Spectroscopic Binary Stars Observed with the 1.52-M Telescope at L'observatoire De Haute Provence

Daniel Underwood PHAS3332 MSci Astronomy

Spectroscopic Binary Stars Observed with the 1.52-m Telescope at l'Observatoire de Haute Provence

By Daniel Underwood PHAS3332 Msci Astronomy

28th April 2010

Daniel Underwood PHAS3332 MSci Astronomy

Abstract

Observations were carried out at the Observatory of Haute Provence on the 1.52-m telescope to observe the properties of a number of spectroscopic binaries; this report focuses on HD112014 and Lambda Tauri. The high resolution spectrograph was not helpful in ascertaining a value for the cyclical variation of either of these stars, but further analysis helped in determining amplitudes of oscillation. For HD112014 the maximum radial velocity was found to be 105.97 km/s for the primary and 117.30 km/s for the secondary. The method used for these calculations could not produce uncertainties, but each value is found to be within 10% of the published values of 108.3 km/s and 128.9 km/s respectively [2]. For Lambda Tauri the situation was found to be similar, with a calculated value of 54.16 km/s, compared to a published value of 56.8 km/s [2]. A mass-ratio for the objects in HD112014 was calculated to be 1.17 ± 0.03 compared to a published value of 1.1902 [2] (no uncertainty attributed), which is an excellent approximation from the experiment.

Introduction

Our aim for this project was to obtain high-resolution spectra of a number of targets that are believed, if not proven, to be spectroscopic binary star systems. By analysing the spectrum of a spectroscopic binary target it is possible to infer radial velocity shifts through phenomena caused by the Doppler Effect. The importance of stellar spectra lies in the spectral lines that arise due to the presence of certain atomic species in the atmospheres of the observed stars; absorption lines develop as a result of the absorption of a discrete wavelength energy (photon) by an electron in a particular molecular species – the wavelength is subsequently absent from the continuum. These lines can vary in position on a star’s spectrum relative to its intrinsic ‘laboratory’ value due to the Doppler Effect, by shifting up or down the Wavelength axis. This depends on the radial component of the star’s velocity relative to the line of sight of observation; a shift of a spectral line towards longer wavelengths, or a “red-shift”, occurs when a star’s velocity is in the direction opposite to the observer, i.e. it is receding, and the opposite is true for “blue-shifts”. This phenomenon is what is investigated when observing spectroscopic binaries. These binary systems cannot be resolved visually, and therefore it is necessary to analyse their spectra, paying particular attention to any evidence of Doppler shifts occurring. If there is any radial component of velocity in a binary system relative to our observation position, then there is a periodicity associated with its recession and advance along the observer’s line of sight, as each star moves towards and away from us in an alternating fashion. Therefore it should be possible to observe a rhythmic behaviour in the shifting of the spectral lines on the star’s spectrum; as the radial velocity changes from a receding direction towards an advancing direction (relative to our observation) the characteristic spectral features should change from being red- shifted to a becoming blue-shifted, from an equilibrium position on the spectrum.

Daniel Underwood PHAS3332 MSci Astronomy

If this shift is observed over time then it is possible to plot the deviation and amplitude of the red/blue-shift from the equilibrium position as a function of time, which should produce a sinusoidal-type variation. It is then possible to extrapolate from this curve the period of rotation for the binary system.

Stars that are determined to be binaries may have different types of spectra from each other, with respect to whether or not features from both stars are present. A double-line spectroscopic binary system has a spectrum where absorption lines from both of the rotating stars are observed, due to the similarities in brightness of both objects within the system. Over a period of time these lines will shift relative to each other, however lines from one star will be blue-shifted as the lines of the other star are red-shifted, which then switches along the period of the rotation. If a characteristic line is observed for both stars then this line will be merged from both stars when one star is directly in front of the other along the line of sight. This merged line will then split into its component parts corresponding to each star when they begin to deviate from this position and their radial component of velocity increases. The lines shift towards the maximum displacement and then go back to merging again, which corresponds to the star that was previously obscured to being the star that is obscuring the other star. Using this characteristic line as a basis, the merged points represent a radial velocity of zero, and their maximum displacement from each other on the spectrum corresponds to the maximum radial velocity of each star. However some spectroscopic binaries are confirmed by the absorption lines of a single star. In most cases, the features from one star are stronger than those of the other star due to a greater luminosity, but sometimes they are so much so that the spectral lines of the secondary object are not observed. These are known as single-line spectroscopic binaries, as the periodic shifting of the lines of the single star show that there is some shift in radial velocity over a period of time.

Other types of binary systems include visual binaries, which are systems that can be visually resolved. Their behaviour is monitored by observing the stars’ motion relative to background sources. The final type of binary system is that know as an eclipsing binary. Binary systems of this nature are aligned such that from our line of sight we will observe one of the system’s stars eclipsing the other star at a certain time in the orbit. Analysis of these binaries is made using photometric studies of their light curves as opposed to their spectra.

The nature of this project is to observe and analyse spectroscopic binaries and therefore obtaining and reducing their spectra will be necessary, by making use of the 1.52-m telescope at OHP.

The radial velocity of an object at a particular point in its cycle is determined using the physics of the Doppler shift. This can be summed up in the following equation:

휆−휆0 푣 = (1) 휆0 푐

Daniel Underwood PHAS3332 MSci Astronomy

Here, λ represents the observed wavelength of a spectral feature, and 휆0 represents the ‘rest’ wavelength of the same feature, measured in a rest frame in the laboratory; 푐 represents the speed of light, and 푣 is the resulting radial velocity. By direct observation of a spectrum the Doppler shift of a spectral feature can be determined by comparing the measured wavelength with the laboratory value, and subsequently the radial velocity can be measured for the object at the specific time corresponding to the observation. Thus for a set of spectra taken over a period of time for a certain binary system, the radial velocities at each point may be measured in this way, and the changes in radial velocity can be plotted as a function of time to determine the period.

