A Search for Dark Matter in an Edge on Spiral Galaxy

Transcription

1 UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 136 PROFESSOR: PROCHASKA A Search for Dark Matter in an Edge on Spiral Galaxy Benjamin Stahl Team: A. Callahan (PI) & J. Gillette August 6, 2014 Abstract The edge-on, bulge-less spiral galaxy IC 1197 was observed using the Shane 120-inch reflecting telescope at Lick Observatory using the Kast instrument in late May, The bias level of the detector was determined to be counts with a read noise of 5.7 counts. The collected spectral images were corrected by removing the bias level and then applying flat field corrections. Additionally, the background sky lines were identified and subtracted from the spectrum of the target galaxy. Using arc lamp spectra and arc line plots, a mapping was found between pixels and wavelengths in angstroms along the horizontal dimension of the images. Next, the mapping was applied and two spectral lines (Hα: ± 1.17 Å and a spectral line of S II: ± 1.17 Å) were identified at the galactic center. These wavelengths were then used to determine the red shift of IC 1197 to be ± , and thus the distance from the Earth to the galaxy to be ± 0.69 Mpc. Using a trigonometric argument, the radial distance was converted from the pixel space of the vertical dimension of the spectral images into physical units of kiloparsecs. For given radial distances away from the galactic center, the rotation speed was found by first determining the wavelength of the spectral line being analyzed at that given radial distance and then using Doppler shift to find the speed. From these results, a rotation curve was plotted for IC 1197 from the experimental data. Next, velocity profiles from theoretical models of just ordinary matter, just dark matter, and a combination of the two were over-plotted on the rotation curve drawn from the experimental observations. The theoretical models were optimized to best fit the observed data, and in doing so it was determined that the observed velocity profile is best explained by a dark matter dominated model. Thus, it is concluded in this paper that IC 1197 contains dark matter.

2 CONTENTS 1 Introduction 4 2 Observations Experimental Apparatus Calibration & Set Up Detector Settings Preparation at Observatory Science Data Collection Other Exposures Observing Conditions Source Observed Timeline of Observations Data Trimming of Images Bias Level & Read Noise Flat Fielding Science Ready Frames Sky Subtraction Mapping from Pixels to Wavelengths Analysis & Results Spectral Line Identification Redshift Distance to Galaxy Mapping from Pixels to Radial Distance Rotational Velocity Discussion Model Development Gravitational Potential Disk Rotation Speed Dark Matter Rotation Curves Rotation Curve Dark Matter Closing Remarks LIST OF FIGURES 1.1 A predicted theoretical rotation curve based on the mass of the ordinary matter that was observed and a rotation curve from experimental observations. There is a clear discrepancy where the observed rotation curve stays higher than the theoretical model [3] An example of the spectrum of an HII region from NGC 7252 [13] The raw spectrum of IC 1197 as observed through the red detector of the KAST instrument using the Shane telescope at Lick Observatory The spectrum of IC 1197 as observed through the red detector of the KAST instrument using the Shane telescope at Lick Observatory. Undesired rows and columns were removed from the raw image to yield this result The observed flat field spectrum after taking the median and flattening is plotted in red as counts (in thousands) versus pixel number. The 17th order polynomial fit to the spectrum is over-plotted in blue. 12 2

3 3.4 The normalized flat field image that was used to remove unwanted equipment signatures from the data The fully reduced and sky line subtracted spectral image of IC 1197 that was used for scientific analysis. Note that the lines that are still present towards the lower part of the image are not trusted and remain as the remnants of sky lines that were not fully removed by the algorithm. This occurred because the rows used to calibrate out the sky lines were taken primarily from the top of the frame, and given that there is a slight tilt across the detector, the correction is more accurate when applied to the top of the frame as compared to the bottom The data points used to find the best fit linear mapping between pixels and wavelength are plotted in red, with the model plotted in blue. Clearly there is strong agreement Spectral plots of the calibration frame. First in terms of pixels and then in terms of wavelengths after applying the mapping The spectra of IC 1197 after the raw image was reduced and the mapping from pixels to wavelength was applied. The two lines selected for scientific analysis are clearly visible above the noise and match well with their known wavelengths This triangle relates the radial distance B from the galactic center to the distance L to the galaxy and and an angle α [20] The rotation curves for a galactic disk as well as a dark matter halo are presented. Additionally, a combined profile is included where the two profiles are weighted equally and added in quadrature. Assumptions made in generating this plot include a stellar mass of M, a dark matter mass of M, a scale-length of 4 kpc, and a concentration parameter of The velocity at a given radial distance of IC 1197 as determined from the experimental data and the analysis conducted on it. A polynomial of third degree is fitted to the data and then over-plotted to make the trend more apparent The velocity at a given radial distance of IC 1197 as determined from the experimental data and the analysis conducted on it is plotted for the segment of data specified in the text directly before the figure The velocity at a given radial distance of IC 1197 as determined from the experimental data and the analysis conducted on it is plotted for the segment of data to be compared. Rotation curves corresponding to a disk only profile, a dark matter only profile, and a combination of the two found by adding in quadrature that have been optimized for the data have been over-plotted. Given the relatively small size of IC 1197, the scale-length was assumed to be 1 kpc as opposed to 4 kpc as assumed for the previous plot of the disk only profile. Error bars were not assigned to the experimental data given their excessive size, which was discussed in Section LIST OF TABLES 2.1 Timeline of observations containing the time, exposure time, slit width, target, declination, right ascension, and any necessary comments. The listing has been abbreviated where possible by indicating when multiple identical frames were taken in the comments as well as compressing all entries not corresponding to IC 1197 or the necessary calibration exposures into as few lines as possible The known wavelengths of spectral lines of Neon appear in the left column of the table, in the middle column are the pixel locations (corresponding to columns of the CCD), and the left column contains an estimate of the uncertainty in the selection of a given pixel to match a spectral line

