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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
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<title>CASt: Distance to the Large Magellanic Cloud</title>
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<td style="vertical-align: top;"> </td>
<td>
<h2 style="text-align: center;">Distance to the LMC datasets</h2>
<h3>The CASt dataset</h3>
<a href="LMC_distance.dat" target="_blank">LMC_distance.dat</a><br>
<br>
<p><span style="font-weight: bold;">Astronomical background
</span></p>
<p>While it is easy to find the position of astronomical objects
in the sky (giving two dimensions of their spatial location), it is
usually very difficult to find their distance (the third
dimension). Throughout the 20th century, a complicated "cosmic
distance ladder" was developed. Starting from the nearest objects
for which distances could be reliably determined, calibrations for
various classes of luminous objects which serve as "standard candles"
were established. A complicated combination of these distance
indicators are applied, each with its own statistical and systematic
uncertainties. Many fundamental results depend on the cosmic
distance ladder: the luminosity of stars, the size of our Milky Way
Galaxy, the expansion rate of the Universe and thus the age of the
Universe, and much more. By the year 2000, most discrepancies in
the cosmic distances had been ironed out, and a consensus distance
scale emerged. However, uncertainties around 5-20% are still
commonly present and systematic discrepancies remain between some
methods.
</p>
<p>One crucial rung of the cosmic distance ladder is the distance
to the Large Magellanic Cloud (LMC), the nearest galaxy with a
significant stellar population. It thus serves as a bridge
between the scale within our Galaxy to the extragalactic
distances. Errors or uncertainties in the LMC distance propagates
throughout all of extragalactic astronomy. Thus, a huge effort
has been made to establish the distance to the LMC. Methods used
include: Cepheid, RR Lyrae and Mira variable stars (three types
of pulsating red giant stars), eclipsing binary stars, horizontal
branch and red clumpgiant stars, white dwarf stars, and the unique
supernova of 1987. The agreement between these methods are
reasonably satisfactory today. The figure below, and the
accompanying dataset give an mean distance modulus of DM = 18.515
±
0.085 mag (Clementini et al. 2003). Similar contemporaneous
meta-analyses give DM = 18.52 +/- 0.10 (Walker 2003), 18.55 +/- 0.06
(Fouque et al., 2003) and 18.48 +/- 0.04 (Feast 2003, all in "Stellar
Candles for the Extragalactic Distance Scale", Lect Notes in Physics
vol 635).
</p>
<p style="text-align: center;"><img
alt="LMC distance (Clementini et al. 2003)" src="LMC_distance.gif"
style="width: 600px; height: 637px;"><br>
</p>
<p> <span style="font-weight: bold;">Dataset</span></p>
<p>Statistically, this is a univariate dataset consistent of
repeated measurements of a single, unchanging physical quantity.
The quantity is known as the Distance Modulum (DM) with units of
magnitudes. It can be converted to linear distance in parsecs (1
parsec ~ 3x10<sup>13</sup> kilometers) with the formula </p>
<div style="text-align: center;">DM =
5*log(Distance) - 5.
</div>
<p>The distance to the LMC is thus about 50 kiloparsecs, with an
uncertainty of 5-10%. Each DM measurement is accompanied by an
individual measurement error; these are heteroscedastic measurement
errors of diverse origins.
</p>
<p>The dataset has 25 rows and the following
4 columns:</p>
<ol>
<li>Distance measurement method</li>
<li>DM (in magnitudes)</li>
<li>Error in DM (in magnitudes). This is the estimated
standard deviation of that measurement assuming Gaussian errors.</li>
<li>Astronomical reference for that measurement</li>
</ol>
<p>The dataset was obtained from the paper "Distance
to the Large
Magellanic Cloud: The RR Lyrae Stars" by Gisella Clementini, Raffaele
Gratton, Angela Bragaglia, Eugenio Carretta, Luca Di Fabrizio, and
Marcella Maio, Astronomical Journal 125, 1309-1329 (2003). </p>
<p> <span style="font-weight: bold;">Statistical exercises</span>
</p>
<ul>
<li>Use the DM values to exercise simple parametric
and nonparametric measures of location and spread. Any method
that can improve the accuracy and precision of the distance to the LMC
is scientifically valuable. <br>
</li>
<li>Use the DM values in the top (Population I distance
indicators) and bottom (Pop II) portions of the table to exercise
simple parametric and nonparametric two sample tests.</li>
<li>Do the same weighting by the heteroscedastic measurement
errors in DM. This is not always trivial (e.g. weighted two
samples tests).</li>
<li>Compare the spread in DM values with the
measurement errors (e.g. using the chi-squared statistic) to evaluate
whether the errors seem reasonable, or to establish whether some
measurements are unreliable outliers. <br>
</li>
</ul>
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