly a 68.2 % credibility interval. We
do not fit a Gaussian to our PPDs to obtain 1-sigma errors.
Our 1-sigma errors in tip magnitude are obtained by finding
the magnitude range spanning 68.2 % of the PPD data points,
on one side of the distribution mode and then the other. It
must be stressed that these quoted errors are merely an indi-
cator of the span of the parameter likelihood distribution and
are no substitute for examining the PPDs themselves. Figures
5 and 6 present the PPD for the RGB tip magnitude based on
the Andromeda I CMD (Fig. 2) and the best fit model to the
LF for the field respectively. The PPD for the LF slope a is
presented in Fig. 7 and a contour map of the distribution of
the tip magnitude vs. a is presented in Fig. 8.
Upon the completion of the algorithm, the RGB tip for
Andromeda I was identified at m = 20.879+0.014
corresponds to an extinction-corrected distance of 731(+5)+18
kpc, where the final errors include contributions from the
extinction and the uncertainty in the absolute magnitude of
the TRGB (see §2.2). The i-band extinction in the direc-
tion of Andromeda I is taken as Aλ = 0.105 magnitudes
(Schlegel, Finkbeiner, & Davis 1998). The parameters a and
f were derived as 0.273 ± 0.011 and 0.083 respectively. This
distance measurement is in excellent agreement with the dis-
tance determined by McConnachie et al. (2004). It is note-
worthy howeverthat our method searches for the TRGB itself
as distinct from the RGB star closest to the TRGB as sort out
by the method of McConnachie et al. (2004), which would
andRGB dm = 1 − f
andBG dm = f
(6)
−0.012.This
(−4)−17

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6Conn et al.
Fig. 5.— The posterior probability distribution for 3 million iterations of the
MCMC on the Andromeda I CMD color-cut presented in Fig 2. The peak
probability is located at i0= 20.88. The distribution is color coded, with red
indicating tip magnitudes within 68.2 % (Gaussian 1-sigma) on either side of
the distribution mode, green those within 90 % and blue those within 99 %.
Fig. 6.— The four magnitude segment of the Andromeda I luminosity func-
tion fitted by our MCMC algorithm. It is built from 3355 stars. The best fit
model is overlaid in red. The bin width for the LF is 0.01 magnitudes.
contribute to our slightly smaller distance measurement. A
similar discrepancy arises in the case of Andromeda II (see
§3).
2.2. A Note on Distance Errors
Despite the small errors in the tip magnitude afforded by
our approach, there are a number of factors that contribute
to produce a somewhat larger error in the absolute distance.
These arise due to uncertainties both in the extinction correc-
tions applied, and in the absolute magnitude of the TRGB in
i-band. Both of these contributions are assumed to be Gaus-
sian, where the 1-σ error in the extinction correction, ∆{Aλ},
is taken as 10% of the correction applied, and the error in the
absolute magnitude of the tip is expressed in equation 7 be-
low:
Fig. 7.— The posterior probability distribution obtained for the slope a of
the Andromeda I luminosity function. The distribution is a clean Gaussian
with the distribution mode at 0.273.
Fig. 8.— A contour map of the distribution of the tip magnitude vs. the
LF slope a. It is noteworthy that there is little correlation between the two
parameters, with the peak of the distribution of a more-or-less independent
of tip magnitude. Regardless of any correlation, the respective PPDs of each
parameter are the result of marginalizing over the other parameter, and thus
take into account any covariance between parameters.
∆{MTRGB
i
} =
=
= ±0.12
?
?
∆2{mTRGB
{0.04}2+ {0.03}2+ {0.11}2
i
}ωCen+ ∆2{Aλ}ωCen+ ∆2{m − M}ωCen
(7)
As we are working in the native CFHT i and g bands, we
adopt this magnitude as MTRGB
i
conversion from MTRGB
I
is based on the absolute magnitude
for the TRGB identified for the SDSS i band (Bellazzini
2008). This is justified by the color equations applying to
the new MegaCam i-band filter (Gwyn 2010). Noting that
the largest contribution to this error is that from the dis-
tance modulus to ω Cen, (m − M)ωCen, derived from the
= −3.44 ± 0.12 where the

