First results from the echelle spectrograph at the Trottier Observatory Howard Trottier February 2016 Overview
This is a report on the first results obtained at the Trottier Observatory using our new echelle spectrograph, which is designed primarily for high-resolution stellar spectroscopy. I’ve tried to write the report assuming relatively little physics and math background, intending for it to be used in some form in a second-year breadth course on observational astronomy that I’m developing for students from both the arts and the sciences. However the level of the report is somewhat uneven, with some parts using more sophisticated physics and math – though the math can usually be skipped entirely in favour of the interpretation of the results. Our spectrograph is made by a small company in France, Shelyak Instruments, and is called the “eShel” because of its “echelle” design.1 The basic design principles of echelles are reviewed later; for now, it’s enough to note that echelles are advantageous compared to other spectrometer designs because of their high resolution. Spectral resolution is characterized by the smallest possible wavelength interval δλ between two neighbouring spectral features that can be individually identified, as illustrated below for two emission lines.
Resolution is usually quoted in terms of a fractional measure R that is defined by
The eShel has a resolution R of about 10,000, meaning that it can distinguish between spectral features separated by about as little as 5,000 Angstroms/10,000 or 0.5Å, at a typical wavelength of optical light. The highest-resolution professional echelles have R above
1 The Shelyak webpage on the eShel is at http://www.shelyak.com/dossier.php?id_dossier=47. The company is owned by two of the world’s top “amateur” spectroscopists, François Cochard and Olivier Thizy. Page 1 of 57 100,000. A tradeoff for the high resolution of the eShel (as with all echelles) is its limited sensitivity, which makes it impractical to use for diffuse objects except for the brightest. I’m stunned by the variety and precision of measurements that can be obtained with relative ease using the eShel (that is, after learning how to use the equipment and software!). High- quality data for bright stars can be collected in minutes, and within about an hour for stars down to about sixth or seventh magnitude; this is very different from acquiring high-quality astronomical images, which require many nights of telescope time. And, unlike astronomical image processing, which demands many hours of effort, with lots of trial-and-error, to obtain the best possible final image, the processing of spectroscopic data requires requires relatively little time and intervention by the user, once the initial setup for a particular spectrometer/ telescope combination has been made. When the unit arrived in late June I knew next-to-nothing about how an echelle works, or how to use one, and I had only a vague sense of what could be done with it. I spent a couple of months learning how to install and calibrate the device, and how to use the data acquisition and analysis software packages. Then from late summer through early winter I spent a good deal of time taking data for a wide range of astrophysical objects, and figuring out how to squeeze as much quantitative information as I could from the results. In addition to measuring the spectra of a few star types, including Vega and the Sun (via the Moon), and closely comparing the results with data from other observatories, I obtained results for the following: the Ring Nebula’s emission spectrum; the orbital velocities2 of the stars in two spectroscopic binaries; the rotational speeds of several stars; the properties of several circumstellar disks; the spectra of stars with fast winds; and, perhaps most exciting of all, extraordinarily precise measurements of the (relative) recessional velocities of some carefully selected stars.3 I also came across some very useful on-line resources that I’ll cite as they come up; some of these, along with the results presented here, should help to suggest projects that others can undertake. And if you want to learn more about how our spectrometer and others work, and about the astrophysics of the systems you might want to study, I’ve included a bibliography of books that run from newbie (without math) to advanced undergrad/grad-level texts. Most of these books will soon be on the shelves of a bookcase at the observatory. Although the data taking and analysis are highly non-trivial procedures, they are made almost routine by some powerful (and free) software – though getting familiar with all of the features and interfaces takes awhile ;). I will eventually write a guide to the software setup and interfaces as used at our observatory,4 here I include information on the capabilities of these packages, emphasizing how the software is used in the analysis of the results. We recently took delivery of a second spectrograph, also made by Shelyak:5 it has lower resolution but higher sensitivity than the eShel, and will provide us with complementary capabilities, notably the ability to take spectra of faint extended objects and measure quantities like galactic redshifts and rotation curves! I hope to install and test the new unit in the spring.
2 Velocities can only be measured along the line of sight of the object.
3 As explained in Section R4, these results show that we should be able to detect a few hot-Jupiter exoplanets! 4 See also Olivier Thizy’s blog http://observatoire-belle-etoile.blogspot.fr/2016/01/eshel-data-reduction.html.
5 The second spectrograph is the Shelyak “LISA”: http://www.shelyak.com/rubrique.php?id_rubrique=12. Page 2 of 57 Organization of the report
The rest of the report is organized as follows. The next section presents more information on the eShel spectrograph, and contains: a short description of how it is set up for an observation run at our observatory; a brief outline of the basic design principles of echelle spectrometers (perhaps too brief to be all that clear or informative!); an overview of how the spectrometer is calibrated; and a brief description of the software that is for data capture and analysis, along with a list of some useful on-line resources. The report then presents a series of detailed sections that analyze and interpret data that I took for a wide variety of astrophysical objects. The results are divided into six sections: Vega as a case study (Section R1); the Ring Nebula and the Moon (R2); Spectroscopic binaries (R3); High-precision radial velocities (R4); Doppler rotational broadening (R5); and last but not least: Circumstellar disks and stellar winds (R6). Note that Section R1 on Vega introduces alot of background information that is used in the later sections. Most of the other sections include a fair amount of background information that is needed to fully analyze and interpret those results. Since I’m still learning about alot of this stuff, I hope that what I’ve written is reasonably clear and accurate, but there are bound to be mistakes, possibly including some whoppers. By all means, skip anything that is too detailed, confusing, or boring! And I welcome any and all feedback, especially any mistakes that you may find.
Page 3 of 57 Introduction to the eShel Spectrograph layout at the observatory
Here are some pictures that show the eShel layout when in use at the observatory.
Incident starlight
Fibre feed from spectrograph
Mirror Guide camera control
50 μm fibre feed to spectrograph Spectrograph
Acquisition unit Acquisition High camera voltage power Control unit & supply calibration sources
Spectrometer
Mirror hole Vega The spectrograph is remarkably compact, and sits on top of a small dolly that is stored in the bathroom (of all places). The dolly should be wheeled into the dome area at least an hour before beginning observations, in order to equilibrate the spectrometer to the ambient temperature. A light-weight acquisition/guiding unit is attached to the periscope. The acquisition unit houses a plane mirror with a small hole: the mirror deflects almost all of the incoming starlight to a
Page 4 of 57 guide camera mounted on the side of the unit, while a fibre optic cable mounted behind the hole in the mirror feeds light to the spectrometer. The spectrograph is positioned beside the telescope control panel, where the ends of the acquisition unit’s fibre optic and control cables come up through the floor, and are attached to the spectrometer and its control unit. The spectrometer’s diffraction grating disperses the light, and a CCD camera mounted on the unit images the resulting spectrum (more on the actual design of the eShel coming up). USB cables connect the equipment to the control room computer. The spectrograph has calibration sources, and the light that they produce when activated is fed to the acquisition unit at the telescope through a separate fibre optic cable. A flip mirror in the acquisition unit redirects the calibration light to the pickup fibre: this ensures that the calibration is done with light that follows the same optical path to the spectrometer as the starlight. More information on the calibration procedure is coming up. The fibre optic cable that transmits the incoming starlight is only 50 microns in diameter! The small size is necessary for the high resolution, but limits the sensitivity of the spectrometer. The above images of the Ring Nebula and Vega (the latter on a night of poor seeing!) were taken through the guide camera, and show the tiny region occupied by the hole in the mirror. This illustrates why this unit is suited to stellar spectroscopy, but not so much to extended sources (though brighter nebulae can be measured). These images also demonstrate the importance of auto-guiding, to keep the target starlight on the hole! Rudimentary principles of an echelle spectrometer6
Figure I-1
Most spectrometers, from cheap to state-of-the art, are made using diffraction gratings as the principle dispersive element. Diffraction gratings consist of reflective or transparent surfaces that are etched with many parallel grooves, as illustrated in Fig. I-1 in the case of a reflection grating. The intensity of the reflected or transmitted light is reinforced in specific directions, and vanishes in others, resulting in a so-called interference or fringe pattern, as seen when white light is reflected from DVDs and other compact disks, producing a spectrum of colours. The diagram in Fig. I-1 illustrates why (we’ll consider reflection gratings exclusively from here on). The reason is that the distances travelled to the reflecting facets by neighbouring incident wavefronts will generally differ by some amount dinc, and likewise the distances travelled after
6 An introduction to spectrographs for amateur astronomers can be found in two short books by Harrison, while a comprehensive treatment can be found in a treatise by Eversberg and Vollman (details in the bibliography). Page 5 of 57 reflection will differ by some amount dref. If the overall path difference dref-dinc (which varies with the angles of incidence and reflection) is a whole number of wavelengths λ, then all the reflected waves add constructively to produce a net reflected wave of maximum intensity, at that wavelength, as illustrated in Fig. I-2. The so-called grating equation expresses the condition for constructive interference, as follows (where a is the spacing between facets): d d = a (sin ✓ sin ✓ ) = n (n =0, 1, 2,...) ref inc ref inc ± ± If the path difference equals a “half-integer” number of wavelengths (n+1/2)λ, then the reflected waves cancel each other out completely (destructive interference), also illustrated in Fig. I-2.