4 1 INTRODUCTION Galaxies are massive collections of stars, gas, dust, and the notoriously mysterious substance known as dark matter. The immense gravitationally bound systems come in a variety of forms, which are known as morphologies. There are elliptical galaxies, which are elliptical in appearance regardless of the viewing angle due to their ellipsoidal profiles. Galaxies of this type can be the largest of all galaxies and they tend to have little structure and instead the appear more diffuse. They also tend to be composed more of older stars than younger stars, reflecting their lower rate of new star formation as compared to other types of galaxies. Another morphological type is the spiral galaxy, where the orbiting stars, gas, and dust have a spiral shape that largely lies within a plane and orbits in spiral arms about the galactic center. Galaxies within this category can vary a good amount, with some being rather bulge-less at the center, while others may contain a bulge at the center which typically will consist of older stars. Further, some spiral galaxies, including the Milky Way, have a bar-shaped band of stars that extends from the core on either side and eventually merges into the spiral structure. In addition to these two main types of galaxies, there many other types such as dwarfs which are small galaxies that are often found to orbit larger spiral or elliptical galaxies. Examples of dwarf galaxies include the Large and Small Magellanic Clouds, which orbit the Milky Way and can be observed from the Southern Hemisphere of the Earth. Additional, so called irregular galactic forms exist and can vary wildly. Even classified galaxies such as spirals and ellipticals can vary a great deal within their categories. The first three galactic constituents mentioned (stars, gas, and dust), and indeed all of the matter that can be directly observed is known as ordinary matter. Composed of quarks and leptons, ordinary matter constitutes the observable objects in a galaxy, from the stars to the planets to the interstellar medium (ISM) of gas and dust. It is possible through modern astronomical techniques to make estimates of the mass of a galaxy from the observed matter that is present [7].Thus, for certain types of galaxies and situations, one can derive the rotational velocity of the observed matter in galaxy at a given radial distance. This relationship can be plotted as a rotation curve where the vertical axis represents the velocity of rotation, and the horizontal axis represents the radial distance from the center of the galaxy. Vera Rubin s work on the rotation curves of galaxies followed this process, and when she compared the rotation curve drawn from a theoretical model to that which was drawn from observed data, a clear discrepancy was uncovered. Figure 1.1 presents an example of what the two rotation curves would have looked like: Figure 1.1: A predicted theoretical rotation curve based on the mass of the ordinary matter that was observed and a rotation curve from experimental observations. There is a clear discrepancy where the observed rotation curve stays higher than the theoretical model [3]. As illustrated in Figure 1.1 the mass observed in the ordinary matter of a galaxy is simply insufficient to warrant the observed rotation speed. In other words, if the gravitational attraction of the ordinary matter was the only thing holding the galaxy together, it would fly apart. This so called galaxy rotation problem has been a topic of active research in astronomy and various solutions have been proposed ranging from modifying Newton s laws to introducing another type of unseen matter to account for the missing mass. Currently, the most widely held solution to the galaxy rotation problem is in a proposed exotic form of matter that is known as dark matter. Though there are many different proposed models as to the true nature and composition 4