Page 7

A Bayesian Approach to locating the TRGB7
eclipsing binary OGLEGC 17, we consider only the contri-
butions from the extinction {Aλ}ωCen, which is taken as 10%
of the Schlegel, Finkbeiner, & Davis (1998) values, and the
apparent tip magnitude determination {mTRGB
that our derived distance modulus may be systematically dis-
placed by up to 0.1 of a magnitude.
MTRGB
i
= −3.44±
principal motive is to obtain relative distances between struc-
tures within the M31 halo rather than the absolute distances
to the structures, this offset is not important. Furthermore, as
measurements for the ω Cen distance modulus improve, our
distances are instantly updatable by applying the necessary
distance shift.
While these external contributions to our distance uncer-
tainties may be taken as Gaussian, the often non-Gaussian
profile of our TRGB (mTRGB
i
) posterior distributions neces-
sitate a more robust treatment then simply addingthe separate
error components in quadrature. Hence to obtain final dis-
tance uncertainties, we produce a Distance Distribution ob-
tained by sampling combinations of mTRGB
from their respective likelihood distributions, thus giving us a
truepictureof the likelihoodspace forthe distance. The result
of this process for Andromeda I is illustrated in Fig. 9. From
this distribution, we determinenot only the quoted1-σ errors,
but also that Andromeda I lies at a distance between 703 and
761 kpc with 90% credibility and between 687 and 778 kpc
with 99% credibility.
i
}ωCenand note
This then gives us
√0.042+ 0.032= −3.44 ± 0.05. Since our
i
, Aλand MTRGB
i
Fig. 9.— A plot of the distribution of possible distances to Andromeda I
obtained through the application of our method. Once again, the colors red,
green and blue denote distances within 68.2%, 90% and 99% credibility in-
tervals respectively.
2.3. Initial Tests
In order to gain a better understanding of the capabilities
of our method when faced with varying levels of luminos-
ity function quality, a series of tests were conducted on ar-
tificial ‘random realization’ data, as well as on sub-samples
of the Andromeda I field utilized above. There are two ma-
jor factors that effect the quality of LF available to work
with, namely the number of stars from which it is built,
and the strength of the background component relative to
the RGB component. Hence to simulate the varying degrees
of LF quality that are likely to be encountered in the M31
halo, artificial LFs were built for 99 combinations of back-
ground height vs. number of stars. Specifically, background
heights of f = 0.1,0.2,...,0.9 were tested against each of
ndata = 10,20,50,100,200,500,1000,2000,5000,10000 &
20000 stars populating the LF.
To achievethis, a model was built as discussed in §2.1, with
a constant tip magnitudeand RGB slope of mTRGB= 20.5 and
a = 0.3 respectively and a background height f set to one of
the nine levels given above. The functional form of the back-
groundwas keptasa horizontallineforthesakeofthetests. A
luminosity function was then built from the model, using one
of the 11 possible values for the number of stars listed above.
This was achieved by assigning to each of the ndata stars a
magnitude chosen at random, but weighted by the model LF
probability distribution - a ‘random realization’ of the model.
The MCMC algorithm was then run on this artificial data set
as described in the previous subsection with mTRGBand a as
free parameters to be recovered. The tests also assume the
photometric errors of the PAndAS survey and further assume
that incompleteness is not an issue in the magnitude range
utilized. The error in the recovered tip position and the offset
of this position from the known tip position in the artificial
data (I = 20.5) were then recorded. The results are presented
in Figures 10 & 11 below. Each pixel represents the average
result of ten 200000iteration MCMC runs for the given back-
ground height vs. number of stars combination. Note that the
kpc distances given correlate to an object distance of 809 kpc
- i.e. mTRGB
I
= 20.5 - which is in keeping with distances to
the central regions of the M31 halo. Furthermore, all stars of
the random realization were generated within a 1 magnitude
range centred on this tip value.
Fig. 10.— A grey-scale map of the one-sigma error in tip magnitude
obtained for different combinations of background height and number of
sources. The actual value recorded for the error (in kpc) is overlaid on each
pixel in red. For these tests, we approximate the one-sigma error as the half-
width of the central 68.2 % of the PPD span.
Figures 10 & 11 are intended to serve as a reference for
future use of the basic method, with regard to the number of
stars required to obtain the distance to within the desired un-
certainty for the available signal-to-noise ratio. The results
follow the inevitable trend of greater performance when the
background height is small and there are many stars popu-
lating the luminosity function. There are some minor devi-
ations from this trend but these result from single outlying