Wave 1
Wave 2
Resultant wave Figure I-2 (Wikipedia) Constructive Interference Destructive Interference Maxima and minima in the intensities of different wavelengths occur at different reflection angles θref (for a given angle of incidence θinc), producing the characteristic spread of colours seen with compact disks. Successive interference bands are referred to by the order Figure I-3 in the sequence, i.e. by the number n in the grating equation, as illustrated in Fig. I-3. A key property of diffraction gratings is that the spread, or dispersion, in wavelength increases with the order, which is depicted in Fig. I-3. This means that to maximize the resolving power of the grating (that is, to split apart neighbouring spectral features to the greatest extent possible), one would like to use the reflected light at the highest possible order. There are however two critical drawbacks with ordinary diffraction gratings. One drawback is that most of the light energy is concentrated in the zeroth-order fringe (n=0), where all wavelengths emerge at the same angle, while the intensity drops rapidly at higher orders. It is useful to note that at zeroth order the angles of incidence and reflection are equal, a situation known as specular reflection; this is the same condition that applies when light is reflected from a single (large) plane mirror. The second drawback with ordinary diffraction gratings is also illustrated in Fig. I-3: light of a given wavelength in one order is overlapped by shorter wavelength light from higher orders.7 In
7 For example, the n=1 maximum for a given wavelength λ overlaps with the n=2 maximum for wavelength λ/2. Page 6 of 57 practice, there will be a range of wavelengths at a given order that does not suffer from an overlap, which is referred to as the free spectral range; this occurs because any detector that records the spectrum will be sensitive only to a limited range of wavelengths. However the free spectral range decreases with increasing order, as shown in Fig. I-3, which illustrates the situation for a sensor like a typical CCD chip that detects mainly optical light. Echelle spectrometers use a two-part strategy to alleviate these drawbacks. One part of the strategy is to use a so-called “blazed” reflection grating, where the reflecting facets are inclined at an angle with respect to the mounting surface, as shown in Fig. I-4. The bias in favour of specular reflection from the inclined facets, combined with Figure I-4 interference due to the underlying grating pattern, concentrates the reflected energy at higher orders. It turns out that at each order there is one particular wavelength that has the most intense maximum, as suggested in Figure I-4 by the red ray emerging at a high order (in the case of the eShel, for example, the red Hα line is near the middle of order 34). To deal with the overlap of wavelengths from different orders, another dispersive element is introduced, known as a cross- disperser. The cross-disperser, which has much lower resolution than the echelle, is used to spread apart the light that Echelle emerges from the echelle, in a direction Cross disperser perpendicular to the initial dispersion, as illustrated in Fig. I-5.8 The cross-disperser can be a reflection grating, as in Fig. I-5, or a prism, as is the case with the eShel. As a result of the cross-disperser, the spectrum produce by an echelle consists of rows of dispersed light, as depicted in Fig. I-5, and in an actual image in Fig. I-6, taken with an eShel, which I lifted from the Shelyak website. Each row in the output corresponds to the spectrum of one order produced by the blazed grating. This Sensor characteristic pattern gives rise to the name “echelle” (French for ladder). Figure I-5
8 Figure I-5 shows just one incident ray of white light, and two overlapped rays (one red, one blue) reflected from the echelle. Other elements of the spectrometer, including a collimator for the incident light, and a lens to focus the dispersed light on the sensor, are not shown (Fig. I-7 on the next page has an accurate drawing of the eShel). Page 7 of 57 8.4 Echelle Systems 353
8.4.3 eShel: A Stable Off-the-Shelf Fiber Echelle
After the great success of Lhires III Littrow in the amateur and professional com- munity, the team around SHELYAK INSTRUMENTS developed and commercialized aready-to-goechellespectrograph,theESHEL.Eshelisacompactstandaloneunit with a prism cross-disperser and fed by fiber optics (Figs. 8.43 and 8.44). Figure 8.45 © Shelyak Instruments Figure I-6 As can be seen in Fig. I-6, the intensity of light is not uniformly distributed across the rungs of the echelle output, because at each order only one specific wavelength of light has the most intense maximum. The bands are also curved; in the case of a prism cross-disperser, the curvature arises in part because the index of refraction in glass varies with wavelength. The image also shows that the wavelengths near the ends of adjacent bands are the same; that is, the colour at one end of a band is the same as the colour on the opposite ends of the bands above and below it. This enables the bands to be “stitched” together to form a complete
8.4 Echelle Systemsspectrum. 353 The drawings in Fig. I-7 illustrate the optical path through the eShel, after the light from the 8.4.3 eShel:fibre A Stable optic Off-t pickuphe-Shelf cable Fiber Echelle enters the unit on the right: a collimator focuses the light on the After the greatdiffraction success of Lhires grating, III Littrow which in the amateur is reflected and professional into com-the prism cross-disperser, and then emerges through munity, the teama aroundlens SthatHELYAK focusesINSTRUMENTS the developeddispersed and commercialized light onto the imaging detector. The spectrum produced by Fig. 8.43 3D schematics of aready-to-goechellespectrograph,theESHEL.Eshelisacompactstandaloneunit the eShel covers more than twentythe eShel spectrograph orders, from wavelengths up to about 7200Å at order 32, with a prism cross-disperser and fed by fiber optics (Figs. 8.43 and 8.44). Figure 8.45 down to wavelengths of about( Courtesy4000Å:Shelyak at order 55. Instruments)
Fig. 8.44 3D side schematics Fig. 8.43 3D schematics of of the eShel spectrograph the eShel spectrograph (Courtesy:Shelyak (Courtesy:Shelyak Instruments) Instruments) © Shelyak Instruments Figure I-7 [email protected]
Page 8 of 57
Fig. 8.44 3D side schematics of the eShel spectrograph (Courtesy:Shelyak Instruments)
[email protected] Overview of the calibration of the eShel Figure I-8 shows a greyscale image taken with our eShel for a continuous source of light. Calibration of the spectrometer is needed for three reasons: to map the geometry of the echelle rungs; to map the pixel position at which light reaches the imaging CCD to the corresponding wavelength; and to correct for the non-uniform response of the spectrometer/ camera to light of different wavelengths. As noted in the previous subsection, the intensity varies across each rung because only one wavelength has its overall maximum intensity at a given order. In addition, the efficiency of the response varies between rungs, due in part to the intrinsic properties of an echelle, as well as the wavelength-dependent quantum efficiency of the CCD imaging chip.
Figure I-8 The shapes of the curved rungs are measured using a source of continuous light (this step also approximately determines the variation of intensity across each rung, know as the “blaze” functions); in the case of the eShel, a lamp with a tungsten filament that is pre-installed in the control unit is used, along with blue LEDs that are used to more fully illuminate the orders at the blue end of the spectrum. The eShel also has a thorium-argon source that is described below. The light produced by the calibration sources is fed via a fibre optic cable to the acquisition unit at the telescope, and then back through the pickup fibre to spectrometer (see the pictures on pg. 4). The image in Fig. I-8 is a one-second exposure with both the tungsten and LED sources turned on (it takes only a few seconds to saturate the camera at the brightest orders with these sources); ten or more images like this are typically averaged together to reduce the noise. These images are often called flat fields, but are very different in the purpose that they serve, and the way in which they are acquired, compared to the flats that are used in astronomical imaging. In the latter case, flats correct for vignetting in the optical path through the telescope to the camera; this is not an issue with the spectrometer, since the 50-micron diameter fibre- optic cable picks off only a tiny portion of the field of view. Spectrometer flats correct instead for the non-uniform wavelength response of the spectrometer and the camera’s CCD chip. The calibration of wavelength with pixel position can be accomplished in many ways, the most precise being the use of a source of light with many discrete and precisely-known wavelengths. The eShel uses a hollow cathode lamp (HCL), shown in Fig. I-9, which is filled
Page 9 of 57 with a mixture of thorium and argon gases. A large voltage applied across the cathode and anode ionizes the atoms in the gas, and the light that is produced when the atoms recombine with the ionized electrons is fed to the acquisition unit, and back to the spectrometer, producing an image like the one in Fig. I-10, which is the result of stacking ten five-second exposures.
Figure I-9 © Shelyak Instruments
Figure I-10 The final step in the calibration procedure is to account for variations in the efficiency of the spectrometer/camera as a function of pixel position. This step can be accomplished by taking the spectrum of a suitably-chosen star, and matching the result to a standard reference spectrum of the same star, or another star of the same spectral type. This will be illustrated in the first set of results below using Vega.
Page 10 of 57 Software and some on-line resources The software used to acquire data with the eShel is part of a package of open-source astronomical software written by French astronomers, called AudeLA,9 which comes with modules written for the eShel. The eShel User Guide10 gives an overview of the data acquisition procedure, and the Installation Manual11 includes details on the calibration methods. While the analysis of data acquired with the eShel can be done using AudeLA, there is a much more comprehensive analysis package called ISIS12 (Integrated Spectrographic Innovative Software – and yes, I know) which is widely used by amateurs, and that works extremely well with eShel data. While not open source, ISIS is free, and I do the data analysis using it almost exclusively. The author Christian Buil is one of the world’s leading “amateur” spectroscopists, and his website13 contains many detailed reports on observations done with a wide variety of spectrometers and telescopes, including the eShel (which he helped to design). In this report I go through how ISIS is used to perform the data analysis, but do not describe the user interface ‒ I’ll save the latter for another document, along with how AudeLA is used for data capture. Incidentally, the eShel website14 has many links to work done by amateurs who have used it to study an impressive range of astrophysical objects; the site also has suggestions for first observation projects.15 There is also a Yahoo group called Astronomical Amateur Spectroscopy,16 with almost 700 members, that is an excellent source of information, including member reports of results obtained with all kinds of spectrometers and telescopes.
9 http://audela.org/dokuwiki/doku.php?id=en:start. We are using the latest beta version of AudeLA, 3.0.0b1, as recommended by Shelyak, available at http://sourceforge.net/projects/audela/files/audela/current-development/.
10 http://thizy.free.fr/shelyak/eshel/DC0009B%20eShel%20User%20Guide.pdf
11 http://thizy.free.fr/shelyak/eshel/DC0010C%20eShel%20Installation%20&%20Maintenance%20Manual.pdf. 12 http://www.astrosurf.com/buil/isis/isis_en.htm
13 http://www.astrosurf.com/buil/index.htm
14 http://www.shelyak.com/rubrique.php?id_rubrique=7. 15 http://www.shelyak.com/dossier.php?id_dossier=13
16 https://groups.yahoo.com/neo/groups/spectro-l/conversations/messages Page 11 of 57 Results Part 1): A case study using Vega Calibration and comparison with other measurements An image of the spectrum of Vega obtained with our eShel is shown in Fig. R1-1.17 This is from less than three minutes of total exposure! Vega is so bright that a single frame can’t exceed about three seconds without saturating the camera response in some regions. This image was produced by stacking about 50 three-second exposures (AudeLA and ISIS both conveniently handle the stacking). The yellow text on the image is produced by the AudeLA package, which nicely identifies the central wavelength in each order. Some absorption lines that standout in Fig. R1-1 are identified with coloured boxes: red identifies the Hydrogen-alpha line at 6562Å; green is the Hydrogen-beta line at 4861Å; cyan is Magnesium at 4481Å; and blue is the Hydrogen-gamma line at 4340Å. The complex series of absorption lines in the orange box at the upper-right are mainly due to oxygen molecules in our own atmosphere; this set of lines is very dense, and runs from about 6850-7000Å. There are other atmospheric absorption lines to contend with that are of greater consequence for our purposes, including H2O absorption lines superimposed on the Hα line. We’ll take another look at atmospheric absorption lines – often referred to as “telluric” lines (from the Latin tellurem for the earth) – further down.
Figure R1-1 To turn an image like this into a graph of intensity versus wavelength requires calibration of the spectrometer/camera system, as outlined in the previous section. In ISIS the calibration is computed on the fly using several user-prepared configuration files, and some user-tuned parameters. The upper panel in Fig. R1-2 on the next page was produced from the above image with two of the three required calibration steps implemented: mapping the geometry of the echelle rungs (which also approximately measures the intensity variations across each order, known as blaze functions), and mapping the wavelength as a function of pixel position.