5 of the dark matter, all share in common the property that dark matter does not interact electromagnetically at any level that is significant. That is, it does not absorb nor transmit electromagnetic radiation such as light to any appreciable or detectable degree, making it dark. Aside from this, the models predict that the dark matter will interact as would any other matter under gravitation. Finer points to the dark matter, such as its proposed composition, while interesting, will not be considered in this paper. Given that the dark matter is assumed to be massive and thus interact with gravity, it can be used to resolve the galaxy rotation problem, because it could cause there to be much more mass in a galaxy than that which is directly observed. It was the goal of this experiment to analyze a galaxy and make an argument for or against the presence of dark matter in this galaxy by plotting theoretical and experimentally determined rotation curves. This was accomplished by completing a series of objectives leading from experimental design, to observation, to data reduction, to analysis, and ultimately to a conclusion. These steps will be briefly outlined now, and then each will be thoroughly detailed throughout the rest of this paper. The design of the experiment was constructed using knowledge of the capabilities of the Shane 120-inch reflecting telescope and the Kast spectrograph, which together served as the experimental apparatus, as well as what sorts of observables would allow for the analysis to yield the desired results. Using high-precision, long-slit spectroscopy as afforded by the experimental apparatus, allows for high resolution spectra to be obtained. If an edge-on spiral galaxy were selected, high resolution spectral measurements could be analyzed to deduce the rotational velocity as a function of radial distance from the galactic center, and thus this condition was set as a constraint on target selection. An additional constraint placed on the target selection was that the edge-on spiral galaxy be bulge-less. This was imposed because a bulge-less galaxy will have a higher rate of star formation, and thus be considered a star forming galaxy. The emission spectrum of star forming galaxies are dominated by H II regions [18]. The benefit of having H II regions dominate the spectra of the scientific target is that H II regions have certain spectral lines that are very intense as shown in the sample H II spectrum presented in Figure 1.2, and thus galaxies dominated by H II emission spectra have spectral lines that are easy to find and analyze. Figure 1.2: An example of the spectrum of an HII region from NGC 7252 [13]. Weighing in these considerations, the galaxy IC 1197 was selected as the scientific target and data was taken using the experimental apparatus. This data was then reduced using processes that will be detailed later in the report. From this resulting reduced and corrected data, analytic processes were conducted to determine various parameters including the red shift of the galaxy and the rotational velocity at a given radial distance, among other things. From this data a rotation curve of the observed galaxy was generated and then compared to theoretical models for the rotation that will be developed in the Discussion Section. 5

6 2 OBSERVATIONS 2.1 EXPERIMENTAL APPARATUS Experimental data was taken using the following primary instruments and devices: Telescope Shane 120-inch reflecting telescope located at Lick Observatory, Mt. Hamilton, California. Detector Kast double spectrograph [17] * Dichroic: D55 * Slit width: 1.0 * Slit length: 145 * Grism: 600/4310 (blue detector) * Grating: 600/7500 (red detector) * Blue CCD: Fairchild 2k x 2k 0.43 arc seconds per pixel Binning 1x1 Read Speed: fast * Red CCD: Reticon 1200 x arc seconds per pixel Binning 1x1 Read Speed: fast 2.2 CALIBRATION & SET UP DETECTOR SETTINGS The settings and equipment used with the Kast spectrograph in the apparatus were decided upon by first making a rough determination of the minimum resolution required to accomplish the scientific goals of the experiment. Assuming that galaxies rotate at roughly 200 km/s, this minimum resolution was determined as follows where R is the resolution, c is the speed of light, and v is the rotation rate as assumed above: R = λ λ = c v = c 1500 (2.1) 200 km/s With this minimum resolution now established, the following configurations for the apparatus were decided upon as follows: First, the spectral lines to be observed in the target were determined to be Hα ( Å) and the Å spectral line of S II due to their high intensity in the spectra of H II regions [15]. These characteristic spectral lines were measured at slightly longer wavelengths due to red shift. Given the wavelengths of these two spectral lines to be observed, the D55 dichroic was selected. With this configuration, both spectral lines were sent to the red camera, allowing for the scientific analysis to be conducted on frames taken with the red camera only. Should additional analysis be desired, frames were collected with the blue camera as well which contain a strong spectral line of O III (5007 Å) [15]. These frames were not analyzed for the scientific purposes of this paper. 6