Another method of determining the radial velocity is known as the cross-correlation method. The theory involves measuring the levels of correlation between two data sets. For an x-y array of data measurements of the tendency of the x-points to correlate with the y-points can be made using a variable r, in simple correlation methods. If x tends to increase with y then the value of r tends towards unity, and r=1 for complete correlation; conversely, r=0 when there is absolutely no correlation. In a similar fashion a correlation methods can be made between two different data sets, provided that they have similar y-values. Shifted spectra are a perfect example of this; the y-values, which represent intensity, are close to identical for the different spectra of the same star, whereas the x-values, representing wavelength, are shifted either way. If a variable k is introduced to represent the correlation between two spectra (k=1 for complete correlation, k= 0 for no correlation) then two data sets can be compared; a constant is added/subtracted from each x-value of one data set, and this is compared with a comparison “zero- point” data set, and the correlation is then evaluated. The program Dipso can perform this measurement over many values of the added constant, and then plot these new x-values against the k variable. This graph should produce a peak corresponding to a correlation between the x-values of the comparison data set and the x-values of the other data set with the a certain ‘lag’ constant added or subtracted (where k=1). The ‘lag’ constant found at this peak corresponds to the difference in Doppler shift between the two data sets. Using this method it is possible to determine the radial velocity that a certain spectrum represents relative to whichever comparison spectrum is used.

For this project I have decided to study two binary star systems, HD112014 and Lambda Tau. These both have fairly short periods of rotation which means that given the length of our duration at the OHP we should be able to obtain a good span of data in order to notice an obvious differential in the radial velocities over the time period. From this we can attempt to find a value for the period of rotation, and even ascertain the amplitude of their radial velocities.

Daniel Underwood PHAS3332 MSci Astronomy

HD112014

This binary system is a double-lined spectroscopic binary. The double line nature arises due to the similar intensities of the lines from each star. This makes it possible to observe any prominent lines merge together and split during the cyclic nature of the variation.

Right Ascension 12h 49m 06.6790s [3] Declination +83° 25’ 04.196” *3+ Spectral Type B9 V [4] Published Period (days) 3.2866 [2]

Figure 1: Field of view of HD112014

Daniel Underwood PHAS3332 MSci Astronomy

Lambda Tauri

This binary system is a single-lined spectroscopic binary, and therefore its nature implies that lines from the primary object can only be observed due to the dominating intrinsic brightness of the primary object.

Right Ascension 04h 00m 40.8157s [3] Declination +12° 29’ 25.248” *3+ Spectral Type B3 V [4] Published Period (days) 3.9529 [2]

Figure 2: Field of view of Lambda Tauri

Daniel Underwood PHAS3332 MSci Astronomy

Observing Procedure and Data Acquisition

It was necessary to plan our observations in advance of arriving at the OHP due to the fact that we had a total of twelve targets of interest that could be observed. The targets, which are suspected to be binary systems, all have different properties concerning their periods. This creates a situation where it is necessary to prioritise our observations, the main factor being precisely what we intend to do with each observation. Our target list was essentially split into two categories; the long period binaries and the short period binaries. This is an intuitive place to start seeing as only six nights of observation would be available to us. Because of this, the shorter period binary star systems would be placed at a higher priority with respect to determining the radial velocity shifts, as attempting to infer the shifts for the binaries with a much larger period would prove to be a worthless endeavour over the period of our visit. The remaining targets with longer periods would yield results that would be more useful in comparison with data from previous years, as the difference in times for these observations would present a notable difference in the spectra for the star in question.

To begin with, the task of researching each of our twelve targets was delegated between the team, such that each student researched one star individually. The purpose of this was to obtain a case for each star which would help us prioritise. Properties of each star were researched, including verifying the published values of period, and also reading into the speculations of whether or not each star was in fact a binary system. These particular ‘unverified’ targets are of as much interest as they would make for an interesting analysis, limited of course by our observation procedure and amount of data sets. Adequate research would ensure that a mixed variety of objects were viewed for analysis at a later date. Another important factor in our prioritisation of observations was to do with the transit times of each target. For the most favourable results we require the highest altitude for the object under observation, when it passes the meridian, to reduce contamination from light pollution as much as possible. The local times for these events were calculated for each object on each night, using the program KStar, which is a mapping program of the night sky that allows the user to determine the whereabouts of a listed object at a certain time and place (Local Time coupled with Latitude/Longitude) with respect to the horizon, and producing a time for meridian-crossings (circumpolar objects are also clarified). These pieces of information are important for organising the observations within each shift, which all run from 19:00 – 05:00 Local Time.

Once these data are determined, then the observations can be planned in a timetable. The observing times for each target are based around the points that they reach their highest altitude, we an exposure would be taken over a time duration that spanned either side of the highest altitude. Clashes between observing two objects under this regime had to be dealt with also, and therefore some observations would have had to be slightly compromised. It was finally decided that we would observe all of the longer period objects on one night, as these data would only be required to be obtained at one instance for our trip, as any comparisons made would only be relative to data sets obtained on previous years.

Daniel Underwood PHAS3332 MSci Astronomy

The rest of the objects of shorter periods (of which obtaining data over the week should produce enough spread to analyse) would be observed on every other night, each at two specific times each night in order to maximise our data set. After the thorough research of our proposed targets decisions were made as to whether or not certain targets could be disregarded if too many clashes and compromises had to be made. This proved to be necessary in some cases, based on the importance of the objects and how highly prioritised they would be, details of which are summed up in the orientation report for this field trip [1].

The observations of our objects were made using the 1.52-m telescope at the OHP, a reflecting telescope equipped with a high resolution spectrograph, ‘Aurélie’. The pass band for which we made our observations was 4070-4130A, which centres upon the H-delta line, a Balmer transition associated with Hydrogen. This is a transition commonly observed within B-type stars, which is what both hd112014 and Lambda Tauri are classed as. The transition occurs at a lab value of 4101.74 Angstroms [7], so this pass band is a perfect range of wavelengths to observe it through. The telescope has a focal length of f/27.6, and the light is focused through a circular aperture. The incident light is split into five separate beams by a Bowen-Walraven splitter, and these beams are then structured into a thin rectangular shape that replicate the effect of a telescope slit. A slit isn’t completely viable in terms of economy for the 1.52-m telescope, so it is much more effective to have this mechanism installed instead. Once the light has been organised into this rectangular nature it is set incident upon the grating. The information is finally projected onto the CCD chip, which has a 2048x1024 array [5].