17 Our imaging camera is monochrome – the low-noise, high quantum efficiency Atik 460ex. There is no reason to use a colour camera, except to produce pretty images, since all the analysis is done using digitized data. Page 12 of 57 The final calibration is to correct for variations in the overall efficiency of the spectrometer/ camera; the upper panel in Fig. R1-2 was produced without applying that final step. As explained in the last section, the overall correction is done by comparing a measured spectrum with a fully-calibrated reference for the same star, or another star of the same spectral type, which in the case of Vega is A0V.18 Spectral classes with relatively few absorption lines are preferred, with early “A”-type stars being a common choice.
Figure R1-2 The lower panel in Fig. R1-2 shows the fully-calibrated spectrum of an A0V spectral class star, zeta Aquilae, taken from a database of standards maintained by the National Optical Astronomy Observatory (NOAO).19 Notice the different vertical scales in the two panels. The NOAO spectra straddle the entire optical band, are cleansed of atmospheric absorption lines, are corrected for Doppler shifts due to the star’s recessional velocity, and have about half the resolution of the eShel, all of which makes NOAO spectra very convenient for the final calibration. Dividing the NOAO spectrum by my uncorrected Vega data produced the black
18 Recall the spectral sequence OBAFGKM running from hot blue to cool red stars, each letter type divided into 10 steps from 0-9. A letter at the end of the spectral type (as in A0V) indicates the luminosity class, which distinguishes between stars of different sizes, with V standing for main sequence stars.
19 https://www.noao.edu/cflib/. ISIS comes with library of spectra from a variety of database standards, including NOAO. Page 13 of 57 curve in the lower panel in Fig. R1-2, known as the spectrometer/camera response function.20 The uncorrected data is multiplied by the response function to produce the “true” spectrum. Note that we are computing the relative intensity, which is defined by setting the intensity at some arbitrarily-chosen wavelength to “1”, and then rescaling the rest of the data. Producing data that is absolutely normalized (i.e., plotting the flux of energy per unit wavelength) is very difficult, and is not necessary in order to measured the wavelengths of spectral lines.
Figure R1-3 Figure R1-3 plots my fully calibrated spectrum for Vega (now including the overall response function), along with two other measurements that provide instructive comparisons. The upper plot is from a professional observatory in France, Observatoire de Haute Provence (OHP, with a 1.93m aperture), taken with an echelle spectrometer called ELODIE21 (now decommissioned) having a resolution R about 5X times larger than the eShel. The lower plot
20 To compute the response function, it is useful to first apply a low-pass filter to smooth out high-frequency noise in the user’s data, and also to “cut” out the deep absorption lines, by interpolation using neighbouring points. ISIS has interactive tools that allow the user to do all of this very handily.
21 http://www.obs-hp.fr/archive/elodie-for-dummies.shtml#archive. The ELODIE website has a huge on-line database of spectra that cover a wavelength range of about 3900-6800Å. ELODIE spectra contain artifacts (spiky discontinuities) that I think occur because they do not bother to impose a continuity condition on data from overlapping regions in adjacent echelle orders. I’ve edited out those discontinuities to make a clearer comparison with my results, which used a smooth matching condition between echelle orders that is built into ISIS. Page 14 of 57 shows data taken by Christian Buil (who I mentioned at the end of the Introduction) using a spectrometer with a resolution about 5X smaller than the eShel, on a small apo-chromatic refractor.22 To more clearly compare the three plots, I’ve shifted the OHP data upwards relative to mine, and shifted Buil’s data lower. Note the four labelled optical Balmer lines. The overall trend in the spectrum, which decreases rapidly with wavelength, makes it hard to compare details in the absorption lines across the whole wavelength range. Spectra are typically presented after fitting the overall trend, known as the “pseudo”-continuum, and dividing it out of the original data: the result is known as the “rectified” intensity. The results of this procedure as applied to my data are illustrated in Fig. R1-4. I used ISIS to fit the overall trend, shown as the red line in the upper plots, and then to divide that out of the original data, to obtain the rectified intensity shown in the lower plot. The word “pseudo” is used to describe this treatment of the “continuum” in part because the measured result differs from the true stellar continuum due to attenuation in our atmosphere.23
Figure R1-4 Figure R1-5 on the next page compares the rectified spectra of the three data sets. It can also be useful to visualize a spectrum in the form of a colourized spectral bar that is synthesized from the data; this is neatly done using another software package called
22 http://www.astrosurf.com/buil/us/vatlas/vatlas.htm.
23 Atmospheric attenuation can be largely compensated by measuring the spectrum of a reference star at the same altitude as the object of interest. ISIS also contains tools that can correct for atmospheric attenuation. Page 15 of 57 VisualSpec.24 The synthesized spectrum corresponding to the rectified intensity of my Vega data is shown in Fig. R1-6, superimposed on the intensity plot. Some sharpening and contrast enhancement were made to bring out the details in the synthesized spectral band; such “aesthetic” image processing step are never applied to the data when used for measurements!
Figure R1-5
TrottObs
Figure R1-6
24 http://www.astrosurf.com/vdesnoux/. Page 16 of 57 Line widths and Vega’s rotation A closer inspection of the pots in Fig. R1-5 on the previous page is very instructive. To begin with, it’s evident that the eShel and OHP data are both of much higher resolution than Buil’s results: notice the intricate series of lines in the former plots that are either reduced in strength, or absent entirely, in Buil's data. This is expected, since Buil’s measurements were made with a spectrometer of 5X lower resolution than the eShel. On the other hand, the OHP data appears to be of only slightly higher resolution than the eShel results, even though the ELODIE spectrometer has an intrinsic resolution that is almost 5X larger.
Figure R1-7 The fact that most of Vega’s spectral lines have comparable widths in the eShel and ELODIE data is well illustrated in Fig. R1-7,25 which zooms into three closely-spaced lines26 that are otherwise representative of the whole spectrum.27 The line widths (FWHM = Full Width at Half Minima) turn out to be about the same as the intrinsic resolution of the eShel, which is about
25 I applied a blueshift to my data for the plot in Fig. R1-7, to account for the relative geocentric motion between the dates of the two datasets (details on that below). 26 I identified the absorption lines in Fig. R1-7 from detailed synthetic spectra for main-sequence B-type and A0 class stars available on-line at http://www.lsw.uni-heidelberg.de/projects/hot-stars/websynspec.php.
27 Notable exceptions are the Hydrogen Balmer lines, which are much deeper and broader than the other lines. The depth of these lines is due to the star’s temperature, which produces photons with energies that are well matched to the Balmer excitation energies, and to the abundance of Hydrogen in the star’s atmosphere. Frequent collisions between these lightest of atoms also generates a wide range of velocities, and the resulting Doppler shifts cause a significant broadening of the spectral lines, which is known as pressure or collisional broadening. Page 17 of 57 0.5Å, compared to about 0.1Å Figure R1-8 for the ELODIE. These large line widths must be due to Vega’s intrinsic properties, and in fact Vega is known to rotate fairly rapidly. (We’ll see a d e f i n i t i v e u n b i a s e d comparison between the resolutions of the eShel and ELODIE in Section R2.) Figure R1-8 illustrates the mechanism of rotational broadening: the parts of a star that rotate away from us redshift spectral lines, while the parts that rotate towards us simultaneously blueshift them. We’ll take a closer look at rotational broadening in Section R5, where I’ll try out a more sophisticated analysis of data that I took for two stars that rotate very rapidly. For now let’s make a quick estimate of Vega’s rotational broadening using the formula for the Doppler shift v = r ⇥ c where Δλ is the wavelength shift for light emitted by a source moving with radial velocity vr (the component of the velocity along our line of sight), and c=300,000 km/sec is the speed of light. This formula is an approximation for source speeds that are much less than the speed of light. The convention is that vr is positive for a source moving away from the observer, resulting in a positive value for Δλ (i.e., a redshift), and negative for a source approaching the observer.
Setting the magnitude of vr equal to Vega’s known equatorial speed, projected along our line sight, of about 25 km/sec,28 gives a rough estimate of the broadening of the spectral lines of around 0.5Å, which is consistent with the observed line widths in Fig. R1-7. Telluric lines At the beginning of this section I pointed out some absorption lines in the eShel image of Vega’s spectrum (Fig. R1-1 on pg. 12), that are caused by molecules in our atmosphere. In the optical band, the telluric lines lie in the red end of the spectrum, starting around 5800Å, and the dominant lines are due to water, except for very strong lines due to oxygen molecules beyond about 6870Å. Telluric lines generally have to be subtracted from the data before spectral features in those regions can be reliably used for analysis. Happily, there is a tool in ISIS that handles the most important part of the job :). A notable stellar absorption line that is contaminated by tellurics is the Hydrogen-alpha line. The upper panel in Fig. R1-9 compares the spectrum around the Hα line in my Vega data before and after the tellurics are removed using the ISIS “H2O” software tool. The subtraction, while generally good enough for subsequent analysis, necessarily accentuates the noise. Telluric lines beyond Hα are strong and complex, starting with a sharp drop due to oxygen molecules at about 6870Å, as can be seen in the TrottObs and Buil data in Fig. R1-5 on pg.16.
28 I took this value from a huge on-line database known as SIMBAD, which pools together published measurements of more than 8 million objects beyond the solar system: http://simbad.u-strasbg.fr/simbad/. Page 18 of 57 This region is shown in closeup in the lower panel in Fig. R1-9, which illustrates the complexity of these effects. While these lines can also be subtracted, I think most amateurs limit themselves to wavelengths not much above the the Hα line, to avoid having to deal with this.
Figure R1-9
Page 19 of 57 Effects of Earth’s motion relative to the Sun One final piece of information that can be wrung out of the Vega data is illustrated in Fig R1-10: the dotted red line in the figure runs through the minimum in the Hα line in my data, but is clearly displaced from the minima in the OHP and Buil data. These offsets are due to differences in the Earth’s orbital velocity around the Sun, in the direction of Vega, on the three dates! The Earth’s orbital velocity is about 30 km/sec, which is easily within the measurement precision of the eShel (more on that in Section R4); for very precise measurements, even Earth’s rotational motion, which is only about 0.3 km/sec at our latitude, must also be taken into account! And, not surprisingly, there is a tool in ISIS that computes the component of the observer’s velocity, relative to the Sun, in the direction of a specified object or celestial coordinate, from any point on the Earth, on any date.