7 The following function to determine the resolution based on the settings and equipment was then used: R = λ S (2.2) W G Where R is the resolution, λ is the wavelength in angstroms of the spectral line to be observed, S is the plate scale of the detector in arc seconds per pixel (0.78 for the red detector and 0.43 for the blue detector), W is the slit width in arc seconds, and G is the angstroms per pixel of the grism (blue detector) or the grating (red detector) [17]. Both the wavelength and the plate scale were fixed for a given detector leaving two free parameters which are determined by the grating or grism used as well as the width of the slit opening. For the red detector, the wavelength was fixed at that of Hα was used as it is lower than the wavelength of the other spectra line and thus would give a lower spectral resolution according to Equation 2.2, and thus prevent the resolution from falling too low for longer wavelengths such as the S II line to analyzed. For the blue detector, the wavelength was fixed at the wavelength of the spectral line of O III. The first of the two free parameters to be decided upon was G which is determined by the grism or grating that is used. For the blue detector, the 600/4310 grism was used because its nominal coverage of Å matched the D55 dichroic that was selected. The value of G for this grism is 1.02 angstroms per pixel [17]. For the red detector, the 600/7500 grating was used because its nominal coverage of Å was the most appropriate for the Hα line to be observed. The value of G for this grating is 2.35 angstroms per pixel [17]. Having determined which grism or grating use, the number of free parameters is reduced to 1 thus allowing for the best value of the slit width W to be determined. Equation 2.2 was used with the parameters already discussed and with various slit widths that were easily configured with the apparatus. From this process, it was decided that a 1 slit width would be utilized because it produced a higher spectral resolution than the minimum of 1500 in each detector (the resolution for the red detector was roughly 2190 and for the blue detector it was roughly 2120) PREPARATION AT OBSERVATORY Set up and calibrations were conducted from the Shane telescope control room at the observatory in the afternoon prior to the evening observing session starting at approximately 00:15:00 UT 05/31/2014. The procedures provided in the Kast manual were followed but will be briefly be outlined now [16]. First the CCD chip temperatures were verified to be sufficiently low (on the order of -110 C) and then it was verified that the D55 dichroic was in place. Next, the software to control the telescope and spectrograph was started. A test exposure was taken with each channel (red and blue) of the detector. Next, a series of exposures were taken to focus the instrument. Then calibration arc lamp exposures were taken which were later used to find a mapping between pixels and wavelengths (this will be discussed in detail in Section 3.6). Then, dome flat field frames were taken followed by bias frames. The dome was opened and then twilight flat field frames were collected. After completing these exposures as well as the detailed set up procedures in the Kast manual, the apparatus was considered ready for scientific data collection. 2.3 SCIENCE DATA COLLECTION After configuring the telescope and taking the necessary exposures to make corrections for instrumental bias, scientific data collection was commenced. The telescope was pointed to the the coordinates of the target, guiding was enabled, and then a single 20 minute exposure was taken. The source that was observed as well as a justification for the exposure time will be provided in Section 2.6. After reading out the exposure and taking a brief look, it was determined that it would be sufficient for accomplishing the scientific goals and so the scientific data collection was considered complete. 2.4 OTHER EXPOSURES The telescope time was shared between a total of three research teams. Both of the other teams had taken their scientific exposures earlier in the observing session, and all of the necessary calibration exposures were complete. 7

8 Thus with extra time, a series of fun exposures were taken of targets that were of additional interest to the investigators. These frames were not collected with the goal of conducting science, but merely for fun at the discretion of the investigators. Upon completing all data collection, the telescope was turned over to the support astronomer and other observers and the observing session was considered complete. The data was retrieved and received by the investigators the following morning. 2.5 OBSERVING CONDITIONS Observations were conducted using the Shane telescope at Lick Observatory, Mt. Hamilton, California on the evening of Friday, May 30, The investigators arrived at the observatory around 16:30 PDT and calibration protocols were commenced roughly 45 minutes later. Scientific data collection was commenced at 03:57 UT, and the exposure of the target used for science in this report was taken at 09:28 UT. During the period of observations, the conditions were very clear with low clouds blocking some of the light from the surrounding cities. The sky above the telescope was quite clear. Temperatures were in the low to mid 50s Fahrenheit, and winds were calm (under 10 MPH) and coming mostly from the North [10]. The moon was a waxing crescent, and thus it is not expected that moonlight contaminated the observations. 2.6 SOURCE OBSERVED Several prominent considerations were taken into account when selecting a target for scientific data collection. The first was that the target be an edge-on spiral galaxy because the rotation rate of such a galaxy as a function of radial distance from the center can readily be determined through spectroscopy. Next, the size of the galaxy along its major axis was desired to be roughly similar to the length of the slit in the apparatus (145 [17]) to allow for efficient use of the apparatus (this configuration would require only one pointing of the telescope). Also, H II regions are common in bulge free galaxies because these galaxies tend to be full of gas and have a high rate of star formation. H II regions contain very characteristic spectral lines making scientific analysis easier, thus the consideration that the galaxy be bulge free was also employed. Another consideration was that the object be accessible by the telescope (18 Twilight: 13h37m Local Sidereal Time (LST), 18 : 19h28m LST [6]). From these considerations, the following list of candidate galaxies was assembled: 1. Designation: IC 1197 [4] RA: 16h08m17.5s DEC: MAG: (g filter) SIZE: 3.34 PA: 55 Z: Designation: UGC 9841 [9] RA: 15h25m33.6s DEC: MAG: (g filter) SIZE: 2.82 PA: 54 Z: Designation: UGC [8] RA: 15h44m49.4s DEC: MAG: (g filter) SIZE: 1.97 PA: 124 Z: Of these options, it was ultimately decided to observe the first one: IC The choice was somewhat arbitrary and was mostly informed by the LST at which the team was able to observe. The exposure time for the selected target was determined using the KAST Exposure Time Calculator [5]. Inputs to this calculator included the dichroic (D55), grism (600/4310), grating (600/7500), and slit width (1.0 ) as were already discussed. Additionally, the CCD binning was chosen as 1x1, which is how the CCD was configured for the data collection. Seeing and Airmass were left at the default values of 1.5 and 1.1, respectively. The magnitude of the object had 1 added to it before input to compensate for the fact that the target had some size and was not a point source like a star. The template was selected as Sa, and then the red shift was input. Finally, the last parameter of exposure time was input, and considered as a free parameter. With the goal of registering counts between 20k and 40k at the wavelengths of the two spectral lines to be considered, it was found that an exposure of 1200 s (20 min) would be suitable. 8