Our observation groups would make transitions between their tasks at midnight, so one team would arrive at the telescope to take over the job of the previous team. This meant that it was essential that we all had copies of the same timetable for a particular night of observation so that everybody knew where they were taking over at a change of shift. At this point we exchanged our procedures and any problems we may have encountered with each other.

The set up for the 1.52-m telescope is such that the readings from the CCD chip are fed onto the computer systems and saved as a Flexible Image Transport System file, or FITS file. This is the file format that we later on work on in our data reduction.

With a list of our targets available, including information for right ascension and declination of each object, we can manoeuvre the telescope to the position we desire. Like the 1.20-m telescope, this requires a rather ‘old-fashioned’ method, as the telescopes are not equipped with the latest up-to-data computer technology. Instead the act of moving the telescopes involves the clamping and un-clamping of motors within the telescope mechanical system. In the telescope dome there is a console that houses the functional equipment in order to do this. There are two axes upon which the telescope can be mechanically moved, corresponding to a change in right ascension and declination, respectively. However each axis has to be slewed separately, so there is a wheel on the console for both right ascension and declination. The direction of movement for each is dependant on the orientation of a switch, i.e. if the switch is turned in a particular way the telescope will slew to increasing RA/Dec, and vice versa.

Daniel Underwood PHAS3332 MSci Astronomy

Once the direction is decided upon, and the switch turned accordingly, the motors for slewing are then operated by the turn of a wheel, which upon turning further will increase the speed of the slew. Next to the wheel on the console is a display of a dial which represents the movement of the RA/Dec through the slew, and this is how we determine where to stop the slew once the display is centred on our desired value of right ascension or declination. This must be done independently for the right ascension and declination. Once the desired values for both are reached it is possible to ascertain whether or not we have reached our target by viewing it through the finding scope attached onto the side of the telescope. This is of course provided that the dome shutter is orientation such that the telescope is pointing towards the night sky. If the object is in the field of view then we can make adjustments for fine-tuning in the control room. The control room contains a small monitor with a console which feeds a video display of the current field of view of the object. Here an object can be focussed onto the monitor using the attached console. The console also allows us to fine-tune the right ascension and declination such that our desired object is directly in the centre of the field of view. We can adjust this if necessary, in case the telescope tracking system fails to maintain a steady track for whatever reason.

Once we have set our object into the centre of the field of view it is possible to start taking readings of its spectrum. For our short period objects we had planned to take two exposures each night, and the majority of this exposure time is planned such that it falls within its transit time of highest altitude. The same was done on the first night for our long-period objects. This also allows us to minimise the effect of cosmic ray hits.

The CCD system is wired to a computer in the control room which is installed with software that allows the user to input the necessary parameters for an observation. Here we enter our exposure times for whichever object we have centred on, and this starts the light acquisition onto the CCD chip. The spectrograph splits up the light into its component wavelengths before hitting the CCD chip, so when the data is finally read into the computer system after the exposure is completed the resulting FITS file is that of a spectrum. This is saved into our data set, and we are careful to note down the file name of the exposure, along with the properties of its contents, i.e. the object name, its exposure time, the data and local time, etc.

This is the main procedure for all of the observations using this telescope. Once the spectra are obtained then they must be reduced and calibrated before they can be analysed.

However we did encounter a few problems when acquiring data on the telescope. There appears to be a slight offset in the telescope which is corrected in the fine-tuning process on the control room. This is most probably due to the fact that the system is rather dated compared to telescope arrangements that can be found today. On the CCD chip there is a ‘dead’ column of pixels around the 780 x-axis position of the array. The data fed into the computer software from the CCD chip for this particular column will therefore be meaningless as far as reading the spectra is concerned, and this particular error will need to be accounted for when reducing the data, as the meaningless readings for this column may compromise our analyses.

Daniel Underwood PHAS3332 MSci Astronomy

Finally, on one of our observing sessions we found that the telescope had its limitations due to the very way it is mounted. Whilst attempting to slew to HD112014 our group found that for the given local time the telescope was unable to reach its right ascension; it was impossible to view this object more than about an hour east of the meridian because of the objects high declination, and therefore would have to be observed at a later time to prevent the rear end of the telescope from colliding with the pier. This wasn’t too much of a problem as it was still possible to observe the object at a later time. It only conflicted with out planning as we wanted to our observations taken over a time that spanned either side of an objects meridian transit.

Overall the acquisition of data did not present too many problems and can, in this respect, be considered a success. However due to unforeseeable weather conditions it turned out that for our trip that was meant to span six nights of observations, only two of them had beneficial observing conditions. So although our acquisitions were fairly well obtained, our data sets were limited to two nights of observation. This will clearly affect our analyses as an insufficient amount of data will mean that there can’t be any solid conclusion guaranteed for our results, which certainly won’t be plentiful.

It is hoped that comparisons with data previously obtained in 2006 using the same configurations and same target objects will help us yield some more valuable results. For this, we must also reduce the previously obtained data as well as our own.

Daniel Underwood PHAS3332 MSci Astronomy

Data Reduction

Before an analysis can be made for a given spectrum certain corrections must be made to it. The most usual corrections made to images obtained with a CCD are to eliminate unwanted effects that arise in an exposure, namely the bias offset and uneven illumination of the CCD chip.