Figure R1-10 Potential confusion arises in the convention for the Earth’s radial velocity with respect to the Sun, which is quoted as positive for Earth motion towards the object, that is, radially outwards from the Sun: Earth’s motion in that direction blueshifts an object’s spectral lines, compared to the expected shift relative to the Sun. The heliocentric velocity correction is negative for motion away from the object, causing a redshift compared to the expected shift relative to the Sun. ISIS gives a heliocentric velocity correction in the direction of Vega, from Vancouver, on the date of my observations (Aug 24 2015) of -10.2 km/sec, and for OHP’s location and date (Feb. 20 2006) of +10.3 km/sec. This means that my data should be Doppler shifted by a relative
Page 20 of 57 velocity of about -20 km/sec compared to the OHP data; that is, my data should appear redshifted relative to theirs by that amount. I did a fit of the relative velocity between the two data sets (I’ll describe how that’s done in Section R4), and found a best fit value of about -16 km/sec. Figure R1-11 zooms into three closely-spaced but otherwise representative lines (the same ones in Fig. R1-7 on pg. 17), and compares my original data, and the data with a compensating blue shift of 16 km/sec, to the OHP results. The plot shows that this shift is easily distinguished by the eShel. The difference with the known value is only 4 km/sec, or about 1/100,000 of the speed of light! This seems really good (perhaps surprisingly so, if one mistakenly compares with the spectral resolution of the eShel, of about 10,000), but we’ll see in Section R4 why we can (and will!) do much better, when comparing two measurements made using the eShel!
Figure R1-11
Page 21 of 57 Results Part 2): Ring around the Moon Emission spectrum of M57
No 33 6801.7Å No 34 6601.6Å No 35 6413.0Å
No 45 4987.9Å No 46 4879.5Å No 47 4775.6Å Figure R2-1 Figure R2-1 shows the acquisition image produced by the eShel for the Ring Nebula, after stacking about ten three-minute exposures. All that shows up against a signal-less background are six prominent spots due to six strong emission lines, along with traces of some weaker lines.29 For convenience, I’ve identified the central wavelengths of some echelle orders near the emission lines. This image looks very different from the acquisition image for a stellar spectrum; compare with Fig. R1-1 on pg. 12. The spectral trace generated from the stacked (but otherwise unprocessed) acquisition image of M57 is shown in Fig. R2-2.30 It is instructive to compare these results with another data set, reproduced in Fig. R2-3, which was taken by an amateur astronomer named Richard Walker.31 Walker’s data was obtained using a spectrograph with about 10X lower resolution than the eShel, but with greater sensitivity (take note again of the tiny size of the eShel acquisition fibre, shown in the figure on pg. 4); the increased sensitivity reveals a number of weaker absorption lines that are not visible in my data.32 On the other hand, the much poorer resolution of Walker’s data is also evident; notice in particular that the three strongest lines at the red end of the spectrum overlap in his data, but are cleanly separated with the eShel.
29 I applied an aggressive low-pass filter to the image after stacking, to filter out the noise in the background, and I also used some contrast enhancement to make the emission lines stand out. Contrast enhancement and other non-linear processing steps should not be applied when the data is used for quantitative analysis.
30 The algorithm used by ISIS to merge adjacent echelle orders, to create an overall spectrum plot, produces unphysical results for a nearly “empty” emission spectrum; it seems to vastly over-scale the signals by imposing continuity on the pure noise in the ends of the orders. I wrote my own script to join the orders to obtain Fig. R2-2. 31 Walker’s web site has an extensive spectroscopic atlas in PDF form (from which I took Fig. R2-3), along with two very useful guides to spectroscopy for amateurs. See http://www.ursusmajor.ch/astrospektroskopie/richard- walkers-page. The web site is in German, but his three hefty PDF documents are in English – look for links about 2/3 of the way down his home page. His atlas will be published by Cambridge University Press in 2017.
32 The small amplitude variations in my data are mostly noise, except for a small spike at 5461Å that is actually mercury (Hg) sky glow, which apparently stands out here against the “empty” sky background, along with small but genuine signals at around 6300Å and 6700Å; compare those regions with Walker’s data in Fig. R2-3. Page 22 of 57 Figure R2-2
Figure R2-3 © Richard Walker Closeups of the absorption lines in the eShel data are shown in Fig. R2-4 on the next page. The full-width at half-maximum of the lines is about 0.5Å, as expected.
Page 23 of 57 Figure R2-4 Page 24 of 57 The Sun via the Moon I took the spectrum of the waxing gibbous Moon on August 25,33 with the goal of comparing it with the spectrum of the Sun. I couldn’t find a convenient plot of the Sun’s optical spectrum on- line, so I compare instead with an ELODIE spectrum of a G2V-class star, in Fig. R2-5; the plots look noisy, but their “rough” appearance is due to very densely-spaced spectral lines. I integrated the Moon for less than two minutes (!), and smoothed out some high-frequency noise (which would not have been visible in this plot), using a low-pass filter built-into ISIS.
Figure R2-5
Figure R2-6 on the next page zooms into three interesting 100Å sections of the spectra, which show that the lunar spectrum agrees very well with the G2V star (as it should!), to within the eShel’s resolution, which is clearly shown to be much lower than that of the ELODIE. It is also useful to examine colourized spectral bars generated synthetically from the two data sets, as shown in Fig. R2-7 on the next page (covering 4300-6600Å). Compared to Vega’s A0V spectral type (depicted in a spectral bar on pg. 16), the G2V spectral-type has much weaker Hydrogen Balmer lines, while overall it is much more complex.
33 The Moon’s spectrum before rectification was too faint at the blue end relative to the red end, when compared with the G2V continuum. This is probably mostly due to strong extinction effects, since the Moon was at a very low altitude (about 200), when I took the data. Page 25 of 57 Figure R2-6
Moon
G2V
Figure R2-7
Page 26 of 57 Results Part 3): Spectroscopic Binaries Spectroscopic binaries come in two broadly-defined varieties. In one type, the spectral lines of only one star are visible (I gather this happens when the other star is too faint to be detected), and these are called single-lined spectroscopic binaries. When the lines of both stars are visible, you’ve got a double-lined spectroscopic binary – and that’s the variety I wanted to look at from the moment the eShel arrived! I’ve shown a crummy animation of the shifting positions of the absorption lines of double-lined spectroscopic binaries in my astronomy class for years, and wanted to finally see the real thing in action for myself. I don’t recall how I came across the double-lined spectroscopic binary 57 Cygni for a first case study, but it has excellent properties: the two stars have large radial velocities, the binary is reasonably bright (visual magnitude about 4.8), and it has a short orbital period of 2.85 days, meaning that one can trace out an interesting part of its time-series in just a few nights – plus it is high in the sky in late summer. I managed to get good-quality data for 57 Cyg on three nights, spaced over a week (on August 18, 24, and 25). I also took data for Mizar, a well-known spectroscopic binary, on one of those nights, although it was not so well positioned by then. I integrated for about 30 minutes for 57 Cyg on each night, and about 10 minutes for Mizar. It happens that the two stars in 57 Cyg have the same spectral type, B5V. Figure R3-1 compares the spectrum of 57 Cyg from the last night of my observations, with the spectrum of a unary B5V class star from the ELODIE database. Figure R3-2 on the next page zooms into a number of prominent absorption lines in the blue-green region of the spectra.
Figure R3-1
Page 27 of 57 Figure R3-2 The HeI (4472Å) and MgII (4481Å) lines on the left side of Fig. R3-2 are shown in close-up in Fig. R3-3a for the three nights; the two HeI lines at the right end of Fig. R3-2 are shown in Fig. R3-3b. I averaged the radial velocities obtained for the four lines from the Doppler formula δλ/ 34 λ=vrad/c, to find the following results (K1/2 is the usual notation for the two velocities):
Aug 19.335 2015 (UT) Aug 25.287 (UT) Aug 26.263 (UT) K1 (km/sec) 33(2) 75(4) Single line: K2 (km/sec) -76(2) -118(5) -18(2)
The numbers in parentheses are standard-error uncertainties [e.g. 33(2)=33±2]: these are only rough estimates, since they are obtained from only four Doppler shifts. It should be noted that the “blending” of nearby absorption lines will tend to shift their minima from their true values, a potential source of systematic uncertainty that I have not tried to account for. On the last night the line pairs were evidently merged into single lines (to within the eSel’s resolution), meaning that the stars were moving at right angles to our line of sight, with their common radial velocity at that point being equal to the overall radial velocity of the system. In fact, the value of -18(2) km/sec that I obtained for the common radial velocity is in good agreement with the results of previous measurements; SIMBAD reports an average of -20(2) km/sec. This fortuitous circumstance is helpful in further analyzing the results.
34 The heliocentric velocity corrections in the direction of 57 Cyg on these nights varied from 3.0 to 1.3 km/sec, and are accounted for in the results; much more information on this kind of analysis is given in Section R4. Page 28 of 57 Figure R3-3a
Figure R3-3b
Page 29 of 57 The results of my measurements are in reasonable agreement with the known radial velocity (RV) curve for 57 Cyg, which is illustrated in Fig. R3-4; I took this diagram from a huge on-line catalogue of spectroscopic binary data.35 The cycle has two points where the stars have a common radial velocity; the results of my three nights of observations matches the cycle when the final night (Aug 26.263 UT) is identified with the 2nd point of common radial velocity, shown by the dotted red line in Fig. R3-4. The relative positions in the cycle of the other two nights can then be identified from their phases relative to the third night, using the known period of 2.85 days. For instance, the 2nd night (Aug 25.287 UT) is earlier than the 3rd night by a phase of (26.263-25.287)/2.85=0.342; I drew the dotted blue vertical line in Fig. R3-4 by displacing it from the red line by that phase. Similarly, the dotted green vertical line identifies the phase of the first night. My estimates of the velocities on those two nights agree with values read off from Fig. R3-4, to within differences of a few times the uncertainties, roughly estimated by combining my uncertainties with those of the K1/2 values of the RV curve.
150.0 9th Catalogue of Spectroscopic Binary Orbits
100.0
50.0
0.0
-50.0 Radial velocity (km/s)
-100.0
-150.0 .0 .2 .4 .6 .8 .0 Phase Figure R3-4 Finally, Fig. R3-5 on the next page shows my results for the spectrum of Mizar, on the one night that I observed it. I compare my results with an ELODIE spectrum of a unary star of the same spectral type (A2V), and the lower plot zooms into a region in the blue-green part of the spectrum with a rich set of absorption lines. The double dips clearly reveal the presence of the two stars, and the inferred radial velocities36 are roughly equal in magnitude, about ±30 km/ sec, to be compared with the known maximum radial velocities of just under 70 km/sec.
35 http://sb9.astro.ulb.ac.be. The database does not quote errors for the maximum radial velocities, but from the cited paper for 57 Cyg the uncertainties in that case appear to be at least a few km/sec.