9 As noted previously, two other research teams collected data of other galaxies, and several fun images were taken with excess time. These observations will be briefly presented in Table 2.1 to account for the use of telescope time but will otherwise not be discussed any further in this report. 2.7 TIMELINE OF OBSERVATIONS A timeline of the observational tasks conducted for this experiment is presented in Table 2.1. All observations were made on UT 2014 May 31. Table 2.1: Timeline of observations containing the time, exposure time, slit width, target, declination, right ascension, and any necessary comments. The listing has been abbreviated where possible by indicating when multiple identical frames were taken in the comments as well as compressing all entries not corresponding to IC 1197 or the necessary calibration exposures into as few lines as possible. UT t exp (s) Slit Width ( ) Target DEC RA Comments 00:18: test 37:20: :45:34.0 beginning of blue exposures 02:01: focus 37:20: :28: exposures 02:11: Ne arc lamp 37:20: :38: :13: Ne arc lamp 37:20: :39: :13: Ne arc lamp 37:20: :40: :15: dome flat 37:20: :42: exposures 02:24: dome flat 37:20: :51: :26: dome flat 37:20: :52: exposures 02:30: dome flat 37:20: :57: exposures 02:34: bias 37:20: :02: exposures 03:28: twilight flat 37:20: :55: :28: twilight flat 37:20: :56: :29: twilight flat 37:20: :56: exposures 03:31: twilight flat 37:20: :59: exposures 03:34: twilight flat 37:20: :01: exposures 03:36: twilight flat 37:20: :03: exposures 08:03: UGC :38: :27: exposures 08:51: UGC :41: :42: :04: UGC :41: :42: :28: IC :31: :08:46.5 science target for this report 09:35: J :45: :54:10.7 possible quasar fun target 09:39: HD :27: :06:12.2 triple star system fun target 09:44: M57 33:04: :53:53.5 fun target 00:18: test 37:20: :45:32.5 beginning of red exposures 00:32: focus 37:20: :59: exposures 01:52: Ne arc lamp 37:20: :19: :53: Ne arc lamp 37:20: :20: :53: Ne arc lamp 37:20: :21: :54: Ne arc lamp 37:20: :21: :54: Ne arc lamp 37:20: :22: :55: Ne arc lamp 37:20: :22: :05: junk 37:20: :32: :15: dome flat 37:20: :42: exposures 02:19: dome flat 37:20: :46: exposures 02:21: dome flat 37:20: :48: exposures 02:34: bias 37:20: :02: exposures 03:28: twilight flat 37:20: :56: exposures 03:31: twilight flat 37:20: :59: exposures Continued on next page... 9

10 Table continued from previous page UT t exp (s) Slit Width ( ) Target DEC RA Comments 03:33: twilight flat 37:20: :01: exposures 03:36: twilight flat 37:20: :03: exposures 08:03: UGC :38: :27: exposures 08:51: UGC :41: :42: :04: UGC :41: :42: :28: IC :31: :08:46.5 science target for this report 09:35: J :45: :54:10.7 possible quasar fun target 09:39: HD :27: :06:12.2 triple star system fun target 09:44: M57 33:04: :53:53.5 fun target 3 DATA Though data was collected through both the blue detector and the red detector, it was decided by the observers that only the data from the red detector would be analyzed to accomplish the scientific goals of the report. This decision was made because the two spectral lines identified in the red science frame discussed in Section 3.6 are sufficient to accomplish these goals. In principle, two spectral lines are required to confirm that they are really what the appear to be (as was done in Section 3.6) and then only one spectral line is needed to determine red shift and rotational velocity as well be explained later in this paper, though two can be used and averaged to provide a stronger result. The data from the blue detector will be kept on reserve should additional spectral lines become necessary for analysis. Thus, the following sections will pertain only to the reduction of the data taken through the red detector. The images collected with the experimental apparatus were delivered in the.fits file format. These raw images were processed into a science ready format by performing a host of processes using a Python script developed by the investigators. The methodologies employed in making these corrections to the data will be outlined in the following sections. 3.1 TRIMMING OF IMAGES A brief view of the spectrum collected through the red detector for the science target shows several things to be corrected. Figure 3.1 shows this spectrum as viewed through a software package known as DS9: Figure 3.1: The raw spectrum of IC 1197 as observed through the red detector of the KAST instrument using the Shane telescope at Lick Observatory. As seen in Figure 3.1, the image contains horizontal strips at the top and bottom of the frame that contain no spectral information. This occurred because the spectrograph does not direct light onto this part of the CCD. To remove these strips of no information, all rows below 59 and above 228 were removed from the image. Though it is not possible to discern from Figure 3.1, the left 2 or so columns registered abnormally low counts and thus it was desired to remove the left 3 columns. Additionally, the entirety of the visible image is 1200 columns 10