The bias offset occurs due to the intrinsic electronic properties of the CCD apparatus, and this is manifested in each and every image that is obtained using the particular set up. It can be thought of as the CCD ‘adding; a bias count to each and every pixel, though this can vary across the image. However the bias for a particular imaging apparatus is almost constant from image to image, so its effects can be subtracted. To subtract the effect from the images it is necessary to obtain bias frames that are then subtracted from the target images themselves. A bias frame is obtained by making a very quick (‘zero length’) exposure with a closed shutter, which then gives a frame that consists of only the bias values for each pixel. Combining a number of bias frames can minimise any readout noise, and this master frame can then be subtracted from every other acquired frame (including flat fields, discussed below) as there is little variation in bias across a range of images .

‘Flat field’ frames are obtained as there is a need to correct the effects of irregular illumination onto a CCD chip, and the varying sensitivity across it. Flat field frames are obtained by taking exposures of an evenly illuminated surface for a given time that causes counts to reach a certain percentage of the saturation level. These frames then give a measure of the variation in light being read onto the CCD. Like bias offsets, frames of the same exposure length can be combined to minimise noise. A normalised master flat field frame that has been bias corrected is then divided through each of the target frames (which are also bias subtracted) in order to correct the irregularity of sensitivity.

Figaro Each target frame has to be corrected for the bias and irregular illumination, so for each observing session bias frames and flat field frames were taken, at the beginning and at the end of each session, which allows for us to create our average master frames. Each target frame obtained on a particular session is corrected by the bias/flat field frames that are also obtained in that session.

We use a UNIX based program known as Figaro to manipulate our frames. These frames are initially saved as FITS files, but we can convert them to Starlink Data Format (.sdf) using Figaro in order to be ale to manipulate and analyse them later on. This is initially done for all of our frames (target, bias, flat field) before we begin to manipulate them. We first obtain a master bias frame for a given data set by summing the individual bias frames and taking the mean, to produce an average bias. The summation of frames as well as the division of an integer (the number of frames) can be done using Figaro also using specified commands.

Daniel Underwood PHAS3332 MSci Astronomy

Once this mean bias frame is obtained, then for a given data set (i.e. a data set obtained on a specific session) the bias is subtracted from the target and flat field frames that have been obtained during that particular observing session. This subtraction is also done using a Figaro command. We rename our bias subtracted files with an indication that they have been bias subtracted, so that we do not get confused when we manipulate them later on.

The flat-field frames taken from each observing session must now be combined into a normalized master frame in order to correct our target frames for the illumination defects. The average is obtained in the same way as the bias offset average is obtained, by summing the individual frames and obtaining a mean. This master flat-field frame is then subsequently divided through all of our target frames using the division command in Figaro. We can rename these corrected frames to indicate that they have indeed been fully corrected using the bias and flat field correction techniques.

The described procedure was done for both the data sets we obtained in 2010 and also for the raw data sets obtained back in 2006 in an identical fashion.

At this point the spectra must be wavelength-calibrated, which involves converting the x- range values of the spectra from pixel number into wavelength values. This is done by obtaining arc frames; an exposure of an arc lamp is made in order to obtain its spectrum, of which the spectral emission features for the particular arc lamp radiation are known. The particular arc lamp used for our calibration process was a Thorium-Argon lamp, and spectra of these were taken for each observing session. The Thorium-Argon lamp is a useful calibration lamp for this experiment as it emits wavelengths within the range of the pass band we are observing through, so calibration of our pixel values can easily be made using the known emission lines from this lamp. The spectra for the Th-Ar frame also need to be bias subtracted like the target frames and flat field frames, but do not require a flat field correction. Once we have a corrected Th-Ar frame its emission lines can be identified by comparisons of charts provided to us, which contain spectral features along with their wavelength values. We then use Figaro to produce a wavelength scale for the x-pixel array. This is done by imaging the Th-Ar emission spectra and labelling each feature using the Figaro command arc. This allows the user to enter the wavelength values for each spectral feature, through comparison with the charts provided to us. Upon doing this across the range of features, a wavelength scale is then produced for the x-value range through which we have our measurements.

Before we can apply this wavelength scale to our various spectra, it is necessary to further manipulate the target frames. Because the exposures were taken using a 2-dimensional CCD ‘array’, the spectrum of light arriving at the CCD chip is observed across the x-axis of the chip, whereas discrete wavelengths at each point in the spectrum are observed across the y-axis. It is therefore intuitive to sum the counts for each column (y-axis) into a single vector to obtain one single spectrum for each frame, which will then produce an array of x-values against relative intensity (pixel count). The extract function in Figaro allows users to do this for each target frame. Once this is done for all of the target frames, the wavelength scale obtained from the calibrated arc frame can be applied to our spectra using the xcopy command. This converts the pixel

Daniel Underwood PHAS3332 MSci Astronomy number axis into a wavelength axis, and this will the help us determine the wavelengths of the spectral features found on our spectra. This in turn will allow us to make the appropriate measurements required to obtain values of radial velocity. Our use of Figaro is now complete; however a few further adjustments need to be made.

Dipso Before we can properly analyse our spectra it is necessary to eliminate bias data, and “bad row” data caused by cosmic rays or dead pixels. Reading the frames into the Dipso program allows the spectra to be represented in graphical form, with wavelength in the x-axis and relative intensity in the y-axis. The ‘dead’ pixel column mentioned previously should present an obvious abnormality in the spectra as the form of an intensity spike, as will the occurrence of a cosmic ray hit. These elements can be ‘snipped’ from the data set by using a Dipso command to eliminate the data within a given x-range, i.e. the x-range in which the abnormalities are found. The ‘missing’ data ranges resulting from the snip can be filled from a further use of a Dipso command.

The spectra are a plot of wavelength against relative intensity (depending on the pixel counts for each column). Thus for each different spectrum the y-range would be different, depending on factors such as exposure time, observed magnitude, and in some cases if there had been elements such as cloud cover that affected the pixel counts. It is therefore necessary to correct for this by normalising the spectra so that each intensity range for the different spectra is relative to the same y-value reading. This is done by fitting a continuum function to the spectrum, which involves drawing a line across the continuum (which must be determined by eye). The spectrum vector is then divided by this function, the result being that the spectrum is normalised such that the continuum is flattened to a value of 1. Doing this for all the spectra ensures that they are calibrated to the same intensity range.