36 The overall radial velocity of Mizar, with respect to Vancouver on that night, was less than 1 km/sec. Page 30 of 57 Figure R3-5
Page 31 of 57 Results Part 4): High-precision radial velocities Motivation Several amateur astronomers have used the eShel and other spectrographs of comparable resolution to detect exoplanets (already discovered by professionals) through the Doppler “wobble” of their parent stars.37 The absorption lines of the star vary in wavelength as it moves around the centre-of-mass of the system, due to the gravitational pull of the exoplanet, in the same way that the absorption lines of spectroscopic binary stars vary in wavelength, and these measurements can be used to infer the mass of the exoplanet. At least one of the exoplanets already studied by amateurs also transits its parent star, and the measured light curve can be used to determine the exoplanet’s size. Together, the two measurements can yield an estimate of the exoplanet density, which is how professionals have distinguished between gas giants and terrestrial planets! The precision of the eShel allows Doppler measurements only for a few hot Jupiters, but even so, working through a sophisticated project of this nature, to establish the likely composition of an exoplanet for oneself, would be very exciting! While I have not yet tried to detect an exoplanet, I’ve come up with a relatively quick and powerful way to test one’s ability to make measurements of the necessary precision. Consider this a proof of principle, before starting the longer-term project of detecting an exoplanet ;). I was initially dubious about the prospects of doing such measurements using the eShel, since the orbital velocities (along our line of sight) of the stars in question are very small, at most about 0.5 km/sec; this is to be compared with spectroscopic binaries, which are routinely measured by amateurs, but where radial velocities above 100 km/sec are not uncommon. My own measurements of the binary 57 Cyg, reported in the previous section, had uncertainties in the radial velocities of about 2-5 km/sec, as much as ten times greater than the largest signal for an exoplanet, which does not seem very promising; however, 57 Cyg's spectral type is not at all well suited to a precision measurement, and I did not try to do a very careful analysis. To make the challenge of these measurement clearer, consider again the Doppler shift formula v = r ⇥ c A radial speed of 1 km/sec implies a fractional wavelength shift Δλ/λ of 1 part in 300,000. At first glance, this might seem impossible for the eShel, given its spectral resolution R of about 10,000 which, if taken at face value, would suggest that measurements are limited to about 30 km/sec in precision. However, the correct interpretation of the spectral resolution is as a measure of how closely spaced two spectral lines can be and still be separated (see the diagram on pg. 1). The precision of a radial velocity measurement, on the other hand, is set by the precision with which the wavelengths of individual spectral features can be determined. The distinction between spectral resolution and wavelength precision is illustrated by Fig. R1-11 on pg. 21, which shows the shift between my spectrum of Vega, and one from ELODIE, due to different geocentric motions on the two dates. The line widths are about 0.7Å, comparable to the spectral resolution of the eShel, while Doppler shifts corresponding to a few
37Christian Buil has two very good webpages with details of exoplanet observations by himself and others: http:// www.astrosurf.com/buil/tauboo/exoplanet.htm and http://www.astrosurf.com/buil/extrasolar/obs.htm. A collaboration called “Spectrashift” produced the first amateur detection of an exoplanet, in 2006, observing tau Boötis using a custom-built spectrograph with smaller resolution than the eShel, on a Meade 16” LX200; see http://arxiv.org/ftp/astro-ph/papers/0609/0609468.pdf. The collaboration’s website is http://www.spectrashift.com. Page 32 of 57 km/sec can arguably can be discerned in the plot just by eye. As we’ll see below, much smaller radial velocities can be resolved if the right approach is used. Strategy The strategy for making radial velocity (RV) measurements of the precision needed to detect hot Jupiters, using the eShel, will be explained in detail as we work through it. The essence of the approach is to consider exoplanets whose parent stars have spectra with many finely- spaced lines, and to measure the spectrum of a reference star of similar spectral type, as close in time as possible to the measurement of the star with the exoplanet. Spectral classes F5-M5 are well suited to this,38 since they have several thousand well-separated absorption lines; hotter stars have too few lines, while the lines in cooler stars are too overlapped. This is partly illustrated by the spectral bands below, generated from data that I took for Vega, which is of spectral type A0V, and � Pegasi, a K4III spectral class star of visual magnitude 4.8.
Vega
� Peg To convincingly demonstrate radial velocity measurements for exoplanets requires many observations; a few measurements will not be sufficient, given their uncertainties, so one really needs to map out a good part of the RV curve to make a clear-cut case (take a look at some results reported by Buil to see what I mean). So I thought it would be useful to find a way to more easily assess the techniques of measurement and analysis at this level of precision. To do this, I used SIMBAD to identify pairs of stars of the same “late” spectral type that could measured on the same night, and which have fairly accurately known radial velocities (to within 200 m/sec or so),39 and that are also reasonably bright (about 5th magnitude or better). In actual exoplanet measurements, one star would have the exoplanet, and the other would serve as a reference star. The spectra of the two stars are taken as close together in time as possible, in order to minimize systematic errors in the spectrometer calibration (such as drift due to small temperature changes). A best fit to the relative radial velocity of the two stars is obtained using a method known as a cross-correlation, which simultaneously compares the full spectrum of the stars. This combination of data acquisition and analysis yields very precise results, since in essence one is exploiting the difference between two measurements, taken with the same instrument, to largely cancel out the systematic uncertainties. I did this investigation in two phases. In the first phase, I took a limited amount of data for three different pairs of stars, with the goal of obtaining an initial estimate of the measurement uncertainties. The reason for measuring three different pairs was to make sure that the excellent result that I had obtained for the first pair was not a fluke! I then followed up by taking much more data for another pair of stars, to do a much more through analysis of systematic uncertainties. Results and analysis follow in the next two subsections.
38 See e.g. Lovis and Fischer, Radial Velocity: http://exoplanets.astro.yale.edu/workshop/EPRV/Bibliography_files/ Radial_Velocity.pdf. Compare also with the G2V spectral band on pg. 26.
39 SIMBAD quotes average values for radial velocities with an uncertainty. However, I haven’t been able to find out how these uncertainties are determined, and I discovered that in some cases the quoted error is much smaller than the spread in values of recent measurements (which are cross-referenced by SIMBAD). I considered only those cases where SIMBAD’s quoted error is consistent with the dispersion in recent measurements. Page 33 of 57 Results and Analysis: Initial study In my initial investigation, I took measurements of three useful pairs of stars, whose properties 40 are listed in the table below. The “known” radial velocities vrad of the stars relative to the Earth-Sun barycentre, including the uncertainties, were taken from SIMBAD. The “heliocentric” velocity correction vhelio is the observer’s velocity relative to the Earth-Sun barycentre, in the direction of the object, on the observation date; the values in the table were obtained from ISIS. I denote the radial velocity of the star relative to the local observer (me!) by vloc, where
vloc = vrad - vhelio The 2nd-to-last column gives the difference in the known local radial velocities of each pair, denoted by �vloc, and this is what will be directly extracted from the measured spectra.41 At the risk of giving away the happy ending for this initial set of results too soon, the last column has my results for �vloc, with the details to follow.
40 Observation Star ID Spectral V. Mag. vrad vhelio vloc �vloc �vloc Date (UT) (constellation) Type SIMBAD ISIS Known Known Measured Sep 30.281 HD211388 K3II-III 4.2 -8.58(8) -5.02 -3.56(8) ___ (Lacerta) HD211073 K3III 4.5 -11.4(2) -4.53 -6.9(2) 3.3(2) 3.5(1) (Lacerta) Oct 4.354 HD218029 K3III 5.2 -8.71(8) 5.87 -14.58(8) ___ (Cepheus) HD216446 K3III 4.8 -31.87(26) 9.83 -41.70(26) 27.1(3) 27.4(1) (Cepheus)
Oct 5.313 � Piscium K4III 4.8 34.19(2) 5.60 28.59(2) ___
� Pegasi K4III 4.8 -18.90(9) -17.48 -1.42(9) 30.0(1) 30.2(1)
To minimize errors arising from potential drifts in the spectrometer calibration, I took images of the spectra in batches: each batch typically consisted of five-to-ten exposures, each of one- or two-minute duration, for both stars, and I took calibration data at the beginning and end of each batch. I averaged the calibration data when extracting the star’s spectrum from a given batch. The final spectrum is an average of the results from each batch. I accumulated between 30 minutes and one hour of data for each star, in three or four calibrated batches. Figure R4-1a on the next page compares my data for two of the stars in my data set with an ELODIE spectrum for � Peg. The spectral plots look noisy, but the rapid variations are actually due to the very dense spectral lines (the signal-to-noise in my data is generally over 100:1, which is conveniently determined using VisualSpec). Figure R4-1b zooms into the spectrum for � Peg near the Sodium doublet: my results are in excellent agreement with the ELODIE data. Figure R4-2a on the page after next compares my data for � Peg and � Psc (the pair of stars with the largest relative velocity in my shopping list), in a wavelength range with many absorption lines that are typical of the rest of the spectrum. Figure R4-2b shows the same spectral range, now with two values of a relative Doppler shift applied to the � Psc data. A
40 All velocities in the table are in km/sec.
41 One could apply the heliocentric corrections to the spectra to begin with, and find the difference in the heliocentric radial velocities, but the local radial velocity is useful because it is encoded in the “raw” data. Page 34 of 57 relative velocity of only a few km/sec can easily be discerned by eye, making it plausible that an error of under one km/sec can be obtained from a mathematical fit to the whole spectrum.
Figure R4-1a
Figure R4-1b
Page 35 of 57 Figure R4-2a
Figure R4-2b
Page 36 of 57 Figures R4-3a and R4-3b similarly plot data for the pair of stars in Lacerta, which has the smallest relative velocity of the three sets, only about 3.5 km/sec. Figure R4-3a already makes it clear that very small relative Doppler shifts can be extracted.
Figure R4-3a
Figure R4-3b
Page 37 of 57 To obtain a best estimate of the relative Doppler velocity �vloc between the two stars in a pair, I followed standard practice, which is to compute a so-called cross-correlation function between 42 their spectra. A “natural” definition of the cross-correlation, which I denote by C12(�vloc), is
I1( )I2 (1 + vloc/c) C12( vloc)= I2 I2 P h 1 ih 2 i where I1(λ) and I2(λ’) are the observed spectralp functions,43 with the wavelength λ’ in one spectrum Doppler-shifted relative to the other using the standard formula λ’=λ(1+�vloc/c), and where the angle brackets in the denominator are normalizing factors, defined by I2 = I2( ),i=1, 2 h i i i One can show that the cross-correlation satisfiesP the so-called Schwarz inequality 1 C ( v ) 1 12 loc If C12 = 1, then the two spectral functions are said to be completely correlated, while if C12 = -1 they are said to be completely anti-correlated.