11 wide, however the header of the.fits file indicated that it was in fact 1232 across (due to the over scan region of the CCD). Thus to correct along the horizontal dimension, all columns below 3 and above 1200 were removed from the image. Figure 3.2 shows the same spectra with the indicated rows and columns removed: Figure 3.2: The spectrum of IC 1197 as observed through the red detector of the KAST instrument using the Shane telescope at Lick Observatory. Undesired rows and columns were removed from the raw image to yield this result. The same rows and columns were removed from all frames that were used in the reduction of the science frame, as well as the science frame itself. 3.2 BIAS LEVEL & READ NOISE The bias level of the CCD is responsible for raising the zero point of the detector to some nominal value, typically around 1000 counts. The CCD will not register negative counts, and thus it is beneficial to implement a bias level to prevent the data from being biased to one side. Though implementing a bias level is critical in preventing the data from being skewed, it must be accounted for and subtracted for the data to be useful for scientific analysis. The fluctuation about the bias level, which is caused by the electronics is defined as the read noise. The bias level can be determined and accounted for by taking bias frames with the experimental apparatus. These frames are taken as 0s exposures with the shutter closed, thus the only counts that should be registered will be due to the bias level. This isn t always the case however, because cosmic rays can be striking the detector simultaneously. To prevent outliers such as cosmic rays from contaminating the determination of the bias level, multiple bias frames are taken and then the median value of counts for each pixel is taken across all of the frames. The resulting frame of median bias level counts for each pixel gives a good representation of the bias level of the detector. The mean value can be taken of this frame to represent the bias level with one number. The read noise of the detector can be found by taking the standard deviation of the counts from each pixel in the median bias frame. Note that the bias level and read noise are functions of the CCD and in no way are affected by the settings of the spectrograph. This process was implemented in a Python script to determine the bias level of the red detector from the 11 bias frames that were collected for it. From this analysis, the bias level was determined to be counts and the read noise was determined to be 5.7 counts. A support astronomer at the observatory found that using the fast read speed with the red detector gives a read noise of 19.6 electrons and a gain of 3.3 electrons per count [2]. Thus in counts the read noise as determined by the astronomer is 5.9 counts, which compares favorably with the result of 5.7 counts which was found from the collected data. 3.3 FLAT FIELDING Most if not all raw data collected with a telescope and associated equipment (spectrograph, in this case) will have unwanted signatures present. While in principle it may be conceivable that these individual signatures could be modeled, predicted, and thereby removed, it is much easier and economical to measure them and then divide them out. This is typically accomplished by flat fielding. In the case of applying flat field corrections to spectroscopic images, it best to take dome flats, where a continuum source of light without strong spectral features is illuminated on a screen and an exposure is taken with the dome 11

12 closed. Thus, 7 dome flats were collected through the red detector with the appropriate slit width (1 ). Each flat field frame that was collected then had the bias level removed by subtracting the median bias frame. Then, the bias corrected flat field frames were stacked and then the median value of counts for each pixel in the stack was taken, resulting a single flat field frame of median values for each pixel. Next, the median value of counts along each column was taken, resulting in a flattened flat field, which was then plotted to show the spectral shape. A 17th order polynomial was fit to this spectrum and then over-plotted on the spectrum as shown in Figure 3.3 Counts (in 1000's) Observed Spectrum Fitted Polynomial Flat Field Spectrum Pixel Number Figure 3.3: The observed flat field spectrum after taking the median and flattening is plotted in red as counts (in thousands) versus pixel number. The 17th order polynomial fit to the spectrum is over-plotted in blue. Having found the spectrum for the flat field as well as having fitted a polynomial to it, the flat field correction was determined by normalizing the continuum flux. This was accomplished by dividing the flat field spectrum by the fitted polynomial along each pixel. The resulting normalized flat field correction is then ready to be used to remove the unwanted signatures of the apparatus from raw data. Figure 3.4 presents this normalized flat field image: Figure 3.4: The normalized flat field image that was used to remove unwanted equipment signatures from the data. 3.4 SCIENCE READY FRAMES After removing the unnecessary rows and columns from each frame, the bias level of the detector and the normalized flat field correction were utilized to convert the raw data into images that were ready for scientific analysis. This conversion was accomplished using the following algorithm: SI = RI B I F F (3.1) Where the science ready image is given by SI, the raw image to be corrected is RI, the normalized flat field correction is F F, and the median bias image is denoted by B I. 12