We now come to assigning the spectra to the time of their exposure. This is used to plot radial velocities as a function of time to create graphs that represent the periodic motion of the binary system. Because date and time are a cyclical measuring system, it is difficult to manipulate their values such that they can be placed in a graph axis; they need to be ordered monotonically. There is a system of measurement that does this – Julian dates. To plot the date-time against the radial velocity of the binary, one needs to convert the date and time of the exposures into Julian dates. The Julian date is a widely used method for representing dates in Astronomy. It presents years, months, days, hours and seconds in a decimalised number, the upshot of this being that subtraction of any Julian date number from another gives a value equal to the time interval in between, which represents a specific date and time difference.

Daniel Underwood PHAS3332 MSci Astronomy

Julian dates are calculated for the corresponding dates and times of which each exposure was taken. For our experiment we took the central point of each exposure to be the time that the exposure was taken, and this exact time was converted to a Julian date for each spectrum.

Daniel Underwood PHAS3332 MSci Astronomy

Data Analysis

The absorption lines from HD112014 and Alpha Tauri are easily resolved such that the method of cross-correlation isn’t necessarily required in order to analyse red shifts. The line positions can be relatively simple to measure by a simple inspection. Since Dipso allows the spectra to be graphically represented, for a given object it should be possible to observe the shifting of a spectral feature by comparing exposures that were taken on different dates for same object. By direct measurement of this spectral feature the radial velocity can be measured. For the purposes of this project the observations of all of our targets were centred upon the H-delta line, and we can therefore use this line a reference for any measurements we make. Comparing our measurements of the wavelength for this prominent feature with the laboratory rest-wavelength will give us an understanding of the magnitude of the shift; we expect radial velocities to be higher the more the line has been shifted, in either direction.

Figure 3: The spectrum of Lambda Tau centred on the H-delta absorption line

Daniel Underwood PHAS3332 MSci Astronomy

Figure 4: Spectrum of HD112104 centred on the H-delta absorption line

The radial velocity can be determined by the use of Equation (1):

휆−휆0 푣 = (1) 휆0 푐

The difference in wavelength in the numerator is the difference between the rest wavelength and the measured wavelength on our spectra.

The wavelengths that we measure on our spectra can be obtained using the Dipso program. Once all of the data sets for an object are loaded they can be graphically represented by the program’s plotting device. The spectra are plotted along the wavelength range through which they are observed (x-axis) against their relative intensities (y-axis), and the user is able to select points on these axes using a cursor. In addition, it is possible to constrain the range covered by the axes, which essentially allows the user to “zoom in” on the graph. Our span of data for a specific star ranges over a number of nights, and for each night we may have a number of exposures. Hence due to the motions of the stars over time we observe shifts over our data set. Making the measurements in Dipso for all of the spectra obtained for a particular star, we can gather a range of values for the radial velocity over the period time of our observations. Attributed to these values are the dates and times of the observations of the spectra they were extrapolated from.

Daniel Underwood PHAS3332 MSci Astronomy

This allows us to create a graphical representation of the periodic motions of the binary system by plotting the radial velocity as a function of time; the radial velocity starts from a zero- point, increases to a maximum displacement then decreases to a minimum, and finally returns to the starting position – this is the motion of the system for one cycle.

To create graphs of these types we can use a spreadsheet program to correlate radial velocities with their corresponding times of exposure, in Julian Dates. Because of our bad luck regarding the amount of data points we were able to obtain, plotting the radial velocity as a function of Julian date will be largely pointless as any plotting program we use will not be able to interpolate a sine curve structure by a best-fitting mechanism. This is because our data points were too few, and since spread over a four year difference, too sparse. Instead, for this project I attempted to fit a model sine curve to my data points by using a method of least squares fitting. In doing so, it should be possible to determine an amplitude of variation for the period of the binary rotation, i.e. the maximum radial velocity for the primary star (and the secondary for a double-line binary).

In order to do this, the data points from both years of the curve are folded into a single phase cycle (arbitrarily from 0 to 1). This converts the Julian dates of each measurement into a phase number, which lies between 0 and 1. This is done using the following equation:

푡−푇 ∅ = (푡 − 푇) 푃 − 푓푙{ } (2) 푃

In this equation, t is the epoch of measurement, in Julian dates, and T is a reference epoch, usually given as a minimum or maximum. P is the published value of the period of the cycle for a given system. The “floor” function of the fraction is subtracted from the fraction; the floor function gives the closest lower integer of the fraction. The values of T and P must be consistent, and for this project have been obtained from published values given by an online binary system catalogue [2]. Unfortunately as our lack of data points is such an issue it isn’t possible for us to calculate the period just from our data using the methods available to us. The reference epochs T are given as primary minima [2].

For a given star, once the phases corresponding to the Julian dates of each measurement have been calculated, a sine wave can be constructed using their values:

퐴 sin 2휋∅ + 퐵 + 퐶 (3)

The process of fitting this curve involves minimising the sum of the squares of the offsets of our data points from this curve. The constant A represents the amplitude of the sinusoidal variation, B represents a shift along the x-axis (phase axis) and the C constant represents a shift in the y-axis. If A is set to 1, and B and C are set to zero, then acting this function on the phases corresponding to our Julian dates will produce a curve that varies between -1 and 1. Our data points for radial velocity obviously vary between the tens of thousands of metres per second, and there therefore there are large offsets between our data points and the curve of this function.

Daniel Underwood PHAS3332 MSci Astronomy

Given a value of phase, upon acting this function on it, a value between -1 and 1 is produced. Subtracting this value from the corresponding radial velocity of that phase (Julian date), and offset is given for that data point. Doing this for all the data points along the phase gives the total range of offsets. The method of least squares fitting requires that the sum of the squares of all of these offset values is minimised. This can be done using the Microsoft Excel spreadsheet program. By squaring the offsets and then summing them, the “Solver” function in Excel can be used to minimise the sum. The function allows the user to minimise a given value by constraining other values that the value to be minimised is dependant upon. In this case, the constrained values are A, B and C of the sine function. The program changes these such that the sum of the squared offsets is minimal, and therefore produces an improved sine curve that is a ‘best fit’ of our data points.

