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In 1893 Solon I Bailey started a program of globular cluster study. He noticed that some clusters (e.g ? Centuri) were extremely rich in variable stars with similar properties - they had periods of less than a day, and light curve amplitudes of around 1 mag. The mean value of apparent magnitude of these stars in a particular cluster was also approximately the same across the sky. Bailey named these "Cluster Type Variables". However an increasing number of stars with these properties were being found outside of clusters - indeed the brightest star of this type ever found was a field variable, RR Lyrae (after which the class is now named). Discoveries then began to come thick and fast, and it is currently estimated that over 85000 exist in the Milky Way alonei. RR Lyrae variables have also been observed in the Andromeda Galaxy, the Large Magellanic Cloud and other Local Group dwarf galaxies.
Measuring the properties of these variables has become increasingly important to astronomers, as it was realised that they could be used to gauge astronomical distances through a period-luminosity (P-L) relation, in a similar way to Cepheids. Various catalogues have studied their properties, for example the General Catalogue of Variable Stars or the more recent Hipparchos Catalogue. Until recently however, no distinct P-L relation had been found, and instead astronomers had to use a relation between metallicity and visual magnitude or the Baade-Wesselink method, the drawbacks of which are discussed later. Currently there is still no P-L relation for V-band observations, although there are now relations for most of the infrared spectral bands.
RR Lyrae variables are also of importance for the study of the population of both the Galactic Bulge (via Baade's Window for example) and the Galactic Halo. Their advanced age and low metallicity combined with distinctive pulsation properties provides an excellent "tracer" for the development of the Milky Way in its early stages, as well as current kinematic analysis. They have also been used as a means of quantifying the interstellar reddening caused by dust in the galactic plane, thanks to the fact that the colour excess is a function of minimum (V-I) colour only. Using this reddening data with other distance indicators (for example red clump stars in the bulge), a meaningful approximation of the distance to the centre of the bulge can be obtained. Clearly then the study of RR Lyrae variables is useful for the understanding of the evolution of both the Milky Way and the rest of the Local Group.
The star to be observed in this study is XX Andromeda (abbr. XX And), an F2 spectral class RRab type variable, located in the constellation of Andromeda at RA: 1h 17m 27.4145s, Dec: +38°57' 02.026". Its moderately high position in the sky at Durham means that it is circumpolar, whilst not exceeding the +65° limit for the telescope fork mount, resulting in minimal atmospheric interference and the maximum possible observing time. The GCVS lists a period of, and a variation in apparent magnitude of 10.8 to 11.13 in V-band. It is also known to exhibit the Blazhko effect, a long-period modulation of the amplitude of an RR Lyrae star (the cause of which is currently under investigation), with a period ofiii, and has an [Fe/H] value of -1.94.
Perhaps the most important advance in astronomy in the last 20 years has been the widespread use of Charge-Coupled Devices (CCDs) to replace photographic plates. Invented in 1969 at Bell Labs by Boyle and Smith, CCDs are a thin piece of semiconductor material (e.g. silicon) upon which lies an grid-like array of metal-oxide semiconductor (MOS) capacitors. During an exposure, if a photon impacts on the silicon an electron/hole pair can be produced, as an electron is pushed up into a higher energy state. The MOS capacitors act as deep potential wells (pixels), which hold the electrons until the exposure is finished. The charge is then read-out to an amplifier at one edge, in a specific order so that that the position of the original pixel can be identified, and related to the magnitude of the detected charge. The charge is converted from a raw number of electrons into ADUs (analogue to digital units), the conversion factor of which is the gain of the CCD.
They are preferred to photographic plates in modern astronomical photometry for several reasons:
- High quantum efficiency (QE) - for each incident photon there is upwards of 90% certainty that an pair will be produced. On the other hand, with photographic plates one can achieve (at best) an efficiency of 3%, so using CCDs will increase the likelihood of detection of distant objects.
- Large dynamic range, allowing them to detect objects with a range of magnitudes in the sky in the same exposure.
- Strong linearity up to the saturation point, so that for longer exposure times the number of electrons produced is proportional to the integration time, whereas photographic plates will experience a drop in their efficiency. Their linearity will also mean that the magnitude of charge in each pixel is linearly proportional to the luminosity of the object.
CCDs have also brought some inherent problems however, for example the noise associated with each image. Because photons obey Gaussian statistics for large counts, there will be a shot noise (uncertainty in the count rate) for each pixel of whereis the number of photons detected. Error in an image also stems from both the bias of the CCD, and the "dark current" present. The bias of a CCD is a systematic voltage offset across the whole CCD to prevent digital underflow during analogue to digital (A-D) conversion. It includes the read-out noise, a result of the manipulation of the pixel charge values during the A-D process and any charge-loss which occurs during the transfer. A CCD's dark current is an unwanted flow of electrons which have been released from the surface of the semiconductor by thermal excitation, and is purely dependant on the surface temperature, rather than being a function of illumination. For this reason the CCD was cooled by both the Peltier method (electrically) and with an active assisting fan, to around 35°C below ambient temperature, as the thermal current is approximately halved for each 7°C reduction in CCD temperaturex.