Page 8

8 Conn et al.
Fig. 11.— A grey-scale map of the offset from the true tip magnitude
obtained for different combinations of background height and number of
sources. The actual (absolute) value recorded for the offset (in kpc) is over-
laid on each pixel in red. These values convey the discrepancy between the
true object distance and that recovered by the MCMC. It was necessary to
remove the direction of the individual offsets before averaging as the values
would otherwise largely cancel out. Examination of the individual offsets
shows no significant bias toward either direction however.
values whose effects would diminish if a higher number of
samples were averaged. It is also noteworthy that the offsets
recorded clearly correlate with the one-sigma errors and are
consistently less than their associated errors.
The results of these random realization tests are borne out
by similar tests conducted on subsamples of the Andromeda
I field. Random samples were drawn containing 335 (10% of
the total sample), 200, 100 and 50 stars. These correspond
approximately to 10, 20, 50 and 100 stars in the 1 magnitude
range centered on the tip. In no case was the derived tip loca-
tion more than 80 kpc from that identified from the full sam-
ple, and the offset grew steadily less as the number of stars
in the sample was increased. Furthermore, the offsets were
almost always less than the 1-sigma errors.
2.4. Algorithm Behavior for Composite Luminosity
Functions
When a field is fed to any RGB tip finding algorithm, it
must be remembered that that field is in fact three dimen-
sions of space projected onto two, and therefore it is possible
that two structures at very different distances may be present
within it. Such a scenario becomes especially likely when
dealing with the busy hive of activity that the PAndAS Sur-
vey has come to reveal around M31. The result of such an
alignment along the line of sight is a luminosity functionbuilt
from two superimposed RGBs with two different - possibly
widely separated - tip magnitudes. Hence it is important to
understand how the TRGB algorithm applied to such a field
will respond.
Unlike other algorithms that have been developed, our
Bayesian approach provides us with a measure for the proba-
bility of the tip being at any given magnitude (the PPD). But
this also leads to an important caveat - the selection crite-
ria imposed on the data that is fed to the algorithm biases it
strongly toward the structure whose distance we are trying to
measure. Taking the Andromeda I measurement of §2.1 for
example, this satellite sits on top of the Giant Stellar Stream
which contributes prominently to the field CMD, yet its con-
tribution to the LF fed to the MCMC is almost eradicated by
our choice of color-cut. Yet if this stringent color-cut is re-
moved, the algorithm remains surprisingly insensitive to the
GSS tip. This is because of another prior constraint we im-
pose on the routine - the background height. With this fixed
background imposed on our fitted model, the MCMC looks
for the first consistent break of the data from the background
- i.e. the tip of the Andromeda I RGB. It is therefore neces-
sary to re-instate the backgroundheight as a free parameter of
the MCMC to give it any chance of finding the tip of the Gi-
ant Stellar Stream’s RGB. By this stage, enough of our prior
constraints have been removed to give the method freedom to
choose the best fit of the unrestricted model to the entire data
set from the field. Nevertheless, the more (correct) prior in-
formation we can feed the algorithm, the better the result we
can expect to receive.
Still, while the method has not been tailored towards com-
posite luminosity functions, it is worth noting that it can be
used successfully to identify more than one object in the line
of sight - a useful ability when the two structures are poorly
separated in color-magnitudespace. The model used assumes
only one RGB and thus one tip; to do otherwise would in-
crease computation times. If two distinct structures are iden-
tified by this method and can not be separated using an ap-
propriate color-cut or altered field boundaries, an appropriate
doubleRGB model should be built to accurately locate the tip
for each structure. But even with the basic single-RGB model
(which will suffice for the vast majority of cases), at least the
presence of a second structure is indicated. If we take the
example of Andromeda I again, the ideal way to obtain a dis-
tance measurement to the portion of the GSS that sits behind
it would be to make a color-cutthat favors it and removes An-
dromeda I, but we can force the algorithm to consider both
structures to demonstrate the extreme case of what might be
encountered in a general halo field. The result is two broad
bumps in the PPD well separated in magnitude. The nature of
the MCMC however is to converge straight onto the nearest
major probability peak, seldom venturing far from that peak.
This is remedied by the addition to the algorithm of Parallel
Tempering.
While an infinite number of iterations of the MCMC would
accurately map probability space in its entirety, Parallel Tem-
peringis awayofachievingthisgoalmuchmorequickly. Par-
allel Tempering involves a simple modification to the MCMC
algorithm, whereby multiple chains are run in parallel. One
chain,the‘coldsampler’runsexactlyas before,but additional
chains have their likelihoods weighted down producinga flat-
ter PPD that is more readily traversed by the MCMC. The
further the chain is from the cold sampler chain, the heavier
the weight that is applied. Every so many iterations, a swap
of parameters is proposed between two random, but adjacent
chains so that even the ‘hottest’ chains eventually affect the
cold sampler chain and allow it to escape any local maximum
it may be stuck in. The result is a cold sampler chain PPD
that is more representative of the full extent of the luminosity
function[see Gregory (2005) for a more detailed discussion].
The result of applying a 4 chain MCMC to the region of An-
dromeda I is summarized in the PPD of Fig. 12.
While the Andromeda I TRGB is found much less accu-
rately by this method as a result of the removal of our prior
constraints for illustrative purposes, it is nevertheless clear
that the addition of Parallel Tempering adds to our algorithm
the facility to identify other structures in the field that may