The most interesting value to arise form this is the value of A. Given that our data points represent points of the actual published period of our system, then the Solver function should vary A to the point were it matches the amplitude of the cycle, and therefore the maximum radial velocity of a star in the system (depending on which lines are measured).

HD112014 is a double-lined spectroscopic binary, and therefore absorption lines from both stars are observed on the spectrum. It is therefore possible to estimate the maximum radial velocities of both the primary and secondary object. The other star studied in this project, Lambda Tauri is a single-lined spectroscopic binary, and therefore it is only possible to measure the maximum radial velocity of the more luminous primary object using this method. As an addition, it is possible to construct a graphical representation of the entire period of each star by performing the sine function on a range of incremental phase values spanning from 0 to 1, using the obtained values of, A, B and C (if C is necessary). Doing this for phases of 0, 0.01, 0.02, 0.03 etc will construct a fairly smooth variation, which can be plotted over our measured values to observe how well they correlate.

However this method does have its drawbacks, as discussed in the following section.

If we have a data set of radial velocity measurements for both stars in a binary system, we can calculate the mass ratio of the stars by plotting the radial velocities of one star against the corresponding radial velocities of the other, taken from the same observation point. We have this option available for HD112014. The line of correlation for the plotted data points will be linear, and it will have an associated gradient; this gradient is equal to the mass ratio of the stars. This can be simply worked out by using Microsoft Excel.

Daniel Underwood PHAS3332 MSci Astronomy

Results & Error Analysis

As previously stated, our sheer lack of data points has limited our investigation quite significantly, and had we had better luck with weather conditions then we may have had a larger range of data to analyse. It is because of this that obtaining a value of cycle period for each object proved an impossible task given our resources and limited use of telescope time.

However, using the methods described above, it was possible to achieve a rough estimate of a specific value that could be compared; the maximum radial velocity of the rotation. However due to the nature of the calculation the propagation of errors is not possible, as the computation if least squares fitting is performed by a computer program in Excel, and not by manual calculation.

So ultimately, the amplitude of the radial velocity will only be an order of magnitude estimation, and will not be a value that can be comparable to published values, as their lack of uncertainty makes them unreliable.

Because of the uncertainty associated with human error in reading the values off the spectra using the Dipso plotting device, I decided to take a range of measurements of the same feature I was monitoring, and then obtain an uncertainty value with the standard deviation of the average value. This included calculating the red shifts and radial velocities of the range of values and computing a standard deviation for each of the obtained values, which should give a measure of the margin of error attributed to human measurement. These computations can be done using Microsoft Excel, given that the range of measurements of the specific feature is input.

The Julian date of each measurement was taken to be the mid-point of each exposure, i.e. the time after which half of the exposure time had elapsed. Therefore our reading of wavelength using Dipso is subject to an uncertainty that falls within the exposure duration of the corresponding measurement. As it is not certain precisely at which point of the exposure our reading corresponds to, the value of the Julian date of our readings is subject to the following uncertainty:

퐸푥푝표푠푢푟푒 푑푢푟푎푡푖표푛 표푓 푚푒푎푠푢푟푒푚푒푛푡 ∆퐽퐷 = ± (4) 2

This means that our reading must fall within this margin of error, which spans either side of the exposure duration from the midpoint, and these must be converted into a Julian date format. Since it was not possible to calculate a period value for the binary systems studied, uncertainties on Julian dates are not necessary as there are no calculations to carry out and therefore no need to propagate the Julian date error margins.

For HD112014 there is a possibility for the mass-ratio of the system to be calculated, i.e. the ratio between the masses of the primary and the secondary. This is done by simply plotting the radial velocities of the primary star against the radial velocities of the secondary, at increments of different measurements that correspond to the same epoch. This should produce a linear

Daniel Underwood PHAS3332 MSci Astronomy correlation, and the gradient of a line representing this straight-line correlation is the ratio of the masses of the two objects.

The red shifts calculated for each line on each exposure were measured using the left-hand side of equation (1), compared to a rest lab wavelength of the H-delta line, which has a value of 4101.74 Angstroms [7].

HD112014

Measurements of the radial velocity were made for each of the spectra using Dipso. This binary is a double-line system, and therefore it is necessary to identify two absorption lines that each correspond to the primary and secondary object. Over our data sets, we observe the H-delta line being separated due to line being partially blended from each star in the system. Due to the intrinsic motions of the stars one of the lines should be red shifted and the other blue shifted, with respect to the laboratory rest wavelength of the transition. Measurements were taken of both of the H-delta lines in this splitting, and there was a clear deviation in their read outs compared with the rest wavelength.

Primary

Julian Date Wavelength Uncertainty on Redshift Uncertainty on Measurement (A) Wavelength Redshift Measurement 2453767.65890 4102.197 2.502E-02 1.113E-04 6.099E-06 2453767.70323 4102.13 1.054E-01 9.508E-05 2.569E-05 2453768.60033 4100.44 1.200E-01 -3.169E-04 2.926E-05 2453768.62236 4100.38 1.200E-01 -3.316E-04 2.926E-05 2453768.71599 4100.44 1.200E-01 -3.169E-04 2.926E-05 2453768.72351 4100.48 1.804E-01 -3.072E-04 4.397E-05 2455240.60556 4100.847 8.527E-02 -2.178E-04 2.079E-05 2455240.66528 4100.683 1.902E-01 -2.576E-04 4.636E-05 2455242.57986 4103.17 2.166E-01 3.486E-04 5.280E-05 2455242.63819 4103.20 1.804E-01 3.559E-04 4.397E-05