The best fit value of the Doppler velocity is found by numerically estimating the value of �vloc that maximizes C12(�vloc). I’m sure you won’t be surprised to learn that ISIS has a tool that does these computations,44 however I wrote my own scripts for this purpose.45,46
To anticipate the potential resolution in �vloc, consider that our eShel/camera pixel scale is about 0.15Å/pixel. Thus a shift in �vloc of 50 m/sec, for example, is equivalent to a shift of about 1/200 of a pixel in the position of any one spectral line in the optical band. By comparison, professionals measure shifts equivalent to 1/1000 of a pixel!47 If this sounds ridiculous, a key point is that the Doppler shift is multiplicative, and in our case there are some 10,000 pixels from one end of the fitted spectral range to the other. Consequently, the cumulative shift across the fit range, for a velocity shift of about 50 m/sec, is non-trivial, amounting to about 50 pixels. Figure R4-4 plots my results for the cross-correlation between the spectra of � Peg and � Psc, which is maximized at �vloc = 30.2 km/sec, in excellent agreement with the known value of
42 If you’d rather skip the math, all you need to know is that the cross-correlation is bounded by “1” in (absolute) value, will precisely equal “1” only for a perfect match, and that the best-fit value for the Doppler shift is the one that numerically maximizes the cross-correlation. You can now skip to the 2nd-to-last paragraph on the page ;). 43 Rectified spectral functions are used, since the smooth continuum behaviour in the relative intensity is not very sensitive to the Doppler shift, while the overall scale of the continuum is very sensitive to the overall calibration.
44 The ISIS tool reads data files for two spectral functions and returns a numerical value for the best-fit Doppler velocity, along with a plot similar the ones I’ve produced. The results of my scripts agree very well with those of the ISIS tool. I wrote my own scripts in part to do more general calculations, such as rotational broadening.
45 ISIS outputs the spectra with a 0.05Å spacing, which I interpolate using cubic splines, to be able to vary �vloc in steps of about 1 km/sec, corresponding to a wavelength spacing of about 0.01Å. Following standard methods, I actually rewrite the shifted λ’=λ(1+�vloc/c) using the log of the wavelengths, lnλ’=lnλ+ln(1+�vloc/c)≈lnλ+�vloc/c, which is a uniform linear shift across the entire spectrum; the cross-correlation and normalizing factors are defined from the outset using sums over lnλ, instead of over λ. Also, the mean-values of the spectral functions are removed before doing the analysis, i.e., an overall constant is subtracted from the spectra such that
47 See e.g. Fischer et al., Exoplanet Detection Techniques, http://www.mpia.de/homes/ppvi/chapter/fischer.pdf. Page 38 of 57 30.0(1) km/sec, and within the bounds suggested by Fig. R4-2b. Similarly, Fig. R4-5 plots the cross-correlation between the stars in Lacerta, which is maximized at �vloc = 3.5 km/sec, in excellent agreement with the known value of 3.3(2) km/sec, and within the bounds suggested by Fig. R4-3b. Finally, the cross-correlation between the stars in Cepheus is maximized at �vloc = 27.4 km/sec, again in excellent agreement with the known value of 27.1(3) km/sec.
Figure R4-4
Figure R4-5
Page 39 of 57 The agreement between my measured values of �vloc and the known values from SIMBAD is very encouraging, and given SIMBAD’s reported errors, suggests that the uncertainties in these results might be at the level of about 100 m/sec. This is confirmed by estimating standard errors. Recall that I took data in batches of 3 or 4 measurements for each pair of stars; computing the cross-correlation separately for each batch produces a set of estimates of �vloc from which the averages and standard errors can be estimated. The estimated standard errors for the three pairs range from about 50 m/sec to 150 m/sec, although these determinations are rough, given the small sample sizes. To be relatively conservative, I quote an uncertainty of 100 m/s in the table on pg. 34 for all three measurements. Results and Analysis: Systematics To get a more reliable estimate of the uncertainties in differential radial velocity measurements, I decided to take alot more data, for just one pair of stars, over several nights. Larger sample sizes not only produce more reliable estimates of the standard error, but measurements taken over a longer period of time can be used to check for potential systematic variations in the results. Systematic drifts in the measurements can be caused by mechanical changes in the spectrometer due to variations in temperature, pressure, and other environmental conditions, as the night progresses. Systematic changes can also be caused by storing the spectrograph after a night of measurements, and then re-connecting it to the telescope on a subsequent night. One expects that systematic effects of these kinds should largely cancel out in cross- correlations between measurements of two stars, taken one after the other, since drifts in the two spectra will be strongly correlated, and this is supported by my initial set of results. By the time I could attempt this followup study, the three pairs of stars that I had measured before were too far to the west after dark. So I found another pair of suitable stars, both in Auriga: 50 Aur (mag 4.8) and 58 Aur (mag 5.0), both of spectral class K3III. I followed the same data acquisition procedure as before, taking exposures in batches, one star immediately after the other, with calibration data taken at the beginning and end of each batch. Each batch consisted of five two-minute exposures for each star, and between 9 and 13 batches were taken on each of three nights: December 30, and January 2 and 8. I computed cross-correlation functions between the two spectra in each batch, to extract the relative Doppler shift from each batch separately. Figure R4-6 shows the results. I plot the difference �vrad in radial velocities, relative to the Earth-Sun barycentre
�vrad = vrad,58Aur - vrad,50Aur rather than using the “local” difference �vloc as I did before; this takes into account a statistically significant change in the relative heliocentric velocity �vhelio = vhelio,58Aur -vhelio,50Aur from the first night to the third (�vhelio = 1.15 km/sec on the first two nights, and �vhelio = 1.13 km/sec on the last night – and yes, that’s a difference of only 20 m/sec!).
As seen in Fig. R4-6, there is no discernible systematic change in the results for �vrad over the course of a night (it took between about five and seven hours to capture the data on a given night). Moreover, there is no statistically significant change in the means over the three nights, although the fluctuations on the second night (and hence the statistical error) were about twice as large as on the other two nights. The poorer quality of the data on the second night is probably due to the seeing conditions, which were extremely poor that night, among the worst that I’ve experienced over many months, with the stellar seeing disks frequently fluctuating off the entrance to the pickup fibre.
Page 40 of 57 Figure R4-6 Here are the results for the means and standard errors on the three nights48 43.54(3), Dec 30 vrad = 843.58(8), Jan 2 (in km/sec) > <4344.62(4), Jan 8 The uncertainties are remarkably small> (though not unreasonably so, as anticipated on pg. 38), ranging from 30 m/sec to 80 m/sec!: These statistical errors are actually comparable in size to a systematic error for this stellar class caused by a phenomenon known as stellar “jitter”. Jitter refers to activity in a target star, such as convective spots in its photosphere, which produce spurious shifts in the results of radial velocity measurements.49 Jitter varies in magnitude with a star’s spectral type, rotation period, and other attributes. For KIII class stars, like those measured here, jitter induces shifts of order 20 m/sec, on times scales comparable to the star’s rotation period (about a week in this class). The upshot, if I’ve read the literature correctly, is that the statistical uncertainties obtained here are about as small as is useful, unless one selects for stars with smaller jitter! Finally, since the whole point of this study was to verify that spectrometer drifts cancel out in correlations between the spectra of two stars, it is very interesting to identify how large a drift occurs in the spectrum of an individual star. To do this, I cross-correlated the two spectra by fixing the data batch used for one star, while stepping through the batches used for the other.
48 The results are also consistent with previous measurements: SIMBAD reports vrad,58Aur =59.53(25) km/sec, and vrad,50Aur =16.09(19) km/sec, for a difference �vrad = 43.44(31) km/sec (also shown in Fig. R4-6), with an error as much as 10 times larger than in my results!
49 M. Perryman, The Exoplanet Handbook (details in the bibliography), Sect. 2.2.6. Page 41 of 57 Figure R4-7 shows results from the 1st and 3rd nights (no clear trend is seen in the 2nd night’s data, consistent with its larger fluctuations). The drift in one spectrum over a night is up to 20 times larger than the error in the relative velocity obtained from the two fully-correlated spectra! It’s not entirely clear what environmental changes actually caused these drifts, but temperature change is a likely factor. Unfortunately we can’t keep the eShel in a temperature-controlled environment: the control room is too small, and the door is opened and closed often (likewise for the bathroom). So I set up the spectrometer and acquisition unit and left them in the dome area, with the dome shutter open, for about an hour before taking data. I then monitored the temperature next to the spectrometer, but the changes were at most a tenth of a degree.
Figure R4-7
Outlook for exoplanet detection The results of this study very clearly confirm that we can use the eShel to detect at least some exoplanets, following the pioneering work of other amateurs. Differential radial velocities with uncertainties of around 50 m/sec are obtainable, to be compared with the largest known parent-star Doppler shift of about 500 m/sec (� Bootis). To summarize the strategy, we pick a hot Jupiter whose parent star has a “late” spectral type (F5-M5), a large enough orbital speed, and is of magnitude about six or brighter. We take the star’s spectrum along with that of a reference star of comparable spectral type. Measurements of the two stars are taken one after the other, with five or six consecutive measurements giving a reliable estimate of the uncertainty in the mean differential radial velocity. Cross-correlations of the two spectra essentially eliminate systematic variations due to mechanical changes in the spectrometer. However, at present there are only a few suitable candidates.50
50 A complete searchable on-line exoplanet database can be found at http://exoplanet.eu. Page 42 of 57 Results Part 5): Doppler rotational broadening I made a rough estimate of Vega’s rotation rate in Section R1, from the Doppler broadening of its absorption lines (see pgs. 16-17). It turns out that hot stars (spectral types A and earlier) often have very fast rotation rates. A short project suggested on the Shelyak website51 is to measure the spectrum of a particularly fast A0V-type star, zeta Aquilae (visual magnitude 2.99); its equatorial speed, projected along our line of sight, is about 320 km/sec, or about 0.1% of the speed of light! Compare this to 20 km/sec for Vega, and a paltry 2 km/sec for the Sun! Since zeta Aquilae was high in the sky in August, when I was starting to take data, it became one of my first targets. Also, one night in early September, I stayed up till dawn to take the spectra of the Pleiades, and later realized that Atlas (the 2nd brightest star in the cluster) also has a fast rotation; the most recent measurement cited by SIMBAD is about 170 km/sec. Figure R5-1 compares my spectra of zeta Aquilae and Vega, and Fig. R5-2 on the next page compares my data for Atlas with the spectrum I took of Maia, another star in the Pleiades with the same spectral type as Atlas (B8III), but having a much slower rotation speed, about 30 km/ sec. It is already apparent from these two plots that many spectral lines in the two very fast rotators are almost completely washed out by the Doppler broadening.