Page 9

A Bayesian Approach to locating the TRGB9
Fig. 12.— The posterior probability distribution for the cold sampler chain
of a4 chain parallel tempering regime. TheMCMC wasrun for 1.5 million it-
erations. The strong peak at m = 20.93 results from the tip of the Andromeda
I RGB, but it has been shifted faint-ward by the presence of the Giant Stellar
Stream, responsible for the peaks at m = 21.29 and m = 21.35. Without the
addition of parallel tempering, the MCMC is liable to spend an inordinate
amount of time stuck in the first major probability peak it encounters.
require separate analysis. Even given a properly constrained
model and data set, the safeguard it provides against a poorly
explored probability space arguably warrants its inclusion.
3. DISTANCES TO TWO MORE SATELLITES
To further illustrate the capabilities of our basic method
as outlined in §2, we have applied it to two more of M31’s
brighter satellites, whose distances have been determined
in past measurements using a range of methods, including
TRGB-finding algorithms. The additional satellites chosen
forthis studyaretherelativelyluminousdwarfspheroidalAn-
dromeda II and the somewhat fainter, newly discovered An-
dromeda XXIII dwarf. The location of both satellites within
the M31 halo can be seen in Figure 3.
3.1. Andromeda II
AndromedaII was discoveredas a result of the same survey
as Andromeda I using the 1.2 m Palomar Schmidt telescope
(van den Bergh 1971). Da Costa et al. (2000) deduce a simi-
lar age for AndromedaII as for AndromedaI but with a wider
spread of metallicities centered on ?Fe/H? = −1.49 ± 0.11
dex. Our Andromeda II luminosity function was built from a
circular field of radius 0.2◦centered on the dwarf spheroidal
with an OBR of 34.0 recorded. This high OBR is not un-
expected with Andromeda II arguably the best populated of
M31’s dwarf spheroidal satellites. The color magnitude dia-
gram for this field is presented in Fig 13.
Application of our algorithm to Andromeda II yields a tip
magnitude of i0= 20.572+0.005
correspondstoanextinction-correcteddistancetoAndromeda
II of 634(+2)+15
as Aλ = 0.121 magnitudes (Schlegel, Finkbeiner, & Davis
1998). This is in good agreement with McConnachie et al.
(2004)’s derived distance of 645± 19 kpc. Values for a and f
were recovered as 0.276 ± 0.009 and 0.028 respectively. The
mTRGB
i
PPD and best fit model found by our method are illus-
trated in Figs. 14 and 15 respectively.
−0.006for the red giant branch which
(−2)−14kpc, where the i-band extinction is taken
3.2. Andromeda XXIII
Fig. 13.— Color-Magnitude Diagram for a circular field of radius 0.2◦cen-
tered on Andromeda II. It is more densely populated than the Andromeda I
CMD (Fig 2) and is very well defined against the stellar background. The
RGB tip is clearly visible at i0∼ 20.6.
Fig. 14.— The posterior probability distribution for 3 million iterations of
the MCMC on a 4 magnitude interval (see Fig 15) of the Andromeda II CMD
selection presented in Fig 13. The peak probability of the distribution is well
defined at i0 ≈ 20.57. The distribution is again color coded as in figure 5,
with red, green and blue corresponding to 68.2 %, 90 % and 99 % credibility
intervals respectively.
Despite its relative brightness among the other satellites
of the M31 system, Andromeda XXIII was only discovered
with the undertaking of the outer portion of the PAndAS sur-
vey in 2009/ 2010, being too faint at MV = −10.2 ± 0.5 to
identify from the Sloan Digital Sky Survey (Richardson et al.
2011). The said paper presents its vital statistics along with
those for the other newly discovered satellites Andromeda
XXIV - XXVII. It is a dwarf spheroidal galaxy and has the
lowest recorded metallicity of the satellites we present with
?Fe/H? = −1.8 ± 0.2. Making use of the deeper coverage of
PAndAS in g-band, Richardson et al. (2011) obtain a distance
measurement of 767 ± 44 kpc from the horizontal branch of
the CMD.
Andromeda XXIII is a more challenging target for our al-
gorithm in its current form, with less than ∼ 50 stars ly-