Julian Date Radial Velocity Uncertainty on (m/s) Radial Velocity 2453767.65890 3.340E+04 1.830E+03 2453767.70323 2.852E+04 7.706E+03 2453768.60033 -9.508E+04 8.777E+03 2453768.62236 -9.947E+04 8.777E+03 2453768.71599 -9.508E+04 8.777E+03 2453768.72351 -9.216E+04 1.319E+04 2455240.60556 -6.534E+04 6.236E+03 2455240.66528 -7.728E+04 1.391E+04 2455242.57986 1.046E+05 1.584E+04 2455242.63819 1.068E+05 1.319E+04

Daniel Underwood PHAS3332 MSci Astronomy

Secondary

Julian Date Average Uncertainty on Redshift Uncertainty on Wavelength Wavelength Redshift Measurement (A) Measurement 2453767.65890 4101.22 4.714E-03 -1.276E-04 1.149E-06 2453767.70323 4101.47 4.950E-02 -6.583E-05 1.207E-05 2453768.6003 4103.33 1.697E-01 3.876E-04 4.137E-05 2453768.62236 4103.43 4.950E-02 4.120E-04 1.207E-05 2453768.71599 4103.34 8.485E-02 3.901E-04 2.069E-05 2453768.72351 4103.54 1.461E-01 4.380E-04 3.563E-05 2455240.60556 4103.08 2.303E-01 3.267E-04 5.614E-05 2455240.66528 4103.3 5.508E-02 3.803E-04 1.343E-05 2455242.57986 4100.27 2.033E-01 -3.592E-04 4.957E-05 2455242.63819 4100.25 3.650E-01 -3.624E-04 8.899E-05

Julian Date Radial Velocity Uncertainty on (m/s) Radial Velocity 2453767.65890 -3.828E+04 3.448E+02 2453767.70323 -1.975E+04 3.620E+03 2453768.60033 1.163E+05 1.241E+04 2453768.62236 1.236E+05 3.620E+03 2453768.71599 1.170E+05 6.206E+03 2453768.72351 1.314E+05 1.069E+04 2455240.60556 9.801E+04 1.684E+04 2455240.66528 1.141E+05 4.028E+03 2455242.57986 -1.078E+05 1.487E+04 2455242.63819 -1.087E+05 2.670E+04

Daniel Underwood PHAS3332 MSci Astronomy

HR112014: Average Velocity as a function of Phase 150000

100000

50000

0 Primary 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -50000 Secondary

Radial Velocity Velocity Radial (m/s) -100000

-150000 Phase

Figure 5: The data points from both 2006 and 2010 are folded into a single phase using equation (2). The phases corresponding to each Julian date are then acted on by equation (3), and the constrained values of A and B are used to create the best-fitting line. The points represent the measured data points.

The graph above is produced by the method of least squares fitting discussed in the Analysis section. The data points for the primary and secondary stars are shown, with the respected fitting lines passing through them. The curves are reflected due to the nature of the cycle; as the primary object moves away (positive radial velocity) the secondary advances (negative radial velocity) with respect to our line of sight, and vice versa.

The minimisation technique constrains the following values of A from equation (3), for both the primary and secondary objects:

Published period: 3.2866 [2] Reference epoch: 2424226.66900 [2]

Values of Amplitude Published Values (km/s) (km/s) Primary 105.97 108.3 [2] Secondary 117.30 128.9 [2]

The ratio of the masses of the two objects in HD112014 can also be calculated using Excel, by plotting the radial velocities of one star against the radial velocities of the other star; the mass ratio is equal to the gradient of the line fitting the correlation between the data sets. Using Excel, a best-fitting line can be attributed to the correlation produced by this plot, along with the uncertainty of the line gradient:

Daniel Underwood PHAS3332 MSci Astronomy

150000

100000

50000

0 V1 vs. V2 -150000 -100000 -50000 0 50000 100000 150000 -50000

-100000

y = -1.1726x + 13146 -150000

Figure 6: The plot of the radial velocity of the primary against the radial velocity of the secondary, at different corresponding intervals (measurement epochs).

Excel calculated the following gradient and uncertainty for the line of best-fit:

Line Gradient -1.1726078 ±0.03431368

This yields a result of 1.17 ± 0.03 for the mass ratio of the system.

The published values of the individual masses of the primary and secondary stars in the system are:

Primary Secondary Mass (Solar Masses) 2.47314 [2] 2.0779 [2] Mass Ratio 1.1902

The calculated value from our data shows a very close estimate, and the published value falls well within the error margin produced by the line of best fit.

Daniel Underwood PHAS3332 MSci Astronomy

Lambda Tauri

The acquisition and analysis of the spectra of this target shows that it is in fact a single-lined spectroscopic binary, at least in the limitations of the equipment used. Therefore analysis could only be made for the primary object in the system (the more luminous one), and the secondary object could not be observed. Due to this only radial velocity measurements and amplitudes can only be made of the primary object. Once again measurements were made centred on the H-delta line.

Julian Date Average Uncertainty on Redshift Uncertainty on Wavelength Wavelength Redshift Measurement (A) Measurement 2453767.32938 4102.007 1.744E-01 6.501E-05 4.252E-05 2453767.43278 4102.02 8.485E-02 6.826E-05 2.069E-05 2453767.44262 4101.84 8.485E-02 2.438E-05 2.069E-05 2453768.37913 4101.86 1.131E-01 2.926E-05 2.758E-05 2453768.42397 4101.89 1.131E-01 3.657E-05 2.758E-05 2453769.31313 4102.67 1.414E-01 2.267E-04 3.448E-05 2453769.31826 4102.73 5.657E-02 2.414E-04 1.379E-05 2453769.32518 4102.81 4.243E-02 2.609E-04 1.034E-05 2453769.33456 4102.79 5.657E-02 2.560E-04 1.379E-05 2453769.34496 4102.73 2.121E-02 2.414E-04 5.172E-06 2453769.35678 4102.85 8.485E-02 2.706E-04 2.069E-05 2455240.34653 4103.34 8.730E-01 3.901E-04 1.170E-04 2455240.35486 4103.13 4.806E-01 3.389E-04 1.172E-04 2455242.30972 4101.65 2.952E-01 -2.275E-05 7.197E-05 2455242.36111 4101.74 4.251E-01 -8.127E-07 1.036E-04