Figure R5-1
A common notation for the equatorial speed of a star along our line of sight is ve.sin(i), where 0 ve is the true equatorial speed, and “i” is the angle of inclination of the rotation axis (i=90 when the axis is perpendicular to our line of sight). I’ll use the short-hand vi =ve.sin(i). It turns out that a very simple model of Doppler rotational broadening can be used to obtain pretty good quantitative estimates of rotation speeds from measured spectra. Figure R5-3 on the next page depicts a star’s hemisphere in the observer’s direction, projected on the sky as a disk. The radial velocity of a point on the disk, projected along our line of sight, increases
51 http://www.shelyak.com/dossier.php?id_dossier=20. Page 43 of 57 linearly with horizontal distance x from the rotation axis; I’ll denote the radial velocity at position x by v(x), where x v(x)=vi ,vi = ve sin(i) a where a is the radius of the star, and with x running from -a to +a, to correctly account for Doppler red-shifting on one side of the disk, and Doppler blue-shifting on the other side.
Figure R5-2
h(x)= a2 x2 p
Figure R5-3 We’ll assume that when a star isn’t rotating, all parts of the projected disk radiate with the same intensity per unit area, I(λ)/Area, where I(λ) is the total intensity at a given wavelength.52
52 A more accurate model would include the effects of limb darkening, which is actually relatively easy to do, but I didn’t bother with that. A really good set of lecture notes on stellar atmospheres, including modelling of Doppler rotational broadening and limb darkening, has been posted on the web by a retired professor at UVic, J.B. Tatum; at http://astrowww.phys.uvic.ca/~tatum/stellatm.html. Chapter 10 has a section on rotational broadening. Page 44 of 57 Then if the star rotates, the radiated light will be Doppler shifted, and instead of contributing to the intensity at some rest wavelength λ, will contribute instead at a shifted wavelength λ’ that depends on the position on the disk according to λ’=λ(1+v(x)/c). Considering the radiation coming from a rectangular area on the disk of infinitesimal width dx at position x, the total intensity as a function of wavelength in this simple model is given by a 1 2 2 I( ,vi)= 2 I 1 v(x)/c 2 a x dx ⇡a a Z ⇣ ⌘ where the factor of 2 accounts for the rectangular strips abovep and below the horizontal line through the diameter. The integral depends on the star’s radius only implicitly, through the overall intensity I(λ), as can be seen by introducing a dimensionless variable ρ=x/a 1 2 2 I( ,vi)= I 1 ⇢ vi/c 1 ⇢ d⇢ ⇡ 1 Z ⇣ ⌘ In practice, the integral is evaluated as a discrete sum, p using spline fits to the digitized spectrum. The best fit to the (projected) equatorial velocity is obtained by cross-correlating the measured spectrum of the target star with the spectrum of a reference star that has been “spun-up” using the above formula. Synthetic reference spectra (idealized versions of stellar spectra without telluric lines or noise) are typically used, but I thought it would be interesting to use my own measured spectrum of a star of similar spectral type as the one whose rotational speed is to be determined. So I substitute the measured spectrum Islow(λ) of a slow rotator (e.g. Vega) in the integral above, to compute a sped-up spectrum Islow(λ,vi), and cross-correlate that with the measured spectrum Ifast(λ) of a fast rotator (e.g. zeta Aquilae). The cross-correlation function Crot(vi) is naturally defined by53
Ifast( )Islow( ,vi) Crot(vi)= I2 I2 P h fastih slowi Figure R5-4 (upper panel) on the next page pplots the cross-correlation between my data for zeta Aquilae and Vega; the best-fit corresponds to a spin-up of Vega by about 280 km/sec, which is remarkably close to the known rotational speed of zeta Aquilae of about 320 km/sec, considering that Vega itself has a rotational speed of about 20 km/sec. Figure R5-2 (lower panel) zooms into two particular spectral lines, comparing the measured spectrum of zeta Aquilae with that of Vega for various spin-up speeds.54 The Hydrogen Balmer lines in A-type stars are not very sensitive probes of the rotation, since they are affected by very strong pressure broadening to begin with. The MgII line on the other hand illustrates the effect of the spin-up very well, and is characteristic of other relatively thin lines. Figure R5-5 on the page-after next has analogous results for the cross-correlation between my data for the fast rotator Atlas, and spin-ups of my data for Maia. The cross-correlation gives a best fit in good agreement with Atlas’ known rotation rate of 170 km/sec, although the overlap of the individual lines does not seem to be as good as before.
53 I actually use the logarithm of the wavelength to define the cross-correlator and normalization factors, in the same way as the recessional velocity cross-correlators were computed in the last section.
54 I applied a low-pass filter to smooth out some high-frequency noise in the spectra for zeta Aquilae and the stars in Taurus, since I took only about 10 minutes of data in these cases. Page 45 of 57 Figure R5-4
Page 46 of 57 Figure R5-5
Page 47 of 57 Results Part 6): Disks and winds Very hot stars (type O and B) can produce very strong emission lines by ionizing material that surrounds the star; emission lines result when the ionized atoms recombine with electrons in the plasma. The emitting material can be in the star’s own “wind”, or in a circumstellar disk. Disks can formed by mass transfer from a companion star, creating an accretion disk, or by material ejected by the star itself, if it has a fast rotation, creating a so-called decretion disk.55 Sometimes the material is so “optically” thick that it completely blocks the visible-light continuum produced by the star’s photosphere and its atmospheric absorption lines. Hot stars with prominent emission lines include so-called Wolf-Rayet stars, which are thought to be highly-evolved O-class stars, which generate very fast stellar winds that shed huge amounts of material (a solar mass in about 100,000 years). Wolf-Rayet stars exhibit broad emission lines of heavier elements (helium, carbon, nitrogen, and oxygen). Since their lifetimes are short (a few hundred thousand years) they are rare, with only about 300 known examples. B-type stars with emission lines are identified generically as Be stars (“e” for emission). The emission lines (usually strongest in the Balmer series) have very distinctive signatures that can distinguish between stellar winds and disks, and interesting quantitative information can sometimes be extracted from the spectra with relative ease, such as wind speed or disk speed and size. Many of these systems change rapidly, sometimes unpredictably (cataclysmic binary stars being one example), and are often monitored by amateur astronomers for research, sometimes in collaboration with professionals (so-called pro-am collaborations). I took spectra of all of these hot-star types, as detailed in the four subsections that follow. Circumstellar disk of VV Cephei B One of the largest stars in the sky is the red-supergiant VV Cephei A, with a radius that is nearly as large as Saturn’s orbit, and which forms an eclipsing binary with a B-class main sequence star (combined visual mag 4.8). Mass blown off by the red supergiant forms an accretion disk around its companion. The orbital period is about 20 years, but it so happens that the next eclipse begins in August 2017. In the upcoming eclipse (which will take about two years to complete), the Be star will pass behind the red supergiant, and spectroscopic measurements will easily track the accretion disk gradually becoming eclipsed and then reemerging! At least one pro-am collaboration has already announced an observing campaign to monitor the system, leading up to and through the eclipse, and invites others to participate.56 I took the spectrum of VV Cep with 30 minutes of integration, results shown in Fig. R6-1 on the next page, including a closeup of the Hα and Hβ emission lines. The spectrum is a blend of the red super-giant (continuous with absorption lines) and the Be star. The doubly-peaked emission-line profile is a distinctive signature of a circumstellar disk, and can be used to make simple estimates of the disk’s rotation speed, its tilt with respect to our line of sight, and even
55 An excellent qualitative introduction to the physics of hot stars can be found in a short book by Robinson. A more thorough introduction is given in the last chapter of the book by Everberg and Vollmann. The Shelyak website also contains a good overview: http://www.shelyak.com/dossier.php?id_dossier=24&lang=2.
56 Hopkins, Bennett and Pollmann, http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?2015SASS... 34...83H&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf. Bennett is a professor at St. Mary’s. An spectroscopy forum that includes pro-am collaborations has more information on the campaign: http://www.spectro-aras.com/forum/viewforum.php?f=2&sid=a48fff0166ea950f3b154f39079538f6. Page 48 of 57 its size (given the star’s mass)! We’ll estimate the speed and size of the VV Cep B disk (on pg. 49), but first we need a short overview of the origins of this intensity profile.
Figure R6-1
Page 49 of 57 The illustration on the right is an artist’s impression of a Be star with a circumstellar disk,57 and Fig. R6-2 shows an example of the characteristic emission line profile of a disk, shown for the Hα line.58 Emission lines are produced when atoms in the disk are ionized by radiation from the star, and then recombine with electrons in the plasma. The lines are broadened analogously to the Doppler broadening of absorption lines of rotating stars: material on one side of the disk rotates away from the observer producing a red-shift, © STScI while material on the other side rotates towards the observer producing a blue- shift. The double-peaked distribution with wings, on the other hand, is unique to a disk. To understand this intensity profile, suppose that particles in the disk rotate more-or-less independently under gravity; then the rotational speed will be smallest at the outer rim of Figure R6-2 the disk, and will increase steadily towards the inner rim. While material in the interior of the disk rotates faster than at the outer rim, the velocities range over all angles to our line of sight; consequently there are two points on every orbit where the radial velocity (i.e., with respect to our line of sight) is the same as the radial velocity at some point on the rim. This is illustrated in Fig. R6-3, which depicts a slice through the middle of a disk, as viewed from above (assume for now that the plane of the disk is aligned with our line of sight). Each curve runs through points in the disk with the same radial velocity. The longest curves start from the two outer edges of the disk, which means that more material is Doppler shifted at the speed of the outer rim than at any other speed. This accounts for the doubly-peaked spectral profile, and shows that the speed of the outer rim can be directly determined from those wavelengths. Figure R6-3 Next, to understand the red and blue wings in the intensity profile, consider that material at the inner ring of the disk has the largest possible orbital speed, which produces radiation with the maximum possible Doppler shift – but if the disk is roughly uniform in density, there will be less and less material as the inner ring is approached, and the intensity will drop as the limits of the
57 Retrieved from https://commons.wikimedia.org/wiki/File:Classical_Be_star.jpg. The object at the lower-left is supposed to be a companion dwarf star.