To remove noise from an image, a set of calibration images may be taken alongside each raw exposure. These are called bias and dark frames. The bias frame is a zero-time exposure which will include both bias and read-out noise. A dark frame can be found by leaving the shutter on the camera closed and taking an exposure seconds long. It can be expressed as
whereis the dark current, andis the thermal noise's statistical variation. Ideally one would be taken before each exposure, as temperature routinely varies slightly with time. A "master dark" frame can be found by taking the average of a large number of dark frames, and will include the equivalent of a master bias. This master dark can then be subtracted from each image to leave a final, processed image with as low a random error as possible.
Subclasses of RR Lyrae Variables
From his observations, Bailey noticed three separate subclasses of variable, which have subsequently been compacted into two subclasses (as subclasses a & b were very similar). The following is paraphrased from Bailey's original description:
- Subclass "ab": Very rapid increase of magnitude, with a moderately rapid decrease in mag. Nearly constant mag for approx one half of the full period. Amplitude of roughly one mag and a period of between 12 and 20 hours.
- Subclass "c": Magnitude always changing, with moderate rapidity. Range generally half a magnitude, with a period of 8 to 10 hours.
As our study concerns an RRab type variable, this class shall be primarily discussed.
RR Lyrae stars are large red stars with a low mass, occupying the area of the instability strip on the H-R Diagram between d-Scuti and Cepheid variables, where it intersects the horizontal branch. They are in the core helium burning stage of their evolution, having exhausted their core hydrogen fuel. Mean physical properties of these variables are under some contention..
It is thought that the progenitor of an RR Lyrae star was a typical low-mass main sequence star, with M* ˜0.8M?. For the first 15 Gyr of its life, the star burns core hydrogen, fusing it into helium. Once the hydrogen supply in the core is exhausted, the star expands to become a red giant, moving off the main sequence and up the giant branch of the Hertzprung-Russell diagram,and shell-burning of hydrogen now occurs around an inert helium core. The helium core eventually collapses, becoming electron degenerate, and increases in temperature until the helium in the core ignites using the triple -a process, causing the "helium flash". The core's degeneracy is lost and the star moves off the giant branch asymptotically, down towards the instability strip. At this point it can develop the pulsational properties of an RR Lyrae star, although this will be dependent on its mass, its chemical composition, and its temperaturei.
Once the helium core is also used up after around 0.1 Gyr, the star begins to expand and cool again, fuelled only by shell burning of hydrogen and helium. The core never becomes hot enough for the fusion of heavier elements. Eventually all the usable fuel is expended and the star will jettison off its outer layers of material to leave a white dwarf star, shining only through the radiation of internal thermal energy.
The study of pulsation theory owes much to Arthur Eddington, who wrote a series of papers detailing a mathematical description of the properties of stars. Having realised that a radial pulsation in a static star would have a decay time of around 8000 years (much shorter than the length of time stars spend in the instability strip), he proposed that stars behaved as thermodynamic heat engines, using some "valve mechanism" to regulate energy flow. In order to fulfil pulsation, this valve would need to make the star more thermally opaque as the star was compressed, and less opaque as it expanded. Effectively this would cause energy to build up when the star was compressed, forcing the star to swell in size until some turning point was reached and the opacity was small enough that energy could escape, leading to the star contracting again. The Rosseland mean opacity shows the overall opacity of a stellar region, and is defined as follows,
Eddington was unable to come up with a particular material that would possess these properties in a star, particularly as during his time it was not believed that hydrogen or helium made up significant proportions of the inside of stars. It is also the case that neutral hydrogen or helium regions cannot be the "valve" region, as for these regions and - i.e. as increases will decrease. This would lead to the pulsation dying out extremely quickly as all the radiative pressure was lost during contraction.
However in 1953 Sergei Zhevakin found that regions of doubly ionised helium would provide an area wherebecomes small or negative, resulting in the desired properties for the gas. It was later shown by R. F. Christy that hydrogen ionisation can play a smaller, but still important, role in the mechanism.
Ionisation zones can make another possible contribution to the "valve" in a star. If the energy from fusion processes cause ionisation in gas regions instead of raising their temperature, then the gas will absorb heat during compression stages, causing a pressure maximum near the minimum volume and thus aiding pulsation. This is known as the mechanism.