13 3.5 SKY SUBTRACTION The spectrum of IC 1197 as collected by the experimental apparatus through the red detector is shown in Figure 3.2 with trimming as the only processing done on the image. The spectrum does indeed show characteristic spectral lines, which differ in terms of relative intensity as well as vertical length. In terms of the science being conducted for this experiment, the relative intensity hardly played a role aside from the requirement that the spectral lines chosen for science be intense enough to be easily observed in the spectrum. The vertical length of the lines was much more important however. Those lines that spanned the entire detector would be from a source whose angular dimension parallel to the slit met or exceeded that of the slit, while the lines that did not cover the entire detector would be from a source smaller than the length of the slit. From this argument, it is easy to reason that the lines that spanned the entire detector were so called sky lines, where the entire sky in the field of view of the telescope acts like a source and emits lines that are of little scientific interest because the do not pertain to the source being observed. Another way in which to determine which lines in the spectrum are background from the sky and which are of scientific interest is to see which ones were not perfectly vertical. Given that the object being observed was an edge-on, rotating spiral galaxy, it was expected (and imperative for the scientific analysis) that the spectral lines from the source have a curve to them. From a computational standpoint, it was easier to automate the removal of skylines by identifying them using the first method discussed. Namely, lines that spanned the entire vertical dimension of the image were removed and those that did not were kept. This was accomplished by identifying rows of pixels in the image that did have the sky lines (every row would, according to the assumption being made) but that did not have the science lines, which tended to be towards the top of the image. Five such rows were identified and then the were stacked and the median was taken such that a single row of median values was obtained. This row was then tiled into an image of the exact same size as the image containing the spectrum as presented in Figure 3.2, then the tiled image was subtracted from the image containing the spectrum. Since the tiled image contained only the signatures of the sky lines, subtracting it from the spectrum removed the sky lines and preserved the lines to be used for scientific analysis. Figure 3.5 presents the spectrum of IC 1197 as observed through the red detector with the sky lines removed as well as after reduction according to the methods presented in previous sections. This was the image used for scientific analysis. Figure 3.5: The fully reduced and sky line subtracted spectral image of IC 1197 that was used for scientific analysis. Note that the lines that are still present towards the lower part of the image are not trusted and remain as the remnants of sky lines that were not fully removed by the algorithm. This occurred because the rows used to calibrate out the sky lines were taken primarily from the top of the frame, and given that there is a slight tilt across the detector, the correction is more accurate when applied to the top of the frame as compared to the bottom. Comparing the sky removed image (Figure 3.5) to the original (Figure 3.2) reveals many lines missing, which were the unwanted sky lines. The remaining lines, are trusted for scientific analysis with particular weight given to those in the middle of the spectrum which were prominent before and after the correction. Given that the correction to remove the sky lines relied largely on rows from the upper portion of the frame, the upper rows in the corrected frame are trusted more than the lower ones. This also explains why there appear to be some lines that go about halfway across the frame starting from the bottom, these are assumed to be sky lines that were not fully removed by the correction (due to the slight tilt across the detector) and are ignored in the analysis. 13

14 3.6 MAPPING FROM PIXELS TO WAVELENGTHS A calibration exposure was taken with a Neon arc lamp using the red camera, which was used to create a mapping from pixels on the CCD to physical wavelengths in angstroms. After reducing this image by removing the bias level and applying flat field corrects (these reduction methods were detailed in Sections 3.2 & 3.3, respectively) the row of pixels corresponding to the galactic center (135) was selected from the calibration image. After verifying that this row was free of any errors, the row and the counts contained within each pixel in this row were plotted to obtain a spectral plot in counts versus pixel location as shown in Figure 3.7a. This plot was then compared to an arc line plot containing the spectral lines of Neon [1]. From this comparison, the known spectral lines of Neon could be found in pixel space, as is presented in Table 3.1: Wavelength (Å) Column (pixel) σ (pixel) Table 3.1: The known wavelengths of spectral lines of Neon appear in the left column of the table, in the middle column are the pixel locations (corresponding to columns of the CCD), and the left column contains an estimate of the uncertainty in the selection of a given pixel to match a spectral line. Given that the columns of pixels on the detector are discrete, there is some uncertainty associated with choosing a column to match to a given spectral line. Assuming this uncertainty to be ± 0.5 pixels is a reasonable choice because if a given spectral line was more than this amount from a given pixel, it would be assigned to the next one instead. Using this data, the following linear mapping was found by finding the best fit linear model to describe the relation between pixel location and wavelength: λ(n) = 2.348n (Å) (3.2) Where λ(n) is the wavelength in angstroms as a function of the pixel number n along the row. Figure 3.6 shows the agreement between the mapping and data points: 14

15 Mapping of Pixel Number to Wavelength Experimental Data Model 7500 Wavelength ( ) Pixel Number Figure 3.6: The data points used to find the best fit linear mapping between pixels and wavelength are plotted in red, with the model plotted in blue. Clearly there is strong agreement. The linear mapping was used to convert the pixel numbers to physical wavelengths in the calibration frame. Figure 3.7 presents the original calibration spectral plot in terms of pixel number and then the corrected calibration spectral plot in terms of wavelength in angstroms: 70 Arc Lamp Spectrum 70 Arc Lamp Spectrum Counts (in 1000's) Counts (in 1000's) Pixel Number Wavelength ( ) (a) Plot of counts vs pixel for the calibration image. (b) Plot of counts vs wavelength for the calibration image. Figure 3.7: Spectral plots of the calibration frame. First in terms of pixels and then in terms of wavelengths after applying the mapping. 4 ANALYSIS & RESULTS Having corrected the science frame according to the methodologies described in detail in Section 3, the frame was then used for scientific analysis. The details of this analysis and the results that follow from it will now be presented in the following sections. 15

