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Study on the Variable Star XX Andromeda

Disclaimer: This work has been submitted by a student. This is not an example of the work written by our professional academic writers. You can view samples of our professional work here.

Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of UK Essays.

Published: Tue, 20 Feb 2018

Abstract

We present the results of a month long V-Band study on the RRab type variable star XX Andromeda. 4526 data points are used to plot a light curve, with 3 maxima observed and added to data from the GEOS database to create an O-C diagram. Three methods of estimating the pulsation period are used, including two Phase Dispersion Minimisation methods and an O-C method, resulting in a best estimate of the period of days. This value is in excellent agreement with the literature values for the period of XX And, from both the Hipparcos catalogue and the GCVS. The distance to XX And is estimated to be pc using a main sequence fitting method to estimate the absolute magnitude, and the mean radius is estimated to be . A flatfielding improvement to the “photom.py” pipeline is suggested to combat dust artefacts on the CCD. Physical reasons are discussed for the distinctive features present in the light curve, namely the “Hump” and the “Bump”.

I. Introduction

In 1893 Solon I Bailey started a program of globular cluster study[i]. He noticed that some clusters (e.g ω 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[ii].

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[iii] or the more recent Hipparchos Catalogue[iv]. 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[v].

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[vi]. 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[vii]. Using this reddening data with other distance indicators (for example red clump stars in the bulge[viii]), 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” (see 1). 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. 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.

II. Theory

Observational Theory

CCD Theory

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[x]. 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[xi].

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[1] that an pair will be produced. On the other hand, with photographic plates one can achieve (at best) an efficiency of 3%[xii], 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[xiii]. 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[xiv], to around 35°C below ambient temperature, as the thermal current is approximately halved for each 7°C reduction in CCD temperaturexii.

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

, [xv](2)

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.

The Automated Photometry Process

Since the experiment involved a large number of images, the photometry processes were automated using several Python scripts and FORTRAN routines.

The script “all.py” was used to iterate the “photom.py” script over a range of images within a directory and print a string detailing which file was currently being processed.

“photom.py” was the main script run, and was used to call several other processes which ran the photometry calculations, among other things. Firstly, it read in the file specified, and split the filename into the file and the extension, by using the find function to search for the full stop as the delimiter.

i=file_name.find(‘.’)

Using the extension to determine the file type, the script then either subtracted the master_dark.sdf frame (if it was a FITS file, and hence a DRACO output file) or converted it to a FITS file (if it was an ST9 file, and hence 14-inch, which had already had the dark frame removed). The conversion is achieved using two separate routines: sbig2ndf, a routine from the SBIG python module which converts compressed output ST9 files created in CCDOPS into NDF files, and ndf2fits, which is a routine from the convert set of variables that converts the NDF files to FITS images. The subtraction of the dark frame is made using the kappa package from Starlink.

“photom.py” then reads the variable star position from a user-created ‘var_sky_position” file. Using this, the script runs “find_astrom.py”. This attempts to match the stars in the image to the USNOA2 catalogue, and produce a new FITS file with the derived header solution. Firstly it takes the given star position as the centre of the image, and runs sextractor to find all the objects in the image. Next, it runs the WCS Tools routine scat at that RA and Dec to attempt to find any known objects in the region from the catalogue and prints it to a new file, usnoa_ref.cat:

commands.getoutput(scat+’ -d -c ua2 -n 200 -m 17 – r 600 ‘+ra+’ ‘+dec+’ j2000 > usnoa_ref.cat’)

The pixel scale is taken from the directory’s automag_driver file, and used by Andrew Pickles’ starfit script to match each object found by sextractor to the catalogue’s objects. This is achieved by the matching of triangles created between sets of objects in the sky to similar triangles created from the catalogue’s objects. Starlink’s astrom routine is then used to correct the solution:

out=commands.getoutput(‘/star/bin/astrom fits=asc’)

print “astrom returns:”, out

Finally, “find_astrom.py” edits the header keys using pyfits to reflect the newly derived solution, and creates a new FITS image with the file ending “_ast.fits”.

“photom.py” then runs sextractor again, to product a new catalogue of the objects from the image, complete with their RA and Dec. The script then performs the aperture photometry using “automag.py”. This measures the relative aperture magnitudes for the objects defined in the new object catalogue, by taking the number of counts within the specified aperture radius from the driver, and applying the formula:

(3)

Here is a constant offset defined in the driver, is the number of counts within the aperture (which is pixels in size) minus the background, and is the integration time. Background errors are calculated by measuring the counts within the two “sky” aperture radii to find the mean and rms sky-counts over pixels,and, and firstly deriving the signal to noise ratio for the star, by applying Equation (4) below[xvi].

(4)

In the above equation, is the gain of the CCD. By using the flux based definition of the magnitude difference and manipulating the logarithm equation, the signal to noise value can be used to find the error on a measured magnitude, as shown in Equation (5).