Different classes of RR Lyrae variable pulsate with different modes. For instance RRab stars all vary in the fundamental mode, whilst RRc stars are pulsating in the first overtone. This is one of the reasons that types "a" and "b" can be separated from type "c" as a separate class. A third class of variables has also been observed, termed RRd type stars, which have a double-mode pulsation, pulsating in the fundamental and first overtone modes simultaneously.
However, some RRab stars show a long-timescale second periodicity, known as the Blazhko effect. This is an overarching period that can be anywhere between 30 days and several years. The cause of this effect is unclear, but is believed to come from either a nonlinear resonance effect between the radial fundamental mode and some non-radial mode, or a cyclical rotating magnetic field that deforms the main radial mode of pulsation.
Estimation of Absolute Magnitude and Distance
RR Lyrae stars are useful for the determination of astronomical distances, especially to regions such as clusters in the Halo, and the Bulge. However, unlike for Cepheids, accurate parallax measurements of distance do not exist for RR Lyrae variables (with the exception of a very few - the star RR Lyrae itself for example), as the majority of stars are simply too far away for resolution currently. Instead, astronomers look to alternative measurement tools, for example main sequence fitting or the Baade-Wesselink method.
Main sequence fitting is the process of determining the distance to a cluster by fitting its colour-magnitude diagram to that of nearby main sequence stars which have a parallax-determined distance. This has produced a wide variety of relations over the last twenty years, but a general relation (that is within error of the majority of current estimates).
The currently favoured method of finding the metallicity is to use the relation, described by Jurcsik & Kovács in their seminal paper "Determination of [Fe/H] from the light curves of RR Lyrae stars". This used a Fourier decomposition of the sixth order the light curve to find multiple properties of an RR Lyrae star. When they plotted the data they found the following linear relation:
This allows the metallicity to be determined accurately, and then used in the main sequence fitting method to find an accurate absolute magnitude for a star.
Finding the absolute magnitudeis important, because it allows for the use of the magnitude equation to determine distance to an object:
The Baade-Wesselink method, originally applied to Cepheid variables, was based on the assumption that a star will have the same surface temperature and brightness at all points of equal colour on the ascending and descending sides of the light curve. This implies that any luminosity variation between two half-phases can be said to be the result of radial differences in the star. Thus a fractional radius change can be measured as. If a radial velocity curve is also plotted for the star, the radius change over the period can be directly measured, and through the combination of these two results a value for the luminosity of the star can be found. This can be used to show the distance to an RR Lyrae star through the relation.
However RR Lyrae variables do not behave exactly like Cepheids; for example during stellar expansion the surface gravity is much greater than when the star is contracting, leading to flux redistribution across the surface. This, combined with shock waves permeating through the stellar atmosphere causing distorted radial velocity curves, means that V band photometry is unfortunately useless for applying the Baade-Wesselink method to RR Lyrae stars. The procedure must instead be carried out in (V-H) or (V-K) colours for example, as infra-red wavelengths are less sensitive to the expansion phase distortions.
Estimation of Radius
From http://arxiv.org/PS_cache/astro-ph/pdf/0503/0503382v1.pdf, use the equation for PRZ for F mode pulsations. When you work it out, period is in days, metallicity is not [Fe/H] value, use the definition in typical properties paperxi,
Use various bits of this website http://www.univie.ac.at/tops/blazhko/Background.html to describe the Blazhko effect.
Oosterhoff described here: http://vger.pa.msu.edu/posters/smith_rev2.pdf
Discuss some stuff here
- Long rise time of XX And: see the table at the bottom of this bulletin - http://www.aavso.org/observing/programs/rrlyrae/number3.pdf
- Couple this with the unusually long period for an RR Lyrae, and ask - "Is this really an RR Lyrae?" Can we apply usual distance relations to it? BLUGH
- Note that Hipparcos was only able to determine parallax distance for one star - RR Lyrae (ref. http://spiff.rit.edu/classes/phys240/lectures/lmc/lmc.html, viewed 06/01/2010)
- Discuss the pitfalls of the [Fe/H] vs Abs Mag relation - Note that this relation depends strongly on ???
- Bluer stars have shorter periods? See page 42 of RR Lyrae stars.
- Perhaps a fourier decomposition would have been better for determining the period?
- Long discussion of the reason for the Hump and Bump in the period
- Could've taken flatfields for draco?
- DONT FORGET TO CHANGE THIS NEXT SECTION INTO SOME DIFFERENT WORDS. Flatfield frames take into account the differences in quantum efficiency between pixels, which would otherwise cause variations in gain levels in the CCD. They are made by pointing the camera at a uniformly illuminated source, and using a long enough exposure to approximately half-fill each potential well. A number of these "raw" flatfield frames are averaged, and then an average dark frame of equal exposure time (a "dark-flat") is subtracted from them. This leaves a "master flat" frame. The raw image then has the scaled master thermal frame subtracted from it before being divided by the master flat, leaving a correctly calibrated image.