16 4.1 SPECTRAL LINE IDENTIFICATION The pixel to wavelength mapping was used to calibrate the scientific data. Figure 4.1 presents a spectrum of the target used for scientific analysis in this report with the wavelength calibration applied. Again, the spectrum was taken from the row corresponding to the galactic center (76). The known wavelengths of the spectral lines to be used in scientific are plotted as vertical lines on top of the spectrum IC 1197 Spectrum IC 1197 Spectrum Hα line S II line Counts Wavelength ( ) Figure 4.1: The spectra of IC 1197 after the raw image was reduced and the mapping from pixels to wavelength was applied. The two lines selected for scientific analysis are clearly visible above the noise and match well with their known wavelengths. To compare the observed spectral lines to the known ones, the following test statistic was used: k = λ 2 λ 1 λ (4.1) Where k is the test statistic, λ 1 and λ 2 are the wavelengths of the two lines, and λ is the mean value of those two wavelengths. This test statistic statistic was calculated with the known wavelengths of the two spectral lines identified and indicated and Figure 4.1. In order to compare the result, the wavelengths of the two lines in question from the observed spectrum needed to be found. To accomplish this, a Gaussian was fit over the rows of the image between two columns of pixels that the desired spectral line was visually verified to be between. The centroid of this Gaussian gives the wavelength of the spectral line, and the associated uncertainty in the wavelength was determined first in pixel space as the standard deviation σ of the Gaussian. The uncertainty in the wavelength was then found by using concepts from differential calculus and the mapping presented in Equation 3.2 to propagate the uncertainty from pixel space into wavelength space as follows: σ λ = 2.348σ pix (4.2) Unfortunately, the uncertainties found using the Gaussian method as described above yielded higher errors than if the pixel uncertainty σ pix in identifying a spectral line was assumed as 0.5 pixels as argued in Section 3.6. Thus, this pixel uncertainty was assigned to the each spectral line that was identified and then propagated using Equation 4.2. Uncertainties in wavelengths were found using this method, not the Gaussian method previously described. After determining the wavelengths of the lines from the observed spectrum, the test statistic was computed. Comparing the test statistic for the known lines to the observed lines yields a percent difference of 1.55 %. Thus, it is concluded that the indicated spectral lines from the observed spectrum of the galactic center at wavelengths of ± 1.17 Å and ± 1.17 Å are indeed those of Hydrogen and singly ionized Sulfur, respectively. 16

17 4.2 REDSHIFT After obtaining the corrected spectrum, the first scientific objective accomplished was to identify two spectral lines and their red-shifted wavelengths from the galactic center. This was accomplished and the details are presented in Section 3.6. Having determined the red-shifted wavelengths of two spectral lines from the galactic center, the next scientific objective of determining the red shift of the galaxy as a whole could be accomplished. Using Equation 4.3, the red shift Z was determined using the known rest frame wavelength of a given spectral line λ and the observed wavelength of the corresponding line λ obs as was found and presented in Section 3.6. Z = λ obs λ 1 (4.3) The uncertainty in the red shift σ Z was found by propagating the uncertainty in the observed wavelengths σ λobs through the functional relationship between the red shift and the observed wavelength presented in Equation 4.3 using the standard rules of differential calculus. Equation 4.4 presents the resulting function for the determination of the uncertainty in the red shift: σ Z = σ λ obs λ The red shift and its uncertainty were calculated for both of the spectral lines identified in Section 3.6 and then averaged to give the result of ± where the uncertainty was found by propagating the individual red shift uncertainty estimates through the mean function. This result for the red shift compares rather poorly with the value of from the literature, which is 4.58σ below the result found from the analysis presented in this paper [4]. This is not of major concern however, because the method used in finding the red shift in the literature was not given and because determining the red shift was not the main purpose of the experiment. (4.4) 4.3 DISTANCE TO GALAXY The distance from the Earth to IC 1197 was found using the experimentally determined red shift and Hubble s law which is presented in the following equation: v = H 0 L (4.5) Where L is the distance to the object from the Earth, v is the red shift velocity, and H 0 is the Hubble constant which has a currently accepted value of 67.3 ± 1.2 km/s/mpc [12]. In order to use the relationship presented in Equation 4.5, the red shift velocity must first be determined. The method used to accomplish this is presented in the following equation: Z v c v Z c (4.6) Solving Equation 4.5 for the desired quantity L and then substituting in the relationship for v in terms of known quantities is accomplished as follows: L = v H 0 L Z c H 0 (4.7) The uncertainty in the distance to the galaxy σ L was found by propagating the uncertainty in the red shift σ Z and the uncertainty in the Hubble constant σ H0 through the functional relationship presented in Equation 4.7. The resulting function for σ L is presented in Equation 4.8: σ L c H 0 σ 2 Z + ( Z H 0 ) 2 σ 2 H 0 (4.8) Using Equations 4.7 & 4.8 with the experimentally found results, the distance to IC 1197 from Earth was determined to be ± 0.69 Mpc. 17