Julian Date Radial Velocity Uncertainty on (m/s) Radial Velocity 2453767.32938 1.950E+04 1.276E+04 2453767.43278 2.048E+04 6.206E+03 2453767.44262 7.314E+03 6.206E+03 2453768.37913 8.777E+03 8.275E+03 2453768.42397 1.097E+04 8.275E+03 2453769.31313 6.802E+04 1.034E+04 2453769.31826 7.241E+04 4.137E+03 2453769.32518 7.826E+04 3.103E+03 2453769.33456 7.680E+04 4.137E+03 2453769.34496 7.241E+04 1.552E+03 2453769.35678 8.119E+04 6.206E+03 2455240.34653 1.170E+05 3.511E+04 2455240.35486 1.017E+05 3.515E+04 2455242.30972 -6.826E+03 2.159E+04 2455242.36111 -2.431E+03 3.109E+04

Daniel Underwood PHAS3332 MSci Astronomy

Lambda Tauri: Average Radial Velocity as a function of Phase 140000 120000 100000 80000 60000 40000 Primary 20000

Radial Velocity Velocity Radial (m/s) 0 -20000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Phase

The minimisation technique constrains the following value of A from equation (3), for the primary object:

Published period: 3.9529 [2] Reference epoch: 2444658.4000 [2]

Values of Amplitude Published Values (km/s) (km/s) Primary 54.16 56.8 [2]

The sinusoidal curve representing the periodic motion of the Lambda Tauri system is not symmetric about the phase axis, as the velocity for our line of sight tends to greater amounts in the direction away from our viewpoint, and advancing velocities (denoted by negative sign) are much less. This suggests some intrinsic motion of the system as a whole away from our Earth-based observing position; because the system has its own intrinsic red shift, the red shifts of the individual spectral lines are enhanced, whereas the blue shifts are constricted.

The mass ratio for the objects in the Lambda Tauri system cannot be determined due to the fact that only lines from the primary object are observed, and therefore the radial velocities for this object can be determined. Thus the same analysis as for HD112014 cannot be carried out.

Daniel Underwood PHAS3332 MSci Astronomy

Conclusion

It is clear from our analysis that a poor set of data spanning such short amounts of time is highly unreliable for determining a value for the period of the binary cycle. Our attempts to find various values to compare with published ones has been greatly hindered by the poor observing conditions that we were subjected to, and therefore it hasn’t been entirely possible to conclude whether the methods used are reliable. Our only derived result that can be used as a basis to determine the success of the experiment is the value of the mass-ratio of HD112014, as it has an uncertainty attributed to it. The result that the experiment yields suggests that the method is a successful analysis, as the result is incredibly close to the published value, which falls within the calculated uncertainty. However the uncertainty is not a product of the uncertainties calculated for the radial velocities through error propagation, but is instead a calculated result based on an Excel programming analysis based on the deviation of points from the line of best-fit.

The least squares fitting of the sinusoidal curves for the periodic variation yields the results for the maximum amplitude when fitted to our data. These however do not have an uncertainty attributed to them as they are computed from the Excel program and not through a manual calculation series, and therefore errors cannot be propagated. Therefore the values obtained can only be viewed as an order of magnitude estimation. Having said this, the results do appear to be close to published values within an accuracy of 10%. It is important to note however that the published values of period were used to carry out the analysis, and therefore the method may tend towards favouring the proximity of measured values to published values of maximum radial velocity.

For HD112014, the radial velocity curves for both objects in the system are plotted against each other, and the minimisation technique has plotted curves which are mirrored in the phase axis (around the zero point for radial velocity). This can be interpreted as the system indeed being of a binary nature, as one object recedes as the other advances, and shows that the measuring analysis has accurately identified the two objects by inspection of the H-delta line splitting, and this is a testament to the resolving power of the OHP 1.52-m telescope.

So although a measurement of period could not be ascertained solely from our data, it has been possible to value the reliability of the method of acquisition and analysis to a certain extent from the results discussed above. A fair judgement of the telescope’s capabilities can then be made, and given better observing conditions I am confident that a larger set of data for these target objects would have helped a period value be determined, and then the subsequent measurements of amplitude and mass-ratio (for HD112014) could be determined with a proper propagation of errors.

The planning carried out beforehand would have still yielded an effective result over the six nights we had available, though a larger span of observations would have improved our certainties as we could view the periodic oscillations of the system over more than a single cycle, instead of having to merge data sets from both years.

Daniel Underwood PHAS3332 MSci Astronomy

References

1) Underwood, D., OHP Observation Orientation Report, University College London (2010) 2) Pourbaix D., Tokovinin A.A., Batten A.H., Fekel F.C., Hartkopf W.I., Levato H., Morrell N.I., Torres G., Udry S.,SB9: The 9th Catalogue of Spectroscopic Binary Orbits, Astronomy & Astrophysics, Universite Libre de Bruxelles (2004), [URL] Available: http://sb9.astro.ulb.ac.be/ 3) SIMBAD Astronomical Database, CDS – Strasbourg, [URL] Available (last viewed April 2010): http://simbad.u-strasbg.fr/ 4) Demande de Temps de Telescope OHP, Septembre 2009 – Fevrier 2010: Program for observations at the Haute-Provence Observatory made available to the UCL students 5) Observatoire de Haute Provence Official Website, [URL] Available: http://www.obs- hp.fr/ 6) Eisberg and Resnick (1985). Quantum Physics. John Wiley and Sons. pp. 97 7) National Institute of Standards and Technology (NIST) Atomic Spectra Database, NIST (2009) [URL] Available (last viewed April 2010): http://www.nist.gov/physlab/data/asd.cfm