(5)

These instrumental magnitudes are appended to the catalogue file, next to each object. “auto_mag2list.py” is subsequently run to pull the calibration stars from the catalogue, by matching the RA and Dec to those in the “cal_sky_positions”. The variable star’s data, as well as the calibration stars’ data and the observation time in Modified Julian Days (MJD) are then appended to a file called “summary.obs”.

Once “photom.py” completes, the raw2dif routine can then be run to perform the differential magnitude calculation. This routine takes each line from the “summary.obs” file and subtracts the average of the two comparison stars’ instrumental magnitudes from the variable star’s instrumental magnitude, . A zero-point constant is then added to put the differential magnitude on the standard scale. This can be measured by taking images of photometric standard stars (from the Tycho catalogue for example), and comparing their instrumental magnitudes to their known apparent magnitudes, as described in the next section.

(6)

raw2dif outputs simply the observation time, variable star’s standardised magnitude, and the error on the magnitude to a user-defined file.

Magnitude Zero-point Measurement

The zero-point is found by measuring the magnitudes of photometric stars with the telescopes, and comparing these to the values found for the stars in the Tycho catalogue. This catalogue uses a separate magnitude system, which can be converted into standard V-band magnitudes using the following formula:

(7)

The difference in these catalogue magnitudes and the observed values can then be used to show the difference that the specific equipment has made. This is the zero-point magnitude.

Period Determination

Two programs are used for the period determination, the routine bforce and PDM win 3.0[xvii].

bforce uses a brute force method to find the period of the variability. It attempts to fit the data onto a user generated model of the light curve (with a phase resolution of 0.1), and wrap (or “fold”) it around a suggested period. The routine then splits the data into a series of bins and estimates the variance in each, as follows;

, (8)

for observations in each bin. If the trial period is incorrect, there will be a large scatter of magnitudes in each bin, i.e. a large variance. This is compared to the variance of the data set as a whole using an F-test, which is achieved by finding the ratioof bin variance (the explained variance) to total variance (the unexplained variance). For an incorrect estimate of the period ≈ 1, whereas for the correct period<< 1. The routine will use a new trial period and re-compute the variances, repeating this until a minimumvalue is obtained.

The PDM program works in a similar, if more refined way, implementing some of the recent changes in the accepted way of calculating a phase dispersion minimisation period. While still using a variation-based method, it finds the period using a beta-distribution method (designated PDM*) rather than an F-test, as this has been shown to be the correct probability distribution to use[xviii]. It also utilises a GUI with a series of user-set options, for example variable phase resolution.

RR Lyrae Theory

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[xix]:

* 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.

Typical characteristics of RRab stars

RR Lyrae stars are large red stars with a low mass, occupying the area of the instability strip on the H-R Diagram (see Fig. 1) between δ-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, but a summary of current approximations is provided in Table 1.

Period

0.2 – 1.1 days

Mv

0.78 ± 0.02

Te

6404 ± 12 K

[Fe/H]

-1.56 ± 0.25

Mass

0.55 ± 0.01 M⨀

Radius

5 ± 1 R⨀

Table 1. Typical properties of RRab variables. All values are mean values of 335 variable stars[xxi], except period which is a typical rangei.

Evolutionary theory

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 (see Fig. 1), 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 -α 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.

Pulsation theory

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[xxii]. 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,

(9)

where is a constant, is the density of the region, and is temperature.

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[xxiii] 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[xxiv].

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[xxv]), as the majority of stars are simply too far away for resolution currently[2]. 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) is given by H. Smithi:

(10)

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”[xxvi]. This used a sixth order Fourier decomposition of the light curve to find multiple properties of an RR Lyrae star. When they plotted the data they found the following linear relation:

(11)

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, taking into account the galactic extinction in the direction of the object due to dust and gas in the galactic plane, :

(12)

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

(13)

whereis Stefan’s Constant, andis the star’s effective temperature.

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[xxvii].

Estimation of Radius

Marconi et alxxv have published an equation relating the period of a fundamental mode RR Lyrae star to its average radius;

(14)

whereis the mean radius (in units of solar radii), is the period (in days), and is the heavier-than-iron metallicity of the star, defined as;

xxi, (15)

whereis the alpha-enhancement with respect to iron, and is taken to be equal to 1. This is derived from their theoretical predictions of the radial oscillations of a metal poor RR Lyrae, and applies to stars with helium abundances of between (0.24 and 0.28).

III. Experimental Methods

Preparing the experiment

Inital sessions were spent becoming aquainted with the computer’s Linux-based operating systems, understanding the basics of photometry and exploring the provided software. Several rooftop sessions were attended to gain knowledge of the telescopes provided, and to learn safety procedures associated with the use of the equipment. Due to initial poor weather, previous year’s data was analysed in order to improve understanding of the provided scripts. A list of RRab targets from the NSVS catalogue[xxviii] was examined to find a suitable object, with a magnitude range visible on the telsecopes available, a period of less than a day, and a high position in the sky.