Page 10

10Conn et al.
Fig. 15.— The four magnitude segment of the Andromeda II luminosity
function fitted by our MCMC algorithm. It is built from 4409 stars. The best
fitmodel isoverlaid inred. Thebin width forthe LFis again 0.01magnitudes.
Fig. 16.— Color-Magnitude Diagram for a circular field of radius 0.1◦cen-
tered on Andromeda XXIII. It is much more sparsely populated than those of
Andromeda I and Andromeda II. The RGB tip appears to lie just bright-ward
of i0= 21.
ing within the one magnitude range centred on the tip and
an OBR of 8.4 for the field and color-cut employed. The
color-magnitude diagram for this circular field of radius 0.1◦
is presented in Fig 16. We find the RGB tip at an i-band
magnitude of 20.885+0.038
of 0.112 magnitudes in the direction of Andromeda XXIII
(Schlegel, Finkbeiner, & Davis 1998), corresponds to a dis-
tance of 733(+13)+23
0.270 ± 0.039 and 0.105 respectively. Curiously, the MCMC
finds several peaks very close to the major peak in the pos-
terior probability distribution (see Fig 17) but these are at-
tributable to the lower star counts available in the luminos-
ity function around the tip. This has the effect of creating
large magnitude gaps between the stars that are just bright-
ward of the tip so that each individual star can mimic the sud-
den increase in star counts associated with the beginning of
the RGB. As a result, there is a range of likely locations for
−0.032, which, given an i-band extinction
(−11)−22kpc. We derive the values of a and f as
the tip, but neglecting any error external to the method, the
PPD shows that the object cannot be more distant than 867.0
kpc nor closer than 605.3 kpc with 99 % confidence.
Fig. 17.— The posterior probability distribution for 3 million iterations of
the MCMC on a 4 magnitude interval (see Fig 18) of the Andromeda XXIII
CMD selection presented in Fig 16. There are several probability peaks in
this instance but the preferred peak lies at 20.885. The distribution is again
color coded as in figure 5, with red, green and blue corresponding to 68.2 %,
90 % and 99 % credibility intervals respectively.
Fig. 18.— The four magnitude segment of the Andromeda XXIII luminosity
function fitted by our MCMC algorithm. It is built from 328 stars. The best fit
model is overlaid in red. Whilst the model LF tested by the MCMC retained
the resolution of 100 bins per magnitude described in §2.1, the data LF is
re-produced here at the lower resolution of 0.04 magnitudes per bin to better
reveal its structure to the eye.
4. CONCLUSIONS
The versatility and robustness of our new method can be
appreciated from §2 and its high level of accuracy is evident
from the measurement errors which are consistently smaller
than those in the literature to date. In addition, it is our hope
thatwith thecorrectpriorsimposed,this newapproachcarries
with it the ability to gauge distances to even the most poorly

Page 11

A Bayesian Approach to locating the TRGB11
populated substructures, bringing a whole new range of ob-
jects with in reach of the TRGB standard candle. In the case
of the M31 halo alone, it will be possible to obtain distances
to all of the new satellites discovered by the PAndAS survey
- a feat previously impractical using the TRGB. Furthermore,
PAndAS has revealed a complicated network of tidal streams
that contain valuableinformationas to the distributionof dark
matter within the M31 halo. With our new method, it will be
possible to systematically obtain distances at multiple points
along these streams, thus providing vital information for con-
straining their orbits.
The great advantage of our new Bayesian method over a
pure maximum likelihood method is the ease with which
prior information may be built into the algorithm, making
it more sensitive to the tip. Here in lies the great power of
the Bayesian approach, whereby the addition of a few care-
fully chosen priors can reduce the measurement errors ten-
fold. The result is an algorithm that is not only very accurate
but highly adaptable and readily applicable to a wide range
of structures within the distance (and metallicity) limitations
of the TRGB standard candle. With instruments such as the
6.5 m Infra-red James Webb Space Telescope and the 42 m
European Extremely Large Telescope expected to be opera-
tional within the decade, these distance limitations will soon
be greatly reduced. This will bring an enormous volume of
space within reach of the TRGB method, includingthe region
of the Virgo Cluster. A tool with which it is possible to apply
the TRGB standard candle to small, sparsely populated struc-
tures and small subsections of large structures alike is hence,
needless to say, invaluable.
ACKNOWLEDGMENTS
A. R. C. would like to thank Sydney University for allow-
ing me the use of their computational and other resources.
In addition, A. R. C. would like to thank fellow student An-
jali Varghese, for her assistance with and practical insights
with regard to the implementation of parallel tempering. A.
R. C. would also like to thank Neil Conn for assistance in
proof reading the document. G. F. L. thanks the Australian
Research Council for support through his Future Fellowship
(FT100100268)and Discovery Project (DP110100678).
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