58 I took this spectrum of Pleione, one of the seven sisters in the Pleiades, as described in the next subsection. Page 50 of 57 Doppler shift are reached. The last spectral feature depicted in Fig. R6-2 is a so-called central reversal, which occurs at the unshifted wavelength of the emission line, corresponding to material that crosses our line of sight, with zero radial velocity. The preceding discussion was for a disk that is aligned along the observer’s line of sight. Figure R6-4 shows how the intensity profile changes with the angle of inclination of the disk.59 If the disk is perpendicular to our line of sight, there is no Doppler broadening, and we have a single peak. The broadening, and the split in wavelength between the peaks, both increase as the disk becomes less inclined, since the radial velocities increase; the central reversal also becomes deeper, as more of the disk becomes blocked by the star (most of that part has small radial velocities). We’ll see examples below of disks of various inclination angles. Now let’s return finally to Fig. R6-1 for the spectrum of the VV Cep system, and make some estimates of the radial speed, and line-of-sight size, of the Be star’s circumstellar disk.60 A rough estimate of the radial speed of the outer rim of the Figure R6-4 disk, obtained from the peak wavelengths of the Hα and Hβ emission lines (relative to the central reversal) is about (70±10) km/sec, averaging over values obtained from the four peaks. An estimate of the radial speed of the inner rim of the disk, obtained from the wavelengths of the ends of the emission line wings, is more difficult to judge in this case, because of the overlap with the supergiant’s absorption lines; a rough estimate is about (250±50) km/sec. The most fun part of the analysis is to use these speeds to estimate the outer and inner radii of the disk! Assuming circular orbits due to the gravity of the Be star, we can relate the orbital speed vorbit of a particle in the disk to the mass M of the star, and the radius Rorbit of the orbit GM v = orbit R r orbit where G=6.67X10-11m3/sec2/kg is Newton’s constant. The VV Cep B mass has been estimated in the range 8-20 solar masses M⊙ (typical for B-type main sequence stars).61 Using M=10XM⊙ 30 (M⊙=2X10 kg), and the disk speeds estimated above, gives Router〜 400R⊙ and Rinner〜 40R⊙, 6 where R⊙=0.7X10 km is the Sun’s radius. The outer radius compares reasonably well (considering the poorly constrained mass) with an old estimate of about 650R⊙, while the inner radius is not much larger than the star’s estimated radius of about 20R⊙, as might be expected. All in all, pretty cool, especially considering the simplicity of the analysis!
59 Figure R6-4 can be found at http://www.astrosurf.com/buil/us/bestar.htm, and many other sources. 60 Note that the stronger intensities of the blue-side peaks, compared to the red sides, can be explained by pile-up of the in-falling material as it swings around the Be-star, which currently lies beyond its super-giant partner.
61 Masses and sizes are cited in http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?2015SASS... 34...83H&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf.. Page 51 of 57 Circumstellar disks among the Seven Sisters It turns out that four of the seven sisters are Be stars with circumstellar disks. I was inspired to take my own spectra by the diagram in the centre of Fig. R6-5, due to Christian Buil,62 which shows the Hα regions of all seven. I’ve included my results (plots in blue) for the Hα regions next to Buil’s, and I compare a more complete spectrum of one of the Be stars, Alcyone, with one of the “regular” B-class stars, Maia, in Fig. R6-6 on the next page. I only integrated for about 10 minutes for each star, and the data are a bit noisy, so I applied a low pass filter to the results (and filtered out telluric lines). My results for Pleione (left-side of Fig. R6-5) and Merope (bottom right) look somewhat different from Buil; in fact, Pleione is thought to have developed a new equatorial disk in the interim,63 and the change between Buil’s spectrum and mine is consistent with that evolution.
© Christian Buil
Figure R6-5 Comparison of the spectra of the four Be stars with the illustrations in Fig. R6-4 on the previous page shows that the circumstellar disks have different orientations with respect to our line of sight. The disk of Alcyone (top-left in Fig. R6-5) appears to be inclined at a very steep angle, while the disk of Pleione appears to be nearly edge-on, as noted above. The disks of Electra and Merope (right-side of the figure) appear to lie at intermediate angles, similar to the 3rd figure from the top in Fig. R6-4.
62 See http://www.astrosurf.com/buil/bestars/m45/img.htm. The data was taken in 2006. The original is in colour, but I converted it to (inverse) greyscale to make it easier to read here.
63 See http://pasj.asj.or.jp/v59/n4/590404/590404.pdf. Page 52 of 57 Figure R6-6 Super-fast winds of Wolf-Rayet 140 The prospect of taking the spectrum of a Wolf-Rayet star, in order to “see” heavy elements that had been dredged up from the star’s interior, and ejected into space in a fast-moving wind, was very appealing64 – and it turns out that one of the brightest, known as WR140, is in Cygnus (it also has an O-type companion). At visual magnitude 6.9, this became the faintest star that I’ve surveyed so far. I only took about 60 minutes of data to produce the spectrum in the upper panel of Fig. R6-7 on the next page. Note the complete absence of a stellar continuum/ absorption-line spectrum. The plot in the lower panel of the figure is from a professional study of the same object,65 and identifies the ranges of emission lines in the wind of the WR star, along with some absorption lines due to its O-class companion, and even some interstellar absorption lines (IS); my spectrum is in excellent agreement with all of these features! The emission lines of WR stars are very broad, due to the Doppler shifts produced by the very fast winds travelling out from the star in all directions (see the P Cygni section below for more on this kind of broadening). Some lines have an unusual “flat-top” shape, like the doubly- ionized CIII line at 5696 Å, which is apparently due to concentric shells of material expanding outwards from the star with a range of velocities.
64 Christian Buil has spectra for many Wolf-Rayet stars: http://www.astrosurf.com/buil/survey/wrstars/wrstars.html.
65 S.V. Marchenko et al., Astrophysical Journal, 565, 59 (2003). This article can be accessed at http:// iopscience.iop.org/article/10.1086/378154/fulltext/. This study included scientists at several Canadian institutions, and part of the data was collected at observatories in Canada (David Dunlap, DAO, Mont Mégantic). Page 53 of 57 Figure R6-7
Marchenko et al. 2003
The wavelengths at the edges of the flat-top spectral lines can be used to make a simple estimate of the wind speed: the blue edge corresponds to the maximum, or terminal, radial velocity towards the observer, and the red edge to the maximum velocity away. Reading off from the edges of the flat-top for the CIII line gives a terminal velocity of about 2,000 km/sec (in good agreement with other estimates)! This is to be compared with the Sun’s paltry thin wind, which has a maximum speed of about 400 km/sec.
Page 54 of 57 P Cygni profile A classic example of a Be star with an optically-thick wind is P Cygni (spectral-type B1Ie), of visual magnitude 4.8 – and another happy choice for fall observing ;)! The profile of this star’s emission lines is actually the prototype for a distinct subclass of Be stars named after this star. Figure R6-8 shows my results for the P Cyg spectrum from just 15 minutes of integration. The star is about 50 times brighter per unit wavelength in the Hα emission line than in the wind’s continuous light (estimated from the total area under the line, known as its equivalent width) and, like the spectrum of WR140, the underlying stellar continuum is completely hidden.
Figure R6-8
Figure R6-9 on the next page zooms into the Hα and Hβ regions of the data. The emission lines are broadened and, perhaps more surprisingly, there are small absorption dips on the blue side of the lines; these blue-side dips are characteristic of all the emission lines in the spectrum, and are associated with the stellar wind, not the star’s photosphere. Figure R6-10 illustrates the origins of these spectral features.66 The emission line is broadened on the red side because of radiation emitted by material in the wind in the region labelled (a), behind the star, which is receding from the observer; similarly, the approaching wind in the region labelled (b), in front of the star, produces radiation that is blue-shifted. Note that radiation from the star in the directions of regions (a) and (b) would be lost to the observer; however, part of the radiation produced by the ionized wind in those regions does travel towards the observer. Conversely, wind material in region (c) mostly radiates away from the observer, and this produces the blue-shifted absorption dip. The blue end of the dip corresponds to the maximum (or “terminal”) wind velocity.
66 Figure R6-10 is taken from a review article by S.P. Owocki: http://www.bartol.udel.edu/%7Eowocki/preprints/ turb98.pdf. The books by Robinson, and Eversberg and Vollmann, have good explanations of these effects. Page 55 of 57 Figure R6-9
S. Owocki, Turbulence in Line-Driven Stellar Winds (b) (a)
(c)
Figure R6-10 In the case of P Cyg, the blueshifts in the two Balmer lines in Fig. R6-9 imply a terminal wind speed of about 275 km/sec, in good agreement with other estimates (which, as near as I can tell, range as high as about 300 km/sec). This is comparable to speed of the Sun’s very thin wind, but much slower than O-class and Wolf-Rayet stars, as we saw with WR140.
Page 56 of 57 THE END! For now that is! An exoplanet measurement is coming by the spring! I should also be able to get our new spectrograph up and running in the spring, which will allow us to do all kinds of fainter, extended objects, with galaxy redshifts and rotation curves at the top of the list! Bibliography The report contains footnotes with many links to on-line databases and other resources not reproduced here; in particular, see the list of resources on pg. 11. Chromey, F. R., To Measure the Sky: An Introduction to Observational Astronomy, Cambridge (2010). An undergraduate text that delivers on the title. I may use this for a course at SFU. The chapter on spectroscopy uses only basic math but explains some core ideas very well. Eversberg T. and K. Vollman, Spectroscopic Instrumentation: Fundamentals and Guidelines for Astronomers, Springer (2015). A comprehensive treatise at a senior undergraduate level, it is nicely written and amazingly clear. In addition to all the technical details on spectrometers, it has an excellent chapter on the physics of massive stars, and details on data analysis and open-source software tools used by professionals. Gray, R. O., and C. J. Corbally, Stellar Spectral Classification, Princeton University Press (2009). Apparently the standard reference on the subject. Harrison K. M., Astronomical Spectroscopy for Amateurs, Springer (2011); and Grating Spectroscopes and How to Use Them, Springer (2012). Two short and informative books written for amateurs. Kaler, J.B., Stars and their Spectra: An Introduction to the Spectral Sequence, Cambridge, 2nd edition (2011). Covers the spectral sequence throughly without math, with lots of examples of specific stars in each category, not as detailed as Gray. Karttunen, H. et al., Fundamental Astronomy, 5th edition, Springer (2011). An excellent introductory undergraduate textbook that is also written with amateur astronomers in mind! Leblanc, F., An Introduction to Stellar Physics, Wiley (2010). An alternative to Tatum’s notes (see below), at about the same undergraduate level, and by another Canadian professor! Perryman, M., The Exoplanet Handbook, Cambridge (2011). An outstanding resource! Robinson, K., Spectroscopy: The Key to the Stars, Springer (2007). An excellent short book written for amateurs that covers a ton of astrophysics, with very clear explanations, and only a tiny bit of optional math. Tennyson, J., Astronomical Spectroscopy: An Introduction to the Atomic and Molecular Physics of Astronomical Spectra, World Scientific, 2nd edition (2011). An undergraduate-level survey that has the physics and mathematics details that are missing from Robinson. Tatum, J.B., http://astrowww.phys.uvic.ca/~tatum/stellatm.html. A really good set of on-line undergraduate lecture notes on stellar atmospheres, by a retired professor at UVic. Walker, R. http://www.ursusmajor.ch/astrospektroskopie/richard-walkers-page. Walker’s web site has an extensive spectroscopic atlas in PDF form, along with two very useful guides to spectroscopy for amateurs.
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