Table 2. Properties of the Telescope and CCD combinations for each dome. Both telescopes were fitted with the same model of V-band filter.

Background information on the chosen star (XX And) was found using the SIMBAD database[xxix], and examined to find previous studies, including estimates of period, metallicity, and star type, as well as dates of previously observed maxima. A plot of the field around the star was taken, and used to identify two calibration stars for the photometry ( 3):

The calibration stars used were USNOA2.0 numbers 1275-00765817 (cal-star 1) and 1275-00761527 (cal-star 2). They were searched for in various catalogues to verify that they were not known to be variable. The best exposure time for our field was estimated to be 30 seconds with the 14-inch telescope, and 60 seconds with DRACO, so as not to saturate the image.

By taking some sample images and viewing them in GAIA, suitable sizes for the apertures were chosen for each telescope. The sizes of the apertures were chosen to enclose the whole star, whilst giving the minimum error. These were then converted from scaled values to numbers of pixels, and entered into seperate “automag_driver” files for each telescope, along with the specific pixel scale, gain and read-out noise.

Telescope

Star

Sky Inner

Sky Outer

14-inch

4.7

14.9

21.4

DRACO

7

25.7

35.1

Table 3. The aperture radii (in pixels) used for each telescope.

Firstly, the “convert” variables were set up. XX And’s RA and Dec in decimal degrees were inserted into a file called “var_sky_position”, and “photom.py” was run on the first frame (called for example “filename.fits”). This produced an output file called “amag.out” which contained the positions of all the recognised stars in the image, as well as a calibrated image “dfilename_ast.fits”. By comparing the (x,y) pixel locations in GAIA for the two calibration stars with the data in “amag.out” the RA and Dec of the calibration stars were noted, and inserted into a text file named “cal_sky_positions”.

Observation of the Variable

Observations of the field containing XX And were then taken over a period of 1 month, using both the 14-inch “Far East” and the 10-inch DRACO telescopes. For the 14-inch, the observing process was as follows: The object was located using the Earth Centre Universe program, the telescope synched and set to track, and the CCD programmed to take around 30 images per sequence at 30 seconds each, with an 8 second dark frame before each new image. For DRACO, the object was found using the provided G.U.I., with care taken to place the variable star and both comparison stars away from dust grains on the CCD. The telescope was set to track, and programmed to take a large number of images with a 60 second exposure. For each new observing session a seperate file was created, containing all the images and the scripts required for automated photometry. For DRACO processing, a master dark file was also copied from the archive. The file “all.py” was then amended to iterate over all the images in the directory, and set running.

Once the photometry had completed, the raw2dif routine was run, and the results viewed by running qplot. The data were adjusted to Heliocentric Julian Days by running the cor2hjd routine, and the final tables were copied across to a main results directory to be added to the full table of data. bfplot was run on the full dataset using an estimate for the period, and the phase values from the output file “fort.30” were killed out and yanked into the dataset file using EMACS. This table was viewed in TOPCAT, and a light curve created. Any clear and accountable anomalies were removed in TOPCAT.

To gain a value for the absolute magnitude of XX And, rather than simply an instrumental magnitude, a series of observations were made of photometric stars which had known magnitudes. These are shown in table 4 below:

Photometric Star

RA

Dec

Apparent V-band Magnitude

1

1h 18m 20.581s

38° 55′ 38.23”

9.847

2

1h 14m 50.729s

38° 29′ 55.80”

9.961

3

1h 15m 12.229s

38° 49′ 10.95”

9.048

4

1h 16m 39.436s

39° 09′ 38.64”

9.735

Table 4: Properties of photometric stars used in the magnitude calibration of XX And.

This gave a value for the correction which had to be made to all the observed values for each telescope. The corrections were then applied to the full dataset.

An O-C diagram was constructed using the data from the Hipparcos mission, the GEOS RR Lyrae Survey, and also archive data from the GEOS database[xxxi]. The period used was the Hipparcos estimate. Since the newly observed data used HJD, and the archive data was in “modified HJD”, an addition of 0.5 HJD has to be made to the new data in order to be comparable. The newly observed data was then added to the diagram, and the input period was altered to give the flattest line possible, thus providing a new estimate of the period. The error on the period is given by the slope of the line[xxxii]. Any historical period changes were searched for in the line of the O-C plot.

The fast_solve routine was run on all of the summary.obs files, and the comparison stars were checked to see whether or not they were varying. The output model file from fast_solve was edited to include estimates of bin values where there was no actual observational data, and then used in the routine bforce. This was run using the period quoted in the Hipparcos catalogue as the initial period to give an estimate of the new period and its error. The period was also estimated using PDMwin, using an output table from TOPCAT.

Errors in the period-finding were estimated using the Jackknife method on both the PDM and bforce programs. This was achieved by recomputing the period, but leaving out one observa


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